. Year
t/°C
Year
t/°C
Year
t/°C
Year
t/°C
Year
t/°C
1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886
13.63 13.54 13.58 13.77 13.61 13.59 13.47 13.75 13.55 13.76 13.79 13.70 13.79 13.71 13.68 13.64 13.79 13.71 13.61 13.58 13.59 13.87 14.00 13.71 13.72 13.75 13.77 13.69 13.64 13.67 13.74
1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917
13.63 13.69 13.83 13.62 13.67 13.58 13.55 13.62 13.64 13.84 13.85 13.66 13.77 13.86 13.76 13.63 13.56 13.51 13.63 13.69 13.50 13.48 13.50 13.54 13.52 13.59 13.59 13.75 13.85 13.63 13.51
1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948
13.62 13.71 13.77 13.78 13.69 13.72 13.66 13.76 13.90 13.80 13.77 13.62 13.84 13.92 13.88 13.75 13.87 13.83 13.87 13.96 14.06 13.98 13.97 14.06 14.04 14.04 14.19 14.06 13.88 13.89 13.89
1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979
13.88 13.79 13.92 14.01 14.07 13.82 13.81 13.74 14.04 14.10 14.03 13.99 14.04 14.02 14.05 13.77 13.84 13.93 13.92 13.90 14.03 13.97 13.81 13.96 14.09 13.82 13.88 13.78 14.06 13.97 14.07
1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
14.11 14.13 14.06 14.25 14.03 14.01 14.10 14.25 14.25 14.19 14.34 14.29 14.14 14.19 14.26 14.38 14.22 14.43 14.59 14.33 14.29
Annual Mean Global Temperature in °C
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SPEED OF SOUND IN VARIOUS MEDIA The speed of sound in various solids, liquids, and gases is given in these tables. While only a single parameter v is needed for liquids and gases, sound propagation in isotropic solids is characterized by three velocity parameters. For a solid of infinite extent (or of finite extent if all dimensions are much larger than a wavelength, there are two relevant quantities, v1: velocity of longitudinal waves vs: velocity of shear waves. For a cylindrical rod with diameter much smaller than a wavelength, vext: velocity of extensional waves along the rod. (Torsional waves in the rod are propagated at the same speed as sheer waves in an infinite solid.) Table 1 lists values for a variety of solid materials. Table 2 covers gases liquids and gases; values for cryogenic liquids are given at the normal boiling point. Table 3 gives the speed of sound in pure water and in seawater of salinity S = 3.5% as a function of temperature. All values are in meters per second and are given for normal atmospheric pressure. REFERENCES 1. 2. 3. 4. 5. 6. 7.
Gray, D.E., Ed., American Institute of Physics Handbook, Third Edition, McGraw Hill, New York, 1972. Anderson, H.L., Ed., A Physicist’s Desk Reference, American Institute of Physics, New York, 1989. Younglove, B.A., Thermophysical Proeprties of Fluids. Part I, J. Phys. Chem. Ref. Data, 11, Suppl. 1, 1982. Younglove, B.A., and Ely, J.F., Thermophysical Properties of Fluids. Part II, J. Phys. Chem. Ref. Data, 16, 577, 1987. Haar, L., Gallagher, J.S., and Kell, G.S., NBS/NRC Steam Tables, Hemisphere Publishing Corp., New York, 1984. Mason, W.P., Physical Acoustics and the Properties of Solids, D. Van Nostrand Co., Princeton, N.J., 1958. Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology, New Series, II/5, Molecular Acoustics, SpringerVerlag, Heidelberg, 1967.
TABLE 1 Speed of Sound in Solids at Room Temperature Name
v1/m s–1
vs/m s–1
vext/m s–1
Metals Aluminum, rolled Beryllium Brass (70 Cu, 30 Zn) Constantan Copper, annealed Copper, rolled Duralumin 17S Gold, hard-drawn Iron, cast Iron, electrolytic Iron, Armco Lead, annealed Lead, rolled Magnesium, annealed Molybdenum Monel metal Nickel Platinum Silver Steel (1% C)
6420 12890 4700 5177 4760 5010 6320 3240 4994 5950 5960 2160 1960 5770 6250 5350 6040 3260 3650 5940
3040 8880 2110 2625 2325 2270 3130 1200 2809 3240 3240 700 690 3050 3350 2720 3000 1730 1610 3220
5000 12870 3480 4270 3810 3750 5150 2030 4480 5120 5200 1190 1210 4940 5400 4400 4900 2800 2680 5180
Name Steel, 347 Stainless Steel, K9 Tin, rolled Titanium Tungsten, annealed Tungsten, drawn Zinc, rolled Other materials Fused silica Glass, heavy silicate flint Glass, light borate crown Glass, pyrex Lucite Nylon 6-6 Polyethylene Polystyrene Rubber, butyl Rubber, gum Rubber, neoprene Tungsten carbide
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v1/m s–1
vs/m s–1
vext/m s–1
5790 5940 3320 6070 5220 5410 4210
3100 3250 1670 3125 2890 2640 2440
5000 5250 2730 5090 4620 4320 3850
5968 3980 5100 5640 2680 2620 1950 2350 1830 1550 1600 6655
3764 2380 2840 3280 1100 1070 540 1120
5760 3720 4540 5170 1840 1800 920 1840
3980
6220
SPEED OF SOUND IN VARIOUS MEDIA (continued) TABLE 2 Speed of Sound in Liquids and Gases Name Liquids Acetone Argon Benzene Bromobenzene Butane 1-Butanol Carbon disulphide Chlorobenzene Cyclohexane 1-Decene Diethyl ether Ethane Ethanol Ethylene Ethylene glycol Fluorobenzene Glycerol Helium Heptane 1-Heptene Hexane Hydrogen Iodobenzene Mercury Methane Methanol Nitrobenzene Nitrogen 1-Nonene Octane 1-Octene Oxygen 1-Pentadecene
t/°C
v/m s–1
20 -185.9 25 20 -0.5 20 25 20 19 20 25 -88.6 20 -103.8 25 20 25 -268.9 20 20 20 -252.9 20 25 -161.5 20 25 -195.8 20 20 20 -183.0 20
1203 813 1310 1169 1034 1258 1140 1311 1280 1250 976 1326 1162 1309 1658 1183 1904 180 1162 1128 1083 1101 1114 1450 1337 1121 1463 939 1218 1197 1184 906 1351
Name Pentane Propane 1-Propanol Tetrachloromethane Trichloromethane 1-Undecene Water Water (sea, S = 3.5%) Gases at 1 atm Air, dry Ammonia Argon Carbon monoxide Carbon dioxide Chlorine Deuterium Ethane Ethylene Helium Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen iodide Hydrogen sulfide Methane Neon Nitric oxide Nitrogen Nitrous oxide Oxygen Sulfur dioxide Water (steam)
t/°C
v/m s–1
20 -42.1 20 25 25 20 25 25
1008 1158 1223 930 987 1275 1497 1535
25 0 27 0 0 0 0 27 27 0 27 0 0 0 0 27 0 10 27 0 27 0 100
346 415 323 338 259 206 890 312 331 965 1310 200 296 157 289 450 435 325 353 263 330 213 473
TABLE 3 Speed of Sound in Water and Seawater (S = 3.5%) at Different Temperatures t/°C 0 10 20 25 30 40 50 60 70 80
v/m s–1 Water Seawater 1401.0 1447.8 1483.2 1497.4 1509.5 1528.4 1541.4 1549.5 1553.2 1552.8
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1449.4 1490.4 1522.2 1535.1 1546.2
ATTENUATION AND SPEED OF SOUND IN AIR AS A FUNCTION OF HUMIDITY AND FREQUENCY This table gives the attenuation and speed of sound as a function of frequency at various values of relative humidity. All values refer to still air at 20°C. REFERENCES 1. Tables of Absorption and Velocity of Sound in Still Air at 68°F (20°C), AD-738576, National Technical Information Service, Springfield, VA. 2. Evans, L. B., Bass, H. E., and Sutherland, L. C., J. Acoust. Soc. Am., 51, 1565, 1972.
Frequency (Hz)
Attenuation (dB/km)
Speed (m/s)
Frequency (Hz)
0.51 1.07 1.26 1.43 1.67 1.84 1.96 2.11 2.27 2.82 4.14 8.84 14.89 26.28 35.81 52.15 75.37 267.01 644.66 1032.14
20 40 50 63 100 200 400 630 800 1250 2000 4000 6300 10000 12500 16000 20000 40000 63000 80000
343.477 343.514 343.525 343.536 343.550 343.559 343.561 343.562 343.562 343.562 343.562 343.564 343.565 343.566 343.566 343.567 343.567 343.567 343.567 343.567
0.03 0.11 0.17 0.25 0.50 1.01 1.59 2.24 2.85 5.09 10.93 38.89 90.61 204.98 294.08 422.51 563.66 1110.97 1639.47 2083.08
0.02 0.06 0.09 0.15 0.34 0.99 1.94 2.57 2.94 4.01 6.55 18.73 42.51 101.84 155.67 247.78 373.78 1195.37 2220.64 2951.71
344.182 344.183 344.183 344.184 344.185 344.190 344.197 344.200 344.201 344.202 344.203 344.204 344.204 344.206 344.208 344.211 344.215 344.238 344.262 344.274
Relative humidity 100%
Relative humidity 30% 20 40 50 63 100 200 400 630 800 1250 2000 4000 6300 10000 12500 16000 20000 40000 63000 80000
Speed (m/s)
Relative humidity 60%
Relative humidity 0% 20 40 50 63 100 200 400 630 800 1250 2000 4000 6300 10000 12500 16000 20000 40000 63000 80000
Attenuation (dB/km)
20 40 50 63 100 200 400 630 800 1250 2000 4000 6300 10000 12500 16000 20000 40000 63000 80000
343.807 343.808 343.810 343.810 343.814 343.821 343.826 343.827 343.828 343.828 343.829 343.831 343.836 343.846 343.854 343.865 343.877 343.911 343.924 343.929
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0.01 0.04 0.06 0.09 0.22 0.77 2.02 3.05 3.57 4.59 6.29 13.58 27.72 63.49 96.63 154.90 237.93 884.28 1973.62 2913.01
344.685 344.685 344.685 344.685 344.686 344.689 344.695 344.699 344.701 344.704 344.705 344.706 344.706 344.706 344.707 344.708 344.709 344.718 344.731 344.742
SPEED OF SOUND IN DRY AIR The values in this table were calculated from the equation of state for dry air (average molecular weight 28.96) treated as a real gas. Values refer to standard atmospheric pressure. The speed of sound varies only slightly with pressure; at two atmospheres and -100°C the value decreases by 0.13%, while at two atmospheres and 80°C the speed increases by 0.04%. REFERENCE Sytchev, V.V., Vasserman, A.A., Kozlov, A.D., Spiridonov, G.A., and Tsymarny, V.A., Thermodynamic Properties of Air, Hemisphere Publishing Corp., New York, 1987. t/°C
vs/m s–1
t/°C
vs/m s–1
t/°C
vs/m s–1
-100 -95 -90 -85 -80 -75 -70 -65 -60 -55 -50 -45 -40
263.5 267.3 271.1 274.8 278.5 282.1 285.7 289.2 292.7 296.1 299.5 302.9 306.2
-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25
309.5 312.7 315.9 319.1 322.3 325.4 328.4 331.5 334.5 337.5 340.4 343.4 346.3
30 35 40 45 50 55 60 65 70 75 80
349.1 352.0 354.8 357.6 360.4 363.2 365.9 368.6 371.3 374.0 376.7
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MUSICAL SCALES EQUAL TEMPERED CHROMATIC SCALE A4 = 440 Hz American Standard pitch. Adopted by the American Standards Association in 1936
Note
Frequency
Note
Frequency
Note
Frequency
Note
Frequency
C0 C#0 D0 D#0 E0 F0 F#0 G0 G#0 A0 A#0 B0 C1 C#1 D1 D#1 E1 F1 F#1 G1 G#1 A1 A#1 B1
16.35 17.32 18.35 19.45 20.60 21.83 23.12 24.50 25.96 27.50 29.14 30.87 32.70 34.65 36.71 38.89 41.20 43.65 46.25 49.00 51.91 55.00 58.27 61.74
C2 C#2 D2 D#2 E2 F2 F#2 G2 G#2 A2 A#2 B2 C3 C#3 D3 D#3 E3 F3 F#3 G3 G#3 A3 A#3 B3
65.41 69.30 73.42 77.78 82.41 87.31 92.50 98.00 103.83 110.00 116.54 123.47 130.81 138.59 146.83 155.56 164.81 174.61 185.00 196.00 207.65 220.00 233.08 246.94
C4 C#4 D4 D#4 E4 F4 F#4 G4 G#4 A4 A#4 B4 C5 C#5 D5 D#5 E5 F5 F#5 G5 G#5 A5 A#5 B5
261.63 277.18 293.66 311.13 329.63 349.23 369.99 392.00 415.30 440.00 466.16 493.88 523.25 554.37 587.33 622.25 659.26 698.46 739.99 783.99 830.61 880.00 932.33 987.77
C6 C#6 D6 D#6 E6 F6 F#6 G6 G#6 A6 A#6 B6 C7 C#7 D7 D#7 E7 F7 F#7 G7 G#7 A7 A#7 B7 C8
1046.50 1108.73 1174.66 1244.51 1318.51 1396.91 1479.98 1567.98 1661.22 1760.00 1864.66 1975.53 2093.00 2217.46 2349.32 2489.02 2637.02 2793.83 2959.96 3135.96 3322.44 3520.00 3729.31 3951.07 4186.01
EQUAL TEMPERED CHROMATIC SCALE A4 = 435 Hz International Pitch, adopted 1891
Note
Frequency
Note
Frequency
Note
Frequency
Note
Frequency
C0 C#0 D0
16.17 17.13 18.15
C2 C#2 D2
64.66 68.51 72.58
C4 C#4 D4
258.65 274.03 290.33
C6 C#6 D6
1034.61 1096.13 1161.31
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D#0 E0 F0 F#0 G0 G#0 A0 A#0 B0 C1 C#1 D1 D#1 E1 F1 F#1 G1 G#1 A1 A#1 B1
19.22 20.37 21.58 22.86 24.22 25.66 27.19 28.80 30.52 32.33 34.25 36.29 38.45 40.74 43.16 45.72 48.44 51.32 54.38 57.61 61.03
D#2 E2 F2 F#2 G2 G#2 A2 A#2 B2 C3 C#3 D3 D#3 E3 F3 F#3 G3 G#3 A3 A#3 B3
76.90 81.47 86.31 91.45 96.89 102.65 108.75 115.22 122.07 129.33 137.02 145.16 153.80 162.94 172.63 182.89 193.77 205.29 217.50 230.43 244.14
D#4 E4 F4 F#4 G4 G#4 A4 A#4 B4 C5 C#5 D5 D#5 E5 F5 F#5 G5 G#5 A5 A#5 B5
307.59 325.88 345.26 365.79 387.54 410.59 435.00 460.87 488.27 517.31 548.07 580.66 615.18 651.76 690.52 731.58 775.08 821.17 870.00 921.73 976.54
D#6 E6 F6 F#6 G6 G#6 A6 A#6 B6 C7 C#7 D7 D#7 E7 F7 F#7 G7 G#7 A7 A#7 B7 C8
1230.37 1303.53 1381.04 1463.16 1550.16 1642.34 1740.00 1843.47 1953.08 2069.22 2192.26 2322.62 2460.73 2607.05 2762.08 2926.32 3100.33 3284.68 3480.00 3686.93 3906.17 4138.44
SCIENTIFIC OR JUST SCALE C4 = 256 Hz
Note
Frequency
Note
Frequency
Note
Frequency
Note
Frequency
C0 D0 E0 F0 G0 A0 B0 C1 D1 E1 F1 G1 A1 B1
16 18 20 21.33 24 26.67 30 32 36 40 42.67 48 53.33 60
C2 D2 E2 F2 G2 A2 B2 C3 D3 E3 F3 G3 A3 B3
64 72 80 85.33 96 106.67 120 128 144 160 170.67 192 213.33 240
C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 G5 A5 B5
256 288 320 341.33 384 426.67 480 512 576 640 682.67 768 853.33 960
C6 D6 E6 F6 G6 A6 B6 C7 D7 E7 F7 G7 A7 B7 C8
1024 1152 1280 1365.33 1536 1706.67 1920 2048 2304 2560 2730.67 3072 3413.33 3840 4096
CHARACTERISTICS OF HUMAN HEARING The human ear is sensitive to sound waves with frequencies in the range from a few hertz to almost 20 kHz. Auditory response is usually expressed in terms of the loudness level of a sound, which is a measure of the sound pressure. The reference level, which is given in the unit phon, is a pure tone of frequency 1000 Hz with sound pressure of 20 mPa (in cgs units, 2 ·10-4 dyn/cm2 ); loudness level is usually expressed in decibels (dB) relative to this reference level. If a normal observer perceives an arbitrary sound to be equally loud as this reference sound, the sound is said to have the loudness level of the reference. The sensitivity of the typical human ear ranges from about 0 dB, the threshold loudness level, to about 140 dB, the level at which pain sets in. The minimum detectable level thus represents a sound wave of pressure 20 mPa and intensity (power density) 10-16 W/cm2. The following figure illustrates the frequency dependence of the threshold for an average young adult.
Frequency in Hz
The relation between loudness level and frequency for a typical person is expressed by the following table:
Sound pressure level in dB relative to 20 mPa 10 20 30 40 50 60 70 80 90 100
125
500
Frequency in Hz 1000 4000
4 17 34 52 70 86 98 108
16 27 39 52 65 76 86 96 105
10 20 30 40 50 60 70 80 90 100
18 28 37 45 54 64 73 83 94 106
8000
10000
11 21 30 38 47 56 66 77 88
17 26 35 44 54 64 74 86
Thus, a 10,000 Hz tone at a pressure level of 50 dB seems equally loud as a 1000 Hz tone at a pressure of 35 dB. The term noise refers to any unwanted sound, either a pure tone or a mixture of frequencies. Since the sensitivity of the ear is frequency dependent, as illustrated by the above table, noise level is expressed in a frequency-weighted scale, known as A-weighting. Decibel readings on this scale are designated as dBa. Typical noise levels from various sources are illustrated in this table:
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CHARACTERISTICS OF HUMAN HEARING (continued)
Source
Noise level in dBa
Rocket engine Jet aircraft engine Light aircraft, cruising Tractor, 150 hp Electric motor, 100 hp at 2600 rpm Pneumatic drill Subway train Vacuum cleaner Heavy automobile traffic Conversational speech Whispered speech Background noise, recording studio
200 160 140 115 105 100 90 85 75 65 40 25-30
Recommended noise thresholds in the workplace have been established by the American Conference of Government Industrial Hygenists. Some examples of the maximum safe levels for different daily exposure times are given below. Duration of exposure
Max. level in dBa
24 h 8h 4h 1h 30 min 15 min 2 min 28 s 0.11 s
80 85 88 94 97 100 109 115 139
No exposure greater than 140 dBa is permitted. Further details may be found in Reference 3.
REFERENCES 1. Anderson, H. L., Editor, A Physicist’s Desk Reference, American Institute of Physics, New York, 1989, chap. 2. 2. Gray, D. E., Ed., American Institute of Physics Handbook, Third Edition, McGraw Hill, New York, 1972, chap. 3. 3. Threshold Limit Values for Chemical Substances and Physical Agents; Biological Exposure Indices, 1999 Edition, American Conference of Governmental Industrial Hygienists, 1330 Kemper Meadow Drive, Cincinnati, OH 45240-1634.
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Section 15: Practical Laboratory Data Standard ITS-90 Thermocouple Tables Secondary Reference Points on the ITS-90 Temperature Scale Laboratory Solvents and other Liquid Reagents Miscibility of Organic Solvents Density of Solvents as a Function of Temperature Dependence of Boiling Point on Pressure Ebullioscopic Constants for Calculation of Boiling Point Elevation Cryoscopic Constants for Calculation of Freezing Point Depression Freezing Point Lowering by Electrolytes in Aqueous Solution Correction of Barometer Readings to 0°C Temperature Determination of Relative Humidity from Dew Point Determination of Relative Humidity from Wet and Dry Bulb Temperatures Constant Humidity Solutions Standard Salt Solutions for Humidity Calibration Low Temperature Baths for Maintaining Constant Temperature Metals and Alloys with Low Melting Temperature Wire Tables Characteristics of Particles and Particle Dispersoids Density of Various Solids Density of Ethanol-Water Mixtures Dielectric Strength of Insulating Materials Coefficient of Friction Flame Temperatures Allocation of Frequencies in the Radio Spectrum
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STANDARD ITS-90 THERMOCOUPLE TABLES The Instrument Society of America (ISA) has assigned standard letter designations to a number of thermocouple types having specified emf-temperature relations. These designations and the approximate metal compositions which meet the required relations, as well as the useful temperature ranges, are given below: Type B Type E Type J Type K Type N Type R Type S Type T
(Pt + 30% Rh) vs. (Pt + 6% Rh) (Ni + 10% Cr) vs. (Cu + 43% Ni) Fe vs. (Cu + 43% Ni) (Ni + 10% Cr) vs. (Ni + 2% Al + 2% Mn + 1% Si) (Ni + 14% Cr + 1.5% Si) vs. (Ni + 4.5% Si + 0. 1% Mg) (Pt + 13% Rh) vs. Pt (Pt + 10% Rh) vs. Pt Cu vs. (Cu + 43% Ni)
0 to 1820°C -270 to 1000°C -210 to 1200°C -270 to 1372°C -270 to 1300°C -50 to 1768°C -50 to 1768°C -270 to 400°C
The compositions are given in weight percent, and the positive leg is listed first. It should be emphasized that the standard letter designations do not imply a precise composition but rather that the specified emf-temperature relation is satisfied. The first set of tables below lists, for each thermocouple type, the emf as a function of temperature on the International Temperature Scale of 1990 (ITS-90). The coefficients in the equation used to generate the table are also given. The second set of tables gives the inverse relationships, i.e., the coefficients in the polynomial equation which expresses the temperature as a function of thermocouple emf. The accuracy of these equations is also stated. Further details and tables at closer intervals may be found in Reference 1. REFERENCES 1. Burns, G. W., Seroger, M. G., Strouse, G. F., Croarkin, M. C., and Guthrie, W.F., Temperature-Electromotive Force Reference Functions and Tables for the Letter-Designated Thermocouple Types Based on the ITS-90, Nat. Inst. Stand. Tech. (U.S.) Monogr. 175, 1993. 2. Schooley, J. F., Thermometry, CRC Press, Boca Raton, FL, 1986.
Type B thermocouples: emf-temperature (°C) reference table and equations Thermocouple emf in Millivolts as a Function of Temperature in Degrees Celsius (ITS-90) Reference Junctions at 0°C t/°C
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400
0.000 -0.002 -0.003 -0.002 -0.000 0.033 0.043 0.053 0.065 0.078 0.178 0.199 0.220 0.243 0.267 0.431 0.462 0.494 0.527 0.561 0.787 0.828 0.870 0.913 0.957
0.002 0.092 0.291 0.596 1.002
0.006 0.107 0.317 0.632 1.048
0.011 0.123 0.344 0.669 1.095
0.017 0.141 0.372 0.707 1.143
0.025 0.159 0.401 0.746 1.192
0.033 0.178 0.431 0.787 1.242
500 600 700 800 900
1.242 1.792 2.431 3.154 3.957
1.293 1.852 2.499 3.230 4.041
1.344 1.913 2.569 3.308 4.127
1.397 1.975 2.639 3.386 4.213
1.451 2.037 2.710 3.466 4.299
1.505 2.101 2.782 3.546 4.387
1.561 2.165 2.854 3.626 4.475
1.617 2.230 2.928 3.708 4.564
1.675 2.296 3.002 3.790 4.653
1.733 2.363 3.078 3.873 4.743
1.792 2.431 3.154 3.957 4.834
1000 1100 1200 1300 1400
4.834 5.780 6.786 7.848 8.956
4.926 5.878 6.890 7.957 9.069
5.018 5.976 6.995 8.066 9.182
5.111 6.075 7.100 8.176 9.296
5.205 6.175 7.205 8.286 9.410
5.299 6.276 7.311 8.397 9.524
5.394 6.377 7.417 8.508 9.639
5.489 6.478 7.524 8.620 9.753
5.585 6.580 7.632 8.731 9.868
5.682 5.780 6.683 6.786 7.740 7.848 8.844 8.956 9.984 10.099
1500 1600 1700 1800
10.099 11.263 12.433 13.591
10.215 11.380 12.549 13.706
10.331 10.447 10.563 10.679 11.497 11.614 11.731 11.848 12.666 12.782 12.898 13.014 13.820
10.796 10.913 11.965 12.082 13.130 13.246
11.029 11.146 11.263 12.199 12.316 12.433 13.361 13.476 13.591
Temperature Ranges and Coefficients of Equations Used to Compute the Above Table The equations are of the form: E = c0 + c1t + c2t2 + c3t3 + ... cntn, where E is the emf in millivolts, t is the temperature in degrees Celsius (ITS-90), and c0, c1, c2, c3, etc. are the coefficients. These coefficients are extracted from NIST Monograph 175. 0°C to 630.615°C c0 c1 c2 c3 c4 c5 c6 c7 c8
= = = = = = = = =
0.000 000 000 0 -2.465 081 834 6 × 10-4 5.904 042 117 1 × 10-6 -1.325 793 163 6 × 10-9 1.566 829 190 1 × 10-12 -1.694 452 924 0 × 10-15 6.299 034 709 4 × 10-19 ............ ............
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630.615°C to 1820°C -3.893 816 862 1 2.857 174 747 0 × 10-2 -8.488 510 478 5 × 10-5 1.578 528 016 4 × 10-7 -1.683 534 486 4 × 10-10 1.110 979 401 3 × 10-13 -4.451 543 103 3 × 10-17 9.897 564 082 1 × 10-21 -9.379 133 028 9 × 10-25
Type E thermocouples: emf-temperature (°C) reference table and equations Thermocouple emf in Millivolts as a Function of Temperature in Degrees Celsius (ITS-90) Reference Junctions at 0°C t/°C -200 -100 0 t/°C
0
-10
-20
-30
-40
-50
-8.825 -9.063 -9.274 -9.455 -9.604 -9.718 -5.237 -5.681 -6.107 -6.516 -6.907 -7.279 0.000 -0.582 -1.152 -1.709 -2.255 -2.787 0
10
20
30
40
50
-60
-70
-80
-90
-100
-9.797 -7.632 -3.306
-9.835 -7.963 -3.811
60
70
80
90
100
-8.273 -8.561 -8.825 -4.302 -4.777 -5.237
0 100 200 300 400
0.000 0.591 1.192 1.801 2.420 3.048 3.685 6.319 6.998 7.685 8.379 9.081 9.789 10.503 13.421 14.164 14.912 15.664 16.420 17.181 17.945 21.036 21.817 22.600 23.386 24.174 24.964 25.757 28.946 29.747 30.550 31.354 32.159 32.965 33.772
4.330 11.224 18.713 26.552 34.579
4.985 11.951 19.484 27.348 35.387
5.648 12.684 20.259 28.146 36.196
6.319 13.421 21.036 28.946 37.005
500 600 700 800 900
37.005 45.093 53.112 61.017 68.787
42.671 50.718 58.659 66.473 74.115
43.479 51.517 59.446 67.246 74.869
44.286 52.315 60.232 68.017 75.621
45.093 53.112 61.017 68.787 76.373
1000
76.373
37.815 45.900 53.908 61.801 69.554
38.624 46.705 54.703 62.583 70.319
39.434 47.509 55.497 63.364 71.082
40.243 48.313 56.289 64.144 71.844
41.053 49.116 57.080 64.922 72.603
41.862 49.917 57.870 65.698 73.360
Temperature Ranges and Coefficients of Equations Used to Compute the Above Table The equations are of the form: E = c0 + c1t + c2t2 + c3t3 + ... cntn, where E is the emf in millivolts, t is the temperature in degrees Celsius (ITS-90), and c0, c1, c2, c3, etc. are the coefficients. These coefficients are extracted from NIST Monograph 175. -270°C 0°C to to 0°C 1000°C c0 = c1 = c2 = c3 = c4 = c5 = c6 = c7 = c8 = c9 = c10 = c11 = c12 = c13 =
0.000 000 000 0 5.866 550 870 8 × 10-2 4.541 097 712 4 × 10-5 -7.799 804 868 6 × 10-7 -2.580 016 084 3 × 10-8 -5.945 258 305 7 × 10-10 -9.321 405 866 7 × 10-12 -1.028 760 553 4 × 10-13 -8.037 012 362 1 × 10-16 -4.397 949 739 1 × 10-18 -1.641 477 635 5 × 10-20 -3.967 361 951 6 × 10-23 -5.582 732 872 1 × 10-26 -3.465 784 201 3 × 10-29
0.000 000 000 0 5.866 550 871 0 × 10-2 4.503 227 558 2 × 10-5 2.890 840 721 2 × 10-8 -3.305 689 665 2 × 10-10 6.502 440 327 0 × 10-13 -1.919 749 550 4 × 10-16 -1.253 660 049 7 × 10-18 2.148 921 756 9 × 10-21 -1.438 804 178 2 × 10-24 3.596 089 948 1 × 10-28 ............ ............ ............
Type J thermocouples: emf-temperature (°C) reference table and equations Thermocouple emf in Millivolts as a Function of Temperature in Degrees Celsius (ITS-90) Reference Junctions at 0°C t/°C -200 -100 0
t/°C
0
-10
-20
-30
-40
-50
-7.890 -8.095 -4.633 -5.037 -5.426 -5.801 -6.159 -6.500 0.000 -0.501 -0.995 -1.482 -1.961 -2.431
0
10
20
30
40
50
-60
-70
-80
-90
-100
-6.821 -2.893
-7.123 -3.344
60
70
80
90
100
-7.403 -7.659 -7.890 -3.786 -4.215 -4.633
0 100 200 300 400
0.000 0.507 1.019 1.537 2.059 2.585 3.116 5.269 5.814 6.360 6.909 7.459 8.010 8.562 10.779 11.334 11.889 12.445 13.000 13.555 14.110 16.327 16.881 17.434 17.986 18.538 19.090 19.642 21.848 22.400 22.952 23.504 24.057 24.610 25.164
3.650 9.115 14.665 20.194 25.720
4.187 9.669 15.219 20.745 26.276
4.726 10.224 15.773 21.297 26.834
5.269 10.779 16.327 21.848 27.393
500 600 700 800 900
27.393 33.102 39.132 45.494 51.877
30.788 36.675 42.919 49.353 55.561
31.362 37.284 43.559 49.989 56.164
31.939 37.896 44.203 50.622 56.763
32.519 38.512 44.848 51.251 57.360
33.102 39.132 45.494 51.877 57.953
57.953 58.545 59.134 59.721 60.307 60.890 61.473 63.792 64.370 64.948 65.525 66.102 66.679 67.255 69.553
62.054 67.831
62.634 63.214 63.792 68.406 68.980 69.553
1000 1100 1200
27.953 33.689 39.755 46.141 52.500
28.516 34.279 40.382 46.786 53.119
29.080 34.873 41.012 47.431 53.735
29.647 35.470 41.645 48.074 54.347
30.216 36.071 42.281 48.715 54.956
Temperature Ranges and Coefficients of Equations Used to Compute the Above Table The equations are of the form: E = c0 + c1t + c2t2 + c3t3 + ... cntn, where E is the emf in millivolts, t is the temperature in degrees Celsius (ITS-90), and c0, c1, c2, c3, etc. are the coefficients. These coefficients are extracted from NIST Monograph 175. -210°C to 760°C c0 c1 c2 c3 c4 c5 c6 c7 c8
= = = = = = = = =
760°C to 1200°C
0.000 000 000 0 5.038 118 781 5 × 10-2 3.047 583 693 0 × 10-5 -8.568 106 572 0 × 10-8 1.322 819 529 5 × 10-10 -1.705 295 833 7 × 10-13 2.094 809 069 7 × 10-16 -1.253 839 533 6 × 10-19 1.563 172 569 7 × 10-23
2.964 562 568 1 × 102 -1.497 612 778 6 3.178 710 392 4 × 10-3 -3.184 768 670 1 × 10-6 1.572 081 900 4 × 10-9 -3.069 136 905 6 × 10-13 ............ ............ ............
TeamLRN
Type K thermocouples: emf-temperature (°C) reference table and equations Thermocouple emf in Millivolts as a Function of Temperature in Degrees Celsius (ITS-90) Reference Junctions at 0°C t/°C -200 -100 0
t/°C
0
-10
-20
-30
-40
-50
-5.891 -6.035 -6.158 -6.262 -6.344 -6.404 -3.554 -3.852 -4.138 -4.411 -4.669 -4.913 0.000 -0.392 -0.778 -1.156 -1.527 -1.889
0
10
20
30
40
50
-60
-70
-6.441 -5.141 -2.243
-6.458 -5.354 -2.587
60
70
-80
-90
-100
-5.550 -5.730 -5.891 -2.920 -3.243 -3.554
80
90
100
0 100 200 300 400
0.000 0.397 0.798 1.203 1.612 2.023 2.436 4.096 4.509 4.920 5.328 5.735 6.138 6.540 8.138 8.539 8.940 9.343 9.747 10.153 10.561 12.209 12.624 13.040 13.457 13.874 14.293 14.713 16.397 16.820 17.243 17.667 18.091 18.516 18.941
2.851 6.941 10.971 15.133 19.366
3.267 3.682 4.096 7.340 7.739 8.138 11.382 11.795 12.209 15.554 15.975 16.397 19.792 20.218 20.644
500 600 700 800 900
20.644 24.905 29.129 33.275 37.326
21.071 25.330 29.548 33.685 37.725
21.497 25.755 29.965 34.093 38.124
21.924 26.179 30.382 34.501 38.522
22.350 26.602 30.798 34.908 38.918
22.776 27.025 31.213 35.313 39.314
23.203 27.447 31.628 35.718 39.708
23.629 27.869 32.041 36.121 40.101
24.055 28.289 32.453 36.524 40.494
1000 1100 1200 1300
41.276 45.119 48.838 52.410
41.665 45.497 49.202 52.759
42.053 45.873 49.565 53.106
42.440 46.249 49.926 53.451
42.826 46.623 50.286 53.795
43.211 46.995 50.644 54.138
43.595 47.367 51.000 54.479
43.978 47.737 51.355 54.819
44.359 44.740 45.119 48.105 48.473 48.838 51.708 52.060 52.410
24.480 28.710 32.865 36.925 40.885
24.905 29.129 33.275 37.326 41.276
Temperature Ranges and Coefficients of Equations Used to Compute the Above Table The equations are of the form: E = c0 + c1t + c2t 2 + c3t 3 + ... cnt n, where E is the emf in millivolts, t is the temperature in degrees Celsius (ITS-90), and c0, c1, c2, c3, etc. are the coefficients. In the 0°C to 1372°C range there is also an exponential term that must be evaluated and added to the equation. The exponential term is of the form: c0exp[c1(t-126.9686)2] , where t is the temperature in °C and c0 and c1 are the coefficients. These coefficients are extracted from NIST Monograph 175.
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10
= = = = = = = = = = =
-270°C to 0°C
0°C to 1372°C
0.000 000 000 0 3.945 012 802 5 × 10-2 2.362 237 359 8 × 10-5 -3.285 890 678 4 × 10-7 -4.990 482 877 7 × 10-9 -6.750 905 917 3 × 10-11 -5.741 032 742 8 × 10-13 -3.108 887 289 4 × 10-15 -1.045 160 936 5 × 10-17 -1.988 926 687 8 × 10-20 -1.632 269 748 6 × 10-23
-1.760 041 368 6 × 10-2 3.892 120 497 5 × 10-2 1.855 877 003 2 × 10-5 -9.945 759 287 4 ×10-8 3.184 094 571 9 × 10-10 -5.607 284 488 9 × 10-13 5.607 505 905 9 × 10-16 -3.202 072 000 3 × 10-19 9.715 114 715 2 × 10-23 -1.210 472 127 5 × 10-26 ...........
TeamLRN
0°C to 1372°C (exponential term) 1.185 976 × 10-1 -1.183 432 × 10-4 ....... ....... ....... ....... ....... ....... ....... ....... .......
Type N thermocouples: emf-temperature (°C) reference table and equations Thermocouple emf in Millivolts as a Function of Temperature in Degrees Celsius (ITS-90) Reference Junctions at 0°C t/°C -200 -100 0 t/°C
0
-10
-20
-30
-40
-50
-3.990 -4.083 -4.162 -4.226 -4.277 -4.313 -2.407 -2.612 -2.808 -2.994 -3.171 -3.336 0.000 -0.260 -0.518 -0.772 -1.023 -1.269 0
10
20
30
40
50
-60
-70
-4.336 -3.491 -1.509
-4.345 -3.634 -1.744
60
70
-80
-90
-100
-3.766 -3.884 -3.990 -1.972 -2.193 -2.407 80
90
100
0 100 200 300 400
0.000 0.261 0.525 0.793 1.065 1.340 2.774 3.072 3.374 3.680 3.989 4.302 5.913 6.245 6.579 6.916 7.255 7.597 9.341 9.696 10.054 10.413 10.774 11.136 12.974 13.346 13.719 14.094 14.469 14.846
1.619 1.902 4.618 4.937 7.941 8.288 11.501 11.867 15.225 15.604
2.189 2.480 2.774 5.259 5.585 5.913 8.637 8.988 9.341 12.234 12.603 12.974 15.984 16.366 16.748
500 600 700 800 900
16.748 20.613 24.527 28.455 32.371
19.059 22.958 26.883 30.807 34.707
19.835 23.742 27.669 31.590 35.482
1000 1100 1200 1300
17.131 21.003 24.919 28.847 32.761
17.515 21.393 25.312 29.239 33.151
17.900 21.784 25.705 29.632 33.541
18.286 22.175 26.098 30.024 33.930
18.672 22.566 26.491 30.416 34.319
36.256 36.641 37.027 37.411 37.795 38.179 40.087 40.466 40.845 41.223 41.600 41.976 43.846 44.218 44.588 44.958 45.326 45.694 47.513
19.447 23.350 27.276 31.199 35.095
38.562 38.944 42.352 42.727 46.060 46.425
20.224 24.134 28.062 31.981 35.869
39.326 39.706 40.087 43.101 43.474 43.846 46.789 47.152 47.513
Temperature Ranges and Coefficients of Equations Used to Compute the Above Table The equations are of the form: E = c0 + c1t + c2t2 + c3t3 + ... cntn, where E is the emf in millivolts, t is the temperature in degrees Celsius (ITS-90), and c0, c1, c2, c3, etc. are the coefficients. These coefficients are extracted from NIST Monograph 175. -270°C to 0°C c0 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10
= = = = = = = = = = =
0.000 000 000 0 2.615 910 596 2 × 10-2 1.095 748 422 8 × 10-5 -9.384 111 155 4 × 10-8 -4.641 203 975 9 × 10-11 -2.630 335 771 6 × 10-12 -2.265 343 800 3 × 10-14 -7.608 930 079 1 × 10-17 -9.341 966 783 5 × 10-20 .......... ..........
20.613 24.527 28.455 32.371 36.256
0°C to 1300°C 0.000 000 000 0 2.592 939 460 1 × 10-2 1.571 014 188 0 × 10-5 4.382 562 723 7 × 10-8 -2.526 116 979 4 × 10-10 6.431 181 933 9 × 10-13 -1.006 347 151 9 × 10-15 9.974 533 899 2 × 10-19 -6.086 324 560 7 × 10-22 2.084 922 933 9 × 10-25 -3.068 219 615 1 × 10-29
Type R thermocouples: emf-temperature (°C) reference table and equations Thermocouple emf in Millivolts as a Function of Temperature in Degrees Celsius (ITS-90) Reference Junctions at 0°C t/°C 0
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
0.000 -0.051 -0.100 -0.145 -0.188 -0.226
t/°C
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400
0.000 0.647 1.469 2.401 3.408
0.054 0.723 1.558 2.498 3.512
0.111 0.800 1.648 2.597 3.616
0.171 0.879 1.739 2.696 3.721
0.232 0.959 1.831 2.796 3.827
0.296 1.041 1.923 2.896 3.933
0.363 1.124 2.017 2.997 4.040
0.431 1.208 2.112 3.099 4.147
0.501 1.294 2.207 3.201 4.255
0.573 1.381 2.304 3.304 4.363
0.647 1.469 2.401 3.408 4.471
500 600 700 800 900
4.471 5.583 6.743 7.950 9.205
4.580 5.697 6.861 8.073 9.333
4.690 5.812 6.980 8.197 9.461
4.800 5.926 7.100 8.321 9.590
4.910 6.041 7.220 8.446 9.720
5.021 6.157 7.340 8.571 9.850
5.133 5.245 6.273 6.390 7.461 7.583 8.697 8.823 9.980 10.111
1000 1100 1200 1300 1400
10.506 11.850 13.228 14.629 16.040
10.638 11.986 13.367 14.770 16.181
10.771 12.123 13.507 14.911 16.323
10.905 12.260 13.646 15.052 16.464
11.039 12.397 13.786 15.193 16.605
11.173 12.535 13.926 15.334 16.746
1500 1600 1700
17.451 17.591 17.732 17.872 18.012 18.152 18.849 18.988 19.126 19.264 19.402 19.540 20.222 20.356 20.488 20.620 20.749 20.877
11.307 12.673 14.066 15.475 16.887
11.442 12.812 14.207 15.616 17.028
18.292 18.431 19.677 19.814 21.003
5.357 5.470 5.583 6.507 6.625 6.743 7.705 7.827 7.950 8.950 9.077 9.205 10.242 10.374 10.506 11.578 12.950 14.347 15.758 17.169
11.714 13.089 14.488 15.899 17.310
11.850 13.228 14.629 16.040 17.451
18.571 18.710 18.849 19.951 20.087 20.222
Temperature Ranges and Coefficients of Equations Used to Compute the Above Table The equations are of the form: E = c0 + c1t + c2t2 + c3t3 + ... cntn, where E is the emf in millivolts, t is the temperature in degrees Celsius (ITS-90), and c0, c1, c2, c3, etc. are the coefficients. These coefficients are extracted from NIST Monograph 175. -50°C to 1064.18°C c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
= = = = = = = = = =
0.000 000 000 00. 5.289 617 297 65 × 10-3 1.391 665 897 82 × 10-5 -2.388 556 930 17 × 10-8 3.569 160 010 63 × 10-11 -4.623 476 662 98 × 10-14 5.007 774 410 34 × 10-17 -3.731 058 861 91 × 10-20 1.577 164 823 67 × 10-23 -2.810 386 252 51 × 10-27
1064.18°C to 1664.5°C 2.951 579 253 16 -2.520 612 513 32 × 10-3 1.595 645 018 65 × 10-5 -7.640 859 475 76 × 10-9 2.053 052 910 24 × 10-12 -2.933 596 681 73 × 10-16 ............. ............. ............. .............
TeamLRN
1664.5°C to 1768.1°C 1.522 321 182 09 × 102 -2.688 198 885 45 × 10-1 1.712 802 804 71 × 10-4 -3.458 957 064 53 × 10-8 -9.346 339 710 46 × 10-15 ............. ............. ............. ............. .............
Type S thermocouples: emf-temperature (°C) reference table and equations Thermocouple emf in Millivolts as a Function of Temperature in Degrees Celsius (ITS-90) Reference Junctions at 0°C
t/°C 0
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
0.000 -0.053 -0.103 -0.150 -0.194 -0.236
t/°C
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400
0.000 0.646 1.441 2.323 3.259
0.055 0.720 1.526 2.415 3.355
0.113 0.795 1.612 2.507 3.451
0.173 0.872 1.698 2.599 3.548
0.235 0.950 1.786 2.692 3.645
0.299 1.029 1.874 2.786 3.742
0.365 1.110 1.962 2.880 3.840
0.433 1.191 2.052 2.974 3.938
0.502 1.273 2.141 3.069 4.036
0.573 1.357 2.232 3.164 4.134
0.646 1.441 2.323 3.259 4.233
500 600 700 800 900
4.233 5.239 6.275 7.345 8.449
4.332 5.341 6.381 7.454 8.562
4.432 5.443 6.486 7.563 8.674
4.532 5.546 6.593 7.673 8.787
4.632 5.649 6.699 7.783 8.900
4.732 5.753 6.806 7.893 9.014
4.833 5.857 6.913 8.003 9.128
4.934 5.961 7.020 8.114 9.242
5.035 6.065 7.128 8.226 9.357
5.137 6.170 7.236 8.337 9.472
5.239 6.275 7.345 8.449 9.587
1000 1100 1200 1300 1400
9.587 10.757 11.951 13.159 14.373
9.703 10.875 12.071 13.280 14.494
9.819 10.994 12.191 13.402 14.615
9.935 11.113 12.312 13.523 14.736
10.051 11.232 12.433 13.644 14.857
10.168 11.351 12.554 13.766 14.978
10.285 11.471 12.675 13.887 15.099
10.403 11.590 12.796 14.009 15.220
10.520 11.710 12.917 14.130 15.341
10.638 11.830 13.038 14.251 15.461
10.757 11.951 13.159 14.373 15.582
1500 1600 1700
15.582 15.702 15.822 15.942 16.062 16.182 16.777 16.895 17.013 17.131 17.249 17.366 17.947 18.061 18.174 18.285 18.395 18.503
16.301 16.420 17.483 17.600 18.609
16.539 16.658 16.777 17.717 17.832 17.947
Temperature Ranges and Coefficients of Equations Used to Compute the Above Table The equations are of the form: E = c0 + c1t + c2t2 + c3t3 + ... cntn, where E is the emf in millivolts, t is the temperature in degrees Celsius (ITS-90), and c0, c1, c2, c3, etc. are the coefficients. These coefficients are extracted from NIST Monograph 175. -50°C to 1064.18°C c0 c1 c2 c3 c4 c5 c6 c7 c8
= = = = = = = = =
0.000 000 000 00 5.403 133 086 31 × 10-3 1.259 342 897 40 × 10-5 -2.324 779 686 89 × 10-8 3.220 288 230 36 × 10-11 -3.314 651 963 89 × 10-14 2.557 442 517 86 × 10-17 -1.250 688 713 93 × 10-20 2.714 431 761 45 × 10-24
1064.18°C to 1664.5°C 1.329 004 440 85 3.345 093 113 44 × 10-3 6.548 051 928 18 × 10-6 -1.648 562 592 09 × 10-9 1.299 896 051 74 × 10-14 ............. ............. ............. .............
TeamLRN
1664.5°C to 1768.1°C 1.466 282 326 36 × 10 2 -2.584 305 167 52 × 10-1 1.636 935 746 41 × 10-4 -3.304 390 469 87 × 10-8 -9.432 236 906 12 × 10-15 ............. ............. ............. .............
Type T thermocouples: emf-temperature (°C) reference table and equations Thermocouple emf in Millivolts as a Function of Temperature in Degrees Celsius (ITS-90) Reference Junctions at 0°C t/°C -200 -100 0
t/°C 0 100 200 300 400
0
-10
-20
-30
-40
-50
-5.603 -5.753 -5.888 -6.007 -6.105 -6.180 -3.379 -3.657 -3.923 -4.177 -4.419 -4.648 0.000 -0.383 -0.757 -1.121 -1.475 -1.819
0
10
20
30
40
50
0.000 0.391 0.790 1.196 1.612 2.036 4.279 4.750 5.228 5.714 6.206 6.704 9.288 9.822 10.362 10.907 11.458 12.013 14.862 15.445 16.032 16.624 17.219 17.819 20.872
-60
-70
-6.232 -4.865 -2.153
-6.258 -5.070 -2.476
60
70
2.468 2.909 7.209 7.720 12.574 13.139 18.422 19.030
-80
-90
-100
-5.261 -5.439 -5.603 -2.788 -3.089 -3.379
80
90
100
3.358 3.814 4.279 8.237 8.759 9.288 13.709 14.283 14.862 19.641 20.255 20.872
Temperature Ranges and Coefficients of Equations Used to Compute the Above Table The equations are of the form: E = c0 + c1t + c2t2 + c3t3 + ... cntn, where E is the emf in millivolts, t is the temperature in degrees Celsius (ITS-90), and c0, c1, c2, c3, etc. are the coefficients. These coefficients are extracted from NIST Monograph 175. -270°C to 0°C c0 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14
= = = = = = = = = = = = = = =
0.000 000 000 0 3.874 810 636 4 × 10-2 4.419 443 434 7 × 10-5 1.184 432 310 5 × 10-7 2.003 297 355 4 × 10-8 9.013 801 955 9 × 10-10 2.265 115 659 3 × 10-11 3.607 115 420 5 × 10-13 3.849 393 988 3 × 10-15 2.821 352 192 5 × 10-17 1.425 159 477 9 × 10-19 4.876 866 228 6 × 10-22 1.079 553 927 0 × 10-24 1.394 502 706 2 × 10-27 7.979 515 392 7 × 10-31
0°C to 400°C 0.000 000 000 0 3.874 810 636 4 × 10-2 3.329 222 788 0 × 10-5 2.061 824 340 4 × 10-7 -2.188 225 684 6 × 10-9 1.099 688 092 8 × 10-11 -3.081 575 877 2 × 10-14 4.547 913 529 0 × 10-17 -2.751 290 167 3 × 10-20 ............ ............ ............ ............ ............ ............
Type B thermocouples: coefficients (ci ) of polynomials for the computation of temperatures in °C as a function of the thermocouple emf in various temperature and emf ranges
Temperature Range:
250°C to 700°C
700°C to 1820°C
emf Range:
0.291 mV to 2.431 mV
2.431 mV to 13.820 mV
c0 c1 c2 c3 c4 c5 c6 c7 c8
9.842 332 1 × 101 6.997 150 0 × 102 -8.476 530 4 × 102 1.005 264 4 × 103 -8.334 595 2 × 102 4.550 854 2 × 102 -1.552 303 7 × 102 2.988 675 0 × 101 -2.474 286 0
2.131 507 1× 102 2.851 050 4 × 102 -5.274 288 7 × 101 9.916 080 4 -1.296 530 3 1.119 587 0 × 10-1 -6.062 519 9 × 10-3 1.866 169 6 × 10-4 -2.487 858 5 × 10-6
= = = = = = = = =
Type E thermocouples: coefficients (ci ) of polynomials for the computation of temperatures in °C as a function of the thermocouple emf in various temperature and emf ranges
Temperature Range:
-200°C to 0°C
0°C to 1000°C
emf Range:
-8.825 mV to 0.0 mV
0.0 mV to 76.373 mV
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
= = = = = = = = = =
0.000 000 0 1.697 728 8 × 101 -4.351 497 0 × 10-1 -1.585 969 7 × 10-1 -9.250 287 1 × 10-2 -2.608 431 4 × 10-2 -4.136 019 9 × 10-3 -3.403 403 0 × 10-4 -1.156 489 0 × 10-5 .......
TeamLRN
0.000 000 0 1.705 703 5 × 101 -2.330 175 9 × 10-1 6.543 558 5 × 10-3 -7.356 274 9 × 10-5 -1.789 600 1 × 10-6 8.403 616 5 × 10-8 -1.373 587 9 × 10-9 1.062 982 3 × 10-11 -3.244 708 7 × 10-14
Type J thermocouples: coefficients (ci ) of polynomials for the computation of temperatures in °C as a function of the thermocouple emf in various temperature and emf ranges
Temperature Range:
-210°C to 0°C
0°C to 760°C
760°C to 1200°C
emf Range:
-8.095 mV to 0.0 mV
0.0 mV to 42.919 mV
42.919 mV to 69.553 mV
c0 c1 c2 c3 c4 c5 c6 c7 c8
= = = = = = = = =
0.000 000 0 1.952 826 8 ×101 -1.228 618 5 -1.075 217 8 -5.908 693 3 × 10-1 -1.725 671 3 × 10-1 -2.813 151 3 × 10-2 -2.396 337 0 × 10-3 -8.382 332 1 × 10-5
0.000 000 1.978 425 × 101 -2.001 204 × 10-1 1.036 969 × 10-2 -2.549 687 × 10-4 3.585 153 × 10-6 -5.344 285 × 10-8 5.099 890 × 10-10 .......
-3.113 581 87 × 103 3.005 436 84 × 102 -9.947 732 30 1.702 766 30 × 10-1 -1.430 334 68 × 10-3 4.738 860 84 × 10-6 ....... ....... .......
Type K thermocouples: coefficients (ci ) of polynomials for the computation of temperatures in °C as a function of the thermocouple emf in various temperature and emf ranges
Temperature Range:
-200°C to 0°C
0°C to 500°C
500°C to 1372°C
emf Range:
-5.891 mV to 0.0 mV
0.0 mV to 20.644 mV
20.644 mV to 54.886 mV
0.000 000 0 2.508 355 × 101 7.860 106 × 10-2 -2.503 131 × 10-1 8.315 270 × 10-2 -1.228 034 × 10-2 9.804 036 × 10-4 -4.413 030 × 10-5 1.057 734 × 10-6 -1.052 755 × 10-8
-1.318 058 × 102 4.830 222 × 101 -1.646 031 5.464 731 × 10-2 -9.650 715 × 10-4 8.802 193 × 10-6 -3.110 810 × 10-8 ....... ....... .......
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
= = = = = = = = = =
0.000 000 0 2.517 346 2 × 101 -1.166 287 8 -1.083 363 8 -8.977 354 0 × 10-1 -3.734 237 7 × 10-1 -8.663 264 3 × 10-2 -1.045 059 8 × 10-2 -5.192 057 7 × 10-4 .......
Type N thermocouples: coefficients (ci ) of polynomials for the computation of temperatures in °C as a function of the thermocouple emf in various temperature and emf ranges
Temperature Range:
-200°C to 0°C
0°C to 600°C
600°C to 1300°C
emf Range:
-3.990 mV to 0.0 mV
0.0 mV to 20.613 mV
20.613 mV to 47.513 mV
0.000 00 3.868 96 × 101 -1.082 67 4.702 05 × 10-2 -2.121 69 × 10-6 -1.172 72 × 10-4 5.392 80 × 10-6 -7.981 56 × 10-8 ......... .........
1.972 485 × 101 3.300 943 × 101 -3.915 159 × 10-1 9.855 391 × 10-3 -1.274 371 × 10-4 7.767 022 × 10-7 ......... ......... ......... .........
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
= = = = = = = = = =
0.000 000 0 3.843 684 7 × 101 1.101 048 5 5.222 931 2 7.206 052 5 5.848 858 6 2.775 491 6 7.707 516 6 × 10-1 1.158 266 5 × 10-1 7.313 886 8 × 10-3
Type R thermocouples: coefficients (ci ) of polynomials for the computation of temperatures in °C as a function of the thermocouple emf in various temperature and emf ranges
Temperature Range:
-50°C to 250°C
250°C to 1200°C
1064°C to 1664.5°C
1664.5°C to 1768.1°C
emf Range:
-0.226 mV to 1.923 mV
1.923 mV to 13.228 mV
11.361 mV to 19.739 mV
19.739 mV to 21.103 mV
c0 = c1 = c2 = c3 = c4 = c5 = c6 = c7 = c8 = c9 = c10 =
0.000 000 0 1.889 138 0 × 102 -9.383 529 0 × 101 1.306 861 9 × 102 -2.270 358 0 × 102 3.514 565 9 × 102 -3.895 390 0 × 102 2.823 947 1 × 102 -1.260 728 1 × 102 3.135 361 1 × 101 -3.318 776 9
1.334 584 505 × 101 1.472 644 573 × 102 -1.844 024 844 × 101 4.031 129 726 -6.249 428 360 × 10-1 6.468 412 046 × 10-2 -4.458 750 426 × 10-3 1.994 710 149 × 10-4 -5.313 401 790 × 10-6 6.481 976 217 × 10-8 ..........
-8.199 599 416 × 101 1.553 962 042 × 102 -8.342 197 663 4.279 433 549 × 10-1 -1.191 577 910 × 10-2 1.492 290 091 × 10-4 .......... .......... .......... .......... ..........
TeamLRN
3.406 177 836 × 104 -7.023 729 171 × 103 5.582 903 813 × 102 -1.952 394 635 × 101 2.560 740 231 × 10-1 .......... .......... .......... .......... .......... ..........
Type S thermocouples: coefficients (ci ) of polynomials for the computation of temperatures in °C as a function of the thermocouple emf in various temperature and emf ranges
Temperature Range:
-50°C to 250°C
250°C to 1200°C
1064°C to 1664.5°C
1664.5°C to 1768.1°C
emf Range:
-0.235 mV to 1.874 mV
1.874 mV to 11.950 mV
10.332 mV to 17.536 mV
17.536 mV to 18.693 mV
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
= = = = = = = = = =
1.291 507 177 × 101 1.466 298 863 × 102 -1.534 713 402 × 101 3.145 945 973 -4.163 257 839 × 10-1 3.187 963 771 × 10-2 -1.291 637 500 × 10-3 2.183 475 087 × 10-5 -1.447 379 511 × 10-7 8.211 272 125 × 10-9
0.000 000 00 1.849 494 60 × 102 -8.005 040 62 × 101 1.022 374 30 × 102 -1.522 485 92 × 102 1.888 213 43 × 102 -1.590 859 41 × 102 8.230 278 80 × 101 -2.341 819 44 × 101 2.797 862 60
-8.087 801 117 × 101 1.621 573 104 × 102 -8.536 869 453 4.719 686 976 × 10-1 -1.441 693 666 × 10-2 2.081 618 890 × 10-4 .......... .......... .......... ..........
5.333 875 126 × 104 -1.235 892 298 × 104 1.092 657 613 × 103 -4.265 693 686 × 101 6.247 205 420 × 10-1 .......... .......... .......... .......... ..........
Type T thermocouples: coefficients (ci ) of polynomials for the computation of temperatures in °C as a function of the thermocouple emf in various temperature and emf ranges
Temperature Range:
-200°C to 0°C
0°C to 400°C
emf Range:
-5.603 mV to 0.0 mV
0.0 mV to 20.872 mV
c0 c1 c2 c3 c4 c5 c6 c7
= = = = = = = =
0.000 000 0 2.594 919 2 × 101 -2.131 696 7 × 10-1 7.901 869 2 × 10-1 4.252 777 7 × 10-1 1.330 447 3 × 10-1 2.024 144 6 × 10-2 1.266 817 1 × 10-3
0.000 000 2.592 800 × 101 -7.602 961 × 10-1 4.637 791 × 10-2 -2.165 394 × 10-3 6.048 144 × 10-5 -7.293 422 × 10-7 ..........
SECONDARY REFERENCE POINTS ON THE ITS-90 TEMPERATURE SCALE The International Temperature Scale of 1990 is described tin Section 1 of this Handbook, where the defining fixed points are listed. The Consultative Committee on Thermometry (CCT) of the International Committee on Weights and Measures (CIPM), which oversees the temperature scale, has recommended a number of secondary reference points whose values have been accurately determined with respect to the primary fixed points. The most accurate of these, referred to as “first quality points”, satisfy several criteria involving purity of the material, reproducibility, and documentation of the measurements. The CCT also lists “second quality points” that do not yet satisfy all the criteria but are still useful. Taken together, these secondary reference points, help fill in the gaps between the primary fixed points. The table below describes these secondary reference points. The best values resulting from the CCT evaluation are listed on both the Kelvin and Celsius scales, along with an estimate of uncertainty. Full details are given in the reference. The entries within each quality group are listed in order of increasing temperature.
REFERENCE Bedford, R. E., Bonnier, G., Maas, H., and Pavese, F., Metrologia 33, 133, 1996. Substance
Type of Transition
T90/K
t90/°C
Uncert.
-272.300 -271.9690 -270.9732 -269.7355 -265.9503 -263.8620 -254.461 -254.426 -248.609 -246.053 -209.999 -195.798 -185.847 -182.953 -182.456 -111.745 -56.558 -38.8290 0 29.7666 99.974 156.5936 271.402 321.069 327.462 630.628 779.63 1554.8 1768.2 1963 2446 2622 3414
0.0030 0.0025 0.0001 0.0025 0.0025 0.0025 0.001 0.001 0.001 0.001 0.001 0.002 0.001 0.001 0.001 0.001 0.001 0.0005
First quality points Zinc Aluminum Helium (4He) Indium Lead Niobium Deuterium (2H2) Deuterium (2H2) Neon (20Ne) Neon Nitrogen Nitrogen Argon Oxygen Methane Xenon Carbon dioxide Mercury Water Gallium Water Indium Bismuth Cadmium Lead Antimony Copper/71.9% silver Palladium Platinum Rhodium Iridium Molybdenum Tungsten
Superconductive transition Superconductive transition Superfluid transition Superconductive transition Superconductive transition Superconductive transition Triple point (equilibrium D2) Triple point (normal D2) Triple point Boiling point Triple point Boiling point Boiling point Condensation point Triple point Triple point Triple point Freezing point Ice point Triple point Boiling point Triple point Freezing point Freezing point Freezing point Freezing point Eutectic melting point Freezing point Freezing point Freezing point Freezing point Melting point Melting point
15-14
TeamLRN
0.8500 1.1810 2.1768 3.4145 7.1997 9.2880 18.689 18.724 24.541 27.097 63.151 77.352 87.303 90.197 90.694 161.405 216.592 234.3210 273.15 302.9166 373.124 429.7436 544.552 594.219 600.612 903.778 1052.78 1828.0 2041.3 2236 2719 2895 3687
0.0001 0.001 0.0002 0.001 0.001 0.001 0.001 0.05 0.1 0.4 3 6 4 7
SECONDARY REFERENCE POINTS ON THE ITS-90 TEMPERATURE SCALE (continued) Substance
Type of Transition
T90/K
t90/°C
Uncert.
-259.198 -252.762 -249.282 -237.536 -229.354 -157.375 -78.464 -49.596 15.650 20.476 26.864 36.315 58.065 97.794 122.336 122.352 356.619 444.614 567.807 567.807 802.018 882.940 1455 1495 1538 1670 1854 2053 2333
0.002 0.002 0.005 0.006 0.001 0.001 0.003 0.005 0.001 0.002 0.001 0.001 0.002 0.005 0.002 0.007 0.004 0.002 0.010 0.002 0.011 0.005 1 3 3 2 8 2 10
Second quality points Hydrogen Hydrogen Oxygen Nitrogen Oxygen Krypton Carbon dioxide Sulfur hexafluoride Gallium/20% indium Gallium/8% tin Diphenyl ether Ethylene carbonate Succinonitrile Sodium Benzoic acid Benzoic acid Mercury Sulfur Copper/66.9% aluminum Silver/30% aluminum Sodium chloride Sodium Nickel Cobalt Iron Titanium Zirconium Aluminum oxide Ruthenium
Triple point (normal H2) Boiling point (normal H2) - transition - transition - transition Triple point Sublimation point Triple point Eutectic melting point Eutectic melting point Triple point Triple point Triple point Freezing point Triple point Freezing point Boiling point Boiling point Eutectic melting point Eutectic melting point Freezing point Boiling point Freezing point Freezing point Freezing point Melting point Melting point Melting point Melting point
15-15
13.952 20.388 23.868 35.614 43.796 115.775 194.686 223.554 288.800 293.626 300.014 309.465 331.215 370.944 395.486 395.502 629.769 717.764 840.957 840.957 1075.168 1156.090 1728 1768 1811 1943 2127 2326 2606
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS This table summarizes the properties of 575 liquids that are commonly used in the laboratory as solvents or chemical reagents. The properties tabulated are: Mr : tm: tb : ρ: η: ε: µ: cp: vp: FP: Fl.Lim: IT TLV
Molecular weight Melting point in ˚C Normal boiling point in ˚C Density in g/mL at the temperature in ˚C indicated by the superscript Viscosity in mPa s (1 mPa s = 1 centipoise) Dielectric constant Dipole moment in D Specific heat capacity of the liquid at constant pressure at 25˚C in J/g K Vapor pressure at 25˚C in kPa (1 kPa = 7.50 mmHg) Flash point in ˚C Flammable (explosive) limit in air in percent by volume Autoignition temperature in ˚C Threshold limit for allowable airborne concentration in parts per million by volume at 25˚C and atmospheric pressure
Data on the temperature dependence of viscosity, dielectric constant, and vapor pressure can be found in the pertinent tables in this Handbook.
REFERENCES 1. 2. 3. 4. 5.
Lide, D. R., Handbook of Organic Solvents, CRC Press, Boca Raton, FL, 1994. Lide, D. R., and Kehiaian, H. V., Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, FL, 1994. Riddick, J. A., Bunger, W. B., and Sakano, T. K., Organic Solvents, Fourth Edition, John Wiley & Sons, New York, 1986. Fire Protection Guide to Hazardous Materials, 11th Edition, National Fire Protection Association, Quincy, MA, 1994. Urben, P. G., Ed., Bretherick’s Handbook of Reactive Chemical Hazards, 5th Edition, Butterworth-Heinemann, Oxford, 1995
Name Acetaldehyde Acetic acid Acetic anhydride Acetone Acetone cyanohydrin Acetonitrile Acetophenone Acetyl bromide Acetyl chloride Acrolein Acrylic acid Acrylonitrile Allyl alcohol Allylamine 2-Amino-2-methyl-1-propanol 3-Amino-1-propanol Aniline Anisole Antimony(V) chloride Antimony(V) fluoride Arsenic(III) chloride Benzaldehyde Benzene Benzeneacetonitrile Benzeneethanamine Benzeneethanol Benzenemethanethiol Benzenesulfonyl chloride Benzenethiol Benzonitrile Benzoyl chloride
Mol. Form. C2H4O C2H4O2 C4H6O3 C3H6O C4H7NO C2H3N C8H8O C2H3BrO C2H3ClO C3H4O C3H4O2 C3H3N C3H6O C3H7N C4H11NO C3H9NO C6H7N C7H8O Cl5Sb F5Sb AsCl3 C7H6O C6H6 C8H7N C8H11N C8H10O C7H8S C6H5ClO2S C6H6S C7H5N C7H5ClO
Mr
tm/°C
44.052 60.052 102.089 58.079 85.105 41.052 120.149 122.948 78.497 56.063 72.063 53.063 58.079 57.095 89.136 75.109 93.127 108.138 299.024 216.752 181.280 106.122 78.112 117.149 121.180 122.164 124.204 176.621 110.177 103.122 140.567
-123.37 16.64 -74.1 -94.7 -19 -43.82 20.5 -96 -112.8 -87.7 12.5 -83.48 -129 -88.2 25.5 12.4 -6.02 -37.13 4 8.3 -16 -57.1 5.49 -23.8 <0 -27 -30 14.5 -14.93 -13.99 -0.4
tb/°C 20.1 117.9 139.5 56.05 95 81.65 202 76 50.7 52.6 141 77.3 97.0 53.3 165.5 187.5 184.17 153.7 140 dec 141 130 178.8 80.09 233.5 195 218.2 194.5 251 dec 169.1 191.1 197.2
/ g mL–1 0.783418 1.044625 1.08220 0.784525 0.93219 0.785720 1.028120 1.662516 1.105120 0.84020 1.051120 0.800725 0.854020 0.75820 0.93420 0.982426 1.021720 0.994020 2.34 3.10 2.150 1.040125 0.876520 1.020515 0.964025 1.020220 1.05820 1.347015 1.077520 1.009315 1.212020
/ mPa s
/D
cp/ J g–1K–1
vp/ kPa
1.056 0.843 0.306
21.0 6.20 22.45 21.01
2.750 1.70 ≈2.8 2.88
2.020 2.053 1.648 2.175
120 2.07 0.680 30.8
0.369 1.681
36.64 17.44
3.92 3.02
2.229 1.703
0.368
15.8
2.72 3.1
1.491
11.9 0.049 16.2 38.4 36.2
1.218
3.85 1.056
0.604
1.267
15-16
TeamLRN
2.022 2.05 2.392
33.0 19.7
3.87 1.60 1.2
7.06 4.30 3.222
1.13 1.38
2.061 1.840
0.090 0.472
1.59 3.0 0 3.5
1.621 1.741
0.169 12.7
17.85 2.2825 17.87 12.31 4.705 28.90 4.26 25.9 23.0
14.1 3.14 33.1
2.068
1.23 4.18
1.572 1.602
FP/ °C -39 39 49 -20 74 6 77 4 -26 50 0 21 -29 67 80 70 52
63 -11 113
Fl. Lim. 4-60% 4-20% 2.7-10.3% 3-13% 2.2-12% 3-16%
2.8-31% 2.4-8% 3-17% 3-18% 2-22%
1.3-11%
1-8%
IT/ °C 175 463 316 465 688 524 570 390 220 438 481 378 374
615 475
192 498
TLV/ ppm 10 5 500 40 10
2 2 0.5
2
0.5
96
0.5 0.11 0.084
72
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS (continued)
Name
Mol. Form.
Mr
tm/°C
tb/°C
/ g mL–1
2.0 -39 12
213 205.31 185 144 207 266.5
1.055020 1.041924 0.981320 1.11320 0.956920
143.012 114.958 390.557 122.186 250.523 117.169 159.808 157.008 137.018 137.018 129.384 163.829 108.965 106.949 137.018 151.045 122.992 122.992 120.976 171.035 198.274 72.106 90.121 90.121 90.121 86.090 69.106 90.187 90.187 88.106 158.195 74.121 74.121
-51.9 -41.5 -55 -10.2 -45 -107 -7.2 -30.72 -112.6 -112.65 -87.9 -57 -118.6 -139.54 -16.2 -88.0 -110.3 -89.0 -119 -27.8 -5.65 -96.86 -77 20.4 7.6 -1.2 -111.9 -115.7 -165 -5.1 -75 -88.6 -88.5
178.5 106 384 282 91 12.65 58.8 156.06 101.6 91.3 68.0 90 38.5 15.8 73.3 129.8 71.1 59.5 70.1 181.7 105 74.8 207.5 235 182.5 88 117.6 98.5 85.0 163.75 200 117.73 99.51
1.2220 1.32315 0.98125 1.179325 2.6
72.106 70.090 86.090 118.174 116.158 116.158 128.169 73.137 73.137 73.137 134.218 134.218 178.228 102.174 90.121 148.245 100.158 86.090 76.141 78.497 92.524 112.942 127.572 127.572 112.557 88.536
-86.64 -76 15 -74.8 -78 -98.9 -64.6 -49.1 <-72 -66.94 -87.85 -57.8 -22.4 -94 6 -52 -92 -43.61 -112.1 -16.3 -44.5 -22 -1.9 -10.28 -45.31 -130
79.59 102.2 169 168.4 126.1 112 145 77.00 62.73 44.04 183.31 169.1 250.3 72.6 89 dec 190 94 204 46 85.5 119 106 208.8 230.5 131.72 59.4
Benzyl acetate Benzyl alcohol Benzylamine 2,2’-Bioxirane Bis(2-aminoethyl)amine N,N’-Bis(2-aminoethyl)-1,2ethanediamine Bis(2-chloroethyl) ether Bis(chloromethyl) ether Bis(2-ethylhexyl) phthalate Bis(2-hydroxyethyl) sulfide Boron tribromide Boron trichloride Bromine Bromobenzene 1-Bromobutane 2-Bromobutane, (±)Bromochloromethane Bromodichloromethane Bromoethane Bromoethene 2-Bromo-2-methylpropane 1-Bromopentane 1-Bromopropane 2-Bromopropane 3-Bromopropene 2-Bromotoluene Bromotrichloromethane Butanal 1,3-Butanediol 1,4-Butanediol 2,3-Butanediol 2,3-Butanedione Butanenitrile 1-Butanethiol 2-Butanethiol Butanoic acid Butanoic anhydride 1-Butanol 2-Butanol 2-Butanone
C9H10O2 C7H8O C7H9N C4H6O2 C4H13N3 C6H18N4
150.174 108.138 107.153 86.090 103.166 146.234
-51.3 -15.4
C4H8Cl2O C2H4Cl2O C24H38O4 C4H10O2S BBr3 BCl3 Br2 C6H5Br C4H9Br C4H9Br CH2BrCl CHBrCl2 C2H5Br C2H3Br C4H9Br C5H11Br C3H7Br C3H7Br C3H5Br C7H7Br CBrCl3 C4H8O C4H10O2 C4H10O2 C4H10O2 C4H6O2 C4H7N C4H10S C4H10S C4H8O2 C8H14O3 C4H10O C4H10O C4H8O
trans-2-Butenal cis-2-Butenoic acid 2-Butoxyethanol Butyl acetate sec-Butyl acetate Butyl acrylate Butylamine sec-Butylamine tert-Butylamine Butylbenzene tert-Butylbenzene Butyl benzoate tert-Butyl ethyl ether tert-Butyl hydroperoxide 1-tert-Butyl-4-methylbenzene Butyl vinyl ether -Butyrolactone Carbon disulfide Chloroacetaldehyde Chloroacetone Chloroacetyl chloride 2-Chloroaniline 3-Chloroaniline Chlorobenzene 2-Chloro-1,3-butadiene
C4H6O C4H6O2 C6H14O2 C6H12O2 C6H12O2 C7H12O2 C4H11N C4H11N C4H11N C10H14 C10H14 C11H14O2 C6H14O C4H10O2 C11H16 C6H12O C4H6O2 CS2 C2H3ClO C3H5ClO C2H2Cl2O C6H6ClN C6H6ClN C6H5Cl C4H5Cl
/ mPa s 5.47 1.624
12.62 10.76
1.9
2.462
0.03
98
21.20 3.51 5.3 28.61
2.6
1.545
0.143
55
2.84
1.804
2.54 3.10
0.799925 0.851620 1.026720 0.901520 0.882520 0.874820 0.889820 0.741420 0.724620 0.695820 0.860120 0.866520 1.00020 0.73625 0.896020 0.861220 0.788820 1.129620 1.263220 1.19 1.1520 1.420220
0.405
1.426
0.685
0.574
0.950
0.352
3.32 1.216120 1.105820 0.95620
15-17
0.753
FP/ °C
0.989 2.015
4.04 24.83 5.204 5.645 2.98 12.8 17.84 17.26
0.553
vp/ kPa
1.22 1.71
0.944 1.074 0.606
0.489 0.458 0.471
cp/ J g–1K–1
5.34 11.916 5.18
3.1028 1.495020 1.275820 1.258520 1.934420 1.98020 1.460420 1.493320 1.427820 1.218220 1.353720 1.314020 1.39820 1.423220 2.01225 0.801620 1.005320 1.017120 1.003320 0.980818 0.793620 0.841620 0.829520 0.952825 0.966820 0.809520 0.806320
0.374
/D
3.1484 5.45 7.315 8.64
9.01 5.63 10.98 6.31 8.09 9.46 7.0 4.641 2.405 13.45 28.8 31.9
0 0 0 1.70 2.08 2.23 1.7
0.911 0.474 0.983 0.798 0.41
2.03 1.42 2.17 2.20 2.18 2.21 ≈1.9
0.925 1.007 1.102 0.875 0.702 1.075
2.72
2.270 2.521 2.220 2.363
2.58
358
1
3%-
369
5 0.001
218 160
298
0.1 0.556 5.26 9.32 19.5 62.5 141 17.7 1.68 18.6 28.9 18.6 5.35 15.7 0.008
51 18 21
2.6-6.6%
565 265 200
7-8% 9-15%
511 530
5 0.5
32 490 -1 79
4.4-7.3%
295
-22 121 121
2-12.5%
218 395 402
1-11% 2.1-15.5%
404 232
200
2.378 1.961
0.15 1.66 0.731 12.2
4-13% 2-8% 1.7-9.8% 1.7-9.9% 2-10%
238 425
1.958 2.450
20 150 200 2 5
2.627 1.813 1.773
48.4 0.150 0.280
69 22 31 29 -12 -9 -9 71 60 107
2-9% 0.8-5.8% 0.7-5.7%
380 410 450
2.13
16.5
2.78 3.67
9.30 5.07 5.135 5.25 4.71
2.1 1.9 1.87
2.316 1.642 1.003
2.23 1.77 1.69
2-7%
-9 13
18.56
13.40 13.3 5.6895 4.914
10
12.6 4.92
1.66 1.8
2.027 1.793 2.391 2.656
39.0 2.6320
460 436
2.201 1.361
1.65
≈0 1.25 4.27 0
TLV/ ppm
0.86 2.32
2.301 1.898
58.5 2.359 2.359 5.52
90 93
IT/ °C
27 24 2 -23 72 54 37 24
3.9 1.53
1.0 1.28 1.3 ≈0 ≈0.83
0.015 0.096
Fl. Lim.
7.45 2.55 6.07 10.8 0.221
0.09 6.65 0.43 48.2
>1.6%
501 0.5
2-10% 0.9-5.8% 1-11% 2-10%
443 279 343 405
292 312
5 27 68 -9 98 -30
1 255 1-50%
90
3.33 0.034 1.558 1.334
1.6 29.5
50 100
10
0.05
28 -20
1-10% 4-20%
705 593
10 10
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS (continued)
Name
Mol. Form.
1-Chlorobutane 2-Chlorobutane Chlorocyclohexane Chlorodibromomethane
C4H9Cl C4H9Cl C6H11Cl CHBr2Cl
Chloroethane 2-Chloroethanol 2-Chloroethyl vinyl ether (Chloromethyl)benzene Chloromethyl methyl ether 1-Chloro-2-methylpropane 2-Chloro-2-methylpropane 1-Chloronaphthalene 1-Chlorooctane 1-Chloropentane 2-Chlorophenol 1-Chloropropane
C2H5Cl C2H5ClO C4H7ClO C7H7Cl C2H5ClO C4H9Cl C4H9Cl C10H7Cl C8H17Cl C5H11Cl C6H5ClO C3H7Cl
2-Chloropropane 3-Chloro-1,2-propanediol 3-Chloropropanenitrile 2-Chloropropene 3-Chloropropene Chlorosulfonic acid 2-Chlorotoluene 4-Chlorotoluene Chromyl chloride trans-Cinnamaldehyde o-Cresol m-Cresol p-Cresol Cyanogen chloride Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclohexylamine 1,3-Cyclopentadiene Cyclopentane Cyclopentanol Cyclopentanone cis-Decahydronaphthalene trans-Decahydronaphthalene Decamethylcyclopentasiloxane Decanal Decane Decanoic acid 1-Decanol 1-Decene Diacetone alcohol Dibenzyl ether Dibromodifluoromethane 1,2-Dibromoethane Dibromomethane 1,2-Dibromotetrafluoroethane Dibutylamine Dibutyl ether Di-tert-butyl peroxide Dibutyl phthalate Dibutyl sebacate Dibutyl sulfide Dichloroacetic acid o-Dichlorobenzene m-Dichlorobenzene trans-1,4-Dichloro-2-butene Dichlorodimethylsilane 1,1-Dichloroethane
C3H7Cl C3H7ClO2 C3H4ClN C3H5Cl C3H5Cl ClHO3S C7H7Cl C7H7Cl Cl2CrO2 C9H8O C7H8O C7H8O C7H8O CClN C4H8 C6H12 C6H12O C6H10O C6H10 C6H13N C5H6 C5H10 C5H10O C5H8O C10H18 C10H18 C10H30O5Si5 C10H20O C10H22 C10H20O2 C10H22O C10H20 C6H12O2 C14H14O CBr2F2 C2H4Br2 CH2Br2 C2Br2F4 C8H19N C8H18O C8H18O2 C16H22O4 C18H34O4 C8H18S C2H2Cl2O2 C6H4Cl2 C6H4Cl2 C4H6Cl2 C2H6Cl2Si C2H4Cl2
Mr
tm/°C
tb/°C
/ g mL–1
/ mPa s 0.422
92.567 92.567 118.604
-123.1 -131.3 -43.81
78.4 68.2 142
0.885720 0.873220 1.00020
208.280 64.514 80.513 106.551 126.584 80.513 92.567 92.567 162.616 148.674 106.594 128.556
-20 -138.4 -67.5 -70 -45 -103.5 -130.3 -25.60 -2.5 -57.8 -99.0 9.4
120 12.3 128.6 108 179 59.5 68.5 50.9 259 183.5 108.4 174.9
2.45120 0.92390 1.201920 1.049520 1.100420 1.06310 0.877320 0.842020 1.188025 0.873420 0.882020 1.263420
78.541 78.541 110.540 89.524 76.525 76.525 116.525 126.584 126.584 154.900 132.159 108.138 108.138 108.138 61.471 56.107 84.159 100.158 98.142 82.143 99.174 66.102 70.133 86.132 84.117 138.250 138.250 370.770 156.265 142.282 172.265 158.281 140.266 116.158 198.260 209.816 187.861 173.835 259.823 129.244 130.228 146.228 278.344 314.461 146.294 128.942 147.002 147.002 124.997 129.061 98.959
-122.9 -117.18
46.5 35.7 213 dec 175.5 22.6 45.1 152 159.0 162.4 117 246 191.04 202.27 201.98 13 12.6 80.73 160.84 155.43 82.98 134 41 49.3 140.42 130.57 195.8 187.3 210 208.5 174.15 268.7 231.1 170.5 167.9 298 22.76 131.6 97 47.35 159.6 140.28 111 340 344.5 185 194 180 173 155.4 70.3 57.3
0.889920 0.861720 1.32518 1.157320 0.901720 0.937620 1.75 1.082520 1.069720 1.91 1.049720 1.032735 1.033920 1.018540 1.18620 0.70380 0.773925 0.962420 0.947820 0.811020 0.819120 0.802120 0.745720 0.948820 0.948720 0.896520 0.865925 0.959320 0.83015 0.726625 0.885840 0.829720 0.740820 0.938720 1.042820
0.334 0.303
2.168325 2.496920 2.14925 0.767020 0.768420 0.70420 1.046520 0.940515 0.838620 1.563420 1.305920 1.288420 1.18325 1.06425 1.175720
-51 -137.4 -134.5 -80 -35.8 7.5 -96.5 -7.5 31.03 12.24 34.77 -6.5 -90.7 6.59 25.93 -27.9 -103.5 -17.8 -85 -93.4 -17.5 -51.90 -42.9 -30.4 -38 -4.0 -29.6 31.4 6.9 -66.3 -44 1.8 -110.1 9.84 -52.5 -110.32 -62 -95.2 -40 -35 -10 -79.7 10 -17.0 -24.8 1.0 -16 -96.9
3.59
/D
cp/ J g–1K–1
vp/ kPa
FP/ °C
Fl. Lim.
IT/ °C
TLV/ ppm
7.276 8.564 7.9505
2.05 2.04 2.1
1.891
13.7 21.0
-12 -10 32
2-10%
240
9.45 25.80
2.05 1.78
1.617
160
4-15% 5-16%
519 425
100
6.854
1.8
1.44
1%-
585
1
7.027 9.663 5.04 5.05 6.654 7.40
2.00 2.13 1.57 2.00 2.16
1.713 1.867 1.307 1.335
0.164 24.9 19.9 42.7
-50 60 27 67
2-8.7%
4.36 0.308
-6 0 121 70 13 64
1.6-8.6%
260
8.588
2.05 2.17
45.8 68.9
<-18 -32
2.6-11% 2.8-11%
520 593
1.635
110 48.9
76 -37 -32
4.5-16% 2.9-11%
485
1.318
0.482
1.468 1.683
>558
31.0
0.314
8.92 8.2
1.647 1.94
0.964 0.837
4.721 6.25
1.56 2.21
1 50 0.025
12.91
17.72 6.76 12.44 13.05
0.894 57.5 2.02 0.625 1.944
2.0243 16.40 16.1 2.2176 4.547
0.413
1.9687 18.5 13.58 2.219 2.184 2.50
3.04 1.948
1.45 1.48 1.48 2.8331 0 ≈0 2.87 0.33 1.3 0.419 ≈0 3.3 ≈0 ≈0
0.838
1.9853
≈0
10.91 0.756 2.80
7.93 2.136 18.2 3.821
≈0 3.2
1.595 0.980
4.9612 7.77 2.34 2.765 3.0830
0.918 0.637 16.63
1.324 1.044
6.58 4.54 4.29 8.33 10.12 5.02
0.66 1.2 1.43
2.160 2.080 2.044
1.841 2.079 1.856 1.805
1.837 2.119 1.84 1.678 1.653
2.210 2.761 2.341 2.144 1.905
0.041 0.019 0.017
81 86 86
>1.4% >1.1% >1.1%
599 558 558
5 5 5
157 13.0 0.10 0.53 11.8 1.20 58.5 42.3 0.294 1.55 0.10 0.164
<10 -20 68 44 -12 31
>1.8% 1-8% 1-9% 1-9% >1.2% 1-9%
245 300 420 310 293
-25 51 26
2%-
361
300 50 25 300 10 75 600
54
1-5%
255
0.170
51
0.8-5.4%
210
0.009 0.210 0.224
82 <55 58 135
2-7%
288 235 643
110 1.55 6.12 43.4 0.34 0.898 3.43
1.0 1.17
0.724 0.61 0.69 2.266 2.136
2.82 2.48 1.61
1.789 1.968 1.943
2.50 1.72
1.105 1.163
0.18 0.252
1.276
18.9 30.5
50 100
47 25 18 157 178 76
1-6% 1.5-7.6%
194
>0.5% >0.4%
402 365
66 72
2-9%
648
<21 -17
3.4-9.5% 5-11%
25 0.005
0.464
15-18
TeamLRN
10.10
2.06
458
100
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS (continued)
Name
Mol. Form.
Mr
tm/°C
98.959 96.943 96.943
-35.7 -122.56 -80.0
96.943 84.933 161.029 112.986 112.986 112.986 110.970 161.029 181.318 105.136 118.174 118.174
-49.8 -97.2 -17
73.137 149.233 134.218 134.218 134.218 118.131 106.120 162.227 134.173 162.227 134.173 176.211
-49.8 -38.8 -31.2 -83.9 -42.83 -43 -10.4 -45 -68 -68
1,2-Dichloroethane 1,1-Dichloroethene cis-1,2-Dichloroethene trans-1,2-Dichloroethene
C2H4Cl2 C2H2Cl2 C2H2Cl2 C2H2Cl2
Dichloromethane (Dichloromethyl)benzene 1,1-Dichloropropane 1,2-Dichloropropane, (±)1,3-Dichloropropane 2,3-Dichloropropene 2,4-Dichlorotoluene Dicyclohexylamine Diethanolamine 1,1-Diethoxyethane 1,2-Diethoxyethane Diethylamine
CH2Cl2 C7H6Cl2 C3H6Cl2 C3H6Cl2 C3H6Cl2 C3H4Cl2 C7H6Cl2 C12H23N C4H11NO2 C6H14O2 C6H14O2 C4H11N
N,N-Diethylaniline o-Diethylbenzene m-Diethylbenzene p-Diethylbenzene Diethyl carbonate Diethylene glycol Diethylene glycol diethyl ether Diethylene glycol dimethyl ether Diethylene glycol monobutyl ether Diethylene glycol monoethyl ether Diethylene glycol monoethyl ether acetate Diethylene glycol monomethyl ether Diethyl ether Diethyl maleate Diethyl malonate Diethyl oxalate Diethyl phthalate Diethyl succinate Diethyl sulfate Diethyl sulfide Diiodomethane Diiodosilane Diisobutylamine Diisopentyl ether Diisopropylamine Diisopropyl ether 1,2-Dimethoxyethane Dimethoxymethane Dimethylacetal N,N-Dimethylacetamide 2,3-Dimethylaniline 2,6-Dimethylaniline N,N-Dimethylaniline 2,2-Dimethylbutane 2,3-Dimethylbutane 3,3-Dimethyl-2-butanone Dimethylcarbamic chloride Dimethyl disulfide N,N-Dimethylethanolamine
C10H15N C10H14 C10H14 C10H14 C5H10O3 C4H10O3 C8H18O3 C6H14O3 C8H18O3 C6H14O3 C8H16O4
C4H10O C8H12O4 C7H12O4 C6H10O4 C12H14O4 C8H14O4 C4H10O4S C4H10S CH2I2 H2I2Si C8H19N C10H22O C6H15N C6H14O C4H10O2 C3H8O2 C4H10O2 C4H9NO C8H11N C8H11N C8H11N C6H14 C6H14 C6H12O C3H6ClNO C2H6S2 C4H11NO
74.121 172.179 160.168 146.141 222.237 174.195 154.185 90.187 267.836 283.911 129.244 158.281 101.190 102.174 90.121 76.095 90.121 87.120 121.180 121.180 121.180 86.175 86.175 100.158 107.539 94.199 89.136
-116.2 -8.8 -50 -40.6 -40.5 -21 -24 -103.91 6.1 -1 -73.5
N,N-Dimethylformamide 2,6-Dimethyl-4-heptanone 1,1-Dimethylhydrazine Dimethyl phthalate 2,6-Dimethylpyridine Dimethyl sulfate Dimethyl sulfide Dimethyl sulfoxide 1,4-Dioxane
C3H7NO C9H18O C2H8N2 C10H10O4 C7H9N C2H6O4S C2H6S C2H6OS C4H8O2
73.094 142.238 60.098 194.184 107.153 126.132 62.134 78.133 88.106
C5H12O3
-100.53 -99.5 10 -13.5 -0.1 28 -100 -74.0
-25
120.147
/ g mL–1
/ mPa s
83.5 31.6 60.1
1.245425 1.21320 1.283720
0.779
48.7 40 205 88.1 96.4 120.9 94 201 256 dec 268.8 102.25 121.2
1.256520 1.326620 1.2625 1.132120 1.156020 1.178525 1.21120 1.247620 0.912320 1.096620 0.825420 0.835125
0.317 0.413
55.5 216.3 184 181.1 183.7 126 245.8 188 162 231 196 218.5
0.705620 0.930720 0.880020 0.860220 0.862020 0.969225 1.119715 0.906320 0.943420 0.955320 0.988520 1.009620
0.319
tb/°C
193
1.035
0.445
30.2 0.989
0.7138 1.066220 1.055120 1.078520 1.23214 1.040220 1.17225 0.836220 3.321120
-61 -85.4 -69.20 -105.1 -113.2 -18.59 <-15 11.2 2.42 -98.8 -128.10 -52.5 -33 -84.67 -59
34.5 223 200 185.7 295 217.7 208 92.1 182 150 139.6 172.5 83.9 68.4 84.5 42 64.5 165 221.5 215 194.15 49.73 57.93 106.1 167 109.74 134
-60.48 -41.5 -57.20 5.5 -6.1 -31.7 -98.24 17.89 11.85
153 169.4 63.9 283.7 144.01 188 dec 37.33 189 101.5
0.944525 0.806220 0.79122 1.190520 0.922620 1.332220 0.848320 1.101025 1.033720
0.422
15-19
0.393 0.379 0.455
FP/ °C
IT/ °C
TLV/ ppm
10.6 80.0 26.8
13 -28 6
6-16% 7-16% 3-15%
413 570 460
10 5 200
2.14 8.93 6.9
0 1.60 2.1
1.205 1.192
2
6-13% 13-23%
460 556
200 50
8.37 10.27
1.8 2.08
1.320
44.2 58.2 0.06 9.09 6.62
21
3-15%
557
75
15
2.6-7.8%
5.68
1.70
25.75 3.80 3.90
2.8 1.4
2.22 2.01 2.195
<0.01 3.68 4.33
>99 172 -21 27
2-13% 2-10%
662 230 205
0.46
3.680 5.15 2.594 2.369 2.259 2.820 31.82 5.70 7.23
0.92
2.313
30.1
2-10%
5
0.7-6%
312 630 395 450 430
1.10 2.3
1.80 2.307 2.104 2.043 2.188 2.243
1.63 0.001
2-17%
224
2.0
0.055
0.315
-23 85 57 56 55 25 124 82 67
0.017 0.029
96 110
425
1.6
2.256
0.024
96
1-23%
240
4.2666 7.560 7.550 8.266 7.86 6.098 29.2 5.723 5.32
1.15
2.369
71.7
2-36%
180 350
2.54 2.49
1.779 1.784 1.647
0.030
-45 121 93 76 161 90 104
>0.7%
457
1.54 1.08
1.900 0.500
7.78 0.172
2.817
1.23 1.15 1.13
2.394
3.805 7.30 2.644
0.972 0.210 10.7 19.9 9.93 53.1 22.9 0.075
1.1-7.1% 1-8%
70 97 96 63 -48 -29
2-12% >1%
3.7
2.016
1.63 1.68 ≈0 ≈0
1.971 1.771 2.227 2.201
9.6
1.8
1.551
3.82
24
0.794
38.25 9.91
3.82 2.7
0.439 0.23
14.36
8.66 7.33 55.0 6.70 47.24 2.2189
1.7
2.060 2.090 2.731 1.561 1.728
1.554 3.96 0
1.901 1.958 1.726
64.4 0.084 4.95
38.85
1.300 0.351 0.361
4.90 1.869 1.889 12.73
42.5 31.3 4.27
436
-1 -28 -2 -32
0.7
0.927
400
29
2.122 2.145 2.129
0.284 1.987 1.177
Fl. Lim.
1.298 1.148 1.201
0.723 0.777720 0.715320 0.719225 0.863725 0.859320 0.850120 0.937225 0.993120 0.984220 0.955720 0.644425 0.661620 0.722925 1.16825 1.062520 0.886620
vp/ kPa
1.8 1.34 1.90
1.6 1.8
0.224
cp/ J g–1K–1
10.42 4.60 9.20
20
20
/D
2-14%
316 443 202 237
5 250
490
10 0.5 0.5 5 500 500
1000
1.2-7% 1.2-7%
371 405 405
58 49 -15 146
2-15% 1-7% 2-95% >0.9%
445 396 249 490
10 25 0.01
83 -37 95 12
188 206 215 180
0.1
2.2-20% 3-42% 2-22%
0.746
20
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS (continued)
Name
Mol. Form.
1,3-Dioxolane Dipentyl ether Dipropylamine Dipropylene glycol monomethyl ether
C3H6O2 C10H22O C6H15N C7H16O3
Dipropyl ether Dodecane
C6H14O C12H26 C12H26O C12H24 C3H5ClO C4H8O C8H12O2
1-Dodecanol 1-Dodecene Epichlorohydrin 1,2-Epoxybutane 1,2-Epoxy-4(epoxyethyl)cyclohexane 1,2-Ethanediamine 1,2-Ethanediol 1,2-Ethanediol, diacetate
C2H8N2 C2H6O2 C6H10O4
1,2-Ethanediol, dinitrate 1,2-Ethanedithiol Ethanethiol Ethanol Ethanolamine 4-Ethoxyaniline Ethoxybenzene 2-Ethoxyethanol 2-Ethoxyethyl acetate Ethyl acetate Ethyl acetoacetate Ethyl acrylate Ethylamine N-Ethylaniline Ethylbenzene Ethyl benzoate Ethyl butanoate 2-Ethyl-1-butanol Ethyl chloroacetate Ethyl chloroformate Ethyl cyanoacetate Ethyleneimine Ethyl formate 2-Ethylhexanal 2-Ethyl-1,3-hexanediol 2-Ethyl-1-hexanol 2-Ethylhexyl acetate Ethyl lactate Ethyl 3-methylbutanoate Ethyl 2-methylpropanoate Ethyl nitrite Ethyl propanoate Ethyl silicate Eucalyptol Fluorobenzene Fluorosulfonic acid Formamide Formic acid Furan Furfural Furfuryl alcohol Germanium(IV) chloride
C2H4N2O6 C2H6S2 C2H6S C2H6O C2H7NO C8H11NO C8H10O C4H10O2 C6H12O3 C4H8O2 C6H10O3 C5H8O2 C2H7N C8H11N C8H10 C9H10O2 C6H12O2 C6H14O C4H7ClO2 C3H5ClO2 C5H7NO2 C2H5N C3H6O2 C8H16O C8H18O2 C8H18O C10H20O2 C5H10O3 C7H14O2 C6H12O2 C2H5NO2 C5H10O2 C8H20O4Si C10H18O C6H5F FHO3S CH3NO CH2O2 C4H4O C5H4O2 C5H6O2 Cl4Ge
Glycerol Glycerol triacetate Glycerol trioleate Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanone
C3H8O3 C9H14O6 C57H104O6 C7H14O C7H16 C7H14O2 C7H16O C7H14O
Mr
tm/°C
tb/°C
/ g mL–1
74.079 158.281 101.190
-97.22 -69 -63
78 190 109.3
1.06020 0.783320 0.740020
148.200
-80
188.3
0.95
/ mPa s
0.517
2.798 2.923
/D 1.19 1.20 1.03
cp/ J g–1K–1 1.593 1.579 2.500
vp/ kPa 14.6 3.21
FP/ °C
Fl. Lim.
2 57 17
IT/ °C 170 299
100
102.174
-114.8
90.08
0.746620
0.396
3.38
1.21
2.169
8.35
21
1.3-7%
188
170.334 186.333 168.319 92.524 72.106 140.180
-9.57 23.9 -35.2 -26 -150 <-55
216.32 260 213.8 118 63.4 227
0.749520 0.830924 0.758420 1.181220 0.829720 1.096620
1.383
2.0120 5.82 2.152 22.6
≈0
2.206 2.351 2.143 1.422 2.039
0.016
74 127 79 31 -22
>0.6%
203 275
4-21% 1.7-19%
411 439
11.14 -12.69
117 197.3
0.8979 1.113520
146.141 152.062 94.199 62.134 46.068 61.083 137.179 122.164 90.121 132.157 88.106 130.141 100.117 45.084 121.180 106.165 150.174 116.158 102.174 122.551 108.524 113.116 43.068 74.079 128.212 146.228 130.228 172.265 118.131 130.185 116.158 75.067 102.132 208.329 154.249 96.102 100.070 45.041 46.026 68.074 96.085 98.101
-31 -22.3 -41.2 -147.88 -114.14 10.5 4.6 -29.43 -70 -61.7 -83.8 -45 -71.2 -80.5 -63.5 -94.96 -34 -98 <-15 -21 -80.6 -22.5 -77.9 -79.6 <-100 -40 -70 -80 -26 -99.3 -88.2
190 198.5 146.1 35.0 78.29 171 254 169.81 135 156.4 77.11 180.8 99.4 16.5 203.0 136.19 212 121.3 147 144.3 95 205 56 54.4 163 244 184.6 199 154.5 135.0 110.1 18 99.1 168.8 176.4 84.73 163 220 101 31.5 161.7 171
1.104320 1.491820 1.23420 0.831525 0.789320 1.018020 1.065216 0.965120 0.925325 0.974020 0.900320 1.036810 0.923420 0.68915 0.962520 0.862625 1.041525 0.873525 0.832620 1.158520 1.135220 1.065420 0.83225 0.920820 0.854020 0.932522 0.831925 0.871820 1.032820 0.865620 0.86820 0.89915 0.884325 0.932020 0.926720 1.022520 1.726 1.133420 1.22020 0.951420 1.159420 1.129620
214.42 92.094 218.203 885.432 114.185 100.202 130.185 116.201 114.185
-51.50 18.1 -78 -4 -43.4 -90.55 -7.17 -33.2 -35
86.55 290 259
1.88 1.261320 1.158320 0.91515 0.813225 0.679525 0.912425 0.821920 0.811120
152.8 98.4 222.2 176.45 151.05
1.20 1.073
≈0 1.8 1.891
0.019 2.2 31.7
0.5 0.1
20
60.098 62.068
-73.9 -82.5 0.8 -42.18 -89 2.49 8.3 -85.61 -38.1 -14.6
TLV/ ppm
16.06
0.287 1.074 21.1 1.197
0.423
2.05 0.631 0.639
13.82 41.4
1.99 2.28
2.872 2.394
1.62 0.01
40 111
3-12% 3-22%
385 398
7.7 28.26 7.26 6.667 25.3 31.94 7.43 4.216 13.38 7.567 6.0814 14.0 6.05 8.7 5.87 2.4463 6.20 5.18 6.19
2.34
2.121
0.030 0.009
88
1.6-8.4%
482
1.898 2.438 3.201
70.3 7.87 0.05
1.870 2.339 2.845 1.937 1.906
0.204 0.71 0.24 12.6 0.095 5.14 141
-17 13 86 116 63 43 56 -4 57 10 -16 85 21 88 24 57 64 16 110 -11 -20 44 127 73 71 46
0.380
9.736 31.62 18.3 8.57
6.27
18.73 7.58 15.4 4.71
0.501
2.03 1.60 1.69 2.3 1.45 2.1 2.2 1.78 1.96 1.22
2.884
0.59 2.00 1.74
1.726 1.638 1.963
2.17 1.90 1.9
1.947
1.74 1.8 2.4
1.74
1.60
3.34 1.607 0.361 1.587
111.0 51.1 2.94 42.1 16.85
3.73 1.425 0.66 3.5 1.9
934
46.53 7.11 3.109 9.07 1.9209 3.04 11.75 11.95
0.387 3.84 5.81 0.714
15-20
TeamLRN
2.01 0.206 0.640
2.015
28.9 32.3
2.438
0.019
2.150 1.07 3.25
5.76 2.50 4.57 5.465
0.550
1.28
0 2.6
≈0
2.6
10 35 0.05
2.8-18% 3-19% 3-24%
300 363 410
0.5 1000 3
3-18% 2-8% 2-12% 1-10% 1.4-14% 4-14%
235 379 426 295 372 385
5 5 400
1-7%
432 490 463
100
5 5
500 3.3-55% 3-16% 0.9-7.2% 0.8-9.7% 1-8% >1.5%
320 455 190 360 231 268 400
4-50% 1.9-11%
90 440
0.5 100
1.523
4.97 1.17 0.260 10.4
13 -35 12 52 48 -15
2.389 2.151 1.686 1.698 2.079
5.75 80.0 0.29 0.097
154 50 -36 60 75
18-57% 2-14% 2-19% 2-16%
434
10 5
316 491
2 10
2.377 1.763
<0.01 <0.01
199 138
3-19% 1%-
370 433
6.09
-4
1-7%
204 275
400
0.49
39
1-8%
393
50
1.920 1.749
2.015 2.242 2.039 2.342 2.037
10
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS (continued)
Name
Mol. Form.
3-Heptanone 4-Heptanone 1-Heptene Hexachloro-1,3-butadiene
C7H14O C7H14O C7H14 C4Cl6
Hexachloro-1,3-cyclopentadiene Hexafluorobenzene Hexamethyldisiloxane Hexamethylphosphoric triamide Hexanal Hexane Hexanedinitrile Hexanoic acid 1-Hexanol 2-Hexanone 1-Hexene Hexyl acetate
C5Cl6 C6F6 C6H18OSi2 C6H18N3OP C6H12O C6H14 C6H8N2 C6H12O2 C6H14O C6H12O C6H12 C8H16O2
Hydrazine Hydrazoic acid Hydrogen cyanide Hydrogen peroxide 3-Hydroxypropanenitrile Indan Indene Iodine bromide Iodine chloride Iodobenzene 1-Iodobutane Iodoethane Iodomethane 1-Iodopropane 2-Iodopropane Iron pentacarbonyl Isobutanal Isobutyl acetate Isobutyl acrylate Isobutylamine Isobutylbenzene Isobutyl formate Isobutyl isobutanoate Isopentane Isopentyl acetate Isophorone Isopropenyl acetate Isopropenylbenzene Isopropyl acetate Isopropylamine Isopropylbenzene Isopropylbenzene hydroperoxide 1-Isopropyl-2-methylbenzene 1-Isopropyl-3-methylbenzene 1-Isopropyl-4-methylbenzene Isoquinoline d-Limonene l-Limonene Mesityl oxide Methacrylic acid Methanol 2-Methoxyaniline 4-Methoxybenzaldehyde 2-Methoxyethanol 2-Methoxyethyl acetate Methyl acetate Methyl acrylate 2-Methylacrylonitrile 2-Methylaniline 3-Methylaniline
H4N2 HN3 CHN H2O2 C3H5NO C9H10 C9H8 BrI ClI C6H5I C4H9I C2H5I CH3I C3H7I C3H7I C5FeO5 C4H8O C6H12O2 C7H12O2 C4H11N C10H14 C5H10O2 C8H16O2 C5H12 C7H14O2 C9H14O C5H8O2 C9H10 C5H10O2 C3H9N C9H12 C9H12O2 C10H14 C10H14 C10H14 C9H7N C10H16 C10H16 C6H10O C4H6O2 CH4O C7H9NO C8H8O2 C3H8O2 C5H10O3 C3H6O2 C4H6O2 C4H5N C7H9N C7H9N
Mr
tm/°C
tb/°C
/ g mL–1
114.185 114.185 98.186
-39 -33 -118.9
147 144 93.64
0.818320 0.817420 0.697020
260.761 272.772 186.054 162.377 179.200 100.158 86.175 108.141 116.158 102.174 100.158 84.159
-21 -9 5.03 -66 7.2 -56 -95.35 1 -3 -47.4 -55.5 -139.76
215 239 80.26 99 232.5 131 68.73 295 205.2 157.6 127.6 63.48
1.55625 1.701925 1.618420 0.763820 1.0320 0.833520 0.660625 0.967620 0.921225 0.813620 0.811320 0.668525
144.212 32.045 43.028 27.026 34.015 71.078 118.175 116.160 206.808 162.357 204.008 184.018 155.965 141.939 169.992 169.992 195.896 72.106 116.158 128.169 73.137 134.218 102.132 144.212 72.149 130.185 138.206 100.117 118.175 102.132 59.110 120.191 152.190 134.218 134.218 134.218 129.159 136.234 136.234 98.142 86.090 32.042 123.152 136.149 76.095 118.131 74.079 86.090 67.090 107.153 107.153
-80.9 1.4 -80 -13.29 -0.43 -46 -51.38 -1.5 40 27.39 -31.3 -103 -111.1 -66.4 -101.3 -90 -20 -65.9 -98.8 -61 -86.7 -51.4 -95.8 -80.7 -159.77 -78.5 -8.1 -92.9 -23.2 -73.4 -95.13 -96.02
171.5 113.55 35.7 26 150.2 221 177.97 182 116 dec 100 dec 188.4 130.5 72.3 42.43 102.5 89.5 103 64.5 116.5 132 67.75 172.79 98.2 148.6 27.88 142.5 215.2 94 165.4 88.6 31.76 152.41 153 178.1 175.1 177.1 243.22 178 178 130 162.5 64.6 224 248 124.1 143 56.87 80.7 90.3 200.3 203.3
0.877915 1.0036
-71.5 -63.7 -67.94 26.47 -74.0 -59 16 -97.53 6.2 0 -85.1 -70 -98.25 <-75 -35.8 -14.41 -31.3
0.687620 1.44 1.040425 0.963920 0.996025 4.3 3.24 1.830820 1.615420 1.935720 2.278920 1.748920 1.704220 1.520 0.789120 0.871220 0.889620 0.72425 0.853220 0.877620 0.854220 0.620120 0.87615 0.925520 0.909020 0.910620 0.871820 0.689120 0.864025 1.0320 0.876620 0.861020 0.857320 1.091030 0.841120 0.84320 0.865320 1.015320 0.791420 1.092320 1.11915 0.964720 1.007419 0.934220 0.953520 0.800120 0.998420 0.988920
15-21
/ mPa s
0.340
12.7 12.60 2.092
/D
cp/ J g–1K–1
vp/ kPa
2.78 ≈0
2.157
0.164 7.52
FP/ °C
Fl. Lim.
46 49 -1
2.029 2.179 31.3
0
0.300
1.8865
≈0
2.600 13.03 14.56 2.077
1.13
4.58 0.583 0.252 0.876
4.42 51.7
0.183
114.9 74.6
5.5
2.7 ≈0 1.75 1.70 2.985 1.573 3.2
1.357
260
1.554 0.556 0.469 0.703 0.653
4.59 6.27 7.82 6.97 7.07 8.19 2.602
0.676
5.068
0.571
4.43 2.318 6.41
0.214
1.845 4.72
0.726 1.24 1.70 1.93 1.976 1.62 2.04 1.95 2.75 1.9 1.3 ≈0 1.88 1.9 0.13 1.9
2.33 2.28 0.325 0.737
5.6268 2.381
1.19 0.79
≈0 2.73
1.47
2.2322 11.0 2.3746 2.3738 15.6
0.602 0.544
0.364
3.82 3.31
1.191 1.918 1.791 2.101 2.270 1.190 1.937 2.353 2.130 2.178
11.3
1.48 20.2 <0.01 0.005 0.11 1.54 24.8
32 -22 93 102 63 25 -26
1-8% 1.2-6.9%
1.961 3.086
0.185
45 38
5-100%
2.612 2.619
98.8
-18
6-40%
0.010
129
1.609 1.609
33.0 5.230 22.0 17.2 8.25 7.07 7.03 6.138 5.816
0.778 0.738 0.888 0.746 0.535 1.228 2.013 2.505 1.793
2.284 1.909 1.834 1.711 1.952 2.771 1.753
1.761 1.519 1.828
2.8 1.65 1.70
2.165 1.871 2.531
2.36 2.1 1.72 1.77 3.69 1.6 1.45
2.249 2.624 1.916 1.845 1.883 1.96 2.118
TLV/ ppm 50 50
2.55 2.79
IT/ °C
1-8% 2-5%
610
0.02 0.01
225 550 380 290 423 253
50 2
5 30 0.01
538 1
0.220
10
0.133 1.85 18.2 53.9 5.75 9.36
2
0.1 23.0 2.39
-18 18 30 -9 55 5 38 -51 25 84 26 54 2 -37 36
1.6-10.6% 1-11%
1-7%
196 421 427 378 427 320 432 420 360 460 432 574 460 402 424
0.19
47
1-6%
436
0.277 0.254 1.47
45
0.7-6.1%
237
31 77 11 118
1-7% 1.6-8.8% 6-36%
344 68 464
15 20 200 0.1
39 49 -10 -3 1 85
2-14% 2-12% 3-16% 2.8-25% 2-6.8%
285 392 454 468
5 5 200 2 1 2 2
19.0 0.257 5.34 0.552 91.7 0.728 0.06 6.02 7.88 78.0 0.61 0.004
16.9
1.31 0.67 28.8 11.0 8.26 0.043 0.036
2-12% 0.8-6% 2-9% 1-8% 1.4-7.6% 1-8% 1-4% 1.9-6.1% 2-8%
482
150
600 50 5 50 250 5 50
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS (continued)
Name
Mol. Form.
N-Methylaniline Methyl benzoate 2-Methyl-1,3-butadiene Methyl butanoate
C7H9N C8H8O2 C5H8 C5H10O2
3-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-2-butanol 3-Methyl-2-butanol, (±)3-Methyl-2-butanone 2-Methyl-1-butene 2-Methyl-2-butene Methyl tert-butyl ether Methyl chloroacetate Methylcyclohexane Methylcyclopentane N-Methylformamide
C5H10O2 C5H12O C5H12O C5H12O C5H10O C5H10 C5H10 C5H12O C3H5ClO2 C7H14 C6H12 C2H5NO
Methyl formate 5-Methyl-2-hexanone Methylhydrazine Methyl isocyanate Methyl lactate, (±)Methyl methacrylate 1-Methylnaphthalene Methyloxirane 2-Methylpentane 3-Methylpentane 2-Methyl-2,4-pentanediol 2-Methyl-1-pentanol 4-Methyl-2-pentanol 4-Methyl-2-pentanone 2-Methylpropanenitrile 2-Methyl-2-propanethiol Methyl propanoate 2-Methylpropanoic acid 2-Methyl-1-propanol 2-Methyl-2-propanol 2-Methylpyridine 3-Methylpyridine 4-Methylpyridine N-Methyl-2-pyrrolidone Methyl salicylate 4-Methylstyrene Morpholine -Myrcene Nickel carbonyl L-Nicotine Nitric acid 2-Nitroanisole Nitrobenzene Nitroethane Nitromethane 1-Nitropropane 2-Nitropropane N-Nitrosodiethylamine N-Nitrosodimethylamine 2-Nitrotoluene 3-Nitrotoluene Nonane Nonanoic acid 1-Nonanol 1-Nonene 4-Nonylphenol cis,cis-9,12-Octadecadienoic acid cis-9-Octadecenoic acid Octane Octanoic acid
C2H4O2 C7H14O CH6N2 C2H3NO C4H8O3 C5H8O2 C11H10 C3H6O C6H14 C6H14 C6H14O2 C6H14O C6H14O C6H12O C4H7N C4H10S C4H8O2 C4H8O2 C4H10O C4H10O C6H7N C6H7N C6H7N C5H9NO C8H8O3 C9H10 C4H9NO C10H16 C4NiO4 C10H14N2 HNO3 C7H7NO3 C6H5NO2 C2H5NO2 CH3NO2 C3H7NO2 C3H7NO2 C4H10N2O C2H6N2O C7H7NO2 C7H7NO2 C9H20 C9H18O2 C9H20O C9H18 C15H24O C18H32O2 C18H34O2 C8H18 C8H16O2
Mr
tm/°C
tb/°C
/ g mL–1
/ mPa s
107.153 136.149 68.118
-57 -12.4 -145.9
196.2 199 34.0
0.989120 1.083725 0.67920
2.04 1.857
5.96 6.642 2.098
102.132 102.132 88.148 88.148 88.148 86.132 70.133 70.133 88.148 108.524 98.186 84.159
-85.8 -29.3 -117.2 -9.1
102.8 176.5 131.1 102.4 112.9 94.33 31.2 38.56 55.0 129.5 100.93 71.8
0.898420 0.93120 0.810420 0.809620 0.818020 0.805120 0.650420 0.662320 0.735325 1.23620 0.769420 0.748620
0.541
5.48
3.69 3.55
0.203
15.63 5.78 12.1 10.37 2.180 1.979
0.679 0.479
12.0 2.024 1.9853
59.067 60.052 114.185 46.072 57.051 104.105 100.117 142.197 58.079 86.175 86.175 118.174 102.174 102.174 100.158 69.106 90.187 88.106 88.106 74.121 74.121 93.127 93.127 93.127 99.131 152.148 118.175 87.120 136.234 170.734 162.231 63.013 153.136 123.110 75.067 61.041 89.094 89.094 102.134 74.081 137.137 137.137 128.255 158.238 144.254 126.239 220.351 280.446 282.462 114.229 144.212
-3.8 -99
199.51 31.7 144 87.5 39.5 144.8 100.5 244.7 35 60.26 63.27 197.1 149 131.6 116.5 103.9 64.2 79.8 154.45 107.89 82.4 129.38 144.14 145.36 202 222.9 172.8 128 167 43 (exp 60) 247 83 272 210.8 114.0 101.19 131.1 120.2 176.9 152 222 232 150.82 254.5 213.37 146.9 ≈295
1.01119 0.971320 0.88820
1.678 0.325
-93.1 -137.53 -133.72 -108.6 -32.1 -126.6 -142.42
-52.36 -45 -47.55 -30.43 -111.9 -153.6 -162.90 -50 -90 -84 -71.5 -0.5 -87.5 -46 -101.9 25.69 -66.68 -18.14 3.67 -23.09 -8 -34.1 -4.8 -19.3 -79 -41.6 10.5 5.7 -89.5 -28.38 -108 -91.3
-10.4 15.5 -53.46 12.4 -5 -81.3 42 -7 13.4 -56.82 16.5
360 125.67 239
/D 1.9 0.25 0.63 1.82
cp/ J g–1K–1 1.933 1.625 2.240 1.941 1.930 2.382 2.803 2.089 2.241 2.179 2.127
≈0 ≈0
1.882 1.886
189.0 9.20 13.53
3.83 1.77
2.096 1.983
21.75
≈2.8
6.32 2.915
1.67 ≈0 2.01 ≈0 ≈0 2.9
2.928 0.923027 1.092820 0.937725 1.020220 0.8590 0.65025 0.659825 0.92315 0.826320 0.807520 0.796525 0.770420 0.794325 0.915020 0.968120 0.801820 0.788720 0.944320 0.956620 0.954820 1.023025 1.18125 0.917325 1.000520 0.801315 1.3125 1.009720 1.55 1.254020 1.203720 1.044825 1.137120 0.996125 0.982125 0.942220 1.004820 1.161119 1.158120 0.719220 0.905220 0.828020 0.725325 0.95020 0.902220 0.893520 0.698625 0.907325
0.286 0.306
4.07 0.545
0.431 1.226 3.33 4.31
2.02
1.886 1.886 23.4
13.11 24.42 5.475 6.200 2.58 17.93 12.47 10.18 11.10 12.2 32.55 8.80 7.42 2.3
1.910 1.578 2.073 2.248 2.213 2.843 2.427 2.672 2.130
4.29 1.66 1.08 1.64 1.7 1.85 2.40 2.70 4.1 2.47
1.943 1.964 2.449 2.949 1.703 1.704 1.707 3.105 1.637
1.55
1.892
vp/ kPa
FP/ °C
Fl. Lim.
IT/ °C
TLV/ ppm 0.5
0.052 73.4
83 -54
4.30 0.067 0.315 2.19 1.20 6.99 81.4 62.1 33.6
14
6.18 18.3 78.1 0.691 6.61 57.7 5.10 0.009 71.7 28.2 25.3 <0.01 0.236 0.698 2.64 24.2 11.5 1.39 5.52 1.5 0.795 0.759 0.04 0.015 0.245 1.34 0.280
43 19 38
1.5-8.9%
395
1.2-9% 1.2-9%
416 350 437
100
200 -20 -20 40 57 -4 -29
7.5-18.5% 1-7% 1-8%
-19 36 -8 -7 49 10
5-23% 1-8% 2.5-92% 5.3-26% >2.2% 1.7-8.2%
-37 <-29 -7 102 54 41 18 8 <-29 -2 56 28 11 39 57 96 96 53 37
3.1-27.5% 1-7% 1.2-7% 1-9% 1.1-9.65% 1-6% 1-8%
250 258
400
449 191 194 534 385
100 50 0.01 0.02 100
529 449 264 278 306 310 448 482
2.5-13% 2-9.2% 2-11% 2-8%
469 481 415 478 538
1-10%
346 454 538 290
0.8-11% 1-11%
20 500 500 25 25 50
50 100
50 20
1.198
0.05
1.744
2
8.937
1.863 0.688 0.630 0.798
45.75 35.6 29.11 37.27 24.70 26.74
0.665 7.01 9.12 0.586
26.26 24.95 1.9722 2.475 8.83 2.180
0.508 5.02
2.754 2.336 1.948 2.85
15-22
TeamLRN
2.17 5.0 4.22 3.23 3.46 3.66 3.73
1.509 1.790 1.746 1.97 1.911
≈0
1.474 1.474 2.217 2.290 2.470 2.142
1.18 ≈0 1.15
2.043 2.229 2.066
≈0 0.79
0.03 2.79 4.79 1.36 2.3
88 28 35 36 24
0.570
106 106 31
0.714
26
1.86
189 13
2-9% 3-17% 7-22% 2%3-11%
0.8-2.9%
482 414 418 421 428
1 100 20 25 10
205
2 2 200
260
1-7%
363 206
300
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS (continued)
Name
Mol. Form.
1-Octanol 2-Octanol 2-Octanone 1-Octene
C8H18O C8H18O C8H16O C8H16
Oxetane 2-Oxetanone Oxirane Oxiranemethanol, (±)Paraldehyde Parathion Pentachloroethane cis-1,3-Pentadiene trans-1,3-Pentadiene Pentanal Pentane Pentanedial
C3H6O C3H4O2 C2H4O C3H6O2 C6H12O3 C10H14NO5PS C2HCl5 C5H8 C5H8 C5H10O C5H12 C5H8O2
1,5-Pentanediol 2,4-Pentanedione 1-Pentanethiol Pentanoic acid 1-Pentanol 2-Pentanol 3-Pentanol 2-Pentanone 3-Pentanone 1-Pentene cis-2-Pentene trans-2-Pentene Pentyl acetate Pentylamine Perchloric acid Peroxyacetic acid Phenol 2-Phenoxyethanol Phenylhydrazine 1-Phenyl-2-propylamine, (±)Phosphinic acid Phosphoric acid Phosphoric trichloride Phosphorothioc trichloride Phosphorus(III) bromide Phosphorus(III) chloride -Pinene -Pinene Piperidine Propanal 1,2-Propanediol 1,3-Propanediol Propanenitrile Propanoic acid Propanoic anhydride 1-Propanol 2-Propanol Propargyl alcohol Propyl acetate Propylamine Propylbenzene Propyl butanoate Propylene carbonate Propyl formate Propyl propanoate Pyridine Pyrrole Pyrrolidine 2-Pyrrolidone Quinoline
C5H12O2 C5H8O2 C5H12S C5H10O2 C5H12O C5H12O C5H12O C5H10O C5H10O C5H10 C5H10 C5H10 C7H14O2 C5H13N ClHO4 C2H4O3 C6H6O C8H10O2 C6H8N2 C9H13N H3O2P H3O4P Cl3OP Cl3PS Br3P Cl3P C10H16 C10H16 C5H11N C3H6O C3H8O2 C3H8O2 C3H5N C3H6O2 C6H10O3 C3H8O C3H8O C3H4O C5H10O2 C3H9N C9H12 C7H14O2 C4H6O3 C4H8O2 C6H12O2 C5H5N C4H5N C4H9N C4H7NO C9H7N
Mr
tm/°C
tb/°C
/ g mL–1
/ mPa s
/D
cp/ J g–1K–1
vp/ kPa
FP/ °C
Fl. Lim.
IT/ °C
130.228 130.228 128.212
-14.8 -31.6 -16
195.16 179.3 172.5
0.826225 0.819320 0.82020
7.29 6.49
10.30 8.13 9.51
1.8 1.71 2.7
2.344 2.535 2.132
0.01
81 88 52
270 265
112.213 58.079 72.063 44.052 74.079 132.157 291.261 202.294 68.118 68.118 86.132 72.149
-101.7 -97 -33.4 -112.5 -45 12.6 6.1 -28.78 -140.8 -87.4 -91.5 -129.67
121.29 47.6 162 10.6 167 dec 124.3 375 162.0 44.1 42 103 36.06
0.714920 0.893025 1.146020 0.882110 1.114325 0.994320 1.268120 1.679620 0.691020 0.671025 0.809520 0.626220
0.447
2.113
≈0 1.94 4.18 1.89
2.148
2.30
21
230
100.117 104.148 100.117 104.214 102.132 88.148 88.148 88.148 86.132 86.132 70.133 70.133 70.133 130.185 87.164 100.459 76.051 94.111 138.164 108.141 135.206 65.997 97.995 153.331 169.398 270.686 137.332 136.234 136.234 85.148 58.079 76.095 76.095 55.079 74.079 130.141 60.095 60.095 56.063 102.132 59.110 120.191 130.185 102.089 88.106 116.158 79.101 67.090 71.121 85.105 129.159
-14 -18 -23 -75.65 -33.6 -77.6 -73 -69 -76.8 -39 -165.12 -151.36 -140.21 -70.8 -55 -112 -0.2 40.89 14 20.6
188 dec 239 138 126.6 186.1 137.98 119.3 116.25 102.26 101.7 29.96 36.93 36.34 149.2 104.3 ≈90 dec 110 181.87 245 243.5 203 130 407 105.5 125 173.2 76.1 156.2 166 106.22 48 187.6 214.4 97.14 141.15 170 97.2 82.3 113.6 101.54 47.22 159.24 143.0 242 80.9 122.5 115.23 129.79 86.56 251 237.16
26.5 42.4 1.18 -36.2 -41.5 -93.6 -64 -61.5 -11.02 -80 -60 -27.7 -92.78 -20.5 -45 -124.39 -87.9 -51.8 -93 -84.75 -99.6 -95.2 -48.8 -92.9 -75.9 -41.70 -23.39 -57.79 25 -14.78
0.991420 0.972125 0.85020 0.933925 0.814420 0.809420 0.820320 0.80920 0.809825 0.640520 0.655620 0.643125 0.875620 0.754420 1.77 1.22615 1.054545 1.10222 1.098620 0.930625 1.49 1.645 1.635 2.8 1.574 0.853925 0.86025 0.860620 0.865725 1.036120 1.053820 0.781820 0.988225 1.011020 0.799725 0.780925 0.947820 0.887820 0.717320 0.859325 0.873020 1.204720 0.907320 0.880920 0.981920 0.969820 0.858620 1.12020 1.097715
15-23
12.42 1.079 2.25
1.43 3.716 2.319
0.224
10.00 1.8371
3.62 3.47 4.15 0.470 0.444 0.195
26.2 26.524 4.847 2.661 15.13 13.71 13.35 15.45 17.00 2.011
0.702
4.79 4.27
12.40 13.03
0.529
1.573 0.321 40.4 0.294 1.030 1.945 2.04 0.544 0.376
0.485 0.879 1.225 0.704 3.34
1.694 1.998
0.92 0.500 0.585
0.859
0.478
≈0
2.317
4.58 68.3
2.5 2.8
3.08 2.08
1.02
1.61 1.7 1.66 1.64 2.7 2.82 ≈0.5 ≈0 ≈0 1.75
2.059 2.361 2.716 2.719 2.137 2.216 2.196 2.163 2.239 2.005 2.501
0.024 0.259 0.804 1.10 4.97 4.72 85.0 66.0 67.4 0.60 4.00
1.224
2.123
1.93 0.055
7.15
2.007
14.1 4.94
2.54
3.498 2.1787 2.4970 4.33 18.5 27.5 35.1 29.7 3.44 18.30 20.8 20.18 20.8 5.62 5.08 2.370 4.3 66.14 6.92 5.249 13.260 8.00 8.30 28.18 9.16
0.56
1.2 2.72 2.2 2.5 4.05 1.75 1.55 1.56 1.13 1.8 1.17 ≈0
74 -20
>2.9% 3-100%
429
36
>1.3%
238
12 -40
2-8%
222 260
129 34 18 96 33 34 41 7 13 -18 <-20 <-20 16 -1 41 79 121 88 <100
TLV/ ppm
0.5 1 2
50 600
335 340
1-10% 1.2-9% 1.2-9% 2-8% >1.6% 1.5-8.7%
400 300 343 435 452 450 275
200 200
1-8% 2.2-22%
360
50
1.8-8.6%
715
5 0.1
1.480 0.905
0.1
0.2 0.64 0.61 4.28 42.2 0.02
33 38 16 -30 99
2.166 2.063 1.806 2.395 2.604
6.14 0.553
1.921 2.776 1.786
4.49 42.1
2 52 63 23 12 36 13 -37 30 37 135 -3 79 20 39 3 129
2.113 2.362 2.507
2.76 6.02
0.618 4.9 1.89
2.141 1.945
2.21 1.74 1.6 3.5 2.29
1.678 1.903 2.202 1.99 1.51
10.9 1.88 2.76 1.10 8.40
255 275 1-10% 2.6-17% 3-13% 3-14% 2.9-12.1% 1.3-9.5% 2-14% 2-13%
207 371 400 512 465 285 412 399
2-8% 2-10% 1-6%
450 318 450
10 200 400 1 200
455 2-12%
482
480
5
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS (continued)
Name
Mol. Form.
Safrole Salicylaldehyde Selenium chloride Selenium oxychloride
C10H10O2 C7H6O2 Cl2Se2 Cl2OSe
Selenium oxyfluoride Styrene Sulfolane Sulfur chloride Sulfur dichloride Sulfuric acid Sulfuryl chloride -Terpinene 1,1,2,2-Tetrabromoethane Tetrabromosilane 1,1,2,2-Tetrachloro-1,2difluoroethane 1,1,1,2-Tetrachloroethane 1,1,2,2-Tetrachloroethane Tetrachloroethene Tetrachloromethane Tetrachlorosilane Tetradecane Tetraethylene glycol Tetrafluoroboric acid Tetrahydrofuran Tetrahydrofurfuryl alcohol 1,2,3,4-Tetrahydronaphthalene Tetrahydropyran Tetrahydrothiophene Tetramethylsilane Tetramethylurea Tetranitromethane Thionyl bromide Thionyl chloride Thiophene Tin(IV) chloride Titanium(IV) chloride Toluene Toluene-2,4-diisocyanate Tribromomethane Tributylamine
F2OSe C8H8 C4H8O2S Cl2S2 Cl2S H2O4S Cl2O2S C10H16 C2H2Br4 Br4Si C2Cl4F2
Tributyl borate Tributyrin Trichloroacetaldehyde 1,2,4-Trichlorobenzene 1,1,1-Trichloroethane 1,1,2-Trichloroethane Trichloroethene Trichloroethylsilane Trichlorofluoromethane Trichloromethane (Trichloromethyl)benzene Trichloromethylsilane Trichloronitromethane 1,2,3-Trichloropropane Trichlorosilane 1,1,2-Trichloro-1,2,2trifluoroethane Tri-o-cresyl phosphate Tridecane 1-Tridecene Triethanolamine Triethylamine Triethylene glycol Triethylene glycol dimethyl ether Triethyl phosphate
C12H27BO3 C15H26O6 C2HCl3O C6H3Cl3 C2H3Cl3 C2H3Cl3 C2HCl3 C2H5Cl3Si CCl3F CHCl3 C7H5Cl3 CH3Cl3Si CCl3NO2 C3H5Cl3 Cl3HSi C2Cl3F3
C2H2Cl4 C2H2Cl4 C2Cl4 CCl4 Cl4Si C14H30 C8H18O5 BF4H C4H8O C5H10O2 C10H12 C5H10O C4H8S C4H12Si C5H12N2O CN4O8 Br2OS Cl2OS C4H4S Cl4Sn Cl4Ti C7H8 C9H6N2O2 CHBr3 C12H27N
C21H21O4P C13H28 C13H26 C6H15NO3 C6H15N C6H14O4 C8H18O4 C6H15O4P
Mr
tm/°C
tb/°C
/ g mL–1
162.185 122.122 228.83
11.2 -7 -85
234.5 197 130 dec
1.100020 1.167420 2.774
165.86 132.96 104.150 120.171 135.037 102.971 98.080 134.970 136.234 345.653 347.702 203.830
8.5 15 -30.65 27.6 -77 -122 10.31 -51
177 125 145 287.3 137 59.6 337 69.4 174 243.5 154 92.8
2.44 2.8 0.901625 1.272318 1.69 1.62 1.8 1.680 0.837519 2.965520 2.8 1.595150
167.849 167.849 165.833 153.823 169.897 198.388 194.226 87.813 72.106 102.132 132.202 86.132 88.172 88.224 116.161 196.033 207.873 118.970 84.140 260.521 189.678 92.139 174.156 252.731 185.349 230.151 302.363 147.387 181.447 133.404 133.404 131.388 163.506 137.368 119.378 195.474 149.480 164.376 147.431 135.452 187.375 368.363 184.361 182.345 149.188 101.190 150.173 178.227 182.154
0 5.39 24.8 -70.2 -42.4 -22.3 -22.62 -68.74 5.82 -6.2
20
-108.44 <-80 -35.7 -49.1 -96.2 -99.06 -0.6 13.8 -50 -101 -38.21 -34.07 -24.12 -94.95 20.5 8.69
130.2 145.2 121.3 76.8 57.65 253.58 328 130 dec 65 178 207.6 88 121.1 26.6 176.5 126.1 140 75.6 84.0 114.15 136.45 110.63 251 149.1
1.5406 1.595320 1.623020 1.594020 1.5 0.759620 1.128515 ≈1.8 0.883325 1.052420 0.964525 0.881420 0.998720 0.64819 0.968720 1.638020
-70 <-70 -75 -57.5 16.92 -30.01 -36.3 -84.7 -105.6 -110.44 -63.41 -4.42 -90 -64 -14.7 -128.2 -36.22
216.5 234 307.5 97.8 213.5 74.09 113.8 87.21 100.5 23.7 61.17 221 65.6 112 157 33 47.7
0.777020 0.856720 1.035020 1.51220 1.45925 1.339020 1.439720 1.464220 1.237320 1.487920 1.478825 1.372320 1.27320 1.655820 1.388920 1.331 1.563525
11 -5.4 -13 20.5 -114.7 -7 -45 -56.4
410 235.47 232.8 335.4 89 285 216 215.5
1.631 1.064920 2.234 1.73 0.866820 1.224420 2.878825
/ mPa s
18.35
/D
cp/ J g–1K–1
2.86
1.818
2.4737 43.26 4.79 2.915
0.123 4.8
1.747 1.498
9.1 2.4526 6.72
1.81
vp/ kPa
FP/ °C
Fl. Lim.
IT/ °C
TLV/ ppm
100 78
46.2 0.695
1.38 0
1.437 0.844 0.908 99.4 2.13
0.456 2.14
8.50 2.268 2.2379 2.0343 20.44 7.52 13.48 2.771 5.66
0.973 1.921 23.10 2.317 9.06 8.675 2.739
0.560 1.857
0.793 0.545 0.421 0.537
1.32 0 0 0 ≈0
1.724 1.50 609 0.347
15-24
TeamLRN
490
20
335
1
0.003
0.852
7.51
0.92 0.967 0.865 0.850 0.855
1.6 0.622 2.42 15.2
47 62 45
5-12% 20-54%
0.002
112 182
>0.5%
200
21.6 0.100 0.05 9.54 2.45 94.2 0.138
-14 75 71 -20
2-12% 1.5-9.7% 1-5%
321 282 385
1.720 1.774 1.645 1.82
1.45 0.55 0
1.017 1.471 0.634 0.766 1.707 1.653 0.517
0.37
2.340 2.23 5.72 6.8
0.8 0.77
7.243 7.1937 3.390
1.76 1.4 0.8 2.04 0.46 1.04 2.03 1.91
0.99
2.313
3.1
1 25 5
200
77
10.6
-1
3.79
4 127 83
1-7% 0.9-9.5%
63 93 180
1-5%
0.726 0.01
1.082 1.131 0.947 0.885 0.957 1.091
6.66 0.057 16.5 3.1 9.91 6.29 106 26.2
1.245
22.5 3.18 0.492
0.908
44.8
0.86
2.87 ≈0 ≈0 3.6 0.66
500
0.005
1.837 1.025
2.41 6.7 2.0213 2.139 29.36 2.418 23.69 7.62 13.20
1-7%
0.479
1.75 2.1 ≈0 1.74 1.90 0 3.5 0
7.319 7.5 0.326 0.656
31 177
1.416 0.993
2.208
2.379 8.433 4.404
3.00 4.8069 6.9
0.81 <0.01
0.36
2.52
20
1.1955 0.756420 0.765820 1.124220 0.727520 1.127415 0.98620 1.069520
1.57 2.206 2.149 2.61 2.173 2.18
0.005 <0.01 7.70
105 -1 32 32 22
480
>0.5%
407
2.5-6.6% 8-13% 6-28% 8-11%
571 500 460 420
50 0.005 0.5
350 10 50 1000 10
127 -9 71 -50
7.6->20%
211 >404 0.1 10
3.2-12.6% 104
1000 225 79 79 179 -7 177 111 115
385
1-10% 1-8% 1-9%
249 371 454
0.5 1
LABORATORY SOLVENTS AND OTHER LIQUID REAGENTS (continued)
Name
Mol. Form.
Trifluoroacetic acid (Trifluoromethyl)benzene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene
C2HF3O2 C7H5F3 C9H12 C9H12
1,3,5-Trimethylbenzene Trimethyl borate Trimethylchlorosilane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane Trimethyl phosphate 2,4,6-Trimethylpyridine Trinitroglycerol Undecane Vanadium(IV) chloride Vanadyl trichloride Vinyl acetate
C9H12 C3H9BO3 C3H9ClSi C8H18 C8H18 C3H9O4P C8H11N C3H5N3O9 C11H24 Cl4V Cl3OV C4H6O2
4-Vinylcyclohexene Water o-Xylene m-Xylene p-Xylene 2,4-Xylenol
C8H12 H2O C8H10 C8H10 C8H10 C8H10O
Mr
tm/°C
tb/°C
/ g mL–1
/ mPa s 0.808
114.023 146.110 120.191
-15.2 -28.95 -25.4
73 102.1 176.12
1.535125 1.188420 0.894420
120.191 120.191 103.912 108.642 114.229 114.229 140.074 121.180 227.087 156.309 192.753 173.299
-43.77 -44.72 -29.3 -40 -107.3 -100.9 -46 -46 13.5 -25.5 -25.7 -79
169.38 164.74 67.5 60 99.22 114.8 197.2 170.6 exp 218 195.9 148 127
0.875820 0.861525 0.91525 0.85625 0.687825 0.726220 1.214420 0.916622 1.593120 0.740220 1.816 1.829
86.090 108.181 18.015 106.165 106.165 106.165 122.164
-93.2 -108.9 0.00 -25.2 -47.8 13.25 24.5
72.8 128 100.0 144.5 139.12 138.37 210.98
0.925625 0.829920 0.9970 0.880210 0.859625 0.856625 0.965020
15-25
1.098
0.890 0.760 0.581 0.603
/D
cp/ J g–1K–1
8.42 9.22 2.656
2.28 2.86 ≈0
2.377 2.279 2.2762
≈0 0
1.789 1.741 1.828
1.943 1.9780 20.6 7.807 19.25 1.9972 3.05 3.4
≈0 ≈0 3.2 2.05
2.093 2.150
≈0
2.207
1.79
1.969
1.8546 0.64 ≈0 0 1.4
4.180 1.753 1.724 1.710
80.100 2.562 2.359 2.2735 5.060
1.289 1.800
vp/ kPa 15.1 5.14 0.30 0.33 17.2 30.7 6.50 3.60 0.11
FP/ °C 12 44 44 50 -8 -28 -12 <21 107
Fl. Lim.
IT/ °C
TLV/ ppm
0.8-6.6%
470
25
1-6% 1-5%
500 559
25 25
395 418 425
300 300
270
0.05
69
15.4 1.87
-8 16
2.6-13.4%
402 269
10 0.1
0.88 1.13 1.19 0.022
32 27 27
1-7% 1-7% 1-7%
463 527 528
100 100 100
MISCIBILITY OF ORGANIC SOLVENTS The chart below gives qualitative information on the miscibility of pairs of organic liquids. Two liquids are considered miscible (indicated by M in the chart) if mixing equal volumes produces a single liquid phase. If two phases separate, they are considered immiscible ( I). An entry of P indicates two phases whose volumes differ appreciably, suggesting a partial miscibility of the components. The symbol R indicates a reaction between the components. All data refer to room temperature. The codes for the columns are: A B C D E F G H I
Acetone Benzaldehyde Benzene Butyl acetate Butyl alcohol Carbon tetrachloride 2-Chloroethanol Chloroform o-Cresol
J K L M N O P Q R
Diethyl ether S N,N-Dimethylaniline T Dipentylamine U Ethyl alcohol V Ethylene glycol W Ethylene glycol monoethyl ether X Formamide Y Furfuryl alcohol Glycerol
Methyl isopropyl ketone Nitromethane 1-Octanol 1,3-Propanediol Pyridine Triethylenetetramine Triethyl phosphate
REFERENCES 1. Drury, J. S., Ind. Eng. Chem. 44, 2744, 1959. 2. Jackson, W. M., and Drury, J. S., Ind. Eng. Chem. 51, 1491, 1959.
Acetone Adiponitrile 2-Amino-2-methyl-1-propanol p-Anisaldehyde Benzaldehyde Benzene Benzonitrile Benzothiazole Benzyl alcohol Benzyl mercaptan 2-Bromoethyl acetate 1,3-Butanediol 2,3-Butanediol Butyl acetate Butyl alcohol Carbon tetrachloride 2-Chloroethanol Chloroform 3-Chloro-1,2-propanediol Cinnamaldehyde o-Cresol Diacetone alcohol Dibenzyl ether Dibutylamine Dibutyl carbonate Dibutyl ether Diethanolamine Diethylacetic acid Diethylene glycol dibutyl ether Diethylene glycol diethyl ether Diethylene glycol monobutyl ether Diethylene glycol monoethyl ether Diethylene glycol monomethyl ether
A B C D - M M M M M M M M M
E F G H M M M M M I M M M M - M M M M M M - M M M M M M M M M M M M M M M M M M M M M M M M P M M I M M P M M M M - M M M M M M - M M M M M M M M M M M M I M M M M M M M M P M M P M M M M R M M M M M M M M M M I I I M I M M M M M M M M M M M M M M M M M M M
I
J M I M M M M M M M
M P M M M M M M M M M M M
M I M M M M
M M M M M
15-26
TeamLRN
K L M N M M M M M M I M M M I M M P M M M I M M I M M M M M M M M I R M M M M M M M P M M M M M I M M M P R M M M I M M M M M I M M M M M I P M M R M M R M M M M M M M M M
O P Q M M M M M M M M M M M I M I M I M M M I M
I M M M I M M M M M M M M I M M M M
R S T U V W I M M M M M I M M M M M M M I I P M M M M I M I M I M I I M I M M M M M M M I I M M M M M M I M M P M M M M M M I M M I M M M I M M M M I M I M M I M M M M I M P M I I M I M M I M M M I M M M M M M M M M M M M
X Y M M
M M M
R M M
M M M M R R M M M M M I
R M M M M M
MISCIBILITY OF ORGANIC SOLVENTS (continued)
Diethylenetriamine Diethyl ether Diethylformamide Dihexyl ether Diisobutyl ketone Diisopropylamine N,N-Dimethylaniline Dipentylamine N,N-Dipropylaniline Dipropylene glycol Ethyl alcohol Ethyl benzoate Ethyl chloroacetate Ethyl cinnamate Ethylene glycol Ethylene glycol monobutyl ether Ethylene glycol monoethyl ether Ethylene glycol monomethyl ether 2-Ethyl-1-hexanol Ethyl phenylacetate Ethyl thiocyanate Formamide Furfuryl alcohol Glycerol 1-Heptadecanol 3-Heptanol Heptyl acetate Hexanenitrile Isobutyl mercaptan Isopentyl acetate Isopentyl alcohol Isopentyl sulfide Methyl disulfide Methyl isobutyl ketone Methyl isopropyl ketone 4-Methylpentanoic acid Nitromethane 1-Octanol o-Phenetidine 1,2-Propanediol 1,3-Propanediol Pyridine Tetradecanol Tributyl phosphate Triethylene glycol Triethylenetetramine Triethyl phosphate 2,6,8-Trimethyl-4-nonanone
A M M M M M M M M M M M M M M M M M M M M M M M I M M M M M M M M M M M M M M M M M M M M M M M M
B C M M M M M M M M M M M M M M M M M M P I M M M M M M M M I M M P I M M M M M M M M M M M M M M M I M M M I M I M M M M P M M M
D E
F G R M M M M M M M R M M M M M M M M M M M M M M P M
I
H M M M M M M M M M M M M M P M M
I
M
M M M M
M M M M M M M I M I M M M I M I
I M M M M M M M M
M M M M M M M M M M
M M M M M M M
M M M M M M M M M P M I M M M M M M M M M M M M M M M
M M
M M
15-27
J
K L I - M M R M M M M M M M M M M M M M M M M M I I P M M M M M M M M M M M I I M M I I P M M M M M M M M M M M M M M M M M M M M M M M P M I I M M M M M I P M I M M M
M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M
N M I M I I M I P I
O P Q R S M M R M I M I M M M M M I M M I M M M M I M I M P M M I M I M M M M M M M M I I M I M I M I M I M I M M M M I M M M I M I I M I M I I M I M - M M M M - M M M M - I M M M I I M I M I M I M I I M I I M I M M M M I I I I I I I M I I P M I I M I M M I M I M M I M I M I M M M M M M M M M M M M M M I M I M P M I M M M M M R M I M I M
T U V W X M M M I M M M R I I I M M M M M I M M I I M M M M M M M M M M M P M M I R I M I M M M M M M M M M M I M M M I M I M M I M M M M M M I I M M M M M M I R I M R M I M M M M M I M R M M M I M M R M R - P I M P - M M M M M M I M - M M M M M P M M M M M M M M I I
Y M M M M M M M M M M
M
M M M M
M M
M M M M M
DENSITY OF SOLVENTS AS A FUNCTION OF TEMPERATURE The table below lists the density of several common solvents in the temperature range from 0°C to 100°C. The values have been calculated from the Rackett Equation using parameters in the reference. Density values refer to the liquid at its saturation vapor pressure; thus entries for temperatures above the normal boiling point are for pressures greater than atmospheric.
REFERENCE Lide, D. R., and Kehiaian, H. V., Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, FL, 1994. Density in g/mL Solvent Acetic acid Acetone Acetonitrile Aniline Benzene 1-Butanol Butylamine Carbon disulfide Chlorobenzene Cyclohexane Decane 1-Decanol Dichloromethane Diethyl ether N,N-Dimethylaniline Ethanol Ethyl acetate Ethylbenzene Ethyl formate Ethyl propanoate Heptane Hexane 1-Hexanol Isopropylbenzene Methanol Methyl acetate N-Methylaniline Methylcyclohexane Methyl formate Methyl propanoate Nitromethane Nonane Octane Pentanoic acid 1-Propanol 2-Propanol Propyl acetate Propylbenzene Propyl formate Tetrachloromethane Toluene Trichloromethane 2,2,4-Trimethylpentane o-Xylene m-Xylene p-Xylene
0°C
10°C
0.8129
0.8016
1.041
1.033 0.8884 0.8200 0.7512 1.277 1.116 0.7872 0.7374
0.8293 0.7606 1.290 1.127 0.7447 1.362 0.7368 0.8121 0.9245 0.8836 0.9472 0.9113 0.7004 0.6774 0.8359 0.8769 0.8157 0.9606 1.0010 0.7858 1.003 0.9383
1.344 0.7254 0.9638 0.8014 0.9126 0.8753 0.9346 0.9005 0.6921 0.6685 0.8278 0.8696 0.8042 0.9478 0.9933 0.7776 0.9887 0.9268
0.7327 0.7185 0.9563 0.8252 0.8092 0.9101 0.8779 0.9275 1.629 0.8846 1.524
0.7252 0.7106 0.9476 0.8151 0.7982 0.8994 0.8700 0.9166 1.611 0.8757 1.507
0.8813
0.8729
20°C 1.051 0.7902 0.7825 1.025 0.8786 0.8105 0.7417 1.263 1.106 0.7784 0.7301 0.8294 1.326 0.7137 0.9562 0.7905 0.9006 0.8668 0.9218 0.8895 0.6837 0.6594 0.8195 0.8615 0.7925 0.9346 0.9859 0.7693 0.9739 0.9150 1.139 0.7176 0.7027 0.9389 0.8048 0.7869 0.8885 0.8619 0.9053 1.593 0.8667 1.489 0.6921 0.8801 0.8644 0.8609
30°C 1.038 0.7785 0.7707 1.016 0.8686 0.8009 0.7320 1.248 1.096 0.7694 0.7226 0.8229 1.307 0.7018 0.9483 0.7793 0.8884 0.8582 0.9087 0.8784 0.6751 0.6502 0.8111 0.8533 0.7807 0.9211 0.9785 0.7608 0.9588 0.9030 1.125 0.7099 0.6945 0.9301 0.7943 0.7755 0.8775 0.8538 0.8938 1.575 0.8576 1.471 0.6836 0.8717 0.8558 0.8523
40°C 1.025 0.7666 0.7591 1.008 0.8584 0.7912 0.7221 1.234 1.085 0.7602 0.7151 0.8162 1.289 0.6896 0.9401 0.7680 0.8759 0.8495 0.8954 0.8671 0.6664 0.6407 0.8027 0.8450 0.7685 0.9074 0.9709 0.7522 0.9433 0.8907 1.111 0.7021 0.6863 0.9211 0.7837 0.7638 0.8662 0.8456 0.8821 1.557 0.8483 1.452 0.6750 0.8633 0.8470 0.8436
15-28
TeamLRN
50°C 1.012 0.7545 0.7473 1.000 0.8481 0.7812 0.7120
60°C 0.9993 0.7421 0.7353 0.9909 0.8376 0.7712 0.7017
70°C 0.9861 0.7293 0.7231 0.9823 0.8269 0.7609 0.6911
80°C 0.9728 0.7163 0.7106 0.9735 0.8160 0.7504 0.6803
90°C 0.9592 0.7029 0.6980 0.9646 0.8049 0.7398 0.6693
100°C 0.9454 0.6890 0.6851 0.9557 0.7935 0.7289 0.6579
1.074 0.7509 0.7074 0.8093 1.269 0.6770 0.9318 0.7564 0.8632 0.8407 0.8818 0.8556 0.6575 0.6311 0.7941 0.8366 0.7562 0.8933 0.9633 0.7435 0.9275 0.8783 1.097 0.6941 0.6779 0.9121 0.7729 0.7519 0.8548 0.8373 0.8702 1.538 0.8389 1.433 0.6663 0.8547 0.8382 0.8347
1.064 0.7414 0.6997 0.8024 1.250 0.6639 0.9234 0.7446 0.8503 0.8318 0.8678 0.8439 0.6485 0.6212 0.7854 0.8280 0.7435 0.8790 0.9556 0.7346 0.9112 0.8656 1.083 0.6861 0.6694 0.9029 0.7619 0.7397 0.8432 0.8289 0.8581 1.518 0.8294 1.414 0.6574 0.8460 0.8292 0.8258
1.053 0.7317 0.6919 0.7955 1.229 0.6505 0.9150 0.7324 0.8370 0.8228 0.8535 0.8319 0.6393 0.6111 0.7766 0.8194 0.7306 0.8643 0.9478 0.7255 0.8945 0.8526 1.069 0.6779 0.6608 0.8937 0.7506 0.7272 0.8313 0.8204 0.8457 1.499 0.8197 1.394 0.6484 0.8372 0.8201 0.8167
1.042 0.7218 0.6839 0.7884 1.208 0.6366 0.9064 0.7200 0.8234 0.8136 0.8389 0.8197 0.6298 0.6006 0.7676 0.8106 0.7174 0.8491 0.9399 0.7163 0.8772 0.8393 1.055 0.6696 0.6520 0.8843 0.7391 0.7143 0.8192 0.8117 0.8330 1.479 0.8098
1.030 0.7117 0.6758 0.7813 1.187 0.6220 0.8978 0.7073 0.8095 0.8043 0.8238 0.8072 0.6202 0.5899 0.7585 0.8017 0.7038 0.8336 0.9319 0.7069 0.8594 0.8257 1.040 0.6611 0.6430 0.8748 0.7273 0.7011 0.8069 0.8030 0.8201 1.458 0.7998
1.019 0.7013 0.6676 0.7740 1.165 0.6068 0.8890 0.6942 0.7952 0.7948 0.8082 0.7944 0.6102 0.5789 0.7492 0.7927 0.6898 0.8176 0.9239 0.6973 0.8409 0.8117 1.026 0.6525 0.6338 0.8652 0.7152 0.6876 0.7942 0.7943 0.8068 1.437 0.7896
0.6391 0.8282 0.8109 0.8075
0.6296 0.8191 0.8015 0.7981
0.6199 0.8099 0.7920 0.7886
DEPENDENCE OF BOILING POINT ON PRESSURE The normal boiling point of a liquid is defined as the temperature at which the vapor pressure reaches standard atmospheric pressure, 101.325 kPa. The change in boiling point with pressure may be calculated from the representation of the vapor pressure by the Antoine Equation, ln p = A1 – A2/(T + A3) where p is the vapor pressure, T the absolute temperature, and A1, A2, and A3 are constants. This table, which has been calculated using the Antoine constants in Reference 1, gives values of ∆t/∆p for a number of liquids, in units of both °C/kPa and °C/mmHg. The correction to the boiling point is generally accurate to 0.1 to 0.2 °C as long as the pressure is within 10% of standard atmospheric pressure. A slightly less accurate estimate of ∆t/∆p may be obtained from the Claussius-Clapeyron equation, with the assumption that the change in volume upon vaporization equals the ideal-gas volume of the vapor. This leads to the equation ∆t/∆p = RTb2 /p0 ∆vapH(Tb) where R is the molar gas constant, p0 is 101.325 kPa, Tb is the normal boiling point temperature (absolute), and ∆vapH(Tb) is the molar enthalpy of vaporization at the normal boiling point. Values of the last quantity may be obtained from the table “Enthalpy of Vaporization” in Section 6. REFERENCE 1. Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, FL, 1994, pp. 4959. tb
Compound
°C
°C/kPa
Acetaldehyde Acetic acid Acetone Acetonitrile Ammonia Aniline Anisole Benzaldehyde Benzene Bromine Butane 1-Butanol Carbon disulfide Chlorine Chlorobenzene 1-Chlorobutane Chloroethane Chloroethylene Cyclohexane Cyclohexanol Cyclohexanone Decane Dibutyl ether Dichloromethane Diethyl ether Dimethyl sulfoxide 1,4-Dioxane Dipropyl ether Ethanol Ethyl acetate Ethylene glycol Heptane Hexafluorobenzene Hexane
20.1 117.9 56.0 81.6 -33.33 184.1 153.7 179.0 80.0 58.8 -0.5 117.7 46.2 -34.04 131.7 78.6 12.3 -13.3 80.7 160.8 155.4 174.1 140.2 39.6 34.5 189.0 101.5 90.0 78.2 77.1 197.3 98.5 80.2 68.7
0.261 0.324 0.289 0.316 0.198 0.378 0.367 0.392 0.321 0.300 0.267 0.278 0.304 0.224 0.365 0.321 0.262 0.241 0.328 0.344 0.382 0.388 0.363 0.276 0.278 0.379 0.321 0.326 0.249 0.300 0.331 0.336 0.305 0.314
∆t/∆p °C/mmHg 0.0348 0.0432 0.0385 0.0421 0.0264 0.0504 0.0489 0.0523 0.0428 0.0400 0.0356 0.0371 0.0405 0.0299 0.0487 0.0428 0.0349 0.0321 0.0437 0.0459 0.0509 0.0517 0.0484 0.0368 0.0371 0.0505 0.0428 0.0435 0.0332 0.0400 0.0441 0.0448 0.0407 0.0419
tb
Compound 1-Hexanol Hydrogen fluoride Iodomethane Isobutane Methanol Methyl acetate Methyl formate N-Methylaniline N-Methylformamide Nitrobenzene Nitromethane 1-Octanol Pentane 1-Pentanol Phenol Propane 1-Propanol 2-Propanol Pyridine Pyrrole Pyrrolidine Styrene Sulfur dioxide Tetrachloroethylene Tetrachloromethane Toluene Trichloroethylene Trichloromethane Trimethylamine Water o-Xylene m-Xylene p-Xylene
15-19
°C
°C/kPa
157.6 20.1 42.5 -11.7 64.6 56.8 31.7 196.2 199.5 210.8 101.1 195.1 36.0 137.9 181.8 -42.1 97.2 82.3 115.2 129.7 86.5 145.1 -10.05 121.3 76.8 110.6 87.2 61.1 2.8 100.0 144.5 139.1 138.3
0.318 0.276 0.291 0.254 0.251 0.282 0.582 0.396 0.371 0.418 0.320 0.360 0.289 0.296 0.349 0.224 0.261 0.247 0.340 0.330 0.309 0.369 0.221 0.354 0.325 0.353 0.330 0.302 0.248 0.276 0.373 0.368 0.369
∆t/∆p °C/mmHg 0.0424 0.0368 0.0388 0.0339 0.0335 0.0376 0.0776 0.0528 0.0495 0.0557 0.0427 0.0480 0.0385 0.0395 0.0465 0.0299 0.0348 0.0329 0.0453 0.0440 0.0412 0.0492 0.0295 0.0472 0.0433 0.0471 0.0440 0.0403 0.0331 0.0368 0.0497 0.0491 0.0492
EBULLIOSCOPIC CONSTANTS FOR CALCULATION OF BOILING POINT ELEVATION The boiling point Tb of a dilute solution of a non-volatile, non-dissociating solute is elevated relative to that of the pure solvent. If the solution is ideal (i.e., follows Raoult’s Law), the amount of elevation depends only on the number of particles of solute present. Hence the change in boiling point ∆Tb can be expressed as ∆Tb = Eb m2 where m2 is the molality (moles of solute per kilogram of solvent) and Eb is the Ebullioscopic Constant, a characteristic property of the solvent. The Ebullioscopic Constant may be calculated from the relation Eb = R Tb2 M/∆vapH where R is the molar gas constant, Tb is the normal boiling point temperature (absolute) of the solvent, M the molar mass of the solvent, and ∆vapH the molar enthalpy (heat) of vaporization of the solvent at its normal boiling point. This table lists Eb values for some common solvents, as calculated from data in the table “Enthalpy of Vaporization” in Section 6. Compound Acetic acid Acetone Acetonitrile Aniline Anisole Benzaldehyde Benzene 1-Butanol Carbon disulfide Chlorobenzene 1-Chlorobutane Cyclohexane Cyclohexanol Decane Dichloromethane Diethyl ether Dimethyl sulfoxide 1,4-Dioxane Ethanol Ethyl acetate Ethylene glycol Heptane
Eb/K kg mol–1
Compound Hexane Iodomethane Methanol Methyl acetate N-Methylaniline N-Methylformamide Nitrobenzene Nitromethane 1-Octanol Phenol 1-Propanol 2-Propanol Pyridine Pyrrole Pyrrolidine Tetrachloroethylene Tetrachloromethane Toluene Trichloroethylene Trichloromethane Water o-Xylene
3.22 1.80 1.44 3.82 4.20 4.24 2.64 2.17 2.42 4.36 3.13 2.92 3.5 6.10 2.42 2.20 3.22 3.01 1.23 2.82 2.26 3.62
15-20
TeamLRN
Eb/K kg mol–1 2.90 4.31 0.86 2.21 4.3 2.2 5.2 2.09 5.06 3.54 1.66 1.58 2.83 2.33 2.32 6.18 5.26 3.40 4.52 3.80 0.513 4.25
CRYOSCOPIC CONSTANTS FOR CALCULATION OF FREEZING POINT DEPRESSION The freezing point Tf of a dilute solution of a non-volatile, non-dissociating solute is depressed relative to that of the pure solvent. If the solution is ideal (i.e., follows Raoult’s Law), this lowering is a function only of the number of particles of solute present. Thus the absolute value of the lowering of freezing point ∆Tf can be expressed as ∆Tf = Ef m2 where m2 is the molality (moles of solute per kilogram of solvent) and Ef is the Cryoscopic Constant, a characteristic property of the solvent. The Cryoscopic Constant may be calculated from the relation Ef = R Tf2 M/∆fusH where R is the molar gas constant, Tb is the freezing point temperature (absolute) of the solvent, M the molar mass of the solvent, and ∆fusH the molar enthalpy (heat) of fusion of the solvent. This table lists cryscopic constants for selected substances, as calculated from data in the table “Enthalpy of Fusion” in Section 6.
Compound Acetamide Acetic acid Acetophenone Aniline Benzene Benzonitrile Benzophenone (+)-Camphor 1-Chloronaphthalene o-Cresol m-Cresol p-Cresol Cyclohexane Cyclohexanol cis-Decahydronaphthalene trans-Decahydronaphthalene Dibenzyl ether p-Dichlorobenzene Diethanolamine Dimethyl sulfoxide
Ef/K kg mol–1 3.92 3.63 5.16 5.23 5.07 5.35 8.58 37.8 7.68 5.92 7.76 7.20 20.8 42.2 6.42 4.70 6.17 7.57 3.16 3.85
15-21
Compound
Ef/K kg mol–1
1,4-Dioxane Diphenylamine Ethylene glycol Formamide Formic acid Glycerol Methylcyclohexane Naphthalene Nitrobenzene Phenol Pyridine Quinoline Succinonitrile 1,1,2,2-Tetrabromoethane 1,1,2,2-Tetrachloro-1,2-difluoroethane Toluene p-Toluidine Tribromomethane Water p-Xylene
4.63 8.38 3.11 4.25 2.38 3.56 2.60 7.45 6.87 6.84 4.26 6.73 19.3 21.4 41.0 3.55 4.91 15.0 1.86 4.31
FREEZING POINT LOWERING BY ELECTROLYTES IN AQUEOUS SOLUTION REFERENCE Forsythe, W. E., Smithsonian Physical Tables, Ninth Edition, Smithsonian Institution, Washington, 1956. Lowering of freezing point of water (in °C) as function of molality (mol/kg) Compound
0.05
0.10
0.25
0.50
0.75
1.00
1.50
2.00
2.50
3.00
CaCl2 CuSO4 HCl HNO3 H2SO4 KBr KCl KNO3 K2SO4 LiCl MgSO4 NH4Cl NaCl NaNO3
0.25 0.13 0.18 0.18 0.20 0.18 0.17 0.17 0.23 0.18 0.13 0.17 0.18 0.18
0.49 0.23 0.36 0.35 0.39 0.36 0.35 0.33 0.43 0.35 0.24 0.34 0.35 0.36
1.27 0.47 0.90 0.88 0.96 0.92 0.86 0.78 1.01 0.88 0.55 0.85 0.85 0.80
2.66 0.96 1.86 1.80 1.95 1.78 1.68 1.47 1.87 1.80 1.01 1.70 1.68 1.62
4.28
6.35
10.78
15.27
20.42
28.08
2.90 2.78 3.04
4.02 3.80 4.28
6.63 5.98 7.35
9.94 8.34 11.35
10.95 16.32
13.92
2.49 2.11
3.29 2.66
4.88
6.50
8.14
9.77
2.08
3.41
2.78 1.50 2.55 2.60 2.63
3.10
TeamLRN
1-22
CORRECTION OF BAROMETER READINGS TO 0°C TEMPERATURE The following corrections are used to reduce the reading of a mercury barometer with a brass scale to 0°C. The number in the table should be subtracted from the observed height of the mercury column to give the true pressure in mmHg (1mmHg = 133.322 Pa). The table is calculated from the formula ∆h = -0.0001634 ht/(1+0.0001818 t), where h is the observed column height in mm and t the Celsius temperature. This relation is based on thermal expansion coefficients of 181.8·10-6 °C-1 for mercury and 18.4·10-6 °C-1 for brass. Observed Height in mm t/°C 620
630
640
650
660
670
680
690
700
710
720
730
740
750
760
770
780
790
800
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
0.00 0.10 0.21 0.31 0.41 0.51 0.62 0.72 0.82 0.92 1.03 1.13 1.23 1.34 1.44 1.54 1.64 1.74 1.85 1.95 2.05 2.15 2.26 2.36 2.46 2.56 2.66 2.77 2.87 2.97 3.07 3.17 3.28 3.38 3.48 3.58 3.68 3.78 3.88 3.99 4.09
0.00 0.10 0.21 0.31 0.42 0.52 0.63 0.73 0.84 0.94 1.04 1.15 1.25 1.36 1.46 1.56 1.67 1.77 1.88 1.98 2.08 2.19 2.29 2.40 2.50 2.60 2.71 2.81 2.91 3.02 3.12 3.22 3.33 3.43 3.53 3.64 3.74 3.84 3.95 4.05 4.15
0.00 0.11 0.21 0.32 0.42 0.53 0.64 0.74 0.85 0.95 1.06 1.17 1.27 1.38 1.48 1.59 1.69 1.80 1.91 2.01 2.12 2.22 2.33 2.43 2.54 2.64 2.75 2.85 2.96 3.06 3.17 3.27 3.38 3.48 3.59 3.69 3.80 3.90 4.01 4.11 4.22
0.00 0.11 0.22 0.32 0.43 0.54 0.65 0.75 0.86 0.97 1.08 1.18 1.29 1.40 1.51 1.61 1.72 1.83 1.93 2.04 2.15 2.26 2.36 2.47 2.58 2.68 2.79 2.90 3.00 3.11 3.22 3.32 3.43 3.54 3.64 3.75 3.86 3.96 4.07 4.18 4.28
0.00 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.87 0.98 1.09 1.20 1.31 1.42 1.53 1.64 1.75 1.86 1.96 2.07 2.18 2.29 2.40 2.51 2.62 2.72 2.83 2.94 3.05 3.16 3.27 3.37 3.48 3.59 3.70 3.81 3.92 4.02 4.13 4.24 4.35
0.00 0.11 0.22 0.33 0.44 0.56 0.67 0.78 0.89 1.00 1.11 1.22 1.33 1.44 1.55 1.66 1.77 1.88 1.99 2.10 2.21 2.32 2.43 2.54 2.66 2.77 2.88 2.99 3.10 3.21 3.32 3.43 3.54 3.64 3.75 3.86 3.97 4.08 4.19 4.30 4.41
0.00 0.11 0.23 0.34 0.45 0.56 0.68 0.79 0.90 1.01 1.13 1.24 1.35 1.46 1.57 1.69 1.80 1.91 2.02 2.13 2.25 2.36 2.47 2.58 2.69 2.81 2.92 3.03 3.14 3.25 3.36 3.48 3.59 3.70 3.81 3.92 4.03 4.14 4.25 4.37 4.48
0.00 0.11 0.23 0.34 0.46 0.57 0.69 0.80 0.91 1.03 1.14 1.26 1.37 1.48 1.60 1.71 1.82 1.94 2.05 2.17 2.28 2.39 2.51 2.62 2.73 2.85 2.96 3.07 3.19 3.30 3.41 3.53 3.64 3.75 3.87 3.98 4.09 4.20 4.32 4.43 4.54
0.00 0.12 0.23 0.35 0.46 0.58 0.70 0.81 0.93 1.04 1.16 1.27 1.39 1.50 1.62 1.74 1.85 1.97 2.08 2.20 2.31 2.43 2.54 2.66 2.77 2.89 3.00 3.12 3.23 3.35 3.46 3.58 3.69 3.81 3.92 4.03 4.15 4.26 4.38 4.49 4.61
0.00 0.12 0.24 0.35 0.47 0.59 0.71 0.82 0.94 1.06 1.17 1.29 1.41 1.53 1.64 1.76 1.88 1.99 2.11 2.23 2.34 2.46 2.58 2.69 2.81 2.93 3.04 3.16 3.28 3.39 3.51 3.63 3.74 3.86 3.98 4.09 4.21 4.32 4.44 4.56 4.67
0.00 0.12 0.24 0.36 0.48 0.60 0.71 0.83 0.95 1.07 1.19 1.31 1.43 1.55 1.67 1.78 1.90 2.02 2.14 2.26 2.38 2.50 2.61 2.73 2.85 2.97 3.09 3.20 3.32 3.44 3.56 3.68 3.79 3.91 4.03 4.15 4.27 4.38 4.50 4.62 4.74
0.00 0.12 0.24 0.36 0.48 0.60 0.72 0.85 0.97 1.09 1.21 1.33 1.45 1.57 1.69 1.81 1.93 2.05 2.17 2.29 2.41 2.53 2.65 2.77 2.89 3.01 3.13 3.25 3.37 3.49 3.61 3.73 3.85 3.97 4.09 4.21 4.32 4.44 4.56 4.68 4.80
0.00 0.12 0.25 0.37 0.49 0.61 0.73 0.86 0.98 1.10 1.22 1.35 1.47 1.59 1.71 1.83 1.96 2.08 2.20 2.32 2.44 2.56 2.69 2.81 2.93 3.05 3.17 3.29 3.41 3.54 3.66 3.78 3.90 4.02 4.14 4.26 4.38 4.50 4.62 4.75 4.87
0.00 0.12 0.25 0.37 0.50 0.62 0.74 0.87 0.99 1.12 1.24 1.36 1.49 1.61 1.73 1.86 1.98 2.10 2.23 2.35 2.47 2.60 2.72 2.84 2.97 3.09 3.21 3.34 3.46 3.58 3.71 3.83 3.95 4.07 4.20 4.32 4.44 4.56 4.69 4.81 4.93
0.00 0.13 0.25 0.38 0.50 0.63 0.75 0.88 1.01 1.13 1.26 1.38 1.51 1.63 1.76 1.88 2.01 2.13 2.26 2.38 2.51 2.63 2.76 2.88 3.01 3.13 3.26 3.38 3.51 3.63 3.75 3.88 4.00 4.13 4.25 4.38 4.50 4.62 4.75 4.87 5.00
0.00 0.13 0.25 0.38 0.51 0.64 0.76 0.89 1.02 1.15 1.27 1.40 1.53 1.65 1.78 1.91 2.03 2.16 2.29 2.41 2.54 2.67 2.79 2.92 3.05 3.17 3.30 3.42 3.55 3.68 3.80 3.93 4.05 4.18 4.31 4.43 4.56 4.68 4.81 4.94 5.06
0.00 0.13 0.26 0.39 0.52 0.64 0.77 0.90 1.03 1.16 1.29 1.42 1.55 1.67 1.80 1.93 2.06 2.19 2.32 2.44 2.57 2.70 2.83 2.96 3.08 3.21 3.34 3.47 3.60 3.72 3.85 3.98 4.11 4.23 4.36 4.49 4.62 4.74 4.87 5.00 5.13
0.00 0.13 0.26 0.39 0.52 0.65 0.78 0.91 1.04 1.17 1.30 1.44 1.57 1.70 1.83 1.96 2.09 2.22 2.35 2.48 2.60 2.73 2.86 2.99 3.12 3.25 3.38 3.51 3.64 3.77 3.90 4.03 4.16 4.29 4.42 4.55 4.68 4.80 4.93 5.06 5.19
0.00 0.10 0.20 0.30 0.40 0.51 0.61 0.71 0.81 0.91 1.01 1.11 1.21 1.31 1.41 1.52 1.62 1.72 1.82 1.92 2.02 2.12 2.22 2.32 2.42 2.52 2.62 2.72 2.82 2.92 3.02 3.12 3.22 3.32 3.42 3.52 3.62 3.72 3.82 3.92 4.02
15-23
DETERMINATION OF RELATIVE HUMIDITY FROM DEW POINT The relative humidity of a water vapor-air mixture is defined as 100 times the partial pressure of water divided by the saturation vapor pressure of water at the same temperature. The relative humidity may be determined from the dew point tdew, which is the temperature at which liquid water first condenses when the mixture is cooled from an initial temperature t. This table gives relative humidity as a function of the dew point depression t - tdew for several values of the dew point. Values are calculated from the vapor pressure table in Section 6.
t - tdew
-10
0
tdew/°C 10
20
30
t - tdew
-10
0.0 0.2 0.4 0.6 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.9 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0
100 99 97 95 94 92 91 90 88 87 86 84 83 82 80 79 78 77 76 75 73 72 71 70 69 69 67 66 65 64 63 62 61 60 60 59 58 57 56 55 54
100 99 97 96 94 93 92 90 89 88 87 85 84 83 82 81 80 79 77 76 75 74 73 72 71 70 69 68 67 66 66 65 64 63 62 61 60 60 59 58 57
100 99 97 96 95 94 92 91 90 89 88 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 69 68 67 66 65 64 63 63 62 61 60 60
100 99 98 96 95 94 93 92 91 90 88 87 86 85 84 83 82 81 80 79 78 77 77 76 75 74 73 72 71 70 70 69 68 67 66 66 65 64 63 63 62
100 99 98 97 96 94 93 92 91 90 89 89 87 86 85 84 83 82 82 81 80 79 78 77 76 75 75 74 73 72 71 71 70 69 68 68 67 66 65 65 64
8.2 8.4 8.6 8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0
54 53 52 51 51 50 49 48 48 47 45 44 42 41 39 38 37 35 34 33 32 31 30 29 28 27 26 25 24 24 22 21 19 18 17 16 15 14 13 12
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0 56 56 55 54 53 53 52 51 51 50 48 47 45 44 42 41 40 38 37 36 35 34 33 32 31 30 29 28 27 26 25 23 22 21 19 18 17 16 15 14
tdew/°C 10
20
59 58 57 57 56 55 55 54 53 53 51 49 48 47 45 44 43 41 40 39 38 37 36 35 34 33 32 31 30 29 27 26 24 23 22 21 20 19 18 17
61 60 60 59 58 58 57 56 56 55 54 52 51 49 48 46 45 44 43 42 40 39 38 37 36 35 34 33 33 32 30 29 27 26 24 23 22 21 20 19
30 63 63 62 61 61 60 59 59 58 57 56 55 53 52 50 49 48 47 45 44
DETERMINATION OF RELATIVE HUMIDITY FROM WET AND DRY BULB TEMPERATURES Relative humidity may be determined by comparing temperature readings of wet and dry bulb thermometers. The following table, extracted from more extensive U.S. National Weather Service tables, gives the relative humidity as a function of air temperature td (dry bulb) and the difference td - tw between dry and wet bulb temperatures. The data assume a pressure near normal atmospheric pressure and an instrumental configuration with forced ventilation.
td/°C
0.5
1.0
1.5
2.0
2.5
(td - tw)/°C 3.0 3.5
4.0
4.5
5.0
5.5
6.0
-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 35 40
83 86 88 89 90 91 91 92 93 93 94 94 95 95 95 96 96 96 96 96 96 97 97
67 71 74 77 79 81 84 85 86 87 88 89 90 90 91 91 92 92 92 93 93 94 94
51 57 61 66 69 72 76 78 79 81 82 83 85 85 86 87 87 88 88 89 89 90 91
35 43 49 55 60 64 68 71 73 75 77 78 79 81 82 83 83 84 85 85 86 87 88
19 29 37 44 50 55 60 63 66 69 71 73 75 76 77 78 80 80 81 82 83 84 85
15 25 33 40 46 52 57 60 63 66 68 70 71 73 74 76 77 78 78 79 81 82
12 22 29 37 43 48 51 55 58 60 63 65 66 68 69 71 72 73 75 77
12 21 29 36 41 46 50 53 56 58 61 63 64 66 67 69 70 72 74
13 22 29 35 40 44 48 51 54 57 59 61 62 64 65 67 69 72
5 14 22 29 35 39 43 47 50 53 55 57 59 61 62 64 67 69
7 16 24 29 34 39 42 46 49 51 54 56 58 59 61 64 67
td/°C
6.5
7.0
7.5
8.0
8.5
(td - tw)/°C 9.0 10.0
11.0
12.0
13.0
14.0
15.0
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
9 17 24 29 34 38 42 45 48 50 53 54 56 58 60 61 62 63 64
11 19 24 29 34 38 41 44 47 49 51 53 55 57 58 59 61 62
5 14 20 25 30 34 38 41 44 46 49 51 52 54 56 57 58 59
8 15 21 26 30 34 37 40 43 46 48 50 51 53 54 56 57
10 16 22 26 30 34 37 40 43 45 47 49 51 52 54 54
6 12 18 23 27 31 34 37 40 42 44 46 48 50 51 53
8 14 18 22 26 29 32 35 37 39 41 43 44
7 12 17 20 24 27 30 32 35 37 39 40
6 11 15 19 22 25 28 30 33 35 36
6 10 14 18 21 24 26 29 31 33
5 10 13 17 20 23 25 27 29
15-24
8 23 31 38 44 49 54 57 60 63 65 67 69 70 72 73 74 75 76 78 80
5 10 15 20 24 28 31 34 37 39 41 43 45 47 48
CONSTANT HUMIDITY SOLUTIONS Anthony Wexler An excess of a water soluble salt in contact with its saturated solution and contained within an enclosed space produces a constant relative humidity and water vapor pressure according to RH = A exp(B/T) where RH is the percent relative humidity (generally accurate to ±2 %), T is the temperature in kelvin, and the constants A and B and the range of valid temperatures are given in the table below. The vapor pressure, p, can be calculated from p = (RH/100) × p0 where p0 is the vapor pressure of pure water at temperature T as given in the table in Section 6 titled “Vapor Pressure of Water from 0 to 370°C”. REFERENCES 1. 2. 3. 4.
Wexler, A. S. and Seinfeld, J. H., Atmospheric Environment, 25A, 2731, 1991. Greenspan, L., J. Res. National Bureau of Standards, 81A, 89, 1977. Broul, et al., Solubility of Inorganic Two-Component Systems, Elsevier, New York, 1981. Wagman, D. D. et al., J. Phys. Chem. Ref. Data, Vol. 11, Suppl. 2, 1982.
Compound
NaOH ⋅ H2O LiBr ⋅ 2H2O ZnBr2 ⋅ 2H2O KOH ⋅ 2H2O LiCl ⋅ H2O CaBr2 ⋅ 6H2O LiI ⋅ 3H2O CaCl2⋅ 6H2O MgCl2⋅ 6H2O NaI⋅ 2H2O Ca(NO3)2 ⋅ 4H2O Mg(NO3)2 ⋅ 6H2O NaBr ⋅ 2H2O NH4NO3 KI SrCl2 ⋅ 6H2O NaNO3 NaCl NH4Cl KBr (NH4)2SO4 KCl Sr(NO3)2 ⋅ 4H2O BaCl2 ⋅ 2H2O CsI KNO3 K2SO4
Temperature range (°C)
RH 25°C
15—60 10—30 5—30 5—30 20—65 11—22 15—65 15—25 5—45 5—45 10—30 5—35 0—35 10—40 5—30 5—30 10—40 10—40 10—40 5—25 10—40 5—25 5—25 5—25 5—25 0—50 10—50
6 6 8 9 11 16 18 29 33 38 51 53 58 62 69 71 74 75 79 81 81 84 85 90 91 92 97
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TeamLRN
A
B
5.48 0.23 1.69 0.014 14.53 0.17 0.15 0.11 29.26 3.62 1.89 25.28 20.49 3.54 29.35 31.58 26.94 69.20 35.67 40.98 62.06 49.38 28.34 69.99 70.77 43.22 86.75
27 996 455 1924 –75 1360 1424 1653 34 702 981 220 308 853 254 241 302 25 235 203 79 159 328 75 75 225 34
STANDARD SALT SOLUTIONS FOR HUMIDITY CALIBRATION Saturated aqueous solutions of inorganic salts are convenient secondary standards for calibration of instruments for measurement of relative humidity. The International Union of Pure and Applied Chemistry has recommended salt solutions for calibrations in the range of 10% to 90% relative humidity, and the American Society for Testing and Materials has published similar standards. The data in this table are taken from the IUPAC recommendations, except for K2CO3 and K2SO4, which are ASTM recommendations. Details on the preparation and use of these standards may be found in References 1 and 2. Data for other salts are given in Reference 3.
REFERENCES 1. Marsh, K. N., Editor, Recommended Reference Materials for the Realization of Physicochemical Properties, Blackwell Scientific Publications, Oxford, 1987, pp.157-162. 2. Standard Practice for Maintaining Constant Relative Humidity by Means of Aqueous Solutions, ASTM Standard E 104-85, Reapproved 1991. 3. Greenspan, L., J. Res. Nat. Bur. Stand., 81A, 89, 1977. Relative Humidity in % t/°C 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
LiCl
MgCl2
K2CO3
Mg(NO3)2
NaCl
KCl
K2SO4
11.31±0.31 11.30±0.27 11.28±0.24 11.25±0.22 11.21±0.21 11.16±0.21 11.10±0.22 11.03±0.23 10.95±0.26 10.86±0.29 10.75±0.33 10.64±0.38 10.51±0.44
33.66±0.33 33.60±0.28 33.47±0.24 33.30±0.21 33.07±0.18 32.78±0.16 32.44±0.14 32.05±0.13 31.60±0.13 31.10±0.13 30.54±0.14 29.93±0.16 29.26±0.18 28.54±0.21 27.77±0.25 26.94±0.29 26.05±0.34
43.1±0.7 43.1±0.5 43.1±0.4 43.2±0.3 43.2±0.3 43.2±0.4 43.2±0.5
60.35±0.55 58.86±0.43 57.36±0.33 55.87±0.27 54.38±0.23 52.89±0.22 51.40±0.24 49.91±0.29 48.42±0.37 46.93±0.47 45.44±0.60
75.51±0.34 75.65±0.27 75.67±0.22 75.61±0.18 75.47±0.14 75.29±0.12 75.09±0.11 74.87±0.12
88.61±0.53 87.67±0.45 86.77±0.39 85.92±0.33 85.11±0.29 84.34±0.26 83.62±0.25 82.95±0.25 82.32±0.25 81.74±0.28 81.20±0.31 80.70±0.35 80.25±0.41 79.85±0.48 79.49±0.57 79.17±0.66 78.90±0.77
98.8±2.1 98.5±0.9 98.2±0.8 97.9±0.6 97.6±0.5 97.3±0.5 97.0±0.4 96.7±0.4 96.4±0.4 96.1±0.4 95.8±0.5
© 2000 CRC Press LLC
LOW TEMPERATURE BATHS FOR MAINTAINING CONSTANT TEMPERATURE A liquid-solid slurry is a convenient means of maintaining a constant temperature environment below room temperature. The following is a list of readily available organic liquids suitable for this purpose, arranged in order of their melting (freezing) points tm. The normal boiling points tb are also given. tm/°C
Compound Isopentane (2-Methylbutane) Methylcyclopentane 3-Chloropropene (Allyl chloride) Pentane Allyl alcohol Ethanol Carbon disulfide Isobutyl alcohol Toluene Acetone Ethyl acetate Dry ice + acetone p-Cymene Trichloromethane (Chloroform) N-Methylaniline Chlorobenzene Anisole Bromobenzene Tetrachloromethane (Carbon tetrachloride) Benzonitrile
© 2000 CRC Press LLC
TeamLRN
-159.9 -142.5 -134.5 -129.7 -129 -114.1 -111.5 -108 -94.9 -94.8 -83.6 -78 -68.9 -63.6 -57 -45.2 -37.5 -30.6 -23 -12.7
tb/°C 27.8 71.8 45.1 36.0 97.0 78.2 46 107.8 110.6 56.0 77.1 177.1 61.1 196.2 131.7 153.7 156.0 76.8 191.1
METALS AND ALLOYS WITH LOW MELTING TEMPERATURE L. I. Berger
Metal or Alloy System Hg Cs-K Cs-Na K-Na Na-Rb Ga-In-Sn Ga-Sn-Zn Cs Ga K-Rb Bi-Cd-In-Pb-Sn Bi-In-Pb-Sn Bi-In-Sn K Bi-Cd-Pb-Sn Bi-In Bi-Cd-Pb Bi-Pb-Sn Na Bi-Cd-Sn In-Sn Cd-In Bi-Pb Bi-Sn-Zn Bi-Sn Bi-Cd In Li Pb-Sn Bi-Tl Sn-Zn Sb-Sn Au-Pb Ag-Sn Bi-Pb-Sb-Sn Cu-Sn Sn
Weight
Melting Temperature °C
Composition, %* Atomic
100 77.0-23.0 94.5-5.5 76.7-23.3 8.0-92.0 62.5-21.5-16.0 82.0-12.0-6.0 100 100 32.0-68.0 44.7-5.3-19.1-22.6-8.3 49.5-21.3-17.6-11.6 32.5-51.0-16.5 100 50.0-12.5-25.0-12.5 33.0-67.0 51.6-8.2-40.2 52.5-32.0-15.5 100 54.0-20.0-26.0 51.8-48.2 25.3-74.7 55.5-44.5 56.0-40.0-4.0 70-30 60.3-39.7 100 100 38.1-61.9 48.0-52.0 91.0-9.0 8.0-92.0 14.6-85.4 3.5-96.5 48.0-28.5-9.0-14.5 0.75-99.25 100
100 50.0-50.0 75.0-25.0 65.9-34.1 24.4-75.6 73.6-15.3-11.1 86.0-7.3-6.7 100 100 50-50 35.1-8.2-27.3-17.9-11.5 39.2-30.7-14.0-16.2 21.1-60.1-18.8 100 41.5-19.3-21.0-18.2 21.3-78.7 48.1-14.2-37.7 46.8-28.7-24.5 100 39.4-27.2-33.4 52.6-47.4 25.7-74.3 55.3-44.7 40.2-50.6-9.2 57.0-43.0 45.0-55.0 100 100 26.1-73.9 47.5-52.5 85.0-15.0 7.8-92.2 15.2-84.8 3.8-96.2 40.8-24.5-13.1-21.6 1.3-98.7 100
-38.84 -37.5 -30.0 -12.65 -5 11 17 28.44 29.77 33 46.7 58.2 60.5 63.38 70 72 91.5 95 97.8 102.5 119 120 124 130 138.5 145.5 156.6 180.5 183 185 198 199 212 221 226 227 231.9
Comments
Ref.
Eutectic (?) Eutectic Eutectic Eutectic Eutectic Eutectic
1 2 3 4 5 5
Eutectic Eutectic Eutectic Eutectic
4 6 6 7
Wood’s alloy Eutectic Eutectic Eutectic
6 8 6 6
Eutectic Eutectic Eutectic Eutectic Eutectic Eutectic Eutectic
6 9 10 11 6, 7 6, 12 13, 14
Eutectic Eutectic Eutectic White Metal Eutectic Eutectic Matrix Alloy Eutectic
6,15 13 14 16 17 13,18 6 13, 19
*The useful expression for correlations between the atomic and weight concentrations of an alloy components are:
f (w, Ak )
f (a, Ak ) =
N
Mk
∑ i =1
f (w, Ai )
and f (w, Ak ) =
M k ⋅ f (a, Ak ) N
∑ M ⋅ f (a , A ) i
Mi
(i = 1,K, k,K, N )
i
i =1
where f(a, Ai) and f(w, Ai) are the atomic and weight concentrations of component Ai, respectively, and Mi is the atomic weight of this component.
15-28
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Zintle, E. and Hauke, W., Z. Electrochem., 44, 104, 1938. Rinck, E., Compt. Rend., 199, 1217, 1934. Krier, C. A., Craign, R. S., and Wallace, W. E., J. Phys. Chem., 61, 522, 1957. Goria, C., Gazz. Chim. Ital., 65, 865, 1935. Baker, H., Ed., ASM Handbook, Volume 3: Alloy Phase Diagrams, ASM Intl., Materials Park, OH, 1992. Sedlacek, V., Non-Ferrous Metals and Alloys, Elsevier, 1986. Villars, P., Prince, A., Okamoto, H., Eds., Handbook of Ternary Alloy Phase Diagrams, ASM Intl., 1994. Palatnik, L. S., Kosevich, V. M., and Tyrina, L. V., Phys. Metals Metallog. (USSR), 11, 75, 1961. Neumann, T. and Alpout, O., J. Less-Common Metals, 6, 108, 1964. Neumann, T. and Predel, B., Z. Metallk., 50, 309, 1959. Roy, P., Orr, R. L., and Hultgren, R., J. Phys. Chem., 64, 1034, 1960. Dobovicek, B. and Smajic, N., Rudarsko-Met. Zbornik, 4, 353, 1962. Massalski, T. B., Okamoto, H., Subramanian, P. R., and Kacprzak, L., Eds., Binary Alloy Phase Diagrams, 2nd ed., ASM Intl., 1990. Dobovicek, B. and Straus, B., Rudarsko-Met. Zbornik, 3, 273, 1960. Schurmann, E. and Gilhaus, F. J., Arch. Eisenhuettenw., 32, 867, 1961. Rosenblatt, G. M. and Birchenall, C. E., Trans. AIME, 224, 481, 1962. Evans, D. S. and Prince, A., in Alloy Phase Diagrams, MRS Simposia Proc., Vol. 19, North-Holland, 1983, p. 383. Umanskiy, M. M., Zh. Fiz. Khim., 14, 846, 1940. Homer, C. E. and Plummer, H., J. Inst. Met., 64, 169, 1939.
15-29
TeamLRN
WIRE TABLES The resistance per unit length of wires of various metals is tabulated here. Values were calculated from resistivity values in the tables “Electrical Resistivity of Pure Metals” and “Electrical Resistivity of Selected Alloys”, which appear in Section 12. In practice, resistance may vary because of differing heat treatments and metal composition. The values in the table refer to 20°C, but values at other temperatures may be calculated from the following resistivity data: Resistivity in 10–8 Ω m at temperature 20°C 25°C
0°C
Metal Aluminum Brass (70% Cu, 30% Zn) Constantan (60% Cu, 40% Ni) Copper Nichrome (79% Ni, 21% Cr) Platinum Silver Tungsten
2.417 5.87 45.43 1.543 107.3 9.6 1.467 4.82
2.650 6.08 45.38 1.678 107.5 10.5 1.587 5.28
2.709 6.13 45.35 1.712 107.6 10.7 1.617 5.39
100°C 3.56 6.91 45.11 2.22 108.3 13.6 2.07 7.18
Resistance per unit length at 20°C in Ω/m B&S Gauge 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Diameter (mm)
Aluminum
Brass
Constantan
Copper
8.252 6.543 5.189 4.115 3.264 2.588 2.053 1.628 1.291 1.024 0.8118 0.6439 0.5105 0.4049 0.3211 0.2548 0.2019 0.1601 0.1270 0.1007 0.07988
0.000495 0.000788 0.00125 0.00199 0.00317 0.00504 0.00800 0.0127 0.0202 0.0322 0.0512 0.0814 0.129 0.206 0.327 0.520 0.828 1.32 2.09 3.33 5.29
0.00114 0.00181 0.00287 0.00457 0.00727 0.0115 0.0184 0.0292 0.0464 0.0738 0.117 0.187 0.297 0.472 0.751 1.19 1.90 3.02 4.80 7.63 12.1
0.00848 0.0135 0.0214 0.0341 0.0542 0.0863 0.137 0.218 0.347 0.551 0.877 1.39 2.22 3.52 5.60 8.90 14.2 22.5 35.8 57.0 90.5
0.000314 0.000499 0.000793 0.00126 0.00200 0.00319 0.00507 0.00806 0.0128 0.0204 0.0324 0.0515 0.0820 0.130 0.207 0.329 0.524 0.833 1.32 2.11 3.35
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Nichrome
Platinum
Silver
Tungsten
0.0201 0.0320 0.0508 0.0808 0.128 0.204 0.325 0.516 0.821 1.30 2.08 3.30 5.25 8.35 13.3 21.1 33.6 53.4 84.9 135 214
0.00196 0.00312 0.00496 0.00789 0.0125 0.0200 0.0317 0.0504 0.0802 0.127 0.203 0.322 0.513 0.815 1.30 2.06 3.28 5.22 8.29 13.2 20.9
0.000297 0.000472 0.000750 0.00119 0.00190 0.00302 0.00479 0.00762 0.0121 0.0193 0.0307 0.0487 0.0775 0.123 0.196 0.311 0.496 0.788 1.25 1.99 3.17
0.00099 0.00157 0.00250 0.00397 0.00631 0.0100 0.0159 0.0254 0.0403 0.0641 0.102 0.162 0.258 0.410 0.652 1.03 1.65 2.62 4.17 6.63 10.5
TeamLRN
DENSITY OF VARIOUS SOLIDS This table gives the range of density for miscellaneous solid materials whose characteristics depend on the source or method of preparation.
REFERENCES 1. Forsythe, W. E., Smithsonian Physical Tables, Ninth Edition, Smithsonian Institution, Washington, 1956. 2. Kaye, G. W. C., and Laby, T. H., Tables of Physical and Chemical Constants, 16th Edition, Longman, London, 1995. 3. Brandrup, J., and Immergut, E. H., Polymer Handbook, Third Edition, John Wiley & Sons, New York, 1989. Material
ρ/ g cm-3
Material
ρ/ g cm-3
Material
ρ/ g cm-3
Agate Alabaster, carbonate sulfate Albite Amber Amphiboles Anorthite Asbestos Asbestos slate Asphalt Basalt Beeswax Beryl Biotite Bone Brasses Brick Bronzes Butter Calamine Calcspar Camphor Cardboard Celluloid Cement, set Chalk Charcoal, oak pine Cinnabar Clay Coal, anthracite bituminous Coke Copal Cork Corundum Diamond Dolomite Ebonite Emery Epidote Feldspar Flint Fluorite Galena Garnet Gelatin Glass, common lead
2.5-2.7
Pyrex Granite Graphite Gum arabic Gypsum Hematite Hornblende Ice Iron, cast Ivory Kaolin Leather, dry Lime, slaked Limestone Linoleum Magnetite Malachite Marble Meerschaum Mica Muscovite Ochre Opal Paper Paraffin Peat blocks Pitch Polyamides Polyethylene Poly(methyl methacrylate) Polypropylene Polystyrene Polytetrafluoroethylene Poly(vinyl acetate) Poly(vinyl chloride) Porcelain Porphyry Pyrite Quartz Resin Rock salt Rubber, hard soft pure gum Neoprene Sandstone Serpentine Silica, fused, Silicon carbide Slag Slate
2.23 2.64-2.76 2.30-2.72 1.3-1.4 2.31-2.33 4.9-5.3 3.0 0.917 7.0-7.4 1.83-1.92 2.6 0.86 1.3-1.4 2.68-2.76 1.18 4.9-5.2 3.7-4.1 2.6-2.84 0.99-1.28 2.6-3.2 2.76-3.00 3.5 2.2 0.7-1.15 0.87-0.91 0.84 1.07 1.15-1.25 0.92-0.97 1.19 0.91-0.94 1.06-1.12 2.28-2.30 1.19 1.39-1.42 2.3-2.5 2.6-2.9 4.95-5.10 2.65 1.07 2.18
Soapstone Solder Starch Steel, stainless Sugar Talc Tallow, beef Tar Topaz Tourmaline Tungsten carbide Wax, sealing Wood (seasoned) alder apple ash balsa bamboo basswood beech birch blue gum box butternut cedar cherry dogwood ebony elm hickory holly juniper larch locust logwood mahogany maple oak pear pine, pitch white yellow plum poplar satinwood spruce sycamore teak, Indian walnut water gum willow Wood’s metal
2.6-2.8 8.7-9.4 1.53 7.8 1.59 2.7-2.8 0.94 1.02 3.5-3.6 3.0-3.2 14.0-15.0 1.8
2.69-2.78 2.26-2.32 2.62-2.65 1.06-1.11 2.9-3.2 2.74-2.76 2.0-2.8 1.8 1.1-1.5 2.4-3.1 0.96-0.97 2.69-2.70 2.7-3.1 1.7-2.0 8.44-8.75 1.4-2.2 8.74-8.89 0.86-0.87 4.1-4.5 2.6-2.8 0.99 0.69 1.4 2.7-3.0 1.9-2.8 0.57 0.28-0.44 8.12 1.8-2.6 1.4-1.8 1.2-1.5 1.0-1.7 1.04-1.14 0.22-0.26 3.9-4.0 3.51 2.84 1.15 4.0 3.25-3.50 2.55-2.75 2.63 3.18 7.3-7.6 3.15-4.3 1.27 2.4-2.8 3-4
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1.19 1.1 0.91-0.93 1.23-1.25 2.14-2.36 2.50-2.65 2.21 3.16 2.0-3.9 2.6-3.3
0.42-0.68 0.66-0.84 0.65-0.85 0.11-0.14 0.31-0.40 0.32-0.59 0.70-0.90 0.51-0.77 1.00 0.95-1.16 0.38 0.49-0.57 0.70-0.90 0.76 1.11-1.33 0.54-0.60 0.60-0.93 0.76 0.56 0.50-0.56 0.67-0.71 0.91 0.66-0.85 0.62-0.75 0.60-0.90 0.61-0.73 0.83-0.85 0.35-0.50 0.37-0.60 0.66-0.78 0.35-0.50 0.95 0.48-0.70 0.40-0.60 0.66-0.98 0.64-0.70 1.00 0.40-0.60 9.70
DENSITY OF ETHANOL-WATER MIXTURES This table gives the density of mixtures of ethanol and water as a function of composition and temperature. The composition is specified in weight percent of ethanol, i.e., mass of ethanol per 100 g of solution. Values from the reference have been converted to true densities.
REFERENCE Washburn, E. W., Ed., International Critical Tables of Numerical Data of Physics, Chemistry, and Technology, Vol. 3, McGraw-Hill, New York, 1926-1932.
Weight % Ethanol 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
10 ∞C
15 ∞C
Density in g/cm3 20 ∞C 25 ∞C
30 ∞C
35 ∞C
40 ∞C
0.99970 0.99095 0.98390 0.97797 0.97249 0.96662 0.95974 0.95159 0.94235 0.93223 0.92159 0.91052 0.89924 0.88771 0.87599 0.86405 0.85194 0.83948 0.82652 0.81276 0.79782
0.99910 0.99029 0.98301 0.97666 0.97065 0.96421 0.95683 0.94829 0.93879 0.92849 0.91773 0.90656 0.89520 0.88361 0.87184 0.85985 0.84769 0.83522 0.82225 0.80850 0.79358
0.99820 0.98935 0.98184 0.97511 0.96861 0.96165 0.95379 0.94491 0.93515 0.92469 0.91381 0.90255 0.89110 0.87945 0.86763 0.85561 0.84341 0.83093 0.81795 0.80422 0.78932
0.99565 0.98667 0.97872 0.97130 0.96392 0.95604 0.94738 0.93787 0.92767 0.91689 0.90577 0.89434 0.88275 0.87097 0.85905 0.84695 0.83470 0.82218 0.80920 0.79553 0.78073
0.99403 0.98498 0.97682 0.96908 0.96131 0.95303 0.94400 0.93422 0.92382 0.91288 0.90165 0.89013 0.87848 0.86664 0.85467 0.84254 0.83027 0.81772 0.80476 0.79112 0.77639
0.99222 0.98308 0.97472 0.96667 0.95853 0.94988 0.94052 0.93048 0.91989 0.90881 0.89747 0.88586 0.87414 0.86224 0.85022 0.83806 0.82576 0.81320 0.80026 0.78668 0.77201
0.99705 0.98814 0.98040 0.97331 0.96636 0.95892 0.95064 0.94143 0.93145 0.92082 0.90982 0.89847 0.88696 0.87524 0.86337 0.85131 0.83908 0.82658 0.81360 0.79989 0.78504
15-33
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DIELECTRIC STRENGTH OF INSULATING MATERIALS L. I. Berger The loss of the dielectric properties by a sample of a gaseous, liquid, or solid insulator as a result of application to the sample of an electric field* greater than a certain critical magnitude is called dielectric breakdown. The critical magnitude of electric field at which the breakdown of a material takes place is called the dielectric strength of the material (or breakdown voltage). The dielectric strength of a material depends on the specimen thickness (as a rule, thin films have greater dielectric strength than that of thicker samples of a material), the electrode shape**, the rate of the applied voltage increase, the shape of the voltage vs. time curve, and the medium surrounding the sample, e.g., air or other gas (or a liquid — for solid materials only). Breakdown in Gases The current carriers in gases are free electrons and ions generated by external radiation. The equilibrium concentration of these particles at normal pressure is about 103 cm-3, and hence the electrical conductivity is very small, of the order of 10-16 - 10-15 S/cm. But in a strong electric field, these particles acquire kinetic energy along their free pass, large enough to ionize the gas molecules. The new charged particles ionize more molecules; this avalanche-like process leads to formation between the electrodes of channels of conducting plasma (streamers), and the electrical resistance of the space between the electrodes decreases virtually to zero. Because the dielectric strength (breakdown voltage) of gases strongly depends on the electrode geometry and surface condition and the gas pressure, it is generally accepted to present the data for a particular gas as a fraction of the dielectric strength of either nitrogen or sulfur hexafluoride measured at the same conditions. In Table 1, the data are presented in comparison with the dielectric strength of nitrogen, which is considered equal to 1.00. For convenience to the reader, a few average magnitudes of the dielectric strength of some gases are expressed in kilovolts per millimeter. The data in the table relate to the standard conditions, unless indicated otherwise. Breakdown in Liquids If a liquid is pure, the breakdown mechanism in it is similar to that in gases. If a liquid contains liquid impurities in the form of small drops with greater dielectric constant than that of the main liquid, the breakdown is the result of formation of ellipsoids from these drops by the electric field. In a strong enough electric field, these ellipsoids merge and form a high-conductivity channel between the electrodes. The current increases the temperature in the channel, liquid boils, and the current along the steam canal leads to breakdown. Formation of a conductive channel (bridge) between the electrodes is observed also in liquids with solid impurities. If a liquid contains gas impurities in the form of small bubbles, breakdown is the result of heating of the liquid in strong electric fields. In the locations with the highest current density, the liquid boils, the size of the gas bubbles increases, they merge and form gaseous channels between the electrodes, and the breakdown medium is again the gas plasma. Breakdown in Solids It is known that the current in solid insulators does not obey Ohm’s law in strong electric fields. The current density increases almost exponentially with the electric field, and at a certain field magnitude it jumps to very high magnitudes at which a specimen of a material is destroyed. The two known kinds of electric breakdown are thermal and electrical breakdowns. The former is the result of material heating by the electric current. Destruction of a sample of a material happens when, at a certain voltage, the amount of heat produced by the current exceeds the heat release through the sample surface; the breakdown voltage in this case is proportional to the square root of the ratio of the thermal conductivity and electrical conductivity of the material. The electrical breakdown results from the tunneling of the charge carriers from electrodes or from the valence band or from the impurity levels into the conduction band, or by the impact ionization. The tunnel effect breakdown happens mainly in thin layers, e.g., in thin p-n junctions. Otherwise, the impact ionization mechanism dominates. For this mechanism, the dielectric strength of an insulator can be estimated using Boltzmann’s kinetic equation for electrons in a crystal. In the following tables, the dielectric strength values are for room temperature and normal atmospheric pressure, unless indicated otherwise.
* The unit of electric field in the SI system is newton per coulomb or volt per meter. ** For example, the U.S. standard ASTM D149 is based on use of symmetrical electrodes, while per U.K. standard BS2918 one electrode is a plane and the other is a rod with the axis normal to the plane.
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Table 1 Dielectric Strength of Gases
Material Nitrogen, N2 Hydrogen, H2 Helium, He Oxygen, O2 Air Air (flat electrodes), kV/mm Air, kV/mm Air, kV/mm Neon, Ne Argon, Ar Chlorine, Cl2 Carbon monoxide, CO Carbon dioxide, CO2
Nitrous oxide, N2O Sulfur dioxide, SO2 Sulfur monochloride, S2Cl2 (at 12.5 Torr) Thionyl fluoride, SOF2 Sulfur hexafluoride, SF6 Sulfur hexafluoride, SF6, kV/mm Perchloryl fluoride, ClO3F Tetrachloromethane, CCl4 Tetrafluoromethane, CF4 Methane, CH4 Bromotrifluoromethane, CF3Br Bromomethane, CH3Br Chloromethane, CH3Cl Iodomethane, CH3I Iodomethane, CH3I, at 370 Torr Dichloromethane, CH2Cl2 Dichlorodifluoromethane, CCl2F2 Chlorotrifluoromethane, CClF3
Dielectric* Strength Ref. 1.00 0.50 0.15 0.92 0.97 3.0 0.4-0.7 1.40 0.25 0.16 0.18 1.55 1.02 1.05 0.88 0.82 0.84 1.24 2.63 2.68 1.02
1,2 1 2 6 3 4 5 1 2 2 1 1 2 1 2 6 2 2 6 1
2.50 2.50 2.63 8.50 9.8 2.73 6.33 6.21 1.01 1.00 1.13 1.35 1.97 0.71 1.29 3.02 2.20 1.92 2.42 2.63 1.43 1.53
1 1 2 7 8 1 1 2 1 1 2 1 2 2 2 2 7 2 1 2,6 1 2
Material Trichlorofluoromethane, CCl3F
Dielectric* Strength Ref.
3.50 4.53 Trichloromethane, CHCl3 4.2 4.39 Methylamine, CH3NH2 0.81 Difluoromethane, CH2F2 0.79 Trifluoromethane, CHF3 0.71 3.84 Bromochlorodifluoromethane, CF2ClBr Chlorodifluoromethane, CHClF2 1.40 1.11 Dichlorofluoromethane, CHCl2F 1.33 2.61 Chlorofluoromethane, CH2ClF 1.03 Hexafluoroethane, C2F6 1.82 2.55 Ethyne (Acetylene), C2H2 1.10 1.11 2.3 Chloropentafluoroethane, C2ClF5 3.0 Dichlorotetrafluoroethane, C2Cl2F4 2.52 Chlorotrifluoroethylene, C2ClF3 1.82 1,1,1-Trichloro-2,2,2-trifluoroethane 6.55 1,1,2-Trichloro-1,2,2-trifluoroethane 6.05 Chloroethane, C2H5Cl 1.00 1,1-Dichloroethane 2.66 Trifluoroacetonitrile, CF3CN 3.5 Acetonitrile, CH3CN 2.11 Dimethylamine, (CH3)2NH 1.04 Ethylamine, C2H5NH2 1.01 Ethylene oxide (oxirane), CH3CHO 1.01 Perfluoropropene, C3F6 2.55 Octafluoropropane, C3F8 2.19 2.47 3,3,3-Trifluoro-1-propene, CH2CHCF3 2.11 Pentafluoroisocyanoethane, C2F5NC 4.5 1,1,1,4,4,4-Hexafluoro-2-butyne, CF3CCCF3 5.84 Octafluorocyclobutane, C4F8 3.34 1,1,1,2,3,4,4,4-Octafluoro-2-butene 2.8 Decafluorobutane, C4F10 3.08 5.5 Perfluorobutanenitrile, C3F7CN Perfluoro-2-methyl-1,3-butadiene, C5F8 5.5 Hexafluorobenzene, C6F6 2.11 Perfluorocyclohexane, C6F12, (saturated vapor) 6.18
*Relative to nitrogen, unless units of kV/mm are indicated.
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1 2 1 2 1 2 2 2 1 2 1 2 1 1 2 1 2 1 6 1 2 2 2 1 2 1 2 1 1 1 2 1 2 2 1 2 2 1 1 1 1 2 2
Table 2 Dielectric Strength of Liquids
Material Helium, He, liquid, 4.2 K Static Dynamic Nitrogen, N2, liquid, 77K Coaxial cylinder electrodes Sphere to plane electrodes Water, H2O, distilled Carbon tetrachloride, CCl4 Hexane, C6H14 Two 2.54 cm diameter spherical electrodes, 50.8 µm space Cyclohexane, C6H12 2-Methylpentane, C6H14 2,2-Dimethylbutane, C6H14 2,3-Dimethylbutane, C6H14 Benzene, C6H6 Chlorobenzene, C6H5Cl 2,2,4-Trimethylpentane, C8H18 Phenylxylylethane Heptane, C7H16 2,4-Dimethylpentane, C7H16 Toluene, C6H5CH3
Octane, C8H18
Dielectric strength kV/mm
Ref.
10 10 5 23
9 11 11 12
20 60 65-70 5.5 16.0 42.0
10 10 13 14 15 16
156 42-48 149 133 138 163 7.1 18.8 140 23.6 166 133 199 46 12.0 20.4 16.6
17,18 16 17,18 17,18 17,18 17,18 14 15 17,18 19 17,18 17,18 17,18 16 14 15 14
Material
Ethylbenzene, C8H10 Propylbenzene, C9H12 Isopropylbenzene, C9H12 Decane, C10H22 Synthetic Paraffin Mixture Synfluid 2cSt PAO Butylbenzene, C10H14 Isobutylbenzene, C10H14 Silicone oils—polydimethylsiloxanes, (CH3)3Si-O-[Si(CH3)2]x-O-Si(CH3)3 Polydimethylsiloxane silicone fluid Dimethyl silicone Phenylmethyl silicone Silicone oil, Basilone M50 Mineral insulating oils Polybutene oil for capacitors Transformer dielectric liquid Isopropylbiphenyl capacitor oil Transformer oil Transformer oil Agip ITE 360 Perfluorinated hydrocarbons Fluorinert FC 6001 Fluorinert FC 77 Perfluorinated polyethers Galden XAD (Mol. wt. 800) Galden D40 (Mol. wt. 2000) Castor oil
Dielectric strength kV/mm
Ref.
20.4 179 226 250 238 192
15 17,18 17,18 17,18 17,18 17,18
29.5 275 222
37 17,18 17,18
15.4 24.0 23.2 10-15 11.8 13.8 28-30 23.6 110.7 9-12.6
20 21,22 22 23 6 6 6 6 24 23
8.0 10.7
23 23
10.5 10.2 65
23 23 25
Table 3 Dielectric Strength of Solids
Material Sodium chloride, NaCl, crystalline Potassium bromide, KBr, crystalline Ceramics Alumina (99.9% Al2O3) Aluminum silicate, Al2SiO5 Berillia (99% BeO) Boron nitride, BN Cordierite, Mg2Al4Si5O18 Forsterite, Mg2SiO4 Porcelain Steatite, Mg3Si4O11•H2O Titanates of Mg, Ca, Sr, Ba, and Pb Barium titanate, glass bonded Zirconia, ZrO2 Glasses Fused silica, SiO2 Alkali-silicate glass Standard window glass Micas Muscovite, ruby, natural
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Dielectric strength kV/mm
Ref
150 80
26 26
13.4 5.9 13.8 37.4 7.9 9.8 35-160 9.1-15.4 20-120 >30 11.4
6,27a 6 6,27b 6 6,27c 28 26 6 3 36 29
470-670 200 9.8-13.8
26 26 28
118
6
Material Phlogopite, amber, natural Fluorophlogopite, synthetic Glass-bonded mica Thermoplastic Polymers Polypropylene Amide polymer nylon 6/6, dry Polyamide-imide copolymer Modified polyphenylene oxide Polystyrene Polymethyl methacrylate Polyetherimide Amide polymer nylon 11(dry) Polysulfone Styrene-acrylonitrile copolymer Acrylonitrile-butadiene-styrene Polyethersulfone Polybutylene terephthalate Polystyrene-butadiene copolymer Acetal homopolymer Acetal copolymer
Dielectric strength kV/mm
Ref
118 118 14.0-15.7
6 6 6
23.6 23.6 22.8 21.7 19.7 19.7 18.9 16.7 16.7 16.7 16.7 15.7 15.7 15.7 15.0 15.0
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
Table 3 Dielectric Strength of Solids (continued)
Material
Dielectric strength kV/mm
Polyphenylene sulfide 15.0 Polycarbonate 15.0 Acetal homopolymer resin (molding resin) 15.0 Acetal copolymer resin 15.0 Thermosetting Molding Compounds Glass-filled allyl 15.7 (Type GDI-30 per MIL-M-14G) Glass-filled epoxy, electrical grade 15.4 Glass-filled phenolic 15.0 (Type GPI-100 per MIL-M-14G) Glass-filled alkyd/polyester 14.8 (Type MAI-60 per MIL-M-14G) Glass-filled melamine 13.4 (Type MMI-30 per MIL-M-14G) Extrusion Compounds for High-Temperature Insulation Polytetrafluoroethylene 19.7 Perfluoroalkoxy polymer 21.7 Fluorinated ethylene-propylene copolymer 19.7 Ethylene-tetrafluoroethylene copolymer 15.7 Polyvinylidene fluoride 10.2 Ethylene-chlorotrifluoroethylene 19.3 copolymer Polychlorotrifluoroethylene 19.7 Extrusion Compounds for Low-Temperature Insulation Polyvinyl chloride Flexible 11.8-15.7 Rigid 13.8-19.7 Polyethylene 18.9 Polyethylene, low-density 21.7 300 Polyethylene, high-density 19.7 Polypropylene/polyethylene copolymer 23.6 Embedding Compounds Basic epoxy resin: 19.7 bisphenol-A/epichlorohydrin polycondensate Cycloaliphatic epoxy: alicyclic 19.7 diepoxy carboxylate Polyetherketone 18.9 Polyurethanes Two-component, polyol-cured 25.4 Two-part solventless, 24.0 polybutylene-based Silicones Clear two-part heat curing eletrical 21.7 grade silicone embedding resin Red insulating enamel (MIL-E-22118) Dry 47.2 Wet 11.8 Enamels Red enamel, fast cure Standard conditions 78.7 Immersion conditions 47.2 Black enamel Standard conditions 70.9 Immersion conditions 47.2
Ref. 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6
30 30 28 6 31 6 6 6
6 30 6 6
6
6 6
6 6 6 6
Material
Dielectric strength kV/mm
Varnishes Vacuum-pressure impregnated baking type solventless polyester varnish Rigid, two-part 70.9 Semiflexible high-bond thixotropic 78.7 Rigid high-bond high-flash 68.9 freon-resistant Baking type epoxy varnish Solventless, rigid, low viscosity, 90.6 one-part Solventless, semiflexible, one-part 82.7 Solventless, semirigid, chemical 106.3 resistant, low dielectric constant Solvable, for hermetic electric motors 181.1 Polyurethane coating Clear conformal, fast cure Standard conditions 78.7 Immersion conditions 47.2 Insulating Films and Tapes Low-density polyethylene film 300 (40 µm thick) Poly-p-xylylene film 410-590 Aromatic polymer films Kapton H (Du Pont) 389-430 Ultem (GE Plastic and Roem AG) 437-565 Hostaphan (Hoechst AG) 338-447 Amorphous Stabar K2000 404-422 (ICI film) Stabar S100 (ICI film) 353-452 Polyetherimide film (26 µm) 486 Parylene N/D (poly-p-xylylene/polydichloro-p-xylylene) 25 µm film 275 Cellulose acetate film 157 Cellulose triacetate film 157 Polytetrafluoroethylene film 87-173 Perfluoroalkoxy film 157-197 Fluorinated ethylene-propylene 197 copolymer film Ethylene-tetrafluoroethylene film 197 Ethylene-chlorotrifluoroethylene 197 copolymer film Polychlorotrifluoroethylene film 118-153.5 High-voltage rubber insulating tape 28 Composites Isophthalic polyester (vinyl toluene monomer) filled with Calcium carbonate, CaCO3 15.0 Gypsum, CaSO4 14.4 Alumina trihydrate 15.4 Clay 14.4 BPA fumarate polyester (vinyl toluene monomer) filled with Calcium carbonate 6.1 Gypsum 5.9 Alumina trihydrate 11.8 Clay 12.6
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Ref.
6 6 6
6 6 6 6
6 6 31 32 33 33 33 33 33 34 6 6 6 6 6 6 6 6 6 6
38 38 38 38
38 38 38 38
Table 3 Dielectric Strength of Solids (continued)
Material Polysulfone resin—30% glass fiber Polyamid resin (Nylon 66)— 30% carbon fiber Polyimide thermoset resin, glass reinforced Polyester resin (thermoplastic)— 40% glass fiber Epoxy resin (diglycidyl ether of bisphenol A), glass reinforced Various Insulators Rubber, natural
Dielectric strength kV/mm
Ref.
16.5-18.7
38
13.0
38
12.0
39
20.0
38
16.0
40
100-215
26
Material Butyl rubber Neoprene Silicone rubber Room-temperature vulcanized silicone rubber Ureas (from carbamide to tetraphenylurea) Dielectric papers Aramid paper, calendered Aramid paper, uncalendered Aramid with Mica
Dielectric strength kV/mm
Ref.
23.6 15.7-27.6 26-36 9.2-10.9
6 6 6 35
11.8-15.7
28
28.7 12.2 39.4
6 6 6
REFERENCES 1. Vijh, A. K. IEEE Trans., EI-12, 313, 1997. 2. Brand, K. P., IEEE Trans., EI-17, 451, 1982. 3. Encyclopedic Dictionary in Physics, Vedensky, B. A. and Vul, B. M., Eds., Vol. 4, Soviet Encyclopedia Publishing House, Moscow, 1965. 4. Kubuki, M., Yoshimoto, R., Yoshizumi, K., Tsuru, S., and Hara, M., IEEE Trans., DEI-4, 92, 1997. 5. Al-Arainy, A. A. Malik, N. H., and Cureshi, M. I., IEEE Trans., DEI-1, 305, 1994. 6. Shugg, W. T., Handbook of Electrical and Electronic Insulating Materials, Van Nostrand Reinhold, New York, 1986. 7. Devins, J. C., IEEE Trans., EI-15, 81, 1980. 8. Xu, X., Jayaram, S., and Boggs, S. A., IEEE Trans., DEI-3, 836, 1996. 9. Okubo, H., Wakita, M., Chigusa, S., Nayakawa, N., and Hikita, M., IEEE Trans., DEI-4, 120, 1997. 10. Hayakawa, H., Sakakibara, H., Goshima, H., Hikita, M., and Okubo, H., IEEE Trans., DEI-4, 127, 1997. 11. Okubo, H., Wakita, M., Chigusa, S., Hayakawa, N., and Hikita, M., IEEE Trans., DEI-4, 220, 1997. 12. Von Hippel, A. R., Dielectric Materials and Applications, MIT Press, Cambridge, MA, 1954. 13. Jones, H. M. and Kunhards, E. E., IEEE Trans., DEI-1, 1016, 1994. 14. Nitta, Y. and Ayhara, Y., IEEE Trans., EI-11, 91, 1976. 15. Gallagher, T. J., IEEE Trans., EI-12, 249, 1977. 16. Wong, P. P. and Forster, E. O., in Dielectric Materials. Measurements and Applications, IEE Conf. Publ. 177, 1, 1979. 17. Kao, K. C. IEEE Trans., EI-11, 121, 1976. 18. Sharbaugh, A. H., Crowe, R. W., and Cox, E. B., J. Appl. Phys., 27, 806, 1956. 19. Miller, R. L., Mandelcorn, L., and Mercier, G. E., in Proc. Intl. Conf. on Properties and Applications of Dielectric Materials, Xian, China, June 24-28, 1985; cited in Ref. 6, p. 492. 20. Hakim, R. M., Oliver, R. G., and St-Onge, H., IEEE Trans., EI-12, 360, 1977. 21. Hosticka, C., IEEE Trans., 389, 1977. 22. Yasufuku, S., Umemura, T., and Ishioka, Y., IEEE Trans., EI-12, 402, 1977. 23. Forster, E. O., Yamashita, H., Mazzetti, C., Pompini, M., Caroli, L., and Patrissi, S., IEEE Trans., DEI-1, 440, 1994. 24. Bell, W. R., IEEE Trans., 281, 1977. 25. Ramu, T. C. and Narayana Rao, Y., in Dielectric Materials. Measurements and Applications, IEE Conf. Publ. 177, 37. 26. Skanavi, G. I., Fizika Dielektrikov; Oblast Silnykh Polei (Physics of Dielectrics; Strong Fields). Gos. Izd. Fiz. Mat. Nauk (State Publ. House for Phys. and Math. Scis.), Moscow, 1958. 27. Kleiner, R. N., in Practical Handbook of Materials Science, Lynch, C. T., Ed., CRC Press, 1989; 27a: p. 304; 27b: p.300; 27c: p. 316. 28. Materials Selector Guide. Materials and Methods, Reinhold Publ., New York, 1973. 29. Flinn, R. A. and Trojan, P. K., Engineering Materials and Their Applications, 2nd ed., Houghton Mifflin, 1981, p. 614. 30. Lynch, C. T., Ed., Practical Handbook of Materials Science, CRC Press, Boca Raton, FL, 1989. 31. Suzuki, H., Mukai, S., Ohki, Y., Nakamichi, Y., and Ajiki, K., IEEE Trans., DEI-4, 238, 1997. 32. Mori, T., Matsuoka, T., and Muzitani, T., IEEE Trans., DEI-1, 71, 1994. 33. Bjellheim, P. and Helgee, B., IEEE Trans., DEI-1, 89, 1994. 34. Zheng, J. P., Cygan, P. J., and Jow, T. R., IEEE Trans., DEI-3, 144, 1996. 35. Danukas, M. G., IEEE Trans., DEI-1, 1196, 1994. 36. Burn, I. and Smithe, D. H., J. Mater. Sci., 7, 339, 1972. 37. Hope, K.D., Chevron Chemical, Private Communication.
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38. Engineering Materials Handbook, vol. 1, Composites, C.A. Dostal, Ed., ASM Intl., 1987. 39. 1985 Materials Selector, Mater. Eng., (12) 1984. 40. Modern Plastics Encyclopedia, McGraw-Hill, v. 62 (No. 10A) 1985–1986. Review Literature on the Subject R1.Kuffel, E. and Zaengl, W. S., HV Engineering Fundamentals, Pergamon, 1989. R2.Kok, J. A., Electrical Breakdown of Insulating Liquids, Phillips Tech. Library, Cleaver-Hum, Longon, 1961. R3.Gallagher, T. J., Simple Dielectric Liquids, Clarendon, Oxford, 1975. R4.Meek, J. M. and Craggs, J. D., Eds., Electric Breakdown in Gases, John Wiley & Sons, 1976. R5.Von Hippel, A. R., Dielectric Materials and Applications, MIT Press, Cambridge, MA, 1954.
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COEFFICIENT OF FRICTION The coefficient of friction between two surfaces is the ratio of the force required to move one over the other to the force pressing the two together. Thus if F is the minimum force needed to move one surface over the other, and W is the force pressing the surfaces together, the coefficient of friction µ is given by µ = F/W. A greater force is generally needed to initiate movement from rest that to continue the motion once sliding has started. Thus the static coefficient of friction µ(static) is usually larger that the sliding or kinetic coefficient µ(sliding). This table gives characteristic values of both the static and sliding coefficients of friction for a number of material combinations. In each case Material 1 is moving over the surface of Material 2. The type of lubrication or any other special condition is indicated in the third column. All values refer to room temperature unless otherwise indicated. It should be emphasized that the coefficient of friction is very sensitive to the condition of the surface, so that these values represent only a rough guide.
REFERENCES 1. Minshall, H., in CRC Handbook of Chemistry and Physics, 73rd Edition, Lide, D. R., ed., CRC Press, Boca Raton, FL, 1992. 2. Fuller, D. D., in American Institute of Physics Handbook, 3rd Edition, Gray, D. E., ed., McGraw-Hill, New York, 1972. Material 1
Material 2
Conditions
µ(static)
µ(sliding)
0.78 0.15 0.005 0.11 0.23
0.42 0.081 0.029 0.084
Metals Hard steel
Hard steel
Hard steel Mild steel
Graphite Mild steel
Mild steel Mild steel Mild steel
Phosphor bronze Cast iron Lead
Mild steel Cast iron Aluminum Aluminum Brass
Brass Cast iron Aluminum Mild steel Mild steel
Brass Bronze Cadmium Copper Copper
Cast iron Cast iron Mild steel Copper Mild steel
Copper Copper Lead Magnesium Magnesium Magnesium Nickel Nickel Tin Zinc
Cast iron Glass Cast iron Magnesium Mild steel Cast iron Nickel Mild steel Cast iron Cast iron
Dry Castor oil Steric acid Lard Light mineral oil Graphite Dry Dry Oleic acid Dry Dry Dry Mineral oil Dry Dry Dry Dry Dry Castor oil Dry Dry Dry Dry Dry Oleic acid Dry Dry Dry Dry Dry Dry Dry Dry Dry Dry
Diamond Diamond
Diamond Metals
Dry Dry
0.058 0.21 0.74
0.95 0.5 0.35 1.10 1.05 0.61 0.51 0.11
0.15 1.4 0.47 0.44 0.30 0.22 0.46
1.6 0.53 1.05 0.68
0.36 0.18 0.29 0.53 0.43
0.6
1.10
0.85
Nonmetals 0.1 0.12
15-50
0.57 0.09 0.34 0.23 0.95 0.3
0.42 0.25 0.53 0.64 0.32 0.21
COEFFICIENT OF FRICTION (continued) Material 1
Material 2
Conditions
Garnet Glass Glass Graphite Mica Nylon Nylon Polyethylene Polyethylene Polystyrene Polystyrene Sapphire Teflon Teflon Tungsten carbide
Mild steel Glass Nickel Graphite Mica Nylon Steel Polyethylene Steel Polystyrene Steel Sapphire Teflon Steel Tungsten carbide
Tungsten carbide Tungsten carbide
Graphite Steel
Dry Dry Dry Dry Freshly cleaved Dry Dry Dry Dry Dry Dry Dry Dry Dry Dry, room temp. Dry, 1000°C Dry, 1600°C Oleic acid Dry Dry Oleic acid
µ(static) 0.94 0.78 0.1 1.0 0.2 0.40 0.2 0.2 0.5 0.3 0.2 0.04 0.04 0.17 0.45 1.8 0.12 0.15 0.5 0.08
µ(sliding) 0.39 0.4 0.56
0.04 0.04
Miscellaneous materials Cotton Leather Leather Oak
Cotton Cast iron Oak Oak
Silk Wood
Silk Wood
Wood Wood
Brick Leather
Ice
Ice
Aluminum
Snow
Brass
Ice
Nylon
Snow
Teflon
Snow
Wax, ski
Snow
Threads Dry Parallel to grain Parallel to grain Perpendicular to grain Clean Dry Wet Dry Dry
0.3 0.6 0.61 0.62 0.54 0.25 0.35 0.2 0.6 0.35
0.56 0.52 0.48 0.32
Various materials on ice and snow Clean, 0°C Clean, -12°C Clean, -80°C Wet, 0°C Dry, 0°C Clean, 0°C Clean, -80°C Wet, 0°C Dry, -10°C Wet, 0°C Dry, 0°C Wet, 0°C Dry, 0°C Dry, -10°C
15-51
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0.1 0.3 0.5 0.4 0.35
0.02 0.035 0.09
0.02 0.15 0.4 0.3 0.05 0.02 0.1 0.04 0.2
FLAME TEMPERATURES This table gives the adiabatic flame temperature for stoichemetric mixtures of various fuels and oxidizers. The temperatures are calculated from thermodynamic and transport properties under ideal adiabatic conditions, using methods described in the reference.
REFERENCE Fristrom, R. M., Flame Structures and Processes, Oxford University Press, New York, 1995.
Adiabatic Flame Temperature in K for Various Fuel-Oxidizer Combinations Oxidizer Fuel
Air
Organic liquids and gases Acetaldehyde 2288 Acetone 2253 Acetylene 2607 Benzene 2363 Butane 2248 Carbon disulfide 2257 Cyanogen 2596 Cyclohexane 2250 Cyclopropane 2370 Decane 2286 Ethane 2244 Ethanol 2238 Ethylene 2375 Hexane 2238 Methane 2236 Methanol 2222 Oxirane 2177 Pentane 2250 Propane 2250 Toluene 2344 Solids Aluminum Lithium Phosphorus (white) Zirconium Other Ammonia Carbon monoxide 1388 Diborane Hydrazine Hydrogen 2169 Hydrogen sulfide 2091 Phosphine Silane
O2
F2
Cl2
N2 O
NO
4006
2493
2965
3127
4855
4005 2711 3242 4278 2845 3350 3037 3000 3414 3139 3043
15-40
ALLOCATION OF FREQUENCIES IN THE RADIO SPECTRUM In the United States the National Telecommunications and Information Administration (NTIA) has responsibility for assigning each portion of the radio spectrum (9 kHz to 300 GHz) for different uses. These assignments must be compatible with the rules of the International Telecommunications Union (ITU), to which the United States is bound by treaty. The current assignments are given in a wall chart (Reference 1) and may also be found on the NTIA web site (Reference 2). The list below summarizes the broad features of the spectrum allocation, with particular attention to those sections of scientific interest. The references should be consulted for details of the allocations in the frequency bands listed here, which in some cases are quite complex. REFERENCES 1. United States Frequency Allocations, 1996 Spectrum Wall Chart, Stock No. 003-000-00652-2, U. S. Government Printing Office, P. O. Box 371954, Pittsburgh, PA 15250-7954. 2. http://www.ntia.doc.gov/osmhome/allochrt.html Frequency range 9 - 19.95 kHz 19.95 - 20.05 kHz 20.05 - 535 kHz 535 - 1605 kHz 1605 - 3500 kHz 3.5 - 4.0 MHz 4.0 - 5.95 MHz 5.95 - 13.36 MHz 13.36 - 13.41 MHz 13.41 - 25.55 MHz 25.55 - 25.67 MHz 25.67 - 37.5 MHz 37.5 -38.25 MHz 38.25 - 50.0 MHz 50.0 - 54.0 MHz 54.0 - 72.0 MHz 72.0 - 73.0 MHz 73.0 - 74.6 MHz 74.6 - 76.0 MHz 76.0 - 88.0 MHz 88.0 - 108.0 MHz 108.0 - 118.0 MHz 118.0 - 174.0 MHz 174.0 - 216.0 MHz 216.0 - 400.05 MHz 400.05 - 400.15 MHz 400.15 - 406.1 MHz 406.1 - 410.0 MHz 410.0 - 470.0 MHz 470.0 - 512.0 MHz 512.0 - 608.0 MHz 608.0 - 614.0 MHz 614.0 - 806.0 MHz 806 -1400 MHz 1400 - 1427 MHz 1427 - 1660 MHz 1660 - 1710 MHz 1710 - 2655 MHz 2655 - 2700 MHz 2.7 - 4.99 GHz 4.99 - 5.0 GHz 5.0 - 10.6 GHz 10.6 - 10.7 GHz 10.7 - 15.35 GHz 15.35 - 15.4 GHz 15.4 - 22.21 GHz
Allocation Maritime communication, navigation Standard frequency and time signal (also at 60 kHz and 2.5, 5, 10, 15, 20, 25 MHz) Maritime and aeronautical communication, navigation AM radio broadcasting Mobile communication and navigation, amateur radio (1800-1900 kHz) Amateur radio Mobile communication Mobile communication, amateur, short-wave broadcasting Radioastronomy Mobile communication, amateur, short-wave broadcasting Radioastronomy Mobile communication, amateur, short-wave broadcasting Radioastronomy Mobile communication Amateur TV channels 2-4 Mobile communication Radioastronomy Mobile communication TV channels 5-6 FM radio broadcasting Aeronautical navigation Mobile communication, space research, meteorological satellites TV channels 7-13 Mobile communication Standard frequency and time satellite (also 20 and 25 GHz) Meteorological aids (radiosonde) Radioastronomy Mobile communication, amateur TV channels 14-20 TV channels 21-36 Radioastronomy TV channels 38-69 Mobile communication, navigation Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research, meteorology Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications
15-36
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ALLOCATION OF FREQUENCIES IN THE RADIO SPECTRUM (continued) Frequency range
Allocation
22.21 - 22.5 GHz 22.25 - 23.6 GHz 23.6 - 24.0 GHz 24.0 - 31.3 GHz 31.3 - 31.8 GHz 31.8 - 42.5 GHz 42.5 - 43.5 GHz 43.5 - 51.4 GHz 51.4 - 54.25 GHz 54.25 - 58.2 GHz 58.2 - 59.0 GHz 59.0 - 64.0 GHz 64.0 - 65.0 GHz 65.0 - 72.77 GHz 72.77 - 72.91 GHz 72.91 - 86.0 GHz 86.0 - 92.0 GHz 92.0 - 105.0 GHz 105.0 - 116.0 GHz 116.0 - 164.0 GHz 164.0 - 168.0 GHz 168.0 - 182.0 GHz 182.0 - 185.0 GHz 185.0 - 217.0 GHz 217.0 - 231.0 GHz 231.0 - 265.0 GHz 265.0 - 275.0 GHz 275.0 - 300.0 GHz
Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy Various navigation and satellite applications Radioastronomy, space research Space research Radioastronomy, space research Satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy, space research Various navigation and satellite applications Radioastronomy Mobile communications
15-37
Section 16: Health and Safety Information Handling and Disposal of Chemicals in Laboratories Flammability of Chemical Substances Threshold Limits for Airborne Contaminants Octanol-Water Partition Coefficients Protection Against Ionizing Radiation Annual Limits on Intakes of Radionuclides Chemical Carcinogens
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FLAMMABILITY OF CHEMICAL SUBSTANCES This table gives properties related to the flammability of about 900 chemical substances. The properties listed are: tB : Normal boiling point in °C ( at 101.325 kPa pressure). FP: Flash point, which is the minimum temperature at which the vapor pressure of a liquid is sufficient to form an ignitable mixture with air near the surface of the liquid. Flash point is not an intrinsic physical property but depends on the conditions of measurement (see Reference 1). Fl. Limits: Flammable limits (often called explosive limits), which specify the range of concentration of the vapor in air (in percent by volume) for which a flame can propagate. Below the lower flammable limit, the gas mixture is too lean to burn; above the upper flammable limit, the mixture is too rich. Values refer to ambient temperature and pressure and are dependent on the precise test conditions. A ? indicates that one of the limits is not known. IT: Ignition temperature (sometimes called autoignition temperature), which is the minimum temperature required for self-sustained combustion in the absence of an external ignition source. As in the case of flash point, the value depends on specified test conditions. Even in cases where very careful measurements of flash point have been replicated in several laboratories, observed values can differ by 3 to 6°C (Reference 4). For more typical measurements, larger uncertainties should be assumed in both flash points and autoignition temperatures. The absence of a flash point entry in this table does not mean that the substance is nonflammable, but only that no reliable value is available. Compounds are listed by molecular formula following the Hill convention. Substances not containing carbon are listed first, followed by those that contain carbon. To locate an organic compound by name or CAS Registry Number when the molecular formula is not known, use the table “Physical Constants of Organic Compounds” in Section 3 and its indexes to determine the molecular formula. REFERENCES 1. Fire Protection Guide to Hazardous Materials, 11th Edition, National Fire Protection Association, Quincy, MA, 1994. 2. Urben, P.G., Editor, Bretherick’s Handbook of Reactive Chemical Hazards, 5th Edition, Butterworth-Heinemann, Oxford, 1995. 3. Daubert, T.E., Danner, R.P., Sibul, H.M., and Stebbins, C.C., Physical and Thermodynamic Properties of Pure Compounds: Data Compilation, extant 1994 (core with 4 supplements), Taylor & Francis, Bristol, PA. 4. Report of Investigation: Flash Point Reference Materials, National Institute of Standards and Technology, Standard Reference Materials Program, Gaithersburg, MD, 1995. Mol. Form.
tB/°C
Name
FP/°C
Fl. Limits
IT/°C
-90 30 <0
1-98% 0.4-?
≈40 35 ≈20 ≈20 36 104 ≈20 ≈50
Compounds not containing carbon B2H6 B5H9 BrH3Si Br3HSi Cl2H2Si Cl3HSi GeH4 Ge2H6 H2 H2S H2S2 H2Te H3N H3P H4N2 H4P2 H4Si H6Si2 H8Si3 P
Diborane Pentaborane(9) Bromosilane Tribromosilane Dichlorosilane Trichlorosilane Germane Digermane Hydrogen Hydrogen sulfide Hydrogen disulfide Hydrogen telluride Ammonia Phosphine Hydrazine Diphosphine Silane Disilane Trisilane Phosphorus (white)
-92.4 60 1.9 109 8.3 33 -88.1 29 -252.8 -59.55 70.7 -2 -33.33 -87.75 113.55 63.5 -111.9 -14.3 52.9 280.5
4.1-99% -50
4-74% 4-44%
260
<22 -50
38 -112 -14 <0
16-25% 1.8-? 5-100% 1.4-?
≈20 ≈20 ≈20 ≈20 38
Compounds containing carbon CHN CH2Cl2 CH2N2
Hydrogen cyanide Dichloromethane Cyanamide
26 40
-18 141
16-16
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6-40% 13-23%
538 556
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form CH2O (CH2O)x CH2O2 CH3Br CH3Cl CH3Cl3Si CH3NO CH3NO2 CH4 CH4Cl2Si CH4O CH4S CH5N CH6N2 CO COS CS2 C2ClF3 C2F4 C2HCl3 C2HCl3O C2H2 C2H2Cl2 C2H2Cl2 C2H2Cl2 C2H2F2 C2H3Br C2H3Cl C2H3ClF2 C2H3ClO C2H3Cl2NO2 C2H3Cl3 C2H3Cl3 C2H3Cl3Si C2H3F C2H3N C2H3NO C2H4 C2H4ClNO2 C2H4Cl2 C2H4Cl2 C2H4O C2H4O C2H4O2 C2H4O2 C2H4O3 C2H5Br C2H5Cl C2H5ClO C2H5Cl3Si C2H5N C2H5NO2 C2H5NO2 C2H5NO3 C2H6 C2H6Cl2Si C2H6O C2H6O C2H6OS C2H6OS C2H6O2
tB/°C
Name Formaldehyde Paraformaldehyde Formic acid Bromomethane Chloromethane Methyltrichlorosilane Formamide Nitromethane Methane Dichloromethylsilane Methanol Methanethiol Methylamine Methylhydrazine Carbon monoxide Carbon oxysulfide Carbon disulfide Chlorotrifluoroethylene Tetrafluoroethylene Trichloroethylene Dichloroacetyl chloride Acetylene 1,1-Dichloroethylene cis-1,2-Dichloroethylene trans-1,2-Dichloroethylene 1,1-Difluoroethylene Bromoethylene Chloroethylene 1-Chloro-1,1-difluoroethane Acetyl chloride 1,1-Dichloro-1-nitroethane 1,1,1-Trichloroethane 1,1,2-Trichloroethane Trichlorovinylsilane Fluoroethylene Acetonitrile Methyl isocyanate Ethylene 1-Chloro-1-nitroethane 1,1-Dichloroethane 1,2-Dichloroethane Acetaldehyde Ethylene oxide Acetic acid Methyl formate Ethaneperoxoic acid Bromoethane Chloroethane Ethylene chlorohydrin Trichloroethylsilane Ethyleneimine Nitroethane Ethyl nitrite Ethyl nitrate Ethane Dichlorodimethylsilane Ethanol Dimethyl ether 2-Mercaptoethanol Dimethyl sulfoxide Ethylene glycol
-19.1 101 3.5 -24.0 65.6 220 101.1 -161.5 41 64.6 5.9 -6.3 87.5 -191.5 -50 46 -27.8 -75.9 87.2 108 -84.7 31.6 60.1 48.7 -85.7 15.8 -13.3 -9.7 50.7 123.5 74.0 113.8 91.5 -72 81.6 39.5 -103.7 124.5 57.4 83.5 20.1 10.6 117.9 31.7 110 38.5 12.3 128.6 100.5 56 114.0 18 87.2 -88.6 70.3 78.2 -24.8 158 189 197.3
16-17
FP/°C 85 70 50
-9 154 35 -9 11 -18 0 -8
-30
Fl. Limits
IT/°C
7.0-73% 7.0-73% 18-57% 10-16% 8.1-17.4% 7.6->20%
424 300 434 537 632 >404
7.3-? 5.0-15.0% 6.0-55% 6.0-36% 3.9-21.8% 4.9-20.7% 2.5-92% 12.5-74% 12-29% 1.3-50.0% 8.4-16.0% 10.0-50.0% 8-10.5%
418 537 316 464 430 194 609 90 200 420
66 2.5-100% 6.5-15.5% 3-15% 6-13% 5.5-21.3% 9-15% 3.6-33.0% 6-18%
305 570 460 460
8-10.5% 6-28%
500 460
6 -7
2.6-21.7% 3.0-16.0% 5.3-26% 2.7-36%
524 534 450
56 -17 13 -39 -20 39 -19 41
5.4-11.4% 6.2-16% 4.0-60% 3.0-100% 4.0-19.9% 4.5-23%
458 413 175 429 463 449
6.8-8.0% 3.8-15.4% 4.9-15.9%
511 519 425
3.3-54.8% 3.4-17% 4.0-50% 4-? 3.0-12.5% 3.4-9.5% 3.3-19% 3.4-27.0%
320 414 90
363 350
2.6-42% 3.2-22%
215 398
-28 6 2
-78 4 76 32 21
-50 60 22 -11 28 -35 10 <21 13 -41 74 95 111
530 472 632 390
472
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C2H6O4S C2H6S C2H6S C2H6S2 C2H7N C2H7N C2H7NO C2H8N2 C2H8N2 C2N2 C3H3Br C3H3N C3H4 C3H4ClN C3H4Cl2 C3H4O C3H4O C3H4O2 C3H4O2 C3H4O3 C3H5Br C3H5Cl C3H5Cl C3H5ClO C3H5ClO C3H5ClO2 C3H5ClO2 C3H5ClO2 C3H5Cl2NO2 C3H5Cl3 C3H5Cl3Si C3H5N C3H5NO C3H5N3O9 C3H6 C3H6 C3H6ClNO2 C3H6ClNO2 C3H6Cl2 C3H6Cl2O C3H6N2 C3H6O C3H6O C3H6O C3H6O C3H6O C3H6O2 C3H6O2 C3H6O2 C3H6O2 C3H6O3 C3H6O3 C3H7Br C3H7Cl C3H7Cl C3H7ClO C3H7ClO C3H7Cl3Si C3H7N C3H7NO C3H7NO2
tB/°C
Name Dimethyl sulfate Ethanethiol Dimethyl sulfide Dimethyl disulfide Ethylamine Dimethylamine Ethanolamine 1,2-Ethanediamine 1,1-Dimethylhydrazine Cyanogen 3-Bromo-1-propyne 2-Propenenitrile Propyne 3-Chloropropanenitrile 2,3-Dichloropropene Propargyl alcohol Acrolein Propenoic acid 2-Oxetanone Ethylene carbonate 3-Bromopropene 2-Chloropropene 3-Chloropropene Epichlorohydrin Propanoyl chloride 2-Chloropropanoic acid Ethyl chloroformate Methyl chloroacetate 1,1-Dichloro-1-nitropropane 1,2,3-Trichloropropane Trichloro-2-propenylsilane Propanenitrile 3-Hydroxypropanenitrile Trinitroglycerol Propene Cyclopropane 1-Chloro-1-nitropropane 2-Chloro-2-nitropropane 1,2-Dichloropropane 1,3-Dichloro-2-propanol Dimethylcyanamide Allyl alcohol Methyl vinyl ether Propanal Acetone Methyloxirane Propanoic acid Ethyl formate Methyl acetate 1,3-Dioxolane Dimethyl carbonate 1,3,5-Trioxane 1-Bromopropane 1-Chloropropane 2-Chloropropane 2-Chloro-1-propanol 1-Chloro-2-propanol Trichloropropylsilane Allylamine N,N-Dimethylformamide 1-Nitropropane
35.1 37.3 109.8 16.5 6.8 171 117 63.9 -21.1 89 77.3 -23.2 175.5 94 113.6 52.6 141 162 248 70.1 22.6 45.1 118 80 185 95 129.5 145 157 117.5 97.1 221 -47.6 -32.8 142 96.4 176 163.5 97.0 5.5 48 56.0 35 141.1 54.4 56.8 78 90.5 114.5 71.1 46.5 35.7 133.5 127 123.5 53.3 153 131.1
16-18
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FP/°C 83 -17 -37 24 -16 20 86 40 -15 10 0 76 15 36 -26 50 74 143 -1 -37 -32 31 12 107 16 57 66 71 35 2 129
62 57 21 74 71 21 -30 -20 -37 52 -20 -10 2 19 45 <-18 -32 52 52 37 -29 58 36
Fl. Limits 2.8-18.0% 2.2-19.7% 3.5-14% 2.8-14.4% 3.0-23.5% 2.5-12.0% 2-95% 6.6-32% 3.0-? 3.0-17.0% 2.1-12.5%
IT/°C 188 300 206 385 400 410 385 249 324 481
2.6-7.8% 2.8-31% 2.4-8.0% 2.9-?
220 438
4.4-7.3% 4.5-16% 2.9-11.1% 3.8-21.0%
295 485 411 500 500
7.5-18.5% 3.2-12.6% 3.1-14%
512
2.0-11.1% 2.4-10.4%
270 455 498
3.4-14.5%
557
2.5-18.0%
378 287 207 465 449 465 455 454
2.6-17% 2.5-12.8% 3.1-27.5% 2.9-12.1% 2.8-16.0% 3.1-16%
3.6-29% 2.6-11.1% 2.8-10.7%
414 490 520 593
2.2-22% 2.2-15.2% 2.2-?
374 445 421
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C3H7NO2 C3H7NO3 C3H8 C3H8O C3H8O C3H8O C3H8O2 C3H8O2 C3H8O2 C3H8O2 C3H8O3 C3H9BO3 C3H9ClSi C3H9N C3H9N C3H9N C3H9NO C3H9NO C3H9NO C3H9O3P C3H9O4P C3H10N2 C4Cl6 C4H2O3 C4H4 C4H4N2 C4H4O C4H4O2 C4H4S C4H5Cl C4H5N C4H5N C4H5N C4H6 C4H6 C4H6O C4H6O C4H6O C4H6O C4H6O C4H6O2 C4H6O2 C4H6O2 C4H6O2 C4H6O2 C4H6O3 C4H6O3 C4H6O6 C4H7Br C4H7BrO2 C4H7Cl C4H7Cl C4H7ClO C4H7ClO2 C4H7N C4H7N C4H7NO C4H7NO C4H8 C4H8 C4H8
tB/°C
Name 2-Nitropropane Propyl nitrate Propane 1-Propanol 2-Propanol Ethyl methyl ether 1,2-Propylene glycol 1,3-Propylene glycol Ethylene glycol monomethyl ether Dimethoxymethane Glycerol Trimethyl borate Trimethylchlorosilane Propylamine Isopropylamine Trimethylamine 3-Amino-1-propanol 1-Amino-2-propanol N-Methyl-2-ethanolamine Trimethyl phosphite Trimethyl phosphate 1,3-Propanediamine Hexachloro-1,3-butadiene Maleic anhydride 1-Buten-3-yne Succinonitrile Furan Diketene Thiophene 2-Chloro-1,3-butadiene 2-Butenenitrile Methylacrylonitrile Pyrrole 1,3-Butadiene 2-Butyne Divinyl ether Ethoxyacetylene trans-2-Butenal 3-Buten-2-one Vinyloxirane Methacrylic acid Vinyl acetate Methyl acrylate 2,3-Butanedione gamma-Butyrolactone Acetic anhydride Propylene carbonate L-Tartaric acid 1-Bromo-2-butene Ethyl bromoacetate 2-Chloro-1-butene 3-Chloro-2-methylpropene 2-Chloroethyl vinyl ether Ethyl chloroacetate Butanenitrile 2-Methylpropanenitrile Acetone cyanohydrin 2-Pyrrolidone 1-Butene cis-2-Butene trans-2-Butene
120.2 110 -42.1 97.2 82.3 7.4 187.6 214.4 124.1 42 290 67.5 60 47.2 31.7 2.8 187.5 159.4 158 111.5 197.2 139.8 215 202 5.1 266 31.5 126.1 84.0 59.4 120.5 90.3 129.7 -4.4 26.9 28.3 50 102.2 81.4 68 162.5 72.5 80.7 88 204 139.5 242 104.5 168.5 58.5 71.5 108 144.3 117.6 103.9 251 -6.2 3.7 0.8
16-19
FP/°C
Fl. Limits
24 20 -104 23 12 -37 99
2.6-11.0% 2-100% 2.1-9.5% 2.2-13.7% 2.0-12.7% 2.0-10.1% 2.6-12.5%
39 -32 199 -8 -28 -37 -37 -5 80 77 74 54 107 24
1.8-14% 2.2-13.8% 3-19%
102 132 -36 34 -1 -20 16 1 39 -31 <-30 <-7 13 -7 <-50 77 -8 -3 27 98 49 135 210
2.0-10.4% 2.0-11.6%
IT/°C 428 175 450 412 399 190 371 400 285 237 370 395 318 402 190 374
1.4-7.1% 21-100%
610 477
2.3-14.3%
4.0-20.0% 2-6.8% 2.0-12.0% 1.4-? 1.7-27%
420 360
2.1-15.5% 2.1-15.6%
232 491
1.6-8.8% 2.6-13.4% 2.8-25%
68 402 468
2.7-10.3%
316 425
4.6-12.0% 48 -19 -12 27 64 24 8 74 129
2.3-9.3% 3.2-8.1%
1.6-? 2.2-12.0%
501 482 688
1.6-10.0% 1.7-9.0% 1.8-9.7%
385 325 324
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C4H8 C4H8 C4H8Cl2 C4H8Cl2 C4H8Cl2O C4H8O C4H8O C4H8O C4H8O C4H8O C4H8O C4H8O C4H8O C4H8OS C4H8O2 C4H8O2 C4H8O2 C4H8O2 C4H8O2 C4H8O2 C4H8O2 C4H8O2 C4H8O2S C4H8O3 C4H8O3 C4H9Br C4H9Br C4H9Cl C4H9Cl C4H9Cl C4H9Cl C4H9Cl3Si C4H9N C4H9NO C4H9NO C4H9NO C4H9NO C4H9NO C4H9NO2 C4H9NO3 C4H10 C4H10 C4H10N2 C4H10O C4H10O C4H10O C4H10O C4H10O C4H10O C4H10O2 C4H10O2 C4H10O2 C4H10O2 C4H10O2 C4H10O2 C4H10O2 C4H10O2S C4H10O3 C4H10O4S C4H10S C4H10S
tB/°C
Name Isobutene Cyclobutane 1,2-Dichlorobutane 1,4-Dichlorobutane Bis(2-chloroethyl) ether 2-Buten-1-ol 2-Methyl-2-propenol Ethyl vinyl ether 1,2-Epoxybutane Butanal Isobutanal 2-Butanone Tetrahydrofuran 1,4-Oxathiane Butanoic acid 2-Methylpropanoic acid Propyl formate Isopropyl formate Ethyl acetate Methyl propanoate 3-Hydroxybutanal 1,4-Dioxane Sulfolane Methyl lactate Ethylene glycol monoacetate 1-Bromobutane 2-Bromobutane 1-Chlorobutane 2-Chlorobutane 1-Chloro-2-methylpropane 2-Chloro-2-methylpropane Butyltrichlorosilane Pyrrolidine N-Ethylacetamide N,N-Dimethylacetamide Butanal oxime 2-Butanone oxime Morpholine N-Acetylethanolamine Butyl nitrate Butane Isobutane Piperazine 1-Butanol 2-Butanol 2-Methyl-1-propanol 2-Methyl-2-propanol Diethyl ether Methyl propyl ether 1,2-Butanediol 1,3-Butanediol 1,4-Butanediol 2,3-Butanediol Ethylene glycol monoethyl ether Ethylene glycol dimethyl ether tert-Butyl hydroperoxide 2,2′-Thiodiethanol Diethylene glycol Diethyl sulfate 1-Butanethiol 2-Butanethiol
-6.9 12.6 124.1 161 178.5 121.5 114.5 35.5 63.4 74.8 64.5 79.5 65 147 163.7 154.4 80.9 68.2 77.1 79.8 101.5 287.3 144.8 188 101.6 91.2 78.6 68.2 68.5 50.9 148.5 86.5 205 165 154 152.5 128 133 -0.5 -11.7 146 117.7 99.5 107.8 82.4 34.5 39.1 190.5 207.5 235 182.5 135 85 282 245.8 208 98.5 85
16-20
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FP/°C <10
Fl. Limits 1.8-9.6% 1.8-?
IT/°C 465 275
52 55 27 33 <-46 -22 -22 -18 -9 -14 42 72 56 -3 -6 -4 -2 66 12 177 49 102 18 21 -12 -10 -6 0 54 3 110 70 58 ≈70 37 179 36 -60 -87 81 37 24 28 11 -45 -20 40 121 121 43 -2 27 160 124 104 2 -23
2.7-? 4.2-35.3%
369 349
1.7-28% 1.7-19% 1.9-12.5% 1.6-10.6% 1.4-11.4% 2-11.8%
202 439 218 196 404 321
2.0-10.0% 2.0-9.2%
2.0-22%
443 481 455 485 426 469 250 180
2.2-?
385
2.6-6.6%
265
1.9-10.1%
240
2.0-11.5% 2.5-13%
2.0-8.7%
1.8-11.5%
490
1.4-11.2%
290 460
1.9-8.5% 1.8-8.4%
287 460
1.4-11.2% 1.7-9.8% 1.7-10.6% 2.4-8.0% 1.9-36.0% 2.0-14.8%
343 405 415 478 180
395
3-18%
402 235 202
2-17%
298 224 436
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C4H10S C4H10S C4H10Se C4H11N C4H11N C4H11N C4H11N C4H11N C4H11NO C4H11NO C4H11NO2 C4H12Sn C4H13N3 C5H4O2 C5H5N C5H6 C5H6N2 C5H6O C5H6O2 C5H7N C5H7NO C5H7NO2 C5H8 C5H8 C5H8 C5H8O C5H8O C5H8O C5H8O2 C5H8O2 C5H8O2 C5H8O2 C5H8O2 C5H8O2 C5H8O3 C5H9NO C5H10 C5H10 C5H10 C5H10 C5H10 C5H10 C5H10 C5H10Cl2 C5H10N2 C5H10O C5H10O C5H10O C5H10O C5H10O C5H10O C5H10O2 C5H10O2 C5H10O2 C5H10O2 C5H10O2 C5H10O2 C5H10O2 C5H10O2 C5H10O2 C5H10O2
tB/°C
Name 2-Methyl-1-propanethiol 2-Methyl-2-propanethiol Diethyl selenide Butylamine sec-Butylamine tert-Butylamine Isobutylamine Diethylamine 2-Amino-1-butanol 2-Amino-2-methyl-1-propanol Diethanolamine Tetramethylstannane Diethylenetriamine Furfural Pyridine 2-Methyl-1-buten-3-yne 2-Methylpyrazine 3-Methylfuran Furfuryl alcohol 1-Methylpyrrole 2-Furanmethanamine Ethyl cyanoacetate 2-Methyl-1,3-butadiene 1-Pentyne Cyclopentene 3-Methyl-3-buten-2-one Cyclopentanone 3,4-Dihydro-2H-pyran Allyl acetate Isopropenyl acetate Vinyl propanoate Ethyl acrylate Methyl methacrylate 2,4-Pentanedione Methyl acetoacetate N-Methyl-2-pyrrolidone 1-Pentene cis-2-Pentene trans-2-Pentene 2-Methyl-1-butene 3-Methyl-1-butene 2-Methyl-2-butene Cyclopentane 1,5-Dichloropentane 3-(Dimethylamino)propanenitrile Cyclopentanol Pentanal 2-Pentanone 3-Pentanone Tetrahydropyran 2-Methyltetrahydrofuran Pentanoic acid 3-Methylbutanoic acid Butyl formate Isobutyl formate Propyl acetate Isopropyl acetate Ethyl propanoate Methyl butanoate 3-Ethoxypropanal Tetrahydrofurfuryl alcohol
88.5 64.3 108 77.0 63.5 44.0 67.7 55.5 178 165.5 268.8 78 207 161.7 115.2 32 137 66 171 115 145.5 205 34.0 40.1 44.2 98 130.5 86 103.5 94 91.2 99.4 100.5 138 171.7 202 29.9 36.9 36.3 31.2 20.1 38.5 49.3 179 173 140.4 103 102.2 101.9 88 78 186.1 176.5 106.1 98.2 101.5 88.6 99.1 102.8 135.2 178
16-21
FP/°C
Fl. Limits
IT/°C
2 <-29 -12 -9 -9 -9 -23 74 67 172 -12 98 60 20 <-7 50 -30 75 16 37 110 -54 <-20 -29
2.5-? 1.7-9.8%
312
1.7-8.9% 2-12% 1.8-10.1%
380 378 312
2-13% 1.9-? 2-6.7% 2.1-19.3% 1.8-12.4%
662 358 316 482
1.8-16.3%
491
1.5-8.9%
395 395
1.8-9.0% 26 -18 22 26 1 10 10 34 77 96 -18 <-20 <-20 -20 -7 -20 -25 >27 65 51 12 7 13 -20 -11 96 18 5 13 2 12 14 38 75
374 432 1.4-14% 1.7-8.2%
372
1-10% 1.5-8.7%
340 280 346 275
1.5-9.1%
365
1.5-?
361
1.5-8.2% 1.6-?
222 452 450
1.7-8.2% 2-9% 1.7-8% 1.8-8% 1.9-11%
400 416 322 320 450 460 440
1.5-9.7%
282
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C5H10O3 C5H10O3 C5H10O3 C5H11Br C5H11Cl C5H11Cl C5H11Cl C5H11Cl3Si C5H11N C5H11N C5H11NO C5H11NO2 C5H12 C5H12 C5H12 C5H12N2 C5H12N2O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O2 C5H12O2 C5H12O2 C5H12O3 C5H12S C5H12S C5H13N C5H13N C6H2Cl4 C6H3ClN2O4 C6H3Cl3 C6H4ClNO2 C6H4Cl2 C6H4Cl2 C6H4Cl2 C6H4Cl2O C6H5Br C6H5Cl C6H5ClO C6H5ClO C6H5Cl2N C6H5Cl3Si C6H5F C6H5NO2 C6H5N3O4 C6H6 C6H6 C6H6N2O2 C6H6O C6H6O2 C6H6O2 C6H6O2 C6H7N C6H7N C6H7N
tB/°C
Name Diethyl carbonate Ethylene glycol monomethyl ether acetate Ethyl lactate 1-Bromopentane 1-Chloropentane 2-Chloro-2-methylbutane 1-Chloro-3-methylbutane Trichloropentylsilane Piperidine N-Methylpyrrolidine 4-Methylmorpholine Isopentyl nitrite Pentane Isopentane Neopentane 1-Methylpiperazine Tetramethylurea 1-Pentanol 2-Pentanol 3-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 2-Methyl-2-butanol 3-Methyl-2-butanol 2,2-Dimethyl-1-propanol Ethyl propyl ether 1,5-Pentanediol 2-Isopropoxyethanol 2,2-Dimethyl-1,3-propanediol Diethylene glycol monomethyl ether 1-Pentanethiol 3-Methyl-2-butanethiol Pentylamine Butylmethylamine 1,2,4,5-Tetrachlorobenzene 1-Chloro-2,4-dinitrobenzene 1,2,4-Trichlorobenzene 1-Chloro-4-nitrobenzene o-Dichlorobenzene m-Dichlorobenzene p-Dichlorobenzene 2,4-Dichlorophenol Bromobenzene Chlorobenzene o-Chlorophenol p-Chlorophenol 3,4-Dichloroaniline Trichlorophenylsilane Fluorobenzene Nitrobenzene 2,4-Dinitroaniline 1,5-Hexadien-3-yne Benzene p-Nitroaniline Phenol 1,2-Benzenediol Resorcinol p-Hydroquinone Aniline 2-Methylpyridine 4-Methylpyridine
126 143 154.5 129.8 107.8 85.6 98.9 172 106.2 81 116 99.2 36.0 27.8 9.4 138 176.5 137.9 119.3 116.2 128 131.1 102.4 112.9 113.5 63.2 239 145 208 193 126.6 104.3 91 244.5 315 213.5 242 180 173 174 210 156.0 131.7 174.9 220 272 201 84.7 210.8 85 80.0 332 181.8 245 287 184.1 129.3 145.3
16-22
TeamLRN
FP/°C 25 49 46 32 13 <21 63 16 -14 24 -40 -51 -65 42 77 33 34 41 50 43 19 38 37 <-20 129 33 129 96 18 3 -1 13 155 194 105 127 66 72 66 114 51 28 64 121 166 91 -15 88 224 <-20 -11 199 79 127 127 165 70 39 57
Fl. Limits
IT/°C
1.5-12.3% 1.5-?
392 400
1.6-8.6% 1.5-7.4% 1.5-7.4%
260 345
1.4-8.0% 1.4-7.6% 1.4-7.5%
1.2-10.0% 1.2-9.0% 1.2-9.0% 1.2-9.0% 1.2-9.0%
210 260 420 450
300 343 435 385 350 437
1.7-9.0% 335
1.38-22.7%
399 240
2.2-22%
2.0-22% 2.5-6.6%
571
2.2-9.2%
648
1.3-9.6%
565 593
1.8-?
482
1.5-? 1.2-7.8%
498
1.8-8.6%
715
1.4-?
608 516 615 538
1.3-11%
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C6H8ClN C6H8Cl2O2 C6H8N2 C6H8N2 C6H8N2 C6H8N2 C6H8O C6H8O4 C6H10 C6H10 C6H10 C6H10 C6H10 C6H10O C6H10O C6H10O C6H10O2 C6H10O2 C6H10O2 C6H10O2 C6H10O3 C6H10O3 C6H10O4 C6H10O4 C6H10O4 C6H11Cl C6H11NO C6H11NO2 C6H11NO2 C6H12 C6H12 C6H12 C6H12 C6H12 C6H12 C6H12 C6H12 C6H12 C6H12 C6H12 C6H12 C6H12 C6H12Cl2O2 C6H12O C6H12O C6H12O C6H12O C6H12O C6H12O C6H12O C6H12O C6H12O C6H12O C6H12O2 C6H12O2 C6H12O2 C6H12O2 C6H12O2 C6H12O2 C6H12O2 C6H12O2
tB/°C
Name Aniline, hydrochloride Hexanedioyl dichloride Adiponitrile o-Phenylenediamine Phenylhydrazine 2,5-Dimethylpyrazine 2,5-Dimethylfuran Dimethyl maleate 1,4-Hexadiene 2-Methyl-1,3-pentadiene 4-Methyl-1,3-pentadiene 2-Hexyne Cyclohexene Diallyl ether Cyclohexanone Mesityl oxide Vinyl butanoate Ethyl 2-butenoate Ethyl methacrylate 2,5-Hexanedione Ethyl acetoacetate Propanoic anhydride Adipic acid Diethyl oxalate Ethylene glycol diacetate Chlorocyclohexane Caprolactam Nitrocyclohexane 4-Acetylmorpholine 1-Hexene cis-2-Hexene 2-Methyl-1-pentene 4-Methyl-1-pentene 4-Methyl-cis-2-pentene 4-Methyl-trans-2-pentene 2-Ethyl-1-butene 2,3-Dimethyl-1-butene 2,3-Dimethyl-2-butene Cyclohexane Methylcyclopentane Ethylcyclobutane 2-Methyl-2-pentene 1,2-Bis(2-chloroethoxy)ethane cis-3-Hexen-1-ol Butyl vinyl ether Isobutyl vinyl ether Hexanal 2-Ethylbutanal 2-Methylpentanal 2-Hexanone 3-Hexanone 4-Methyl-2-pentanone Cyclohexanol Hexanoic acid 2-Methylpentanoic acid Diethylacetic acid Pentyl formate Butyl acetate sec-Butyl acetate Isobutyl acetate Propyl propanoate
295 257 243.5 155 93.5 202 65 75.8 76.5 84.5 82.9 94 155.4 130 116.7 136.5 117 194 180.8 170 337.5 185.7 190 142 270 205 63.4 68.8 62.1 53.9 56.3 58.6 64.7 55.6 73.3 80.7 71.8 70.8 67.3 232 156.5 94 83 131 117 127.6 123.5 116.5 160.8 205.2 195.6 194 130.4 126.1 112 116.5 122.5
16-23
FP/°C 193 72 93 156 88 64 7 113 -21 -12 -34 -10 -12 -7 44 31 20 2 20 79 57 63 196 76 88 32 125 88 113 -26 -21 -28 -7 -32 -29 <-20 <-20 <-20 -20 -29 -15 <-7 121 54 -9 -9 32 21 17 25 35 18 68 102 107 99 26 22 31 18 79
Fl. Limits
1.0-? 1.5-?
IT/°C
550
2.0-6.1%
1.2-?
310
1.1-9.4% 1.4-7.2% 1.4-8.8%
420 344
1.4-9.5% 1.3-9.5%
499 295 285 420
1.6-8.4%
482
1.2-6.9%
253 300 300
1.3-8% 1.0-8.35% 1.2-7.7%
315 360 401 245 258 210
255
1.2-7.7% 1-8% 1-8% 1.2-8.0% 1-9%
1.7-7.6% 1.7-9.8% 1.3-10.5%
199 423 448 300 380 378 400 425 421
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C6H12O2 C6H12O2 C6H12O2 C6H12O3 C6H12O3 C6H12S C6H13Cl C6H13N C6H13NO C6H13NO C6H13NO C6H13NO2 C6H14 C6H14 C6H14 C6H14 C6H14 C6H14N2O C6H14O C6H14O C6H14O C6H14O C6H14O C6H14O C6H14O C6H14O2 C6H14O2 C6H14O2 C6H14O2 C6H14O2 C6H14O3 C6H14O3 C6H14O3 C6H14O3 C6H14O4 C6H15N C6H15N C6H15N C6H15N C6H15N C6H15NO2 C6H15NO3 C6H15N3 C6H15O4P C6H16N2 C7H3ClF3NO2 C7H4ClF3 C7H4F3NO2 C7H5ClO C7H5ClO C7H5Cl3 C7H5F3 C7H6N2O4 C7H6O C7H6O2 C7H6O2 C7H6O3 C7H7Br C7H7Br C7H7Cl C7H7NO2
tB/°C
Name Ethyl butanoate Ethyl 2-methylpropanoate Diacetone alcohol Ethylene glycol monoethyl ether acetate Paraldehyde Cyclohexanethiol 1-Chlorohexane Cyclohexylamine N-Butylacetamide 2,6-Dimethylmorpholine N-Ethylmorpholine 4-Morpholineethanol Hexane 2-Methylpentane 3-Methylpentane 2,2-Dimethylbutane 2,3-Dimethylbutane 1-Piperazineethanol 1-Hexanol 2-Methyl-1-pentanol 4-Methyl-2-pentanol 2-Ethyl-1-butanol Dipropyl ether Diisopropyl ether Butyl ethyl ether 2,5-Hexanediol 2-Methyl-2,4-pentanediol Ethylene glycol monobutyl ether 1,1-Diethoxyethane Ethylene glycol diethyl ether 1,2,6-Hexanetriol Diethylene glycol monoethyl ether Diethylene glycol dimethyl ether Trimethylolpropane Triethylene glycol Hexylamine Butylethylamine Dipropylamine Diisopropylamine Triethylamine Diisopropanolamine Triethanolamine 1-Piperazineethanamine Triethyl phosphate N,N-Diethylethylenediamine 1-Chloro-4-nitro-2-(trifluoromethyl)benzene 1-Chloro-2-(trifluoromethyl)benzene 1-Nitro-3-(trifluoromethyl)benzene Benzoyl chloride 4-Chlorobenzaldehyde (Trichloromethyl)benzene (Trifluoromethyl)benzene 1-Methyl-2,4-dinitrobenzene Benzaldehyde Benzoic acid Salicylaldehyde Salicylic acid o-Bromotoluene p-Bromotoluene (Chloromethyl)benzene o-Nitrotoluene
121.5 110.1 167.9 156.4 124.3 158.9 135 134 229 146.6 138.5 227 68.7 60.2 63.2 49.7 57.9 246 157.6 149 131.6 147 90.0 68.5 92.3 218 197.1 168.4 102.2 119.4 196 162 285 132.8 107.5 109.3 83.9 89 250 335.4 220 215.5 144 232 152.2 202.8 197.2 213.5 221 102.1 179.0 249.2 197 181.7 184.3 179 222
16-24
TeamLRN
FP/°C 24 13 58 56 36 43 35 31 116 44 32 99 -22 <-29 -7 -48 -29 124 63 54 41 57 21 -28 4 110 102 69 -21 27 191 96 67 149 177 29 18 17 -1 -7 127 179 93 115 46 135 59 103 72 88 127 12 207 63 121 78 157 79 85 67 106
Fl. Limits
IT/°C 463
1.8-6.9% 2-8% 1.3-?
643 379 238
1.9-9.4%
293
1.1-7.5% 1.0-7.0% 1.2-7.0% 1.2-7.0% 1.2-7.0%
225 264 278 405 405
1.1-9.65% 1.0-5.5%
310
1.3-7.0% 1.4-7.9%
188 443
1-9% 4-13% 1.6-10.4%
306 238 230 205
0.9-9.2%
371
1.1-7.1% 1.2-8.0%
299 316 249 374
1-10% 454
211
192 570 1.1-?
540
1.1-?
585
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C7H7NO2 C7H7NO2 C7H8 C7H8 C7H8O C7H8O C7H8O C7H8O C7H8O C7H8O2 C7H8O3S C7H9N C7H9N C7H9NO C7H10O C7H10O4 C7H12 C7H12O2 C7H12O2 C7H12O2 C7H12O4 C7H14 C7H14 C7H14 C7H14 C7H14 C7H14O C7H14O C7H14O C7H14O C7H14O C7H14O C7H14O C7H14O C7H14O C7H14O C7H14O2 C7H14O2 C7H14O2 C7H14O2 C7H14O2 C7H15NO2 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16N2O C7H16O C7H16O C7H16O C7H16O C7H17N C7H18N2 C8H4O3 C8H6O4 C8H6O4 C8H7ClO C8H7N C8H8
tB/°C
Name m-Nitrotoluene p-Nitrotoluene Toluene Bicyclo[2.2.1]hepta-2,5-diene o-Cresol m-Cresol p-Cresol Benzyl alcohol Anisole 4-Methoxyphenol p-Toluenesulfonic acid o-Methylaniline p-Methylaniline o-Anisidine 3-Cyclohexene-1-carboxaldehyde 3,3-Diacetoxy-1-propene 4-Methylcyclohexene Butyl acrylate Isobutyl acrylate Cyclohexyl formate Diethyl malonate 1-Heptene trans-2-Heptene Cycloheptane Methylcyclohexane Ethylcyclopentane 2-Heptanone 3-Heptanone 4-Heptanone 5-Methyl-2-hexanone cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol cis-3-Methylcyclohexanol trans-3-Methylcyclohexanol cis-4-Methylcyclohexanol trans-4-Methylcyclohexanol Pentyl acetate Isopentyl acetate sec-Pentyl acetate Butyl propanoate Propyl butanoate Ethyl N-butylcarbamate Heptane 2-Methylhexane 3-Methylhexane 2,3-Dimethylpentane 2,4-Dimethylpentane 2,2,3-Trimethylbutane 4-Morpholinepropanamine 2-Heptanol 3-Heptanol 2,4-Dimethyl-3-pentanol 2,3,3-Trimethyl-2-butanol Heptylamine N,N-Diethyl-1,3-propanediamine Phthalic anhydride Phthalic acid Terephthalic acid α-Chloroacetophenone Benzeneacetonitrile Styrene
232 238.3 110.6 89.5 191.0 202.2 201.9 205.3 153.7 243 200.3 200.4 224 105 180 102.7 145 132 162 200 93.6 98 118.4 100.9 103.5 151.0 147 144 144 165 167.5 174.5 174.5 173 174 149.2 142.5 130.5 146.8 143.0 202 98.5 90.0 92 89.7 80.4 80.8 220 159 157 138.7 131 156 168.5 295
247 233.5 145
16-25
FP/°C 106 106 4 -21 81 86 86 93 52 132 184 85 87 118 57 82 -1 29 30 51 93 -1 <0 <21 -4 <21 39 46 49 36 65 65 70 70 70 70 16 25 32 32 37 92 -4 -1 -4 -56 -12 <0 104 71 60 49 <0 54 59 152 168 260 118 113 31
Fl. Limits
IT/°C
1.1-7.1%
480
1.4-? 1.1-? 1.1-?
599 558 558 436 475 421 482 482
1.7-9.9%
292 427
260 1.1-6.7% 1.2-6.7% 1.1-6.7% 1.1-7.9%
1.0-8.2%
1.1-7.5% 1.0-7.5%
250 260 393
191 296 296 295 295 295 295 360 360 426
1.05-6.7% 1.0-6.0% 1.1-6.7%
204 280 280 335 412
375
1.7-10.5%
570 496
0.9-6.8%
490
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C8H8O C8H8O C8H8O C8H8O2 C8H8O2 C8H8O2 C8H8O2 C8H8O3 C8H9Cl C8H9NO C8H9NO2 C8H10 C8H10 C8H10 C8H10 C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O2 C8H11N C8H11N C8H11N C8H11N C8H11N C8H11N C8H11NO C8H11NO C8H11NO C8H12 C8H12 C8H12O4 C8H12O4 C8H14O2 C8H14O2 C8H14O3 C8H14O3 C8H14O3 C8H14O4 C8H14O5 C8H14O6 C8H15ClO C8H16 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16O C8H16O C8H16O C8H16O2 C8H16O2 C8H16O2 C8H16O2 C8H16O2
tB/°C
Name Phenyloxirane Benzeneacetaldehyde Acetophenone Benzeneacetic acid Phenyl acetate Methyl benzoate 2-Methoxybenzaldehyde Methyl salicylate 1-Chloro-4-ethylbenzene Acetanilide Methyl 2-aminobenzoate Ethylbenzene o-Xylene m-Xylene p-Xylene p-Ethylphenol Benzeneethanol α-Methylbenzyl alcohol Phenetole Benzyl methyl ether 4-Methylanisole 2-Phenoxyethanol N-Ethylaniline N,N-Dimethylaniline 2,3-Xylidine 2,6-Xylidine α-Methylbenzylamine 5-Ethyl-2-picoline N-Phenylethanolamine o-Phenetidine p-Phenetidine 1,5-Cyclooctadiene 4-Vinylcyclohexene Diethyl maleate Diethyl fumarate Cyclohexyl acetate Butyl methacrylate Butanoic anhydride 2-Methylpropanoic anhydride Butyl acetoacetate Ethyl succinate Diethylene glycol diacetate Diethyl tartrate Octanoyl chloride 1-Octene 2,4,4-Trimethyl-1-pentene 2,4,4-Trimethyl-2-pentene Ethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane cis-1,4-Dimethylcyclohexane Propylcyclopentane Octanal 2-Ethylhexanal 2-Octanone Hexyl acetate sec-Hexyl acetate 2-Ethylbutyl acetate Pentyl propanoate Butyl butanoate
194.1 195 202 265.5 196 199 243.5 222.9 184.4 304 256 136.1 144.5 139.1 138.3 217.9 218.2 205 169.8 170 175.5 245 203.0 194.1 221.5 215 187 178.3 279.5 232.5 254 150.8 128 223 214 173 160 200 183 217.7 200 281 195.6 121.2 101.4 104.9 131.9 129.8 123.5 124.4 131 171 163 172.5 171.5 147.5 162.5 168.6 166
16-26
TeamLRN
FP/°C 74 71 77 >100 80 83 118 96 64 169 >100 21 32 27 27 104 96 93 63 135 60 121 85 63 97 96 79 68 152 115 116 35 16 121 104 58 52 54 59 85 90 135 93 82 21 -5 2 35 16 11 16
Fl. Limits
IT/°C 498 570
454 530 0.8-6.7% 0.9-6.7% 1.1-7.0% 1.1-7.0%
432 463 527 528
371 1.0-?
1.1-6.6%
269 350 335 0.9-5.8% 1.0-6.2%
0.8-4.8% 0.9-6.6%
279 329
230 391 305 238 304 304 269
52 44 52 45 45 54 41 53
0.85-7.2%
190
378
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C8H16O2 C8H16O2 C8H16O2 C8H16O2 C8H16O3 C8H16O4 C8H17Cl C8H17Cl C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18O C8H18O C8H18O C8H18O C8H18O2 C8H18O2 C8H18O2 C8H18O3 C8H18O4 C8H18O5 C8H18S C8H18S C8H19N C8H19N C8H19N C8H19N C8H20O4Si C8H23N5 C9H6N2O2 C9H7N C9H10 C9H10 C9H10 C9H10 C9H10O C9H10O C9H10O2 C9H10O2 C9H10O2 C9H11NO C9H12 C9H12 C9H12 C9H12 C9H12 C9H12 C9H12 C9H12 C9H12O C9H12O2 C9H12O3S C9H13N C9H14O C9H14O C9H14O6 C9H16
tB/°C
Name Isobutyl butanoate Isobutyl isobutanoate Ethyl hexanoate 1,4-Cyclohexanedimethanol Pentyl lactate Diethylene glycol monoethyl ether acetate 1-Chlorooctane 3-(Chloromethyl)heptane Octane 2,3-Dimethylhexane 2,4-Dimethylhexane 3-Ethyl-2-methylpentane 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1-Octanol 2-Octanol 2-Ethyl-1-hexanol Dibutyl ether 2-Ethyl-1,3-hexanediol 2,2,4-Trimethyl-1,3-pentanediol Di-tert-butyl peroxide Diethylene glycol diethyl ether 2,5,8,11-Tetraoxadodecane Tetraethylene glycol 1-Octanethiol Dibutyl sulfide Octylamine Dibutylamine Diisobutylamine 2-Ethylhexylamine Ethyl silicate Tetraethylenepentamine Toluene-2,4-diisocyanate Quinoline o-Methylstyrene m-Methylstyrene p-Methylstyrene Isopropenylbenzene 1-Phenyl-1-propanone 4-Methylacetophenone Ethyl benzoate Benzyl acetate Methyl 2-phenylacetate 4-Methylacetanilide Propylbenzene Isopropylbenzene o-Ethyltoluene m-Ethyltoluene p-Ethyltoluene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene α−Ethylbenzyl alcohol Ethylene glycol monobenzyl ether Ethyl p-toluenesulfonate Amphetamine Phorone Isophorone Triacetin Octahydroindene
156.9 148.6 167 283 218.5 181.5 172 125.6 115.6 109.5 115.6 110 99.2 114.8 195.1 180 184.6 140.2 244 235 111 188 216 328 199.1 185 179.6 159.6 139.6 169.2 168.8 341.5 251 237.1 169.8 164 172.8 165.4 217.5 226 212 213 216.5 307 159.2 152.4 165.2 161.3 162 176.1 169.3 164.7 219 256 203 197.5 215.2 259 167
16-27
FP/°C 50 38 49 167 79 110 70 60 13 7 10 <21 <21 -12 <21 81 88 73 25 127 113 18 82 111 182 69 76 60 47 29 60 52 163 127
Fl. Limits 0.96-7.59%
IT/°C 432 316 425
1.0-6.5%
206 438 460 346 418 425
0.88-9.7% 1.5-7.6%
231 194 360 346
1.1-6%
321 0.9-9.5%
53 53 53 54 99 96 88 90 91 168 30 36
0.8-11.0% 0.8-11.0% 0.8-11.0% 1.9-6.1%
44 44 50 100 129 158 <100 85 84 138
0.8-6.6% 0.9-6.4% 1-5%
480 538 538 538 574
490 460
0.8-6.0% 0.9-6.5%
450 424 440 480 475 470 500 559 352
0.8-3.8% 1.0-?
460 433 296
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C9H16O2 C9H18 C9H18 C9H18 C9H18 C9H18O C9H18O C9H18O2 C9H18O2 C9H18O2 C9H20 C9H20 C9H20 C9H20 C9H20 C9H20 C9H20 C9H20 C9H21BO3 C9H21N C9H21NO3 C10H7Cl C10H8 C10H8O C10H9N C10H10O2 C10H10O4 C10H10O4 C10H10O4 C10H11NO2 C10H12 C10H12O2 C10H12O2 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14O C10H14O2 C10H15N C10H15N C10H15NO2 C10H16 C10H16 C10H16 C10H16 C10H16 C10H16O C10H18 C10H18O C10H18O C10H18O C10H18O
tB/°C
Name Allyl hexanoate 1-Nonene Propylcyclohexane Isopropylcyclohexane Butylcyclopentane 2-Nonanone Diisobutyl ketone Pentyl butanoate Isopentyl butanoate Butyl 3-methylbutanoate Nonane 3-Ethyl-4-methylhexane 4-Ethyl-2-methylhexane 2,2,5-Trimethylhexane 3,3-Diethylpentane 3-Ethyl-2,4-dimethylpentane 2,2,3,3-Tetramethylpentane 2,2,3,4-Tetramethylpentane Triisopropyl borate Tripropylamine Triisopropanolamine 1-Chloronaphthalene Naphthalene 2-Naphthol 1-Naphthalenamine Safrole Dimethyl phthalate Dimethyl isophthalate Dimethyl terephthalate Acetoacetanilide 1,2,3,4-Tetrahydronaphthalene Isopropyl benzoate Ethyl phenylacetate Butylbenzene sec-Butylbenzene tert-Butylbenzene Isobutylbenzene p-Cymene 1,2,3,4-Tetramethylbenzene 1,2,3,5-Tetramethylbenzene 1,2,4,5-Tetramethylbenzene o-Diethylbenzene m-Diethylbenzene p-Diethylbenzene Butyl phenyl ether 4-tert-Butyl-1,2-benzenediol N-Butylaniline N,N-Diethylaniline N-Phenyl-N,N-diethanolamine Dipentene d-Limonene α-Pinene β-Pinene β-Phellandrene Camphor trans-Decahydronaphthalene Borneol Linalol α-Terpineol Cineole
186 146.9 156.7 154.8 156.6 195.3 169.4 186.4 179 150.8 140 133.8 124.0 146.3 136.7 140.2 133.0 140 156 259 217.9 285 300.8 234.5 283.7 282 288 207.6 216 227 183.3 173.3 169.1 172.7 177.1 205 198 196.8 184 181.1 183.7 210 285 243.5 216.3 178 178 156.2 166 171.5 207.4 187.3 198 220 176.4
16-28
TeamLRN
FP/°C
Fl. Limits
IT/°C
66 26
60 49 57 59 53 31 24 <21 13 390 <21 <21 28 41 160 121 79 153 157 100 146 138 153 185 71 99 99 71 52 60 55 47 74 71 54 57 56 55 82 130 107 85 196 45 45 33 38 49 66 54 66 71 90 48
0.9-5.9% 0.8-7.1%
248 283 250 360 396
0.8-2.9%
205
0.7-?
280
0.7-5.7%
290
0.8-4.9%
430
0.9-5.9%
320 >558 526
0.9-?
490 518
0.8-5.0%
385
0.8-5.8% 0.8-6.9% 0.7-5.7% 0.8-6.0% 0.7-5.6%
410 418 450 427 436 427 427
0.7-6.0%
0.7-? 0.7-6.1%
0.6-3.5% 0.7-5.4%
395 450 430
630 387 237 237 255 275 466 255
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C10H18O C10H18O4 C10H19NO2 C10H20 C10H20 C10H20 C10H20 C10H20O C10H20O2 C10H20O2 C10H21N C10H22 C10H22 C10H22 C10H22 C10H22O C10H22O C10H22O2 C10H22O5 C10H22S C10H23N C10H23N C11H10 C11H12O3 C11H14O2 C11H16 C11H16 C11H16 C11H16 C11H16O C11H17N C11H20O2 C11H22 C11H22O C11H22O2 C11H24 C11H24 C11H24O C12H9Br C12H10 C12H10Cl2Si C12H10O C12H10O C12H11N C12H11N C12H12 C12H14O4 C12H14O4 C12H16 C12H16O3 C12H17NO C12H18 C12H20O4 C12H22O4 C12H22O6 C12H23N C12H24 C12H24O2 C12H25Br C12H26 C12H26O
tB/°C
Name trans-Geraniol Dibutyl oxalate N-tert-Butylaminoethyl methacrylate 1-Decene Butylcyclohexane Isobutylcyclohexane tert-Butylcyclohexane Citronellol 2-Ethylhexyl acetate Ethyl octanoate N-Butylcyclohexanamine Decane 2-Methylnonane 3-Ethyloctane 4-Ethyloctane 1-Decanol Dipentyl ether Ethylene glycol dibutyl ether Tetraethylene glycol dimethyl ether Dipentyl sulfide Decylamine Dipentylamine 1-Methylnaphthalene Ethyl benzoylacetate Butyl benzoate p-tert-Butyltoluene Pentylbenzene 1,3-Diethyl-5-methylbenzene Pentamethylbenzene 4-tert-Butyl-2-methylphenol p-tert-Pentylaniline 2-Ethylhexyl acrylate Pentylcyclohexane 2-Undecanone Nonyl acetate Undecane 2-Methyldecane 2-Undecanol 4-Bromo-1,1'-Biphenyl Biphenyl Dichlorodiphenylsilane o-Phenylphenol Diphenyl ether 2-Aminobiphenyl Diphenylamine 1-Ethylnaphthalene Diethyl phthalate Diethyl terephthalate Cyclohexylbenzene Pentyl salicylate N-Butyl-N-phenylacetamide 1,5,9-Cyclododecatriene Dibutyl maleate Dimethyl sebacate Dibutyl tartrate Dicyclohexylamine 1-Dodecene Ethyl decanoate 1-Bromododecane Dodecane 1-Dodecanol
230 241 170.5 180.9 171.3 171.5 224 199 208.5 174.1 167.1 166.5 163.7 231.1 190 203.3 275.3 220.5 202.5 244.7 250.3 190 205.4 205 232 237 260.5 203.7 231.5 210 195.9 189.3 228 310 256.1 305 286 258.0 299 302 258.6 295 302 240.1 270 281 240 280 320 213.8 241.5 276 216.3 259
16-29
FP/°C
Fl. Limits
>100 104 96 <55
96 71 79 93 51
IT/°C
235 246 274 342 0.76-8.14%
268
0.8-5.4%
210 210 230 229 288 170
82 57 85 141 85 99 51
529 141 107 68 66 455 427
93 118 102 82
252 239
89 68 69 225 113 144 113 142 124 112
0.6-5.8%
0.8-1.5%
153 161 117 99 132 141 71 141 145 91 >99 79 >100 144 74 127
0.7-?
540 530 618 450 634 480 457
284
0.6-?
203 275
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form.
Name
C12H26O C12H26O3 C12H26S C12H27BO3 C12H27N C12H27O4P C13H12 C13H12 C13H14N2 C13H26 C13H26O C13H28 C13H28O C14H8O2 C14H10 C14H10 C14H12O2 C14H12O3 C14H14 C14H14O C14H16 C14H16N2O2 C14H23N C14H28 C14H30 C14H30O C15H18 C15H24 C15H24O C15H26O6 C15H33N C16H14O C16H18 C16H22O4 C16H26 C16H34 C16H34O C16H35N C17H20N2O C17H34O C17H36O C18H14 C18H14 C18H15O3P C18H15O4P C18H15P C18H30 C18H32O7 C18H34O2 C18H34O4 C18H36O2 C18H37Cl3Si C18H38 C18H38O C19H16 C19H38O C19H38O2 C19H40 C20H14O4 C20H28 C20H42
2-Butyl-1-octanol Diethylene glycol dibutyl ether 1-Dodecanethiol Tributyl borate Tributylamine Tributyl phosphate 2-Methylbiphenyl Diphenylmethane p,p′-Diaminodiphenylmethane 1-Tridecene 2-Tridecanone Tridecane 1-Tridecanol 9,10-Anthracenedione Anthracene Phenanthrene Benzyl benzoate Benzyl salicylate 1,1-Diphenylethane Dibenzyl ether 1-Butylnaphthalene o-Dianisidine N,N-Dibutylaniline 1-Tetradecene Tetradecane 1-Tetradecanol 1-Pentylnaphthalene Nonylbenzene 2,6-Di-tert-butyl-4-methylphenol Tributyrin Tripentylamine 1,3-Diphenyl-2-buten-1-one 2-Butyl-1,1'-biphenyl Dibutyl phthalate Decylbenzene Hexadecane Dioctyl ether Bis(2-ethylhexyl)amine N,N′-Diethylcarbanilide 2-Heptadecanone 1-Heptadecanol o-Terphenyl m-Terphenyl Triphenyl phosphite Triphenyl phosphate Triphenylphosphine Dodecylbenzene Butyl citrate Oleic acid Dibutyl sebacate Stearic acid Trichlorooctadecylsilane Octadecane 1-Octadecanol Triphenylmethane 2-Nonadecanone Methyl stearate Nonadecane Diphenyl phthalate 1-Decylnaphthalene Eicosane
tB/°C 246.5 256 277 234 216.5 289 255.5 265.0 398 232.8 263 235.4 377 339.9 340 323.5 320 272.6 298 289.3 274.8 233 253.5 289 307 280.5 265 307.5 242.5 342.5 340 298 286.8 283
320 333 332 363 360
328 360 344.5
316.3 359 443 329.9 379 343
16-30
TeamLRN
FP/°C 110 118 128 93 63 146 137 130 220 79 107 79 121 185 121 171 148 >100 >100 135 360 206 110 110 112 141 124 99 127 180 102 177 >100 157 107 136 >100 132 150 120 154 163 191 218 220 180 140 157 189 178 196 89 >100 >100 124 153 >100 224 177 >100
Fl. Limits
IT/°C 310
502 485
0.6-?
540 480 440
0.5-?
235 200
0.5-?
407
0.5-?
430 402 202 205
0.4-?
368 363 365 395 227 450
230
232
FLAMMABILITY OF CHEMICAL SUBSTANCES (continued) Mol. Form. C21H21O4P C21H26O3 C21H32O2 C22H42O2 C22H42O4 C22H44O2 C23H46O2 C24H20Sn C24H38O4 C25H48O4
tB/°C
Name Tri-o-cresyl phosphate 4-Octylphenyl salicylate Methyl abietate Butyl oleate Bis(2-ethylhexyl) adipate Butyl stearate Pentyl stearate Tetraphenylstannane Bis(2-ethylhexyl) phthalate Bis(2-ethylhexyl) azelate
410
343 420 384
16-31
FP/°C 225 216 180 180 206 160 185 232 218 227
Fl. Limits
IT/°C 385 416
0.4-?
377 355
0.3-?
374
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS Several organizations recommend limits of exposure to airborne contaminants in the workplace. These include the Occupational Safety and Health Administration (OSHA), the National Institute for Occupational Safety and Health (NIOSH), and the non-governmental organization, American Conference of Governmental Industrial Hygienists (ACGIH). The threshold limit value (TLV) for a substance is defined as the concentration level under which the majority of workers may be repeatedly exposed, day after day, without adverse effects. The TLV recommendations are given in two forms: • Time-weighted average (TWA) concentration for a normal 8-h workday and 40-h workweek. • Short-term exposure limit (STEL), which should not be exceeded for more than 15 min. Both kinds of limits are specified for some substances. The following table gives threshold limit values for a number of substances that may be encountered in the atmosphere of a chemical laboratory or industrial facility. All values refer to the concentration in air at 25°C and normal atmospheric pressure. Data for gases are given both in parts per million by volume (ppm) and in mass concentration (mg/m3). Values for liquids refer to mists or aerosols, and those for solids to dusts or fumes; both are stated in mg/m3. A “C” following a value indicates a ceiling limit which should not be exceeded even for very brief periods because of acute toxic effects of the substance. Substances are listed by systematic name, which is followed by molecular formula in the Hill format and Chemical Abstracts Service Registry Number. Common synonyms are given in brackets [ ] for some compounds.
REFERENCES 1. 2000 TLV’s and BEI’s, American Conference of Governmental Industrial Hygienists, 1330 Kemper Meadow Drive, Cincinnati, OH 452401634, 2000. 2. NIOSH Pocket Guide to Chemical Hazards, U.S. Department of Health and Human Services, National Institute for Occupational Health and Safety, U.S. Government Printing Office, Washington, DC, 1994. 3. Chemical Information Manual, U.S. Department of Labor, Occupational Safety and Health Administration, Washington, DC, 1991.
Substance Abate [Temephos] Acetaldehyde Acetic acid Acetic anhydride Acetone Acetone cyanohydrin Acetonitrile Acetophenone 2-(Acetyloxy)benzoic acid [Aspirin] Acrolein [2-Propenal] Acrylamide Acrylic acid [2-Propenoic acid] Acrylonitrile [Propenenitrile] Adipic acid Adiponitrile Aldrin Allyl alcohol [2-Propen-1-ol] Allyl glycidyl ether Allyl propyl disulfide Aluminum (metal dust) Aluminum oxide 4-Amino-3,5,6-trichloropyridinecarboxlic acid [Picloram] Ammonia Ammonium chloride Ammonium perfluorooctanoate Ammonium sulfamate Aniline Antimony Arsenic Arsine
Molecular Formula
CAS Reg. No.
C16H20O6P2S3 C2H4O C2H4O2 C4H6O3 C3H6O C4H7NO C2H3N C8H8O C9H8O4 C3H4O C3H5NO C3H4O2 C3H3N C6H10O4 C6H8N2 C12H8Cl6 C3H6O C6H10O2 C6H12S2 Al Al2O3
3383-96-8 75-07-0 64-19-7 108-24-7 67-64-1 75-86-5 75-05-8 98-86-2 50-78-2 107-02-8 79-06-1 79-10-7 107-13-1 124-04-9 111-69-3 309-00-2 107-18-6 106-92-3 2179-59-1 7429-90-5 1344-28-1
C6H3Cl3N2O2 H3N ClH4N C8H4F15NO2 H6N2O3S C6H7N Sb As AsH3
1918-02-1 7664-41-7 12125-02-9 3825-26-1 7773-06-0 62-53-3 7440-36-0 7440-38-2 7784-42-1
16-32
TeamLRN
Time-Weighted Average ppm mg/m3
Short-Term Exposure Limit ppm mg/m3
10 10 5 500
25 21 1188
40 10
67 49 5
2 2 2 0.5 1 2
25
2
0.05
0.03 5.9 4.3 5 9 0.25 1.2 5 12 10 10 10 17 10 0.01 10 7.6 0.5 0.01 0.16
25 C 15
45 C 37
750 4.7 C 60
1780 5C 101
0.1 C
0.23 C
3
18
35
24 20
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance Atrazine Azinphos-methyl Barium Barium sulfate Benomyl Benzene 1,3-Benzenedimethanamine [m-Xylene diamine] Benzenethiol [Phenyl mercaptan] p-Benzoquinone [Quinone] Benzoyl chloride Benzoyl peroxide Benzyl acetate Beryllium Biphenyl Bis(4-amino-3-chlorophenyl)methane [4,4-Methylene bis(2-chloroaniline)] Bis(2-chloroethyl) ether [2,2'-Dichlorethyl ether] Bis(chloromethyl) ether Bis(2-dimethylaminoethyl) ether [DMAEE] Bis(2-ethylhexyl) phthalate [Di-sec-octyl phthalate] Bismuth telluride Boron oxide Boron tribromide Boron trifluoride Bromacil Bromine Bromine pentafluoride Bromochloromethane [Halon 1011] 2-Bromo-2-chloro-1,1,1-trifluoroethane [Halothane] Bromoethane [Ethyl bromide] Bromoethene [Vinyl bromide] Bromomethane [Methyl bromide] Bromotrifluoromethane 1,3-Butadiene Butane 1-Butanethiol [Butyl mercaptan] 1-Butanol 2-Butanol [sec-Butyl alcohol] 2-Butanone [Methyl ethyl ketone] trans-2-Butenal [Crotonaldehyde] 3-Buten-2-one Butyl acetate sec-Butyl acetate tert-Butyl acetate Butyl acrylate Butylamine tert-Butyl chromate Butyl glycidyl ether Butyl lactate o-sec-Butylphenol p-tert-Butyltoluene Cadmium Calcium carbonate
Time-Weighted Average ppm mg/m3
Molecular Formula
CAS Reg. No.
C8H14ClN5 C10H12N3O3PS2 Ba BaO4S C14H18N4O3 C6H6
1912-24-9 86-50-0 7440-39-3 7727-43-7 17804-35-2 71-43-2
C8H12N2 C6H6S C6H4O2 C7H5ClO C14H10O4 C9H10O2 Be C12H10
1477-55-0 108-98-5 106-51-4 98-88-4 94-36-0 140-11-4 7440-41-7 92-52-4
0.2
5 61 0.002 1.3
C13H12Cl2N2
101-14-4
0.01
0.11
C4H8Cl2O C2H4Cl2O C8H20N20
111-44-4 542-88-1 3033-62-3
5 0.001 0.05
29 0.0047 0.33
C24H38O4 Bi2Te3 B2O3 BBr3 BF3 C9H13BrN2O2 Br2 BrF5 CH2BrCl
117-81-7 1304-82-1 1303-86-2 10294-33-4 7637-07-2 314-40-9 7726-95-6 7789-30-2 74-97-5
C2HBrClF3 C2H5Br C2H3Br CH3Br CBrF3 C4H6 C4H10 C4H10S C4H10O C4H10O C4H8O C4H6O C4H6O C6H12O2 C6H12O2 C6H12O2 C7H12O2 C4H11N C8H18CrO4 C7H14O2 C7H14O3 C10H14O C11H16 Cd CCaO3
151-67-7 74-96-4 593-60-2 74-83-9 75-63-8 106-99-0 106-97-8 109-79-5 71-36-3 78-92-2 78-93-3 4170-30-3 78-94-4 123-86-4 105-46-4 540-88-5 141-32-2 109-73-9 1189-85-1 2426-08-6 138-22-7 89-72-5 98-51-1 7440-43-9 1317-65-3
16-33
0.84 0.5
5 0.2 0.5 10 10 1.6
0.5 0.1
2.3 0.44
Short-Term Exposure Limit ppm mg/m3
2.5
8 0.1 C
0.5 C 10
0.01
10
58
0.15
1.0
5 10 10
0.1 0.1 200
10 0.66 0.72 1060
50 5 0.5 1 1000 2 800 0.5
404 22 2.2 3.9 6090 4.4 1900 1.8
100 200
303 590
150 200 200 2
713 950 950 10
25 5 5 1
133 30 31 6.1 0.01 10
2.8 C
10
1C 1C
10 C 2.8 C
0.2
1.3
50 C
152 C
300 0.3 C 0.2 C 200
885 0.9 C 0.6 C 950
5C
15 C 0.1 C
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance
Molecular Formula
CAS Reg. No.
Calcium chromate Calcium cyanamide Calcium hydroxide Calcium metasilicate Calcium oxide Calcium sulfate Camphor Caprolactam Captafol Captan Carbaryl Carbofuran Carbon black Carbon dioxide Carbon disulfide Carbon monoxide Carbonyl chloride [Phosgene] Carbonyl fluoride Cesium hydroxide Chlordane Chlorine Chlorine dioxide Chlorine trifluoride Chloroacetaldehyde Chloroacetone α-Chloroacetophenone Chloroacetyl chloride Chlorobenzene o-Chlorobenzylidene malononitrile 2-Chloro-1,3-butadiene [Chloroprene] Chlorodifluoromethane Chloroethane [Ethyl chloride] 2-Chloroethanol [Ethylene chlorohydrin] Chloroethene [Vinyl chloride] Chloromethane [Methyl chloride] (Chloromethyl)benzene [Benzyl chloride] 1-Chloro-4-nitrobenzene 1-Chloro-1-nitropropane Chloropentafluoroethane 2-Chloropropanoic acid 3-Chloropropene [Allyl chloride] 2-Chlorostyrene o-Chlorotoluene Chlorpyrifos Chromium Chromyl chloride Clopidol Cobalt Cobalt carbonyl Cobalt hydrocarbonyl Copper Cresol (all isomers) Crufomate Cyanamide Cyanogen Cyanogen chloride Cyclohexane
CaCrO4 CCaN2 CaH2O2 CaO3Si CaO CaO4S C10H16O C6H11NO C10H9Cl4NO2S C9H8Cl3NO2S C12H11NO2 C12H15NO3 C CO2 CS2 CO CCl2O CF2O CsHO C10H6Cl8 Cl2 ClO2 ClF3 C2H3ClO C3H5ClO C8H7ClO C2H2Cl2O C6H5Cl C10H5ClN2 C4H5Cl CHClF2 C2H5Cl C2H5ClO C2H3Cl CH3Cl C7H7Cl C6H4ClNO2 C3H6ClNO2 C2ClF5 C3H5ClO2 C3H5Cl C8H7Cl C7H7Cl C9H11Cl3NO3PS Cr Cl2CrO2 C7H7Cl2NO Co C8Co2O8 C4HCoO4 Cu C7H8O C12H19ClNO3P CH2N2 C2N2 CClN C6H12
13765-19-0 156-62-7 1305-62-0 1344-95-2 1305-78-8 7778-18-9 76-22-2 105-60-2 2425-06-1 133-06-2 63-25-2 1563-66-2 1333-86-4 124-38-9 75-15-0 630-08-0 75-44-5 353-50-4 21351-79-1 57-74-9 7782-50-5 10049-04-4 7790-91-2 107-20-0 78-95-5 532-27-4 79-04-9 108-90-7 2698-41-1 126-99-8 75-45-6 75-00-3 107-07-3 75-01-4 74-87-3 100-44-7 100-00-5 600-25-9 76-15-3 598-78-7 107-05-1 2039-87-4 95-49-8 2921-88-2 7440-47-3 14977-61-8 2971-90-6 7440-48-4 10210-68-1 16842-03-8 7440-50-8 1319-77-3 299-86-5 420-04-2 460-19-5 506-77-4 110-82-7
16-34
TeamLRN
Time-Weighted Average ppm mg/m3
0.5 0.1
0.003 0.5 5 10 2 10 12 1 (solid) 0.1 5 5 0.1 3.5 9000 31 29 0.40 5.4 2 0.5 1.5 0.28
0.05 0.05 10
0.32 0.23 46
10 1000 100
36 3540 264
1 50 1 0.1 2 1000 0.1 1 50 50
10
2.5 103 5.2 0.64 10 6320 0.44 3 283 259 0.2 0.5 0.16 10 0.02 0.1 0.1 0.2 22 5 2 21
300
1030
2 5 (gas)
5000 10 25 0.1 2
0.025
5
Short-Term Exposure Limit ppm mg/m3
4 10 (gas)
24 3 (solid)
30,000
54,000
5
13
1 0.3 0.1 C 1C 1C
2.9 0.83 0.38 C 3.2 C 3.8 C
0.15
0.69
0.05 C
0.39 C
1C
3.3 C
100
207
2 75
6 425
0.3 C
0.75 C
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance Cyclohexanol Cyclohexanone Cyclohexene Cyclohexylamine Cyclonite [Hexahydro-1,3,5-trinitro1,3,5-triazine] 1,3-Cyclopentadiene Cyclopentane Cyhexatin Decaborane(14) Diacetone alcohol 4,4'-Diaminodiphenylmethane [4,4-Methylene dianiline] Diazinon Diazomethane Diborane Dibromodifluoromethane 2-Dibutylaminoethanol 2,6-Di-tert-butyl-4-methylphenol Dibutylphenyl phosphate Dibutyl phosphate Dibutyl phthalate Dichloroacetylene o-Dichlorobenzene p-Dichlorobenzene 1,4-Dichloro-2-butene (unspecified isomer) Dichlorodifluoromethane 1,3-Dichloro-5,5-dimethyl hydantoin Dichlorodiphenyltrichloroethane [DDT] 1,1-Dichloroethane [Ethylidene dichloride] 1,2-Dichloroethane [Ethylene dichloride] 1,1-Dichloroethene [Vinylidene chloride] 1,2-Dichloroethylene (both isomers) Dichlorofluoromethane Dichloromethane [Methylene chloride] 1,1-Dichloro-1-nitroethane (2,4-Dichlorophenoxy)acetic acid 1,2-Dichloropropane 2,2-Dichloropropanoic acid 1,3-Dichloropropene (both isomers) 1,2-Dichloro-1,1,2,2-tetrafluoroethane Dichlorvos Dicrotophos m-Dicyanobenzene [m-Phthalodinitrile] Dicyclopentadiene Dieldrin Diethanolamine Diethylamine 2-Diethylaminoethanol Diethylenetriamine [Bis(2-amimoethyl)amine] Diethyl ether Diethyl phthalate 1,1-Difluoroethene Diglycidyl ether Diisopropylamine Diisopropyl ether Dimethoxymethane [Methylal]
Molecular Formula
CAS Reg. No.
Time-Weighted Average ppm mg/m3
C6H12O C6H10O C6H10 C6H13N
108-93-0 108-94-1 110-83-8 108-91-8
50 25 300 10
C3H6N6O6 C5H6 C5H10 C18H34OSn B10H14 C6H12O2
121-82-4 542-92-7 287-92-3 13121-70-5 17702-41-9 123-42-2
C13H14N2 C12H21N2O3PS CH2N2 B2H6 CBr2F2 C10H23NO C15H24O C14H23O4P C8H19O4P C16H22O4 C2Cl2 C6H4Cl2 C6H4Cl2 C4H6Cl2 CCl2F2 C5H6Cl2N2O2 C14H9Cl5 C2H4Cl2 C2H4Cl2 C2H2Cl2 C2H2Cl2 CHCl2F CH2Cl2 C2H3Cl2NO2 C8H6Cl2O3 C3H6Cl2 C3H4Cl2O2 C3H4Cl2 C2Cl2F4 C4H7Cl2O4P C8H16NO5P C8H4N2 C10H12 C12H8Cl6O C4H11NO2 C4H11N C6H15NO C4H13N3 C4H10O C12H14O4 C2H2F2 C6H10O3 C6H15N C6H14O C3H8O2
101-77-9 333-41-5 334-88-3 19287-45-7 75-61-6 102-81-8 128-37-0 2528-36-1 107-66-4 84-74-2 7572-29-4 95-50-1 106-46-7 764-41-0 75-71-8 118-52-5 50-29-3 75-34-3 107-06-2 75-35-4 540-59-0 75-43-4 75-09-2 594-72-9 94-75-7 78-87-5 75-99-0 542-75-6 76-14-2 62-73-7 141-66-2 626-17-5 77-73-6 60-57-1 111-42-2 109-89-7 100-37-8 111-40-0 60-29-7 84-66-2 75-38-7 2238-07-5 108-18-9 108-20-3 109-87-5
16-35
75 600 0.05 50 0.1 0.2 0.1 100 0.5 0.3 1
25 10 0.005 1000
100 10 5 200 10 50 2 75 1 1000 0.1
5 0.46 5 2 1 400 500 0.1 5 250 1000
Short-Term Exposure Limit ppm mg/m3
206 100 1010 41 0.5 203 1720 5 0.25 238 0.81 0.1 0.34 0.11 858 3.5 10 3.5 8.6 5 150 60 0.026 4950 0.2 1 405 40 20 793 42 174 12 10 347 5 4.5 7000 0.90 0.25 5 27 0.25 2 15 9.6 4.2 1210 5 1310 0.53 21 1040 3110
0.15
0.75
2
17
0.1 C 50
0.39 C 301
0.4
110
508
15
45
500
1520
310
1300
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance Dimethyl mercury N,N-Dimethylacetamide Dimethylamine N,N-Dimethylaniline 2,2-Dimethylbutane 2,3-Dimethylbutane N,N-Dimethylformamide 2,6-Dimethyl-4-heptanone [Diisobutyl ketone] 1,1-Dimethylhydrazine Dimethyl phthalate Dimethyl sulfate Dinitrobenzene (all isomers) Dinitrotoluene (all isomers) 1,4-Dioxane Dioxathion Diphenylamine Diphenyl ether 4,4'-Diphenylmethane diisocyanate Dipropylene glycol monomethyl ether Diquat Disulfiram Disulfoton Diuron Divinyl benzene (all isomers) Endosulfan Endrin Enflurane Epichlorohydrin [(Chloromethyl)oxirane] 1,2-Epoxy-4-(epoxyethyl)cyclohexane [Vinylcyclohexene dioxide] 1,2-Ethanediamine [Ethylenediamine] Ethanethiol [Ethyl mercaptan] Ethanol Ethanolamine Ethion Ethoxydimethylsilane Ethyl acetate Ethyl acrylate Ethylamine Ethylbenzene Ethyl tert-butyl ether [ETBE] Ethylene glycol Ethylene glycol dinitrate Ethylene glycol monobutyl ether [2-Butoxyethanol] Ethylene glycol monoethyl ether [2-Ethoxyethanol] Ethylene glycol monoethyl ether acetate [2-Ethoxyethyl acetate] Ethylene glycol monomethyl ether [2-Methoxyethanol] Ethylene glycol monomethyl ether acetate [2-Methoxyethyl acetate] Ethyleneimine Ethylene oxide [Oxirane] Ethyl formate Ethylidene norbornene
Molecular Formula
CAS Reg. No.
C2H6Hg C4H9NO C2H7N C8H11N C6H14 C6H14 C3H7NO C9H18O C2H8N2 C10H10O4 C2H6O4S C6H4N2O4 C7H6N2O4 C4H8O2 C12H26O6P2S4 C12H11N C12H10O C15H10N2O2 C7H16O3 C12H12N2 C10H20N2S4 C8H19O2PS3 C9H10Cl2N2O C10H10 C9H6Cl6O3S C12H8Cl6O C3H2ClF5O C3H5ClO
593-74-8 127-19-5 124-40-3 121-69-7 75-83-2 79-29-8 68-12-2 108-83-8 57-14-7 131-11-3 77-78-1 25154-54-5 25321-14-6 123-91-1 78-34-2 122-39-4 101-84-8 101-68-8 34590-94-8 231-36-7 97-77-8 298-04-4 330-54-1 1321-74-0 115-29-7 72-20-8 13838-16-9 106-89-8
C8H12O2 C2H8N2 C2H6S C2H6O C2H7NO C9H22O4P2S4 C4H12OSi C4H8O2 C5H8O2 C2H7N C8H10 C6H14O C2H6O2 C2H4N2O6
106-87-6 107-15-3 75-08-1 64-17-5 141-43-5 563-12-2 14857-34-2 141-78-6 140-88-5 75-04-7 100-41-4 637-92-3 107-21-1 628-96-6
C6H14O2
Time-Weighted Average ppm mg/m3
10 5 5 500 500 10 25 0.01 0.1 0.15 20
1 0.005 100
10
75 0.5
0.5 400 5 5 100 5
0.57 25 1.3 1880 7.5 0.4 2.1 1440 20 9.2 434 20
0.05
0.31
111-76-2
20
97
C4H10O2
110-80-5
5
18
C6H12O3
111-15-9
5
27
C3H8O2
109-86-4
5
16
C5H10O3 C2H5N C2H4O C3H6O2 C9H12
110-49-6 151-56-4 75-21-8 109-94-4 16219-75-3
5 0.5 1 100
24 0.88 1.8 303
16-36
TeamLRN
0.1 10 0.5 1000 3
0.01 36 9.2 25 1760 1760 30 145 0.025 5 0.52 1.0 0.2 72 0.2 10 7 0.051 600 0.5 2 0.1 10 53 0.1 0.1 566 1.9
Short-Term Exposure Limit ppm mg/m3 0.03 15 10 1000 1000
27.6 50 3500 3500
2
14
150
900
6
15
1.5
6.4
15 15 125
61 27.6 543 100 C
5C
25 C
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance N-Ethylmorpholine Ethyl p-nitrophenyl benzenethiophosphate [EPN] Ethyl silicate Fenamiphos Fensulfothion Fenthion Ferbam Ferrocene [Dicyclopentadienyl iron] Fluorine Fluorine monoxide [Oxygen difluoride] Fonofos Formaldehyde Formamide Formic acid Furfural [2-Furaldehyde] Furfuryl alcohol [2-Furanmethanol] Germane [Germanium tetrahydride] Glycerol Graphite Hafnium Heptachlor Heptane 2-Heptanone [Methyl pentyl ketone] 3-Heptanone [Ethyl butyl ketone] 4-Heptanone [Dipropyl ketone] Hexachlorobenzene Hexachloro-1,3-butadiene 1,2,3,4,5,6-Hexachlorocyclohexane [Lindane] Hexachloro-1,3-cyclopentadiene Hexachloroethane [Perchloroethane] Hexachloronaphthalene (all isomers) Hexamethylene diisocyanate Hexane 1,6-Hexanediamine [Hexamethylenediamine] 2-Hexanone [Butyl methyl ketone] 1-Hexene sec-Hexyl acetate Hydrazine Hydrazoic acid Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen peroxide Hydrogen selenide Hydrogen sulfide p-Hydroquinone [1,4-Benzenediol] 2-Hydroxypropyl acrylate Indene Indium Iodine Iodomethane [Methyl iodide] Iron(III) oxide Iron pentacarbonyl Isobutyl acetate Isopentane
Molecular Formula
CAS Reg. No.
Time-Weighted Average ppm mg/m3
C6H13NO
100-74-3
5
C14H14NO4PS C8H20O4Si C13H22NO3PS C11H17O4PS2 C10H15O3PS2 C9H18FeN3S6 C10H10Fe F2 F2O C10H15OPS2 CH2O CH3NO CH2O2 C5H4O2 C5H6O2 GeH4 C3H8O3 C Hf C10H5Cl7 C7H16 C7H14O C7H14O C7H14O C6Cl6 C4Cl6 C6H6Cl6 C5Cl6 C2Cl6 C10H2Cl6 C8H12N2O2 C6H14 C6H16N2 C6H12O C6H12 C8H16O2 H4N2 HN3 BrH ClH CHN FH H2O2 H2Se H2S C6H6O2 C6H10O3 C9H8 In I2 CH3I Fe2O3 C5FeO5 C6H12O2 C5H12
2104-64-5 78-10-4 22224-92-6 115-90-2 55-38-9 14484-64-1 102-54-5 7782-41-4 7783-41-7 944-22-9 50-00-0 75-12-7 64-18-6 98-01-1 98-00-0 7782-65-2 56-81-5 7440-44-0 7440-58-6 76-44-8 142-82-5 110-43-0 106-35-4 123-19-3 118-74-1 87-68-3 58-89-9 77-47-4 67-72-1 1335-87-1 822-06-0 110-54-3 124-09-4 591-78-6 592-41-6 108-84-9 302-01-2 7782-79-8 10035-10-6 7647-01-0 74-90-8 7664-39-3 7722-84-1 7783-07-5 7783-06-4 123-31-9 999-61-1 95-13-6 7440-74-6 7553-56-2 74-88-4 1309-37-1 13463-40-6 110-19-0 78-78-4
16-37
10
1
Short-Term Exposure Limit ppm mg/m3
24 0.1 85 0.1 0.1 0.2 10 10 1.6
2 0.05 C
3.1 0.11 C
0.3 C
0.37 C
10
19
15
60
500
2050
75
350
10
40
0.11 C 3C 5C 4.7 C 3C
0.19 C 9.9 C 7.5 C 5C 2.3 C
15
21
0.1 C
1.0 C
0.2
0.45
0.1 10 5 2 10 0.2
400 50 50 50 0.02 0.01 1 0.005 50 0.5 5 30 50 0.01
1 0.05 10 0.5 10
2 0.1 150 600
18 9.4 7.9 40 0.63 10 2 0.5 0.05 1640 233 233 233 0.002 0.21 0.5 0.11 9.7 0.2 0.034 176 2.3 20 103 295 0.013
1.4 0.16 14 2 2.8 48 0.1 12 5 0.23 713 1770
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance Isopentyl acetate [Isoamyl acetate] Isophorone Isophorone diisocyanate Isopropenylbenzene [α-Methyl styrene] 2-Isopropoxyethanol Isopropyl acetate Isopropylamine N-Isopropylaniline Isopropylbenzene [Cumene] Isopropyl glycidyl ether Kaolin Ketene Lead Lead(II) arsenate Lead(II) chromate Lithium hydride Magnesium carbonate [Magnesite] Magnesium oxide Malathion Maleic anhydride Manganese Manganese cyclopentadienyl tricarbonyl Mercury Mesityl oxide Methacrylic acid [2-Methylpropenoic acid] Methanethiol [Methyl mercaptan] Methanol Methomyl o-Methoxyaniline [o-Anisidine] p-Methoxyaniline [p-Anisidine] Methoxychlor 4-Methoxyphenol Methyl acetate Methyl acrylate 2-Methylacrylonitrile Methylamine o-Methylaniline [o-Toluidine] m-Methylaniline [m-Toluidine] p-Methylaniline [p-Toluidine] N-Methylaniline 3-Methyl-1-butanol [Isoamyl alcohol] 3-Methyl-2-butanone [Methyl isopropyl ketone] Methyl tert-butyl ether [MTBE] Methyl 2-cyanoacrylate Methylcyclohexane Methylcyclohexanol (all isomers) 2-Methylcyclohexanone 2-Methylcyclopentadienyl manganese tricarbonyl Methyl demeton 2-Methyl-3,5-dinitrobenzamide [Dinitolmide] 2-Methyl-4,6-dinitrophenol [Dinitro-o-cresol] Methylene bis(4-cyclohexylisocyanate) Methyl ethyl ketone peroxide Methyl formate 6-Methyl-1-heptanol [Isooctyl alcohol] 5-Methyl-3-heptanone
Molecular Formula
CAS Reg. No.
C7H14O2 C9H14O C12H18N2O2 C9H10 C5H12O2 C5H10O2 C3H9N C9H13N C9H12 C6H12O2 C2H2O Pb As2O8Pb3 CrO4Pb HLi CMgO3 MgO C10H19O6PS2 C4H2O3 Mn C8H5MnO3 Hg C6H10O C4H6O2 CH4S CH4O C5H10N2O2S C7H9NO C7H9NO C16H15Cl3O2 C7H8O2 C3H6O2 C4H6O2 C4H5N CH5N C7H9N C7H9N C7H9N C7H9N C5H12O C5H10O C5H12O C5H5NO2 C7H14 C7H14O C7H12O
123-92-2 78-59-1 4098-71-9 98-83-9 109-59-1 108-21-4 75-31-0 768-52-5 98-82-8 4016-14-2 1332-58-7 463-51-4 7439-92-1 7784-40-9 7758-97-6 7580-67-8 546-93-0 1309-48-4 121-75-5 108-31-6 7439-96-5 12079-65-1 7439-97-6 141-79-7 79-41-4 74-93-1 67-56-1 16752-77-5 90-04-0 104-94-9 72-43-5 150-76-5 79-20-9 96-33-3 126-98-7 74-89-5 95-53-4 108-44-1 106-49-0 100-61-8 123-51-3 563-80-4 1634-04-4 137-05-3 108-87-2 25639-42-3 583-60-8
C9H7MnO3 C6H15O3PS2 C8H7N3O5 C7H6N2O5 C15H22N2O2 C8H18O2 C2H4O2 C8H18O C8H16O
12108-13-3 8022-00-2 148-01-6 534-52-1 5124-30-1 1338-23-4 107-31-3 26952-21-6 541-85-5
16-38
TeamLRN
Time-Weighted Average ppm mg/m3 100
532
0.005 50 25 250 5 2 50 50
200 2 1 5 2 2 2 0.5 100 200 40 0.2 400 50 50
0.045 242 106 1040 12 11 246 238 2 0.86 0.05 0.15 0.075 0.025 10 10 10 4 0.2 0.4 0.025 60 70 0.98 262 2.5 0.5 0.5 10 5 606 7 2.7 6.4 8.8 8.8 8.8 2.2 361 705 144 0.9 1610 234 229
0.005
0.8 0.5 5 0.2 0.054
100 50 25
246 266 131
0.5
0.1
15 20 0.5 200 0.1 0.1
Short-Term Exposure Limit ppm mg/m3
5C
28 C
100
483
310 10
1290 24
75
356
1.5
2.6
25
100
250
328
250
757
15
19
125
452
75
344
0.2 C 150
1.5 C 368
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance 5-Methyl-2-hexanone [Methyl isopentyl ketone] Methylhydrazine Methyl isocyanate Methyl methacrylate Methyloxirane [1,2-Propylene oxide] Methyl parathion 2-Methylpentane 3-Methylpentane 2-Methyl-2,4-pentanediol [Hexylene glycol] 4-Methyl-2-pentanol [Methyl isobutyl carbinol] 4-Methyl-2-pentanone [Isobutyl methyl ketone] 2-Methyl-1-propanol [Isobutyl alcohol] 2-Methyl-2-propanol [tert-Butyl alcohol] Methylstyrene (all isomers) N-Methyl-N,2,4,6-tetranitroaniline [Tetryl] Metribuzin Mevinphos Mica Molybdenum Monocrotophos Morpholine Naled Naphthalene 1-Naphthalenylthiourea [ANTU] Neopentane Nickel Nickel carbonyl Nickel(III) sulfide Nicotine Nitrapyrin Nitric acid Nitric oxide p-Nitroaniline Nitrobenzene Nitroethane Nitrogen dioxide Nitrogen trifluoride Nitromethane 1-Nitropropane 2-Nitropropane Nitrotoluene (all isomers) Nitrous oxide Nonane (all isomers) Octachloronaphthalene Octane (all isomers) Osmium(VIII) oxide [Osmium tetroxide] Oxalic acid 2-Oxetanone [β-Propiolactone] Oxiranemethanol [Glycidol] Ozone Paraquat Parathion Pentaborane(9) Pentachloronaphthalene (unspecified isomer) Pentachloronitrobenzene Pentachlorophenol Pentaerythritol
Molecular Formula
CAS Reg. No.
C7H14O CH6N2 C2H3NO C5H8O2 C3H6O C8H10NO5PS C6H14 C6H14 C6H14O2 C6H14O C6H12O C4H10O C4H10O C9H10 C7H5N5O8 C8H14N4OS C7H13O6P
110-12-3 60-34-4 624-83-9 80-62-6 75-56-9 298-00-0 107-83-5 96-14-0 107-41-5 108-11-2 108-10-1 78-83-1 75-65-0 25013-15-4 479-45-8 21087-64-9 7786-34-7 12001-26-2 7439-98-7 6923-22-4 110-91-8 300-76-5 91-20-3 86-88-4 463-82-1 7440-02-0 13463-39-3 12035-72-2 54-11-5 1929-82-4 7697-37-2 10102-43-9 100-01-6 98-95-3 79-24-3 10102-44-0 7783-54-2 75-52-5 108-03-2 79-46-9 1321-12-6 10024-97-2 111-84-2 2234-13-1 111-65-9 20816-12-0 144-62-7 57-57-8 556-52-5 10028-15-6 4685-14-7 56-38-2 19624-22-7 1321-64-8 82-68-8 87-86-5 115-77-5
Mo C7H14NO5P C4H9NO C4H7Br2Cl2O4P C10H8 C11H10N2S C5H12 Ni C4NiO4 Ni3S2 C10H14N2 C6H3Cl4N HNO3 NO C6H6N2O2 C6H5NO2 C2H5NO2 NO2 F3N CH3NO2 C3H7NO2 C3H7NO2 C7H7NO2 N2O C9H20 C10Cl8 C8H18 O4Os C2H2O4 C3H4O2 C3H6O2 O3 C12H14N2 C10H14NO5PS B5H9 C10H3Cl5 C6Cl5NO2 C6HCl5O C5H12O4
16-39
Time-Weighted Average ppm mg/m3 50 0.01 0.02 50 20 500 500 25 50 50 100 50
0.01
20 10 600 0.05
2 25 1 100 3 10 20 25 10 2 50 200 300 0.0002 0.5 2 0.1
0.005
234 0.019 0.047 205 48 0.2 1760 1760 104 205 152 303 242 1.5 5 0.092 3 10 0.25 71 3 52 0.3 1770 1.5 0.12 0.14 0.5 10 5.2 31 3 5 307 5.6 29 50 91 36 11 90 1050 0.1 1400 0.0016 1 1.5 6.1 0.2 0.5 0.1 0.013 0.5 0.5 0.5 10
Short-Term Exposure Limit ppm mg/m3
100
410
1000 1000 25 C 40 75
3500 3500 121 C 167 307
100
483
0.03
0.27
15
79
4
20 10
5
9.4
375 0.0006
0.3 1750 0.0047 2
0.015
0.039
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance Pentanal [Valeraldehyde] Pentane Pentanedial [Glutaraldehyde] 2-Pentanone [Methyl propyl ketone] 3-Pentanone [Diethyl ketone] Pentyl acetate (all isomers) Perchloromethyl mercaptan Perchloryl fluoride Perfluoroacetone [Hexafluoroacetone] Perfluoroisobutene Phenol 10H-Phenothiazine Phenylenediamine (all isomers) Phenyl glycidyl ether Phenylhydrazine Phenylphosphine Phorate Phosphine Phosphoric acid Phosphorus (white) Phosphorus(III) chloride [Phosphorus trichloride] Phosphorus(V) chloride [Phosphorus pentachloride] Phosphorus(V) oxychloride [Phosphoryl chloride] Phosphorus(V) sulfide Phthalic anhydride Piperazine dihydrochloride 2-Pivaloyl-1,3-indandione [Pindone] Platinum Potassium hydroxide Propane Propanoic acid 1-Propanol 2-Propanol [Isopropyl alcohol] Propargyl alcohol [2-Propyn-1-ol] Propoxur Propyl acetate 1,2-Propylene glycol dinitrate Propylene glycol monomethyl ether Propyleneimine Propyl nitrate Propyne [Methylacetylene] 2-Pyridinamine [2-Aminopyridine] Pyridine Pyrocatechol [Catechol] Resorcinol Rhodium Ronnel Rotenone Selenium Selenium hexafluoride Sesone Silane Silicon Silicon carbide
Molecular Formula
CAS Reg. No.
C5H10O C5H12 C5H8O2 C5H10O C5H10O C7H14O2 CCl4S ClFO3 C3F6O C4F8 C6H6O C12H9NS C6H8N2 C9H10O2 C6H8N2 C6H7P C7H17O2PS3 H3P H3O4P P
110-62-3 109-66-0 111-30-8 107-87-9 96-22-0 628-63-7 594-42-3 7616-94-6 684-16-2 382-21-8 108-95-2 92-84-2 25265-76-3 122-60-1 100-63-0 638-21-1 298-02-2 7803-51-2 7664-38-2 7723-14-0
Cl3P
Time-Weighted Average ppm mg/m3 50 600
176 1770
200 200 50 0.1 3 0.1
705 700 265 0.76 13 0.68
5
19 5 0.1 0.6 0.44
0.1 0.1
0.02
0.05 0.42 1 0.1
7719-12-2
0.2
1.1
Cl5P
10026-13-8
0.1
0.85
Cl3OP P2S5 C8H4O3 C4H12Cl2N2 C14H14O3 Pt HKO C3H8 C3H6O2 C3H8O C3H8O C3H4O C11H15NO3 C5H10O2 C3H6N2O6 C4H10O2 C3H7N C3H7NO3 C3H4 C5H6N2 C5H5N C6H6O2 C6H6O2 Rh C8H8Cl3O3PS C23H22O6 Se F6Se C8H7Cl2NaO5S H4Si Si CSi
10025-87-3 1314-80-3 85-44-9 142-64-3 83-26-1 7440-06-4 1310-58-3 74-98-6 79-09-4 71-23-8 67-63-0 107-19-7 114-26-1 109-60-4 6423-43-4 107-98-2 75-55-8 627-13-4 74-99-7 504-29-0 110-86-1 120-80-9 108-46-3 7440-16-6 299-84-3 83-79-4 7782-49-2 7783-79-1 136-78-7 7803-62-5 7440-21-3 409-21-2
0.1
0.63 1 6.1 5 0.1 1 2C 4500 30 492 983 2.3 0.5 835 0.34 369 4.7 107 1640 1.9 16 23 45 1 10 5 0.2 0.95 10 6.6 10 10
16-40
TeamLRN
0.3
1
2500 10 200 400 1 200 0.05 100 2 25 1000 0.5 5 5 10
0.12 5
Short-Term Exposure Limit ppm mg/m3
750 0.05 C 250 300 100
2210 0.2 C 881 1050 530
6
25
0.01 C
0.082 C
0.05 C
0.23 C 0.2 1.4 3
1
0.5
2.8
3
250 500
614 1230
250
1040
150
553
40
172
20
90
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance Silicon dioxide (α-quartz) Silicon dioxide (tridymite) Silicon dioxide (cristobalite) Silicon dioxide (vitreous) Silver Sodium azide Sodium fluoroacetate Sodium hydrogen sulfite Sodium hydroxide Sodium metabisulfite Sodium pyrophosphate Sodium tetraborate decahydrate Stibine Strontium chromate Strychnine Styrene Sucrose Sulfotep Sulfur chloride Sulfur decafluoride Sulfur dioxide Sulfur hexafluoride Sulfur tetrafluoride Sulfuric acid Sulfuryl fluoride Sulprofos Talc Tantalum Tantalum(V) oxide Tellurium Tellurium hexafluoride Terephthalic acid Terphenyl (all isomers) 1,1,2,2-Tetrabromoethane [Acetylene tetrabromide] Tetrabromomethane [Carbon tetrabromide] 1,1,1,2-Tetrachloro-2,2-difluoroethane 1,1,2,2-Tetrachloro-1,2-difluoroethane 1,1,2,2-Tetrachloroethane Tetrachloroethene [Perchloroethylene] Tetrachloromethane [Carbon tetrachloride] Tetrachloronaphthalene (all isomers) Tetraethyl lead Tetraethyl pyrophosphate [TEPP] Tetrahydrofuran [Oxolane] Tetramethyl lead Tetramethyl silicate Tetramethyl succinonitrile Tetranitromethane Thallium 4,4'-Thiobis(6-tert-butyl-m-cresol) Thioglycolic acid Thionyl chloride Thiram Tin Titanium(IV) oxide [Titanium dioxide] Toluene
Molecular Formula
CAS Reg. No.
O2Si O2Si O2Si O2Si Ag N3Na C2H2FNaO2 HNaO3S HNaO Na2O5S2 Na4O7P2 B4H20Na2O17 H3Sb CrO4Sr C21H22N2O2 C8H8 C12H22O11 C8H20O5P2S2 Cl2S2 F10S2 O2S F6S F4S H2O4S F2O2S C12H19O2PS3 Ta O5Ta2 Te F6Te C8H6O4 C18H14
14808-60-7 15468-32-3 14464-46-1 60676-86-0 7440-22-4 26628-22-8 62-74-8 7631-90-5 1310-73-2 7681-57-4 7722-88-5 1303-96-4 7803-52-3 7789-06-2 57-24-9 100-42-5 57-50-1 3689-24-5 10025-67-9 5714-22-7 7446-09-5 2551-62-4 7783-60-0 7664-93-9 2699-79-8 35400-43-2 14807-96-6 7440-25-7 1314-61-0 13494-80-9 7783-80-4 100-21-0 26140-60-3
C2H2Br4 CBr4 C2Cl4F2 C2Cl4F2 C2H2Cl4 C2Cl4 CCl4 C10H4Cl4 C8H20Pb C8H20O7P2 C4H8O C4H12Pb C4H12O4Si C8H12N2 CN4O8 Tl C22H30O2S C2H4O2S Cl2OS C6H12N2S4 Sn O2Ti C7H8
79-27-6 558-13-4 76-11-9 76-12-0 79-34-5 127-18-4 56-23-5 1335-88-2 78-00-2 107-49-3 109-99-9 75-74-1 681-84-5 3333-52-6 509-14-8 7440-28-0 96-69-5 68-11-1 7719-09-7 137-26-8 7440-31-5 13463-67-7 108-88-3
16-41
Time-Weighted Average ppm mg/m3
Short-Term Exposure Limit ppm mg/m3
0.05 0.05 0.05 0.1 0.1 0.29 C 0.05 5 2C
0.1
20
2 1000
5
0.02
1 0.1 500 500 1 25 5
5 5 5 0.51 0.002 0.15 85 10 0.2
5.2 6000 1 21 1 2 5 5 0.1 0.10 10
1
14 1.4 4170 4170 6.9 170 31 2 0.1 0.05 590 0.15 6 2.8 0.04 0.1 10 3.8
50
1 2 10 188
200 1 0.5 0.005
40
170
1C 0.01 C 5
5.5 C 0.10 C 13
0.1 C 10
0.44 C 3 42
0.53 C
5C
0.3
4.1
100 10
685 63
250
737
1C
4.9 C
THRESHOLD LIMITS FOR AIRBORNE CONTAMINANTS (continued)
Substance Toluene-2,4-diisocyanate 1H-1,2,4-Triazol-3-amine Tribromomethane [Bromoform] Tributyl phosphate Trichloroacetic acid 1,2,4-Trichlorobenzene 1,1,1-Trichloroethane [Methyl chloroform] 1,1,2-Trichloroethane Trichloroethene Trichlorofluoromethane Trichloromethane [Chloroform] (Trichloromethyl)benzene [Benzotrichloride] Trichloronaphthalene (all isomers) Trichloronitromethane [Chloropicrin] 2,4,5-Trichlorophenoxyacetic acid 1,2,3-Trichloropropane 1,1,2-Trichloro-1,2,2-trifluoroethane Tri-o-cresyl phosphate Triethanolamine Triethylamine Triiodomethane [Iodoform] Trimellitic anhydride [1,2,4-Benzenetricarboxylic anhydride] Trimethylamine Trimethylbenzene (all isomers) Trimethyl phosphite Trinitroglycerol [Nitroglycerin] 2,4,6-Trinitrophenol [Picric acid] 2,4,6-Trinitrotoluene [TNT] Triphenylamine Triphenyl phosphate Tungsten Uranium Vanadium(V) oxide Vinyl acetate 4-Vinylcyclohexene Warfarin Xylene (all isomers) Xylidine (all isomers) Yttrium Zinc chloride Zinc chromate, basic Zinc oxide Zirconium
Molecular Formula
CAS Reg. No.
C9H6N2O2 C2H4N4 CHBr3 C12H27O4P C2HCl3O2 C6H3Cl3 C2H3Cl3 C2H3Cl3 C2HCl3 CCl3F CHCl3 C7H5Cl3 C10H5Cl3 CCl3NO2 C8H5Cl3O3 C3H5Cl3 C2Cl3F3 C21H21O4P C6H15NO3 C6H15N CHI3
584-84-9 61-82-5 75-25-2 126-73-8 76-03-9 120-82-1 71-55-6 79-00-5 79-01-6 75-69-4 67-66-3 98-07-7 1321-65-9 76-06-2 93-76-5 96-18-4 76-13-1 78-30-8 102-71-6 121-44-8 75-47-8
C9H4O5 C3H9N C9H12 C3H9O3P C3H5N3O9 C6H3N3O7 C7H5N3O6 C18H15N C18H15O4P W U O5V2 C4H6O2 C8H12 C19H16O4 C8H10 C8H11N Y Cl2Zn CrH2O4Zn OZn Zr
552-30-7 75-50-3 25551-13-7 121-45-9 55-63-0 88-89-1 118-96-7 603-34-9 115-86-6 7440-33-7 7440-61-1 1314-62-1 108-05-4 100-40-3 81-81-2 1330-20-7 1300-73-8 7440-65-5 7646-85-7 13530-65-9 1314-13-2 7440-67-7
16-42
TeamLRN
Time-Weighted Average ppm mg/m3
Short-Term Exposure Limit ppm mg/m3
0.005
0.02
0.14
5C 450
37 C 2460
100 1000 C
537 5620 C
0.01 C
0.08 C
1250
9590
3
12
15
0.04 C 36
0.5 0.2 1
0.036 0.2 5.2 2.2 6.7
350 10 50
1910 55 269
10
49
0.1 10 1000
1 0.6
5 25 2 0.05
10 0.1 100 0.5
5 0.67 10 60 7670 0.1 5 4.1 10
12 123 10 0.46 0.1 0.1 5 3 5 0.2 0.05 35 0.44 0.1 434 2.5 1 1 0.045 5 5
10 0.6 15
53
150
651
2 10 10
OCTANOL-WATER PARTITION COEFFICIENTS The octanol-water partition coefficient, P, is a widely used parameter for correlating biological effects of organic substances. It is a property of the two-phase system in which water and 1-octanol are in equilibrium at a fixed temperature and the substance is distributed between the water-rich and octanol-rich phases. P is defined as the ratio of the equilibrium concentration of the substance in the octanol-rich phase to that in the water-rich phase, in the limit of zero concentration. In general, P tends to be large for compounds with extended non-polar structures (such as long chain or multiring hydrocarbons) and small for compounds with highly polar groups. Thus P (or, in its more common form of expression, log P) provides a measure of the lipophilic vs. hydrophilic nature of a compound, which is an important consideration in assessing the potential toxicity. A discussion of methods of measurement and accuracy considerations for log P may be found in Reference 1. This table gives selected values of log P for about 450 organic compounds, including many of environmental importance. All values refer to a nominal temperature of 25°C. The source of each value is indicated in the last column. These references contain data on many more compounds than are included here. Compounds are listed by molecular formula following the Hill convention. To locate a compound by name or CAS Registry Number when the molecular formula is not known, use the table “Physical Constants of Organic Compounds” in Section 3 and its indexes to determine the molecular formula. REFERENCES 1. Sangster, J., J. Phys. Chem. Ref. Data, 18, 1111, 1989. 2. Mackay, D., Shiu, W.Y., and Ma, K.C., Illustrated Handbook of Physical-Chemical Properties and Environmental Fate for Organic Chemicals, Lewis Publishers/CRC Press, Boca Raton, FL, 1992. 3. Shiu, W.Y., and Mackay, D., J. Phys. Chem. Ref. Data, 15, 911, 1986. 4. Pinsuwan, S., Li, L., and Yalkowsky, S.H., J. Chem. Eng. Data, 40, 623, 1995. 5. Solubility Data Series, International Union of Pure and Applied Chemistry, Vol. 20, Pergamon Press, Oxford, 1985. 6. Solubility Data Series, International Union of Pure and Applied Chemistry, Vol. 38, Pergamon Press, Oxford, 1985. 7. Miller, M.M., Ghodbane, S., Wasik, S.P., Tewari, Y.B., and Martire, D.E., J. Chem. Eng. Data, 29, 184, 1984. Mol. Form. CCl2F2 CCl3F CCl4 CHBr3 CHCl3 CH2BrCl CH2Br2 CH2Cl2 CH2F2 CH2I2 CH2O CH2O2 CH3Br CH3Cl CH3F CH3I CH3NO CH3NO2 CH4O CH5N C2Cl3F3 C2Cl4 C2Cl6 C2HCl3 C2HCl5 C2H2Cl2 C2H2Cl2 C2H2Cl2 C2H2Cl4 C2H3Cl C2H3Cl3 C2H3Cl3 C2H3N C2H4Cl2 C2H4Cl2
Name Dichlorodifluoromethane Trichlorofluoromethane Tetrachloromethane Tribromomethane Trichloromethane Bromochloromethane Dibromomethane Dichloromethane Difluoromethane Diiodomethane Formaldehyde Formic acid Bromomethane Chloromethane Fluoromethane Iodomethane Formamide Nitromethane Methanol Methylamine 1,1,2-Trichlorotrifluoroethane Tetrachloroethylene Hexachloroethane Trichloroethylene Pentachloroethane 1,1-Dichloroethylene cis-1,2-Dichloroethylene trans-1,2-Dichloroethylene 1,1,2,2-Tetrachloroethane Chloroethylene 1,1,1-Trichloroethane 1,1,2-Trichloroethane Acetonitrile 1,1-Dichloroethane 1,2-Dichloroethane
© 2000 CRC Press LLC
log P
Ref.
Mol. Form.
Name
log P
Ref.
2.16 2.53 2.64 2.38 1.97 1.41 2.3 1.25 0.20 2.5 0.35 -0.54 1.19 0.91 0.51 1.5 -1.51 -0.33 -0.74 -0.57 3.16 2.88 4.00 2.53 2.89 2.13 1.86 1.93 2.39 1.38 2.49 2.38 -0.34 1.79 1.48
2 2 2 2 2 2 2 2 1 2 1 1 2 2 1 2 1 1 1 1 2 2 4 2 2 2 2 2 2 2 2 2 1 2 2
C2H4O C2H4O C2H4O2 C2H5Br C2H5Cl C2H5I C2H5NO C2H5NO2 C2H6O C2H6O C2H6OS C2H6O2S C2H7N C2H7N C3H3N C3H4Cl2 C3H4O C3H4O C3H5Br C3H5ClO C3H5Cl3 C3H5N C3H5NO C3H6Cl2 C3H6O C3H6O C3H6O C3H6O C3H6O2 C3H6O2 C3H7Br C3H7Br C3H7Cl C3H7Cl C3H7I
Acetaldehyde Ethylene oxide Acetic acid Bromoethane Chloroethane Iodoethane Acetamide Nitroethane Ethanol Dimethyl ether Dimethyl sulfoxide Dimethyl sulfone Ethylamine Dimethylamine 2-Propenenitrile cis-1,3-Dichloropropene Propargyl alcohol Acrolein 3-Bromopropene Epichlorohydrin 1,2,3-Trichloropropane Propanenitrile Acrylamide 1,2-Dichloropropane Allyl alcohol Propanal Acetone Methyloxirane Propanoic acid Methyl acetate 1-Bromopropane 2-Bromopropane 1-Chloropropane 2-Chloropropane 1-Iodopropane
0.45 -0.30 -0.17 1.6 1.43 2 -1.26 0.18 -0.30 0.10 -1.35 -1.41 -0.13 -0.38 0.25 2.03 -0.38 -0.01 1.79 0.30 2.63 0.16 -0.78 2.0 0.17 0.59 -0.24 0.03 0.33 0.18 2.1 1.9 2.04 1.90 2.5
1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 2 1 1 1 1 1 1 2 2 1 1 2
OCTANOL-WATER PARTITION COEFFICIENTS (continued) Mol. Form. C3H7N C3H7NO C3H7NO C3H7NO2 C3H8O C3H8O C3H8S C3H9N C3H9N C3H9N C3H9N C4H4O C4H4S C4H5N C4H6 C4H6 C4H6O C4H6O2 C4H6O2 C4H6O2 C4H7N C4H8 C4H8 C4H8 C4H8Cl2O C4H8O C4H8O C4H8O C4H8O C4H8O2 C4H8O2 C4H8O2 C4H9Br C4H9Cl C4H9F C4H9I C4H9N C4H9NO C4H9NO C4H9NO2 C4H10 C4H10O C4H10O C4H10O C4H10O C4H10O C4H10S C4H10S C4H11N C4H11N C4H11N C5H5N C5H6O C5H7N C5H8 C5H8 C5H8O2 C5H8O2 C5H9N C5H10
Name Allylamine N,N-Dimethylformamide N-Methylacetamide 1-Nitropropane 1-Propanol 2-Propanol 1-Propanethiol Propylamine Isopropylamine Ethylmethylamine Trimethylamine Furan Thiophene Pyrrole 1,3-Butadiene 2-Butyne 2,5-Dihydrofuran Methacrylic acid Vinyl acetate Methyl acrylate Butanenitrile cis-2-Butene trans-2-Butene Isobutene Bis(2-chloroethyl) ether Ethyl vinyl ether Butanal 2-Butanone Tetrahydrofuran Butanoic acid Propyl formate Ethyl acetate 1-Bromobutane 1-Chlorobutane 1-Fluorobutane 1-Iodobutane Pyrrolidine Butanamide N,N-Dimethylacetamide 1-Nitrobutane Isobutane 1-Butanol 2-Butanol 2-Methyl-1-propanol 2-Methyl-2-propanol Diethyl ether 1-Butanethiol Diethyl sulfide Butylamine tert-Butylamine Diethylamine Pyridine 2-Methylfuran 1-Methylpyrrole 1,4-Pentadiene 1-Pentyne Methyl methacrylate Ethyl acrylate Pentanenitrile 1-Pentene
log P
Ref.
Mol. Form.
0.03 -1.01 -1.05 0.87 0.25 0.05 1.81 0.48 0.26 0.15 0.16 1.34 1.81 0.75 1.99 1.46 0.46 0.93 0.73 0.80 0.60 2.33 2.31 2.35 1.12 1.04 0.88 0.29 0.46 0.79 0.83 0.73 2.75 2.64 2.58 3 0.46 -0.21 -0.77 1.47 2.8 0.84 0.65 0.76 0.35 0.89 2.28 1.95 0.86 0.40 0.58 0.65 1.85 1.21 2.48 1.98 1.38 1.32 0.94 2.2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2
C5H10 C5H10O C5H10O C5H10O C5H10O C5H10O C5H10O2 C5H10O2 C5H10O2 C5H10O3 C5H11Br C5H11F C5H11N C5H11NO2 C5H12 C5H12 C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H13N C6Cl6 C6HCl5 C6HCl5O C6H2Cl4 C6H2Cl4 C6H2Cl4 C6H3Cl3 C6H3Cl3 C6H3Cl3 C6H4Cl2 C6H4Cl2 C6H4Cl2 C6H4Cl2O C6H5Br C6H5Cl C6H5F C6H5I C6H5NO2 C6H6 C6H6O C6H6S C6H7N C6H7N C6H7N C6H7N C6H8 C6H8O C6H8O C6H8O C6H10 C6H10 C6H10 C6H10O C6H10O C6H10O2
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TeamLRN
Name Cyclopentane 2-Pentanone 3-Pentanone 3-Methyl-2-butanone Tetrahydropyran 2-Methyltetrahydrofuran Pentanoic acid Propyl acetate Ethyl propanoate Diethyl carbonate 1-Bromopentane 1-Fluoropentane Piperidine 1-Nitropentane Pentane Neopentane 1-Pentanol 2-Pentanol 3-Pentanol 3-Methyl-1-butanol 2-Methyl-2-butanol 3-Methyl-2-butanol 2,2-Dimethyl-1-propanol Methyl tert-butyl ether Pentylamine Hexachlorobenzene Pentachlorobenzene Pentachlorophenol 1,2,3,4-Tetrachlorobenzene 1,2,3,5-Tetrachlorobenzene 1,2,4,5-Tetrachlorobenzene 1,2,3-Trichlorobenzene 1,2,4-Trichlorobenzene 1,3,5-Trichlorobenzene o-Dichlorobenzene m-Dichlorobenzene p-Dichlorobenzene 2,4-Dichlorophenol Bromobenzene Chlorobenzene Fluorobenzene Iodobenzene Nitrobenzene Benzene Phenol Benzenethiol Aniline 2-Methylpyridine 3-Methylpyridine 4-Methylpyridine 1,4-Cyclohexadiene 5-Hexyn-2-one 2-Cyclohexen-1-one 2-Ethylfuran 1,5-Hexadiene 1-Hexyne Cyclohexene 5-Hexen-2-one Cyclohexanone Ethyl methacrylate
log P
Ref.
3.00 0.84 0.82 0.56 0.82 1.85 1.39 1.24 1.21 1.21 3.37 2.33 0.84 2.01 3.45 3.11 1.51 1.25 1.21 1.28 0.89 1.28 1.31 0.94 1.49 5.47 5.03 5.07 4.55 4.65 4.51 4.04 3.98 4.02 3.38 3.48 3.38 3.23 2.99 2.84 2.27 3.28 1.85 2.13 1.48 2.52 0.90 1.11 1.20 1.22 2.3 0.58 0.61 2.40 2.8 2.73 2.86 1.02 0.81 1.94
1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 5 4 5 5 5 5 5 5 5 5 5 4 2 1 2 2 1 1 4 1 1 1 1 1 2 1 1 1 2 2 1 1 1 1
OCTANOL-WATER PARTITION COEFFICIENTS (continued) Mol. Form.
Name
log P
Ref.
Mol. Form.
C6H11Br C6H11N C6H12 C6H12 C6H12 C6H12 C6H12O C6H12O C6H12O C6H12O C6H12O2 C6H12O2 C6H13Br C6H13N C6H14 C6H14 C6H14 C6H14 C6H14O C6H14O C6H14O C6H14O C6H14O C6H14O C6H15N C6H15N C6H15N C7H5BrO2 C7H5BrO2 C7H5BrO2 C7H5N C7H6O C7H6O2 C7H6O2 C7H6O3 C7H7Br C7H7Cl C7H7Cl C7H7Cl C7H7Cl C7H7NO2 C7H8 C7H8 C7H8O C7H8O C7H8O C7H8O C7H8O C7H9N C7H9N C7H9N C7H9N C7H9N C7H14 C7H14 C7H14O C7H14O C7H15Br C7H15Cl C7H15I
Bromocyclohexane Hexanenitrile 1-Hexene 4-Methyl-1-pentene Cyclohexane Methylcyclopentane Cyclohexanol Hexanal 2-Hexanone 4-Methyl-2-pentanone Hexanoic acid Butyl acetate 1-Bromohexane Cyclohexylamine Hexane 3-Methylpentane 2,2-Dimethylbutane 2,3-Dimethylbutane 1-Hexanol 2-Hexanol 3-Hexanol 3,3-Dimethyl-2-butanol Dipropyl ether Diisopropyl ether Hexylamine Dipropylamine Triethylamine 2-Bromobenzoic acid 3-Bromobenzoic acid 4-Bromobenzoic acid Benzonitrile Benzaldehyde Benzoic acid Phenyl formate Salicylic acid (Bromomethyl)benzene o-Chlorotoluene m-Chlorotoluene p-Chlorotoluene (Chloromethyl)benzene p-Nitrotoluene Toluene 1,3,5-Cycloheptatriene o-Cresol m-Cresol p-Cresol Benzyl alcohol Anisole Benzylamine o-Methylaniline m-Methylaniline p-Methylaniline N-Methylaniline 1-Heptene Methylcyclohexane 2-Heptanone 5-Methyl-2-hexanone 1-Bromoheptane 1-Chloroheptane 1-Iodoheptane
3.20 1.66 3.40 2.5 3.44 3.37 1.23 1.78 1.38 1.31 1.92 1.82 3.80 1.49 4.00 3.60 3.82 3.85 2.03 1.76 1.65 1.48 2.03 1.52 2.06 1.67 1.45 2.20 2.87 2.86 1.56 1.48 1.88 1.26 2.20 2.92 3.42 3.28 3.33 2.30 2.42 2.73 2.63 1.98 1.98 1.97 1.05 2.11 1.09 1.32 1.40 1.39 1.66 3.99 3.88 1.98 1.88 4.36 4.15 4.70
1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 4 4 4 1 1 4 1 4 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
C7H16 C7H16O C7H16O C7H16O C7H16O C7H17N C8H6 C8H6O C8H6S C8H7N C8H7N C8H8 C8H8O C8H8O C8H8O C8H8O C8H8O C8H8O2 C8H8O2 C8H8O2 C8H8O2 C8H8O2 C8H8O2 C8H10 C8H10 C8H10 C8H10 C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H10O C8H11N C8H11N C8H11N C8H14O2 C8H15N C8H16 C8H16 C8H16O C8H16O2 C8H17Br C8H18 C8H18O C8H18O C8H18O C8H18O C9H7N
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Name Heptane 1-Heptanol 2-Heptanol 3-Heptanol 4-Heptanol Heptylamine Phenylacetylene Benzofuran Benzo[b]thiophene Benzeneacetonitrile Indole Styrene Acetophenone 2-Methylbenzaldehyde Benzeneacetaldehyde 2,3-Dihydrobenzofuran Phenyloxirane o-Toluic acid m-Toluic acid p-Toluic acid Benzeneacetic acid Phenyl acetate Methyl benzoate Ethylbenzene o-Xylene m-Xylene p-Xylene o-Ethylphenol m-Ethylphenol p-Ethylphenol 2,4-Xylenol 2,5-Xylenol 2,6-Xylenol 3,4-Xylenol 3,5-Xylenol Benzeneethanol α-Methylbenzyl alcohol 3-Methylbenzenemethanol 4-Methylbenzenemethanol Phenetole Benzyl methyl ether 2-Methylanisole 3-Methylanisole 4-Methylanisole p-Ethylaniline N,N-Dimethylaniline Benzeneethanamine Butyl methacrylate Octanenitrile 1-Octene Cyclooctane 2-Octanone Octanoic acid 1-Bromooctane Octane 1-Octanol 2-Octanol 4-Octanol Dibutyl ether Quinoline
log P
Ref.
4.50 2.62 2.31 2.24 2.22 2.57 2.40 2.67 3.12 1.56 2.14 3.05 1.63 2.26 1.78 2.14 1.61 2.32 2.37 2.34 1.41 1.49 2.20 3.15 3.12 3.20 3.15 2.47 2.50 2.50 2.35 2.34 2.36 3.23 2.35 1.36 1.42 1.60 1.58 2.51 1.35 2.74 2.66 2.81 1.96 2.31 1.41 2.88 2.75 4.57 4.45 2.37 3.05 4.89 5.15 3.07 2.90 2.68 3.21 2.03
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1
OCTANOL-WATER PARTITION COEFFICIENTS (continued) Mol. Form. C9H7N C9H8 C9H8O2 C9H9N C9H10 C9H10O C9H10O C9H10O C9H10O2 C9H10O2 C9H10O2 C9H10O2 C9H12 C9H12 C9H12 C9H12 C9H12 C9H12 C9H12 C9H12O C9H12O C9H12O C9H12O C9H12O C9H13N C9H13N C9H18 C9H18O C9H18O C9H20 C9H20O C9H21N C10H7Cl C10H7Cl C10H8 C10H8 C10H8O C10H8O C10H12O2 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14O C10H20O C10H20O2 C10H22 C10H22O C11H9N C11H10 C11H10 C11H16 C11H16 C11H22O C11H22O2 C12Cl10 C12HCl9
Name Isoquinoline Indene trans-Cinnamic acid Benzenepropanenitrile Indan 1-Phenyl-1-propanone 1-Phenyl-2-propanone 4-Methylacetophenone 2-Phenylpropanoic acid Benzyl acetate 4-Methylphenyl acetate Ethyl benzoate Propylbenzene Isopropylbenzene o-Ethyltoluene p-Ethyltoluene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene 2-Propylphenol 4-Propylphenol 2,3,6-Trimethylphenol 2,4,6-Trimethylphenol Benzenepropanol N,N-Dimethylbenzylamine Amphetamine 1-Nonene 2-Nonanone 5-Methyl-2-octanone Nonane 1-Nonanol Tripropylamine 1-Chloronaphthalene 2-Chloronaphthalene Naphthalene Azulene 1-Naphthol 2-Naphthol Isopropyl benzoate Butylbenzene tert-Butylbenzene Isobutylbenzene p-Cymene 1,2,4,5-Tetramethylbenzene 1,2,3,4-Tetramethylbenzene 1,2,3,5-Tetramethylbenzene 4-Butylphenol 2-Decanone Decanoic acid Decane 1-Decanol 4-Phenylpyridine 1-Methylnaphthalene 2-Methylnaphthalene Pentylbenzene Pentamethylbenzene 2-Undecanone Methyl decanoate Decachlorobiphenyl 2,2′,3,3′,4,5,5′,6,6′Nonachlorobiphenyl
log P
Ref.
Mol. Form.
2.08 2.92 2.13 1.72 3.33 2.19 1.44 2.19 1.80 1.96 2.11 2.64 3.69 3.66 3.53 3.63 3.60 3.63 3.42 2.93 3.20 2.67 2.46 1.88 1.98 1.76 5.15 3.16 2.92 5.65 4.02 2.79 3.90 3.98 3.34 3.22 2.84 2.70 3.18 4.26 4.11 4.01 4.10 4.10 4.00 4.10 3.65 3.77 4.09 6.25 4.57 2.59 3.87 4.00 4.90 4.56 4.09 4.41 8.26
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3
C12H2Cl8
8.16
3
C12H3Cl7 C12H4Cl6 C12H4Cl6 C12H4Cl6 C12H5Cl5 C12H5Cl5 C12H6Cl4 C12H6Cl4 C12H7Cl3 C12H7Cl3 C12H8Cl2 C12H8Cl2 C12H8O C12H9Cl C12H9Cl C12H9Cl C12H9N C12H10 C12H10 C12H10N2 C12H10O C12H10S C12H11N C12H12 C12H12 C12H12 C12H14O C12H18 C12H18 C12H22O C12H24O2 C12H26O C13H8O C13H9N C13H10 C13H10O C13H10O2 C13H11NO C13H12 C13H12 C13H12O C13H12O C14H10 C14H10 C14H12 C14H12 C14H12O C14H12O2 C14H14 C14H14 C14H22 C14H28O2 C15H12 C15H12 C15H12 C16H10 C16H10 C16H14 C16H32O2
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TeamLRN
Name 2,2′,3,3′,5,5′,6,6′Octachlorobiphenyl 2,2′,3,3′,4,4′,6-Heptachlorobiphenyl 2,2′,3,3′,4,4′-Hexachlorobiphenyl 2,2′,4,4′,6,6′-Hexachlorobiphenyl 2,2′,3,3′,6,6′-Hexachlorobiphenyl 2,3,4,5,6-Pentachlorobiphenyl 2,2′,4,5,5′-Pentachlorobiphenyl 2,3,4,5-Tetrachlorobiphenyl 2,2′,4′,5-Tetrachlorobiphenyl 2,4,5-Trichlorobiphenyl 2,4,6-Trichlorobiphenyl 2,5-Dichlorobiphenyl 2,6-Dichlorobiphenyl Dibenzofuran 2-Chlorobiphenyl 3-Chlorobiphenyl 4-Chlorobiphenyl Carbazole Acenaphthene Biphenyl Azobenzene Diphenyl ether Diphenyl sulfide Diphenylamine 1-Ethylnaphthalene 1,2-Dimethylnaphthalene 1,4-Dimethylnaphthalene 4-Phenylcyclohexanone Hexylbenzene Hexamethylbenzene Cyclododecanone Dodecanoic acid 1-Dodecanol 9H-Fluoren-9-one Acridine 9H-Fluorene Benzophenone Phenyl benzoate N-Phenylbenzamide Diphenylmethane 4-Methylbiphenyl Diphenylmethanol Benzyl phenyl ether Anthracene Phenanthrene trans-Stilbene 1-Methylfluorene 2-Phenylacetophenone Benzyl benzoate 1,2-Diphenylethane 4,4′-Dimethylbiphenyl Octylbenzene Tetradecanoic acid 2-Methylanthracene 9-Methylanthracene 1-Methylphenanthrene Fluoranthene Pyrene 9,10-Dimethylanthracene Hexadecanoic acid
log P
Ref.
7.10 6.70 7.00 7.00 6.70 6.30 6.40 5.72 5.73 5.60 5.47 5.10 5.00 4.12 4.52 4.58 4.61 3.72 3.96 3.76 3.82 4.21 4.45 3.44 4.40 4.31 4.37 2.45 5.52 4.69 4.10 4.6 5.13 3.58 3.40 4.20 3.18 3.59 2.62 4.14 4.63 2.67 3.79 4.56 4.52 4.81 4.97 3.18 3.97 4.70 5.09 6.30 6.1 5.15 5.07 5.14 5.07 5.08 5.69 7.17
3 3 3 3 3 3 3 3 7 3 3 3 3 1 1 1 1 1 4 6 1 1 1 4 1 1 1 1 1 4 1 1 1 1 1 4 1 1 1 1 1 1 1 4 4 1 1 1 1 1 1 1 1 2 1 2 4 4 1 1
OCTANOL-WATER PARTITION COEFFICIENTS (continued) Mol. Form.
Name
log P
Ref.
Mol. Form.
C17H12 C17H12 C18H12 C18H12 C18H12 C18H12 C18H15N C18H30O2 C18H32O2 C18H34O2
11H-Benzo[a]fluorene 11H-Benzo[b]fluorene Benz[a]anthracene Chrysene Naphthacene Triphenylene Triphenylamine Linolenic acid Linoleic acid Oleic acid
5.40 5.75 5.91 5.73 5.76 5.49 5.74 6.46 7.05 7.64
1 1 1 4 1 4 1 1 1 1
C18H36O2 C19H16O C20H12 C20H12 C20H32O2 C20H40O2 C21H16
© 2000 CRC Press LLC
C22H12 C24H12
Name Stearic acid Triphenylmethanol Perylene Benzo[a]pyrene Arachidonic acid Arachidic acid 1,2-Dihydro-3-methylbenz[j] aceanthrylene Benzo[ghi]perylene Coronene
log P
Ref.
8.23 3.68 6.25 6.20 6.98 9.29
1 1 1 4 1 1
6.75 6.90 6.05
1 1 4
PROTECTION AGAINST IONIZING RADIATION The following data and rules of thumb are helpful in estimating the penetrating capability of and danger of exposure to various types of ionizing radiation. More precise data should be used for critical applications.
Alpha Particles Alpha particles of at least 7.5 MeV are required to penetrate the epidermis, the protective layer of skin, 0.07 mm thick.
Electrons Electrons of at least 70 keV are required to penetrate the epidermis, the protective layer of skin, 0.07 mm thick. The range of electrons in g/cm2 is approximately equal to the maximum energy (E) in MeV divided by 2. The range of electrons in air is about 3.65 m per MeV; for example, a 3 MeV electron has a range of about 11 m in air. A chamber wall thickness of 30 mg/cm2 will transmit 70% of the initial fluence of 1 MeV electrons and 20% of that of 0.4 MeV electrons. When electrons of 1 to 2 MeV pass through light materials such as water, aluminum, or glass, less than 1% of their energy is dissipated as bremsstrahlung. The bremsstrahlung from 1 Ci of 32P aqueous solution in a glass bottle is about 1 mR/h at 1 meter distance. When electrons from a 1 Ci source of 90Sr - 90Y are absorbed, the bremsstrahlung hazard is approximately equal to that presented by the gamma radiation from 12 mg of radium. The average energy of the bremsstrahlung is about 300 keV.
Gamma Rays The air-scattered radiation (sky-shine) from a 100 Ci 60Co source placed 1 ft behind a 4 ft high shield is about 100 mrad/h at 6 ft from the outside of the shield. Within ±20% for point source gamma emitters with energies between 0.07 and 4 MeV, the exposure rate (R/h) at 1 ft is 6C⋅E⋅n where C is the activity in curies, E is the energy in MeV, and n is the number of gammas per disintegration.
Neutrons An approximate HVL (thickness of absorber for which the neutron flux falls to half its initial value) for 1 MeV neutrons is 3.2 cm of paraffin; that for 5 MeV neutrons is 6.9 cm of paraffin).
Miscellaneous The activity of any radionuclide is reduced to less than 1% after 7 half-lives (i.e., 2-7 = 0.8%). For nuclides with a half-life greater than 6 days, the change in activity in 24 hours will be less than 10%. 10 HVL (half-value layers) attenuates approximately by 10-3. There is 0.64 mm3 of radon gas at STP in transient equilibrium with 1 Ci of radium. The natural background from all sources in most parts of the world leads to an equivalent dose rate of about 0.04 to 4 mSv per year for the average person. About 84% of this comes from terrestrial sources, the remainder from cosmic rays. The U. S. average is about 3.6 mSv/yr but can range up to 50 mSv/yr in some areas. A passenger in a plane flying at 12,000 meters receives 5 µSv/hr from cosmic rays (as compared to about 0.03 µSv/hr at sea level). The ICRP recommended exposure limit to man-made sources of ionizing radiation (Reference 2) is 20 mSv/yr averaged over 5 years, with the dose in any one year not to exceed 50 mSv. A whole-body dose of about 3 Gy over a short time interval will typically lead to 50% mortality in 30 days assuming no medical treatment.
Units The gray (Gy) is the SI unit of absorbed dose; it is a measure of the mean energy imparted to a sample of irradiated matter, divided by the mass of the sample. Gy is a special name for the SI unit J/kg. The sievert (Sv) is the SI unit of equivalent dose, which is defined as the absorbed dose multiplied by a weighting factor that expresses the longterm biological risk from low-level chronic exposure to a specified type of radiation. The Sv is another special name for J/kg. 1 curie (Ci) = 3.7⋅1010 becquerel (Bq); i.e., 3.7⋅1010 disintegrations per second. 1 roentgen (R) = 2.58⋅10-4 coulomb per kilogram (C/kg); a measure of the charge (positive or negative) liberated by x-ray or gamma radiation in air, divided by the mass of air. 1 rad = 0.01 Gy 1 rem = 0.01 Sv REFERENCES 1. Padikal, T.N., and Fivozinsky, S.P., Medical Physics Data Book, National Bureau of Standards Handbook 138, U. S. Government Printing Office, Washington, D.C., 1981. 2. 1990 Recommendations of the International Commission on Radiological Protection, ICRP Publication 60, Annals of the ICRP, Pergamon Press, Oxford, 1991. 3. Radiation: Doses, Effects, Risks, United Nations Sales No. E.86.III.D.4, 1985. 4. Review of Particle Properties, Phys. Rev. D, 50, 1173, 1994 (p. 1268).
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ANNUAL LIMITS ON INTAKES OF RADIONUCLIDES K. F. Eckerman The following table lists, for workers, the annual limits on oral and inhalation intakes (ALI) for selected radionuclides based on the occupational radiation protection guidance of the International Commission on Radiological Protection (References 1 and 2). An intake of one ALI corresponds to an annual whole body dose of 0.02 Sv (2 rem). The ALI is expressed in the SI unit of activity, the becquerel (Bq), and in the conventional unit, the microcurie (µCi); 1 µCi = 3.7⋅104 Bq. The chemical form of inhaled radionuclides is, in most instances, stated in terms of the rate of absorption to blood from the lungs and the fractional absorption from the small intestine. Type F, M, and S denote chemical forms which are absorbed from the lungs at rates characterized as fast, moderate, and slow, respectively. The time to absorb 90% of the deposited radionuclide, in the absence of radioactive decay, corresponds to about 10 minutes, 150 days, and 7000 days for Type F, M, and S compounds, respectively. Type F compounds can be considered to be more soluble than M or S, S being the most insoluble. Chemical form consideration for ingestion is specified by the fractional absorption from the small intestine, denoted as f1. The f1 values range from 10–5 to 1. Higher fractional absorption is associated with greater solubility of the compound. REFERENCES 1. 1990 Recommendations of the International Commission on Radiological Protection, ICRP Publication 60, Annals of the ICRP 21, (1—3), Pergamon Press, Oxford, 1991. 2. Dose Coefficients for Intakes of Radionuclides by Workers, ICRP Publication 68, Annals of the ICRP, 24(4), Pergamon Press, Oxford, 1995.
Inhalation intakes Physical half-life 3H
12.3 y
11C
0.340 h
14C
5730 y
18F
1.83 h
22Na 32P
2.60 y 15.0 h 14.3 d
35S
87.4 d
24Na
42K
51Cr
12.4 h 22.6 h 163 d 4.53 d 27.7 d
54Mn
312 d
52Fe
8.28 h
55Fe
2.70 y
43K 45Ca 47Ca
Chemical form Type/f1 HT gas HTO vapor CO CO2 Organic compounds CO CO2 Organic compounds F 1.000 M 1.000 S 1.000 F 1.000 F 1.000 F 0.800 M 0.800 Inorganic compounds F 0.800 M 0.800 Vapor Organic compounds F 1.000 F 1.000 M 0.300 M 0.300 F 0.100 M 0.100 S 0.100 F 0.100 M 0.100 F 0.100 M 0.100 F 0.100 M 0.100
Oral intakes
ALI
Chemical form f1
Bq
µCi
ALI
Bq
µCi
1.1E+13 1.1E+09 1.7E+10 9.1E+09 6.2E+09
3.0E+08 3.0E+04 4.5E+05 2.5E+05 1.7E+05
1.000
1.1E+13
3.0E+08
1.000
8.3E+08
2.3E+04
2.5E+10 3.1E+09 3.4E+07
6.8E+05 8.3E+04 9.3E+02
1.000
3.4E+07
9.3E+02
3.7E+08 2.2E+08 2.2E+08 1.0E+07 3.8E+07 1.8E+07 6.9E+06
1.0E+04 6.1E+03 5.8E+03 2.7E+02 1.0E+03 4.9E+02 1.9E+02
1.000
4.1E+08
1.1E+04
1.000 1.000 0.800
6.3E+06 4.7E+07 8.3E+06
1.7E+02 1.3E+03 2.3E+02
2.5E+08 1.8E+07 1.7E+08
6.8E+03 4.9E+02 4.5E+03
0.800 0.100
1.4E+08 1.1E+08
3.9E+03 2.8E+03
1.000
2.6E+07
7.0E+02
1.000 1.000 0.300 0.300 0.100 0.010
4.7E+07 8.0E+07 2.6E+07 1.3E+07 5.3E+08 5.4E+08
1.3E+03 2.2E+03 7.1E+02 3.4E+02 1.4E+04 1.5E+04
0.100
2.8E+07
7.6E+02
0.100
1.4E+07
3.9E+02
0.100
6.1E+07
1.6E+03
1.0E+08 7.7E+07 8.7E+06 9.5E+06 6.7E+08 5.9E+08 5.6E+08 1.8E+07 1.7E+07 2.9E+07 2.1E+07 2.2E+07 6.1E+07
2.7E+03 2.1E+03 2.4E+02 2.6E+02 1.8E+04 1.6E+04 1.5E+04 4.9E+02 4.5E+02 7.8E+02 5.7E+02 5.9E+02 1.6E+03
16-48
ANNUAL LIMITS ON INTAKES OF RADIONUCLIDES (continued) Inhalation intakes Physical half-life 59Fe
44.5 d
57Co
271 d
58Co
70.8 d
60Co
5.27 y
64Cu
12.7 h
59Ni
75000 y
63Ni
96.0 y
65Zn 67Ga
244 d 3.26 d
68Ga
1.13 h
68Ge
288 d
75Se
120 d
79Se
65000 y
86Rb 85Sr
18.6 d 64.8 d
87mSr
2.80 h
89Sr
50.5 d
90Sr
29.1 y
99Mo
2.75 d
99mTc
6.02 h
99Tc
213000 y
106Ru
1.01 y
111In
2.83 d
113mIn
1.66 h
113Sn
115 d
123I
13.2 h
125I
60.1 d
129I
1.57⋅107 y
Chemical form Type/f1 F 0.100 M 0.100 M 0.100 S 0.050 M 0.100 S 0.050 M 0.100 S 0.050 F 0.500 M 0.500 S 0.500 F 0.050 M 0.050 Vapor F 0.050 M 0.050 Vapor S 0.500 F 0.001 M 0.001 F 0.001 M 0.001 F 1.000 M 1.000 F 0.800 M 0.800 F 0.800 M 0.800 F 1.000 F 0.300 S 0.010 F 0.300 S 0.010 F 0.300 S 0.010 F 0.300 S 0.010 F 0.800 S 0.050 F 0.800 M 0.800 F 0.800 M 0.800 F 0.050 M 0.050 S 0.050 F 0.020 M 0.020 F 0.020 M 0.020 F 0.020 M 0.020 F 1.000 Vapor F 1.000 Vapor F 1.000
Oral intakes
ALI Bq
µCi
6.7E+06 6.3E+06 5.1E+07 3.3E+07 1.4E+07 1.2E+07 2.8E+06 1.2E+06 2.9E+08 1.3E+08 1.3E+08 9.1E+07 2.1E+08 2.4E+07 3.8E+07 6.5E+07 1.0E+07 7.1E+06 1.8E+08 7.1E+07 4.1E+08 2.5E+08 2.4E+07 2.5E+06 1.4E+07 1.2E+07 1.3E+07 6.5E+06 1.5E+07 3.6E+07 3.1E+07 9.1E+08 5.7E+08 1.4E+07 3.6E+06 6.7E+05 2.6E+05 5.6E+07 1.8E+07 1.0E+09 6.9E+08 5.0E+07 6.3E+06 2.0E+06 1.2E+06 5.7E+05 9.1E+07 6.5E+07 1.1E+09 6.3E+08 2.5E+07 1.1E+07 1.8E+08 9.5E+07 2.7E+06 1.4E+06 3.9E+05
1.8E+02 1.7E+02 1.4E+03 9.0E+02 3.9E+02 3.2E+02 7.6E+01 3.2E+01 7.9E+03 3.6E+03 3.6E+03 2.5E+03 5.8E+03 6.5E+02 1.0E+03 1.7E+03 2.7E+02 1.9E+02 4.9E+03 1.9E+03 1.1E+04 6.7E+03 6.5E+02 6.8E+01 3.9E+02 3.2E+02 3.4E+02 1.7E+02 4.2E+02 9.7E+02 8.4E+02 2.5E+04 1.5E+04 3.9E+02 9.7E+01 1.8E+01 7.0E+00 1.5E+03 4.9E+02 2.7E+04 1.9E+04 1.4E+03 1.7E+02 5.5E+01 3.2E+01 1.5E+01 2.5E+03 1.7E+03 2.8E+04 1.7E+04 6.8E+02 2.8E+02 4.9E+03 2.6E+03 7.4E+01 3.9E+01 1.1E+01
16-49
TeamLRN
Chemical form f1
Bq
µCi
0.100
1.1E+07
3.0E+02
0.100 0.050 0.100 0.050 0.100 0.050 0.500
9.5E+07 1.1E+08 2.7E+07 2.9E+07 5.9E+06 8.0E+06 1.7E+08
2.6E+03 2.8E+03 7.3E+02 7.7E+02 1.6E+02 2.2E+02 4.5E+03
0.050
3.2E+08
8.6E+03
0.050
1.3E+08
3.6E+03
0.500 0.001
5.1E+06 1.1E+08
1.4E+02 2.8E+03
0.001
2.0E+08
5.4E+03
1.000
1.5E+07
4.2E+02
0.800 0.050 0.800 0.050 1.000 0.300 0.010 0.300 0.010 0.300 0.010 0.300 0.010 0.800 0.050 0.800
7.7E+06 4.9E+07 6.9E+06 5.1E+07 7.1E+06 3.6E+07 6.1E+07 6.7E+08 6.1E+08 7.7E+06 8.7E+06 7.1E+05 7.4E+06 2.7E+07 1.7E+07 9.1E+08
2.1E+02 1.3E+03 1.9E+02 1.4E+03 1.9E+02 9.7E+02 1.6E+03 1.8E+04 1.6E+04 2.1E+02 2.4E+02 1.9E+01 2.0E+02 7.3E+02 4.5E+02 2.5E+04
0.800
2.6E+07
6.9E+02
0.050
2.9E+06
7.7E+01
0.020
6.9E+07
1.9E+03
0.020
7.1E+08
1.9E+04
0.020
2.7E+07
7.4E+02
1.000
9.5E+07
2.6E+03
1.000
1.3E+06
3.6E+01
1.000
1.8E+05
4.9E+00
ALI
ANNUAL LIMITS ON INTAKES OF RADIONUCLIDES (continued) Inhalation intakes Physical half-life 131I
8.04 d
129Cs
141Ce
1.34 d 2.06 y 13.1 d 30.0 y 32.5 d
144Ce
284 d
133Ba 169Yb
10.7 y 12.7 d 32.0 d
198Au
2.69 d
198mAu
2.30 d
197Hg
2.67 d
134Cs 136Cs 137Cs
140Ba
203Hg
201Tl
46.6 d
207Bi
3.04 d 22.3 y 38.0 y
210Po
138 d
224Ra
228Th
3.66 d 1600 y 5.75 y 1.91 y
230Th
77000 y
232Th
1.40⋅1010 y
234U
2.44⋅105 y
210Pb
226Ra 228Ra
Chemical form Type/f1
Oral intakes
ALI
Chemical form f1
Bq
µCi
1.000
9.1E+05
2.5E+01
1.000 1.000 1.000 1.000 5.0E-04
3.3E+08 1.1E+06 6.7E+06 1.5E+06 2.8E+07
9.0E+03 2.8E+01 1.8E+02 4.2E+01 7.6E+02
5.0E-04
3.8E+06
1.0E+02
0.100 0.100 5.0E-04
2.0E+07 8.0E+06 2.8E+07
5.4E+02 2.2E+02 7.6E+02
0.100
2.0E+07
5.4E+02
0.100
1.5E+07
4.2E+02
1.000 0.400
2.0E+08 1.2E+08
5.5E+03 3.2E+03
ALI
Bq
µCi
Vapor F 1.000 Vapor F 1.000 F 1.000 F 1.000 F 1.000 M 5.0E-04 S 5.0E-04 M 5.0E-04 S 5.0E-04 F 0.100 F 0.100 M 5.0E-04 S 5.0E-04 F 0.100 M 0.100 S 0.100 F 0.100 M 0.100 S 0.100 Inorganic compounds F 0.400
2.1E+05 1.8E+06 1.0E+06 2.5E+08 2.1E+06 1.1E+07 3.0E+06 7.4E+06 6.5E+06 8.7E+05 6.9E+05 1.1E+07 1.3E+07 9.5E+06 8.3E+06 5.1E+07 2.0E+07 1.8E+07 3.4E+07 1.0E+07 1.1E+07
5.6E+00 4.9E+01 2.7E+01 6.7E+03 5.6E+01 2.8E+02 8.1E+01 2.0E+02 1.7E+02 2.4E+01 1.9E+01 3.0E+02 3.4E+02 2.6E+02 2.3E+02 1.4E+03 5.5E+02 4.9E+02 9.2E+02 2.7E+02 2.8E+02
2.4E+08
6.4E+03
Vapor Organic compounds F 0.020 M 0.020 Inorganic compounds F 0.400
4.5E+06
1.2E+02
2.0E+08 7.1E+07
5.4E+03 1.9E+03
0.020
8.7E+07
2.4E+03
2.7E+07
7.2E+02
1.000 0.400
1.1E+07 1.8E+07
2.8E+02 4.9E+02
Vapor Organic compounds F 0.020 M 0.020 F 1.000 F 0.200 F 0.050 M 0.050 F 0.100 M 0.100 M 0.200 M 0.200 M 0.200 M 5.0E-04 S 2.0E-04 M 5.0E-04 S 2.0E-04 M 5.0E-04 S 2.0E-04 F 0.020 M 0.020 S 0.002
2.9E+06
7.7E+01
3.4E+07 1.1E+07 2.6E+08 1.8E+04 2.4E+07 6.3E+06 2.8E+04 9.1E+03 8.3E+03 1.7E+03 1.2E+04 8.7E+02 6.3E+02 7.1E+02 2.8E+03 6.9E+02 1.7E+03 3.1E+04 9.5E+03 2.9E+03
9.2E+02 2.8E+02 7.1E+03 4.9E-01 6.4E+02 1.7E+02 7.6E-01 2.5E-01 2.3E-01 4.5E-02 3.2E-01 2.4E-02 1.7E-02 1.9E-02 7.5E-02 1.9E-02 4.5E-02 8.4E-01 2.6E-01 7.9E-02
0.020
3.7E+07
1.0E+03
1.000 0.200 0.050
2.1E+08 2.9E+04 1.5E+07
5.7E+03 7.9E-01 4.2E+02
0.100
8.3E+04
2.3E+00
0.200 0.200 0.200 5.0E-04 2.0E-04 5.0E-04 2.0E-04 5.0E-04 2.0E-04 0.020 0.002
3.1E+05 7.1E+04 3.0E+04 2.9E+05 5.7E+05 9.5E+04 2.3E+05 9.1E+04 2.2E+05 4.1E+05 2.4E+06
8.3E+00 1.9E+00 8.1E-01 7.7E+00 1.5E+01 2.6E+00 6.2E+00 2.5E+00 5.9E+00 1.1E+01 6.5E+01
16-50
ANNUAL LIMITS ON INTAKES OF RADIONUCLIDES (continued) Inhalation intakes Physical half-life
Oral intakes
Chemical form Type/f1
Bq
µCi
F 0.020 M 0.020 S 0.002 F 0.020 M 0.020 S 0.002 M 5.0E-04 M 5.0E-04 M 5.0E-04 S 1.0E-05
3.3E+04 1.1E+04 3.3E+03 3.4E+04 1.3E+04 3.5E+03 1.3E+03 1.8E+07 6.7E+02 1.8E+03
9.0E-01 3.0E-01 8.9E-02 9.3E-01 3.4E-01 9.5E-02 3.6E-02 4.9E+02 1.8E-02 4.9E-02
ALI
235U
7.04⋅108 y
238U
4.47⋅109 y
237Np 238Pu
2.14⋅106 y 2.36 d 87.7 y
239Pu
24100 y
M 5.0E-04 S 1.0E-05
6.3E+02 2.4E+03
1.7E-02 6.5E-02
241Pu
14.4 y
M 5.0E-04 S 1.0E-05
3.4E+04 2.4E+05
9.3E-01 6.4E+00
241Am
432 y 18.1 y 2.64 y
M 5.0E-04 M 5.0E-04 M 5.0E-04
7.4E+02 1.2E+03 1.5E+03
2.0E-02 3.2E-02 4.2E-02
239Np
244Cm 252Cf
16-51
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Chemical form f1
Bq
µCi
0.020 0.002
4.3E+05 2.4E+06
1.2E+01 6.5E+01
0.020 0.002
4.5E+05 2.6E+06
1.2E+01 7.1E+01
5.0E-04 5.0E-04 5.0E-04 1.0E-05 1.0E-04 5.0E-04 1.0E-05 1.0E-04 5.0E-04 1.0E-05 1.0E-04 5.0E-04 5.0E-04 5.0E-04
1.8E+05 2.5E+07 8.7E+04 2.3E+06 4.1E+05 8.0E+04 2.2E+06 3.8E+05 4.3E+06 1.8E+08 2.1E+07 1.0E+05 1.7E+05 2.2E+05
4.9E+00 6.8E+02 2.4E+00 6.1E+01 1.1E+01 2.2E+00 6.0E+01 1.0E+01 1.2E+02 4.9E+03 5.6E+02 2.7E+00 4.5E+00 6.0E+00
ALI
CHEMICAL CARCINOGENS The following substances are listed in the 10th Report on Carcinogens, released in December 2002 by the National Institute of Environmental Health Sciences (NIEHS) under the National Toxicology Program (NTP). Substances are grouped in two classes: - Known to be human carcinogens: There is sufficient evidence of carcinogenicity from studies in humans which indicates a causal relationship between exposure to the substance and human cancer. - Reasonably anticipated to be human carcinogens: There is limited evidence of carcinogenicity from studies in humans which indicates that causal interpretation is credible, but that alternative explanations, such as chance, bias, or confounding factors, could not be adequately excluded; or there is sufficient evidence of carcinogenicity from studies in experimental animals. The NTP report also lists many poorly defined materials such as soots, tars, mineral oils, coke oven emissions, etc. These materials are not included here. The table lists the name normally used in the Handbook of Chemistry and Physics, followed by additional names by which the substance is known. In many cases the primary name given here is different from that used in the NTP report; however, names used in the NTP report appear in the Other names column. The Chemical Abstracts Service Registry Number (CAS RN), given in the last column, is taken from the NTP report. Extensive details on each substance are given in the reference. REFERENCE Public Health Service, National Toxicology Program, 10th Report on Carcinogens, available on the Internet at http://ehp.niehs.nih.gov/roc/toc10.html
Substance Known to be Human Carcinogens Aflatoxins 4-Aminobiphenyl Arsenic compounds, inorganic Asbestos Azathioprine Benzene p-Benzidine Beryllium and beryllium compounds Bis(2-chloroethyl) sulfide Bis(chloromethyl) ether 1,3-Butadiene 1,4-Butanediol dimethylsulfonate Cadmium and cadmium compounds Chlorambucil Chloroethene 1-(2-Chloroethyl)-3-(4-methylcyclohexyl)1-nitrosourea Chloromethyl methyl ether Chromium hexavalent compounds Cyclophosphamide Cyclosporin A Diethylstilbestrol Erionite Estrogens, steroidal Melphalan Methoxsalen (with UV therapy) 2-Naphthylamine Nickel compounds Oxirane Radon Silicon dioxide (respirable size) Silicon dioxide (respirable size) Silicon dioxide (respirable size) Tamoxifen 2,3,7,8-Tetrachlorodibenzo-p-dioxin Thorium(IV) oxide Triethylenethiophosphoramide
Other names
CAS RN
1402-68-2 92-67-1
p-Biphenylamine
1H-Purine, 6-[(1-methyl-4-nitro-1H-imidazol-5-yl)thio][1,1'-Biphenyl]-4,4'-diamine Mustard gas
Myleran; Busulfan
Vinyl chloride; Chloroethylene
1332-21-4 446-86-6 71-43-2 92-87-5 7440-41-7 505-60-2 542-88-1 106-99-0 55-98-1 7440-43-9 305-03-3 75-01-4
MeCCNU
13909-09-6 107-30-2
2H-1,3,2-Oxazaphosphorin-2-amine, N,N-bis(2-chloroethyl)tetrahydro-, 2-oxide Cyclosporine
50-18-0 59865-13-3 56-53-1 66733-21-9
L-Phenylalanine, 4-[bis(2-chloroethyl)amino]PUVA; 9-Methoxy-7H-furo[3,2-g][1]benzopyran-7-one 2-Aminonaphthalene; β -Naphthylamine
148-82-3 298-81-7 91-59-8
Ethylene oxide
75-21-8 10043-92-2 14808-60-7 14464-46-1 15468-32-3 10540-29-1 1746-01-6 1314-20-1 52-24-4
Quartz; Silica Cristobalite; Silica Tridymite; Silica TCDD; Dioxin Thorium dioxide Thiotepa; Tris(1-aziridinyl)phosphine, sulfide
16-53
CHEMICAL CARCINOGENS (continued) Substance
Other names
Reasonably Anticipated to be Human Carcinogens Acetaldehyde Ethanal 2-(Acetylamino)fluorene Acrylamide 2-Propenamide Acrylonitrile Propenenitrile Adriamycin Doxorubicin 2-Amino-9,10-anthracenedione 2-Aminoanthraquinone 1-Amino-2-methyl-9,10-anthracenedione 1-Amino-2-methylanthraquinone 2-Amino-3-methyl-3H-imidazo[4,5-f]quinoline IQ Azacitidine 5-Azacytidine; 1,3,5-Triazine-2(1H)-one, 4-amino-1-beta-D-ribofuranosylBenz[a]anthracene Benzo[b]fluoranthene Benz[e]acephenanthrylene Benzo[j]fluoranthene Benzo[k]fluoranthene 2,3,1',8'-Binaphthylene Benzo[a]pyrene 2,2'-Bioxirane Diepoxybutane Bis(4-amino-3-chlorophenyl)methane 4,4-Methylene-bis(2-chloraniline); MBOCA 2,2-Bis(bromomethyl)-1,3-propanediol BBMP; Pentaerythritol dibromide Bis(2-chloroethyl)methylamine hydrochloride Nitrogen mustard hydrochloride N,N’-Bis(2-chloroethyl)-N-nitrosourea BCNU; Carmustine Bis[4-(dimethylamino)phenyl]methane Michler’s Base; 4,4-Methylenebis(N,N-dimethylbenzenamine) 1,3-Bis(2,3-epoxypropoxy)benzene Diglycidyl resorcinol ether Bis(2-ethylhexyl) phthalate DEHP; Di(2-ethylhexyl) phthalate Bromodichloromethane Bromoethene Vinyl bromide tert-Butyl-4-hydroxyanisole BHA; Butylated hydroxyanisole Chloramphenicol Chlorendic acid 5-Norbornene-2,3-dicarboxylic acid, 1,4,5,6,7,7-hexachloroChlorinated paraffins (C12, 60% Cl) 4-Chloro-1,2-benzenediamine 4-Chloro-o-phenylenediamine 2-Chloro-1,3-butadiene Chloroprene 1-(2-Chloroethyl)-3-cyclohexyl-1-nitrosourea CCNU; Lomustine; Belustine 4-Chloro-2-methylaniline p-Chloro-o-toluidine 4-Chloro-2-methylaniline hydrochloride p-Chloro-o-toluidine hydrochloride 1-Chloro-2-methylpropene Dimethylvinyl chloride 3-Chloro-2-methylpropene Chlorozotocin D-Glucose, 2-[[[(2-chloroethyl)nitrosoamino]carbonyl]amino]-2-deoxyC.I. Basic Red 9, monohydrochloride Cupferron Dacarbazine 1H-Imidazole-4-carboxamide, 5-(3,3-dimethyl-1-triazenyl)cis-Diamminedichloroplatinum Cisplatin 2,4-Diaminoanisole sulfate 1,3-Benzenediamine, 4-methoxy, sulfate 4,4'-Diaminodiphenyl ether 4,4-Oxydianiline 4,4'-Diaminodiphenylmethane 4,4'-Methylenedianiline Dibenz[a,h]acridine Dibenz[a,j]acridine Dibenz[a,h]anthracene 7H-Dibenzo[c,g]carbazole Dibenzo[a,e]pyrene Naphtho[1,2,3,4-def]chrysene Dibenzo[a,h]pyrene Dibenzo[b,def]chrysene Dibenzo[a,i]pyrene Benzo[rst]pentaphene Dibenzo[a,l]pyrene Dibenzo[def,p]chrysene 1,2-Dibromo-3-chloropropane 1,2-Dibromoethane Ethylene dibromide; EDB 2,3-Dibromo-1-propanol DBP 2,3-Dibromo-1-propanol, phosphate (3:1) Tris(2,3-dibromopropyl) phosphate p-Dichlorobenzene 1,4-Dichlorobenzene 3,3'-Dichloro-p-benzidine [1,1'-Biphenyl]-4,4'-diamine, 3,3'-dichloro-
16-54
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CAS RN
75-07-0 53-96-3 79-06-1 107-13-1 23214-92-8 117-79-3 82-28-0 76180-96-6 320-67-2 56-55-3 205-99-2 205-82-3 207-08-9 50-32-8 1464-53-5 101-14-4 3296-90-0 55-86-7 154-93-8 101-61-1 101-90-6 117-81-7 75-27-4 593-60-2 25013-16-5 56-75-7 115-28-6 108171-26-2 95-83-0 126-99-8 13010-47-4 95-69-2 3165-93-3 513-37-1 563-47-3 54749-90-5 569-61-9 135-20-6 4342-03-4 15663-27-1 39156-41-7 101-80-4 101-77-9 226-36-8 224-42-0 53-70-3 194-59-2 192-65-4 189-64-0 189-55-9 191-30-0 96-12-8 106-93-4 96-13-9 126-72-7 106-46-7 91-94-1
CHEMICAL CARCINOGENS (continued) Substance 3,3'-Dichloro-p-benzidine dihydrochloride 1,2-Dichloroethane Dichloromethane 1,3-Dichloropropene (unspecified isomer) Diethyl sulfate 2,3-Dihydro-6-propyl-2-thioxo4(1H)-pyrimidinone 1,8-Dihydroxy-9,10-anthracenedione 1,2-Dimethoxy-4-allylbenzene 3,3'-Dimethoxybenzidine p-(Dimethylamino)azobenzene 2',3-Dimethyl-4-aminoazobenzene Dimethylcarbamic chloride 1,1-Dimethylhydrazine Dimethyl sulfate 1,6-Dinitropyrene 1,8-Dinitropyrene 1,4-Dioxane 1,2-Diphenylhydrazine Disperse Blue No. 1 Epichlorohydrin 1,2-Epoxy-4-(epoxyethyl)cyclohexane N-(4-Ethoxyphenyl)acetamide Ethyl carbamate Ethyl methanesulfonate N-Ethyl-N-nitrosourea Fluoroethene Formaldehyde Furan Hexachlorobenzene 1,2,3,4,5,6-Hexachlorocyclohexane, (1α ,2α ,3β ,4α ,5α ,6β ) 1,2,3,4,5,6-Hexachlorocyclohexane, (1α ,2α ,3β ,4α ,5β ,6β ) 1,2,3,4,5,6-Hexachlorocyclohexane, (1α ,2β ,3α ,4β ,5α ,6β ) Hexachlorocyclohexane (other isomers) Hexachloroethane Hexamethylphosphoric triamide Hydrazine Hydrazine sulfate 2-Imidazolidinethione Indeno[1,2,3-cd]pyrene Kepone Lead(II) acetate Lead(II) phosphate o-Methoxyaniline hydrochloride 2-Methoxy-5-methylaniline o-Methylaniline o-Methylaniline hydrochloride 2-Methyl-1,3-butadiene 5-Methylchrysene 4,4-Methylenedianiline dihydrochloride Methyl methanesulfonate N-Methyl-N’-nitro-N-nitrosoguanidine N-Methyl-N-nitrosourea Methyloxirane Metronidazole
Other names 3,3'-Dichloro-[1,1'-biphenyl]-4,4'-diamine dihydrochloride Ethylene dichloride Methylene chloride
Propylthiouracil Danthron; 1,8-Dihydroxyanthraquinone Methyleugenol Dianisidine
CAS RN 612-83-9 107-06-2 75-09-2 542-75-6 64-67-5 51-52-5
Perchlorobenzene Lindane; γ -Hexachlorocyclohexane
117-10-2 93-15-2 119-90-4 60-11-7 97-56-3 79-44-7 57-14-7 77-78-1 42397-64-8 42397-65-9 123-91-1 122-66-7 2475-45-8 106-89-8 106-87-6 62-44-2 51-79-6 62-50-0 759-73-9 75-02-5 50-00-0 110-00-9 118-74-1 58-89-9
α -Hexachlorocyclohexane
319-84-6
β -Hexachlorocyclohexane
319-85-7
o-Aminoazotoluene; 4-o-Tolylazo-o-toluidine Dimethylcarbamoyl chloride UDMH
Hydrazobenzene 9,10-Anthracenedione, 1,4,5,8-tetraamino(Chloromethyl)oxirane 4-Vinyl-1-cyclohexene dioxide Phenacetin Urethane ENU; N-Nitroso-N-ethylurea Vinyl fluoride
Perchloroethane Hexamethylphosphoramide; Tris(dimethylamino)phosphine oxide
Ethylene thiourea 1,10-(1,2-Phenylene)pyrene Chlordecone
o-Anisidine hydrochloride p-Cresidine; 5-Methyl-o-anisidine o-Toluidine o-Toluidine hydrochloride Isoprene Benzenamine, 4,4'-methylenedi-, dihydrochloride
N-Nitroso-N-methylurea 1,2-Propylene oxide 2-Methyl-5-nitro-1H-imidazole-1-ethanol
16-55
608-73-1 67-72-1 680-31-9 302-01-2 10034-93-2 96-45-7 193-39-5 143-50-0 301-04-2 7446-27-7 134-29-2 120-71-8 95-53-4 636-21-5 78-79-5 3697-24-3 13552-44-8 66-27-3 70-25-7 684-93-5 75-56-9 443-48-1
CHEMICAL CARCINOGENS (continued) Substance Mirex Nickel (metallic) Nitrilotriacetic acid 2-Nitroanisole 6-Nitrochrysene Nitrofen 2-Nitropropane 1-Nitropyrene 4-Nitropyrene N-Nitrosodibutylamine N-Nitrosodiethanolamine N-Nitrosodiethylamine N-Nitrosodimethylamine 4-(N-Nitrosomethylamino)-1(3-pyridyl)-1-butanone N-Nitroso-N-methylvinylamine 4-Nitrosomorpholine N-Nitrosonornicotine N-Nitrosopiperidine N-Nitroso-N-propyl-1-propanamine N-Nitrosopyrrolidine N-Nitrososarcosine Norethisterone Ochratoxin A 2-Oxetanone Oxiranemethanol Oxymetholone Phenazopyridine hydrochloride Phenolphthalein Phenoxybenzamine hydrochloride Phenyloxirane Phenytoin Polybrominated biphenyls Polychlorinated biphenyls Procarbazine hydrochloride Progesterone 1,3-Propane sultone Propyleneimine Reserpine Safrole Selenium sulfide Streptozotocin Sulfallate Tetrachloroethene Tetrachloromethane Tetrafluoroethene N,N,N’,N’-Tetramethyl-4,4'diaminobenzophenone Tetranitromethane Thioacetamide Thiourea o-Tolidine Toluene-2,4-diamine Toluene diisocyanate (unspecified isomer) Toxaphene 1H-1,2,4-Triazol-3-amine 1,1,1-Trichloro-2,2-bis(4-chlorophenyl)ethane
Other names
CAS RN
1,3,4-Metheno-1H-cyclobuta[cd]pentalene, 1,1a,2,2,3,3a,4,5,5,5a,5b,6-dodecachlorooctahydro-
2385-85-5
N,N-Bis(carboxymethyl)glycine 1-Methoxy-2-nitrobenzene Benzene, 2,4-dichloro-1-(4-nitrophenoxy)-
Ethanol, 2,2'-(nitrosoimino)DEN; Diethylnitrosamine DMN; Dimethylnitrosamine NNK; Ketone, 3-pyridyl-3-(N-methyl-N-nitrosamino)propyl Ethenamine, N-methyl-N-nitrosoN-Nitrosomorpholine 1-Nitrosopiperidine N-Nitrosodipropylamine Glycine, N-methyl-N-nitroso19-Norpregn-4-en-20-yn-3-one, 17-hydroxy-, (17 α )β -Propiolactone Glycidol Androstan-3-one, 17-hydroxy-2-(hydroxymethylene)-17-methyl2,6-Pyridinediamine, 3-(phenylazo)-, monohydrochloride 3,3-Bis(4-hydroxyphenyl)-1(3H)-isobenzofuranone Benzenemethanamine, N-(2-chloroethyl)-N(1-methyl-2-phenoxyethyl)-, hydrochloride Styrene-7,8-oxide 5,5-Diphenyl-2,4-imidazolidinedione PBBs PCBs Pregn-4-ene-3,20-dione 1,2-Oxathiolane, 2,2-dioxide 2-Methylaziridine 5-(2-Propenyl)-1,3-benzodioxole D-Glucopyranose, 2-deoxy-2-[[(methylnitrosoamino)carbonyl]amino]N,N-Diethyldithiocarbamic acid, 2-chloroallyl ester Perchloroethylene Carbon tetrachloride Tetrafluoroethylene Bis(dimethylamino)benzophenone; Michler’s Ketone
3,3-Dimethylbenzidine 2,4-Diaminotoluene Polychlorocamphene Amitrole DDT; Dichlorodiphenyltrichloroethane
16-56
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7440-02-0 139-13-9 91-23-6 7496-02-8 1836-75-5 79-46-9 5522-43-0 57835-92-4 924-16-3 1116-54-7 55-18-5 62-75-9 64091-91-4 4549-40-0 59-89-2 16543-55-8 100-75-4 621-64-7 930-55-2 13256-22-9 68-22-4 303-47-9 57-57-8 556-52-5 434-07-1 136-40-3 77-09-8 63-92-3 96-09-3 57-41-0 1336-36-3 366-70-1 57-83-0 1120-71-4 75-55-8 50-55-5 94-59-7 7446-34-6 18883-66-4 95-06-7 127-18-4 56-23-5 116-14-3 90-94-8 509-14-8 62-55-5 62-56-6 119-93-7 95-80-7 26471-62-5 8001-35-2 61-82-5 50-29-3
CHEMICAL CARCINOGENS (continued) Substance Trichloroethene Trichloromethane (Trichloromethyl)benzene 2,4,6-Trichlorophenol 1,2,3-Trichloropropane
Other names Trichloroethylene Chloroform Benzotrichloride
CAS RN 79-01-6 67-66-3 98-07-7 88-06-2 96-18-4
16-57
Appendix A: Mathematical Tables Miscellaneous Mathematical Constants Decimal Equivalents of Common Fractions Quadratic Formula Exponential and Hyperbolic Functions and their Common Logarithms Natural Trigonometric Functions to Four Places Relation of Angular Functions in Terms of One Another Derivatives Integration Integrals Differential Equations Fourier Series Fourier Expansions for Basic Periodic Functions The Fourier Transforms Series Expansion Vector Analysis Orthogonal Curvilinear Coordinates Transformation of Integrals Moment of Inertial for Various Bodies of Mass Bessel Functions The Factorial Function The Gamma Function The Beta Function The Error Function Orthogonal Polynomials Tables of Orthogonal Polynomials Clebsch-Gordan Coefficients Normal Probability Function Percentage Points, Student's t-Distribution Percentage Points, Chi-Square Distribution Percentage Points, F-Distribution
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MISCELLANEOUS MATHEMATICAL CONSTANTS π CONSTANTS π
=
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37511
1/π
=
0.31830 98861 83790 67153 77675 26745 02872 40689 19291 48091
π2
=
9.86960 44010 89358 61883 44909 99876 15113 53136 99407 24079
loge π
=
1.14472 98858 49400 17414 34273 51353 05871 16472 94812 91531
log10 π √ log10 2π
=
0.49714 98726 94133 85435 12682 88290 89887 36516 78324 38044
=
0.39908 99341 79057 52478 25035 91507 69595 02099 34102 92128
CONSTANTS INVOLVING e e
=
2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69996
1/e
=
0.36787 94411 71442 32159 55237 70161 46086 74458 11131 03177
e2
=
7.38905 60989 30650 22723 04274 60575 00781 31803 15570 55185
M
=
log10 e = 0.43429 44819 03251 82765 11289 18916 60508 22943 97005 80367
1/M
=
loge 10 = 2.30258 50929 94045 68401 79914 54684 36420 76011 01488 62877
log10 M
=
9.63778 43113 00536 78912 29674 98645 − 10
π e AND eπ CONSTANTS πe
=
22.45915 77183 61045 47342 71522
eπ
=
23.14069 26327 79269 00572 90864
e−π
=
0.04321 39182 63772 24977 44177
e1/2π
=
4.81047 73809 65351 65547 30357
ii
=
e−1/2π = 0.20787 95763 50761 90854 69556
NUMERICAL CONSTANTS √ √ 3
2
=
1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37695
2
=
1.25992 10498 94873 16476 72106 07278 22835 05702 51464 70151
loge 2
=
0.69314 71805 59945 30941 72321 21458 17656 80755 00134 36026
log10 2 √ 3 √ 3 3
=
0.30102 99956 63981 19521 37388 94724 49302 67881 89881 46211
=
1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81039
=
1.44224 95703 07408 38232 16383 10780 10958 83918 69253 49935
loge 3
=
1.09861 22886 68109 69139 52452 36922 52570 46474 90557 82275
log10 3
=
0.47712 12547 19662 43729 50279 03255 11530 92001 28864 19070
OTHER CONSTANTS Euler’s Constant γ
=
loge γ
=
0.57721 56649 01532 86061 −0.54953 93129 81644 82234
Golden Ratio φ
=
1.61803 39887 49894 84820 45868 34365 63811 77203 09180
A-1
1/16
1/8
3/16
1/4
5/16
3/8
7/16
1/2
DECIMAL EQUIVALENTS OF COMMON FRACTIONS 1/64 0.015625 33/64 0.515625 1/32 2/64 0.03125 17/32 34/64 0.53125 3/64 0.046875 35/64 0.546875 2/32 4/64 0.0625 9/16 18/32 36/64 0.5625 5/64 0.078125 37/64 0.578125 3/32 6/64 0.09375 19/32 38/64 0.59375 7/64 0.109375 39/64 0.609375 4/32 8/64 0.125 5/8 20/32 40/64 0.625 9/64 0.140625 41/64 0.640625 5/32 10/64 0.15625 21/32 42/64 0.65625 11/64 0.171875 43/64 0.671875 6/32 12/64 0.1875 11/16 22/32 44/64 0.6875 13/64 0.203125 45/64 0.703125 7/32 14/64 0.21875 23/32 46/64 0.71875 15/64 0.234375 47/64 0.734375 8/32 16/64 0.25 3/4 24/32 48/64 0.75 17/64 0.265625 49/64 0.765625 9/32 18/64 0.28125 25/32 50/64 0.78125 19/64 0.296875 51/64 0.796875 10/32 20/64 0.3125 13/16 26/32 52/64 0.8125 21/64 0.328125 53/64 0.828125 11/32 22/64 0.34375 27/32 54/64 0.84375 23/64 0.359375 55/64 0.859375 12/32 24/64 0.375 7/8 28/32 56/64 0.875 25/64 0.390625 57/64 0.890625 13/32 26/64 0.40625 29/32 58/64 0.90625 27/64 0.421875 59/64 0.921875 14/32 28/64 0.4375 15/16 30/32 60/64 0.9375 29/64 0.453125 61/64 0.953125 15/32 30/64 0.46875 31/32 62/64 0.96875 31/64 0.484375 63/64 0.984375 16/32 32/64 0.5 1/1 32/32 64/64 1
A-2
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QUADRATIC FORMULA The solution of the equation ax2 + bx + c = 0, where a = 0, is given by √ −b ± b2 − 4ac x= . 2a
EXPONENTIAL AND HYPERBOLIC FUNCTIONS AND THEIR COMMON LOGARITHMS ex x
Value
log10
e−x (value)
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71
1.0000 1.0101 1.0202 1.0305 1.0408 1.0513 1.0618 1.0725 1.0833 1.0942 1.1052 1.1163 1.1275 1.1388 1.1503 1.1618 1.1735 1.1853 1.1972 1.2092 1.2214 1.2337 1.2461 1.2586 1.2712 1.2840 1.2969 1.3100 1.3231 1.3364 1.3499 1.3634 1.3771 1.3910 1.4049 1.4191 1.4333 1.4477 1.4623 1.4770 1.4918 1.5063 1.5220 1.5373 1.5527 1.5683 1.5841 1.6000 1.6161 1.6323 1.6487 1.6653 1.6820 1.6989 1.7160 1.7333 1.7507 1.7683 1.7860 1.8040 1.8221 1.8404 1.8589 1.8776 1.8965 1.9155 1.9348 1.9542 1.9739 1.9937 2.0138 2.0340
0.00000 .00434 .00869 .01303 .01737 .02171 .02606 .03040 .03474 .03909 .04343 .04777 .05212 .05646 .06080 .06514 .06949 .07383 .07817 .08252 .08686 .09120 .09554 .09989 .10423 .10857 .11292 .11726 .12160 .12595 .13029 .13463 .13897 .14332 .14766 .15200 .15635 .16069 .16503 .16937 .17372 .17806 .18240 .18675 .19109 .19543 .19978 .20412 .20846 .21280 .21715 .22149 .22583 .23018 .23452 .23886 .24320 .24755 .25189 .25623 .26058 .26492 .26926 .27361 .27795 .28229 .28664 .29098 .29532 .29966 .30401 .30835
1.00000 0.99005 .98020 .97045 .96079 .95123 .94176 .93239 .92312 .91393 .90484 .89583 .88692 .87809 .86936 .86071 .85214 .84366 .83527 .82696 .81873 .81058 .80252 .79453 .78663 .77880 .77105 .76338 .75578 .74826 .74082 .73345 .72615 .71892 .71177 .70469 .69768 .69073 .68386 .67706 .67032 .66365 .65705 .65051 .64404 .63763 .63128 .62500 .61878 .61263 .60653 .60050 .59452 .58860 .58275 .57695 .57121 .56553 .55990 .55433 .54881 .54335 .53794 .53259 .52729 .52205 .51685 .51171 .50662 .50158 .49659 .49164
sinh x
cosh x
Value
log10
Value
log10
tanh x (value)
0.0000 .0100 .0200 .0300 .0400 .0500 .0600 .0701 .0801 .0901 .1002 .1102 .1203 .1304 .1405 .1506 .1607 .1708 .1810 .1911 .2013 .2115 .2218 .2320 .2423 .2526 .2629 .2733 .2837 .2941 .3045 .3150 .3255 .3360 .3466 .3572 .3678 .3785 .3892 .4000 .4108 .4216 .4325 .4434 .4543 .4653 .4764 .4875 .4986 .5098 .5211 .5324 .5438 .5552 .5666 .5782 .5897 .6014 .6131 .6248 .6367 .6485 .6605 .6725 .6846 .6967 .7090 .7213 .7336 .7461 .7586 .7712
-∞ −2.00001 −2.30106 −2.47719 −2.60218 −2.69915 −2.77841 −2.84545 −2.90355 −2.95483 −1.00072 −1.04227 −1.08022 −1.11517 −1.14755 −1.17772 −1.20597 −1.23254 −1.25762 −1.28136 −1.30392 −1.32541 −1.34592 −1.36555 −1.38437 −1.40245 −1.41986 −1.43663 −1.45282 −1.46847 −1.48362 −1.49830 −1.51254 −1.52637 −1.53981 −1.55290 −1.56564 −1.57807 −1.59019 −1.60202 −1.61358 −1.62488 −1.63594 −1.64677 −1.65738 −1.66777 −1.67797 −1.68797 −1.69779 −1.70744 −1.71692 −1.72624 −1.73540 −1.74442 −1.75330 −1.76204 −1.77065 −1.77914 −1.78751 −1.79576 −1.80390 −1.81194 −1.81987 −1.82770 −1.83543 −1.84308 −1.85063 −1.85809 −1.86548 −1.87278 −1.88000 −1.88715
1.0000 1.0001 1.0002 1.0005 1.0008 1.0013 1.0018 1.0025 1.0032 1.0041 1.0050 1.0061 1.0072 1.0085 1.0098 1.0113 1.0128 1.0145 1.0162 1.0181 1.0201 1.0221 1.0243 1.0266 1.0289 1.0314 1.0340 1.0367 1.0395 1.0423 1.0453 1.0484 1.0516 1.0549 1.0584 1.0619 1.0655 1.0692 1.0731 1.0770 1.0811 1.0852 1.0895 1.0939 1.0984 1.1030 1.1077 1.1125 1.1174 1.1225 1.1276 1.1329 1.1383 1.1438 1.1494 1.1551 1.1609 1.1669 1.1730 1.1792 1.1855 1.1919 1.1984 1.2051 1.2119 1.2188 1.2258 1.2330 1.2402 1.2476 1.2552 1.2628
0.00000 .00002 .00009 .00020 .00035 .00054 .00078 .00106 .00139 .00176 .00217 .00262 .00312 .00366 .00424 .00487 .00554 .00625 .00700 .00779 .00863 .00951 .01043 .01139 .01239 .01343 .01452 .01564 .01681 .01801 .01926 .02054 .02187 .02323 .02463 .02607 .02755 .02907 .03063 .03222 .03385 .03552 .03723 .03897 .04075 .04256 .04441 .04630 .04822 .05018 .05217 .05419 .05625 .05834 .06046 .06262 .06481 .06703 .06929 .07157 .07389 .07624 .07861 .08102 .08346 .08593 .08843 .09095 .09351 .09609 .09870 .10134
0.00000 .01000 .02000 .02999 .03998 .04996 .05993 .06989 .07983 .08976 .09967 .10956 .11943 .12927 .13909 .14889 .15865 .16838 .17808 .18775 .19738 .20697 .21652 .22603 .23550 .24492 .25430 .26362 .27291 .28213 .29131 .30044 .30951 .31852 .32748 .33638 .34521 .35399 .36271 .37136 .37995 .33847 .39693 .40532 .41364 .42190 .43008 .43820 .44624 .45422 .46212 .46995 .47770 .48538 .49299 .50052 .50798 .51536 .52267 .52990 .53705 .54413 .55113 .55805 .56490 .57167 .57836 .58498 .59152 .59798 .60437 .61068
A-3
TeamLRN
EXPONENTIAL AND HYPERBOLIC FUNCTIONS AND THEIR COMMON LOGARITHMS ex x 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43
Value 2.0544 2.0751 2.0959 2.1170 2.1383 2.1598 2.1815 2.2034 2.2255 2.2479 2.2705 2.2933 2.3164 2.3396 2.3632 2.3869 2.4100 2.4351 2.4596 2.4843 2.5093 2.5345 2.5600 2.5857 2.6117 2.6379 2.6645 2.6912 2.7183 2.7456 2.7732 2.8011 2.8292 2.8577 2.8864 2.9154 2.9447 2.9743 3.0042 3.0344 3.0659 3.0957 3.1268 3.1582 3.1899 3.2220 3.2544 3.2871 3.3201 3.3535 3.3872 3.4212 3.4556 3.4903 3.5254 3.5609 3.5996 3.6328 3.6693 3.7062 3.7434 3.7810 3.8190 3.8574 3.8962 3.9354 3.9749 4.0149 4.0552 4.0960 4.1371 4.1787
log10 .31269 .31703 .32138 .32572 .33006 .33441 .33875 .34309 .34744 .35178 .35612 .36046 .36481 .36915 .37349 .37784 .38218 .38652 .39087 .39521 .39955 .40389 .40824 .41258 .41692 .42127 .42561 .42995 .43429 .43864 .44298 .44732 .45167 .45601 .46035 .46470 .46904 .47338 .47772 .48207 .48641 .49075 .49510 .49944 .50378 .50812 .51247 .51681 .52115 .52550 .52984 .53418 .53853 .54287 .54721 .55155 .55590 .56024 .56458 .56893 .57327 .57761 .58195 .58630 .59064 .59498 .59933 .60367 .60801 .61236 .61670 .62104
e−x (value) .48675 .48191 .47711 .47237 .46767 .46301 .45841 .45384 .44933 .44486 .44043 .43605 .43171 .42741 .42316 .41895 .41478 .41066 .40657 .40242 .39852 .39455 .39063 .38674 .38289 .37908 .37531 .37158 .36788 .36422 .36060 .35701 .35345 .34994 .34646 .34301 .33960 .33622 .33287 .32956 .32628 .32303 .31982 .31644 .31349 .31037 .30728 .30422 .30119 .29820 .29523 .29229 .28938 .28650 .28365 .28083 .27804 .27527 .27253 .26982 .26714 .26448 .26185 .25924 .25666 .25411 .25158 .24908 .24660 .24414 .24171 .23931
sinh x Value .7838 .7966 .8094 .8223 .8353 .8484 .8615 .8748 .8881 .9015 .9150 .9286 .9423 .9561 .9700 .9840 .9981 1.0122 1.0265 1.0409 1.0554 1.0700 1.0847 1.0995 1.1144 1.1294 1.1446 1.1598 1.1752 1.1907 1.2063 1.2220 1.2379 1.2539 1.2700 1.2862 1.3025 1.3190 1.3356 1.3524 1.3693 1.3863 1.4035 1.4208 1.4382 1.4558 1.4735 1.4914 1.5095 1.5276 1.5460 1.5645 1.5831 1.6019 1.6209 1.6400 1.6593 1.6788 1.6984 1.7182 1.7381 1.7583 1.7786 1.7991 1.8198 1.8406 1.8617 1.8829 1.9043 1.9259 1.9477 1.9697
A-4
cosh x log10 −1.89423 −1.90123 −1.90817 −1.91504 −1.92185 −1.92859 −1.93527 −1.94190 −1.94846 −1.95498 −1.96144 −1.96784 −1.97420 −1.98051 −1.98677 −1.99299 −1.99916 0.00528 .01137 .01741 .02341 .02937 .03530 .04119 .04704 .05286 .05864 .06439 .07011 .07580 .06146 .08708 .09268 .09825 .10379 .10930 .11479 .12025 .12569 .13111 .13649 .14186 .14720 .15253 .15783 .16311 .16836 .17360 .17882 .18402 .18920 .19437 .19951 .20464 .20975 .21485 .21993 .22499 .23004 .23507 .24009 .24509 .25008 .25505 .26002 .26496 .26990 .27482 .27974 .28464 .28952 .29440
Value 1.2706 1.2785 1.2865 1.2947 1.3030 1.3114 1.3199 1.3286 1.3374 1.3464 1.3555 1.3647 1.3740 1.3835 1.3932 1.4029 1.4128 1.4229 1.4331 1.4434 1.4539 1.4645 1.4753 1.4862 1.4973 1.5085 1.5199 1.5314 1.5431 1.5549 1.5669 1.5790 1.5913 1.6038 1.6164 1.6292 1.6421 1.6552 1.6685 1.6820 1.6956 1.7083 1.7233 1.7374 1.7517 1.7662 1.7808 1.7957 1.8107 1.8258 1.8412 1.8568 1.8725 1.8884 1.9045 1.9208 1.9373 1.9540 1.9709 1.9880 2.0053 2.0228 2.0404 2.0583 2.0764 2.0947 2.1132 2.1320 2.1509 2.1700 2.1894 2.2090
log10 .10401 .10670 .10942 .11216 .11493 .11773 .12055 .12340 .12627 .12917 .13209 .13503 .13800 .14099 .14400 .14704 .15009 .15317 .15627 .15939 .16254 .16570 .16888 .17208 .17531 .17855 .18181 .18509 .18839 .19171 .19504 .19839 .20176 .20515 .20855 .21197 .21541 .21886 .22233 .22582 .22931 .23283 .23636 .23990 .24346 .24703 .25062 .25422 .25784 .26146 .26510 .26876 .27242 .27610 .27979 .28349 .28721 .29093 .29467 .29842 .30217 .30594 .30972 .31352 .31732 .32113 .32495 .32878 .33262 .33647 .34033 .34420
tanh x (value) .61691 .62307 .62915 .63515 .64108 .64693 .65721 .65841 .66404 .66959 .67507 .68048 .68581 .69107 .69626 .70137 .70642 .71139 .21630 .72113 .72590 .73059 .73522 .73978 .74428 .74870 .75307 .75736 .76159 .76576 .76987 .77391 .77789 .78181 .78566 .78946 .79320 .79688 .80050 .80406 .80757 .81102 .81441 .81775 .82104 .82427 .82745 .83058 .83365 .83668 .83965 .84258 .83546 .84828 .85106 .85380 .85648 .85913 .86172 .86428 .86678 .86925 .87167 .87405 .87639 .87869 .88095 .88317 .88535 .88749 .88960 .89167
EXPONENTIAL AND HYPERBOLIC FUNCTIONS AND THEIR COMMON LOGARITHMS ex x 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15
Value 4.2207 4.2631 4.3060 4.3492 4.3929 4.4371 4.4817 4.5267 4.5722 4.6182 4.6646 4.7115 4.7588 4.8066 4.8550 4.9037 4.9530 5.0028 5.0531 5.1039 5.1552 5.2070 5.2593 5.3122 5.3656 5.4195 5.4739 5.5290 5.5845 5.6407 5.6973 5.7546 5.8124 5.8709 5.9299 5.9895 6.0496 6.1104 6.1719 6.2339 6.2965 6.3598 6.4237 6.4383 6.5535 6.6194 6.6859 6.7531 6.8210 6.8895 6.9588 7.0287 7.0993 7.1707 7.2427 7.3155 7.3891 7.4633 7.5383 7.6141 7.6906 7.7679 7.8460 7.9248 8.0045 8.0849 8.1662 8.2482 8.3311 8.4149 8.4994 8.5849
log10 .62538 .62973 .63407 .63841 .64276 .64710 .65144 .65578 .66013 .66447 .66881 .67316 .67750 .68184 .68619 .69053 .69487 .69921 .70356 .70790 .71224 .71659 .72093 .72527 .72961 .73396 .73830 .74264 .74699 .75133 .75567 .76002 .76436 .76870 .77304 .77739 .78173 .78607 .79042 .79476 .79910 .80344 .80779 .81213 .81647 .82082 .82516 .82950 .83385 .83819 .84253 .84687 .85122 .85556 .85990 .86425 .86859 .87293 .87727 .88162 .88596 .89030 .89465 .89899 .90333 .90768 .91202 .91636 .92070 .92505 .92939 .93373
e−x (value) .23693 .23457 .23224 .22993 .22764 .22537 .22313 .22091 .21871 .21654 .21438 .21225 .21014 .20805 .20598 .20393 .20190 .19989 .19790 .19593 .19398 .19205 .19014 .18825 .18637 .18452 .18268 .18087 .17907 .17728 .17552 .17377 .17204 .17033 .16864 .16696 .16530 .16365 .16203 .16041 .15882 .15724 .15567 .15412 .15259 .15107 .14957 .14808 .14661 .14515 .14370 .14227 .14086 .13946 .13807 .13670 .13534 .13399 .13266 .13134 .13003 .12873 .12745 .12619 .12493 .12369 .12246 .12124 .12003 .11884 .11765 .11648
sinh x Value 1.9919 2.0143 2.0369 2.0597 2.0827 2.1059 2.1293 2.1529 2.1768 2.2008 2.2251 2.2496 2.2743 2.2993 2.3245 2.3499 2.3756 2.4015 2.4276 2.4540 2.4806 2.5075 2.5346 2.5620 2.5896 2.6175 2.6456 2.6740 2.7027 2.7317 2.7609 2.7904 2.8202 2.8503 2.8806 2.9112 2.9422 2.9734 3.0049 3.0367 3.0689 3.1013 3.1340 3.1671 3.2005 3.2341 3.2682 3.3025 3.3372 3.3722 3.4075 3.4432 3.4792 3.5156 3.5923 3.5894 3.6269 3.6647 3.7028 3.7414 3.7803 3.8196 3.8593 3.8993 3.9398 3.9806 4.0219 4.0635 4.1056 4.1480 4.1909 4.2342
A-5
TeamLRN
cosh x log10 .29926 .30412 .30896 .31379 .31862 .32343 .32823 .33303 .33781 .34258 .34735 .35211 .35686 .36160 .36633 .37105 .37577 .38048 .38518 .38987 .39456 .39923 .40391 .40857 .41323 .41788 .42253 .42717 .43180 .43643 .44105 .44567 .45028 .45488 .45948 .46408 .46867 .47325 .47783 .48241 .48698 .49154 .49610 .50066 .50521 .50976 .51430 .51884 .52338 .52791 .53244 .53696 .54148 .54600 .55051 .55502 .55953 .56403 .56853 .57303 .57753 .58202 .58650 .59099 .59547 .59995 .60443 .60890 .61337 .61784 .62231 .62677
Value 2.2288 2.2488 2.2691 2.2896 2.3103 2.3312 2.3524 2.3738 2.3955 2.4174 2.4395 2.4619 2.4845 2.5073 2.5305 2.5538 2.5775 2.6013 2.6255 2.6499 2.6746 2.6995 2.7247 2.7502 2.7760 2.8020 2.8283 2.8549 2.8818 2.9090 2.9364 2.9642 2.9922 3.0206 3.0492 3.0782 3.1075 3.1371 3.1669 3.1972 3.2277 3.2585 3.2897 3.3212 3.3530 3.3852 3.4177 3.4506 3.4838 3.5173 3.5512 3.5855 3.6201 3.6551 3.6904 3.7261 3.7622 3.7987 3.8335 3.8727 3.9103 3.9483 3.9867 4.0255 4.0647 4.1043 4.1443 4.1847 4.2256 4.2669 4.3085 4.3507
log10 .34807 .35196 .35585 .35976 .36367 .36759 .37151 .37545 .37939 .38334 .38730 .39126 .39524 .39921 .40320 .40719 .41119 .41520 .41921 .42323 .42725 .43129 .43532 .43937 .44341 .44747 .45153 .45559 .45966 .46374 .46782 .47191 .47600 .48009 .48419 .48830 .49241 .49652 .50064 .50476 .50889 .51302 .51716 .52130 .52544 .52959 .53374 .53789 .54205 .54621 .55038 .55455 .55872 .56290 .56707 .57126 .57544 .57963 .58382 .58802 .59221 .59641 .60061 .60482 .60903 .61324 .61745 .62167 .62589 .63011 .63433 .63856
tanh x (value) .89370 .89569 .89765 .89958 .90147 .90332 .90515 .90694 .90870 .91042 .91212 .91379 .91542 .91703 .91860 .92015 .92167 .92316 .92462 .92606 .92747 .92886 .93022 .93155 .93286 .93415 .93541 .93665 .93786 .93906 .94023 .94138 .94250 .94361 .94470 .94576 .94681 .94783 .94884 .94983 .95080 .95175 .95268 .95359 .95449 .95537 .95624 .95709 .95792 .95873 .95953 .96032 .96109 .96185 .96259 .96331 .96403 .96473 .96541 .96609 .96675 .96740 .96803 .96865 .96926 .96986 .97045 .97103 .97159 .97215 .97269 .97323
EXPONENTIAL AND HYPERBOLIC FUNCTIONS AND THEIR COMMON LOGARITHMS ex x 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80 2.81 2.82 2.83 2.84 2.85 2.86
Value 8.6711 8.7583 8.8463 8.9352 9.0250 9.1157 9.2073 9.2999 9.3933 9.4877 9.5831 9.6794 9.7767 9.8749 9.9742 10.074 10.176 10.278 10.381 10.486 10.591 10.697 10.805 10.913 11.023 11.134 11.246 11.359 11.473 11.588 11.705 11.822 11.941 12.061 12.182 12.305 12.429 12.554 12.680 12.807 12.936 13.066 13.197 13.330 13.464 13.599 13.736 13.874 14.013 14.154 14.296 14.440 14.585 14.732 14.880 15.029 15.180 15.333 15.487 15.643 15.800 15.959 16.119 16.281 16.445 16.610 16.777 16.945 17.116 17.288 17.462
log10 .93808 .94242 .94676 .95110 .95545 .95979 .96413 .96848 .97282 .97716 .98151 .98585 .99019 .99453 .99888 1.00322 1.00756 1.01191 1.01625 1.02059 1.02493 1.02928 1.03362 1.03796 1.04231 1.04665 1.05099 1.05534 1.05968 1.06402 1.06836 1.07271 1.07705 1.08139 1.08574 1.09008 1.09442 1.09877 1.10311 1.10745 1.11179 1.11614 1.12048 1.12482 1.12917 1.13351 1.13785 1.14219 1.14654 1.15008 1.15522 1.15957 1.16391 1.16825 1.17260 1.17694 1.18128 1.18562 1.18997 1.19431 1.19865 1.20300 1.20734 1.21168 1.21602 1.22037 1.22471 1.22905 1.23340 1.23774 1.24208
e−x (value) .11533 .11418 .11304 .11192 .11080 .10970 .10861 .10753 .10646 .10540 .10435 .10331 .10228 .10127 .10026 .09926 .09827 .09730 .09633 .09537 .09442 .09348 .09255 .09163 .09072 .08982 .08892 .08804 .08716 .08629 .08543 .08458 .08374 .08291 .08208 .08127 .08046 .07966 .07887 .07808 .07730 .07654 .07577 .07502 .07427 .07353 .07280 .07208 .07136 .07065 .06995 .06925 .06856 .06788 .06721 .06654 .06587 .06522 .06457 .06393 .06329 .06266 .06204 .06142 .06081 .06020 .05961 .05901 .05843 .05784 .05727
sinh x Value 4.2779 4.3221 4.3666 4.4116 4.4571 4.5030 4.5494 4.5962 4.6434 4.6912 4.7394 4.7880 4.8372 4.8868 4.9370 4.9876 5.0387 5.0903 5.1425 5.1951 5.2483 5.3020 5.3562 5.4109 5.4662 5.5221 5.5785 5.6354 5.6929 5.7510 5.8097 5.8689 5.9288 5.9892 6.0502 6.1118 6.1741 6.2369 6.3004 6.3645 6.4293 6.4946 6.5607 6.6274 6.6947 6.7628 6.8315 6.9008 6.9709 7.0417 7.1132 7.1854 7.2583 7.3319 7.4063 7.4814 7.5572 7.6338 7.7112 7.7894 7.8683 7.9480 8.0285 8.1098 8.1919 8.2749 8.3586 8.4432 8.5287 8.6150 8.7021
A-6
cosh x log10 .63123 .63569 .64015 .64460 .64905 .65350 .65795 .66240 .66684 .67128 .67572 .68016 .68459 .68903 .69346 .69789 .70232 .70675 .71117 .71559 .72002 .72444 .72885 .73327 .73769 .74210 .74652 .75093 .75534 .75975 .76415 .76856 .77296 .77737 .78177 .78617 .79057 .79497 .79937 .80377 .80816 .81256 .81695 .82134 .82573 .83012 .83451 .83890 .84329 .84768 .85206 .85645 .86083 .86522 .86960 .87398 .87836 .88274 .88712 .89150 .89588 .90026 .90463 .90901 .91339 .91776 .92213 .92651 .93088 .93525 .93963
Value 4.3932 4.4362 4.4797 4.5236 4.5679 4.6127 4.6580 4.7037 4.7499 4.7966 4.8437 4.8914 4.9395 4.9881 5.0372 5.0868 5.1370 5.1876 5.2388 5.2905 5.3427 5.3954 5.4487 5.5026 5.5569 5.6119 5.6674 5.7235 5.7801 5.8373 5.8951 5.9535 6.0125 6.0721 6.1323 6.1931 6.2545 6.3166 6.3793 6.4426 6.5066 6.5712 6.6365 6.7024 6.7690 6.8363 6.9043 6.9729 7.0423 7.1123 7.1831 7.2546 7.3268 7.3998 7.4735 7.5479 7.6231 7.6991 7.7758 7.8533 7.9316 8.0106 8.0905 8.1712 8.2527 8.3351 8.4182 8.5022 8.5871 8.6728 8.7594
log10 .64278 .64701 .65125 .65548 .65972 .66396 .66820 .67244 .67668 .68093 .68518 .68943 .69368 .69794 .70219 .70645 .71071 .71497 .71923 .72349 .72776 .73203 .73630 .74056 .74484 .74911 .75338 .75766 .76194 .76621 .77049 .77477 .77906 .78334 .78762 .79191 .79619 .80048 .80477 .80906 .81335 .81764 .82194 .82623 .83052 .83482 .83912 .84341 .84771 .85201 .85631 .86061 .86492 .86922 .87352 .87783 .88213 .89644 .89074 .89505 .89936 .90367 .90798 .91229 .91660 .92091 .92522 .92953 .93385 .93816 .94247
tanh x (value) .97375 .97426 .97477 .97526 .97574 .97622 .97668 .97714 .97759 .97803 .97846 .97888 .97929 .97970 .98010 .98049 .98087 .98124 .98161 .98197 .98233 .98267 .98301 .98335 .98367 .98400 .98431 .98462 .98492 .98522 .98551 .98579 .98607 .98635 .98661 .98688 .98714 .98739 .98764 .98788 .98812 .98835 .98858 .98881 .98903 .98924 .98946 .98966 .98987 .99007 .99026 .99045 .99064 .99083 .99101 .99118 .99136 .99153 .99170 .99186 .99202 .99218 .99233 .99248 .99263 .99278 .99292 .99306 .99320 .99333 .99346
EXPONENTIAL AND HYPERBOLIC FUNCTIONS AND THEIR COMMON LOGARITHMS ex x 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.25 6.50 6.75 7.00 7.50 8.00 8.50 9.00 9.50 10.00
Value 17.637 17.814 17.993 18.174 18.357 18.541 18.728 18.916 19.106 19.298 19.492 19.688 19.886 20.086 21.115 22.198 23.336 24.533 25.790 27.113 28.503 29.964 31.500 33.115 34.813 36.598 38.475 40.447 42.521 44.701 46.993 49.402 51.935 54.598 60.340 66.686 73.700 81.451 90.017 99.484 109.95 121.51 134.29 148.41 164.02 181.27 200.34 221.41 244.69 270.43 298.87 330.30 365.04 403.43 518.01 665.14 854.06 1096.6 1808.0 2981.0 4914.8 8103.1 13360. 22026.
log10 1.24643 1.25077 1.25511 1.25945 1.26380 1.26814 1.27248 1.27683 1.28117 1.28551 1.28985 1.29420 1.29854 1.30288 1.32460 1.34631 1.36803 1.38974 1.41146 1.43317 1.45489 1.47660 1.49832 1.52003 1.54175 1.56346 1.58517 1.60689 1.62860 1.65032 1.67203 1.69375 1.71546 1.73718 1.78061 1.82404 1.86747 1.91090 1.95433 1.99775 2.04118 2.08461 2.12804 2.17147 2.21490 2.25833 2.30176 2.34519 2.38862 2.43205 2.47548 2.51891 2.56234 2.60577 2.71434 2.82291 2.93149 3.04006 3.25721 3.47436 3.69150 3.90865 4.12580 4.34294
e−x (value) .05670 .05613 .05558 .05502 .05448 .05393 .05340 .05287 .05234 .05182 .05130 .05079 .05029 .04979 .04736 .04505 .04285 .04076 .03877 .03688 .03508 .03337 .03175 .03020 .02872 .02732 .02599 .02472 .02352 .02237 .02128 .02024 .01925 .01832 .01657 .01500 .01357 .01227 .01111 .01005 .00910 .00823 .00745 .00674 .00610 .00552 .00499 .00452 .00409 .00370 .00335 .00303 .00274 .00248 .00193 .00150 .00117 .00091 .00055 .00034 .00020 .00012 .00007 .00005
sinh x Value 8.7902 8.8791 8.9689 9.0596 9.1512 9.2437 9.3371 9.4315 9.5268 9.6231 9.7203 9.8185 9.9177 10.018 10.534 11.076 11.647 12.246 12.876 13.538 14.234 14.965 15.734 16.543 17.392 18.286 19.224 20.211 21.249 22.339 23.486 24.691 25.958 27.290 30.162 33.336 36.843 40.719 45.003 49.737 54.969 60.751 67.141 74.203 82.008 90.633 100.17 110.70 122.34 135.21 149.43 165.15 182.52 201.71 259.01 332.57 427.03 548.32 904.02 1490.5 2457.4 4051.5 6679.9 11013.
A-7
TeamLRN
cosh x log10 .94400 .94837 .95274 .95711 .96148 .96584 .97021 .97458 .97895 .98331 .98768 .99205 .99641 1.00078 1.02259 1.04440 1.06620 1.08799 1.10977 1.13155 1.15332 1.17509 1.19685 1.21860 1.24036 1.26211 1.28385 1.30559 1.32733 1.34907 1.37081 1.39254 1.41427 1.43600 1.47946 1.52291 1.56636 1.60980 1.65324 1.69668 1.74012 1.78355 1.82699 1.87042 1.91389 1.95729 2.00074 2.04415 2.08758 2.13101 2.17444 2.21787 2.26130 2.30473 2.41331 2.52188 2.63046 2.73904 2.95618 3.17333 3.39047 3.60762 3.82477 4.04191
Value 8.8469 8.9352 9.0244 9.1146 9.2056 9.2976 9.3905 9.4844 9.5791 9.6749 9.7716 9.8693 9.9680 10.068 10.581 11.122 11.690 12.287 12.915 13.575 14.269 14.999 15.766 16.573 17.421 18.313 19.250 20.236 21.272 22.362 23.507 24.711 25.977 27.308 30.178 33.351 36.857 40.732 45.014 49.747 54.978 60.759 67.149 74.210 82.014 90.639 100.17 110.71 122.35 135.22 149.44 165.15 182.52 201.72 259.01 332.57 427.03 548.32 904.02 1490.5 2457.4 4051.5 6679.9 11013.
log10 .94679 .95110 .95542 .95974 .96405 .96837 .97269 .97701 .98133 .98565 .98997 .99429 .99861 1.00293 1.02454 1.04616 1.06779 1.08943 1.11108 1.13273 1.15439 1.17605 1.19772 1.21940 1.24107 1.26275 1.28444 1.30612 1.32781 1.34951 1.37120 1.39290 1.41459 1.43629 1.47970 1.52310 1.56652 1.60993 1.65335 1.69677 1.74019 1.78361 1.82704 1.87046 1.91389 1.95731 2.00074 2.04417 2.08760 2.13103 2.17445 2.21788 2.26131 2.30474 2.41331 2.52189 2.63046 2.73903 2.95618 3.17333 3.39047 3.60762 3.82477 4.04191
tanh x (value) .99359 .99372 .99384 .99396 .99408 .99420 .99431 .99443 .99454 .99464 .99475 .99485 .99496 0.99505 0.99552 0.99595 0.99633 0.99668 0.99700 0.99728 0.99754 0.99777 0.99799 0.99818 0.99835 0.99851 9.99865 0.99878 0.99889 0.99900 0.99909 0.99918 0.99926 0.99933 0.99945 0.99955 0.99963 0.99970 0.99975 0.99980 0.99983 0.99986 0.99989 0.99991 0.99993 0.99994 0.99995 0.99996 0.99997 0.99997 0.99998 0.99998 0.99998 0.99999 0.99999 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
NATURAL TRIGONOMETRIC FUNCTIONS TO FOUR PLACES x radians
x degrees
sin x
cos x
tan x
cot x
sec x
csc x
.0000 .0029 .0058 .0087 .0116 .0145 .0175 .0204 .0233 .0262 .0291 .0320 .0349 .0378 .0407 .0436 .0465 .0495 .0524 .0553 .0582 .0611 .0640 .0669 .0698 .0727 .0756 .0785 .0814 .0844 .0873 .0902 .0931 .0960 .0989 .1018 .1047 .1076 .1105 .1134 .1164 .1193 .1222 .1251 .1280 .1309 .1338 .1367 .1396 .1425 .1454 .1484 .1513 .1542 .1571 .1600 .1629 .1658 .1687 .1716 .1745 .1774 .1804 .1833 .1862 .1891 .1920 .1949 .1978 .2007
0◦ 00 10 20 30 40 50 ◦ 1 00 10 20 30 40 50 ◦ 2 00 10 20 30 40 50 ◦ 3 00 10 20 30 40 50 ◦ 4 00 10 20 30 40 50 ◦ 5 00 10 20 30 40 50 ◦ 6 00 10 20 30 40 50 ◦ 7 00 10 20 30 40 50 ◦ 8 00 10 20 30 40 50 ◦ 9 00 10 20 30 40 50 ◦ 10 00 10 20 30 40 50 ◦ 11 00 10 20 30
.000 .0029 .0058 .0087 .0116 .0145 .0175 .0204 .0233 .0262 .0291 .0320 .0349 .0378 .0407 .0436 .0465 .0494 .0523 .0552 .0581 .0610 .0640 .0669 .0698 .0727 .0756 .0785 .0814 .0843 .0872 .0901 .0929 .0958 .0987 .1016 .1045 .1074 .1103 .1132 .1161 .1190 .1219 .1248 .1276 .1305 .1334 .1363 .1392 .1421 .1449 .1478 .1507 .1536 .1564 .1593 .1622 .1650 .1679 .1708 .1736 .1765 .1794 .1822 .1851 .1880 .1908 .1937 .1965 .1994
1.0000 1.0000 1.0000 1.0000 .9999 .9999 .9998 .9998 .9997 .9997 .9996 .9995 .9994 .9993 .9992 .9990 .9989 .9988 .9986 .9985 .9983 .9981 .9980 .9978 .9976 .9974 .9971 .9969 .9967 .9964 .9962 .9959 .9957 .9954 .9951 .9948 .9945 .9942 .9939 .9936 .9932 .9929 .9925 .9922 .9918 .9914 .9911 .9907 .9903 .9899 .9894 .9890 .9886 .9881 .9877 .9872 .9868 .9863 .9858 .9853 .9848 .9843 .9838 .9833 .9827 .9822 .9816 .9811 .9805 .9799
.0000 .0029 .0058 .0087 .0116 .0145 .0175 .0204 .0233 .0262 .0291 .0320 .0349 .0378 .0407 .0437 .0466 .0495 .0524 .0553 .0582 .0612 .0641 .0670 .0699 .0729 .0758 .0787 .0816 .0846 .0875 .0904 .0934 .0963 .0992 .1022 .1051 .1080 .1110 .1139 .1169 .1198 .1228 .1257 .1284 .1317 .1346 .1376 .1405 .1435 .1465 .1495 .1524 .1554 .1584 .1614 .1644 .1673 .1703 .1733 .1763 .1793 .1823 .1853 .1883 .1914 .1944 .1974 .2004 .2035
– 343.8 171.9 114.6 85.94 68.75 57.29 49.10 42.96 38.19 34.37 31.24 28.64 26.43 24.54 22.90 21.47 20.21 19.08 18.07 17.17 16.35 15.60 14.92 14.30 13.73 13.20 12.71 12.25 11.83 11.43 11.06 10.71 10.39 10.08 9.788 9.514 9.255 9.010 8.777 8.556 8.345 8.144 7.953 7.770 7.596 7.429 7.269 7.115 6.968 6.827 6.691 6.561 6.435 6.314 6.197 6.084 5.976 5.871 5.769 5.671 5.576 5.485 5.396 5.309 5.226 5.145 5.066 4.989 4.915
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.001 1.001 1.001 1.001 1.001 1.001 1.001 1.001 1.002 1.002 1.002 1.002 1.002 1.002 1.003 1.003 1.003 1.003 1.004 1.004 1.004 1.004 1.005 1.005 1.005 1.006 1.006 1.006 1.006 1.007 1.007 1.008 1.008 1.008 1.009 1.009 1.009 1.010 1.010 1.011 1.011 1.012 1.012 1.012 1.013 1.013 1.014 1.014 1.015 1.015 1.016 1.016 1.017 1.018 1.018 1.019 1.019 1.020 1.020
– 343.8 171.9 114.6 85.95 68.76 57.30 49.11 42.98 38.20 34.38 31.26 28.65 26.45 24.56 22.93 21.49 20.23 19.11 18.10 17.20 16.38 15.64 14.96 14.34 13.76 13.23 12.75 12.29 11.87 11.47 11.10 10.76 10.43 10.13 9.839 9.597 9.309 9.065 8.834 8.614 8.405 8.206 8.016 7.834 7.661 7.496 7.337 7.185 7.040 6.900 6.765 6.636 6.512 6.392 6.277 6.166 6.059 5.955 5.855 5.759 5.665 5.575 5.487 5.403 5.320 5.241 5.164 5.089 5.016
cos x
sin x
cot x
tan x
csc x
sec x
A-8
90◦ 00 50 40 30 20 10 ◦ 89 00 50 40 30 20 10 ◦ 88 00 50 40 30 20 10 ◦ 87 00 50 40 30 20 10 ◦ 86 00 50 40 30 20 10 ◦ 85 00 50 40 30 20 10 ◦ 84 00 50 40 30 20 10 ◦ 83 00 50 40 30 20 10 ◦ 82 00 50 40 30 20 10 ◦ 81 00 50 40 30 20 10 ◦ 80 00 50 40 30 20 10 ◦ 79 00 50 40 30 x degree
1.5708 1.5679 1.5650 1.5621 1.5592 1.5563 1.5533 1.5504 1.5475 1.5446 1.5417 1.5388 1.5359 1.5330 1.5301 1.5272 1.5243 1.5213 1.5184 1.5155 1.5126 1.5097 1.5068 1.5039 1.5010 1.4981 1.4952 1.4923 1.4893 1.4864 1.4835 1.4806 1.4777 1.4748 1.4719 1.4690 1.4661 1.4632 1.4603 1.4573 1.4544 1.4515 1.4486 1.4457 1.4428 1.4399 1.4370 1.4341 1.4312 1.4283 1.4254 1.4224 1.4195 1.4166 1.4137 1.4108 1.4079 1.4050 1.4021 1.3992 1.3963 1.3934 1.3904 1.3875 1.3846 1.3817 1.3788 1.3759 1.3730 1.3701 x radians
NATURAL TRIGONOMETRIC FUNCTIONS TO FOUR PLACES x radians .2036 .2065 .2094 .2123 .2153 .2182 .2211 .2240 .2269 .2298 .2327 .2356 .2385 .2414 .2443 .2473 .2502 .2531 .2560 .2589 .2618 .2647 .2676 .2705 .2734 .2763 .2793 .2822 .2851 .2880 .2909 .2938 .2967 .2996 .3025 .3054 .3083 .3113 .3142 .3171 .3200 .3229 .3258 .3287 .3316 .3345 .3374 .3403 .3432 .3462 .3491 .3520 .3599 .3578 .3607 .3636 .3665 .3694 .3723 .3752 .3782 .3811 .3840 .3869 .3898 .3927 .3956 .3985 .4014 .4043
x degrees
sin x
cos x
tan x
cot x
sec x
csc x
40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10 20 30 40 50 00 10
.2022 .2051 .2079 .2108 .2136 .2164 .2193 .2221 .2250 .2278 .2306 .2334 .2363 .2391 .2419 .2447 .2476 .2404 .2532 .2560 .2588 .2616 .2644 .2672 .2700 .2728 .2756 .2784 .2812 .2840 .2868 .2896 .2924 .2952 .2979 .3007 .3035 .3062 .3090 .3118 .3145 .3173 .3201 .3228 .3256 .3283 .3311 .3338 .3365 .3393 .3420 .3448 .3475 .3502 .3529 .3557 .3584 .3611 .3638 .3665 .3692 .3719 .3746 .3773 .3800 .3827 .3854 .3881 .3907 .3934
.9793 .9787 .9781 .9775 .9769 .9763 .9757 .9750 .9744 .9737 .9730 .9724 .9717 .9710 .9703 .9696 .9689 .9681 .9674 .9667 .9659 .9652 .9644 .9636 .9628 .9621 .9613 .9605 .9596 .9588 .9580 .9572 .9563 .9555 .9546 .9537 .9528 .9520 .9511 .9502 .9492 .9483 .9474 .9465 .9455 .9446 .9436 .9426 .9417 .9407 .9397 .9387 .9377 .9367 .9356 .9346 .9336 .9325 .9315 .9304 .9293 .9283 .9272 .9261 .9250 .9239 .9228 .9216 .9205 .9194
.2065 .2095 .2126 .2156 .2186 .2217 .2247 .2278 .2309 .2339 .2370 .2401 .2432 .2462 .2493 .2524 .2555 .2586 .2617 .2648 .2679 .2711 .2732 .2773 .2805 .2836 .2867 .2899 .2931 .2962 .2994 .3026 .3057 .3089 .3121 .3153 .3185 .3217 .3249 .3281 .3314 .3346 .3378 .3411 .3443 .3476 .3508 .3541 .3574 .3607 .3640 .3673 .3706 .3739 .3772 .3805 .3839 .3872 .3906 .3939 .3973 .4006 .4040 .4074 .4108 .4142 .4176 .4210 .4245 .4279
3.843 4.773 4.705 4.638 4.574 4.511 4.449 4.390 4.331 4.275 4.219 4.165 4.113 4.061 4.011 3.962 3.914 3.867 3.821 3.776 3.732 3.689 3.647 3.606 3.566 3.526 3.487 3.450 3.412 3.376 3.340 3.305 3.271 3.237 3.204 3.172 3.140 3.108 3.078 3.047 3.018 2.989 2.960 2.932 2.904 2.877 2.850 2.824 2.798 2.773 2.747 2.723 2.699 2.675 2.651 2.628 2.605 2.583 2.560 2.539 2.517 2.496 2.475 2.455 2.434 2.414 2.394 2.375 2.356 2.337
1.021 1.022 1.022 1.023 1.024 1.025 1.025 1.026 1.026 1.027 1.028 1.028 1.029 1.030 1.031 1.031 1.032 1.033 1.034 1.034 1.035 1.036 1.037 1.038 1.039 1.039 1.040 1.041 1.042 1.043 1.044 1.045 1.046 1.047 1.048 1.049 1.049 1.050 1.051 1.052 1.053 1.054 1.056 1.057 1.058 1.059 1.060 1.061 1.062 1.063 1.064 1.065 1.066 1.068 1.069 1.070 1.071 1.072 1.074 1.075 1.076 1.077 1.079 1.080 1.081 1.082 1.084 1.085 1.086 1.088
4.945 4.876 4.810 4.745 4.682 4.620 4.560 4.502 4.445 4.390 4.336 4.284 4.232 4.182 4.134 4.086 4.039 3.994 3.950 3.906 3.864 3.822 3.782 3.742 3.703 3.665 3.628 3.592 3.556 3.521 3.487 3.453 3.420 3.388 3.356 3.326 3.295 3.265 3.236 3.207 3.179 3.152 3.124 3.098 3.072 3.046 3.021 2.996 2.971 2.947 2.924 2.901 2.878 2.855 2.833 2.812 2.790 2.769 2.749 2.729 2.709 2.689 2.669 2.650 2.632 2.613 2.595 2.577 2.559 2.542
cos x
sin x
cot x
tan x
csc x
sec x
12◦
13◦
14◦
15◦
16◦
17◦
18◦
19◦
20◦
21◦
22◦
23◦
A-9
TeamLRN
78◦
77◦
76◦
75◦
74◦
73◦
72◦
71◦
70◦
69◦
68◦
67◦
20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50
x degree
1.3672 1.3643 1.3614 1.3584 1.3555 1.3526 1.3497 1.3468 1.3439 1.3410 1.3381 1.3352 1.3323 1.3294 1.3265 1.3235 1.3206 1.3177 1.3148 1.3119 1.3090 1.3061 1.3032 1.3003 1.2974 1.2945 1.2915 1.2886 1.2857 1.2828 1.2799 1.2770 1.2741 1.2712 1.2683 1.2654 1.2625 1.2595 1.2566 1.2537 1.2508 1.2479 1.2450 1.2421 1.2392 1.2363 1.2334 1.2305 1.2275 1.2246 1.2217 1.2188 1.2159 1.2130 1.2101 1.2072 1.2043 1.2014 1.1985 1.1956 1.1926 1.1897 1.1868 1.1839 1.1810 1.1781 1.1752 1.1723 1.1694 1.1665 x radians
NATURAL TRIGONOMETRIC FUNCTIONS TO FOUR PLACES x radians
x degrees
sin x
cos x
tan x
cot x
sec x
csc x
.4072 .4102 .4131 .4160 .4189 .4218 .4247 .4276 .4305 .4334 .4363 .4392 .4422 .4451 .4480 .4509 .4538 .4567 .4596 .4625 .4654 .4683 .4712 .4741 .4771 .4800 .4829 .4858 .4887 .4916 .4945 .4974 .5003 .5032 .5061 .5091 .5120 .5149 .5178 .5207 .5236 .5265 .5294 .5323 .5352 .5381 .5411 .5440 .5469 .5498 .5527 .5556 .5585 .5614 .5643 .5672 .5701 .5730 .5760 .5789 .5818 .5847 .5876 .5905 .5934 .5963 .5992 .6021 .6050 .6080
20 30 40 50 24◦ 0 10 20 30 40 50 25◦ 00 10 20 30 40 50 26◦ 00 10 20 30 40 50 27◦ 00 10 20 30 40 50 28◦ 00 10 20 30 40 50 29◦ 00 10 20 30 40 50 30◦ 00 10 20 30 40 50 31◦ 00 10 20 30 40 50 32◦ 00 10 20 30 40 50 33◦ 00 10 20 30 40 50 34◦ 00 10 20 30 40 50
.3961 .3987 .4014 .4041 .4067 .4094 .4120 .4147 .4173 .4200 .4226 .4253 .4279 .4305 .4331 .4358 .4384 .4410 .4436 .4462 .4488 .4514 .4540 .4566 .4592 .4617 .4643 .4669 .4695 .4720 .4746 .4772 .4797 .4823 .4848 .4874 .4899 .4924 .4950 .4975 .5000 .5025 .5050 .5075 .5100 .5125 .5150 .5175 .5200 .5225 .5250 .5275 .5299 .5324 .5348 .5373 .5398 .5422 .5446 .5471 .5495 .5519 .5544 .5568 .5592 .5616 .5640 .5664 .5688 .5712
.9182 .9171 .9159 .9147 .9135 .9124 .9112 .9100 .9088 .9075 .9063 .9051 .9038 .9026 .9013 .9001 .8988 .8975 .8962 .8949 .8936 .8923 .8910 .8897 .8884 .8870 .8857 .8843 .8829 .8816 .8802 .8788 .8774 .8760 .8746 .8732 .8718 .8704 .8689 .8675 .8660 .8646 .8631 .8616 .8601 .8587 .8572 .8557 .8542 .8526 .8511 .8496 .8480 .8465 .8450 .8434 .8418 .8403 .8397 .8371 .8355 .8339 .8323 .8307 .8290 .8274 .8258 .8241 .8225 .8208
.4314 .4348 .4383 .4417 .4452 .4487 .4522 .4557 .4592 .4628 .4663 .4699 .4734 .4770 .4806 .4841 .4877 .4913 .4950 .4986 .5022 .5059 .5095 .5132 .5169 .5206 .5243 .5280 .5317 .5354 .5392 .5430 .5467 .5505 .5543 .5581 .5619 .5658 .5696 .5735 .5774 .5812 .5851 .5890 .5930 .5969 .6009 .6048 .6088 .6128 .6168 .6208 .6249 .6289 .6330 .6371 .6412 .6453 .6494 .6536 .6577 .6619 .6661 .6703 .6745 .6787 .6830 .6873 .6916 .6959
2.318 2.300 2.282 2.264 2.246 2.229 2.211 2.194 2.177 2.161 2.145 2.128 2.112 2.097 2.081 2.066 2.050 2.035 2.020 2.006 1.991 1.977 1.963 1.949 1.935 1.921 1.907 1.894 1.881 1.868 1.855 1.842 1.829 1.816 1.804 1.792 1.780 1.767 1.756 1.744 1.732 1.720 1.709 1.698 1.686 1.675 1.664 1.653 1.643 1.632 1.621 1.611 1.600 1.590 1.580 1.570 1.560 1.550 1.540 1.530 1.520 1.511 1.501 1.492 1.483 1.473 1.464 1.455 1.446 1.437
1.089 1.090 1.092 1.093 1.095 1.096 1.097 1.099 1.100 1.102 1.103 1.105 1.106 1.108 1.109 1.111 1.113 1.114 1.116 1.117 1.119 1.121 1.122 1.124 1.126 1.127 1.129 1.131 1.133 1.134 1.136 1.138 1.140 1.142 1.143 1.145 1.147 1.149 1.151 1.153 1.155 1.157 1.159 1.161 1.163 1.165 1.167 1.169 1.171 1.173 1.175 1.177 1.179 1.181 1.184 1.186 1.188 1.190 1.192 1.195 1.197 1.199 1.202 1.204 1.206 1.209 1.211 1.213 1.216 1.218
2.525 2.508 2.491 2.475 2.459 2.443 2.427 2.411 2.396 2.381 2.366 2.352 2.337 2.323 2.309 2.295 2.281 2.268 2.254 2.241 2.228 2.215 2.203 2.190 2.178 2.166 2.154 2.142 2.130 2.118 2.107 2.096 2.085 2.074 2.063 2.052 2.041 2.031 2.020 2.010 2.000 1.990 1.980 1.970 1.961 1.951 1.942 1.932 1.923 1.914 1.905 1.896 1.887 1.878 1.870 1.861 1.853 1.844 1.836 1.828 1.820 1.812 1.804 1.796 1.788 1.781 1.773 1.766 1.758 1.751
cos x
sin x
cot x
tan x
csc x
sec x
A-10
66◦
65◦
64◦
63◦
62◦
61◦
60◦
59◦
58◦
57◦
56◦
40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10 00 50 40 30 20 10
x degree
1.1636 1.1606 1.1577 1.1548 1.1519 1.1490 1.1461 1.1432 1.1403 1.1374 1.1345 1.1316 1.1286 1.1257 1.1228 1.1199 1.1170 1.1141 1.1112 1.1083 1.1054 1.1025 1.0996 1.0966 1.0937 1.0908 1.0879 1.0850 1.0821 1.0792 1.0763 1.0734 1.0705 1.0676 1.0647 1.0617 1.0588 1.0559 1.0530 1.0501 1.0472 1.0443 1.0414 1.0385 1.0356 1.0327 1.0297 1.0268 1.0239 1.0210 1.0181 1.0152 1.0123 1.0094 1.0065 1.0036 1.0007 .9977 .9948 .9919 .9890 .9861 .9832 .9803 .9774 .9745 .9716 .9687 .9657 .9628 x radians
NATURAL TRIGONOMETRIC FUNCTIONS TO FOUR PLACES x radians
x degrees
sin x
cos x
tan x
cot x
sec x
csc x
.6109 .6138 .6167 .6196 .6225 .6254 .6283 .6312 .6341 .6370 .6400 .6429 .6458 .6487 .6516 .6545 .6574 .6603 .6632 .6661 .6690 .6720 .6749 .6778 .6807 .6836 .6865 .6894 .6923 .6952 .6981 .7010 .7039 .7069 .7098 .7127 .7156 .7185 .7214 .7243 .7272 .7301 .7330 .7359 .7389 .7418 .7447 .7476 .7505 .7534 .7563 .7592 .7621 .7650 .7679 .7709 .7738 .7767 .7796 .7825 .7854
35◦ 00 10 20 30 40 50 36◦ 00 10 20 30 40 50 37◦ 00 10 20 30 40 50 38◦ 00 10 20 30 40 50 39◦ 00 10 20 30 40 50 40◦ 00 10 20 30 40 50 41◦ 00 10 20 30 40 50 42◦ 00 10 20 30 40 50 43◦ 00 10 20 30 40 50 44◦ 00 10 20 30 40 50 45◦ 00
.5736 .5760 .5783 .5807 .5831 .5854 .5878 .5901 .5925 .5948 .5972 .5995 .6018 .6041 .6065 .6088 .6111 .6134 .6157 .6180 .6202 .6225 .6248 .6271 .6293 .6316 .6338 .6361 .6383 .6406 .6428 .6450 .6472 .6494 .6517 .6539 .6561 .6583 .6604 .6626 .6648 .6670 .6691 .6713 .6734 .6756 .6777 .6799 .6820 .6841 .6862 .6884 .6905 .6926 .6947 .6967 .6988 .7009 .7030 .7050 .7071
.8192 .8175 .8158 .8141 .8124 .8107 .8090 .8073 .8056 .8039 .8021 .8004 .7986 .7969 .7951 .7934 .7916 .7898 .7880 .7862 .7844 .7826 .7808 .7790 .7771 .7753 .7735 .7716 .7698 .7679 .7660 .7642 .7623 .7604 .7585 .7566 .7547 .7528 .7509 .7490 .7470 .7451 .7431 .7412 .7392 .7373 .7353 .7333 .7314 .7294 .7274 .7254 .7234 .7214 .7193 .7173 .7153 .7133 .7112 .7092 .7071
.7002 .7046 .7089 .7133 .7177 .7221 .7265 .7310 .7355 .7400 .7445 .7490 .7536 .7581 .7627 .7673 .7720 .7766 .7813 .7860 .7907 .7954 .8002 .8050 .8098 .8146 .8195 .8243 .8292 .8342 .8391 .8441 .8491 .8541 .8591 .8642 .8693 .8744 .8796 .8847 .8899 .8952 .9004 .9057 .9110 .9163 .9217 .9271 .9325 .9380 .9435 .9490 .9545 .9601 .9657 .9713 .9770 .9827 .9884 .9942 1.0000
1.428 1.419 1.411 1.402 1.393 1.385 1.376 1.368 1.360 1.351 1.343 1.335 1.327 1.319 1.311 1.303 1.295 1.288 1.280 1.272 1.265 1.257 1.250 1.242 1.235 1.228 1.220 1.213 1.206 1.199 1.192 1.185 1.178 1.171 1.164 1.157 1.150 1.144 1.137 1.130 1.124 1.117 1.111 1.104 1.098 1.091 1.085 1.079 1.072 1.066 1.060 1.054 1.048 1.042 1.036 1.030 1.024 1.018 1.012 1.006 1.0000
1.221 1.223 1.226 1.228 1.231 1.233 1.236 1.239 1.241 1.244 1.247 1.249 1.252 1.255 1.258 1.260 1.263 1.266 1.269 1.272 1.275 1.278 1.281 1 284 1.287 1.290 1.293 1.296 1.299 1.302 1.305 1.309 1.312 1.315 1.318 1.322 1.325 1.328 1.332 1.335 1.339 1.342 1.346 1.349 1.353 1.356 1.360 1.364 1.367 1.371 1.375 1.379 1.382 1.386 1.390 1.394 1.398 1.402 1 406 1.410 1.414
1.743 1.736 1.729 1.722 1.715 1.708 1.701 1.695 1.688 1.681 1.675 1.668 1.662 1.655 1.649 1.643 1.636 1.630 1.624 1.618 1.612 1.606 1.601 1.595 1.589 1.583 1.578 1.572 1.567 1.561 1.556 1.550 1.545 1.540 1.535 1.529 1.524 1.519 1.514 1.509 1.504 1 499 1.494 1.490 1.485 1.480 1.476 1.471 1.466 1.462 1.457 1.453 1.448 1.444 1.440 1.435 1.431 1.427 1.423 1.418 1.414
cos x
sin x
cot x
tan x
csc x
sec x
A-11
TeamLRN
55◦ 00 50 40 30 20 10 54◦ 00 50 40 30 20 10 53◦ 00 50 40 30 20 10 52◦ 00 50 40 30 20 10 51◦ 00 50 40 30 20 10 50◦ 00 50 40 30 20 10 49◦ 00 50 40 30 20 10 48◦ 00 50 40 30 20 10 47◦ 00 50 40 30 20 10 46◦ 00 50 40 30 20 10 45◦ 00 x degree
.9599 .9570 .9541 .9512 .9483 .9454 .9425 .9396 .9367 .9338 .9308 .9279 .9250 .9221 .9192 .9163 .9134 .9105 .9076 .9047 .9018 .8988 .8959 .8930 .8901 .8872 .8843 .8814 .8785 .8756 .8727 .8698 .8668 .8639 .8610 .8581 .8552 .8523 .8494 .8465 .8436 .8407 .8378 .8348 .8319 .8290 .8261 .8232 .8203 .8174 .8145 .8116 .8087 .8058 .8029 .7999 .7970 .7941 .7912 .7883 .7854 x radians
RELATION OF ANGULAR FUNCTIONS IN TERMS OF ONE ANOTHER TRIGONOMETRIC FUNCTIONS Function
sin α
sin α cos α tan α cot α sec α cscα
±
cos α √ 2
sin α √
±
±
±√1−sin2 α ±
1−sin2 α sin α
√
±
√
±
√
1 cos α
±
√
±
1 +cot2 α
1
cot α √
1+tan2 α
±
1+tan2 α tan α
√
1−cos2 α
sec2 α −1 sec α
1 +cot2 α
1 cot α
±
√sec α
±
√cot α
1 1 +tan2 α
1 tan α
1−cos2 α
1
±
1
tan α
√cos α
±
cot α √
1 +tan2 α
1−cos2 α cos α
1−sin2 α
1 sin α
±
cos α √
1 −sin2 α
√sin α
tan α √tan α
1 −cos α
±
±
1 sec α
√
±
√
±
1+cot2 α cot α
√
1+cot2 α
sec2 α−1 1 sec2 α−1
sec α √sec α
±
sec2 α−1
Note: The choice of sign depends upon the quadrant in which the angle terminates.
HYPERBOLIC FUNCTIONS Function sinh x = cosh x = tanh x =
Sinh x Sinh x
1 + sinh2 x √
sinh x 1+sinh2 x 1 sinh x
cosech x = sech x =
√ 1 2 √1+sinh x
coth x =
1+sinh2 x sinh x
Function
cosech x
sinh x =
1 cosechx
cosh x = tanh x = cosech x = sech x = coth x =
±
√
√
+
cosh x cosh2 x − 1
±
1
cosech x ± √ cosecx 2 cosech x+1 2 cosech x + 1
√
cosh x √
cosh2 x−1 cosh x
±√
√
1
ñ cosh x
±
sech x √
sech x ±√
±1 coth2 x−1
±√
coth x
coth2 x−1 1 cothx
1 + sech2 x 1−sech1 zsx
1−tanh2 x tanh x
√
1 sechx
sechx
tanh x
coth x
1−sech2 x sechx
±√
1−tanh2 x
1 tanh x
cosh2 x−1
±
1−tanh2 x 1
1 − tanh2 x
cosh2 x−1 1 cosh x
cosech2 x+1 cosechx
cosech1 x+1
tanh x √ tanh x
± ±
1
√
√
coth2 x−1 1 coth2 x−1 coth x
coth x
1−sech2 x
Whenever two signs are shown, choose + sign if x is positive, −sign if x is negative.
A-12
csc α 1 csc α
√
±
csc2 α−1 csc α
√
1
±
csc2 α−1
±
csc2 α−1
√
√csc α ± csc2 α−1
cscα
DERIVATIVES In the following formulas u, v, w represent functions of x, while a, c, n represent fixed real numbers. All arguments in the trigonometric functions are measured in radians, and all inverse trigonometric and hyperbolic functions represent principal values 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
d (a) = 0 dx d (x) = 1 dx d du (au) = a dx dx d du dv dw (u + v − w) = + − dx dx dx dx d dv du (uv) = u +v dx dx dx d dw du dv (uvw) = uv + vw + uw dx dx dx dx − u dv d u v du 1 du u dv = dx 2 dx = − dx v v v dx v 2 dx d n du (u ) = nun−1 dx dx d √ 1 du u = √ dx 2 u dx d 1 1 du =− 2 dx u u dx d n du 1 = − n+1 dx un u dx n n−1 d u du dv u = m+1 nv − mu dx v m v dx dx d n m du dv n−1 m−1 nv v (u v ) = u + mu dx dx dx d d du [f (u)] = [f (u)] · dx du dx (x)] dy *Let y = f (x) and dx = d[fdx = f (x) define respectively a function and its derivative for any value x in their common domain. The differential for the function at such a value x is accordingly defined as
dy = d[f (x)] =
d[f (x)] dy dx = dx = f (x)dx dx dx
Each derivative formula has an associated differential formula. For example, formula 6 above has the differential formula d(uvw) = uv dw + vw du + uw dv
A-13
TeamLRN
DERIVATIVES (Continued) 15. 16.
17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
2 d2 df (u) d2 u d2 f (u) du [f (u)] = · · 2+ dx2 du dx du2 dx n 2 n−2 n n−1 u u d n d u n dv d n d vd [uv] = + v n+ n n−1 2 n−2 dx 0 dx 1 dx dx 2 dx dx k n−k n u n d vd n d v +··· + + ··· + u k n−k k dx dx n dxn n! where nr = r!(n−r)! is the binomial coefficient, n non-negative integer, and n0 = 1. du dx 1 if = dx = 0 dx du du d 1 du (loga u) = (loga e) dx u dx d 1 du (loge u) = dx u dx d u du (a ) = au (loge a) dx dx d u du (e ) = eu dx dx d v du dv (u ) = vuv−1 + (loge u) uv dx dx dx d du (sin u) = (cos u) dx dx d du (cos u) = − (sin u) dx dx d du (tan u) = (sec2 u) dx dx d du (cot u) = − (csc2 u) dx dx d du (sec u) = sec u · tan u dx dx d du (csc u) = − csc u · cot u dx dx d du (vers u) = sin u dx dx π d du 1 π (arcsin u) = √ , − ≤ arcsin u ≤ dx 2 2 1 − u2 dx
A-14
DERIVATIVES (Continued) 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
d du 1 (arccos u) = − √ , (0 ≤ arccos u ≤ π) 2 dx 1 − u dx π d 1 du π (arctan u) = , − < arctan u < dx 1 + u2 dx 2 2 d 1 du (arc cot u) = − , (0 ≤ arc cot u ≤ π) dx 1 + u2 dx d du 1 π π (arc sec u) = √ , 0 ≤ arc sec u < , −π ≤ arc sec u < − dx 2 2 u u2 − 1 dx d du 1 π π (arc csc u) = − √ , 0 < arc csc u ≤ , −π < arc csc u ≤ − dx 2 2 u u2 − 1 dx d du 1 (arc vers u) = √ , (0 ≤ arc vers u ≤ π) dx 2u − u2 dx d du (sinh u) = (cosh u) dx dx d du (cosh u) = (sinh u) dx dx d du (tanh u) = (sech2 u) dx dx d du (coth u) = − (csch2 u) dx dx d du (sech u) = − (sech u · tanh u) dx dx d du (csch u) = − (csch u · coth u) dx dx d du d 1 (sinh−1 u) = [log(u + u2 + 1)] = √ 2 dx dx u + 1 dx d d 1 du (cosh−1 u) = [log(u + u2 − 1)] = √ , (u > 1, cosh−1 u > 0) dx dx u2 − 1 dx d 1+u 1 du d 1 (tanh−1 u) = log = , (u2 < 1) dx dx 2 1−u 1 − u2 dx d u+1 1 du d 1 (coth−1 u) = log = , (u2 > 1) dx dx 2 u−1 1 − u2 dx
√ d du d 1 + 1 − u2 1 −1 (sech u) = , (0 < u < 1, sech−1 u > 0) log =− √ dx dx u u 1 − u2 dx
√ d d du 1 + 1 + u2 1 −1 (csch u) = log =− √ 2 dx dx u |u| 1 + u dx
49. 50. 51.
d dq
q
f (x) dx = f (q),
[p constant]
p
q
f (x) dx = −f (p), [q constant] q q ∂ dq dp d f (x, a) dx = [f (x, a)] dx + f (q, a) − f (p, a) da da da p p ∂a d dp
p
A-15
TeamLRN
INTEGRATION The following is a brief discussion of some integration techniques. A more complete discussion can be found in a number of good text books. However, the purpose of this introduction is simply to discuss a few of the important techniques which may be used, in conjunction with the integral table which follows, to integrate particular functions. No matter how extensive the integral table, it is a fairly uncommon occurrence to find in the table the exact integral desired. Usually some form of transformation will have to be made. The simplest type of transformation, and yet the most general, is substitution. Simple forms of substitution, such as y = ax, are employed almost unconsciously by experienced users of integral tables. Other substitutions may require more thought. In some sections of the tables, appropriate substitutions are suggested for integrals which are similar to, but not exactly like, integrals in the table. Finding the right substitution is largely a matter of intuition and experience. Several precautions must be observed when using substitutions: 1. Be sure to make the substitution in the dx term, as well as everywhere else in the integral. 2. Be sure that the function substituted is one-to-one and continuous. If this is not the case, the integral must be restricted in such a way as to make it true. See the example following. 3. With definite integrals, the limits should also be expressed in terms of the new dependent variable. With indefinite integrals, it is necessary to perform the reverse substitution to obtain the answer in terms of the original independent variable. This may also be done for definite integrals, but it is usually easier to change the limits. Example:
x4 dx − x2 Here we make the substitution x = |a| sin θ. Then dx = |a| cos θ dθ, and √ a2 − x2 = a2 − a2 sin2 θ = |a| 1 − sin2 θ = |a cos θ| √
a2
Notice the absolute value signs. It is very important to keep in mind that a square √ root radical always denotes the positive square root, and to assure the sign is always kept positive. Thus x2 = |x|. Failure to observe this is a common cause of errors in integration. Notice also that the indicated substitution is not a one-to-one function, that is, it does not have a unique inverse. Thus we must restrict the range of θ in such a way as to make the function one-to-one. Fortunately, this is easily done by solving for θ x θ = sin−1 |a| and restricting the inverse sine to the principal values, − π2 ≤ θ ≤
Thus the integral becomes
π 2
.
a4 sin4 θ|a| cos θ dθ |a| | cos θ|
Now, however, in the range of values chosen for θ, cos θ is always positive. Thus we may remove the absolute value signs from cos θ in the denominator. (This is one of the reasons that the principal values of the inverse trigonometric functions are defined as they are.) Then the cos θ terms cancel, and the integral becomes a4
sin4 θ dθ
By application of integral formulas 299 and 296, we integrate this to −a4
sin3 θ cos θ 3a4 3a4 − cos θ sin θ + θ+C 4 8 8 A-16
INTEGRATION (Continued) We now must perform the inverse substitution to get the result in terms of x. We have θ = sin−1 sin θ =
x |a|
x |a|
√ a2 − x2 x2 2 cos θ = ± 1 − sin θ = ± 1 − 2 = ± . a |a|
Then
Because of the previously mentioned fact that cos θ is positive, we may omit the ± sign. The reverse substitution then produces the final answer x4 x 1 3 3 √ dx = − x3 a2 − x2 − a2 x a2 − x2 + a4 sin−1 + C. 2 2 4 8 8 |a| a −x Any rational function of x may be integrated, if the denominator is factored into linear and irreducible quadratic factors. The function may then be broken into partial fractions, and the individual partial fractions integrated by use of the appropriate formula from the integral table. See the section on partial fractions for further information. Many integrals may be reduced to rational functions by proper substitutions. For example, z = tan
x 2
will reduce any rational function of the six trigonometric functions ofxto a rational function of z. (Frequently there are other substitutions which are simpler to use, but this one will always work. See integral formula number 484.) √ Any rational function of x and ax + b may be reduced to a rational function of z by making the substitution √ z = ax + b. Other likely substitutions will be suggested by looking at the form of the integrand. The other main method of transforming integrals is integration by parts. This involves applying formula number 5 or 6 in the accompanying integral table. The critical factor in this method is the choice of the functions u and v. In order for the method to be successful, v = ∫ dv and ∫ v du must be easier to integrate than the original integral. Again, this choice is largely a matter of intuition and experience. Example:
x sin x dx
Two obvious choices are u = x, dv = sin x dx, or u = sin x, dv = x dx. Since a preliminary mental calculation indicates that ∫ v du in the second choice would be more, rather than less, complicated than the original integral (it would contain x2 ), we use the first choice.
u=x dv = sin x dx x sin x dx =
du = dx v = − cos x
u dv = uv −
v du = −x cos x +
cos x dx
= sin x − x cos x Of course, this result could have been obtained directly from the integral table, but it provides a simple example of the method. In more complicated examples the choice of u and v may not be so
A-17
TeamLRN
INTEGRATION (Continued) obvious, and several different choices may have to be tried. Of course, there is no guarantee that any of them will work. Integration by parts may be applied more than once, or combined with substitution. A fairly common case is illustrated by the following example. Example:
ex sin x dx
Let
u = ex dv = sin x dx
ex sin x dx =
Then du = ex dx v = − cos x
u dv = uv −
v du = −ex cos x +
ex cos x dx
In this latter integral, Let u = ex Then du = ex dx dv = cos x dx v = sin x
ex sin x dx = −ex cos x +
ex cos x dx = −ex cos x +
u dv x = −e cos x + uv − v du = −ex cos x + ex sin x − ex sin x dx
This looks as if a circular transformation has taken place, since we are back at the same integral we started from. However, the above equation can be solved algebraically for the required integral: 1 ex sin x dx = ex sin x − ex cos x 2 In the second integration by parts, if the parts had been chosen as u = cos x, dv = ex dx, we would indeed have made a circular transformation, and returned to the starting place. In general, when doing repeated integration by parts, one should never choose the function u at any stage to be the same as the function v at the previous stage, or a constant times the previous v The following rule is called the extended rule for integration by parts. It is the result of n + 1 successive applications of integration by parts. If g1 (x) = g(x) dx, g2 (x) = g1 (x) dx, g3 (x) = g2 (x) dx, . . . , gm (x) = gm−1 (x) dx, . . . , then
f (x) · g(x) dx = f (x) · g1 (x) − f (x) · g2 (x) + f (x) · g3 (x) − + · · · +(−1)n f (n) (x)gn+1 (x) + (−1)n+1 f (n+1) (x)gn+1 (x) dx.
A-18
INTEGRATION (Continued) A useful special case of the above rule is when f (x) is a polynomial of degree n. Thenf (n+1) (x) = 0, and f (x) · g(x) dx = f (x) · g1 (x) − f (x) · g2 (x) + f (x) · g3 (x) − + · · · + (−1)n f (n) (x)gn+1 (x)+C Example:
If f (x) = x2 , g(x) = sin x x2 sin x dx = −x2 cos x + 2x sin x + 2 cos x + C
Another application of this formula occurs if f (x) = af (x)
and
g (x) = bg(x),
where a and b are unequal constants. In this case, by a process similar to that used in the above example for ∫ ex sin x dx, we get the formula f (x) · g (x) − f (x) · g(x) f (x)g(x) dx = +C b−a This formula could have been used in the example mentioned. Here is another example. Example: If f (x) = e2x , g(x) = sin 3x, then a = 4, b = −9, and e2x 3 e2x cos 3x − 2 e2x sin 3x e2x sin 3x dx = +C = (2 sin 3x − 3 cos 3x) + C −9 − 4 13 The following additional points should be observed when using this table. 1. A constant of integration is to be supplied with the answers for indefinite integrals. 2. Logarithmic expressions are to base e = 2.71828. . ., unless otherwise specified, and are to be evaluated for the absolute value of the arguments involved therein. 3. All angles are measured in radians, and inverse trigonometric and hyperbolic functions represent principal values, unless otherwise indicated. 4. If the application of a formula produces either a zero denominator or the square root of a negative number in the result, there is usually available another form of the answer which avoids this difficulty. In many of the results, the excluded values are specified, but when such are omitted it is presumed that one can tell what these should be, especially when difficulties of the type herein mentioned are obtained. 5. When inverse trigonometric functions occur in the integrals, be sure that any replacements made for them are strictly in accordance with the rules for such functions. This causes little difficulty when the argument of the inverse trigonometric function is positive, since then all angles involved are in the first quadrant. However, if the argument is negative, special care must be used. Thus if u > 0, 1 sin−1 u = cos−1 1 − u2 = csc−1 , etc. u However, if u < 0, 1 sin−1 u = − cos−1 1 − u2 = −π − csc−1 , etc. u See the section on inverse trigonometric functions for a full treatment of the allowable substitutions.
A-19
TeamLRN
INTEGRATION (Continued) 6. In integrals 340–345 and some others, the right side includes expressions of the form A tan−1 [B + C tan f (x)]. In these formulas, the tan−1 does not necessarily represent the principal value. Instead of always employing the principal branch of the inverse tangent function, one must instead use that branch of the inverse tangent function upon which f (x) lies for any particular choice of x. Example:
4π
0
dx 2 + sin x
= = =
Here tan−1
4π 2 tan(x/2 + 1) 2 √ tan−1 √ 3 3 0 2 tan 0 + 1 2 −1 2 tan 2π + 1 √ tan √ √ − tan−1 3 3 3 √ 2 13π π 4 3π 4π √ − =√ = 6 6 3 3 3 2 tan 2π + 1 1 13π √ = tan−1 √ = , 6 3 3
sincef (x) = 2π; and tan−1
2 tan 0 + 1 1 π √ = tan−1 √ = , 6 3 3
since f (x) = 0. 7. Bn and En where used in Integrals represents the Bernoulli and Euler numbers as defined in tables of Bernoulli and Euler polynomials contained in certain mathematics reference and hand-books.
A-20
INTEGRALS
ELEMENTARY FORMS 1. 2. 3. 4 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15.
16. 17. 18.
19. 20.
21. 22.
a dx = ax
a · f (x) dx = a f (x) dx dy φ(y) φ(y) dx = dy, where y = dx y (u + v) dx = u dx + v dx, where u and v are any functions of x u dv = u dv − v du = uv − v du dv du u dx = uv− v dx dx dx n+1 x xn dx = , except n = −1 n+1 f (x) dx = log f (x), (df (x) = f (x) dx) f (x) dx = log x x f (x) dx = f (x), (df (x) = f (x) dx) 2 f (x) ex dx = ex eax dx = eax /a bax dx =
bax , a log b
(b > 0)
log x dx = x log x − x ax log a dx = ax ,
(a > 0)
dx 1 x = tan−1 a2 + x2 a a 1 −1 x tanh a a dx or = 2 2 a −x 1 a+x log a−x , (a2 > x2 ) 2a1 −1 x − a coth a dx or = x2 − a2 1 log x−a , (x2 > a2 ) x+a 2a −1 x sin |a| dx or √ = a2 − x2 x − cos−1 |a| , (a2 > x2 ) √ dx √ = log(x + x2 ± a2 ) x2 ± a2 1 x dx √ = sec−1 |a| a √ x x2 − a2 dx 1 a + a2 ± x2 √ = − log a x x a2 ± x2
A-21
TeamLRN
INTEGRALS (Continued) FORMS CONTAINING (a + bx) For forms containing a + bx, but not listed in the table, the substitution u =
23. 24. 25.
26.
27. 28. 29.
30.
31. 32. 33. 34. 35. 36.
37. 38. 39. 40. 41.
n+1
a+bx x
may prove helpful.
(a + bx) (a + bx) dx = , (n = −1) (n + 1)b 1 a (a + bx)n+2 − 2 (a + bx)n+1 , (n = −1, −2) x(a + bx)n dx = 2 b (n + 2) b (n + 1) n+1 1 (a + bx)n+3 (a + bx)n+2 2 n 2 (a + bx) − 2a +a x (a + bx) dx = 3 b n+3 n + 2 n+1 m+1 n m n−1 x (a+bx) an + m+n+1 x (a + bx) dx m+n+1 or m n+1 m+1 n+1 1 (a + bx) + (m + n + 2) x (a + bx) dx −x xm (a + bx)n dx = a(n+1) or m−1 n m n+1 1 x (a + bx) − ma x (a + bx) dx b(m+n+1) dx 1 = log (a + bx) b a + bx dx 1 =− 2 (a + bx) b(a + bx) dx 1 =− (a + bx)3 2b(a + bx)2 1 b2 [a + bx − a log(a + bx)] x dx = or a + bx x − a log(a + bx) b b2 x dx 1 a = log (a + bx) + 2 b2 a + bx (a + bx) 1 a −1 x dx = + , n = 1, 2 2 n−2 (a + bx)n (n − 1)(a + bx) n−1 b (n − 2) (a + bx) 2 x dx 1 1 = 3 (a + bx)2 − 2a(a + bx) + a2 log (a + bx) a + bx b 2 x2 dx 1 a2 = a + bx − 2a log (a + bx) − (a + bx)2 b3 a + bx 2 1 2a a2 x dx = 3 log (a + bx) + − 3 (a + bx) b a + bx 2(a + bx)2 2 x dx 1 2a −1 = 3 + (a + bx)n b (n − 3) (a + bx)n−3 (n − 2) (a + bx)n−2 a2 − , n = 1, 2, 3 (n − 1) (a + bx)n−1 dx 1 a + bx = − log x(a + bx) a x dx 1 a + bx 1 = − 2 log x(a + bx)2 a(a + bx) a x 2 dx 1 1 2a + bx x = 3 + log x(a + bx)3 a 2 a + bx a + bx a + bx 1 b dx = − + 2 log x2 (a + bx) ax a x dx b2 x 2bx − a + 3 log = x3 (a + bx) 2a2 x2 a a + bx n
A-22
INTEGRALS (Continued) 42.
dx a + 2bx a + bx 2b =− 2 + 3 log x2 (a + bx)2 a x(a + bx) a x
FORMS CONTAINING c2 ± x2 or x2 − c2
43. 44. 45. 46. 47. 48. 49. 50. 51.
dx 1 x = tan−1 2 2 c c c +x 1 c+x dx = log , (c2 > x2 ) 2 − x2 c 2c c − x dx 1 x−c = log , (x2 > c2 ) 2 − c2 x 2c x +c x dx 1 = ± log (c2 ± x2 ) 2 2 2 c ±x x dx 1 =∓ 2 ± x2 )n+1 2 ± x2 )n (c 2n(c dx dx 1 x = + (2n − 3) 2 2 n 2 2 n−1 2c2 (n − 1) (c2 ± x2 )n−1 (c ± x ) (c ± x ) dx 1 dx x = − (2n − 3) − 2 2 n 2c2 (n − 1) (x2 − c2 )n−1 (x2 − c2 )n−1 (x − c ) x dx 1 = log (x2 − c2 ) 2 2 2 x −c x dx 1 =− (x2 − c2 )n+1 2n (x2 − c2 )n
FORMS CONTAINING a + bx AND c + dx, c If k = 0, then v = au
52. 53. 54. 55. 56. 57.
58.
59.
v dx 1 = · log u·v k u x dx 1 a c = log(u) − log(v) u·v k b d dx 1 1 d v = + log u2 · v k u k u v x dx −a c = − 2 log u2 · v bku k u x2 dx a2 1 c2 a(k − bc) log(u) = + log(v) + u2 · v b2 ku k 2 d b2 dx dx 1 −1 = − (m + n − 2)b un · v m k(m − 1) un−1 · v m−1 un · v m−1 u bx k log(v) dx = + v d d2 um −1 um+1 dx k(n−1) vn−1 + b(n − m − 2) v n−1 or m−1 um dx u −1 um = + mk dx d(n−m−1) v n−1 vn vn or m−1 u −1 um − mb dx d(n−1) vn−1 v n−1
A-23
TeamLRN
INTEGRALS (Continued)
60. 61. 62. 63. 64. 65. 66.
67.
68. 69. 70. 71.
72.
73. 74. 75. 76. 77. 78. 79. 80. 81. 82.
FORMS CONTAINING (a + bxn )
√ dx 1 −1 x ab √ tan = , (ab > 0) a + bx2 a √ ab a+x −ab 1 2√−ab log a−x√−ab , (ab < 0) dx √ = or a + bx2 tanh−1 x a−ab , (ab < 0) √ 1 −ab dx 1 bx = tan−1 2 + b2 x2 a ab a x dx 1 = log(a + bx2 ) bx2 2b a+ x2 dx dx x a = − 2 b b a + bx2 a + bx dx x 1 dx = + 2 )2 2) (a + bx 2a(a + bx 2a a + bx2 dx 1 a + bx = log a2 − b2 x2 2ab a − bx x 1 2m−1 dx (a+bx2 )m 2ma (a+bx2 )m + 2ma dx or = dx (a + bx2 )m+1 r!(r−1)! 1 (2m)! x m r=1 (4a)m−r (2r)!(a+bx2 )r + (4a)m (m!)2 2a a+bx2 x dx 1 =− (a + bx2 )m+1 2bm(a + bx2 )m 2 x dx −x 1 dx = + (a + bx2 )m+1 2mb(a + bx2 )m 2mb (a + bx2 )m 1 x2 dx = log 2 2a a+ bx2 x(a + bx ) dx 1 b dx =− − x2 (a + bx2 ) ax a a + bx2 1 1 dx 2am(a+bx2 )m + a x(a+bx2 )m dx or = m x(a + bx2 )m+1 1 x2 ar + log 2 r 2 m+1 r=1 r(a+bx ) a+bx 2a dx 1 b dx dx = − 2 (a + bx2 )m 2 )m+1 x2 (a + bx2 )m+1 a x a (a + bx
dx k 1 (k + x)3 √ −1 2x − k 3 a √ , k = = + 3 tan log a + bx3 3a 2 a + bx3 k 3
b √ a + bx3 1 1 a x dx −1 2x − k √ , k= 3 log = + 3 tan a + bx3 3bk 2 (k + x)3 b k 3 x2 dx 1 3 = log(a + bx ) 3 3b
a + bx x2 + 2kx + 2k2 k 1 2kx a dx −1 log 2 = + tan , ab > 0, k = 4 a + bx4 2a 2 x − 2kx + 2k2 2k2 − x2 4b dx k 1 a x+k −1 x 4 = − log + tan , ab < 0, k = a + bx4 2a 2 x−k k b
2 1 a x dx −1 x = tan , ab > 0, k = a + bx4 2bk k
b x dx x2 − k 1 a log , ab < 0, k = = − a + bx4 4bk x2 + k b
x2 dx 1 1 x2 − 2kx + 2k2 2kx −1 4 a = + tan , ab > 0, k = log a + bx4 4bk 2 x2 + 2kx + 2k2 2k2 − x2 4b 1 a x x−k x2 dx = + 2 tan−1 log , ab < 0, k = 4 − a + bx4 4bk x+k k b
A-24
INTEGRALS (Continued) 83. 84.
85. 86.
87.
88.
x3 dx 1 = log(a + bx4 ) a + bx4 4b dx 1 xn = log n x(a + bx ) an a + bxn dx 1 b dx xn dx = − n m+1 n m (a + bx ) a (a + bx ) a (a + bxn )m+1 m m−n x dx x xm−n dx 1 a dx = − n p+1 n p n p+1 (a + bx ) b (a b (a + + bx ) bx ) dx 1 b dx dx = − m (a + bxn )p m−n (a + bxn )p+1 xm (a + bxn )p+1 a x a x m−n+1 1 (a + bxn )p+1 − a(m − n + 1) xm−n (a + bxn )p dx b(np+m+1) x or m+1 1 (a + bxn )p + anp xm (a + bxn )p−1 dx np+m+1 x or xm (a + bxn )p dx = 1 xm+1 (a + bxn )p+1 − (m + 1 + np + n)b xm+n (a + bxn )p dx a(m+1) or m+1 1 (a + bxn )p+1 + (m + 1 + np + n) xm (a + bxn )p+1 dx −x an(p+1)
89. 90.
91. 92. 93. 94. 95.
96. 97. 98. 99. 100. 101.
FORMS CONTAINING c3 ± x3 dx 1 (c ± x)3 1 2x ∓ c = ± 2 log 3 + √ tan−1 √ 3 ±x 6c c ± x3 c2 3 c 3 dx dx x 2 = 3 3 + 3 (c3 ± x3 )2 3c (c ±x3 ) 3c c3 ± x3 dx dx 1 x = + (3n − 1) (c3 ± x3 )n+1 3nc3 (c3 ± x3 )n (c3 ± x3 )n 3 3 x dx 1 1 c ±x 2x ∓ c = ± √ tan−1 √ log c3 ± x3 6c (c ± x)3 c 3 c 3 x dx x dx x2 1 = 3 3 + 3 (c3 ± x3 )2 3c (c ±x3 ) 3c c3 ± x3 x dx 1 x dx x2 = + (3n − 2) (c3 ± x3 )n+1 3nc3 (c3 ± x3 )n (c3 ± x3 )n 2 x dx 1 = ± log(c3 ± x3 ) c3 ± x3 3 x2 dx 1 =∓ (c3 ± x3 )n+1 3n(c3 ± x3 )n dx x3 1 = 3 log 3 3 3 x(c ± x ) 3c c ± x3 dx 1 x3 1 = 3 3 + 6 log 3 3 3 2 3 x(c ± x ) 3c (c ± x ) 3c c ± x3 dx dx 1 1 = + 3 x(c3 ± x3 )n+1 3nc3 (c3 ± x3 )n c x(c3 ± x3 )n x dx dx 1 1 =− 3 ∓ 3 x2 (c3 ± x3 ) c x c c3 ± x3 dx x dx dx 1 1 = ∓ x2 (c3 ± x3 )n+1 c3 x2 (c3 ± x3 )n c3 (c3 ± x3 )n+1 c3
FORMS CONTAINING c4 ± x4
A-25
TeamLRN
INTEGRALS (Continued) 102.
103. 104.
105. 106. 107. 108.
dx c4 + x4 dx c4 − x4 x dx c4 + x4 x dx c4 − x4 x2 dx c4 + x4 x2 dx c4 − x4 x3 dx c4 ± x4
√ √ 1 1 x2 + cx 2 + c2 cx 2 √ √ + tan−1 2 log c − x2 2c3 2 2 x2 − cx 2 + c2 1 1 c+x x = 3 log + tan−1 2c 2 c−x c 2 1 −1 x = 2 tan 2c c2 2 1 c + x2 = 2 log 2 2 4c √ √ c − x2 1 1 x − cx 2 + c2 −1 cx 2 √ = √ + tan log c2 − x2 2c 2 2 x2 + cx 2 + c2 1 1 c+x −1 x = log − tan 2c 2 c−x c 1 4 4 = ± log (c ± x ) 4 =
FORMS CONTAINING (a + bx + cx2 ) X = a + bx + cx2 and q = 4ac − b2 If q = 0, then X = c x + 109. 110. 111.
112. 113. 114. 115. 116. 117. 118.
119.
120.
b 2 2c ,
and formulas starting with 23 should be used in place of these.
dx 2 2cx + b = √ tan−1 √ , (q > 0) X q q −2 −1 2cx+b √ √−q tanh −q dx or = √ X √1 log 2cx+b−√−q , (q < 0) −q 2cx+b+ −q dx 2cx + b dx 2c = + X2 qX q X 2cx + b 3c dx 6c2 dx 1 = + + X3 q 2X 2 qX q 2 X 2(2n − 1)c dx 2cx + b + n qn Xn nqX dx or = n
n X n+1 2cx + b q r (r − 1)!r! dx (2n)! c + (n!)2 q q cX (2r)! X r=1 x dx dx 1 b = log X − X 2c 2c X x dx bx + 2a dx b = − X2 qX q X x dx 2a + bx b(2n − 1) dx = − − n X n+1 nqX n nq X dx x2 b2 − 2ac x b dx = − 2 log X + X c 2c 2c2 X x2 dx (b2 − 2ac)x + ab 2a dx = + X2 cqX q X m−1 xm−1 n−m+1 b x dx xm dx = − − · X n+1 (2n − m + 1)cX n 2n − m + 1 c X n+1 m−2 m−1 x a dx + · 2n − m+1 c X n+1 dx dx 1 x2 b = log − xX 2a X 2a X
A-26
INTEGRALS (Continued) 2 dx X 1 c b b dx log − − = + 2 2 2a2 x2 ax 2a a x X X dx dx dx 1 b 1 = − + n n−1 n n−1 xX 2a(n − 1)X 2a X a xX dx 1 n+m−1 b dx = − − · xm X n+1 (m − 1)axm−1 X n m−1 a xm−1 X n+1 2n + m − 1 c dx − · m−1 a xm−2 X n+1
121. 122. 123.
124. 125. 126. 127. 128. 129. 130. 131.
132. 133. 134.
136. 137.
138.
√
a + bx
2 a + bx dx = (a + bx)3 3b √ 2(2a − 3bx) (a + bx)3 x a + bx dx = − 15b2 2 √ 2(8a − 12abx + 15b2 x2 ) (a + bx)3 x2 a + bx dx = 105b3 √ m 2 b(2m+3) x (a + bx)3 − ma xm−1 a + bx dx √ xm a + bx dx = or 2 √ m!(−a)m−r r+1 a + bx m r=0 r!(m−r)!(2r+3) (a + bx) bm+1 √ √ dx a + bx √ dx = 2 a + bx + a x x a + bx √ √ dx b a + bx a + bx √ dx = + x2 x 2
x a + bx √ √ (a + bx)3 1 (2m − 5)b a + bx a + bx dx = − + dx xm (m − 1)a xm−1 2 xm−1 √ dx 2 a + bx √ = b a + bx x dx 2(2a − bx) √ √ a + bx =− 3b2 a + bx x2 dx 2(8a2 − 4abx − 3b2 x2 ) √ √ a + bx = 3 15b a + bx m−1 √ x dx 2 (2m+1)b xm a + bx − ma √ m x dx a + bx √ = or √ a + bx m r r √
2(−a) a+bx bm+1√
(−1) m!(a+bx) m r=0 (2r+1)r!(m−r)!ar
√ dx 1 a + bx − a √ = √ log √ √ , (a > 0) a x a + bx a + bx + a 2 dx a + bx √ , (a < 0) = √ tan−1 −a −a x a + bx √ dx b dx a + bx √ √ − =− 2 ax 2a x a + bx x a + bx √ dx (2n−3)b a+bx − √ − (2n−2)a (n−1)axn−1 xn−1 a + bx or
√ n−r−1 n−1 dx b a + bx r!(r − 1)! (2n−2)! √ = − − [(n−1)!]2 xn a + bx a xr 2(r)! 4a r=1 n−1 dx b √ + − 4a x a + bx
135.
FORMS CONTAINING
A-27
TeamLRN
INTEGRALS (Continued) 139. 140.
141. 142.
143.
2(a + bx)22±n b(2 ± n) 2 (a + bx)24±n a(a + bx)2±n ±n 2 2 − x(a + bx) dx = 2 4±n b 2±n 1 b dx dx dx m = m m−2 − a a x(a + bx)2 (a + bx)2 x(a + bx)2 (a + bx)n/2 dx (a + bx) (n−2) /2 = b (a + bx) (n−2) /2 dx + a dx x x 2 √ √ 2 z −a f (x, a + bx) dx = , z z dz, (z = a + bx) f b b n
(a + bx)±2 dx =
FORMS CONTAINING u = a + bx
√
a + bx and
v = c + dx
√
c + dx
k = ad − bc
If k = 0, then, v = ( ac )u, and formulas starting with 124 should be used in place of these.
144.
145. 146. 147. 148. 149. 150.
151. 152. 153. 154.
√ 2 √ tanh−1 bduv , bd > 0, k < 0 bv bd or √ dx −1 2 bduv , bd > 0, k > 0. √ = √bd tanh du uv or √ (bυ+ bduv)2 √1 log , (bd > 0) υ bd √ −1 −bduv √2 tan bv −bd dx or √ = uv − √ 1 sin−1 2bdx+ad+bc , (bd < 0) |k| −bd √ k + 2bv √ k2 dx √ uv dx = uv − uv 4bd √ √ 8bd d u− kd √1 log √ √ kd d u+ kd dx or √ = v u √ √ √1 log (d u− kd)2 , (kd > 0) υ kd √ 2 d u dx √ = √ tan−1 √ , (kd < 0) v u −kd √−kd dx ad + bc uv x dx √ √ = − bd √ 2bd uv uv −2 uv dx √ = v uv √ kv υ dx dx k uυ √ √ = − b 2b uυ uυ
v v dx v √ dx = u |v| uv m √ √ 1 v dx m √ u dx = 2v m+1 u + k v (2m + 3)d √ u 1 dx 3 dx u √ √ = − + m − b (m − 1)k v m−1 2 vm u v m−1 u
A-28
INTEGRALS (Continued) 155.
√ m−1 m 2 u − mk v √u dx b(2m+1) v v dx √ = or √ u 2(m!)2 u m − 4k m−r (2r)!2 v r m
b(2m+1)!
156. 157.
√
158. 159. 160. 161. 162. 163. 164.
165. 166. 167. 168. 169. 170.
171. 172. 173. 174. 175. 176. 177.
r=0
b
(r!)
FORMS CONTAINING
√
x2 ± a2
√ 1 √ 2 x2 ± a2 dx = x x ± a2 ± a2 log (x + x2 ± a2 ) 2 √ dx √ = log (x + x2 ± a2 ) 2 2 x ±a dx 1 x √ = sec−1 |a| a √ x x2 − a2 dx 1 a + x2 + a2 √ = − log a x x x2 + a2 √ √ √ a + x2 + a2 x2 + a2 dx = x2 + a2 − a log x √ x √ x2 − a2 −1 x 2 2 dx = x − a − |a| sec x a √ x dx √ = x2 ± a2 x2 ± a2 √ 1 2 x x2 ± a2 dx = (x ± a2 )3 3 √ 1 2 3a2 x √ 2 3a4 (x2 ± a2 )3 dx = x ± a2 + log(x + x2 ± a2 ) x (x ± a2 )3 ± 4 2 2 dx ±x = √ a2 x2 ± a2 (x2 ± a2 )3 x dx −1 = √ x2 ± a2 (x2 ± a2 )3 1 2 x (x2 ± a2 )3 dx = (x ± a2 )5 5 √ √ x 2 a2 √ 2 a4 x2 x2 ± a2 dx = (x ± a2 )3 ∓ x x ± a2 − log (x + x2 ± a2 ) 4 8 8 √ 1 2 2 2 x3 x2 + a2 dx = ( x2 − a ) (a + x2 )3 5 15 √ 1 2 a2 2 x3 x2 − a2 dx = (x − a2 )5 + (x − a2 )3 5 3 √ x2 dx x√ 2 a2 √ = x ± a2 ∓ log (x + x2 ± a2 ) 2 2 2 2 x ±a √ x3 dx 1 2 2 3 √ = (x ± a ) ∓ a2 x2 ± a2 2 2 3 x ±a √ dx x2 ± a2 √ =∓ 2 x2 x2 ± a2 √ a x √ dx 1 a + x2 + a2 x2 + a2 √ = + log 2 2 2a3 x x3 x2 + a2 √2a x 2 2 dx 1 x −a −1 x √ sec = + 2a2 x2 2|a3 | a x3 x2 − a2 x 2 a2 x 2 a4 x √ 2 2 2 2 3 2 5 x (x ± a ) dx = (x ± a ) ∓ (x ± a2 )3 − x ± a2 6 24 16 √ a6 ∓ log (x + x2 ± a2 ) 16 1 2 a2 2 (x ± a2 )7 ∓ (x ± a2 )5 x3 (x2 ± a2 )3 dx = 7 5
A-29
TeamLRN
INTEGRALS (Continued) √ √ x2 ± a2 dx x2 ± a2 = − + log (x + x2 ± a2 ) 2 √ x √ √ 2x 2 1 a + x2 + a2 x +a x2 + a2 dx = − − log 3 2 2a x √ 2x √ 2x 2 2 2 1 x −a x −a −1 x dx = − + sec 2 x3 2|a| a 2x √ 2 (x2 ± a2 )3 x ± a2 dx = ∓ x4 3a2 x3 √ x2 dx −x = √ + log (x + x2 ± a2 ) 2 2 2 2 3 x ±a (x ± a ) √ x3 dx a2 = x2 ± a2 ± √ 2 2 2 3 x ± a2 (x ± a ) √ dx 1 1 a + x2 + a2 = √ − 3 log 2 2 3 a x a2 x2 + a2 x (x + a ) dx 1 1 −1 x =− √ − 3 sec |a | a a2 x2 − a2 x (x2 − a2 )3 √ 2 1 dx x x ± a2 =− 4 +√ a x x2 ± a2 x2 (x2 ± a2 )3 √ dx 1 3 3 a + x2 + a2 √ √ =− − + 5 log 3 2 2 3 2a x 2a2 x2 x2 + a2 2a4 x2 + a2 x (x + a ) dx 1 3 3 −1 x √ √ = − − sec 2|a5 | a 2a2 x2 x2 − a2 2a4 x2 − a2 x3 (x2 − a2 )3 m−2 x 1 m−1 √ 2 m − 1 xm √ √ dx = x x ± a2 ∓ dx a2 m m x2 ± a2 x2 ± a2
m 2m r!(r − 1)! √ (2m)! x √ dx = 2m x2 ± a2 (∓a2 )m−r (2x)2r−1 2 2 2 2 (m!) (2r)! x ±a r=1 √ +(∓a2 )m log (x + x2 ± a2 ) m √ x2m+1 (2r)!(m!)2 √ dx = x2 ± a2 (∓4a2 )m−r x2r 2 2 2 (2m + 1)!(r!) x ±a √ r=0 dx dx (m − 2) x2 ± a2 √ √ =∓ ∓ (m − 1)a2 xm−1 (m − 1)a2 xm x2 ± a2 xm−2 x2 ± a2 m−1 (m − 1)!m!(2r)!22m−2r−1 √ dx √ = x2 ± a2 (r!)2 (2m)!(∓a2 )m−r x2r+1 x2m x2 ± a2 √ r=0 2 2 x +a m (2m)! r!(r−1)! m−r+1 dx √ = (m!) 2 r=1 (−1) a2 2(2r)!(4a2 )m−r x2r 2m+1 2 2 x x +a √ m+1 x2 +a2 +a + 2(−1) 2m a2m+1 log x
√ m dx (2m)! 1 x2 − a2 r!(r − 1)! −1 x √ = + 2m 2m+1 sec (m!)2 a2 2(2r)!(4a2 )m−r x2r 2 |a| a x2m+1 x2 − a2 r=1 √ 2 2 dx x −a √ =− a(x − a) (x − a) x2 − a2 √ dx x2 − a2 √ = a(x a) x2 − a2 (x +√ + a) x 2 2 f (x, x + a ) dx = a f (a tan u, a sec u) sec2 u du, u = tan−1 , a > 0 a √ x u = sec−1 , a > 0 f (x, x2 − a2 ) dx = a f (a sec u, a tan u) sec u tan u du, a √
178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190.
191. 192. 193.
194.
195. 196. 197. 198. 199.
A-30
INTEGRALS (Continued)
200.
202. 203. 204. 205. 206.
207. 208. 209. 210. 211.
212. 213.
214. 215. 216. 217. 218.
219. 220. 221. 222. 223.
√
a2 − x2
1 √ 2 x a2 − x2 dx = x a − x2 + a2 sin−1 2 |a| −1 x sin |a| dx √ or = a2 − x2 x − cos−1 |a| √ dx 1 a + a2 − x2 √ = − log a x x a2 − x2 √ √ √ a + a2 − x2 a2 − x2 2 2 dx = a − x − a log x x √ x dx √ = − a2 − x2 a2 − x2 √ 1 2 x a2 − x2 dx = − (a − x2 )3 3 1 2 3a2 x √ 2 3a4 x (a2 − x2 )3 dx = a − x2 + sin−1 x (a − x2 )3 + 4 2 2 |a| x dx = √ a2 a2 − x2 (a2 − x2 )3 x dx 1 = √ a2 − x2 (a2 − x2 )3 1 2 x (a2 − x2 )3 dx = − (a − x2 )5 5 √ √ x 2 a2 x x2 a2 − x2 dx = − (a − x2 )3 + x a2 − x2 + a2 sin−1 4 8 |a| √ 1 2 3 2 2 x a2 − x2 dx = (− x − a ) (a2 − x2 )3 5 15 1 a2 x 2 a4 x √ 2 a6 x x2 (a2 − x2 )3 dx = − x (a2 − x2 )5 + (a − x2 )3 + a − x2 + sin−1 6 24 16 16 |a| 1 2 a2 2 x3 (a2 − x2 )3 dx = (a − x2 )7 − (a − x2 )5 7 5 x2 dx x√ 2 a2 x √ =− a − x2 + sin−1 2 2 2 √ 2 |a| a −x dx a2 − x2 √ =− 2 2 2 2 x a −x √a x √ a2 − x2 a2 − x2 x dx = − − sin−1 2 x x |a| √ √ √ 1 a + a2 − x2 a2 − x2 a2 − x2 dx = − + log 3 2 2a x 2x √ x 2 2 3 2 2 (a − x ) a −x dx = − x4 3a2 x3 2 x dx x x = √ − sin−1 2 2 2 2 3 |a| a −x (a − x ) x3 dx 2 2 1√ 2 √ = − (a − x2 )3/2 − x2 (a2 − x2 )1/2 = − a − x2 (x2 + 2a2 ) 2 2 3 3 a −x √ x3 dx x2 a2 √ = 2(a2 − x2 )1/2 + 2 = − + a2 − x2 2 )1/2 2 2 2 2 3 (a − x a −x (a − x ) √ √ dx 1 a + a2 − x2 a2 − x2 √ =− − 3 log 2a2 x2 2a x x3 a2 − x2 √ dx 1 1 a + a2 − x2 = √ − 3 log a x a2 a2 − x2 x (a2 − x2 )3
√
201.
FORMS CONTAINING
A-31
TeamLRN
INTEGRALS (Continued) 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235.
√ 2 dx 1 x a − x2 = 4 − +√ a x a2 − x2 x2 (a2 − x2 )3 √ dx 1 3 3 a + a2 − x2 √ √ =− + − 5 log 2a x 2a2 x2 a2 − x2 2a4 a2 − x2 x3 (a2 − x2 )3 √ m m−1 2 2 x x xm−2 (m − 1)a2 a −x √ √ dx = − dx + m a2 − x2 a2 − x2
m m 2m r!(r − 1)! √ x (2m)! a2m 2m−2r 2r−1 −1 x 2 2 √ dx = x + 2m sin a − a −x (m!)2 22m−2r+1 (2r)! 2 |a| a2 − x2 r=1 m 2m+1 2 √ (2r)!(m!) x √ dx = − a2 − x2 (4a2 )m−r x2r 2 (2m + 1)!(r!) a2 − x2 r=0 √ dx dx m−2 a2 − x2 √ √ =− + (m − 1)a2 xm−1 (m − 1)a2 xm a2 − x2 xm−2 a2 − x2 m−1 (m − 1)!m!(2r)!22m−2r−1 √ ax √ = − a2 − x2 (r!)2 (2m)!a2m−2r x2r+1 x2m a2 − x2
r=0 √ √ m dx (2m)! 1 a − a2 − x2 a2 − x2 r!(r − 1)! √ = + log − (m!)2 a2 2(2r)!(4a2 )m−r x2r 22m a2m+1 x x2m+1 a2 − x2 r=1 √ √ dx 1 (b a2 − x2 + x a2 − b2 )2 √ = √ log , (a2 > b2 ) b2 − x2 (b2 − x2 ) a2 − x2 2b a2 − b2 √ 1 x b2 − a2 dx √ = √ tan−1 √ , (b2 > a2 ) (b2 − x2 ) a2 − x2 b b2 − a2 b √a2 − x2 1 x a2 + b2 dx √ = √ tan−1 √ 2 2 2 2 2 2 (b + x ) a − √ x b a +b √ b a2 − x2 √ 2 2 2 2 2 a −x a +b x a + b2 x dx = − sin−1 sin−1 √ 2 2 b +x |b| |a| |a| x2 + b2 √ x f (x, a2 − x2 ) dx = a f (a sin u, a cos u) cos u du, u = sin−1 , a > 0 a
224.
236.
FORMS CONTAINING
√
a + bx + cx2
X = a + bx + cx2 , q = 4ac − b2 , and k = If q = 0, then 237. 238.
239. 240. 241. 242.
√
X=
√ c x+
b 2c
√ 1 √c log(2 cX + 2cx + b) dx √ = or X √1 sinh−1 2cx+b √ , (c > 0) q c dx 1 −1 2cx + b √ = −√ √ , (c < 0) sin −q −c X 2(2cx + b) dx √ = √ X X q X 2(2cx + b) 1 dx √ √ = + 2k X2 X X √ X 3q 2(2cx+b) X dx√ (2n−1)qX n + 2k(n−1) 2n−1 X n−1 X dx or √ = n n−1 Xn X (2cx+b)(n!)(n−1)!4 k (2r)! n−1 √ r=0 (4kX)r (r!)2 q[(2n)!] X √ √ (2cx + b) X dx 1 √ Xdx = + 4c 2k X
A-32
4c q
INTEGRALS (Continued) 243. 244. 245. 246.
247. 248. 249. 250. 251. 252. 253. 254. 255. 256. 257. 258.
259. 260. 261. 262. 263.
264.
√ √ dx (2cx + b) X 3 3 √ X Xdx = X+ + 2 8c √ 2k 8k X √ dx (2cx + b) X 5X 5 15 2 2 √ + 2 + X Xdx = X + 12c 4k 8k 16k3 X √ √ n−1 (2cx+b)X n X 2n+1 X + Xdx 4(n+1)c 2(n+1)k √ n or X Xdx = √ dx (2n+2)! k(2cx+b) X n r!(r+1)!(4kX)r √ + r=0 c (2r+2)! [(n+1)!]2 (4k)n+1 X √ b dx X x dx √ = √ − c 2c X X 2(bx + 2a) x dx √ =− √ X X q √ X x dx b dx X √ =− √ − n n n (2n − 1)cX 2c X X X X 2 2 x dx 3b − 4ac x dx 3b √ √ √ = X+ − 2 2c 4c 8c2 X X x2 dx (2b2 − 4ac)x + 2ab 1 dx √ = √ √ + c X X cq X X (2b2 − 4ac)x + 2ab 4ac + (2n − 3)b2 dx x2 dx √ = √ + √ X n X (2n − 1)cq X n−1 X (2n −1)cq Xn−1 X x3 dx x2 dx 5bx 5b2 2a √ 3ab 5b3 √ √ = − + 3 − 2 X+ − 2 2 3c 12c 8c 3c 4c 16c3 X X n−1 n−2 xn dx 1 n−1 √ (n − 1)a (2n − 1)b dx dx x x √ √ √ = − X− x nc 2nc nc X X X √ √ X X b b(2cx + b) √ dx √ X− x Xdx = − 3c √ 8c2 4ck X 2 √ √ X X b xX Xdx = − X Xdx 5c √ 2c n+1 √ √ X b X − X n X dx xX n X dx = (2n + 3)c √ 2c √ 5b X X 5b2 − 4ac √ 2 x X dx = x − X dx + 6c√ 4c 16c2 2 aX + bx + 2a 1 dx √ = − √ log , (a > 0) x a x X dx 1 bx + 2a √ = √ √ , (a < 0) sin−1 |x| −q −a x X √ 2 X dx √ =− , (a = 0) bx x X √ b dx dx X √ √ =− − 2 X ax 2a x x X √ √ dx b dx Xdx √ √ +a = X+ x 2 X x √ √ X dx b dx Xdx X √ +c √ =− + x2 x 2 x X X
√
FORMS INVOLVING
√
2ax − x2
√ 1 x−a 2ax − x2 dx = (x − a) 2ax − x2 + a2 sin−1 2 |a|
A-33
TeamLRN
INTEGRALS (Continued) 265.
266. 267. 268.
269. 270.
271.
272.
273. 274. 275. 276. 277.
278. 279. 280. 281.
−1 a−x cos |a| dx √ or = −1 x−a 2ax − x2 sin |a|n−1 n−1 √ (2ax−x2 )3/2 −x + (2n+1)a 2ax − x2 dx x n+2 n+2 or √ n+1 √ xn 2ax − x2 dx = (2n+1)!(r!)2 an−r+1 r 2ax − x2 xn+2 − n r=0 2n−r (2r+1)!(n+2)!n! x n+2 sin−1 x−a + (2n+1)!a 2n n!(n+2)! |a| √ √ 2 1/2 2 (2ax − x ) n−3 2ax − x 2ax − x2 dx = + dx n n x (3 − 2n)ax (2n − 3)a xn−1 n−1 √ 2 n−1 −x 2ax−x √x + a(2n−1) dx n n 2ax−x2 xn dx √ = or 2ax − x2 n √ (2n)!r!(r−1)!an−r r−1 − 2ax − x2 n x + (2n)!a sin−1 r=1 2n (n!)2 2n−r (2r)!(n!)2 √ 2ax − x2 n−1 a(1−2n)xn + (2n−1)a √dx xn−1 2ax−x2 dx √ = or xn 2ax − x2 √ 2n−r (n−1)!n!(2r)! − 2ax − x2 n−1 r=0 (2n)!(r!)2 an−r xr+1 dx x−a = √ (2ax − x2 )3/2 a2 2ax − x2 x dx x = √ (2ax − x2 )3/2 a 2ax − x2
x−a |a|
MISCELLANEOUS ALGEBRAIC FORMS √ dx √ = log(x + a + 2ax + x2 ) 2ax + x2 √ √ √ x√ 2 c ax2 + c dx = ax + c + √ log x a + ax2 + c , (a > 0) 2 2 a √ x√ 2 c a 2 ax + c dx = ax + c + √ , (a < 0) sin−1 x − 2 c 2 −a
√ 1+x dx = sin−1 x − 1 − x2 1−x √ n √ ax +c− c 1 n√c log √axn +c+√c dx √ = or √ n √ x axn + c c √ 2 log ax√+c− , (c > 0) n n c
x n dx ax 2 √ , (c < 0) = √ sec−1 − c x axn + c n −c √ √ dx 1 √ = √ log(x a + ax2 + c), (a > 0) a ax2 + c a 1 dx √ , (a < 0) = √ sin−1 x − c −a ax2 + c x(ax2 +c)m+1/2 + (2m+1)c (ax2 + c)m−1/2 dx 2(m+1) 2(m+1) or √ m (ax2 + c)m+1/2 dx = (2m+1)!(r!)2 cm−r 2 r 2 +c x ax r=0 22m−2r+1 m!(m+1)!(2r+1)! (ax + c) m+1 √ dx + (2m+1)!c 22m+1 m!(m+1)!
m+ 1 2
x(ax2 + c)
2
3
(ax + c)m+ 2 dx = (2m + 3)a
A-34
ax2 +c
INTEGRALS (Continued) 282.
283. 284. 285. 286. 287.
288. 289.
290. 291. 292. 293. 294. 295. 296. 297. 298.
299. 300. 301. 302. 303.
(ax2 + c)m−1/2 (ax2 +c)m+1/2 + c dx 2m+1 x
(ax2 + c)m+1/2 dx = or x √ m cm−r (ax2 +c)r dx m+1 ax2 + c √ + c r=0 2r+1 x ax2 + c dx x 2m−2 (2m−1)c(ax2 +c)m−1/2 + (2m−1)c (ax2 + c)m−1/2 dx = or (ax2 + c)m+1/2 m−1 22m−2r−1 (m−1)!m!(2r)! √ x r=0 (2m)!(r!)2 cm−r (ax2 +c)r ax2 +c √ 2 dx (m − 2)a dx ax + c √ √ − =− m−2 ax2 + c (m − 1)cxm−1√ (m − 1)c xm ax2 + c x √ x 2 + 1 + x4 1 + x2 1 √ dx = √ log 1 − x2 (1 − x2 ) 1 + x4 2 √ 2 1−x 1 x 2 √ dx = √ tan−1 √ (1 + x2 ) 1 + x4 1 + x4 2 √ n dx 2 a + x + a2 √ √ =− log n 2 na xn x x +a dx 2 a −1 √ =− sin √ n na x x xn − a2
3/2 2 x −1 x dx = sin a3 − x3 3 a
FORMS INVOLVING TRIGONOMETRIC FUNCTIONS 1 (sin ax) dx = − cos ax a 1 (cos ax) dx = sin ax a 1 1 (tan ax) dx = − log cos ax = log sec ax a a 1 1 (cot ax) dx = log sin ax = − log csc ax a a π 1 1 ax (sec ax) dx = log(sec ax + tan ax) = log tan + a a 4 2 1 ax 1 (csc ax) dx = log(csc ax − cot ax) = log tan a a 2 1 1 1 1 (sin2 ax) dx = − cos ax sin ax + x = x − sin 2ax 2a 2 2 4a 1 (sin3 ax) dx = − (cos ax)(sin2 ax + 2) 3a 3x sin 2ax sin 4ax (sin4 ax) dx = − + 8 4a 32a sinn−1 ax cos ax n−1 n (sin ax) dx = − + (sinn−2 ax) dx na n m−1 cos ax (2m)! (2m)!(r!)2 sin2r+1 ax + 2m x (sin2m ax) dx = − 2m−2r 2 a 2 (2r + 1)!(m!) 2 (m!)2 r=0 m cos ax 22m−2r (m!)2 (2r)! (sin2m+1 ax) dx = − sin2r ax a r=0 (2m + 1)!(r!)2 1 1 1 1 (cos2 ax) dx = sin ax cos ax + x = x + sin 2ax 2a 2 2 4a 1 (sin ax)(cos2 ax + 2) (cos3 ax) dx = 3a
A-35
TeamLRN
INTEGRALS (Continued) 304.
305. 306. 307. 308.
309. 310. 311.
312.
313. 314. 315.
316. 317. 318. 319. 320.
321. 322. 323.
3x sin 2ax sin 4ax + + 8 4a 32a 1 n−1 n n−1 ax sin ax + (cos ax) dx = cos (cosn−2 ax) dx na n m−1 sin ax (2m)! (2m)!(r!)2 cos2r+1 ax + 2m x (cos2m ax) dx = 2m−2r a r=0 2 (2r + 1)!(m!)2 2 (m!)2 m sin ax 22m−2r (m!)2 (2r)! (cos2m+1 ax) dx = cos2r ax a r=0 (2m + 1)!(r!)2 dx 1 = (csc2 ax)dx = − cot ax 2 a sin ax dx m−2 1 cos ax dx m + ax) dx = − = (csc · sinm ax (m − 1)a sinm−1 ax m−1 sinm−2 ax m−1 22m−2r−1 (m − 1)!m!(2r)! 1 dx = (csc2m ax) dx = − cos ax 2m a sin ax (2m)!(r!)2 sin2r+1 ax r=0 dx = (csc2m+1 ax) dx sin2m+1 ax m−1 1 (2m)!(r!)2 = − cos ax 2m−2r a 2 (m!)2 (2r + 1)! sin2r+2 ax r=0 (cos4 ax) dx =
ax 1 (2m)! log tan · 2m 2 a 2 (m!) 2 dx 1 2 = (sec ax) dx = tan ax cos2 ax a dx 1 sin ax n−2 dx n ax) dx = = (sec · + cosn ax (n − 1)a cosn−1 ax n−1 cosn−2 ax m−1 22m−2r−1 (m − 1)!m!(2r)! 1 dx = (sec2m ax) dx = sin ax 2m cos ax a (2m)!(r!)2 cos2r+1 ax r=0 dx = (sec2m+1 ax) dx cos2m+1 ax m−1 1 (2m)!(r!)2 = sin ax 2m−2r a 2 (m!)2 (2r + 1)! cos2r+2 ax r=0 +
1 (2m)! log(sec ax + tan ax) · a 22m (m!)2 sin(m − n)x sin(m + n)x (sin mx) (sin nx) dx = − , (m2 = n2 ) 2(m − n) 2(m + n) sin(m − n)x sin(m + n)x (cos mx) (cos nx) dx = + , (m2 = n2 ) 2(m − n) 2(m + n) 1 (sin ax) (cos ax) dx = sin2 ax 2a cos(m − n)x cos(m + n)x (sin mx) (cos nx) dx = − − , (m2 = n2 ) 2(m − n) 2(m + n) 1 x (sin2 ax) (cos2 ax) dx = − sin 4ax + 32a 8 cosm+1 ax m (sin ax) (cos ax) dx = − (m + 1)a sinm+1 ax m (sin ax) (cos ax) dx = (m + 1)a cosm−1 ax sinn+1 ax m−1 (cosm−2 ax) (sinn ax) dx + (m+n)a m+n or (cosm ax) (sinn ax) dx = − sinn−1 ax cosm+1 ax + n−1 (cosm ax) (sinn−2 ax) dx +
(m+n)a
A-36
m+n
INTEGRALS (Continued) 324.
325. 326. 327. 328. 329. 330. 331. 332.
333. 334. 335. 336. 337. 338.
339. 340.
341.
cosm ax cosm+1 ax m−n+2 − − (n−1)a dx n−1 n−1 sin ax sinn−2 ax cosm ax or dx = sinn ax cosm−2 ax cosm−1 ax m−1 + dx n−1 n m−n a(m−n) sin ax sin max m+1 sin ax sin ax m−n+2 dx a(n−1) cosn−1 ax − n−1 cosn−2 ax m sin ax or dx = cosn ax sinm−2 ax sinm−1 ax m−1 dx − a(m−n) cosn−1 ax + m−n cosn ax sin ax 1 sec ax dx = = cos2 ax a cos ax a π sin2 ax 1 1 ax dx = − sin ax + log tan + cos ax a a 4 2 cos ax 1 csc ax dx = − =− a sin ax a sin2 ax dx 1 = log tan ax (sin ax) (cos ax) a 1 ax dx = sec ax + log tan 2 (sin ax) (cos ax) a 2 dx 1 dx = + (sin ax) (cosn ax) a(n − 1) cosn−1 ax (sin ax) (cosn−2 ax) π dx 1 1 ax = − csc ax + log tan + 2 a a 4 2 (sin ax) (cos ax) dx 2 = − cot 2ax (sin2 ax) (cos2 ax) a − a(m−1) (sinm−11 ax) (cosn−1 ax) dx m+n−2 + m−2 m−1 dx (sin ax) (cosn ax) = sinm ax cosn ax or dx a(n−1) sinm−11 ax cosn−1 ax + m+n−2 n−1 sinm ax cosn−2 ax 1 sin(a + bx) dx = − cos(a + bx) b 1 cos(a + bx) dx = sin(a + bx) b π dx 1 ax = ∓ tan ∓ 1 ± sin ax a 4 2 dx 1 ax = tan 1 + cos ax a 2 dx 1 ax = − cot 1 − cos ax a 2 a tan x +b 2 √ tan−1 √ 2 2 2 2 2 a −b a −b dx or = √ a + b sin x a tan x +b− b2 −a2 2 √ √ 21 2 log 2 2 b −a a tan x √2 +b+ b −a 2 −b2 tan x a 2 √ 22 2 tan−1 a+b a −b dx or = √ a + b cos x b2 −a2 tan x +a+b 2 √ 1 log √ x b2 −a2
b2 −a2 tan 2 −a−b
A-37
TeamLRN
INTEGRALS (Continued)
342.
343. 344. 345. 346. 347. 348.
349.
350. 351. 352. 353. 354. 355. 356. 357. 358. 359. 360.
361.
dx a+ b sin x + c cos x √ b− b2 +c2 −a2 +(a−c) tan x 2 √ 1 log √ 2 2 2 , if a2 < b2 + c2 , a = c 2 +c2 −a2 b b+ b +c −a +(a−c) tan x 2 or b+(a−c) tan x = √ 2 tan−1 √ 2 2 22 , if a2 > b2 + c2 2 2 2 a −b −c a −b −c or 1 a−(b+c) cos x−(b−c) sin x , if a2 = b2 + c2 , a = c. a a−(b−c) cos x+(b+c) sin x
sin2 x dx 1 a+b x a −1 = tan tan x − , (ab > 0, or |a| > |b|) 2x a + b cos b a a + b b 1 dx b tan x −1 = tan a a2 cos2 x + b2 sin2 x √ ab √ 2 2 cos cx x a + b2 a2 + b2 tan cx −1 dx = tan − 2 2 2 2 2 ab c a b a + b sin cx 1 sin cx cos cx dx = log(a cos2 cx + b sin2 cx) 2 cx + b sin2 cx 2c(b a cos − a) cos cx dx dx = a cos cx + b sin cx a + b tan cx = c(a21+b2 ) [acx + b log(a cos cx + b sin cx)] sin cx dx 1 dx = = [acx − b log (a sin cx + b cos cx)] a sin cx + b cos cx a + b cot cx c(a2 + b2 ) √ c tan x+b− b2 −ac 1 √ √ log , (b2 > ac) 2 b2 −ac c tan x+b+ b2 −ac or dx = √ 1 tan−1 c√tan x+b , (b2 < ac) a cos2 x + 2b cos x sin x + c sin2 x ac−b2 ac−b2 or − c tan1x+b , (b2 = ac) π sin ax 1 ax dx = ±x + tan ∓ a 4 2 1 ± sin ax π dx 1 ax 1 ax = tan ∓ + log tan (sin ax) (1 ± sin ax) a 4 2 a 2 π dx 1 ax ax 1 3 π = − tan − tan − − 2 2a 4 2 6a 4 2 (1 + sin ax) dx 1 π ax ax 1 3 π = cot − cot − + 2 2a 4 2 6a 4 2 (1 − sin ax) sin ax 1 π ax ax 1 3 π dx = − tan − tan − + 2 2a 4 2 6a 4 2 (1 + sin ax) π sin ax 1 ax ax 1 3 π dx = − cot − cot − + 2 4 2 6a 4 2 2a (1 − sin ax) x a dx sin x dx = − b b a + b sin x a + b sin x dx 1 x b dx = log tan − (sin x) (a + b sin x) a 2 a a + b sin x dx dx b cos x a = 2 + 2 2 2 ) (a + b sin x) 2 (a + b sin x) (a − b a − b a + b sin x sin xdx dx a cos x h = 2 + (a + b sin x)2 (b − a2 )(a + b sin x)√ b2 − a2 a + b sin x dx 1 a2 + b2 tan cx −1 tan = √ a a2 + b2 sin2 cx ac a2 + b2 √ a2 −b2 tan cx −1 1 √ tan , (a2 > b2 ) a ac a2 −b2 dx or = √ a2 − b2 sin2 cx 2 2 √1 √b −a tan cx+a , (a2 < b2 ) log 2 2 2 2 2ac
b −a
b −a tan cx−a
A-38
INTEGRALS (Continued) 362. 363. 364. 365. 366. 367. 368. 369. 370. 371. 372. 373. 374.
375. 376. 377. 378. 379. 380. 381. 382. 383. 384. 385. 386. 387. 388.
cos ax 1 ax dx = x − tan 1 + cos ax a 2 cos ax 1 ax dx = −x − cot 1 − cos ax a 2 π dx 1 ax 1 ax = log tan + − tan (cos ax)(1 + cos ax) a 4 2 a 2 π dx 1 ax 1 ax = log tan + − cot (cos ax)(1 − cos ax) a 4 2 a 2 dx 1 ax 1 ax 3 = tan + tan (1 + cos ax)2 2a 2 6a 2 dx 1 ax 1 3 ax = − cot − cot (1 − cos ax)2 2a 2 6a 2 cos ax 1 ax 1 3 ax dx = tan − tan (1 + cos ax)2 2a 2 6a 2 cos ax 1 ax 1 3 ax dx = cot − cot (1 − cos ax)2 2a 2 6a 2 cos x dx x a dx = − a + b cos x b b a + b cos x x 1 π b dx dx = log tan + − (cos x)(a + b cos x) a 2 4 a a + b cos x dx dx b sin x a = 2 − 2 (a + b cos x)2 (b − a2 )(a + b cos x) b − a2 a + b cos x cos x dx a sin x b dx = 2 − (a + b cos x)2 (a − b2 )(a + b cos x) a2 − b2 a + b cos x dx a+b 2 cx −1 = tan tan a2 + b2 − 2ab cos cx c(a2 − b2 ) a−b 2 dx 1 −1 a tan cx √ √ = tan a2 + b2 cos2 cx ac a2 + b2 a2 + b2 −1 √ a tan cx √1 tan , (a2 > b2 ) a2 −b2 ac a2 −b2 dx or = √ a2 − b2 cos2 cx a tan cx− b2 −a2 √1 √ log , (b2 > a2 ) 2 2 2 2 2ac
b −a
a tan cx+
b −a
sin ax 1 dx = ∓ log(1 ± cos ax) 1 ± cos ax a cos ax 1 dx = ± log (1 ± sin ax) 1 ± sin ax a dx 1 1 ax =± + log tan (sin ax)(1 ± cos ax) 2a(1 ± cos ax) 2a 2 π dx 1 1 ax =∓ + log tan + (cos ax)(1 ± sin ax) 2a(1 ± sin ax) 2a 4 2 sin ax 1 dx = log(sec ax ± 1) (cos ax)(1 ± cos ax) a 1 cos ax dx = − log(csc ax ± 1) (sin ax)(1 ± sin ax) a π sin ax 1 1 ax dx = ± log tan + (cos ax)(1 ± sin ax) 2a(1 ± sin ax) 2a 4 2 cos ax 1 1 ax dx = − ± log tan (sin ax)(1 ± cos ax) 2a(1 ± cos ax) 2a 2 dx 1 π ax = √ log tan ± sin ax ± cos ax 2 8 a 2 dx 1 π = tan ax ∓ (sin ax ± cos ax)2 2a 4 dx 1 ax = ± log 1 ± tan 1 + cos ax ± sin ax a 2 b tan cx + a 1 dx log = 2abc b tan cx − a a2 cos2 cx − b2 sin2 cx
A-39
TeamLRN
INTEGRALS (Continued)
389. 390. 391.
392.
393. 394. 395.
396.
397. 398. 399. 400. 401. 402. 403. 404. 405. 406. 407. 408. 409. 410. 411.
1 x sin ax − cos ax 2 a a 2x a2 x2 − 2 2 x (sin ax) dx = 2 sin ax − cos ax a a3 2 2 2 3 3a x − 6 a x − 6x x3 (sin ax) dx = sin ax − cos ax 4 3 a am−1 1 m m cos ax dx − a x cos ax + a x or [ m m−2r r+1 xm sin ax dx = m! 2 ] cos ax · xa2r+1 r=0 (−1) (m−2r)! m−1 [ m−2r−1 ] m! + sin ax r=02 (−1)r (m−2r−1)! · x a2r+2 Note: [s] means greatest integer ≤ s; Thus [3.5] means 3; [5] = 5, 12 = 0. 1 x x(cos ax) dx = 2 cos ax + sin ax a a 2x cos ax a2 x2 − 2 2 x (cos ax) dx = + sin ax 2 a a3 2 2 2 3 3a x − 6 a x − 6x x3 (cos ax) dx = cos ax + sin ax 4 3 a m−1a xm sin ax m − a x sin ax dx a or xm (cos ax)dx = r xm−2r m! sin ax |m/2| r=0 (−1) (m−2r)! · a2r+1 m−2r−1 m! + cos ax |(m−1)/2| (−1)r (m−2r−1)! ·x r=0 1 a2r+2 Note: [s] means greatest integer ≤ s; Thus [3.5] means 3; [5] = 5, 2 = 0. r sin ax (ax)2n+1 (−1)n dx = x (2n + 1)(2n + 1)! n=0 r cos ax (ax)2n (−1)n dx = log x + x 2n(2n)! n=1 2 x x sin 2ax cos 2ax − − x(sin2 ax) dx = 2 4 4a 2 8a x 1 x3 x cos 2ax 2 2 − − 3 sin 2ax − x (sin ax) dx = 6 4a 8a 4a2 x cos 3ax 3x cos ax sin 3ax 3 sin ax x(sin3 ax) dx = − − + 2 12a 36a 4a 4a2 x2 x sin 2ax cos 2ax 2 x(cos ax) dx = + + 2 4 4a 8a 3 2 x x cos 2ax x 1 x2 (cos2 ax) dx = + − 3 sin 2ax + 6 4a 8a 4a2 x sin 3ax 3x sin ax cos 3ax 3 cos ax 3 x(cos ax) dx = + + + 2 12a 36a 4a 4a2 sin ax cos ax sin ax a dx = − + dx m (m − 1)xm−1 m − 1 xm−1 x cos ax cos ax a sin ax dx = − − dx m (m − 1)xm−1 m−1 xm−1 x x x cos ax 1 dx = ∓ + 2 log(1 ± sin ax) 1 ± sin ax a(1 ± sin ax) a x ax x ax 2 dx = tan + 2 log cos 1 + cos ax a 2 a 2 x ax x ax 2 dx = − cot + 2 log sin a 2 a 2 1 − cosax x + sin x x dx = x tan 2 1 + cos x x − sin x x dx = −x cot 1 − cos x 2 x(sin ax) dx =
A-40
INTEGRALS (Continued) √ 2 sin ax ax 2 2 1 − cos ax dx = − √ cos( ) =− a 2 a 1 − cos ax √ √ 2 sin ax 2 2 ax 1 + cos ax dx = √ = sin( ) a 2 a 1 + cos ax √ x x 1 + sin x dx = ±2 sin − cos , 2 2 π π [use + if (8k − 1) 2 < x ≤ (8k + 3) 2 , otherwise − ; k an integer] √ x x 1 − sin x dx = ±2 sin + cos , 2 2 π π [use + if (8k − 3) 2 < x ≤ (8k + 1) 2 , otherwise −; k an integer] √ x dx √ = ± 2 log tan , 4 1 − cos x [use + if 4kπ < x < (4k + 2)π, otherwise −; k an integer] x + π √ dx √ = ± 2 log tan , 4 1 + cos x [use+ if (4k − 1)π < x < (4k + 1)π, otherwise −; k an integer] x √ dx π √ = ± 2 log tan − , 4 8 1 − sin x π π [use+ if (8k + 1) 2 < x < (8k + 5) 2 , otherwise −; k an integer] x √ π dx √ = ± 2 log tan + , 4 8 1 + sin x π π [use+ if (8k − 1) 2 < x < (8k + 3) 2 , otherwise −; k an integer] 1 (tan2 ax) dx = tan ax − x a 1 1 (tan3 ax) dx = tan2 ax + log cos ax 2a a tan3 ax 1 (tan4 ax) dx = − tan ax + x 3a a tann−1 ax (tann ax) dx = − (tann−2 ax) dx a(n − 1) 1 2 (cot ax) dx = − cot ax − x a 1 1 (cot3 ax) dx = − cot2 ax − log sin ax 2a a 1 1 (cot4 ax) dx = − cot3 ax + cot ax + x 3a a cotn−1 ax (cotn ax) dx = − − (cotn−2 ax) dx a(n − 1) x x cot ax 1 dx = x(csc2 ax) dx = − + 2 log sin ax 2 a a sin ax x x cos ax dx = x(cscn ax) dx = − sinn ax a(n − 1) sinn−1 ax 1 − 2 a (n − 1)(n − 2) sinn−2 ax (n − 2) x + dx n−2 (n − 1) sin ax x 1 1 dx = x(sec2 ax) dx = x tan ax + 2 log cos ax 2 a a cos ax x x sin ax dx = x(secn ax) dx = cosn ax a(n − 1) cosn−1 ax 1 − 2 a (n − 1)(n − 2) cosn−2 ax x n−2 dx + n−1 cosn−2 ax
412. 413. 414.
415.
416.
417.
418.
419.
420. 421. 422. 423. 424. 425. 426. 427. 428. 429.
430. 431.
√
A-41
TeamLRN
INTEGRALS (Continued)
432. 433. 434. 435.
436. 437. 438.
439. 440.
1 sin ax b cos ax dx = − sin−1 √ 2 2 ab 1 + b2 1 + b sin ax 1 sin ax dx = − log(b cos ax + 1 − b2 sin2 ax) ab 1 − b2 sin2 ax cos ax 1 + b2 b cos ax (sin ax) 1 + b2 sin2 ax dx = − 1 + b2 sin2 ax − sin−1 √ 2a 2ab 1 + b2 cos ax 2 2 2 2 (sin ax) 1 − b sin ax dx = − 1 − b sin ax 2a 1 − b2 log(b cos ax + 1 − b2 sin2 ax) − 2ab 1 cos ax dx = log(b sin ax + 1 + b2 sin2 ax) 2 sin2 ax ab 1 + b 1 cos ax dx = sin−1 (b sin ax) 2 2 ab 1 − b sin ax sin ax (cos ax) 1 + b2 sin2 ax dx = 1 + b2 sin2 ax 2a 1 log(b sin ax + 1 + b2 sin2 ax) + 2ab sin ax 1 1 − b2 sin2 ax + sin−1 (b sin ax) (cos ax) 1 − b2 sin2 ax dx = 2a 2ab dx ±1 a−b √ = √ sin cx , (a > |b|) sin−1 2 a c a − b a + b tan cx [use+ if (2k − 1) π2 < x ≤ (2k + 1) π2 , otherwise −; k an integer]
441. 442. 443. 444. 445. 446. 447. 448. 449. 450. 451. 452.
453. 454.
FORMS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS √
1 − a2 x2 √ a 1 − a2 x2 −1 −1 (cos ax) dx = x cos ax − a 1 −1 −1 (tan ax) dx = x tan ax − log (1 + a2 x2 ) 2a 1 (cot−1 ax) dx = x cot−1 ax + log (1 + a2 x2 ) 2a √ 1 (sec−1 ax) dx = x sec−1 ax − log (ax + a2 x2 − 1) a √ 1 (csc−1 ax) dx = x csc−1 ax + log (ax + a2 x2 − 1) a x x √ sin−1 (a > 0) dx = x sin−1 + a2 − x2 , a a √ x x cos−1 (a > 0) dx = x cos−1 − a2 − x2 , a a a x x dx = x tan−1 − log (a2 + x2 ) tan−1 a a 2 a −1 x −1 x cot + log(a2 + x2 ) dx = x cot a a 2 √ 1 −1 2 2 x [sin (ax)] dx = 2 [(2a x − 1) sin−1 (ax) + ax 1 − a2 x2 ] 4a √ 1 x [cos−1 (ax)] dx = 2 [(2a2 x2 − 1) cos−1 (ax) − ax 1 − a2 x2 ] 4a xn+1 a xn+1 dx n −1 √ x [sin (ax)] dx = , (n = −1) sin−1 (ax) − n+1 n+1 a2 x2 1− n+1 n+1 x a dx x √ xn [cos−1 (ax)] dx = , (n = −1) cos−1 (ax) + n+1 n+1 1 − a2 x2 (sin−1 ax) dx = x sin−1 ax +
A-42
INTEGRALS (Continued)
455. 456. 457. 458. 459. 460. 461. 462. 463. 464.
465.
466.
467. 468.
469.
1 + a2 x2 x tan−1 ax − 2a2 2a xn+1 a xn+1 xn (tan−1 ax) dx = dx tan−1 ax − n+1 n+1 1 + a2 x2 2 2 1+a x x x(cot−1 ax) dx = cot−1 ax + 2 2a 2a xn+1 a xn+1 xn (cot−1 ax) dx = dx cot−1 ax + n+ n+ 1 1 + a2 x2 1 √ −1 −1 2 2 sin (ax) 1− 1−a x sin (ax) dx = a log − x2 x √ x cos−1 (ax) dx 1 1 + 1 − a2 x2 −1 = − cos (ax) + alog x2 x x −1 tan (ax) dx 1 a 1 + a2 x2 −1 = − tan (ax) − log x2 x 2 x2 −1 cot ax 1 a x2 −1 dx = − cot ax − log 2 2 x2 x 2 a x√ + 1 2 1 − a2 x2 −1 2 −1 2 (sin ax) dx = x(sin ax) − 2x + sin−1 ax √ a 2 1 − a2 x2 (cos−1 ax)2 dx = x(cos−1 ax)2 − 2x − cos−1 ax a √ n 1 − a2 x2 −1 n x(sin ax) + (sin−1 ax)n−1 a − n(n − 1) (sin−1 ax)n−2 dx or −1 n (sin ax) dx = [n/2] n! (−1)r x(sin−1 ax)n−2r (n − 2r)! r=0 √ [n−1/2] 1 − a2 x2 r n! + (−1) (sin−1 ax)n−2r−1 (n − 2r − 1)!a r=0 Note: [s] means greatest means 3; [5] = 5, 12 = 0. integer ≤ s. Thus [3.5] √ n 1 − a2 x2 x(cos−1 ax)n − (cos−1 ax)n−1 a − n(n − 1) (cos−1 ax)n−2 dx or −1 n (cos ax) dx = [n/2] n! (−1)r x(cos−1 ax)n−2r (n − 2r)! r=0 √ [n−1/2] 2 2 r n! 1 − a x × (−1) (cos−1 ax)n−2r−1 (n − 2r − 1)!a r=0 1 1 √ (sin−1 ax)dx = (sin−1 ax)2 2 x2 2a 1 − a xn xn−1 √ xn 2 x2 sin−1 ax + √ (sin−1 ax) dx = − 1 − a na2 n2 a 1 − a2 x2 xn−2 n−1 √ sin−1 ax dx + na2 1 − a2 x2 1 1 √ (cos−1 ax) dx = − (cos−1 ax)2 2a 1 − a2 x2 x(tan−1 ax) dx =
A-43
TeamLRN
INTEGRALS (Continued)
471. 472. 473.
474. 475. 476.
477. 478.
479. 480. 481. 482.
xn xn−1 √ xn (cos−1 ax) dx = − 1 − a2 x2 cos−1 ax − 2 2 2 2 na n a 1−a x n−1 xn−2 −1 √ + cos ax dx na2 1 − a2 x2 −1 1 tan ax dx = (tan−1 ax)2 a2 x2 + 1 2a cot−1 ax 1 dx = − (cot−1 ax)2 a2 x2 + 1 2a x2 1 √ x sec−1 ax dx = sec−1 ax − 2 a2 x2 − 1 2 2a xn+1 1 xn dx √ xn sec−1 ax dx = sec−1 ax − n+1 n+1 a2 x2 − 1 √ −1 −1 2 2 sec ax sec ax a x −1 dx = − + x2 x x x2 1 √ 2 2 −1 −1 x csc ax dx = csc ax + 2 a x − 1 2 2a n+1 x 1 xn dx n −1 −1 √ x csc ax dx = csc ax + n+1 n+1 a2 x2 − 1 √ csc−1 ax csc−1 ax a2 x2 − 1 dx = − − x2 x x √
470.
483. 484.
FORMS INVOLVING TRIGONOMETRIC SUBSTITUTIONS
x 2z dz f (sin x) dx = 2 f , z = tan 2 2 1+z 1+z 2 1 − z2 x dz f (cos x) dx = 2 f , z = tan 1 + z2 1 + z2 2 du , (u = sin x) f (sin x) dx = f (u) √ 1 − u2 du f (cos x) dx = − f (u) √ , (u = cos x) u2 1 − du √ f (sin x, cos x) dx = f u, 1 − u2 √ , (u = sin x) 1− u2 1 − z2 x dz 2z , , z = tan f (sin x, cos x) dx = 2 f 2 2 2 1+z 1+z 1+z 2
LOGARITHMIC FORMS
485. 486. 487. 488. 489.
490.
(log x) dx = x log x − x
x2 x2 log x − 2 4 x3 x3 2 x (log x) dx = log x − 3 9 xn+1 xn+1 n x (log ax) dx = log ax − n+1 (n + 1)2 (log x)2 dx = x(log x)2 − 2x log x + 2x x(log x)n − n (log x)n−1 dx, (n = −1) (log x)n dx = or (− log x)r (−1)n n!x n r=0 r! x(log x) dx =
A-44
INTEGRALS (Continued) 491. 492. 493.
494. 495. 496.
497. 498. 499. 500. 501.
502.
503. 504.
505.
506.
507. 508.
(log x)n 1 dx = (log x)n+1 x n+1 (log x)2 (log x)3 dx = log(log x) + log x + + + ··· log x 2 · 2! 3 · 3! dx = log(log x) x log x 1 dx =− x(log x)n (n − 1)(log x)n−1 xm dx xm dx xm+1 m+1 =− + n n−1 (log x) (n − n − 1 (log x)n−1 1)(log x) m+1 n (log x) n x m+1 xm (log x)n−1 dx − m+1 m n x (log x) dx = or (− log x)r (−1)n n! xm+1 n r=0 r!(m+1)n−r m+1
xp+1 [b sin(b ln x) + (p + 1) cos(b ln x)] + c (p + 1)2 + b2 p+1 x [(p + 1) sin(b ln x) − b cos(b ln x)] + c xp sin(b ln x) dx = (p + 1)2 + b2 ax + b [log(ax + b)] dx = log(ax + b) − x a a log(ax + b) ax + b dx = log x − log(ax + b) x2 b
bx m+1 1 b m m+1 − − log(ax + b) x [log(ax + b)] dx = x m+1 a m+1 m+1 1 ax r b 1 − − − m+1 a r b r=1 log(ax + b) 1 log(ax + b) 1 ax + b a m−1 dx = − + log − m m−1 x m−1 x m−1 b x r m−2 1 a m−1 1 b + , (m > 2) − − m−1 b r ax r=1 x+a log dx = (x + a) log(x + a) − (x − a) log(x − a) x − a xm+1 − am+1 x+a xm+1 − (−a)m+1 xm log log(x + a) − log(x − a) dx = x−a m+1 m+1 [ m+1 2 ] x m−2r+2 2am+1 1 + m + 1 r=1 m − 2r + 2 a Note: [s] means greatest integer ≤ s; Thus [3.5] means 3; [5] = 5, 12 = 0. 1 x+a x−a 1 x2 − a2 1 log − log dx = log 2 x x−a x x+a a √ x2 4ac−b2 b x + 2c log X − 2x + tan−1 √2cx+b 2 , (b2 − 4ac < 0) c 4ac−b or √ b2 −4ac (log X) dx = b log X − 2x + tanh−1 √2cx+b , (b2 − 4ac > 0) x + 2c c b2 −4ac where X = a + bx + cx2 n+2 n+1 xn+1 x x 2c b xn (log X) dx = log X − dx − dx n+1 n+1 X n+1 X 2 where X = a + bx + cx x [log(x2 + a2 )] dx = x log(x2 + a2 ) − 2x + 2a tan−1 a xp cos(b ln x) dx =
A-45
TeamLRN
INTEGRALS (Continued)
509. 510. 511. 512. 513. 514. 515. 516.
x+a x −a 1 2 1 2 2 2 2 2 2 x [log(x ± a )] dx = (x ± a ) log(x ± a ) − x 2 2 √ √ √ 2 2 2 2 [log(x + x ± a )] dx = x log(x + x ± a ) − x2 ± a2 √ 2 √ √ x x2 ± a2 a2 x 2 2 2 2 x [log(x + x ± a )] dx = ± log(x + x ± a ) − 2 4 4 m+1 √ √ x 1 xm+1 m √ x [log(x + x2 ± a2 )] dx = dx log(x + x2 ± a2 ) − m + 1√ m√+ 1 x2 ± a2 √ 2 2 2 2 2 2 log(x + x + a ) 1 a+ x +a log(x + x + a ) dx = − − log 2 x x a x √ √ log(x + x2 − a2 ) log(x + x2 − a2 ) 1 −1 x dx = − + sec x2 x |a| a
1 xn log(x2 − a2 ) dx = xn+1 log(x2 − a2 ) − an+1 log(x − a) n+1 [n/2] 2r n−2r+1 a x −(−a)n+1 log(x + a) − 2 n − 2r + 1 r=0 Note: [s] means greatest integer ≤ s; Thus [3.5] means 3; [5] = 5, 12 = 0. [log(x2 − a2 )] dx = x log(x2 − a2 ) − 2x + a log
EXPONENTIAL FORMS
517. 518. 519. 520.
521.
522. 523. 524. 525. 526. 527.
528.
529.
x
x
e dx = e
e−x dx = −e−x
eax a eax x eax dx = 2 (ax − 1) a xm eax m xm−1 eax dx − a a xm eax dx = or eax m (−1)r m!xm−r r=0 (m−r)!ar+1 ax e dx ax a2 x2 a3 + x3 = log x + + + + ··· 1! 2 · 2! 3 · 3! axx ax ax e e 1 a e dx = − + dx m m−1 m−1 x m − 1 x m − 1 x ax ax e 1 e log x eax log x dx = − dx a a x dx ex = x − log(1 + ex ) = log x 1 + ex 1+e dx x 1 px = − log(a + be ) px a ap
a + be dx 1 a −1 mx √ tan = e , (a > 0, b > 0) aemx + be−mx b m ab √ mx √ ae − b 1 2m√ab log √a emx +√b dx = or a mx aemx − be−mx −1 √ , (a > 0, b > 0) tanh−1 e b m ab ax + a−x x −x (a − a ) dx = log a eax dx =
A-46
INTEGRALS (Continued) 530. 531. 532. 533.
534.
535.
536. 537. 538.
539.
540. 541. 542.
543.
eax 1 dx = log(b + ceax ) b + ceax ac ax xe eax dx = (1 + ax)2 a2 (1 + ax) 1 −x2 −x2 xe dx = − e 2 eax [a sin(bx) − b cos(bx)] ax e [sin(bx)] dx = a2 + b2 ax e [(b − c) sin(b − c)x + a cos(b − c)x] eax [sin(bx)][sin(cx)] dx = 2[a2 + (b − c)2 ] ax e [(b + c) sin(b + c)x + a cos(b + c)x] − 2[a2 + (b + c)2 ] eax [a sin(b−c)x−(b−c) cos(b−c)x] 2[a2 +(b−c)2 ] eax [a sin(b+c)x−(b+c) cos(b+c)x] + 2[a2 +(b+c)2 ] or eax [(a sin bx − b cos bx)[cos(cx − α)] ρ eax [sin(bx)][cos(cx)] dx = −c(sin bx) sin(cx − α)] where 2 2 c2 )2 + 4a2 c2 , ρ = (a + b − ρ cos α = a2 + b2 − c2 , ρ sin α = 2ac eax cos c eax [sin(bx)][sin(bx + c)] dx = 2a eax [a cos(2bx + c) + 2b sin(2bx + c)] − 2(a2 + 4b2 ) ax e sin c eax [sin(bx)][cos(bx + c)] dx = − 2a eax [a sin(2bx + c) − 2b cos(2bx + c)] + 2(a2 + 4b2 ) ax e eax [cos(bx)] dx = 2 [a cos(bx) + b sin(bx)] a + b2 eax [(b − c) sin(b − c)x + a cos(b − c)x] eax [cos(bx)][cos(cx)]dx = 2[a2 + (b − c)2 ] ax e [(b + c) sin(b + c)x + a cos(b + c)x] + 2[a2 + (b + c)2 ] eax cos c ax e [cos(bx)][cos(bx + c)] dx = 2a eax [a cos(2bx + c) + 2b sin(2bx + c)] + 2(a2 + 4b2 ) eax sin c ax e [cos(bx)][sin(bx + c)] dx = 2a eax [a sin(2bx + c) − 2b cos(2bx + c)] + 2(a2 + 4b2 ) 1 eax [sinn bx] dx = 2 (a sin bx − nb cos bx)eax sinn−1 bx a + n2 b2 +n(n − 1)b2 eax [sinn−2 bx] dx 1 eax [cosn bx] dx = 2 (a cos bx + nb sin bx)eax cosn−1 bx a + n2 b2 +n(n − 1)b2 eax [cosn−2 bx] dx
A-47
TeamLRN
INTEGRALS (Continued)
545.
546.
547.
548.
549.
550.
551.
1 m x x e (sin x − cos x) 2 m m xm−1 ex sin x dx + xm−1 ex cos x dx − 2 m ax a sin bx−b cos bx 2 x e a2 +b2 m xm−1 eax (a sin bx − b cos bx) dx − a2 +b2 or xm eax [sin bx] dx = (−1)r m!xm−r ax m r=0 ρr+1 (m−r)! sin[bx − (r + 1)α] e where ρ = a2 + b2 , ρ cos α = a, ρ sin α = b 1 xm ex cos x dx = xm ex (sin x + cos x) 2 m m xm−1 ex sin x dx − xm−1 ex cos x dx − 2 2 m ax a cos bx+b sin bx x e a2 +b2 m xm−1 eax (a cos bx + b sin bx) dx − a2 +b2 m ax x e cos bx dx = or (−1)r m!xm−r ax m cos[bx − (r + 1)α] e √r=0 ρr+1 (m−r)! ρ = a2 + b2 , ρ cos α = a, ρ sin α = b eax cosm−1 x sinn x[a cos x+(m+n) sin x] (m+n) 2 +a2 na eax (cosm−1 x)(sinn−1 x) dx − (m+n)2 +a2 + (m−1)(m+n) eax (cosm−2 x)(sinn x) dx (m+n)2 +a2 or ax m n−1 x[a sin x−(m+n) cos x] e cos x sin(m+n) 2 +a2 ma eax (cosm−1 x)(sinn−1 x) dx + (m+n) 2 +a2 + (n−1)(m+n) eax (cosm x)(sinn−2 x) dx (m+n)2 +a2 or eax (cosm x)(sinn x)dx = eax (cosm−1 x)(sinn−1 x)(a sin x cos x+m sin2 x−n cos2 x) (m+n)2 +a2 m(m−1) eax (cosm−2 x)(sinn x) dx + (m+n)2 +a2 n(n−1) + (m+n) eax (cosm x)(sinn−2 x) dx 2 +a2 or eax (cosm−1 x)(sinn−1 x)(a cos x sin x+m sin2 x−n cos2 x) (m+n)2 +a2 m(m−1) eax (cosm−2 x)(sinn−2 x) dx + (m+n)2 +a2 + (n−m)(n+m−1) eax (cosm x)(sinn−2 x) dx 2 2 (m+n) +a xm ex sin x dx =
544.
xeax (a sin bx − b cos bx) + b2 eax [(a2 − b2 ) sin bx − 2ab cos bx] − 2 (a + b2 )2 xeax xeax (cos bx) dx = 2 (a cos bx − b sin bx) a + b2 eax [(a2 − b2 ) cos bx − 2ab sin bx] − 2 (a + b2 )2 eax a2 + (n − 2)2 eax [a sin x + (n − 2) cos x] eax + dx dx = − n−1 sinn x (n − 1)(n − 2) (n − 1)(n − 2) sin x sinn−2 x xeax (sin bx) dx =
a2
A-48
INTEGRALS (Continued) eax eax eax [a cos x − (n − 2) sin x] a2 + (n − 2)2 dx = − + dx n n−1 cos x (n − 1)(n − 2) cos x (n − 1)(n − 2) cosn−2 x n−1 a tan x eax tann x dx = eax − eax tann−1 x dx − eax tann−2 x dx n−1 n−1
552. 553.
HYPERBOLIC FORMS
554. 555. 556. 557. 558. 559. 560. 561. 562. 563. 564. 565. 566.
567.
568.
569. 570. 571. 572.
(sinh x) dx = cosh x (cosh x) dx = sinh x (tanh x) dx = log cosh x (coth x) dx = log sinh x
(sech x) dx = tan−1 (sinh x) x (csch x) dx = log tanh 2 x(sinh x) dx = x cosh x − sinh x xn (sinh x) dx = xn cosh x − n xn−1 (cosh x)dx x(cosh x) dx = x sinh x − cosh x xn (cosh x) dx − xn sinh x − n xn−1 (sinh x) dx (sech x)(tanh x) dx = −sechx (csch x)(coth x) dx = −cschx sinh 2x x (sinh2 x)dx = − 4 2 1 (sinh m+1 x)(coshn−1 x) m+n n−1 (sinhm x)(coshn−2 x) dx + m+n or (sinhm x)(coshn x) dx = m−1 1 sinh x coshn+1 x m+n m−1 (sinhm−2 x)(coshn x)dx, (m + n = 0) − m+n 1 − (m−n)(sinhm−1 x)(coshn−1 x) dx − m+n−2 , (m = 1) m−2 m−1 (sinh x)(coshn x) dx or (sinhm x)(coshn x) 1 (n−1) sinhm−1 x coshn−1 x dx m+n−2 + n−1 , (n = 1) m n−2 (sinh x)(cosh x) (tanh2 x) dx = x − tanh x tanhn−1 x (tanhn x) dx = − + (tanhn−2 x)dx, (n = 1) n−1
(sech2 x) dx = tanh x (cosh2 x) dx =
sinh 2x x + 4 2
A-49
TeamLRN
INTEGRALS (Continued) 573. 574. 575. 576. 577. 578. 579.
580. 581. 582. 583.
584. 585. 586. 587. 588. 589. 590. 591. 592.
593. 594. 595. 596.
(coth2 x) dx = x − coth x (cothn x) dx = −
cothn−1 x + n−1
cothn−2 x dx,
(n = 1)
(csch2 x) dx = −ctnh x sinh(m + n)x sinh(m − n)x − , (m2 = n2 ) 2(m + n) 2(m − n) sinh(m + n)x sinh(m − n)x (cosh mx)(cosh nx) dx = + , (m2 = n2 ) 2(m + n) 2(m − n) cosh(m + n)x cosh(m − n)x (sinh mx)(cosh nx) dx = + , (m2 = n2 ) 2(m + n) 2(m − n) x x √ sinh−1 dx = x sinh−1 − x2 + a2 , (a > 0) a a 2 a2 x√ 2 x x −1 x x sinh x + a2 , (a > 0) + sinh−1 − dx = a 2 4 a 4 n+1 1 x xn+1 xn sinh−1 x dx = (n = −1) sinh−1 x − 1 dx, n+1 n+1 (1 + x2 ) 2 √ −1 x −1 x x cosh a − x2 − a2 , cosh a > 0 −1 x or dx = cosh a x cosh−1 xa + x2 − a2 , cosh−1 xa < 0 , (a > 0) 1 x 2x2 − a2 x x x cosh−1 cosh−1 − (x2 − a2 ) 2 dx = a 4 a 4 xn+1 1 xn+1 xn (cosh−1 x) dx = (n = −1) cosh−1 x − 1 dx, n+1 n+1 (x2 − 1) 2 a x x x dx = x tanh−1 + log(a2 − x2 ), tanh−1 <1 a a 2 a a x −1 x −1 x 2 2 + log(x − a ), coth dx = x coth >1 a a 2 a x ax x2 − a2 −1 x −1 x x tanh tanh + , dx = <1 a 2 a 2 a xn+1 1 xn+1 n −1 −1 dx, (n = −1) x (tanh x) dx = tanh x − n+1 n+1 1 − x2 2 2 ax x x x x −a x coth−1 coth−1 + , >1 dx = a 2 a 2 a n+1 x 1 xn+1 xn coth−1 x dx = coth−1 x + dx, (n = −1) n+1 n+1 x2 − 1 (sinh mx)(sinh nx) dx =
(sech−1 x) dx = xsech−1 x + sin−1 x x2 1√ 1 − x2 sech−1 x − 2 2 xn xn+1 1 xn sech−1 x dx = (n = −1) sech−1 x + 1 dx, n+1 n+1 (1 − x2 ) 2 x sinh−1 x csch−1 x dx = xcsch−1 x + |x| x2 1 x √ 1 + x2 csch−1 x + xcsch−1 x dx = 2 2 |x| xn+1 1 x xn xn csch−1 x dx = (n = −1) csch−1 x + 1 dx, n+1 n + 1 |x| (x2 + 1) 2 xsech−1 x dx =
A-50
INTEGRALS (Continued) DEFINITE INTEGRALS
n 1 ∞ n−1 ∞ n−1 −x 1 m = Γ(n) x e dx = 0 log x1 dx = n1 n 0 m=1 1+ m for n = 0, −1, −2, −3, . . . (This is the Gamma function) ∞ n! tn p−t dt = , (n = 0, 1, 2, 3, . . . and p > 0) (log p)n+1 0 ∞ Γ(n) tn−1 e−(a+1)t dt = , (n > 0, a > −1) (a + 1)n n 0 1 Γ(n + 1) 1 xm log dx = , (m > −1, n > −1) x (m + 1)n+1 0 Γ(n) is finite if n > 0; Γ(n + 1) = nΓ(n) Γ(n) · Γ(1 − n) = sinπnπ Γ(n) = (n − 1)! if n = integer > 0 ∞ √ 2 1 1 e−t dt = π = 1.7724538509 · · · = (− )! Γ( ) = 2 2 2 0 √ Γ(n + 12 ) = 1·3·5...(2n−1) π n = 1, 2, 3, . . . 2n √ (−1)n 2n π Γ(−n + 12 ) = 1·3·5...(2n−1) n = 1, 2, 3, . . . 1 ∞ Γ(m)Γ(n) xm−1 xm−1 (1 − x)n−1 dx = dx = = B(m, n) (1 + x)m+n Γ(m + n) 0 0 (This is the Beta function) B(m, n) = B(n, m) = Γ(m)Γ(n) , where m and n are any positive real numbers. Γ(m+n) b Γ(m + 1) · Γ(n + 1) (x − a)m (b − x)n dx = (b − a)m+n+1 , (m > −1, n > −1, b > a) Γ(m + n + 2) a∞ 1 dx = , [m > 1] m x m −1 1 ∞ dx = π csc pπ, [0 < p < 1] (1 + x)xp 0 ∞ dx = −π cot pπ, [0 < p < 1] (1 − x)xp 0 ∞ p−1 π x dx = = B(p, 1 − p) = Γ(p)Γ(1 − p), [0 < p < 1] (1 + x) sin pπ 0 ∞ m−1 π x dx = , [0 < m < n] 1 + xn n sin mπ 0 n
a+1−bc ∞ Γ c − a+1 Γ a+1 m b xa dx b b = (m + xb )c b Γ(c) 0 a+1 > −1, b > 0, m > 0, c > b a∞ dx √ =π (1 + x) x 0 ∞ π π a dx = , if a > 0; 0, if a = 0; − , if a < 0 2 + x2 a 2 2 0 a 1 a 2 2 2 n/2 2 n/2 (a − x ) dx = (a − x ) dx 2 −a 0 π 1 · 3 · 5...n · · an+1 (n odd, a > 0) = 2 · 4 · 6 . . . (n + 1) 2 1 m+n+1 B m+1 , n+2 (a > 0, m > −1, n > −2) 2a a 2 2 m 2 2 n/2 or x (a − x ) dx = m+1 )Γ( n+2 2 ) 0 1 am+n+1 Γ( 2m+n+3 (a > 0, m > −1, n > −2) 2 Γ( ) 2 1+
597.
598. 599. 600. 601. 602. 603. 604. 605. 606. 607. 608. 609. 610. 611. 612. 613. 614. 615.
616. 617. 618.
619.
A-51
TeamLRN
INTEGRALS (Continued)
620.
621. 622. 623.
π/2 (cosn x) dx 0 or π π/2 1·3·5·7...(n−1) , (n an even integer, n = 0) 2·4·6·8...(n) 2 (sinn x) dx = or 0 1·3·5·7...(n−1) , (n an odd integer, n = 0) 2·4·6·8...(n) n+1 √ π Γ( 2 ) , (n > −1) 2 Γ( n +1) 2 ∞ π π sin mx dx = ; if m > 0; 0, if m = 0; − , if m < 0 x 2 2 0 ∞ cos x dx =∞ x 0 ∞ π tan x dx = x 2 0 π
π
sin ax · sin bx dx =
624.
0 π/a 625. 626. 627. 628.
0 π
0
[sin(ax)][cos(bx)] dx =
0 ∞ 0 ∞ 0 π
632. 633. 634. 635. 636. 637. 638. 639. 640. 641. 642. 643. 644.
2a , if a − b is odd, or 0 if a − b is even a 2 − b2 π π if m < −1 or m > 1; , if m = ±1; , if m2 < 1 4 2 (a ≤ b)
π
0 ∞
631.
sin x cos mx dx = 0, x πa sin ax sin bx dx = , x2 2
sin2 mx dx =
629. 630.
cos ax · cos bx dx = 0, (a = b; a, b integers) π [sin(ax)][cos(ax)] dx = [sin(ax)][cos(ax)]dx = 0 0
2
cos2 mx dx = 0
π 2
(m is a non-zero integer)
π|p| sin (px) dx = x2 2 ∞ π sin x dx = , 0 p > 0; , p > q > 0; , p = q > 0 x 2 4 0 ∞ π −|ma| cos(mx) dx = e x2 + a2 2 |a|
0 ∞ ∞ 1 π cos(x2 ) dx = sin(x2 ) dx = 2 2 0 0 ∞ 1 π sin axn dx = Γ(1/n) sin , n>1 na1/n 2n 0 ∞ 1 π cos axn dx = Γ(1/n) cos , n>1 1/n na 2n
∞ 0 ∞ sin x cos x π √ dx = √ dx = 2 x x 0 0 ∞ 3 ∞ 3 (a) 0 sinx x dx = π4 (b) 0 sinx2 x dx 34 log 3 ∞ 3π sin3 x dx = 3 x 8 0 ∞ π sin4 x dx = x4 3 0 π/2 cos−1 a dx , (|a| < 1) = √ 1 − a2 0 π 1 + a cos x π dx , (a > b ≥ 0) = √ 2 − b2 a + b cos x a 0 0
A-52
INTEGRALS (Continued)
645. 646. 647. 648. 649. 650. 651. 652. 653. 654. 655. 656. 657.
658.
659. 660.
661.
662. 663. 664. 665.
2π
2π dx , (a2 < 1) = √ 1 + a cos x 1 − a2 ∞ b cos ax − cos bx dx = log x a 0 π/2 π dx = 2 sin2 x + b2 cos2 x 2|ab| a 0 π/2 π(a2 + b2 ) dx = , (a, b > 0) 2 2 2 2 2 4a3 b3 (a sin x + b cos x) 0 π/2 1 n m sinn−1 x cosm−1 x dx = B , , m and n positive integers 2 2 2 0 π/2 2 · 4 · 6 . . . (2n) (sin2n+1 θ) dθ = , (n = 1, 2, 3, . . . ) 1 · 3 · 5 . . . (2n + 1) 0 π/2 1 · 3 · 5 . . . (2n − 1) π (sin2n θ) dθ = , (n = 1, 2, 3, . . . ) 2 · 4 . . . (2n) 2 # " 0 π/2 1 1 1 1 x − + − + · · · dx = 2 sin x 12 32 52 72 0 π/2 dx π = 1 + tanm x 4 0 π/2 √ 3 (2π) 2 cos θ dθ = 1 2 Γ( 4 ) 0 π/2 π hπ , (0 < h < 1) (tanh θ) dθ = 2 cos 2 0 ∞ −1 −1 π a tan (ax) − tan (bx) dx = log , (a, b > 0) x 2 b 0 b b b The area enclosed by a curve defined through the equation x c + y c = a c where a > 0, c 2 c [Γ( )] 2ca2 a positive odd integer and b a positive even integer is given by Γ b2c b (b) 0
I =
xh−1 y m−1 z n−1 dv, where R denotes the region of space bounded by the co-
R p q k ordinate planes and that portion of the surface xa + yb + zc = 1, which lies in the first octant and where h, m, n, p, q, k, a, b, c, denote positive real numbers is given by a h c x p 1 x p y q 1 e m e n−1 h−1 x dx y dy − z dz 1− 1− a a b 0 0 0 h m n ah bm cn Γ p Γ q Γ k = pqk Γ h + m + n + 1 p q k ∞ 1 −ax e dx = , (a > 0) a 0 ∞ −ax b e − e−bx dx = log , (a, b > 0) x a 0 Γ(n+1) n > −1, a > 0 ∞ an+1 xn e−ax dx = or 0 n! (a > 0, n positive integer) an+1 ∞ Γ(k) n+1 n p x exp(−ax )dx = , n > −1, p > 0, a > 0, k = pak p 0 ∞ √ 2 2 1 1 1 −a x e dx = π= Γ , (a > 0) 2a 2a 2 0 ∞ 2 1 xe−x dx = 2 0 √ ∞ 2 π x2 e−x dx = 4 0
A-53
TeamLRN
INTEGRALS (Continued)
1 · 3 · 5 . . . (2n − 1) π (a > 0, n > − 12 ) 2n+1 an a 0 ∞ 2 n! x2n+1 e−ax dx = n+1 , (a > 0, n > −1) 2a
0 1 m m! ar m −ax −a x e dx = m+1 1 − e a r! 0 r=0 √ ∞ 2 a2 −2a −x − 2 e π x e dx = , (a ≥ 0) 2
0 ∞ √ 1 π e−nx x dx = (n > 0) 2n n 0
∞ −nx e π √ dx = (n > 0) n x 0 ∞ a e−ax (cos mx) dx = 2 , (a > 0) a + m2 0 ∞ m e−ax (sin mx) dx = 2 , (a > 0) a + m2 0 ∞ 2ab xe−ax [sin(bx)] dx = 2 , (a > 0) (a + b2 )2 0 ∞ a2 − b2 xe−ax [cos(bx)] dx = 2 , (a > 0) (a + b2 )2 0 ∞ n![(a + ib)n+1 − (a − ib)n+1 ] xn e−ax [sin(bx)] dx = , (i2 = −1, a > 0) 2i(a2 + b2 )n+1 0 ∞ n![(a − ib)n+1 + (a + ib)n+1 ] xn e−ax [cos(bx)] dx = , (i2 = −1, a > 0, n > −1) 2 + b2 )n+1 2(a 0 ∞ −ax e sin x dx = cot−1 a, (a > 0) x 0 √ ∞ b2 π −a2 x2 e cos bx dx = exp − 2 , (ab = 0) 2|a| 4a 0 ∞ π π −t cos φ b−1 e t [sin(t sin φ)] dt − [Γ(b)] sin(bφ), b > 0, − < φ < 2 2 0 ∞ π π e−t cos φ tb−1 [cos(t sin φ)] dt − [Γ(b)] cos(bφ), b > 0, − < φ < 2 2 0 ∞ bπ b−1 t cos t dt = [Γ(b)] cos , (0 < b < 1) 2 0 ∞ bπ tb−1 (sin t) dt = [Γ(b)] sin , (0 < b < 1) 2 0 1 (log x)n dx = (−1)n · n! (n > −1) 0 1 √ 1 1 2 π dx = log x 2 0 − 1 1 2 √ 1 dx = π log x 0 1 n 1 dx = n! log x 0 1 3 x log(1 − x) dx = − 4 0 1 1 x log(1 + x) dx = 4 0 1 (−1)n n! xm (log x)n dx = , m > −1, n = 0, 1, 2, . . . (m + 1)n+1 0 If n = 0, 1, 2, . . . replace n! by Γ(n + 1).
666. 667. 668. 669. 670. 671. 672. 673. 674. 675. 676. 677. 678. 679. 680. 681. 682. 683. 684. 685. 686. 687. 688. 689. 690.
∞
2
x2n e−ax dx =
A-54
INTEGRALS (Continued)
1
691. 0 1 692. 0 1 693. 0 1 694. 0 1 695. 0 1 696. 0 1
697. 698. 699. 700. 701. 702. 703. 704. 705. 706. 707. 708. 709. 710. 711. 712. 713. 714. 715. 716.
π2 log x dx = − 1+x 12 π2 log x dx = − 1−x 6 π2 log(1 + x) dx = x 12 π2 log(1 − x) dx = − x 6 (log x)[log(1 + x)] dx = 2 − 2 log 2 − (log x)[log(1 − x)] dx = 2 −
π2 12
π2 6
π2 log x dx = − 2 1 −x 8 0 1 1+x π2 dx log = · 1 − x x 4 0 1 π log x dx √ = − log 2 2 2 0 1 1 − x n Γ(n + 1) 1 m x log dx = , if m + 1 > 0, n + 1 > 0 x (m +1)n+1 0 1 p p+1 (x − xq )dx = log , (p + 1 > 0, q + 1 > 0) log x q+1 0 1 √ dx = π, (same as integral 686) 0 log x1 x ∞ π2 e +1 log dx = x e −1 4 0 π/2 π/2 π (log sin x) dx = log cos x dx = − log 2 2 0 0 π/2 π/2 π (log sec x) dx = log csc x dx = log 2 2 0 0 π π2 x(log sin x) dx = − log 2 2 0 π/2 (sin x)(log sin x) dx = log 2 − 1 0 π/2 (log tan x) dx = 0 √ 0π a + a 2 − b2 log(a ± b cos x) dx = π log , (a ≥ b) 2 0 $ π 2π log a a≥b>0 log(a2 − 2ab cos x + b2 ) dx = 2π log b b≥a>0 0 ∞ π aπ sin ax dx = tanh 2|b| 2b 0 ∞ sinh bx π aπ cos ax dx = sech 2|b| 2b 0 ∞ cosh bx π dx = cosh ax 2|a| 0 ∞ π2 x dx = 2 (a > 0) 4a 0 ∞ sinh ax a −ax e (cosh bx) dx = 2 , (0 ≤ |b| < a) a − b2 0 ∞ b e−ax (sinh bx) dx = 2 , (0 ≤ |b| < a) a − b2 0
A-55
TeamLRN
INTEGRALS (Continued)
717. 718. 719.
∞
π aπ 1 sinh ax dx = csc − (b > 0) bx + 1 e 2b b 2a 0 ∞ 1 π aπ sinh ax dx = − cot (b > 0) ebx − 1 2a 2b b 0 2 2 2 π/2 π 1·3 1·3·5 dx 1 2 4 6 = k + k + k + ··· , 1+ 2 2 2·4 2·4·6 0 1 − k2 sin2 x if k2 < 1
2 4 6 2 π/2 π 1·3 1·3·5 k k 1 2 2 2 1 − k sin x dx = k − − − ··· , 2 1− 2 2 2·4 3 2·4·6 5 0
720.
721. 722. 723. 724.
725. 726.
727. 728.
if k2 < 1
∞
e−x log x dx = −γ = −0.5772157 . . . √ ∞ 2 π e−x log x dx = − (γ + 2 log 2) 4 0 ∞ 1 1 − [Euler’s Constant] e−x dx = γ = 0.5772157 . . . −x 1 − e x 0 ∞ 1 1 − e−x dx = γ = 0.5772157 . . . x 1 + x 0 For n even : n/2−1 n sin(n − 2k)x 1 n 1 + n cosn x dx = n−1 x k 2 (n − 2k) 2 n/2 k=0 n/2−1 n sin[(n − 2k)( π −x)] 1 1 n n 2 + n x sin x dx = n−1 k 2 2k − n 2 n/2 k=0 For n odd: (n−1)/2 n sin(n − 2k)x 1 n cos x dx = n−1 k 2 n − 2k k=0 π (n−1)/2 n sin (n − 2k) 2 −x 1 sinn x dx = n−1 k 2 2k–n
0
k=0
A-56
2k − n
DIFFERENTIAL EQUATIONS
SPECIAL FORMULAS Certain types of differential equations occur sufficiently often to justify the use of formulas for the corresponding particular solutions. The following set of tables I to XIV covers all first, second, and nth order ordinary linear differential equations with constant coefficients for which the right members are of the form P (x)erx sin sx or P (x)erx cos sx, where r and s are constants and P (x), is a polynomial of degree n. When the right member of a reducible linear partial differential equation with constant coefficients is not zero, particular solutions for certain types of right members are contained in tables XV to XXI. In these tables both F and P are used to denote polynomials, and it is assumed that no denominator is zero. In any formula the roles of x and y may be reversed throughout, changing a formula in which x dominates to one in which y dominates. XXI are applicable Tables XIX, XX, m! st whether the equations are reducible or not. The symbol m stands for n (m−n)!n! and is the n + 1 coefficient in the expansion of (a + b)m . Also 0! = 1 by definition.
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DIFFERENTIAL EQUATIONS (Continued) Solution of Linear Differential Equations with Constant Coefficients Any linear differential equation with constant coefficients may be written in the form p(D)y = R(x) where • • • • •
D is the differential operation: Dy = p(D) is a polynomial in D, y is the dependent variable, x is the independent variable, R(x) is an arbitrary function of x.
dy dx
A power of D represents repeated differentiation, that is dn y dxn For such an equation, the general solution may be written in the form Dn y =
y = yc + yp where yp is any particular solution, and yc is called the complementary function. This complementary function is defined as the general solution of the homogeneous equation, which is the original differential equation with the right side replaced by zero, i.e. p(D)y = 0 The complementary function yc may be determined as follows: 1. Factor the polynomial p(D) into real and complex linear factors, just as if D were a variable instead of an operator. 2. For each nonrepeated linear factor of the form (D - a), where a is real, write down a term of the form ceax where c is an arbitrary constant. 3. For each repeated real linear factor of the form (D − a)n , write down n terms of the form c1 eax + c2 xeax + c3 x2 eax + · · · + cn xn−1 eax where the ci ’s are arbitrary constants. 4. For each non-repeated conjugate complex pair of factors of the form (D − a + ib)(D − a − ib), write down 2 terms of the form c1 eax cos bx + c2 eax sin bx 5. For each repeated conjugate complex pair of factors of the form (D − a + ib)n (D − a − ib)n , write down 2n terms of the form c1 eax cos bx + c2 eax sin bx + c3 xeax cos bx + c4 xeax sin bx + · · · + c2n−1 xn−1 eax cos bx + c2n xn−1 eax sin bx 6. The sum of all the terms thus written down is the complementary function yc . To find the particular solution yp , use the following tables, as shown in the examples. For cases not shown in the tables, there are various methods of finding yp . The most general method is called variation of parameters. The following example illustrates the method:
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DIFFERENTIAL EQUATIONS (Continued) Example: Find yp for (D2 − 4)y = ex . This example can be solved most easily by use of equation 63 in the tables following. However it is given here as an example of the method of variation of parameters. The complementary function is yc = c1 e2x + c2 e−2x To find yp , replace the constants in the complementary function with unknown functions, yp = ue2x + ve−2x We now prepare to substitute this assumed solution into the original equation. We begin by taking all the necessary derivatives: yp = ue2x + ve−2x yp = 2ue2x − 2ve−2x + u e2x + v e−2x For each derivative of yp except the highest, we set the sum of all the terms containing u and v to 0. Thus the above equation becomes u e2x + v e−2x = 0
and yp = 2ue2x − 2ve−2x
Continuing to differentiate, we have yp = 4ue2x + 4ve−2x + 2u e2x − 2v e−2x When we substitute into the original equation, all the terms not containing u or v cancel out. This is a consequence of the method by which yp was set up. Thus all that is necessary is to write down the terms containing u or v in the highest order derivative of yp , multiply by the constant coefficient of the highest power of D in p(D), and set it equal to R(x). Together with the previous terms in u and v which were set equal to 0, this gives us as many linear equations in the first derivatives of the unknown functions as there are unknown functions. The first derivatives may then be solved for by algebra, and the unknown functions found by integration. In the present example, this becomes u e2x + v e−2x = 0 2u e2x − 2v e−2x = ex We eliminate v and u separately, getting 4u e2x = ex 4v e−2x = −ex Thus
Therefore, by integrating
u = 14 e−x v = − 14 e3x u = − 14 e−x 1 3x v = − 12 e
A constant of integration is not needed, since we need only one particular solution. Thus 1 1 yp = ue2x + ve−2x = − e−x e2x − e3x e−2x 4 12 1 1 1 = − ex − ex = − ex 4 12 13
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DIFFERENTIAL EQUATIONS (Continued) and the general solution is y = yc + yp = c1 e2x + c2 e−2x −
1 x e 3
The following samples illustrate the use of the tables. Example 1:
Solve (D2 − 4)y = sin 3x. Substitution of q = −4, s = 3 in formula 24 gives yp =
sin 3x −9 − 4
wherefore the general solution is y = c1 e2x + c2 e−2x −
sin 3x 13
Example 2: Obtain a particular solution of (D2 − 4D + 5)y = x2 e3x sin x. Applying formula 40 with a = 2, b = 1, r = 3, s = 1, P (x) = x2 , s + b = 2, s − b = 0, a − r = −1, (a − r)2 +(s + b)2 =5, (a − r)2 +(s − b)2 = 1, we have
yp
=
=
2(−1)2 0 2(−1)0 3·1·0−0 2 3 · 1 · 2 − 23 2 − − − x + 2x + 2 5 1 25 1 125 1 1−4 −1 1−0 −1 − 3(−1)0 −1 −1 − 3(−1)4 e3x cos x − − − x2 + 2x + 2 − 2 5 1 25 1 125 1 1 2 4 2 28 2 136 x − x− − e3x sin x + − x2 + e3x cos x 5 25 125 5 25 125 e3x sin x 2
The special formulas effect a very considerable saving of time in problems of this type. Example 3: Obtain a particular solution of (D2 − 4D + 5)y = x2 e2x cos x. (Compare with Example 2.) Formula 40 is not applicable here since for this equation r = a, s = b, wherefore the denominator (a − r)2 +(s − b)2 = 0. We turn instead to formula 44. Substituting a = 2, b = 1, P (x) = x2 and replacing sin by cos, cos by -sin, we obtain e2x cos x 2 2 e2x sin x yp = (x −4 ) + (x2 −12 ) dx 4 2 3 2 x 1 x x − − e2x cos x + e2x sin x = 4 8 6 4 which is the required solution. Example 4: Find zp for (Dx − 3Dy ) z = ln(y + 3x). Referring to Table XV we note that formula 69 (not 68) is applicable. This gives zp = x ln(y + 3x) It is easily seen that −y/3 ln(y + 3x) would serve equally well. Example 5: Solve (Dx + 2Dy − 4) z = y cos(y − 2x). Since R in formula 76 contains a polynomial in x, not y, we rewrite the given equation in the form (Dy +12 Dx − 2) z =12 y cos(y − 2x). Then zc = e2y F (x − 1 y) = e2x f (2x − y) 2 and by the formula 1 zp = − cos(y − 2x) · 2
1 y + 2 2 2
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1 = − (2y + 1) cos(y − 2x) 8
DIFFERENTIAL EQUATIONS (Continued) Example 6: Find zp for (Dx + 4Dy )3 z = (2x − y)2 . Using formula 79, we obtain 2 3 u du u5 (2x − y)5 zp = = =− 3 [2 + 4(−1)] 5 · 4 · 3 · (−8) 480 Example 7:
Find zp for (Dx3 + 5Dx2 Dy − 7Dx + 4)z = e2x+3y . By formula 87 zp =
Example 8:
e2x+3y e2x+3y = 23 + 5 · 22 · 3 − 7 · 2 + 4 58
Find zp for (Dx4 + 6Dx3 Dy + Dx Dy + Dy2 + 9)z = sin(3x + 4y)
Since every term in the left number is of even degree in the two operators Dx and Dy , formula 90 is applicable. It gives sin(3x + 4y) (−9)2 + 6(−9)(−12) + (−12) + (−16) + 9 sin(3x + 4y) = 710
zp =
Table I: (D − a)y = R R 1. erx 2. sin sx∗ 3. P (x) 4. erx sin sx∗ 5. P (x) erx 6. P (x) sin sx∗
yp
erx r−a cos sx − a sin sx+s a2 +s2
− a1 P (x) +
= √
P (x) a
+
1 a2 +s2
P (x) a2
sin sx + tan−1 as (n) + · · · + P an(x)
Replace a by a − r in formula 2 and multiply by erx . rx Replace a by a − r in formula 3 and multiply by e .
a2 −s2 a3 −3as2 a 2 +s2 P (x) + (a2 +s2 )2 P (x) + (a2 +s2 )3 P (x) a k k k k−2 2 k−4 4 a − 2 a s + 4 a s −··· P (k−1) (x) + · · · (a2 +s2 )k
+ ···
3a2 s−s3 s 2as 2 2 P (x) + (a2 +s2 )2 P (x) + (a2 +s2 )3 P (x) a +s k k ak−1 s− 3 ak−3 s3 +··· + 1 P (k−1) (x) + · · · (a2 +s2 )k
+ ···
− sin sx +
− cos sx
7. P (x)erx sin sx∗ Replace a by a − r in formula 6 and multiply by erx . 8. eax xeax ax sx 9. eax sin sx∗ − e cos s 10. P (x)eax eax P (x) dx 11. P (x)eax sin sx
v ax sx P (x) − − P s3(x) + P s5(x) − · · · − e cos s *For cos sx in R replace “sin” by “cos” and “cos” by “−sin” in yp . eax sin sx s
P (x) s3
Dn =
dn dxn
(m n )=
m! (m − n)!n!
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0! = 1
P (x) s2
+
P iv (x) s4
− ···
DIFFERENTIAL EQUATIONS (Continued) Table II: (D − a)2 y = R R 12. erx
yp
erz (r−a)2 1 [(a2 (a2 +s2 )
13. sin sx∗
− s2 ) sin sx + 2as cos sx] = 2P (x) a
3P (x) a2
1 a2 +s2
sin sx + tan−1
(n+1)P (n) (x) an multiply by erx .
1 a2
15. erx sin sx∗ 16. P (x)erx
Replace a by a − r in formula 13 and Replace a by a − r in formula 14 and multiply by erx .
17. P (x) sin sx∗
+
a2 −s2 a3 −3as2 a4 −6a2 s2 +s4 P (x) + 2 (a P (x) 2 +s2 )3 P (x) + 3 (a2 +s2 )2 (a2 +s2 )4 ak − k ak−2 s2 + k ak−4 s4 −··· 2 4 (k−2) (x) + · · · +(k − 1) P (a2 +s2 )k
sin sx
+ cos sx
2as P (x) + (a2 +s2 )2
+(k − 1)
k 1
2
3
3
ak−1 s− k ak−3 s3 +··· 3 (a2 +s2 )k
3
P (k−2) (x) + · · ·
iv vi sin sx P (x) − 3Ps2(x) + 5P s4(x) − 7P s6(x) s2 v ax sx 2P (x) − e scos + 4P s3(x) − 6Ps5(x) − · · · 2 s
−e
+ ···
3a s−s 4a s−4as 2 (a 2 +s2 )3 P (x) + 3 (a2 +s2 )4 P (x) + · · ·
18. P (x)erx sin sx∗ Replace a by a − r in formula 17 and multiply by erx . 1 2 ax 19. eax x e 2 ax sx 20. eax sin sx∗ − e ssin 2 P (x) dx 21. P (x)eax eax dx 22. P (x)eax sin sx∗
+ ··· +
14. P (x)
P (x) +
2as a2 −s2
ax
+ ···
* For cos sx in R replace “sin” by “cos” and “cos” by “-sin” in yp .
Table III: (D 2 + q)y = R R 23. erx 24. sin sx∗ 25. P (x) 26. erx sin sx 27. P (x)erx
28. P (x) sin sx∗
yp
erx r 2 +q sin sx −s2 +q iv (2k) 1 P (x) − P q(x) + P q2(x) − · · · + (−1)k P qk (x) · · · q (r 2 −s2 +q)erx sin sx−2rserx cos sx erx √ = sin sx 2 2 2 2 (r −s +q) +(2rs) (r 2 −s2 +q)2 +(2rs)2
− tan−1
2rs r 2 −s2 +q
3r 2 −q 4r 3 −4qr 2r P (x) + (r2+q) (x) + · · · 2 P (x) − (r 2 +q)3 P r 2 +q k k−1 k−3 k k−5 2 −(k )r q+( )r q −··· k−1 (1 )r (k−1) 3 5 P (x) + · · · + · · · + (−1) 2 +q)k−1 (r 3s2 +q 5s4 +10s2 q+q 2 iv sin sx P (x) + · · · P (x) − (−s 2 +q)2 P (x) + (−s2 +q) (−s2 +q)4 erx r 2 +q
P (x) −
(2k+1 )s2k +(2k+1 )s2k−2 q+(2k+1 )s2k−4 q2 +··· (2k) 3 5 +(−1)k 1 P (x) + · · · (−s2 +q)2k 2 2P (x) 4s +4q s cos sx − (−s (x) + · · · − (−s 2 +q) 2 +q)3 P (−s2 +q) 2k 2k−2 2k 2k−4 s + s q+··· ( ) ( ) k+1 1 (2k−1) 3 P (x) + · · · +(−1) (−s2 +q)2k−1
Table IV: (D 2 +b2 )y = R 29. sin bx∗ 30. P (x) sin bx∗
bx − x cos 2b sin bx (2b)2
P (x) −
P (x) (2b)2
+
P iv (x) (2b)4
− ··· −
cos bx 2b
P (x) −
P (x) (2b)2
+ · · · dx
* For cos sx in R replace “sin” by “cos” and “cos” by “− sin” in yp.
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DIFFERENTIAL EQUATIONS (Continued) Table V: (D 2 + pD+q)y = R R 31. erx
yp
erx r 2 +pr+q (q−s2 ) sin sx−ps cos sx (q−s2 )2 +(ps)2
32. sin sx∗
1 q
33. P (x) 34. erx sin sx∗ 35. P (x)erx
= √
1 (q−s2 )2 +(ps)2
sin sx − tan−1
ps q−s2
3 p2 −q P (x) − p −2pq P (x) + · · · q2 q3 n−1 n−2 n−2 n−4 2 n p −(1 )p q+(2 )p q −··· (n) P (x) +(−1)n qn 2
P (x) − pq P (x) +
Replace p by p + 2r, q by q+pr+r in formula 32 and multiply by erx . Replace p by p + 2r, q by q+pr+r2 in formula 33 and multiply by erx .
Table VI: (D − b)(D − a)y = R
a a2 − s2 b b 2 − s2 P (x) P (x) + − − a2 + s2 b 2 + s2 (a2 + s2 )2 (b2 + s2 )2 b3 − 3bs2 a3 − 3as2 − (x) + · · · P + (a2 + s2 )3 (b2 + s2 )3 cos sx s 2as s 2bs + P (x) + P (x) − 2 − 2 2 2 2 2 2 2 2 2 b−a a +s b +s (a + s ) (b + s ) † 3a2 s − s2 3b2 s − s3 + − (x) + · · · P (a2 + s2 )3 (b2 + s2 )3
sin sx b−a
36. P (x) sin sx∗
37. P (x)erx sin sx∗ Replace and multiply by erx . a by a–r, b by b − r in formula 36 ax P (x) P (x) P (x) P (n) (x) e + (b−a) 38. P (x)eax P (x) dx + (b−a) 2 + (b−a)3 + · · · + (b−a)n+1 a−b * For cos sx in R replace “sin” by “cos” and “cos” by “-sin” in yp . † For additional terms, com e with formula 6.
Table VII: (D 2 − 2aD + a2 + b2 )y = R R
yp ∗
P (x) 3a (s−b)−(s−b) + 3a[a(s+b)−(s+b) P − (x) + · · · 3 3 2 +(s+b)2 ] [a22+(s−b)22] a −(s+b) a2 −(s−b)2 cos sx a a − 2b P (x) + P (x) − − 2 2 2 2 2 2 2 2 2 2 a +(s+b) a +(s−b) [a +(s+b) ] [a +(s−b) ] 2 † 2 3 2 + a[a2−3a(s+b) P (x) + · · · − a[a2−3a(s−b) +(s+b)2 ]3 +(s−b)2 ]3 rx 40. P (x)erx sin sx∗ Replace a by a − r in formula 39 and multiply by e . 39. P (x) sin sx
sin sx 2b
s+b
a2 +(s+b)2
−
s−b
a2 +(s−b)2
P (x) +
2
3
2a(s+b)
[a2 +(s+b)2 ]2 2
−
2a(s−b)
[a2 +(s−b)2 ]2
3
iv P (x) eax + P b4(x) − · · · b2 P (x) − b2 eax sin sx eax sin sx∗ −s2ax +b2 eax sin bx∗ − xe 2bcos bx P (x) P iv (x) eax sin bx P (x) − P (x)eax sin bx∗ + − · · · (2b)2 (2b)2 (2b)4 iv P (x) eax cos bx − 2b P (x) − (2b)2 + P(2b)(x) 4 − · · · dx
41. P (x)eax
42. 43. 44.
* For cos sx in R replace “sin’ by “cos’ and “cos” by “-sin” in yp . † For additional terms, com e with formula 6.
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DIFFERENTIAL EQUATIONS (Continued) Table VIII: f (D)y = [D n + an−1 D n−1 + · · · + a1 D + a0 ]y = R R 45. erx 46. sin sx∗
yp
erx f (r) [a0 −a2 s2 +a4 s4 −··· ] sin sx−[a1 s−a3 s3 +a5 s5 +··· ] cos sx [a0 −a2 s2 +a4 s4 −··· ]2 +[a1 s−a3 s3 +a5 s5 −··· ]2
Table IX: f (D 2 )y = R 47. sin sx∗
sin sx f (−s2 )
=
sin sx a0 −a2 s2 +···±s2n
Table X: (D − a)n y = R 48. erx 49. sin sx∗
erx (r−a)n n n−2 2 4 n−4 4 (−1)n n s + n a s − · · · ] sin sx (a2 +s2 )2 {[a − 2 a n n−3 3 n n−1 +[ s − s + · · · ] cos sx} a a 1 n P (x) 3 n+1 P (x) n+2 P (x) (−1)n P (x) + 1 a + 2 an a2 + a2 3 rx
50. P (x) + ··· 51. erx sin sx∗ Replace a by a − r in formula 49 and multiply by e . Replace a by a − r in formula 50 and multiply by erx . 52. erx P (x) n n + 1 53. P (x) sin sx∗ (−1)n sin sx[An P (x) + An+1 P (x) + An+2 P (x) + n + 2
1
2
(x) + · · · ] 3 n n + 1 n Bn+1 P (x) + Bn+2 P (x) + + (−1) cos sx[Bn P (x) + 1 2 n + 2 Bn+3 P (x) + · · · ] 3 k−2 2 k k−4 4 s + 4 a s − ··· ak − k a a2 − s2 2 a A1 = 2 , A2 = , . . . , Ak = 2 2 2 2 a +s (a + s ) (a2 + s2 )k k−3 3 k k−1 s− k s + ··· a 2as 1 a 3 a B1 = 2 , B = , . . . , B = 2 k a + s2 (a2 + s2 )2 (a2 + s2 )k rx An+3 P
54. erx sin sx∗
Replace a by a − r in formula 53 and multiply by e .
55. eax P (x)
eax
56. P (x)eax sin sx∗
···
(−1)
n−1 2
P (x) dxn ax
e sn
sin sx
P (x) n s n−1
−
n+2 P (x) n−1
s3
+
n+4 P v (x) n−1
s5
− ···
n+1 P (x) n+3 P iv (x) P (x) − n−1 + − · · · (n odd) 2 4 n−1 s s n n+1 P (x) n+3 P iv (x) (−1) 2 eax sin sx n−1 P (x) − n−1 s2 + n−1 s4 − · · · n−1 sn n n+2 P (x) n+4 P v (x) (−1) 2 eax cos sx n P (x) + − n−1 + n−1 s5 − · · · (n even) sn s n−1 s3 n+1
+
(−1) 2
eax cos sx sn
n−1 n−1
* For cos sx in Rreplace “sin” by “cos” and “cos” by “− sin” in yp.
Table XI: (D − a)n f (D)y = R 57. eax
xn n!
ax
· fe(a) *For cos sxin Rreplace “sin” by “cos” and “cos” by “− sin” in yp.
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DIFFERENTIAL EQUATIONS (Continued) Table XII: (D 2 + q)n y = R R 58. erx 59. sin sx∗ 60. P (x)
yp erx /(r2 + q)n sin sx/(q − s2 )n n P (x) n+1 P iv (x) n+2 P vi (x) 1 + − + · · · P (x) − n 2 2 3 q 1 2 3 q q q erx (A2 +B 2 )n
61. erx sin sx∗
" n n n n−2 2 n−4 4 A − A B + A B − · · · sin sx 2 4 # n n n−1 n−3 3 − A B− A B + · · · cos sx 1 3
2
2
A = r − s + q,
B = 2rs
Table XIII: (D 2 + b2 )n y = R 62. sin bx∗
n
cos bx (−1)n+1/2 xn!(2b) n
n
sin bx (n odd), (−1)n/2 xn!(2b) n
(n even)
Table XIV: (D n − q)y = R n erx /(r − q)
63. erx
(n)
64. P (x)
− 1q P (x) P
65. sin sx∗
− q sin sx+(−1) q 2 +s2n
rx
66. e
sin sx∗
(x)
q n−1 2
Ae
rx
+
P (2n) (x) q2
sn cos sx
+ ···
(n odd),
rx
rx sin sr−Be cos sx = √ e2 2 sin A2 +B 2 A n n n−2 2 n +B r − 2 r s + 4 rn−4 s4 − n n−1 s − n3 rn−3 s3 + · · · r 1
sx
sin sx (−s2 )n/2 −q − tan−1 B A
(n even)
· · · − q, A= B= *For cos sx in R replace “sin” by “cos” and cos by “− sin” in yp.
Table XV: (Dx + mDy )z = R R zp eax+by 67. eax+by a+mb f (u)du 68. f (ax + by) ∫ a+mb , u = ax + by 69. f (y − mx) xf (y − mx) 70. φ(x, y)f (y − mx) f (y − mx) ∫ φ(x, a + mx)dx (a = y − mx after integration)
Table XVI: (Dx + mDy − k)z = R 71. eax+by 72. sin(ax + by)∗ 73. eαx+βy sin(ax + by)∗ 74. exk f (ax + by) 75. f (y − mx) 76. p(x)f (y − mx) 77. ekx f (y − mx)
eax+by a+mb−k sin(ax+by) − (a+bm) cos(ax+by)+k (a+bm)2 +k2
Replace k in 72 by k − α − mβand multiply by eαx+βy
ekx f (u)du , u = ax + by a+mb − f (y−mx) k − k1 f (y − mx) p(x) + p k(x) kx
+
p (x) k2
+ ··· +
p(n) (x) kn
xe f (y − mx) *For cos(ax + by) replace “sin” by “cos” and “cos” by “-sin” in zp . k+r ∂ ∂ ; Dy = ∂y ; Dxk Dyr = ∂∂k ∂ r Dx = ∂x x
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y
DIFFERENTIAL EQUATIONS (Continued) Table XVII: (Dz + mDy )n z = R R 78. eax+by
zp eax+by n (a+mb) ··· f (u)dun , (a+mb)n xn f (y − mx) n!
79. f (ax + by) 80. f (y − mx) 81. φ(x, y)f (y + mx)
f (y − mx)
u = ax + by ···
φ(x, a + mx)dxn (a = y − mx after integration)
Table XVIII:(Dx + mDy − k)n z = R eax+by (a+mb−k)n (−1)n f (y−mx) kn (−1)n f (y − mx) p(x) + n1 p k(x) kn ekx ∫ ∫ ··· ∫ f (u)dun , u = ax + by (a+mb)n xn kx e f (y − mx) n!
82. eax+by 83. f (y − mx) 84. P (x)f (y − mx) 85. ekz f (ax + by) 86. ekx f (y − mx)
+
n+1 p (x) 2
k3
+
n+2 p (x) 3
k3
+ ···
Table XIX: [Dxn + a1 Dxn−1 Dy + a2 Dxn−2 Dy2 + · · · + an Dyn ]z = R 87. eax+by
eax+by a + a1 an−1 b + a2 an−2 b2 + · · · + an bn
88. f (ax + by)
· · · f (u)dun , (u = ax + by) an + a1 an−1 b + a2 an−2 b2 + · · · + an bn Table XX: F (Dx , Dy )z = R
89. eax+by
eax+by F (a,b)
Table XXI:F (Dx2 , Dx Dy , Dy2 )z = R 90. sin(ax + by)∗
sin(ax+by) F (−a2 ,−ab,−b2 )
*For cos(ax + by)replace “sin ” by “cos”, and “cos” by “-sin” in zp .
A-65
DIFFERENTIAL EQUATIONS (Continued) DIFFERENTIAL EQUATIONS G(v) dv yF (xy) dx + x G(xy) dy = 0 ln x = +c v{G(v) − F (v)} where v = xy. If G(v) = F (v), the solution is xy = c. Linear, homogeneous, second order equation d2 y dy + cy = 0 +b dx2 dx b, c are real constants
Let m1 , m2 be the roots of m2 + bm + c = 0. Then there are 3 cases: Case 1.
m1 , m2 real and distinct:
Case 2.
y = c1 em1 x + c2 em2 x m1 , m2 real and equal:
Case 3.
y = c1 em1 x + c2 xem1 x m1 = p + qi, m2 = p − qi :
y = epx (c1 cos qx + c2 sin qx) √ where p = −b/2, q = 4c − b2 /2 Linear, nonhomogeneous, second order equation d2 y dy + cy = R(x) +b 2 dx dx b, c are real constants
There are 3 cases corresponding to those immediately above: Case 1.
Case 2.
Case 3.
y = c1 em1 x + c2 em2 x em1 x + e−m1 x R(x) dx m1 − m 2 em2 x e−m2 x R(x) dx + m2 − m 1 y = c1 em1 x + c2 xem1 x + xem1 x e−m1 x R(x) dx − em1 x xe−m1 x R(x) dx y = epx (c1 cos qx + c2 sin qx) epx sin qx + e−px R(x) cos qx dx q epx cos qx − e−px R(x) sin qx dx q
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DIFFERENTIAL EQUATIONS (Continued) Putting x = et , the equation becomes
Euler or Cauchy equation x2
dy d2 y + bx + cy = S(x) dx dx
dy d2 y + (b − 1) + cy = S(et ) dt2 dt and can then be solved as a linear second order equation. y = c1 Jn (λx) + c2 Yn (λx)
Bessel’s equation x2
dy d2 y +x + (λ2 x2 − n2 )y = 0 dx2 dx y = x−p c1 Jq/r
Transformed Bessel’s equation x2
dy d2 y + (2p + 1)x + (α2 x2r + β 2 )y = 0 dx2 dx
where q =
α ! xr + c2 Yq/r xr r r
p2 − β 2 .
y = c1 Pn (x) + c2 Qn (x)
Legendre’s equation (1 − x2 )
α
d2 y dy − 2x + n(n + 1)y = 0 dx2 dx
Differential equation
Method of solution f1 (x) g2 (y) dx + dy = c f2 (x) g1 (y)
Separation of variables f1 (x)g1 (y) dx + f2 (x)g2 (y) dy = 0 Exact equation M (x, y) dx + N (x, y) dy = 0 where ∂M/∂y = ∂N/∂x
Linear first order equation dy + P (x)y = Q(x) dx Bernoulli’s equation dy + P (x)y = Q(x)y n dx Homogeneous equation y dy =F dx x Reducible to homogeneous (a1 x + b1 y + c1 )dx +(a2 x + b2 y + c2 ) dy = 0 b1 a1 = a2 b2 Reducible to separable (a1 x + b1 y + c1 ) dx +(a2 x + b2 y + c2 ) dy = 0 b1 a1 = a2 b2
M ∂x+
n−
∂ ∂y
M ∂x dy = c
where ∂x indicates that the integration is to be performed with respect to x keeping y constant. ye P dx = Qe P dx dx + c ve(1−n)
P dx =
where v = y
1−n
Qe(1−n)
P dx
dx + c
if n = 1, the solution is
ln y = (Q −dvP ) dx + c ln x = F (v)−v +c where v = y/x.If F (v) = v, the solution is y = cx Set u = a1 x + b1 y + c1 v = a 2 x + b 2 y + c2 Eliminate x and y and the equation becomes homogenous
Set u = a1 x + b1 y Eliminate x or y and equation becomes separable
A-67
FOURIER SERIES 1. If f (x) is a bounded periodic function of period 2L (i.e. f x + 2L) = f (x), and satisfies the Dirichlet conditions: (a) In any period f (x) is continuous, except possibly for a finite number of jump discontinuities. (b) In any period f (x) has only a finite number of maxima and minima. Then f (x) may be represented by the Fourier series a0 nπx nπx an cos + + bn sin 2 L L n=1 ∞
where an and bn are as determined below. This series will converge to f (x) at every point where f (x) is continuous, and to f (x+ ) + f (x− ) 2 (i.e., the average of the left-hand and right-hand limits) at every point where f (x) has a jump discontinuity.
an
=
bn
=
1 L 1 L
L
f (x) cos
nπx dx, L
n = 0, 1, 2, 3, . . . ,
f (x) sin
nπx dx, L
n = 1, 2, 3, . . .
−L L −L
we may also write 1 an = L
α+2L
α
nπx 1 f (x) cos dx and bn = L L
α+2L
f (x) sin α
nπx dx L
where α is any real number. Thus if α = 0,
an bn
= =
1 L 1 L
2L
f (x) cos
nπx dx, L
n = 0, 1, 2, 3, . . . ,
f (x) sin
nπx dx, L
n = 1, 2, 3, . . .
0
0
2L
2. If in addition to the restrictions in (1), f (x) is an even function (i.e., f (−x) = f (x)), then the Fourier series reduces to ∞ nπx a0 + an cos 2 L n=1 That is, bn = 0. In this case, a simpler formula for an is 2 L nπx an = f (x) cos dx, n = 0, 1, 2, 3, . . . L 0 L
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3. If in addition to the restrictions in (1), f (x) is an odd function (i.e., f (−x) = −f (x)), then the Fourier series reduces to ∞ nπx bn sin L n=1 That is, an = 0. In this case, a simpler formula for the bn is bn =
2 L
L
f (x) sin 0
nπx dx, L
n = 1, 2, 3, . . .
4. If in addition to the restrictions in (2) above, f (x) = −f (L − x), then an will be 0 for all even values of n, including n = 0. Thus in this case, the expansion reduces to ∞
a2m−1 cos
m=1
(2m − 1)πx L
5. If in addition to the restrictions in (3) above, f (x) = f (L − x), then bn will be 0 for all even values of n. Thus in this case, the expansion reduces to ∞
b2m−1 sin
m=1
(2m − 1)πx L
(The series in (4) and (5) are known as odd-harmonic series, since only the odd harmonics appear. Similar rules may be stated for even-harmonic series, but when a series appears in the even-harmonic form, it means that 2L has not been taken as the smallest period of f (x). Since any integral multiple of a period is also a period, series obtained in this way will also work, but in general computation is simplified if 2L is taken to be the smallest period.) 6. If we write the Euler definitions for cos θ and sin θ, we obtain the complex form of the Fourier Series known either as the “Complex Fourier Series” or the “Exponential Fourier Series” of f (x). It is represented as n=+∞ 1 f (x) = cn eiωn x 2 n=−∞ where 1 cn = L
L
f (x) e−iωn x dx,
−L
n = 0, ±1, ±2, ±3, . . .
with ωn = nπ L for n = 0, ±1, ±2, . . . The set of coefficients cn is often referred to as the Fourier spectrum. 7. If both sine and cosine terms are present and if f (x) is of period 2L and expandable by a Fourier series, it can be represented as nπx a0 cn sin + + φn , 2 L n=1 ∞
f (x) =
an = cn sin φn ,
bn = cn cos φn ,
where cn =
A-69
a2n
+
b2n ,
φn = arctan
an bn
It can also be represented as nπx a0 f (x) = + + φn , cn cos 2 L n=1 ∞
an = cn cos φn ,
where
bn = −cn sin φn ,
cn =
a2n + b2n ,
bn φn = arctan − an
where φn is chosen so as to make an , bn , and cn hold. 8. The following table of trigonometric identities should be helpful for developing Fourier series. n 0 (−1)n
sin nπ cos nπ ∗ sin nπ 2 ∗ cos nπ 2 sin nπ 4
neven 0 +1 0 (−1)n/2
*A useful formula for sin sin
nπ 2
nodd 0 −1 (−1)(n−1)/2 0 √ (n2 +4n+11)/8 2 (−1) 2
and cos
nπ 2
nπ (i)n+1 = [(−1)n − 1] 2 2
n/2 odd 0 +1 0 −1 (−1)(n−2)/4
n/2 even 0 +1 0 +1 0
is given by
and
cos
nπ (i)n = [(−1)n + 1], 2 2
where i2 = −1.
AUXILIARY FORMULAS FOR FOURIER SERIES 4 1 3πx 1 5πx πx + sin + sin + ··· [0 < x < k] sin π k 3 k 5 k 2k 1 2πx 1 3πx πx x= − sin + sin − ··· [−k < x < k] sin π k 2 k 3 k k 3πx 5πx πx 4k 1 1 x = − 2 cos + 2 cos + 2 cos + ··· [0 < x < k] 2 π k 3 k 5 k 2k2 3πx π2 πx 4 π2 2πx 4 π2 − sin − sin + − 3 sin x2 = 3 π 1 1 k 2 k 3 3 k 2 2 π 5πx 4πx 4 π − sin + − 3 sin + ··· [0 < x < k] 4 k 5 5 k k2 2πx 3πx 4πx πx 4k2 1 1 1 − 2 cos − 2 cos + 2 cos − 2 cos + ··· x2 = 3 π k 2 k 3 k 4 k 1=
[−k < x < k]
1− 1 22 1 1− 2 2 1 1+ 2 3 1 1 + 2 22 4 1−
1 1 1 + − 3 5 7 1 1 + 2 + 2 3 4 1 1 + 2 − 2 3 4 1 1 + 2 − 2 5 7 1 1 + 2 + 2 6 8
+ ··· = + ··· = + ··· = + ··· = + ··· =
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π 4 π2 6 π2 12 π2 8 π2 24
FOURIER EXPANSIONS FOR BASIC PERIODIC FUNCTIONS f (x) =
f (x) =
4 π
2 π
n=1,3,5...
∞ n=1
∞
c 2 f (x)= L +π
f (x) =
f (x) =
f (x) =
f (x) =
f (x) =
A-71
n=1
∞
2 L
2 π
1 2
n=1
∞ n=1
−
8 π2
1 2
1 n
(−1)n n
(−1)n n
sin nπ 2
nπx cos nπc L − 1 sin L
nπc L
n=1,3,5,...
∞ n=1
1 n
nπx L
sin nπx L
sin nπx L
n=1,3,5,...
cos
sin( 12 nπc/L) 1 2 nπc/L
(−1)n+1 n
4 π2
1 π
sin
−
sin nπx L
1 n2
cos nπx L
(−1)(n−1)/2 n2
sin nπx L
sin nπx L
FOURIER EXPANSIONS FOR BASIC PERIODIC FUNCTIONS (Continued) f (x) = 12 (1 + a) +
f (x) =
f (x) =
f (x) =
f (x) π4
2 π
1 2
2 π
∞ n=1
−
n=1
∞ n=1
f (x) =
f (x) =
f (x) =
9 π2
1 n
1 π
(−1)n n
sin
∞ n=1
32 3π 2
n=1
1 2
nπ 4
1 n2
∞
+
(−1)n−1 n
4 π 2 (1−2a)
∞
1+
n=1
n=1,3,5,...
sin nπa sin
nπ 3
sin
sin ωt −
sin
nπ 4
2 π
1 n2
sin
nπx ; L
nπx ; L
n=2,4,6,...
a=
a=
c 2L
c 2L
a=
nπx ; L
sin nπa sin
nπx ; L
nπx ; L
sin
cos nπa cos
1+(−1)n nπ(1−2a)
1+
cos nπa − 1] cos nπx ; L c a = 2L
1 [(−1)n n2
sin nπa nπ(1−a)
sin
1 n2
∞
2 π 2 (1−a)
c 2L
nπx ; L
a=
c 2L
a=
a=
c 2L
1 n2 −1
cos nωt
Extracted from graphs and formulas, pages 372, 373, Differential Equations in Engineering Problems, Salvadori and Schwarz, published by Prentice-Hall, Inc.,1954.
A-72
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c 2L
THE FOURIER TRANSFORMS For a piecewise continuous function F (x) over a finite interval 0 ≤ x ≤ π; the finite Fourier cosine transform of F (x) is π fc (n) = F (x) cos nx dx (n = 0, 1, 2, . . . ) 0
If x ranges over the interval 0 ≤ x ≤ L, the substitution x = πx/L allows the use of this definition, also. The inverse transform is written. x 1 2 F (x) = fc (0) − fc (n) cos nx (0 < x < π) π π n=1 where F (x) = formula
F (x+')+F (x−') . 2
We observe that F (x+) = F (x−) = F (x) at points of continuity. The
π
fc(2) (n) =
F (x) cos nx dx (1)
0
= −n2 fc (n) − F (0) + (−1)n F (π) makes the finite Fourier cosine transform useful in certain boundary value problems. Analogously, the finite Fourier sine transform of F (x) is π fs (n) = F (x) sin nx dx (n = 1, 2, 3, . . . ) 0
and F (x) = Corresponding to (1) we have fs(2) (n)
∞ 2 fs (n) sin nx (0 < x < π) π n=1
=
π
F (x) sin nx dx
(2)
0
=
−n2 fs (n) − n F (0) − n(−1)n F (π)
Fourier Transforms If F (x) is defined for x ≤ 0 and is piecewise continuous over any finite interval, and if absolutely convergent, then
x 2 fc (α) = F (x) cos(αx) dx π 0
x 0
F (x) dx is
is the Fourier cosine transform of F (x). Furthermore,
x 2 F (x) = fc (α) cos(αx) dα. π 0 If limx→∞ dn F/dxn = 0, then an important property of the Fourier cosine transform is
x 2r d F 2 (2r) fc (α) = cos(αx) dx π 0 dx2r
r−1 2 (−1)n a2r−2n−1 α2n + (−1)r α2r fc (α) =− π n=0 where limx→∞ dr F/dxr = ar, makes it useful in the solution of many problems. Under the same conditions.
x 2 fs (α) = F (x) sin(αx) dx π 0
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(3)
defines the Fourier sine transform of F (x), and
F (x) = Corresponding to (3) we have
fs(2r) (α)
2 π
= =
−
2 π
x
fs (α) sin(αx) dα 0
∞
d2r F sin(αx) dx dx2r 0 r 2 (−1)n α2n−1 a2r−2n + (−1)r−1 α2r fs (α) π n=1
∞ F (x)dx is absolutely convergent, then Similarly, if F (x) is defined for −∞ < x < ∞, and if ∫−∞ ∞ 1 F (x)eiax dx f (α) = √ 2π −∞
is the Fourier transform of F (x), and 1 F (x) = √ 2π Also, if
∞
f (α)e−iax dα
−∞
n d F lim n = 0 (n = 1, 2, . . . , r − 1) |x|→∞ dx
then 1 f (r) (α) = √ 2π
∞
F (r) (x)eiαx dx = (−iα)r f (α)
−∞
Finite Sine Transforms π
1 fs (n) =
0
n+1
2 (−1)
fs (n) F (x) sin nx dx (n = 1, 2, . . . )
F (x) F (x) F (π − x)
fs (n)
3
1 n
π−x π
4
(−1)n+1 n
x π
5
1−(−1)n n
6
2 n2
7
(−1)n+1 n3
x(π 2 −x2 ) 6π
8
1−(−1)n n3
x(π−x) 2
9
π 2 (−1)n−1 n
sin
1 x
nπ 2
10 π(−1)n 11
n [1 n2 +c2
12
n n2 +c2
when 0 < x < π/2
π − x when π/2 < x < π
−
6 n3
2[1−(−1)n ] n3
−
π2 n
x2
x3
− (−1)n ecπ ]
ecx sinh c(π−x) sinh cπ
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(4)
fs (n) (k = 0, 1, 2, . . . )
n
13
n2 −k2
14
π2
when n = m
0
when n = m
F (x) sin k(π−x) sin kπ
(m = 1, 2, . . . )
n [1 − (−1)n cos kπ] n2 − k 2 (k = 1, 2, . . . ) n [1 − (−1)n+m ] n2 −m2 when n = m = 1, 2, . . . 0 when n = m n (k = 0, 1, 2, . . . ) 2 2 2 (n −k ) bn (|b| ≤ 1) n 1−(−1)n n b (|b| ≤ 1) n
15
16 17 18 19
sin mx cos kx
cos mx k(π−x) π sin kx − x cos 2k sin kπ 2k sin2 kπ 2 b sin x arctan 1−b cos x π 2 sin x arctan 2b1−b 2 π
Finite Cosine Transforms 1 2 3
fc (n) π fc (n) = 0 F (x) cos nx dx (n = 0, 1, 2, . . . ) (−1)n fc (n) 0 when n = 1, 2, · · · ;fc (0) = π
4
2 n
sin
nπ ; fc (0) 2
=0
n
2
; fc (0) = π2 5 − 1−(−1) n2 2 (−1)n 6 ; fc (0) = π6 n2 7 n12 ; fc (0) = 0 n n 4 8 3π 2 (−1) − 6 1−(−1) ; fc (0) = π4 n2 n4 (−1)n ec π−1 9 n2 +c2 1 10 n2 +c 2 k 11 [( − 1)n cos πk − 1] n2 − k 2 (k = 0, 1, 2, · · · ) (−1)n+m −1 12 ; fc (m) = 0 (m = 1, 2, · · · ) n2 −m2 1 (k = 0, 1, 2, . . . ) 13 n2 −k 2 14 0 when n = 1, 2, · · · ; π fc (m) = (m = 1, 2, · · · ) 2
F (x) F (x) F (π − x) 1 $ 1 when 0 < x < π/2 −1 when π/2 < x < π x x2 2π (π−x)2 2π 3
−
π 6
x
1 cx e c coshc(π−x) csinhcπ
sin kx
1 m
sin mx k(π−x) − cosk sin kπ cos mx
Fourier Sine Transforms
1 2 3
F (x) $ 1 (0 < x < a) 0 (x > a)
e−x
5
xe−x
7
xp−1 (0 < p < 1) $ sin x (0 < x < a) 0 (x > a)
4 6
2
cos sin
fs (α) 2 π
1−cos α α
2 Γ(p) π αp
√1 2π
sin
sin[a(1−α)] 1−α
2 α π 1+α2 −α2 /2
/2
pπ 2
−
sin[a(1+α)] 1+α
αe 2 2 ∗ √ 2 2 2 sin α2 c α2 − cos α2 S α2 2 2 ∗ √ 2 2 2 cos α2 C α2 + sin α2 S α2
2
x 2
x2 2
A-75
Here C(y) and S(y) are the Fresnel integrals: y 1 1 √ cos t dt, C(y) = √ t 2π 0
1 S(y) = √ 2π
y
0
1 √ sin t dt t
*More extensive tables of the Fourier sine and cosine transforms can be found in Fritz Oberhettinger, Tabellen zur-Fourier Transformation, Springer, 1957.
Fourier Cosine Transforms F (x) $ 1 (0 < x < a) 0 (x > a)
1
xp−1 (0 < p < 1) $ cos x (0 < x < a) 0 (x > a)
2 3 4
e−x
5
e−x
2
6
cos
7
sin
x2 2
2 sin aα π α 2 Γ(p) π αp
√1 2π
cos
e
cos cos
pπ 2
sin[a(1−α)] 1−α
2 1 π 1+α2 1 −α /2
/2
x2 2
fc (α)
+
α2 2
−
π 4
α2 2
+
π 4
sin[a(1+α)] 1+α
Fourier Transforms F (x) 1
2
0
"
4
eiwx (p < x < q) 0 (x " −cx+iwx < p, x > q) (x > 0) e (c > 0) 0 (x < 0) 2 e−px R(p) > 0
5
cos px2
6
sin px2
7
|x|−p
8
−a|x| e√ |x|
9 10
cosh ax cosh πx sinh ax sinh πx $
2 3
12 13 14
15
ip(w+α) −eiq(w+α) √i e (w+α) 2π
i √ 2π(w+α+ic) 2 √1 e−α /4p 2p 2 √1 cos α 4p 2p
(0 < p < 1)
√ 2 2 √ a +x ]
cos
2 cos 2 cosh 2 π cosh α+cos a 1 sin a √ 2π cosh α+cos a
π
(|x| > a)
"
sin[b
"
π 4
√ 2 2 (a +α )+a √ 2 2 a +α a α
a −x
0
−
α2 + π4 4p pπ 2 Γ(1−p) sin 2 (1−p) π |α| √1 2p
(−π < a < π) (−π < a < π) √ 1 (|x| < a) 2 2
11
f (α) |α| < a |α| > a
" π
sin ax x
a2 +x2
pn (x) (|x| < 1) 0 √ (|x| > 1) 2 2 cos[b √ a −x ] (|x| < a) a2 −x2 0 (|x| > a) √ a2 −x2 ] cosh[b √ (|x| < a) a2 −x2 0 (|x| > a)
2
J0 (aα) 0 √ J (a b2 − α2 ) 0 2
π
in √ J 1 (α) α n+ 2
π 2
π 2
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√ J0 (a a2 + b2 ) √ J0 (a α2 − b2 )
(|α| > b) (|α| < b)
*More extensive tables of Fourier transforms can be found in W . Magnus and F . Oberhettinger, Formulas and Theorems of the Special Functions of Mathematical Physics. Chelsea, 1949, 116–120. The following functions appear among the entries of the tables on transforms. Function Ei(x) Si(x) Ci(x) erf(x) erfc(x) Ln (x)
Definition x ev dv; or sometimes defined as −x v x −v = x e v dv −Ei(−x) x sin v dv v 0 x
cos v v
dv; or sometimes defined as negative of this integral x −v2 √2 e dv π 0 ∞ 2 1− erf(x) = √2π x e−v dv x n e d (xn e−x ), n = 0, 1, . . . n! dxn x
Name Sine, Cosine, and Exponential Integral tables pages 548–556 Sine, Cosine, and Exponential Integral tables pages 548–556 Sine, Cosine, and Exponential Integral tables pages 548–556 Error function Complementary function to error function
Laguerre polynomial of degree n
n
SERIES EXPANSION The expression in parentheses following certain of the series indicates the region of convergence. If not otherwise indicated it is to be understood that the series converges for all finite values of x.
(x + y)n
=
(1 ± x)n
=
(1 ± x)−n
=
(1 ± x)−1
=
BINOMIAL SERIES n(n − 1) n−2 2 n(n − 1)(n − 2) n−3 3 xn + nxn−1 y + y + y + · · · (y 2 < x2 ) x x 2! 3! n(n − 1)(n − 2)x3 n(n − 1)x2 ± + · · · (x2 < 1) 1 ± nx + 2! 3! n(n + 1)x2 n(n + 1)(n + 2)x3 1 ∓ nx + ∓ + · · · (x2 < 1) 2! 3! 1 ∓ x + x2 ∓ x3 + x4 ∓ x5 + · · · (x2 < 1)
−2
=
1 ∓ 2x + 3x2 ∓ 4x3 + 5x4 ∓ 6x5 + · · ·
(1 ± x)
(x2 < 1)
REVERSION OF SERIES Let a series be represented by y = a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + · · · with a1 = 0. The coefficients of the series x = A 1 y + A 2 y 2 + A 3 y 3 + A4 y 4 + · · · are A1 =
1 a1
A2 = − A4 =
A5
=
A6
=
A7
=
a2 a31
A3 =
1 (2a22 − a1 a3 ) a51
1 (5a1 a2 a3 − a21 a4 − 5a32 ) a71
1 (6a21 a2 a4 + 3a21 a23 + 14a42 − a31 a5 − 21a1 a22 a3 ) a91 1 (7a31 a2 a5 + 7a31 a3 a4 + 84a1 a32 a3 − a41 a6 − 28a21 a22 a4 − 28a21 a2 a33 − 42a52 ) a11 1 1 (8a41 a2 a6 + 8a41 a3 a5 + 4a41 a24 + 120a21 a32 a4 + 180a21 a22 a23 + 132a62 − a51 a7 a13 1 −36a31 a22 a5 − 72a31 a2 a3 a4 − 12a31 a33 − 330a1 a42 a3 )
A-77
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TAYLOR SERIES 1. f (x) =f (a) + (x − a)f (a) +
(x − a)2 (x − a)3 f (a) + f (a) 2! 3!
(x − a)n (n) f (a) + · · · n! (Increment form) + ··· +
(Taylor’s Series)
h2 f (x) + 2! x2 = f (h) + xf (h) + f (h) + 2!
2. f (x + h) = f (x) + hf (x) +
h3 f (x) + · · · 3! x3 f (h) + · · · 3!
3. If f (x) is a function possessing derivatives of all orders throughout the interval a ≤ x ≤ b, then there is a value X, with a < X < b, such that f (b) = f (a) + (b − a)f (a) +
(b − a)2 (b − a)n−1 (n−1) (b − a)n (n) f (a) + · · · + f f (X) (a) + 2! (n − 1)! n!
f (a + h) = f (a) + hf (a) +
h2 hn−1 (n−1) hn (n) (a) + f (a) + · · · + f f (a + θh) 2! (n − 1)! n!
where b = a + h and 0 < θ < 1. Or f (x) = f (a) + (x − a)f (a) +
(x − a)2 f (n−1) (a) f (a) + · · · + (x − a)n−1 + Rn , 2! (n − 1)!
where
f (n) [a + θ · (x − a)] (x − a)n , 0 < θ < 1. n! The above forms are known as Taylor’s series with the remainder term. Rn =
4. Taylor’s series for a function of two variables ∂ ∂ ∂f (x, y) ∂f (x, y) If h +k f (x, y) = h +k ; ∂x ∂y ∂x ∂y 2 2 ∂ 2 f (x, y) ∂ ∂ 2 f (x, y) ∂ 2 ∂ f (x, y) h f (x, y) = h2 + 2hk +k + k ∂x ∂y ∂x2 ∂x∂y ∂y 2 y=b n ∂ ∂ etc., and if h ∂x + k ∂y f (x, y) where the bar and subscripts means that after differx=a entiation we are to replace x by a and y by b, then y=b y=b n ∂ 1 ∂ ∂ ∂ f (a+h, b+k) = f (a, b)+ h +· · ·+ f (x, y) +· · · +k f (x, y) h +k ∂x ∂y n! ∂x ∂y x=a x=a MACLAURIN SERIES f (n−1) (0) x2 x3 f (x) = f (0) + xf (0) + f (0) + f (0) + · · · + xn−1 + Rn , 2! 3! (n − 1)!
where Rn =
xn f (n) (θx) , n!
A-78
0 < θ < 1.
EXPONENTIAL SERIES 1 1 1 1 + + + + ··· 1! 2! 3! 4! x3 x4 x2 + + + ··· ex = 1 + x + 2! 3! 4! (x loge a)2 (x loge a)3 ax = 1 + x loge a + + + ··· 2! 3! (x − a)2 (x − a)3 x a e = e 1 + (x − a) + + + ··· 2! 3! e=1+
LOGARITHMIC SERIES 3 x−1 1 x−1 2 + + 13 x−1 + ··· x 2 x x (x− 1) − 12 (x − 1)2 + 13 (x − 1)3 − · · · 3 5 1 x−1 1 x−1 loge x = 2 x−1 + + + ··· x+1 3 x+1 5 x+1
(x > 12 ) (2 ≥ x > 0)
loge x = loge x =
(x > 0)
loge (1 + x) = x − 12 x2 + 13 x3 − 14 x4 + (−1 < x ≤ 1) ··· loge (n + 1) − loge (n − 1) = 2 n1 + 3n1 3+ 5n1 5 + · · · 3 x x loge (a + x) = loge a + 2 2a+x + 13 2a+x 5 x + 15 2a+x + ··· (a > 0, −a < x < +∞) 3 5 2n−1 1+x = 2 x + x3 + x5 + · · · + x2n−1 + · · · −1 < x < 1 loge 1−x loge x =
loge a +
(x−a) a
−
(x−a)2 2a2
+
(x−a)3 3a3
− +···
0 < x ≤ 2a
TRIGONOMETRIC SERIES 3 5 7 sin x = x − x3! + x5! − x7! + · · · (all real values of x) 2 4 6 cos x = 1 − x2! + x4! − x6! + · · · (all real values of x) n−1 2n 3 5 7 9 (22n −1)B2n 2n−1 tan x = x + x3 + 2x + 17x + 62x + · · · + (−1) 2(2n)! x + ··· , 15 315 2835 2 x2 < π4 and Bn represents the nth Bernoulli number x3 45
5
n+1 2n
7
(−1) 2 x − 2x B2n x2n−1 − · · · , 945 − 4725 − · · · − (2n)! 2 2 th x < π and Bn represents the n Bernoulli number n 2 5 4 61 6 277 8 2n x + 720 x + 8064 x + · · · + (−1) + ··· , sec x = 1 + x2 + 24 (2n)! E2n x 2 π 2 th x < 4 and En represents the n Euler number 7 31 127 csc x = x1 + x6 + 360 x3 + 15,120 x5 + 604,800 x7 + · · ·
cot x =
1 x
−
x 3
+ (−1)
−
n+1
2(22n−1 −1)
(2n)! x2 < π 2
B2n x2n−1 + · · · , and Bn represents the nth Bernoulli number
2 2 2 sin x = x 1 − πx2 1 − 22xπ2 1 − 32xπ2 · · · 2 4x2 4x2 cos x = 1 − 4x 1 − 1 − ··· 2 2 2 2 2 π 3 π 5 π x 1·3 5 1·3·5 7 sin−1 x = x + 2·3 x + 2·4·6·7 x + ··· + 2·4·5 3 7 π x 1·3 5 −1 + · · · cos x = 2 − x + 2·3 + 2·4·5 x + 1·3·5x 2·4·6·7 3
tan−1 x tan−1 x tan−1 x cot−1 x
3
5
7
= x − x3 + x5 − x7 + · · · = π2 − x1 + 3x1 3 − 5x1 5 + 7x1 7 − · · · = − π2 − x1 + 3x1 3 − 5x1 5 + 7x1 7 − · · · 3 5 7 = π2 − x + x3 − x5 + x7 − · · · A-79
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(x2 < ∞)
x2 < 1, − π2
(x2 < ∞) < sin−1 x < π2
(x2 < 1, 0 < cos−1 x < π) (x2 < 1) (x > 1) (x < −1) (x2 < 1)
loge sin x =
loge x − 2
loge cos x = − x2 − loge tan x = esin x
=
cos x
e
=
etan x
=
sin x =
x2 6
4
x 12
−
−
x4 180
6
x 45
−
x6 2835 − · · · 17x8 2520 − · · ·
−
4 62x6 loge x + x3 + 7x 90 + 2835 + · · · 2 4 8x5 3x6 56x7 1 + x + x2! − 3x 4! − 5! − 6! + 7! 2 4 6 31x e 1 − x2! + 4x 4! − 6! + · · ·
2 2 (x < π2 ) x2 < π4 2 x2 < π4
2
+
3x3 3!
9x4 37x5 4! + 5! + · · · 2 sin a + (x − a) cos a − (x−a) sin a 2! (x−a)3 (x−a)4 − 3! cos a + 4! sin a + · · ·
1+x+
x2 2!
+ ···
+
x2 <
π2 4
1 2
v, |V −1 /V V ˆ V0 v 1 V2 V 21 V r rV=Vr r r r r s1
|=v . /v v
= 0) V1 -V2 − V2 . 1
1 +V2 .
,
23
V1 + V2 = V2 + V1 (r + s)V1 = rV1 + sV1 ; r(V1 + V2 ) = rV1 + rV2 V1 + (V2 + V3 ) = (V1 + V2 ) + V3 = V1 + V2 + V3 . V = rV1 + sV2
12
12
ABOP V = rV1 + (1 − r)V2
A-80
4!
sin a + · · ·
VECTOR ANALYSIS Definitions Any quantity which is completely determined by its magnitude is called a scalar. Examples of such are mass, density, temperature, etc. Any quantity which is completely determined by its magnitude and direction is called a vector. Examples of such are velocity, acceleration, force, etc. A vector quantity is represented by a directed line segment, the length of which represents the magnitude of the vector. A vector quantity is usually represented by a boldfaced letter such as V. Two vectors V1 and V2 are equal to one another if they have equal magnitudes and are acting in the same directions. A negative vector, written as -V, is one which acts in the opposite direction to V, but is of equal magnitude to it. If we represent the magnitude of V by v, we write |V| = v. A vector parallel to V, but equal to the reciprocal of its magnitude is written as V−1 or as 1/V. The unit vector V/v (when v = 0) is that vector which has the same direction as V, but has a ˆ ). magnitude of unity (sometimes represented as V0 or v Vector Algebra The vector sum of V1 and V2 is represented by V1 +V2 . The vector sum of V1 and -V2 , or the difference of the vector V2 from V1 is represented by V1 − V2 . If r is a scalar, then rV=Vr, and represents a vector r times the magnitude of V, in the same direction as V if r is positive, and in the opposite direction if r is negative. If r and s are scalars, V1 , V2 , V3 , vectors, then the following rules of scalars and vectors hold: V1 + V2 = V2 + V1 (r + s)V1 = rV1 + sV1 ; r(V1 + V2 ) = rV1 + rV2 V1 + (V2 + V3 ) = (V1 + V2 ) + V3 = V1 + V2 + V3 Vectors in Space A plane is described by two distinct vectors V1 and V2 . Should these vectors not intersect each other, then one is displaced parallel to itself until they do (Fig. 1). Any other vector V lying in this plane is given by V = rV1 + sV2 A position vector specifies the position in space of a point relative to a fixed origin. If therefore V1 and V2 are the position vectors of the points A and B, relative to the origin O, then any point P on the line AB has a position vector V given by V = rV1 + (1 − r)V2
A-80
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The scalar “r” can be taken as the metric representation of P since r = 0 implies P = B and r = 1 implies P = A (Fig. 2). If P divides the line AB in the ratio r:s then r s V= V1 + V2 r+s r+s
Figure 1. Figure 2. The vectors V1 , V2 , V3 , . . . ,Vn are said to be linearly dependent if there exist scalars r1 , r2 , r3 , . . . ,rn , not all zero, such that r1 V 1 + r2 V 2 + · · · + rn V n = 0 A vector V is linearly dependent upon the set of vectors V1 , V2 , V3 , . . . ,Vn if V = r 1 V 1 + r 2 V 2 + r3 V 3 + · · · + r n V n Three vectors are linearly dependent if and only if they are co-planar. All points in space can be uniquely determined by linear dependence upon three base vectors i.e., three vectors any one of which is linearly independent of the other two. The simplest set of base vectors are the unit vectors along the coordinate Ox, Oy and Oz axes. These are usually designated by i, j and k respectively. If V is a vector in space, and a, b and c are the respective magnitudes of the projections of the vector along the axes then V = ai + bj + ck and v=
a2 + b2 + c2
and the direction cosines of V are cos α = a/v,
cos β = b/v,
cos γ = c/v.
The law of addition yields V1 + V2 = (a1 + a2 )i + (b1 + b2 )j + (c1 + c2 )k The Scalar, Dot, or Inner Product of Two Vectors This product is represented as V1 · V2 and is defined to be equal to v1 v2 cos θ, where θ is the angle from V1 to V2 , i.e., V1 · V2 = v1 v2 cos θ The following rules apply for this product: V1 · V2 = a1 a2 + b1 b2 + c1 c2 = V2 · V1 It should be noted that this verifies that scalar multiplication is commutative. (V1 + V2 ) · V3 = V1 · V3 + V2 · V3 V1 · (V2 + V3 ) = V1 · V2 + V1 · V3 A-81
If V1 is perpendicular to V2 then V1 · V2 = 0, and if V1 is parallel to V2 then V1 · V2 = v1 v2 = rw12 In particular i · i = j · j = k · k = 1, and i·j=j·k=k·i=0 The Vector or Cross Product of Two Vectors This product is represented as V1 × V2 and is defined to be equal to v1 v2 (sin θ)1, where θ is the angle from V1 to V2 and 1 is a unit vector perpendicular to the plane of V1 and V2 and so directed that a right-handed screw driven in the direction of 1 would carry V1 into V2 , i.e., V1 × V2 = v1 v2 (sin θ)1 |V1 × V2 | V1 · V2 The following rules apply for vector products:
and tan θ =
V1 × V2
= −V2 × V1
V1 × (V2 + V3 )
= V1 × V2 + V1 × V3
(V1 + V2 ) × V3
= V1 × V3 + V2 × V3
V1 × (V2 × V3 ) = V2 (V3 · V1 ) − V3 (V1 · V2 ) i × i = j × j = k × k = 0 (the zero vector) i × j = k,
j × k = i,
k×i
= j
If V1 = a1 i + b1 j + c1 k, V2 = a2 i + b2 j + c2 k, and V3 = a3 i + b3 j + c3 k, then i j k V1 × V2 = a1 b1 c1 = (b1 c2 − b2 c1 )i + (c1 a2 − c2 a1 )j + (a1 b2 − a2 b1 )k a2 b2 c2 It should be noted that, since V1 × V2 = −V2 × V1 , the vector product is not commutative. Scalar Triple Product There is only one possible interpretation of the expression V1 · V2 × V3 and that is V1 · (V2 × V3 ) which is obviously a scalar. Further V1 · (V2 × V3 ) = (V1 × V2 ) · V3 = V2 · (V3 × V1 ) a1 b1 c1 = a2 b2 c2 a3 b3 c3 = r1 r2 r3 cos φ sin θ, Where θ is the angle between V2 and V3 and φ is the angle between V1 and the normal to the plane of V2 and V3 . This product is called the scalar triple product and is written as [V1 V2 V3 ]. The determinant indicates that it can be considered as the volume of the parallelepiped whose three determining edges are V1 , V2 and V3 . It also follows that cyclic permutation of the subscripts does not change the value of the scalar triple product so that [V1 V2 V3 ] = [V2 V3 V1 ] = [V3 V1 V2 ] but
[V1 V2 V3 ] = −[V2 V1 V3 ]
etc.
A-82
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and
[V1 V1 V2 ] ≡ 0
etc.
Given three non-coplanar reference vectors V1 , V2 and V3 , the reciprocal system is given by V1∗ , V2∗ and V3∗ , where 1 = v1 v1∗ = v2 v2∗ = v3 v3∗ 0 = v1 v2∗ = v1 v3∗ = v2 v1∗ etc. V2 × V3 V3 × V1 V1∗ = , V2∗ = , [V1 V2 V3 ] [V1 V2 V3 ]
V3∗ =
V1 × V2 [V1 V2 V3 ]
The system i, j, k is its own reciprocal. Vector Triple Product The product V1 × (V2 × V3 ) defines the vector triple product. Obviously, in this case, the brackets are vital to the definition. V1 × (V2 × V3 ) = (V1 · V3 )V2 − (V1 · V2 )V3 i j k b c a 1 1 1 = c2 a2 a2 b2 c b 2 2 b3 c3 c3 a3 a3 b3
i.e. it is a vector, perpendicular to V1 , lying in the plane of V2 , V3 . Similarly i j k b1 c1 c1 a1 a1 b1 (V1 × V2 ) × V3 = c2 a2 a2 b2 b2 c2 b3 c3 a3 V1 × (V2 × V3 ) + V2 × (V3 × V1 ) + V3 × (V1 × V2 ) ≡ 0 If V1 × (V2 × V3 ) = (V1 × V2 ) × V3 then V1 , V2 , V3 form an orthogonal set. Thus i, j, k form an orthogonal set. Geometry of the Plane, Straight Line and Sphere The position vectors of the fixed points A, B, C, D relative to O are V1 , V2 , V3 , V4 and the position vector of the variable point P is V. The vector form of the equation of the straight line through A parallel to V2 is V = V1 + rV2 or
(V − V1 ) = rV2
or
(V − V1 ) × V2 = 0
while that of the plane through A perpendicular to V2 is (V − V1 ) · V2 = 0 The equation of the line AB is V = rV1 + (1 − r)V2 and those of the bisectors of the angles between V1 and V2 are V1 V2 or V=r ± v1 v2 V = r(ˆ v1 ± v ˆ2 ) The perpendicular from C to the line through A parallel to V2 has as its equation ˆ2 · (V1 − V3 )ˆ v2 . V = V1 − V 3 − v A-83
The condition for the intersection of the two lines, V = V1 + rV3 and V = V2 + sV4 is [(V1 − V2 )V3 V4 ] = 0. The common perpendicular to the above two lines is the line of intersection of the two planes [(V − V1 )V3 (V3 × V4 )] = 0
and
[(V − V2 )V4 (V3 × V4 )] = 0
and the length of this perpendicular is [(V1 − V2 )V3 V4 ] . |V3 × V4 | The equation of the line perpendicular to the plane ABC is V = V1 × V2 + V2 × V3 + V3 × V1 and the distance of the plane from the origin is [V1 V2 V3 ] . |(V2 − V1 ) × (V3 − V1 )| In general the vector equation V · V2 = r defines the plane which is perpendicular to V2 , and the perpendicular distance from A to this plane is r − V1 · V2 v2 The distance from A, measured along a line parallel to V3 , is r − V1 · V2 V2 · v ˆ3
or
r − V1 · V2 v2 cos θ
where θ is the angle between V2 and V3 . (If this plane contains the point C then r = V3 · V2 and if it passes through the origin then r = 0.) Given two planes V · V1 = r V · V2 = s then any plane through the line of intersection of these two planes is given by V · (V1 + λV2 ) = r + λs where λ is a scalar parameter. In particular λ = ±v1 /v2 yields the equation of the two planes bisecting the angle between the given planes. The plane through A parallel to the plane of V2 , V3 is V = V1 + rV2 + sV3 or
(V − V1 ) · V2 × V3 = 0
or
[VV2 V3 ] − [V1 V2 V3 ] = 0
so that the expansion in rectangular Cartesian coordinates yields (where V ≡ xi + yj + zk): (x − a1 ) (y − b1 ) (z − c1 ) a2 b2 c2 = 0 a3 b3 c3
A-84
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which is obviously the usual linear equation in x, y, and z. The plane through AB parallel to V3 is given by [(V − V1 )(V1 − V2 )V3 ] = 0 or
[VV2 V3 ] − [VV1 V3 ] − [V1 V2 V3 ] = 0
The plane through the three points A, B and C is V = V1 + s(V2 − V1 ) + t(V3 − V1 ) or V = rV1 + sV2 + tV3 (r + s + t ≡ 1) or
[(V − V1 )(V1 − V2 )(V2 − V3 )] = 0
or
[VV1 V2 ] + [VV2 V3 ] + [VV3 V1 ] − [V1 V2 V3 ] = 0
For four points A, B, C, D to be coplanar, then rV1 + sV2 + tV3 + uV4 ≡ 0 ≡ r + s + t + u The following formulae relate to a sphere when the vectors are taken to lie in three dimensional space and to a circle when the space is two dimensional. For a circle in three dimensions take the intersection of the sphere with a plane. The equation of a sphere with center O and radius OA is V · V = v12 or
(notV = V1 )
(V − V1 ) · (V + V1 ) = 0
while that of a sphere with center B radius v1 is (V − V2 ) · (V − V2 ) = v12 or V · (V − 2V2 ) = v12 − v22 If the above sphere passes through the origin then V · (V − 2V2 ) = 0 (note that in two dimensional polar coordinates this is simply) r = 2a · cos θ while in three dimensional Cartesian coordinates it is x2 + y 2 + z 2 − 2 (a2 x + b2 y + c2 x) = 0. The equation of a sphere having the points A and B as the extremities of a diameter is (V − V1 ) · (V − V2 ) = 0. The square of the length of the tangent from C to the sphere with center B and radius v1 is given by (V3 − V2 ) · (V3 − V2 ) = v12 The condition that the plane V · V3 = s is tangential to the sphere (V − V2 ) · (V − V2 ) = v12 is (s − V3 · V2 ) · (s − V3 · V2 ) = v12 v32 . A-85
The equation of the tangent plane at D, on the surface of sphere (V − V2 ) · (V − V2 ) = v12 , is (V − V4 ) · (V4 − V2 ) = 0 or V · V4 − V2 · (V + V4 ) = v12 − v22 The condition that the two circles (V − V2 ) · (V − V2 ) = v12 and (V − V4 ) · (V − V4 ) = v32 intersect orthogonally is clearly (V2 − V4 ) · (V2 − V4 ) = v12 + v32 The polar plane of D with respect to the circle (V − V2 ) · (V − V2 ) = v12 is V · V4 − V2 · (V + V4 ) = v12 − v22 Any sphere through the intersection of the two spheres (V − V2 ) · (V − V2 ) = v12 and (V − V4 ) · (V − V4 ) = v32 is given by (V − V2 ) · (V − V2 ) + λ(V − V4 ) · (V − V4 ) = v12 + λv32 while the radical plane of two such spheres is 1 V · (V2 − V4 ) = − (v12 − v22 − v32 + v42 ) 2 Differentiation of Vectors If V1 = a1 i + b1 j + c1 k, and V2 = a2 i + b2 j + c2 k, and if V1 and V2 are functions of the scalar t, then d dV1 dV2 (V1 + V2 + · · · ) = + + ··· dt dt dt dV1 da1 db1 dc1 = i+ j+ k, etc dt dt dt dt d dV1 dV2 (V1 · V2 ) = · V2 + V1 · dt dt dt dV1 dV2 d (V1 × V2 ) = × V2 + V1 × dt dt dt dV dv V· =v· dt dt In particular, if V is a vector of constant length then the right hand side of the last equation is identically zero showing that V is perpendicular to its derivative. The derivatives of the triple products are
dV2 dV3 V2 V3 + V1 V3 + V1 V2 and dt dt dV1 dV2 dV3 d {V1 × (V2 × V3 )} = × (V2 × V3 ) + V1 × × V3 + V1 × V2 × dt dt dt dt d [V1 V2 V3 ] = dt
dV1 dt
A-86
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Geometry of Curves in Space s = the length of arc, measured from some fixed point on the curve (Fig. 3). V1 = the position vector of the point A on the curve V1 + δV1 = the position vector of the point P in the neighborhood of A ˆ t =the unit tangent to the curve at the point A, measured in the direction of s increasing. The normal plane is that plane which is perpendicular to the unit tangent. The principal normal is defined as the intersection of the normal plane with the plane defined by V1 and V1 +δV1 in the limit as δV1 − 0. n ˆ = the unit normal (principal) at the point A. The plane defined by ˆ t and n ˆ is called the osculating plane (alternatively plane of curvature or local plane). ρ = the radius of curvature at A. δθ=the angle subtended at the origin by δV1 . κ=
dθ 1 = ds ρ
ˆ =the unit binormal i.e. the unit vector which is parallel to ˆ b t×n ˆ at the point A: λ = the torsion of the curve at A
Figure 3. Frenet’s Formulae: dˆ t = κˆ n ds dˆ n ˆ = −κˆ t + λb ds ˆ db = −λˆ n ds The following formulae are also applicable: 1 ˆ Unit tangent t = dV ds Equation of the tangent (V − V1 ) × ˆ t=0 1d2 V1 Unit normal n ˆ = κds2 Equation of the normal plane (V − V1 ) · ˆ t=0 Equation of the normal (V − V1 ) × n ˆ=0 ˆ =ˆ Unit binormal b t×n ˆ ˆ=0 Equation of the binormal (V − V1 ) × b ˆ or V = V1 + ub
or
V = V1 + qˆ t
or
V = V1 + rˆ n
2
Equation of the osculating plane:
d V1 1 or V = V1 + w dV ds × ds2 ˆ [(V − V1 )tn ˆ] = 0 1 d2 V1 or (V − V1 ) dV =0 ds ds2
A-87
Differential Operators—Rectangular Coordinates dS =
∂S ∂S ∂S · dx + · dy + · dz ∂x ∂y ∂z
By definition ∂ ∂ ∂ ∇ ≡ del ≡ i ∂x + j ∂y + k ∂z 2 2 ∂ ∂ ∇2 ≡ Laplacian ≡ ∂x 2 + ∂y 2 +
∂2 ∂z 2
∂S ∂S If S is a scalar function, then ∇S ≡ grad S ≡ ∂S dx i + dy j + dz k Grad S defines both the direction and magnitude of the maximum rate of increase of S at any point. Hence the name gradient and also its vectorial nature. ∇S is independent of the choice of rectangular coordinates.
Figure 4.
∇S =
∂S n ˆ ∂n
(5)
where n ˆ is the unit normal to the surface S =constant, in the direction of S increasing. The total derivative of S at a point having the position vector V is given by (Fig. 4) ∂S n ˆ · dV ∂n = dV · ∇S
dS =
and the directional derivative of Sin the direction of U is U · ∇S = U · (∇S) = (U · ∇)S Similarly the directional derivative of the vector V in the direction of U is (U · ∇)V The distributive law holds for finding a gradient. Thus if S and T are scalar functions ∇(S + T ) = ∇S + ∇T The associative law becomes the rule for differentiating a product: ∇(ST ) = S∇T + T ∇S If V is a vector function with the magnitudes of the components parallel to the three coordinate axes Vx , Vy , Vz , then ∂Vx ∂Vy ∂Vz ∇ · V ≡ div V ≡ + + ∂x ∂y ∂z
A-88
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The divergence obeys the distributive law. Thus, if V and U are vector functions, then ∇ · (V + U) = ∇ · V + ∇ · U ∇ · (SV) = (∇S) · V + S(∇ · V) ∇ · (U × V) = V · (∇ × U) − U · (∇ × V) As with the gradient of a scalar, the divergence of a vector is invariant under a transformation from one set of rectangular coordinates to another. ∇ × V ≡ curl V ( sometimes ∇ΛV or rot V) ∂Vx ∂Vy ∂Vz ∂Vx ∂Vx ∂Vy ≡ − i+ − j+ − k ∂y ∂z ∂z ∂x ∂x ∂y i j k ∂ ∂ ∂ = ∂x ∂y ∂z Vx Vy Vz The curl (or rotation) of a vector is a vector which is invariant under a transformation from one set of rectangular coordinates to another. ∇ × (U + V) = ∇ × U + ∇ × V ∇ × (SV) = (∇S) × V + S(∇ × V) ∇ × (U × V) = (V · ∇)U − (U · ∇)V + U(∇ · V) − V(∇ · U) If V = Vx i + Vy j + Vz k then ∇ · V = ∇Vx · i + ∇Vy · j + ∇Vz · k and ∇ × V = ∇Vx × i + ∇Vy × j + ∇Vz × k The operator ∇ can be used more than once. The possibilities where ∇ is used twice are: ∇ · (∇θ) ≡ div grad θ ∇ × (∇θ) ≡ curl grad θ ∇(∇ · V) ≡ grad div V ∇ · (∇ × V) ≡ div curl V ∇ × (∇ × V) ≡ curl curl V Thus, if S is a scalar and V is a vector: div grad S ≡ ∇ · (∇S) ≡ Laplacian S ≡ ∇2 S ≡
∂2S ∂2S ∂2S + + ∂x2 ∂y 2 ∂z 2
curl grad S ≡ 0 curl curl V ≡ grad div V − ∇2 V; div curl V ≡ 0 Taylor’s expansion in three dimensions can be written f (V + ε) = eε·∇ f (V)
where V = xi + yj + zk and ε = hi + lj + mk
A-89
Orthogonal Curvilinear Coordinates If at a point P there exist three uniform point functions u, vand w so that the surfaces u =const., v =const., and w =const., intersect in three distinct curves through P then the surfaces are called the coordinate surfaces through P . The three lines of intersection are referred to as the coordinate lines and their tangents a, b, and c as the coordinate axes. When the coordinate axes form an orthogonal set the system is said to define orthogonal curvilinear coordinates at P . Consider an infinitesimal volume enclosed by the surfaces u, v, w, u + du, v + dv, and w + dw (Fig. 5).
Figure 5. The surface P RS ≡ u = constant, and the face of the curvilinear figure immediately opposite this is u + du =constant, etc. In terms of these surface constants P = P (u, v, w) Q = Q(u + du, v, w)
and P Q = h1 du
R = R(u, v + dv, w)
and P R = h2 dv
S = S(u, v, w + dw)
and P S = h3 dw
where h1 , h2 , and h3 are functions of u, v, and w. • In rectangular Cartesians i, j, k h1 = 1, h2 = 1, h3 = 1. ˆ ∂ ˆ ∂ ∂ b Φ ˆ c ∂ ˆ a ∂ ˆ∂ . =i , = , =k h1 ∂u ∂x h2 ∂v r ∂φ h3 ∂w ∂z ˆΦ ˆ • In cylindrical Cartesians ˆ r, θ, h1 = 1, h2 = 1, h3 = 1. ˆ ∂ ˆ ∂ ∂ b Φ ˆ c ∂ ˆ a ∂ ˆ∂ . = rˆ , = , =k h1 ∂u ∂r h2 ∂v r ∂φ h3 ∂w ∂z ˆΦ ˆ • In spherical coordinates ˆ r, θ, h1 = 1, ˆ a ∂ ∂ =ˆ r , h1 ∂u ∂r
h2 = r, ˆ ∂ b ∂ Φ = , h2 ∂v r ∂θ
h3 = r sin θ ˆ ˆ c ∂ Φ ∂ = h3 ∂w r sin θ ∂φ
The general expressions for grad, div and curl together with those for ∇2 and the directional derivative are, in orthogonal curvilinear coordinates, given by:
A-90
TeamLRN
∇S = (V · ∇)S = ∇·V = ∇×V =
∇2 S =
ˆ ∂S b ˆ c ∂S ˆ a ∂S + + h1 ∂u h2 ∂v h3 ∂w V1 ∂S V2 ∂S V3 ∂S + + h1 ∂u h ∂v h3 ∂w " 2 # 1 ∂ ∂ ∂ (h2 h3 V1 ) + (h3 h1 V2 ) + (h1 h2 V3 ) . h1 h2 h3 ∂u ∂v ∂w " # # ˆ " ∂ ˆ a ∂ b ∂ ∂ (h3 V3 ) − (h2 V2 ) + (h1 V1 ) − (h3 V3 ) h2 h3 ∂v ∂w h3 h1 ∂w ∂u " # ˆ c ∂ ∂ + (h2 V2 ) − (h1 V1 ) h1 h2 ∂u ∂v " # 1 ∂ h3 h1 ∂S ∂ ∂ h2 h3 ∂S h1 h2 ∂S + + h1 h2 h3 ∂u h1 ∂u ∂v h2 ∂v ∂w h3 ∂w FORMULAS OF VECTOR ANALYSIS
Rectangular coordinates
Cylindrical coordinates
Conversion to rectangular coordinates
x = r cos ϕ
Gradient . . .
∇φ =
Divergence ...
∇·A =
Curl . . .
Laplacian ...
∂φ ∂x i
+
∂Ax ∂x
∂φ ∂y j
+
i ∂ ∇ × A = ∂x Ax
∇2 φ =
∂2 φ ∂x2
+
∂Ay ∂y
+
k
∂ ∂y
∂ ∂z
Ay
Az
+
∂φ ∂r r
∇·A= z + ∂A ∂z
∂Az ∂z
j
∂2 φ ∂y 2
∇φ =
∂φ ∂z k
+
y = r sin ϕ z = z
1 ∂φ r ∂ϕ Φ
+
1 ∂(rAr ) r ∂r
1 r∂ r ∇ × A = ∂r Ar
∇2 φ =
∂2 φ ∂z 2
1 ∂ r ∂r
+
1 rk ∂ ∂z
rAϕ
Az
r ∂φ ∂r
+
2
(c)
F (c)
(∇φ) · ˆ t ds =
(c)
dφ (c)
A-91
z = r cos θ
∂φ ∂r r
+
1 ∂φ r ∂θ θ
2 1 ∂(r Ar ) ∂r r2 ∂A ϕ 1 + r sin θ ∂ϕ
+
∂φ 1 r sin θ ∂ϕ Φ
+
∂(Aθ sin θ) 1 r sin θ ∂θ
r Φ θ r2 sin θ r sin θ r ∂ ∂ ∂ ∇ × A = ∂r ∂θ ∂ϕ A rAθ rAϕ sin θ r
1 ∂2 φ r 2 ∂ϕ2
∇2 φ =
1 ∂ r 2 ∂φ ∂r r 2 ∂r ∂2 φ 1 + r2 sin2 θ ∂ϕ2
ds ) dS= n ˆ
y = r sin ϕ sin θ
∇·A=
1 ∂Aϕ r ∂ϕ
∂ ∂ϕ
x = r cos ϕ sin θ
∇φ =
∂φ ∂z k
Φ
F ·ˆ t ds = = ∇φ
+
+ ∂∂zφ 2
ˆ ˆ F ds= ˆ t . sC. S. V t C P n
F
Spherical coordinates
+
1 ∂ r 2 sin θ ∂θ
sin θ ∂φ ∂θ
dS )
∂ϕ2
Transformation of Integrals If 1. 2. 3. 4. 5. 6. 7. 8. then
s is the distance along a curve “C” in space and is measured from some fixed point. S is a surface area V is a volume contained by a specified surface ˆ t =the unit tangent to C at the point Pn ˆ =the unit outward pointing normal F is some vector function ds is the vector element of curve (= ˆ t ds ) dS is the vector element of surface (= n ˆ dS ) ˆ F · t ds = F (c)
and when F = ∇φ
(c)
(∇φ) · ˆ t ds =
(c)
dφ (c)
Gauss’ Theorem (Green’s Theorem) When S defines a closed region having a volume V : (∇ · F) dV = (F · n ˆ ) dS = F · dS (v) (s)
(∇φ) dV = (∇ × F) dV =
also
φn ˆ dS
(v)
and
(s)
(v)
(s)
(ˆ n × F) dS
(s)
Stokes’ Theorem When C is closed and bounds the open surface S: n ˆ · (∇ × F) dS = F · ds (s) (c)
(ˆ n × ∇φ) dS =
also
φ ds
(s) (c)
Green’s Theorem
(∇φ · ∇θ) dS =
φn ˆ · (∇θ) dS =
φ(∇2 θ) dV (v)
(s)
(s)
θ·n ˆ (∇φ)dS =
=
φ(∇2 θ) dV (v)
m
(s)
A-92
TeamLRN
5
MOMENT OF INERTIA FOR VARIOUS BODIES OF MASS The mass of the body is indicated by m Body
Axis
Moment of inertia
Uniform thin rod
m l3
The rectangular sheet, sides a and b
Normal to the length, at one end Normal to the length, at the center Through the center parallel to b
Thin rectangular sheet, sides a and b Thin circular sheet of radius r
Through the center perpendicular to the sheet Normal to the plate through the center
ma
Thin circular sheet of radius r
Along any diameter
m r4
Thin circular ring. Radii r1 and r2
Through center normal to plane of ring
m
2 +r 2 r1 2 2
Thin circular ring. Radii r1 and r2
Any diameter
m
2 +r 2 r1 2 4
Rectangular parallelepiped, edges a, b, and c Sphere, radius r
Through center perpendicular to face ab, (parallel to edge c) Any diameter
ma
Spherical shell, external radius, r1 , internal radius r2
Any diameter
Uniform thin rod
2
2
l m 12 2
m a12 2 +b2 12
2
m r2
2
2 +b2 12
m 25 r 2 m 25
5 −r 5 ) (r1 2 3 −r 3 ) (r1 2
Body
Axis
Moment of inertia
Spherical shell, very thin, mean radius, r Right circular cylinder of radius r, length l Right circular cylinder of radius r, length l
Any diameter
m 23 r 2
The longitudinal axis of the solid Transverse diameter
m r2
Hollow circular cylinder, length l, radii r1 and r2 Thin cylindrical shell, length l, mean radius, r
The longitudinal axis of the figure The longitudinal axis of the figure
m
Hollow circular cylinder, length l, radii r1 and r2
Transverse diameter
m
Hollow circular cylinder, length l, very thin, mean radius
Transverse diameter
m
Elliptic cylinder, length l, transverse semiaxes a and b
Longitudinal axis
m
Right cone, altitude h, radius of base r
Axis of the figure
3 2 m 10 r
Spheroid of revolution, equatorial radius r
Polar axis
m 2r5
Ellipsoid, axes 2a, 2b, 2c
Axis 2a
m (b
A-92
2
m
r2 4
+
l2 12
2 +r 2 ) (r1 2 2
mr2
2 +r 2 r1 2 4 r2 2
+
a2 +b2 4
2
2 +c2 ) 5
+ l2 12
l2 12
SPECIAL FUNCTIONS Bessel Functions 1. Bessel’s differential equation for a real variable x is x2
d2 y dy +x + (x2 − n2 )y = 0 2 dx dx
2. When n is not an integer, two independent solutions of the equation are Jn (x), J−n (x), where Jn (x) =
∞ k=0
x (−1)k k!Γ(n + k + 1) 2
n+2k
3. If n is an integer Jn (x) = (−1)n Jn (x), where # " xn x4 x6 x2 Jn (x) = n + 4 + 6 + ... 1− 2 2 n! 2 · 1!(n + 1) 2 · 2!(n + 1) (n + 2) 2 · 3!(n + 1) (n + 2) (n + 3) 4. For n = 0 and n = 1, this formula becomes x2 22 (1!)2 x x3 2 − 23 ·1!2!
J0 (x) = 1 −
+
J1 (x) =
+
x4 24 (2!)2 x5 25 ·2!3!
− −
x6 x8 26 (3!)2 + 28 (4!)2 − · · · x7 x9 27 ·3!4! + 29 ·4!5! − . . .
5. When x is large and positive, the following asymptotic series may be used 1 π π ! 2 2 − Q0 (x) sin x − P0 (x) cos x − πx 4 4 & 2 12 % 3π 3π J1 (x) = πx P1 (x) cos x − 4 − Q1 (x) sin x − 4
J0 (x) =
where 1 2 · 32 12 · 32 · 52 · 72 12 · 32 · 52 · 72 · 92 · 112 + − + ··· 2!(8x)2 4!(8x)4 6!(8x)6 12 12 ·32 ·52 12 ·32 ·52 ·72 ·92 Q0 (x) ∼ − 1!8x + 3!(8x)3 − + −··· 5!(8x)5
P0 (x) ∼ 1 −
2 2 2 2 2 12 ·3·5 12 ·32 ·52 ·7·9 ·7 ·9 ·11·13 + 1 ·3 ·56!(8x) 6 2!(8x)2 − 4!(8x)4 2 2 2 2 2 2 1·3 1 ·3 ·5·7 1 ·3 ·5 ·7 ·9·11 − + − · · · 3 5 1!8x 3!(8x) 5!(8x)
P1 (x) ∼ 1 + Q1 (x) ∼
− +···
[In P1 (x) the signs alternate from+to-after the first term] 6. The zeros of J0 (x) and J1 (x). If j0s and j1s are the sth zeros of J0 (x) and J1 (x) respectively, and if a = 4s − 1, b = 4s + 1 # " 1 62 15, 116 12, 554, 474 8, 368, 654, 292 2 j0,s ∼ πa 1 + 2 2 − 4 4 + − + − +··· 10 a10 4 % π a 3π a 15π 6 a6 105π 8 a8 315π & 3,902,418 895,167,324 4716 j1,s ∼ 14 πb 1 − π26b2 + π46b4 − 5π 6 b6 + 35π 8 b8 − 35π 10 b10 + · · · & % s+1 3 2 (−1) 2 56 9664 J1 (j0,s ) ∼ 1 − 3π4 a4 + 5π6 a6 − 7,381,280 1 21π 8 a8 + · · · πa 23 & s 2 % 19,584 2,466,720 J0 (j1,s ) ∼ (−1) 12 1 + π24 4 b4 − 10π 6 b6 + 7π 8 b8 − · · · πb 2
A-93
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SPECIAL FUNCTIONS 7. Table of zeros for J0 (x) and J1 (x) Define {αn , βn } by J1 (αn ) = 0 and J0 (βn ) = 0. Roots αn 2.4048 5.5201 8.6537 11.7915 14.9309 18.0711 21.2116
J1 (αn ) 0.5191 -0.3403 0.2715 -0.2325 0.2065 -0.1877 0.1733
Roots β n 0.0000 3.8317 7.0156 10.1735 13.3237 16.4706 19.6159
J0 (βn ) 1.0000 -0.4028 0.3001 -0.2497 0.2184 -0.1965 0.1801
8. Recurrence formulas 2n Jn (x) nJn (x) + xJn (x) = xJn−1 (x) x Jn−1 (x) − Jn+1 (x) = 2Jn (x) nJn (x) − xJn (x) = xJn+1 (x)
Jn−1 (x) + Jn+1 (x) =
(k)
9. If Jn is written for Jn (x) and Jn relationships are important (r)
J0 (2) J0 (3) J0 (4) J0
is written for
(r−1)
J 12 (x) =
2
πx
J− 12 (x) = Jn+ 32 (x) = Jn− 12 (x) =
sin x
2 πx cos x 1 d 1 −xn+ 2 dx {x−(n+ 2 ) Jn+ 12 (x)} 1 1 d x−(n+ 2 ) dx {xn+ 2 Jn+ 12 (x)}
πx 12
n 0
2
x3
2
− cos x
− 1 sin x − x3 cos x − x6 sin x − x152 − 1 cos x etc.
15
πx 12
Jn+ 12 (x) sin x
sin x x
1 2
then the following derivative
= −J1 = −J0 + x1 J1 = 12 (J2 − J0 ) = x1 J0 + 1 − x22 J1 = 14 (−J3 + 3J1 ) = 1 − x32 J0 − x2 − x63 J1 = 18 (J4 − 4J2 + 3J0 ), etc.
10. Half order Bessel functions
3
dk {Jn (x)}, dxk
3 x2
−
15 x3
J−(n+ 12 ) (x) cos x
3 x2
− cosx x − sin x − 1 cos x + x3 sin x
−
6 x
cos x −
15 x2
− 1 sin x
11. Additional solutions to Bessel’s equation are Yn (x) (also called Weber’s function, and sometimes denoted by Nn (x)) (1) (2) Hn (x) and Hn (x) (also called Hankel functions) These solutions are defined as follows Jn (x) cos (nπ) − J−n (x) sin(nπ) Yn (x) = −v (x) lim Jv (x) cos(vπ)−J sin(vπ) v→n
n
not an integer n
A-94
an integer
(1)
Hn (x) = Jn (x) + iYn (x) (2) Hn (x) = Jn (x) − iYn (x)
SPECIAL FUNCTIONS The additional properties of these functions may all be derived from the above relations and the known properties of Jn (x). 12. Complete solutions to Bessel’s equation may be written as c1 Jn (x) + c2 J−n (x)
if n is not an integer
or, for any value of n, c1 Jn (x) + c2 Yn (x)
c1 Hn(1) x + c2 Hn(2) (x)
or
13. The modified (or hyperbolic) Bessel’s differential equation is x2
dy d2 y +x − (x2 + n2 )y = 0 dx2 dx
14. When n is not an integer, two independent solutions of the equation are In (x) and I−n (x), where ∞ x n+2k 1 In (x) = k!Γ(n + k + 1) 2 k=0
15. If n is an integer, In (x) = I−n (x) =
xn 2n n!
"
x2 x4 + 22 · 1!(n + 1) 24 · 2!(n + 1)(n + 2) # x6 + 6 + ··· 2 · 3!(n + 1) (n + 2) (n + 3) 1+
16. For N = 0 and n = 1, this formula becomes I0 (x) = 1 + I1 (x) =
x 2
+
x2 22 (1!)2 3
x 23 ·1!2!
+
+
x4 24 (2!)2 5
x 25 ·2!3!
+
+
x6 26 (3!)2 7
x 27 ·3!4!
17. Another solution to the modified Bessel’s equation is $ 1 I−n (x)−In (x) 2π sin (nπ) Kn (x) = (x)−Iv (x) lim 21 π I−vsin (vπ) v→n
+
+ 9
x8 28 (4!)2
x 29 ·4!5!
+ ···
+ ···
n not an integer n an integer
This function is linearly independent of In (x) for all values of n. Thus the complete solution to the modified Bessel’s equation may be written as c1 In (x) + c2 I−n (x) n not an integer or c1 In (x) + c2 Kn (x)
any n
18. The following relations hold among the various Bessel functions: In (z) = i−m Jm (iz) Yn (iz) = (i)n+1 In (z) − π2 i−n Kn (z) Most of the properties of the modified Bessel function may be deduced from the known properties of Jn (x) by use of these relations and those previously given. 19. Recurrence formulas In−1 (x) − In+1 (x) = 2n x In (x) In−1 (x) + In+1 (x) = 2In (x) n n In (x) = In+1 (x) + x In (z) In−1 (x) − x In (x) = In (x)
A-95
TeamLRN
SPECIAL FUNCTIONS The Factorial Function For non-negative integers n, the factorial of n, denoted n!, is the product of all positive integers less than or equal to n; n! = n · (n − 1) · (n − 2) · · · 2 · 1. If n is a negative integer (n = −1, −2, . . . ) then n! = ±∞. Approximations to n! for large n include Stirling’s formula n n+ 12 √ n! ≈ 2πe , e and Burnsides’s formula n+ 12 √ n + 12 n! ≈ 2π . e n 0 2 4 6 8 10 12 14 16 18 20 30 50 70 90 110 130 500
n! 1 2 24 720 40320 3.6288 × 106 4.7900 × 108 8.7178 × 1010 2.0923 × 1013 6.4024 × 1015 2.4329 × 1018 2.6525 × 1032 3.0414 × 1064 1.1979 × 10100 1.4857 × 10138 1.5882 × 10178 6.4669 × 10219 1.2201 × 101134
Definition: Γ(n) =
∞
log10 n! 0.00000 0.30103 1.38021 2.85733 4.60552 6.55976 8.68034 10.94041 13.32062 15.80634 18.38612 32.42366 64.48307 100.07841 138.17194 178.20092 219.81069 1134.0864
n 1 3 5 7 9 11 13 15 17 19 25 40 60 80 100 120 150 1000
n! 1 6 120 5040 3.6288 × 105 3.9917 × 107 6.2270 × 109 1.3077 × 1012 3.5569 × 1014 1.2165 × 1017 1.5511 × 1025 8.1592 × 1047 8.3210 × 1081 7.1569 × 10118 9.3326 × 10157 6.6895 × 10198 5.7134 × 10262 4.0239 × 102567
log10 n! 0.00000 0.77815 2.07918 3.70243 5.55976 7.60116 9.79428 12.11650 14.55107 17.08509 25.19065 47.91165 81.92017 118.85473 157.97000 198.82539 262.75689 2567.6046
The Gamma Function t
n−1 −t
e
dt
n>0
0
Recursion Formula: Γ(n + 1) = nΓ(n) Γ(n + 1) = n! if n = 0, 1, 2, . . . where 0! = 1 For n < 0 the gamma function can be defined by using Γ(n) = Γ(n+1) n Graph:
A-96
SPECIAL FUNCTIONS Special Values:
Γ(1/2) =
√
π
1 · 3 · 5 · · · (2m − 1) √ π m = 1, 2, 3, . . . 2m √ (−1)m 2m π m = 1, 2, 3, . . . Γ(−m + 1/2) = 1 · 3 · 5 · · · (2m − 1)
Γ(m + 1/2) =
Definition:
1 · 2 · 3···k kx k→∞ (x + 1) (x + 2) · · · (x + k) ∞ % −x/m & x 1+ m e = xeγx
Γ(x + 1) = lim 1 Γ(x)
m=1
This is an infinite product representation for the gamma function where γ is Euler’s constant. Properties: ∞ Γ (1) = eγx ln x dx = −γ 0 Γ (x) 1 1 1 1 1 1 = −γ + − − − + + ... + + ··· Γ(x) 1 x " 2 x+1 n x + n#− 1 √ 1 139 1 Γ(x + 1) = 2πx xx e−x 1 + − + ... + 2 12x 288x 51, 840x3 This is called Stirling’s asymptotic series. ∞ Values of Γ(n) = 0 e−x xn−1 dx; n 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24
Γ(n) 1.00000 .99433 .98884 .98355 .97844 .97350 .96874 .96415 .95973 .95546 .95135 .94740 .94359 .93993 .93642 .93304 .92980 .92670 .92373 .92089 .91817 .91558 .91311 .91075 .90852
n 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49
Γ(n) .90640 .90440 .90250 .90072 .89904 .89747 .89600 .89464 .89338 .89222 .89115 .89018 .88931 .88854 .88785 .88726 .88676 .88636 .88604 .88581 .88566 .88560 .88563 .88575 .88595
n 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74
A-97
TeamLRN
Γ(n + 1) = nΓ(n) Γ(n) .88623 .88659 .88704 .88757 .88818 .88887 .88964 .89049 .89142 .89243 .89352 .89468 .89592 .89724 .89864 .90012 .90167 .90330 .90500 .90678 .90864 .91057 .91258 .91466 .91683
n 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00
Γ(n) .91906 .92137 .92376 .92623 .92877 .93138 .93408 .93685 .93969 .94261 .94561 .94869 .95184 .95507 .95838 .96177 .96523 .96877 .97240 .97610 .97988 .98374 .98768 .99171 .99581 1.00000
SPECIAL FUNCTIONS Definition: B(m, n) =
The Beta Function
1
t
m−1
(1 − t)
m−1
dt
m > 0, n > 0
0
Γ(m)Γ(n) Γ(m + n) B(m, n) = B(n, m) π/2 B(m, n) = 2 0 sin2m−1 θ cos2n−1 θ dθ ∞ tm−1 B(m, n) = 0 (1+t)m+n dt 1 m−1 (1−t)n−1 B(m, n) = rn (r + 1)m 0 t (r+t) dt m+n
Relationship with Gamma function: B(m, n) = Properties:
The Error Function x 2 −t2 Definition: erf(x) = √ e dt π 0 3 x 1 x5 1 x7 2 x− + − + ··· Series: erf (x) = √ 3 2! 5 3! 7 π Property: erf(x) = − erf(−x)
1 x erf √ 2 2 0 √ To evaluate erf(2.3), one proceeds as follows: For √x2 = 2.3, one finds x = (2.3) ( 2) = 3.25. In the normal probability function table (page A-104), one finds the entry 0.4994 opposite the value 3.25. Thus erf(2.3) = 2(0.4994) = 0.9988. x
Relationship with Normal Probability Function f (t) :
f (t) dt =
2 erfc(z) = 1 − erf(z) = √ π
∞
e−t dt 2
z
is known as the complementary error function. Orthogonal Polynomials I: Legendre Name: Legendre Symbol: Pn (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y − 2 xy + n(n + 1)y = 0 y = Pn (x) [n/2] 1 n 2n − 2m n−2m Explicit Expression: Pn (x) = n (−1)m x 2 m=0 m n Recurrence Relation: (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x) Weight: 1 Standardization: Pn (1)=1 +1 2 [Pn (x)]2 dx = Norm: 2n +1 −1 (−1)n dn {(1 − x2 )n } Rodrigues’ Formula: Pn (x) = n 2 n! dxn ∞ Generating Function: R−1 = Pn (x)z n ; −1 < x < 1, |z| < 1, √ n=0 R = 1 − 2xz + z 2 Inequality: |Pn (x)| ≤ 1, −1 ≤ x ≤ 1. II: Tschebysheff, First Kind Name: Tschebysheff, First Kind Symbol: Tn (x) Interval:[-1, 1] Differential Equation: (1 − x2 )y − xy + n2 y = 0 y = Tn (x) [n/2] n (n − m − 1)! Explicit Expression: (2x)n−2m = cos(n arccos x) = Tn (x) (−1)m 2 m=0 m!(n − 2m)! A-98
SPECIAL FUNCTIONS Recurrence Relation: Tn+1 (x) = 2xTn (x) − Tn−1 (x) Weight: (1 − x2 )−1/2 Standardization: Tn (1) = 1 " +1 π/2, n = 0 Norm: −1 (1 − x2 )−1/2 [Tn (x)]2 dx = π, n = 0 √ (−1)n (1 − x2 )1/2 π dn {(1 − x2 )n−(1/2) } = Tn (x) Rodrigues’ Formula: dxn 2n+1 Γ(n + 12 ) ∞ 1 − xz Generating Function: = Tn (x) z n , −1 < x < 1, |z| < 1 1 − 2xz − z 2 n=0 Inequality: |Tn (x)| ≤ 1, −1 ≤ x ≤ 1. III: Tschebysheff, Second Kind Name: Tschebysheff, Second Kind Symbol Un (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y − 3 xy + n(n + 2)y = 0 y = Un (x) [n/2] (m − n)! Explicit Expression: Un (x) = (2x)n−2m (−1)m m!(n − 2m)! m=0 sin[(n + 1)θ] Un (cos θ) = sin θ Recurrence Relation: Un+1 (x) = 2xUn (x) − Un−1 (x) Standardization: Un (1) = n + 1 Weight: (1 − x2 )1/2 +1 π 2 1/2 2 Norm: (1 − x ) [Un (x)] dx = 2 −1 √ (−1)n (n + 1) π dn 2 n+(1/2) {(1 − x ) } Rodrigues’ Formula: Un (x) = (1 − x2 )1/2 2n+1 Γ(n + 32 ) dxn ∞ 1 n Generating Function: = Un (x)z , − 1 < x < 1, |z| < 1 1 − 2xz + z 2 n=0 Inequality: |Un (x)| ≤ n + 1, −1 ≤ x ≤ 1.
IV: Jacobi (α,β) (x) Interval: [-1, 1] Name: Jacobi Symbol: Pn Differential Equation: (1 − x2 )y + [β − α − (α + β + 2)x]y + n(n + α + β + 1)y = 0 (α,β) y = Pn (x) n 1 n + α n + β Explicit Expression: Pn(α,β) (x) = n (x − 1)n−m (x + 1)m 2 m=0 m n−m Recurrence Relation: (α,β)
2(n + 1) (n + α + β + 1) (2n + α + β)Pn+1 (x) = (2n + α + β + 1)[(α2 − β 2 ) + (2n + α + β + 2) × (2n + α + β)x]Pn(α,β) (x) (α,β)
− 2(n + α) (n + β) (2n + α + β + 2)Pn−1 (x) (α,β) Weight: (1 − x)α (1 + x)β ; α, β > 1 Standardization: Pn (x) = n+α n +1 2α+β+1 Γ(n + α + 1)Γ(n + β + 1) Norm: (1 − x)α (1 + x)β [Pn(α,β) (x)]2 dx = (2n + α + β + 1)n!Γ(n + α + β + 1) −1 dn (−1)n (α,β) Rodrigues’ Formula: Pn (x) = n {(1 − x)n+α (1 + x)n+β } α β 2 n!(1 − x) (1 + x) dxn
A-99
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SPECIAL FUNCTIONS ∞ Generating Function: R−1 (1 − z + R)−α (1 + z + R)−β = 2−α−β Pn(α,β) (x)z n , n=0 √ R = 1 − 2xz+ z 2 , |z| < 1 n+q 1 ∼ nq if q = max(α, β) ≥ − n 2 (α,β) Inequality: max |Pn(α,β) (x)| = (x )| ∼ n−1/2 if q < − 12 |Pn −1≤x≤1 x is one of the two maximum points nearest β−α α+β+1
V: Generalized Laguerre (α) Name: Generalized Laguerre Symbol: Ln (x) Interval: [0, ∞] xy + (α + 1 − x)y + ny = 0 Differential Equation: (α) y = Ln (x) n 1 m (α) m n+α (−1) x Explicit Expression: Ln (x) = n − m m! m=0 (α)
(α)
(α)
Recurrence Relation: (n + 1)Ln + 1(x) = [(2n + α + 1) − x]Ln (x) − (n + α)Ln − 1(x) n (α) Weight: xα e−x , α > −1 Standardization: Ln (x) = (−1) xn + · · · n! ∞ Γ(n + α + 1) 2 Norm: xα e−x [L(α) n (x)] dx = n! 0 dn 1 (α) {xn+α e−x } Rodrigues’ Formula: Ln (x) = n!xα e−xdxn∞ Generating Function: (1 − z)−α−1 exp
xz z−1
=
(α)
Ln (x)z n
n=0
Γ(n + α + 1) x/2 x≥0 e ; α>0 n!Γ(α + 1) x≥0 (a) Γ(α+n+1) x/2 |Ln (x)| ≤ 2 − n!Γ(α+1) e ; −1 < α < 0
Inequality: |L(α) n (x) ≤
VI: Hermite Name: Hermite Symbol:Hn (x) Interval: [−∞, ∞] Differential Equation: y − 2xy + 2ny = 0 [n/2] (−1)m n!(2x)n−2m Explicit Expression: Hn (x) = m!(n − 2m)! m=0 Recurrence Relation:Hn+1 (x) = 2xHn (x) − 2nHn−1 (x) 2 Weight: e−x Standardization: Hn (1) = 2n xn + · · · ∞ 2 √ 2 e−x [Hn (x)] dx = 2n n! π Norm: −∞
2
n
2
−x d ) Rodrigues’ Formula: Hn (x) = (−1)n ex dx n (e ∞ n 2 z Generating Function: e−x +2zx = Hn (x) n! n=0 √ x2 /2 n/2 k2 n! k ≈ 1.086435 Inequality: |Hn (x)|e
Tables of Orthogonal Polynomials
H0 = 1 H1 = 2x H2 = 4x2 − 2
x10 = (30240H0 + 75600H2 + 25200H4 + 2520H6 + 90H8 + H10 )/1024 x9 = (15120H1 + 10080H3 + 1512H5 + 72H7 + H9 )/512 x8 = (1680H0 + 3360H2 + 840H4 + 56H6 + H8 )/256
A-100
H3 = 8x3 − 12x x7 = (840H1 + 420H3 + 42H5 + H7 )/128 4 2 H4 = 16x − 48x + 12 x6 = (120H0 + 180H2 + 30H4 + H6 )/64 5 3 H5 = 32x − 160x + 120x x5 = (60H1 + 20H3 + H5 )/32 H6 = 64x6 − 480x4 + 720x2 − 120 x4 = (12H0 + 12H2 + H4 )/16 H7 = 128x7 − 1344x5 + 3360x3 − 1680x x3 = (6H1 + H3 )/8 8 6 4 2 H8 = 256x − 3584x + 13440x − 13440x + 1680 x2 = (2H0 + H2 )/4 9 7 5 3 H9 = 512x − 9216x + 48384x − 80640x + 30240x x = (H1 )/2 H10 = 1024x10 − 23040x8 + 161280x6 − 403200x4 + 302400x2 − 30240 1 = H0 L0 L1 L2 L3 L4 L5 L6
=1 x6 = 720L0 − 4320L1 + 10800L2 − 14400L3 + 10800L4 − 4320L5 + 720L6 = −x + 1 x5 = 120L0 − 600L1 + 1200L2 − 1200L3 + 600L4 − 120L5 = (x2 − 4x + 2)/2 x4 = 24L0 − 96L1 + 144L2 − 96L3 + 24L4 = (−x3 + 9x2 − 18x + 6)/6 x3 = 6L0 − 18L1 + 18L2 − 6L3 4 3 2 = (x − 16x + 72x − 96x + 24)/24 x2 = 2L0 − 4L1 + 2L2 5 4 3 2 = (−x + 25x − 200x + 600x − 600x + 120)/120 x = L0 − L1 = (x6 − 36x5 + 450x4 − 2400x3 + 5400x2 − 4320x + 720)/720 1 = L0
P0 = 1 x10 = (4199P0 + 16150P2 + 15504P4 + 7904P6 + 2176P8 + 256P10 )/46189 P1 = x x9 = (3315P1 + 4760P3 + 2992P5 + 960P7 + 128P9 )/12155 2 P2 = (3x − 1)/2 x8 = (715P0 + 2600P2 + 2160P4 + 832P6 + 128P8 )/6435 P3 = (5x3 − 3x)/2 x7 = (143P1 + 182P3 + 88P5 + 16P7 )/429 4 2 P4 = (35x − 30x + 3)/8 x6 = (33P0 + 110P2 + 72P4 + 16P6 )/231 5 3 P5 = (63x − 70x + 15x)/8 x5 = (27P1 + 28P3 + 8P5 )/63 6 4 2 P6 = (231x − 315x + 105x − 5)/16 x4 = (7P0 + 20P2 + 8P4 )/35 P7 = (429x7 − 693x5 + 315x3 − 35x)/16 x3 = (3P1 + 2P3 )/5 8 6 4 2 P8 = (6435x − 12012x + 6930x − 1260x + 35)/128 x2 = (P0 + 2P2 )/3 9 7 5 3 P9 = (12155x − 25740x + 18018x − 4620x + 315x)/128 x = P1 P10 = (46189x10 − 109395x8 + 90090x6 − 30030x4 + 3465x2 − 63)/256 1 = P0 T0 = 1 x10 = (126T0 + 210T2 + 120T4 + 45T6 + 10T8 + T10 )/512 T1 = x x9 = (126T1 + 84T3 + 36T5 + 9T7 + T9 )/256 2 T2 = 2x − 1 x8 = (35T0 + 56T2 + 28T4 + 8T6 + T8 )/128 3 T3 = 4x − 3x x7 = (35T1 + 21T3 + 7T5 + T7 )/64 T4 = 8x4 − 8x2 + 1 x6 = (10T0 + 15T2 + 6T4 + T6 )/32 T5 = 16x5 − 20x3 + 5x x5 = (10T1 + 5T3 + T5 )/16 6 4 2 T6 = 32x − 48x + 18x − 1 x4 = (3T0 + 4T2 + T4 )/8 T7 = 64x7 − 112x5 + 56x3 − 7x x3 = (3T1 + T3 )/4 T8 = 128x8 − 256x6 + 160x4 − 32x2 + 1 x2 = (T0 + T2 )/2 9 7 5 3 T9 = 256x − 576x + 432x − 120x + 9x x = T1 T10 = 512x10 − 1280x8 + 1120x6 − 400x4 + 50x2 − 1 1 = T0 U0 = 1 x10 = (42U0 + 90U2 + 75U4 + 35U6 + 9U8 + U10 )/1024 U1 = 2x x9 = (42U1 + 48U3 + 27U5 + 8U7 + U9 )/512 2 U2 = 4x − 1 x8 = (14U0 + 28U2 + 20U4 + 7U6 + U8 )/256 3 U3 = 8x − 4x x7 = (14U1 + 14U3 + 6U5 + U7 )/128 U4 = 16x4 − 12x2 + 1 x6 = (5U0 + 9U2 + 5U4 + U6 )/64 5 3 U5 = 32x − 32x + 6x x5 = (5U1 + 4U3 + U5 )/32 6 4 2 U6 = 64x − 80x + 24x − 1 x4 = (2U0 + 3U2 + U4 )/16 7 5 3 U7 = 128x − 192x + 80x − 8x x3 = (2U1 + U3 )/8 U8 = 256x8 − 448x6 + 240x4 − 40x2 + 1 x2 = (U0 + U2 )/4 9 7 5 3 U9 = 512x − 1024x + 672x − 160x + 10x x = (U1 )/2 U10 = 1024x10 − 2304x8 + 1792x6 − 560x4 + 60x2 − 1 1 = U0
A-101
TeamLRN
Clebsch–Gordan coefficients
j1 m1
×
j2 j m2 m
k
'
= δm,m1 +m2
(j1 + j2 − j)!(j + j1 − j2 )!(j + j2 − j1 )!(2j + 1) (j + j1 + j2 + 1)!
(−1)k (j1 + m1 )!(j1 − m1 )!(j2 + m2 )!(j2 − m2 )!(j + m)!(j − m)! . k!(j1 + j2 − j − k)!(j1 − m1 − k)!(j2 + m2 − k)!(j − j2 + m1 + k)!(j − j1 − m2 + k)!
1. Conditions: (a) Each of {j1 , j2 , j, m1 , m2 , m} may be an integer, or half an integer. Additionally: j > 0, j1 > 0, j2 > 0 and j + j1 + j2 is an integer. (b) j1 + j2 − j ≥ 0. (c) j1 − j2 + j ≥ 0. (d) −j1 + j2 + j ≥ 0. (e) |m1 | ≤ j1 , |m2 | ≤ j2 , |m| ≤ j. 2. Special values: j2 j j1 = 0 if m1 + m2 = m. (a) m m2 m 1 j1 0 j = δj1 ,j δm1 ,m . (b) m1 0 m j 1 j2 j = 0 when j1 + j2 + j is an odd integer. (c) 0 0 0 j1 j j1 = 0 when 2j1 + j is an odd integer. (d) m1 m1 m j1 j2 j : 3. Symmetry relations: all of the following are equal to m1 m2 m j2 j1 j (a) , −m2 −m1 −m j2 j1 j , (b) (−1)j1 +j2 −j m m2 m 1 j1 j2 j , (c) (−1)j1 +j2 −j −m1 −m2 −m j j2 j1 2j+1 j2 +m2 (d) (−1) , 2j1 +1 −m m2 −m1 j j2 j1 2j+1 (−1)j1 −m1 +j−m , (e) 2j1 +1 m −m2 m1 j2 j j1 2j+1 , (−1)j−m+j1 −m1 (f) 2j1 +1 m2 −m −m1 j1 j j2 2j+1 (−1)j1 −m1 , (g) 2j2 +1 m1 −m −m2 j j1 j2 2j+1 (h) (−1)j1 −m1 . 2j2 +1 m −m1 m2 By use of the symmetry relations, Clebsch–Gordan coefficients may be put in the standard form j1 ≤ j2 ≤ j and m ≥ 0.
A-102
m2
m
j1
j
− 12
0
1 2
1
0
1 2
1 2
1
1 2
0
1 2
1
1 2
1 2
1 2
1
1
1 2
1 2
j1 m1
√
1
m2
m
j1
j
−1
0
1
1
−1
0
1
2
− 12
0
1 2
3 2
− 12
1 2
1
1
− 12
1 2
1
2
0
0
1
2
0
0
1 2
3 2
0
1 2
1 2
3 2
0
1 2
1
1
0
1 2
1
2
0
1
1
1
2 2 √ 3 2 √ 2 2 √ 3 2
1 2 m2
j m
m2
m
j1
j
≈ 0.707107
0
1
1
2
≈ 0.866025
1 2
0
1 2
3 2
≈ 0.707107
1 2
1 2
1
1
≈ 0.866025
1 2
1 2
1
2
1 2
1
1 2
3 2
1 2
3 2
1
2
1
0
1
1
√
1 ≈ 1.000000 j1 1 j m1 m2 m √ 2 2 √ 6 6 √ 2 2 3 4 √ 5 4 √ 6 3 √ 3 2 √ 6 3 √ 2 4 √ 10 4 √ 2 2
j1 m1
≈ 0.707107 ≈ 0.408248 ≈ 0.707107 ≈ 0.750000 ≈ 0.559017 ≈ 0.816496 ≈ 0.866025 ≈ 0.8164967 ≈ 0.353553 ≈ 0.790569 ≈ 0.707107
1
0
1
2
1
1 2
1 2
3 2
2 2 √ 2 2 √ − 42 √ 10 4 √ 30 6 √ 105 12 √ − 22 √ 6 6 √ 3 3
1
1 2
1
1
− 34
1
1 2
1
2
1
1
1 2
3 2
1
1
1
1
√
≈ 0.707107 ≈ 0.707107 ≈ −0.353553 ≈ 0.790569 ≈ 0.912871 ≈ 0.853913 ≈ −0.707107 ≈ 0.408248 ≈ 0.577350 ≈ −0.750000 ≈ 0.559017
10 4 √ − 22 √ 2 2
≈ 0.790569
1
≈ 1.000000
√
1
1
2
1
3 2
1 2
3 2
1
3 2
1
2
√ 105 12
1
2
1
2
1
TeamLRN
5 4
1
A-103
1 j m2 m
≈ −0.707107 ≈ 0.707107
≈ 0.853913 ≈ 1.000000
NORMAL PROBABILITY FUNCTION Table of the normal distribution For a standard normal random variable (Φ(z) is the area under the Standard Normal Curve from −∞ to z). Proportion of the total area (%) 68.27 90 95 95.45 99.0 99.73 99.8 99.9
Limits µ − λσ µ−σ µ − 1.65σ µ − 1.96σ µ − 2σ µ − 2.58σ µ − 3σ µ − 3.09σ µ − 3.29σ x Φ(x) 2[1 − Φ(x)] x 1 − Φ(x)
3.09 10−3
µ + λσ µ+σ µ + 1.65σ µ + 1.96σ µ + 2σ µ + 2.58σ µ + 3σ µ + 3.09σ µ + 3.29σ 1.282 0.90 0.20 3.72 10−4
1.645 0.95 0.10 4.26 10−5
1.960 0.975 0.05 4.75 10−6
Remaining area (%) 31.73 10 5 4.55 0.99 0.27 0.2 0.1
2.326 0.99 0.02 5.20 10−7
2.576 0.995 0.01 5.61 10−8
3.090 0.999 0.002 6.00 10−9
6.36 10−10
Areas under the Standard Normal Curve from 0 to z z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9
0 .0000 .0398 .0793 .1179 .1554 .1915 .2258 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 4990 4993 4995 4997 4998 4998 4999 4999 5000
1 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2612 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .4991 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .5000
2 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2652 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .4991 .4994 .4995 .4997 .4998 .4999 .4999 .4999 .5000
3 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .4991 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000
4 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2996 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000
A-104
5 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000
6 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000
7 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .4992 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000
8 .0319 .0714 .1103 .1480 .1844 .2190 .2518 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .4993 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000
9 .0359 .0754 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .4999 .5000
Common sample size calculations Parameter
Estimate
Sample size
µ
x ¯
n=
p
pˆ
n=
µ2 − µ2
x ¯1 − x ¯2
n1 = n2 =
(zα/2 )2 (σ12 + σ22 ) E2
p1 − p2
pˆ1 − pˆ2
n1 = n2 =
(zα/2 )2 (p1 q1 + p2 q2 ) E2
z
α/2
· σ 2
E (zα/2 )2 · pq E2
Common one sample confidence intervals Parameter
Assumptions
100(1 − α)% Confidence interval
µ
n large, σ 2 known, or normality, σ 2 known
σ x ¯ ± zα/2 · √ n
µ
normality, σ 2 unknown
σ2
normality
p
binomial experiment, n large
s x ¯ ± tα/2,n−1 · √ n (n − 1)s2 (n − 1)s2 , χ2α/2,n−1 χ21−α/2,n−1
pˆ(1 − pˆ) pˆ ± zα/2 · n
Common two sample confidence intervals 100(1 − α)% Confidence interval
Parameter
Assumptions
µ1 − µ2
normality, independence, σ12 , σ22 known or n1 , n2 large, independence, σ12 , σ22 known
(¯ x1 − x ¯2 ) ± zα/2 · (¯ x1 − x ¯2 ) ±
σ12
=
σ22
µ1 − µ2
normality, independence, unknown
µ1 − µ2
normality, independence, σ12 = σ22 unknown
σ2 σ12 + 2 n1 n2
1 1 + n1 n2 (n1 − 1)s21 + (n2 − 1)s22 s2p = n1 + n2 −2 s2 s21 (¯ x1 − x ¯2 ) ± tα/2,ν · + 2 n n 1 2
t α2 ,n1 +n2 −2 · sp
ν≈
s2 1 n1
2 (s2 1 /n1 ) n1 −1
+ +
s2 2 n2
2
2 (s2 2 /n2 ) n2 −1
µ1 − µ2
normality, n pairs, dependence
sd d¯ ± tα/2,n−1 · √ n
p1 − p2
binomial experiments, n1 , n2 large, independence
(ˆ p1 − pˆ2 )±
pˆ2 (1 − pˆ2 ) pˆ1 (1 − pˆ1 ) zα/2 · + n1 n2
A-105
TeamLRN
PERCENTAGE POINTS, STUDENT’S t-DISTRIBUTION This table gives values of t such that F (t) =
Γ n+1 x2 n+1 2 1 + dx − √ n n 2 nπΓ 2 −∞ for n, the number of degrees of freedom, equal to 1, 2, . . . , 30, 40, 60, 120, ∞; and for F (t) = 0.60, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, and 0.9995. The t-distribution is symmetrical, so that F (−t) = 1 − F (t) n/F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
.60 .325 .289 .277 .271 .267 .265 .263 .262 .261 .260 .260 .259 .259 .258 .258 .258 .257 .257 .257 .257 .257 .256 .256 .256 .256 .256 .256 .256 .256 .256 .255 .254 .254 .253
.75 1.000 .816 .765 .741 .727 .718 .711 .706 .703 .700 .697 .695 .694 .692 .691 .690 .689 .688 .688 .687 .686 .686 .685 .685 .684 .684 .684 .683 .683 .683 .681 .679 .677 .674
t
.90 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.296 1.289 1.282
.95 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.671 1.658 1.645
.975 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000 1.980 1.960
.99 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.390 2.358 2.326
.995 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660 2.617 2.576
.9995 636.619 31.598 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.767 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.460 3.373 3.291
*This table is abridged from the “Statistical Tables” of R. A. Fisher and Frank Yates published by Oliver & Boyd. Ltd., Edinburgh and London, 1938. It is here published with the kind permission of the authors and their publishers.
PERCENTAGE POINTS, CHI-SQUARE DISTRIBUTION This table gives values of χ2 such that
χ2
1 x(n−2)/2 e−x/2 dx n/2 Γ n 2 0 2 for n, the number of degrees equal to 1, 2, . . ., 30. For n > 30, a normal approximation is quite √ of freedom, √ accurate. The expression 2x2 − 2n − 1 is approximately normally distributed as the standard normal distribution. Thus χ2α , the α-point of the distribution, may be computed by the formula √ 1 χ2α = [xα + 2n − 1]2 , 2 F (χ)2 =
where xα is the α-point of the cumulative normal distribution. For even values of n, F (χ2 ) can be written as 1 − F (χ2 ) =
x −1
x=0
A-106
e−λ λx x!
with λ = 12 χ2 and x = 12 n. Thus the cumulative Chi-Square distribution is related to the cumulative Poisson distribution. Another approximate formula for large n
3 2 2 2 χα = n 1 − + zα 9n 9n n = degrees of freedom zα = the normal deviate (the value of x for which F (x) = the desired percentile). x 1.282 1.645 1.960 2.326 2.576 3.090 F (x) .90 .95 .975 .99 .995 .999 χ2.99 = 60[1 − 0.00370 + 2.326(0.06086)]3 = 88.4 is the 99th percentile for 60 degrees of freedom.
χ2
1 xn−2/2 e−x/2 dx 2n/2 Γ n2
F (χ2 ) = 0
( n F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
.005 .0000393 .0100 .0717 .207 .412 .676 .989 1.34 1.73 2.16 2.60 3.07 3.57 4.07 4.60 5.14 5.70 6.26 6.84 7.43 8.03 8.64 9.26 9.89 10.5 11.2 11.8 12.5 13.1 13.8
.010 .000157 .0201 .115 .297 .554 .872 1.24 1.65 2.09 2.56 3.05 3.57 4.11 4.66 5.23 5.81 6.41 7.01 7.63 8.26 8.90 9.54 10.2 10.9 11.5 12.2 12.9 12.6 14.3 15.0
.025 .000982 .0506 .216 .484 .831 1.24 1.69 2.18 2.70 3.25 3.82 4.40 5.01 5.63 6.26 6.91 7.56 8.23 8.91 9.59 10.3 11.0 11.7 12.4 13.1 13.8 14.6 15.3 16.0 16.8
.050 .00393 .103 .352 .711 1.15 1.64 2.17 2.73 3.33 3.94 4.57 5.23 5.89 6.57 7.26 7.96 8.67 9.39 10.1 10.9 11.6 12.3 13.1 13.8 14.6 15.4 16.2 16.9 17.7 18.5
.100 .0158 .211 .584 1.06 1.61 2.20 2.83 3.49 4.17 4.87 5.58 6.30 7.04 7.79 8.55 9.31 10.1 10.9 11.7 12.4 13.2 14.0 14.8 15.7 16.5 17.3 18.1 18.9 19.8 20.6
.250 .102 .575 1.21 1.92 2.67 3.45 4.25 5.07 5.90 6.74 7.58 8.44 9.30 10.2 11.0 11.9 12.8 13.7 14.6 15.5 16.3 17.2 18.1 19.0 19.9 20.8 21.7 22.7 23.6 24.5
.500 .455 1.39 2.37 3.36 4.35 5.35 6.35 7.34 8.34 9.34 10.3 11.3 12.3 13.3 14.3 15.3 16.3 17.3 18.3 19.3 20.3 21.3 22.3 23.3 24.3 25.3 26.3 27.3 28.3 29.3
.750 1.32 2.77 4.11 5.39 6.63 7.84 9.04 10.2 11.4 12.5 13.7 14.8 16.0 17.1 18.2 19.4 20.5 21.6 22.7 23.8 24.9 26.0 27.1 28.2 29.3 30.4 31.5 32.6 33.7 34.8
.900 2.71 4.61 6.25 7.78 9.24 10.6 12.0 13.4 14.7 16.0 17.3 18.5 19.8 21.1 22.3 23.5 24.8 26.0 27.2 28.4 29.6 30.8 32.0 33.2 34.4 35.6 36.7 37.9 39.1 40.3
.950 3.84 5.99 7.81 9.49 11.1 12.6 14.1 15.5 16.9 18.3 19.7 21.0 22.4 23.7 25.0 26.3 27.6 28.9 30.1 31.4 32.7 33.9 35.2 36.4 37.7 38.9 40.1 41.3 42.6 43.8
.975 5.02 7.38 9.35 11.1 12.8 14.4 16.0 17.5 19.0 20.5 21.9 23.3 24.7 26.1 27.5 28.8 30.2 31.5 32.9 34.2 35.5 36.8 38.1 39.4 40.6 41.9 43.2 44.5 45.7 47.0
.990 6.63 9.21 11.3 13.3 15.1 16.8 18.5 20.1 21.7 23.2 24.7 26.2 27.7 29.1 30.6 32.0 33.4 34.8 36.2 37.6 38.9 40.3 41.6 43.0 44.3 45.6 47.0 48.3 49.6 50.9
PERCENTAGE POINTS, F -DISTRIBUTION This table gives values of F such that F Γ m+n m 2 n mm/2 nn/2 xm−2/2 (n + mx)−(m+n)/2 dx F (F ) = Γ 2 Γ 2 0 for selected values of m, the number of degrees of freedom of the numerator of F ; and for selected values of n, the number of degrees freedom of the denominator of F . The table also provides values corresponding to F (F )=.10,.05,.025,.01,.005,.001 since F1−α for m and n degrees of freedom is the reciprocal of Fα for n and m degrees of freedom. Thus 1 1 = = .164 F.05 (4, 7) = F.95 (7, 4) 6.09
A-107
TeamLRN
.995 7.88 10.6 12.8 14.9 16.7 18.5 20.3 22.0 23.6 25.2 26.8 28.3 29.8 31.3 32.8 34.3 35.7 37.2 38.6 40.0 41.4 42.8 44.2 45.6 46.9 48.3 49.6 51.0 52.3 53.7
F
F (F ) = 0 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
F =
s2 1 s2 2
Γ m+n 2 n mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .90 Γ m Γ 2 2
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
39.86 8.53 5.54 4.54 4.06 3.78 3.59 3.46 3.36 3.29 3.23 3.18 3.14 3.10 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94 2.93 2.92 2.91 2.90 2.89 2.89 2.88 2.84 2.79 2.75 2.71
49.50 9.00 5.46 4.32 3.78 3.46 3.26 3.11 3.01 2.92 2.86 2.81 2.76 2.73 2.70 2.67 2.64 2.62 2.61 2.59 2.57 2.56 2.55 2.54 2.53 2.52 2.51 2.50 2.50 2.49 2.44 2.39 2.35 2.30
53.59 9.16 5.39 4.19 3.62 3.29 3.07 2.92 2.81 2.73 2.66 2.61 2.56 2.52 2.49 2.46 2.44 2.42 2.40 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.28 2.23 2.18 2.13 2.08
55.83 9.24 5.34 4.11 3.52 3.18 2.96 2.81 2.69 2.61 2.54 2.48 2.43 2.39 2.36 2.33 2.31 2.29 2.27 2.25 2.23 2.22 2.21 2.19 2.18 2.17 2.17 2.16 2.15 2.14 2.09 2.04 1.99 1.94
57.24 9.29 5.31 4.05 3.45 3.11 2.88 2.73 2.61 2.52 2.45 2.39 2.35 2.31 2.27 2.24 2.22 2.20 2.18 2.16 2.14 2.13 2.11 2.10 2.09 2.08 2.07 2.06 2.06 2.05 2.00 1.95 1.90 1.85
58.20 9.33 5.28 4.01 3.40 3.05 2.83 2.67 2.55 2.46 2.39 2.33 2.28 2.24 2.21 2.18 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.04 2.02 2.01 2.00 2.00 1.99 1.98 1.93 1.87 1.82 1.77
58.91 9.35 5.27 3.98 3.37 3.01 2.78 2.62 2.51 2.41 2.34 2.28 2.23 2.19 2.16 2.13 2.10 2.08 2.06 2.04 2.02 2.01 1.99 1.98 1.97 1.96 1.95 1.94 1.93 1.93 1.87 1.82 1.77 1.72
59.44 9.37 5.25 3.95 3.34 2.98 2.75 2.59 2.47 2.38 2.30 2.24 2.20 2.15 2.12 2.09 2.06 2.04 2.02 2.00 1.98 1.97 1.95 1.94 1.93 1.92 1.91 1.90 1.89 1.88 1.83 1.77 1.72 1.67
59.86 9.38 5.24 3.94 3.32 2.96 2.72 2.56 2.44 2.35 2.27 2.21 2.16 2.12 2.09 2.06 2.03 2.00 1.98 1.96 1.95 1.93 1.92 1.91 1.89 1.88 1.87 1.87 1.86 1.85 1.79 1.74 1.68 1.63
60.19 9.39 5.23 3.92 3.30 2.94 2.70 2.54 2.42 2.32 2.25 2.19 2.14 2.10 2.06 2.03 2.00 1.98 1.96 1.94 1.92 1.90 1.89 1.88 1.87 1.86 1.85 1.84 1.83 1.82 1.76 1.71 1.65 1.60
60.71 9.41 5.22 3.90 3.27 2.90 2.67 2.50 2.38 2.28 2.21 2.15 2.10 2.05 2.02 1.99 1.96 1.93 1.91 1.89 1.87 1.86 1.84 1.83 1.82 1.81 1.80 1.79 1.78 1.77 1.71 1.66 1.60 1.55
61.22 9.42 5.20 3.87 3.24 2.87 2.63 2.46 2.34 2.24 2.17 2.10 2.05 2.01 1.97 1.94 1.91 1.89 1.86 1.84 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.72 1.66 1.60 1.55 1.49
61.74 9.44 5.18 3.84 3.21 2.84 2.59 2.42 2.30 2.20 2.12 2.06 2.01 1.96 1.92 1.89 1.86 1.84 1.81 1.79 1.78 1.76 1.74 1.73 1.72 1.71 1.70 1.69 1.68 1.67 1.61 1.54 1.48 1.42
62.00 9.45 5.18 3.83 3.19 2.82 2.58 2.40 2.28 2.18 2.10 2.04 1.98 1.94 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.72 1.70 1.69 1.68 1.67 1.66 1.65 1.64 1.57 1.51 1.45 1.38
62.26 9.46 5.17 3.82 3.17 2.80 2.56 2.38 2.25 2.16 2.08 2.01 1.96 1.91 1.87 1.84 1.81 1.78 1.76 1.74 1.72 1.70 1.69 1.67 1.66 1.65 1.64 1.63 1.62 1.61 1.54 1.48 1.41 1.34
62.53 9.47 5.16 3.80 3.16 2.78 2.54 2.36 2.23 2.13 2.05 1.99 1.93 1.89 1.85 1.81 1.78 1.75 1.73 1.71 1.69 1.67 1.66 1.64 1.63 1.61 1.60 1.59 1.58 1.57 1.51 1.44 1.37 1.30
62.79 9.47 5.15 3.79 3.14 2.76 2.51 2.34 2.21 2.11 2.03 1.96 1.90 1.86 1.82 1.78 1.75 1.72 1.70 1.68 1.66 1.64 1.62 1.61 1.59 1.58 1.57 1.56 1.55 1.54 1.47 1.40 1.32 1.24
63.06 9.48 5.14 3.78 3.12 2.74 2.49 2.32 2.18 2.08 2.00 1.93 1.88 1.83 1.79 1.75 1.72 1.69 1.67 1.64 1.62 1.60 1.59 1.57 1.56 1.54 1.53 1.52 1.51 1.50 1.42 1.35 1.26 1.17
63.33 9.49 5.13 3.76 3.10 2.72 2.47 2.29 2.16 2.06 1.97 1.90 1.85 1.80 1.76 1.72 1.69 1.66 1.63 1.61 1.59 1.57 1.55 1.53 1.52 1.50 1.49 1.48 1.47 1.46 1.38 1.29 1.19 1.00
=
S1 S2 /n, m
where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common
2
variance σ and based on mandndegrees of freedom, respectively. F Γ m+n 2 n mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .95 F (F ) = Γ 2 Γ m 0 2 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
F =
s2 1 s2 2
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
161.4 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.32 4.30 4.28 4.26 4.24 4.23 4.21 4.20 4.18 4.17 4.08 4.00 3.92 3.84
199.5 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.47 3.44 3.42 3.40 3.39 3.37 3.35 3.34 3.33 3.32 3.23 3.15 3.07 3.00
215.7 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 3.07 3.05 3.03 3.01 2.99 2.98 2.96 2.95 2.93 2.92 2.84 2.76 2.68 2.60
224.6 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.73 2.71 2.70 2.69 2.61 2.53 2.45 2.37
230.2 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.59 2.57 2.56 2.55 2.53 2.45 2.37 2.29 2.21
234.0 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.47 2.46 2.45 2.43 2.42 2.34 2.25 2.17 2.10
236.8 19.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.49 2.46 2.44 2.42 2.40 2.39 2.37 2.36 2.35 2.33 2.25 2.17 2.09 2.01
238.9 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.18 2.10 2.02. 1.94
240.5 19.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.12 2.04 1.96 1.88
241.9 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.22 2.20 2.19 2.18 2.16 2.08 1.99 1.91 1.83
243.9 19.41 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.00 1.92 1.83 1.75
245.9 19.43 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.18 2.15 2.13 2.11 2.09 2.07 2.06 2.04 2.03 2.01 1.92 1.84 1.75 1.67
248.0 19.45 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.10 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.93 1.84 1.75 1.66 1.57
249.1 19.45 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.51 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.05 2.03 2.01 1.98 1.96 1.95 1.93 1.91 1.90 1.89 1.79 1.70 1.61 1.52
250.1 19.46 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.57 2.47 2.38 2.31 2.25 2.19 2.15 2.11 2.07 2.04 2.01 1.98 1.96 1.94 1.92 1.90 1.88 1.87 1.85 1.84 1.74 1.65 1.55 1.46
251.1 19.47 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.96 1.94 1.91 1.89 1.87 1.85 1.84 1.82 1.81 1.79 1.69 1.59 1.50 1.39
252.2 19.48 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 2.49 2.38 2.30 2.22 2.16 2.11 2.06 2.02 1.98 1.95 1.92 1.89 1.86 1.84 1.82 1.80 1.79 1.77 1.75 1.74 1.64 1.53 1.43 1.32
253.3 19.49 8.55 5.66 4.40 3.70 3.27 2.97 2.75 2.58 2.45 2.34 2.25 2.18 2.11 2.06 2.01 1.97 1.93 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.71 1.70 1.68 1.58 1.47 1.35 1.22
254.3 19.50 8.53 5.63 4.36 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71 1.69 1.67 1.65 1.64 1.62 1.51 139 1.25 1.00
=
S1 S2 /n, m
where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common
2
variance σ and based on m and n degrees of freedom, respectively.
A-108
F
F (F ) = 0 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
F =
s2 1 s2 2
Γ m+n 2 n mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .975 Γ m Γ 2 2
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
647.8 38.51 17.44 12.22 10.01 8.81 8.07 7.57 7.21 6.94 6.72 6.55 6.41 6.30 6.20 6.12 6.04 5.98 5.92 5.87 5.83 5.79 5.75 5.72 5.69 5.66 5.63 5.61 5.59 5.57 5.42 5.29 5.15 5.02
799.5 39.00 16.04 10.65 8.43 7.26 6.54 6.06 5.71 5.46 5.26 5.10 4.97 4.86 4.77 4.69 4.62 4.56 4.51 4.46 4.42 4.38 4.35 4.32 4.29 4.27 4.24 4.22 4.20 4.18 4.05 3.93 3.80 3.69
864.2 39.17 15.44 9.98 7.76 6.60 5.89 5.42 5.08 4.83 4.63 4.47 4.35 4.24 4.15 4.08 4.01 3.95 3.90 3.86 3.82 3.78 3.75 3.72 3.69 3.67 3.65 3.63 3.61 3.59 3.46 3.34 3.23 3.12
899.6 39.25 15.10 9.60 7.39 6.23 5.52 5.05 4.72 4.47 4.28 4.12 4.00 3.89 3.80 3.73 3.66 3.61 3.56 3.51 3.48 3.44 3.41 3.38 3.35 3.33 3.31 3.29 3.27 3.25 3.13 3.01 2.89 2.79
921.8 39.30 14.88 9.36 7.15 5.99 5.29 4.82 4.48 4.24 4.04 3.89 3.77 3.66 3.58 3.50 3.44 3.38 3.33 3.29 3.25 3.22 3.18 3.15 3.13 3.10 3.08 3.06 3.04 3.03 2.90 2.79 2.67 2.57
937.1 39.33 14.73 9.20 6.98 5.82 5.12 4.65 4.32 4.07 3.88 3.73 3.60 3.50 3.41 3.34 3.28 3.22 3.17 3.13 3.09 3.05 3.02 2.99 2.97 2.94 2.92 2.90 2.88 2.87 2.74 2.63 2.52 2.41
948.2 39.36 14.62 9.07 6.85 5.70 4.99 4.53 4.20 3.95 3.76 3.61 3.48 3.38 3.29 3.22 3.16 3.10 3.05 3.01 2.97 2.93 2.90 2.87 2.85 2.82 2.80 2.78 2.76 2.75 2.62 2.51 2.39 2.29
956.7 39.37 14.54 8.98 6.76 5.60 4.90 4.43 4.10 3.85 3.66 3.51 3.39 3.29 3.20 3.12 3.06 3.01 2.96 2.91 2.87 2.84 2.81 2.78 2.75 2.73 2.71 2.69 2.67 2.65 2.53 2.41 2.30 2.19
963.3 39.39 14.47 8.90 6.68 5.52 4.82 4.36 4.03 3.78 3.59 3.44 3.31 3.21 3.12 3.05 2.98 2.93 2.88 2.84 2.80 2.76 2.73 2.70 2.68 2.65 2.63 2.61 2.59 2.57 2.45 2.33 2.22 2.11
968.6 39.40 14.42 8.84 6.62 5.46 4.76 4.30 3.96 3.72 3.53 3.37 3.25 3.15 3.06 2.99 2.92 2.87 2.82 2.77 2.73 2.70 2.67 2.64 2.61 2.59 2.57 2.55 2.53 2.51 2.39 2.27 2.16 2.05
976.7 39.41 14.34 8.75 6.52 5.37 4.67 4.20 3.87 3.62 3.43 3.28 3.15 3.05 2.96 2.89 2.82 2.77 2.72 2.68 2.64 2.60 2.57 2.54 2.51 2.49 2.47 2.45 2.43 2.41 2.29 2.17 2.05 1.94
984.9 39.43 14.25 8.66 6.43 5.27 4.57 4.10 3.77 3.52 3.33 3.18 3.05 2.95 2.86 2.79 2.72 2.67 2.62 2.57 2.53 2.50 2.47 2.44 2.41 2.39 2.36 2.34 2.32 2.31 2.18 2.06 1.94 1.83
993.1 39.45 14.17 8.56 6.33 5.17 4.47 4.00 3.67 3.42 3.23 3.07 2.95 2.84 2.76 2.68 2.62 2.56 2.51 2.46 2.42 2.39 2.36 2.33 2.30 2.28 2.25 2.23 2.21 2.20 2.07 1.94 1.82 1.71
997.2 39.46 14.12 8.51 6.28 5.12 4.42 3.95 3.61 3.37 3.17 3.02 2.89 2.79 2.70 2.63 2.56 2.50 2.45 2.41 2.37 2.33 2.30 2.27 2.24 2.22 2.19 2.17 2.15 2.14 2.01 1.88 1.76 1.64
1001 39.46 14.08 8.46 6.23 5.07 4.36 3.89 3.56 3.31 3.12 2.96 2.84 2.73 2.64 2.57 2.50 2.44 2.39 2.35 2.31 2.27 2.24 2.21 2.18 2.16 2.13 2.11 2.09 2.07 1.94 1.82 1.69 1.57
1006 39.47 14.04 8.41 6.18 5.01 4.31 3.84 3.51 3.26 3.06 2.91 2.78 2.67 2.59 2.51 2.44 2.38 2.33 2.29 2.25 2.21 2.18 2.15 2.12 2.09 2.03 2.05 2.03 2.01 1.88 1.74 1.61 1.48
1010 39.48 13.99 8.36 6.12 4.96 4.25 3.78 3.45 3.20 3.00 2.85 2.72 2.61 2.52 2.45 2.38 2.32 2.27 2.22 2.18 2.14 2.11 2.08 2.05 2.03 2.00 1.98 1.96 1.94 1.80 1.67 1.53 1.39
1014 39.49 13.95 8.31 6.07 4.90 4.20 3.73 3.39 3.14 2.94 2.79 2.66 2.55 2.46 2.38 2.32 2.26 2.20 2.16 2.11 2.08 2.04 2.01 1.98 1.95 1.93 1.91 1.89 1.87 1.72 1.58 1.43 1.27
1018 39.50 13.90 8.26 6.02 4.85 4.14 3.67 3.33 3.08 2.88 2.72 2.60 2.49 2.40 2.32 2.25 2.19 2.13 2.09 2.04 2.00 1.97 1.94 1.91 1.88 1.85 1.83 1.81 1.79 1.64 1.48 1.31 1.00
=
S1 S2 /n, m
where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common
2
variance σ and based on mandndegrees of freedom, respectively. F Γ m+n 2 n mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .99 F (F ) = Γ 2 Γ m 0 2 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
F =
s2 1 s2 2
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
4052 98.50 34.12 21.20 16.26 13.75 12.25 11.26 10.56 10.04 9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.18 8.10 8.02 7.95 7.88 7.82 7.77 7.72 7.68 7.64 7.60 7.56 7.31 7.08 6.85 6.63
4999.5 99.00 30.82 18.00 13.27 10.92 9.55 8.65 8.02 7.56 7.21 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.78 5.72 5.66 5.61 5.57 5.53 5.49 5.45 5.42 5.39 5.18 4.98 4.79 4.61
5403 99.17 29.46 16.69 12.06 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.87 4.82 4.76 4.72 4.68 4.64 4.60 4.57 4.54 4.51 4.31 4.13 3.95 3.78
5625 99.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.37 4.31 4.26 4.22 4.18 4.14 4.11 4.07 4.04 4.02 3.83 3.65 3.48 3.32
5764 99.30 28.24 15.52 10.97 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.69 4.56 4.44 4.34 4.25 4.17 4.10 4.04 3.99 3.94 3.90 3.85 3.82 3.78 3.75 3.73 3.70 3.51 3.34 3.17 3.02
5859 99.33 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.81 3.76 3.71 3.67 3.63 3.59 3.56 3.53 3.50 3.47 3.29 3.12 2.96 2.80
5928 99.36 27.67 14.98 10.46 8.26 6.99 6.18 5.61 5.20 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.77 3.70 3.64 3.59 3.54 3.50 3.46 3.42 3.39 3.36 3.33 3.30 3.12 2.95 2.79 2.64
5982 99.37 27.49 14.80 10.29 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 3.71 3.63 3.56 3.51 3.45 3.41 3.36 3.32 3.29 3.26 3.23 3.20 3.17 2.99 2.82 2.66 2.51
6022 99.39 27.35 14.66 10.16 7.98 6.72 5.91 5.35 4.94 4.63 4.39 4.19 4.03 3.89 3.78 3.68 3.60 3.52 3.46 3.40 3.35 3.30 3.26 3.22 3.18 3.15 3.12 3.09 3.07 2.89 2.72 2.56 2.41
6056 99.40 27.23 14.55 10.05 7.87 6.62 5.81 5.26 4.85 4.54 4.30 4.10 3.94 3.80 3.69 3.59 3.51 3.43 3.37 3.31 3.26 3.21 3.17 3.13 3.09 3.06 3.03 3.00 2.98 2.80 2.63 2.47 2.32
6106 99.42 27.05 14.37 9.89 7.72 6.47 5.67 5.11 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.17 3.12 3.07 3.03 2.99 2.96 2.93 2.90 2.87 2.84 2.66 2.50 2.34 2.18
6157 99.43 26.87 14.20 9.72 7.56 6.31 5.52 4.96 4.56 4.25 4.01 3.82 3.66 3.52 3.41 3.31 3.23 3.15 3.09 3.03 2.98 2.93 2.89 2.85 2.81 2.78 2.75 2.73 2.70 2.52 2.35 2.19 2.04
6209 99.45 26.69 14.02 9.55 7.40 6.16 5.36 4.81 4.41 4.10 3.86 3.66 3.51 3.37 3.26 3.16 3.08 3.00 2.94 2.88 2.83 2.78 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.37 2.20 2.03 1.88
6235 99.46 26.60 13.93 9.47 7.31 6.07 5.28 4.73 4.33 4.02 3.78 3.59 3.43 3.29 3.18 3.08 3.00 2.92 2.86 2.80 2.75 2.70 2.66 2.62 2.58 2.55 2.52 2.49 2.47 2.29 2.12 1.95 1.79
6261 99.47 26.50 13.84 9.38 7.23 5.99 5.20 4.65 4.25 3.94 3.70 3.51 3.35 3.21 3.10 3.00 2.92 2.84 2.78 2.72 2.67 2.62 2.58 2.54 2.50 2.47 2.44 2.41 2.39 2.20 2.03 1.86 1.70
6287 99.47 26.41 13.75 9.29 7.14 5.91 5.12 4.57 4.17 3.86 3.62 3.43 3.27 3.13 3.02 2.92 2.84 2.76 2.69 2.64 2.58 2.54 2.49 2.45 2.42 2.38 2.35 2.33 2.30 2.11 1.94 1.76 1.59
6313 99.48 26.32 13.65 9.20 7.06 5.82 5.03 4.48 4.08 3.78 3.54 3.34 3.18 3.05 2.93 2.83 2.75 2.67 2.61 2.55 2.50 2.45 2.40 2.36 2.33 2.29 2.26 2.23 2.21 2.02 1.84 1.66 1.47
6339 99.49 26.22 13.56 9.11 6.97 5.74 4.95 4.40 4.00 3.69 3.45 3.25 3.09 2.96 2.84 2.75 2.66 2.58 2.52 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.17 2.14 2.11 1.92 1.73 1.53 1.32
6366 99.50 26.13 13.46 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.17 3.00 2.87 2.75 2.65 2.57 2.49 2.42 2.36 2.31 2.26 2.21 2.17 2.13 2.10 2.06 2.03 2.01 1.80 1.60 1.38 1.00
=
S1 S2 /n, m
where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common
2
variance σ and based on m and n degrees of freedom, respectively.
A-109
TeamLRN
SPECIAL FUNCTIONS Bessel Functions 1. Bessel’s differential equation for a real variable x is x2
d2 y dy +x + (x2 − n2 )y = 0 2 dx dx
2. When n is not an integer, two independent solutions of the equation are Jn (x), J−n (x), where Jn (x) =
∞ k=0
x (−1)k k!Γ(n + k + 1) 2
n+2k
3. If n is an integer Jn (x) = (−1)n Jn (x), where # " xn x4 x6 x2 Jn (x) = n + 4 + 6 + ... 1− 2 2 n! 2 · 1!(n + 1) 2 · 2!(n + 1) (n + 2) 2 · 3!(n + 1) (n + 2) (n + 3) 4. For n = 0 and n = 1, this formula becomes x2 22 (1!)2 x x3 2 − 23 ·1!2!
J0 (x) = 1 −
+
J1 (x) =
+
x4 24 (2!)2 x5 25 ·2!3!
− −
x6 x8 26 (3!)2 + 28 (4!)2 − · · · x7 x9 27 ·3!4! + 29 ·4!5! − . . .
5. When x is large and positive, the following asymptotic series may be used 1 π π ! 2 2 − Q0 (x) sin x − P0 (x) cos x − πx 4 4 & 2 12 % 3π 3π J1 (x) = πx P1 (x) cos x − 4 − Q1 (x) sin x − 4
J0 (x) =
where 12 · 32 12 · 32 · 52 · 72 12 · 32 · 52 · 72 · 92 · 112 + − + ··· 2!(8x)2 4!(8x)4 6!(8x)6 12 12 ·32 ·52 12 ·32 ·52 ·72 ·92 Q0 (x) ∼ − 1!8x + 3!(8x)3 − + −··· 5!(8x)5
P0 (x) ∼ 1 −
2 2 2 2 2 12 ·3·5 12 ·32 ·52 ·7·9 ·7 ·9 ·11·13 + 1 ·3 ·56!(8x) 6 2!(8x)2 − 4!(8x)4 2 2 2 2 2 2 1·3 1 ·3 ·5·7 1 ·3 ·5 ·7 ·9·11 − + − · · · 3 5 1!8x 3!(8x) 5!(8x)
P1 (x) ∼ 1 + Q1 (x) ∼
− +···
[In P1 (x) the signs alternate from+to-after the first term] 6. The zeros of J0 (x) and J1 (x). If j0s and j1s are the sth zeros of J0 (x) and J1 (x) respectively, and if a = 4s − 1, b = 4s + 1 # " 1 62 15, 116 12, 554, 474 8, 368, 654, 292 2 j0,s ∼ πa 1 + 2 2 − 4 4 + − + − +··· 10 a10 4 % π a 3π a 15π 6 a6 105π 8 a8 315π & 3,902,418 895,167,324 4716 j1,s ∼ 14 πb 1 − π26b2 + π46b4 − 5π 6 b6 + 35π 8 b8 − 35π 10 b10 + · · · & % s+1 3 2 (−1) 2 56 9664 J1 (j0,s ) ∼ 1 − 3π4 a4 + 5π6 a6 − 7,381,280 1 21π 8 a8 + · · · πa 23 & s 2 % 19,584 2,466,720 J0 (j1,s ) ∼ (−1) 12 1 + π24 4 b4 − 10π 6 b6 + 7π 8 b8 − · · · πb 2
A-93
SPECIAL FUNCTIONS 7. Table of zeros for J0 (x) and J1 (x) Define {αn , βn } by J1 (αn ) = 0 and J0 (βn ) = 0. Roots αn 2.4048 5.5201 8.6537 11.7915 14.9309 18.0711 21.2116
J1 (αn ) 0.5191 -0.3403 0.2715 -0.2325 0.2065 -0.1877 0.1733
Roots β n 0.0000 3.8317 7.0156 10.1735 13.3237 16.4706 19.6159
J0 (βn ) 1.0000 -0.4028 0.3001 -0.2497 0.2184 -0.1965 0.1801
8. Recurrence formulas 2n Jn (x) nJn (x) + xJn (x) = xJn−1 (x) x Jn−1 (x) − Jn+1 (x) = 2Jn (x) nJn (x) − xJn (x) = xJn+1 (x)
Jn−1 (x) + Jn+1 (x) =
(k)
9. If Jn is written for Jn (x) and Jn relationships are important (r)
J0 (2) J0 (3) J0 (4) J0
is written for
(r−1)
J 12 (x) =
2
πx
J− 12 (x) = Jn+ 32 (x) = Jn− 12 (x) =
sin x
2 πx cos x 1 d 1 −xn+ 2 dx {x−(n+ 2 ) Jn+ 12 (x)} 1 1 d x−(n+ 2 ) dx {xn+ 2 Jn+ 12 (x)}
πx 12
n 0
2
x3
2
− cos x
− 1 sin x − x3 cos x − x6 sin x − x152 − 1 cos x etc.
15
πx 12
Jn+ 12 (x) sin x
sin x x
1 2
then the following derivative
= −J1 = −J0 + x1 J1 = 12 (J2 − J0 ) = x1 J0 + 1 − x22 J1 = 14 (−J3 + 3J1 ) = 1 − x32 J0 − x2 − x63 J1 = 18 (J4 − 4J2 + 3J0 ), etc.
10. Half order Bessel functions
3
dk {Jn (x)}, dxk
3 x2
−
15 x3
J−(n+ 12 ) (x) cos x
3 x2
− cosx x − sin x − 1 cos x + x3 sin x
−
6 x
cos x −
15 x2
− 1 sin x
11. Additional solutions to Bessel’s equation are Yn (x) (also called Weber’s function, and sometimes denoted by Nn (x)) (1) (2) Hn (x) and Hn (x) (also called Hankel functions) These solutions are defined as follows Jn (x) cos (nπ) − J−n (x) sin(nπ) Yn (x) = −v (x) lim Jv (x) cos(vπ)−J sin(vπ) v→n
n
not an integer n
an integer
A-94
TeamLRN
(1)
Hn (x) = Jn (x) + iYn (x) (2) Hn (x) = Jn (x) − iYn (x)
SPECIAL FUNCTIONS The additional properties of these functions may all be derived from the above relations and the known properties of Jn (x). 12. Complete solutions to Bessel’s equation may be written as c1 Jn (x) + c2 J−n (x)
if n is not an integer
or, for any value of n, c1 Jn (x) + c2 Yn (x)
c1 Hn(1) x + c2 Hn(2) (x)
or
13. The modified (or hyperbolic) Bessel’s differential equation is x2
d2 y dy +x − (x2 + n2 )y = 0 dx2 dx
14. When n is not an integer, two independent solutions of the equation are In (x) and I−n (x), where ∞ x n+2k 1 In (x) = k!Γ(n + k + 1) 2 k=0
15. If n is an integer, In (x) = I−n (x) =
xn 2n n!
"
x2 x4 + 22 · 1!(n + 1) 24 · 2!(n + 1)(n + 2) # x6 + 6 + ··· 2 · 3!(n + 1) (n + 2) (n + 3) 1+
16. For N = 0 and n = 1, this formula becomes I0 (x) = 1 + I1 (x) =
x 2
+
x2 22 (1!)2 3
x 23 ·1!2!
+
+
x4 24 (2!)2 5
x 25 ·2!3!
+
+
x6 26 (3!)2 7
x 27 ·3!4!
17. Another solution to the modified Bessel’s equation is $ 1 I−n (x)−In (x) 2π sin (nπ) Kn (x) = (x)−Iv (x) lim 21 π I−vsin (vπ) v→n
+
+ 9
x8 28 (4!)2
x 29 ·4!5!
+ ···
+ ···
n not an integer n an integer
This function is linearly independent of In (x) for all values of n. Thus the complete solution to the modified Bessel’s equation may be written as c1 In (x) + c2 I−n (x) n not an integer or c1 In (x) + c2 Kn (x)
any n
18. The following relations hold among the various Bessel functions: In (z) = i−m Jm (iz) Yn (iz) = (i)n+1 In (z) − π2 i−n Kn (z) Most of the properties of the modified Bessel function may be deduced from the known properties of Jn (x) by use of these relations and those previously given. 19. Recurrence formulas In−1 (x) − In+1 (x) = 2n x In (x) In−1 (x) + In+1 (x) = 2In (x) n n In (x) = In+1 (x) + x In (z) In−1 (x) − x In (x) = In (x)
A-95
SPECIAL FUNCTIONS The Factorial Function For non-negative integers n, the factorial of n, denoted n!, is the product of all positive integers less than or equal to n; n! = n · (n − 1) · (n − 2) · · · 2 · 1. If n is a negative integer (n = −1, −2, . . . ) then n! = ±∞. Approximations to n! for large n include Stirling’s formula n n+ 12 √ n! ≈ 2πe , e and Burnsides’s formula n+ 12 √ n + 12 n! ≈ 2π . e n 0 2 4 6 8 10 12 14 16 18 20 30 50 70 90 110 130 500
n! 1 2 24 720 40320 3.6288 × 106 4.7900 × 108 8.7178 × 1010 2.0923 × 1013 6.4024 × 1015 2.4329 × 1018 2.6525 × 1032 3.0414 × 1064 1.1979 × 10100 1.4857 × 10138 1.5882 × 10178 6.4669 × 10219 1.2201 × 101134
log10 n! 0.00000 0.30103 1.38021 2.85733 4.60552 6.55976 8.68034 10.94041 13.32062 15.80634 18.38612 32.42366 64.48307 100.07841 138.17194 178.20092 219.81069 1134.0864
n 1 3 5 7 9 11 13 15 17 19 25 40 60 80 100 120 150 1000
A-96
TeamLRN
n! 1 6 120 5040 3.6288 × 105 3.9917 × 107 6.2270 × 109 1.3077 × 1012 3.5569 × 1014 1.2165 × 1017 1.5511 × 1025 8.1592 × 1047 8.3210 × 1081 7.1569 × 10118 9.3326 × 10157 6.6895 × 10198 5.7134 × 10262 4.0239 × 102567
log10 n! 0.00000 0.77815 2.07918 3.70243 5.55976 7.60116 9.79428 12.11650 14.55107 17.08509 25.19065 47.91165 81.92017 118.85473 157.97000 198.82539 262.75689 2567.6046
Definition: Γ(n) =
∞
The Gamma Function t
n−1 −t
e
dt
n>0
0
Recursion Formula: Γ(n + 1) = nΓ(n) Γ(n + 1) = n! if n = 0, 1, 2, . . . where 0! = 1 For n < 0 the gamma function can be defined by using Γ(n) = Γ(n+1) n Graph:
A-96
SPECIAL FUNCTIONS Special Values:
Γ(1/2) =
√
π
1 · 3 · 5 · · · (2m − 1) √ π m = 1, 2, 3, . . . 2m √ (−1)m 2m π m = 1, 2, 3, . . . Γ(−m + 1/2) = 1 · 3 · 5 · · · (2m − 1)
Γ(m + 1/2) =
Definition:
1 · 2 · 3···k kx k→∞ (x + 1) (x + 2) · · · (x + k) ∞ % −x/m & x 1+ m e = xeγx
Γ(x + 1) = lim 1 Γ(x)
m=1
This is an infinite product representation for the gamma function where γ is Euler’s constant. Properties: ∞ Γ (1) = eγx ln x dx = −γ 0 Γ (x) 1 1 1 1 1 1 = −γ + − − − + + ... + + ··· Γ(x) 1 x " 2 x+1 n x + n#− 1 √ 1 139 1 − + ... Γ(x + 1) = 2πx xx e−x 1 + + 2 12x 288x 51, 840x3 This is called Stirling’s asymptotic series. ∞ Values of Γ(n) = 0 e−x xn−1 dx; n 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24
Γ(n) 1.00000 .99433 .98884 .98355 .97844 .97350 .96874 .96415 .95973 .95546 .95135 .94740 .94359 .93993 .93642 .93304 .92980 .92670 .92373 .92089 .91817 .91558 .91311 .91075 .90852
n 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49
Γ(n) .90640 .90440 .90250 .90072 .89904 .89747 .89600 .89464 .89338 .89222 .89115 .89018 .88931 .88854 .88785 .88726 .88676 .88636 .88604 .88581 .88566 .88560 .88563 .88575 .88595
n 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74
A-97
TeamLRN
Γ(n + 1) = nΓ(n) Γ(n) .88623 .88659 .88704 .88757 .88818 .88887 .88964 .89049 .89142 .89243 .89352 .89468 .89592 .89724 .89864 .90012 .90167 .90330 .90500 .90678 .90864 .91057 .91258 .91466 .91683
n 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00
Γ(n) .91906 .92137 .92376 .92623 .92877 .93138 .93408 .93685 .93969 .94261 .94561 .94869 .95184 .95507 .95838 .96177 .96523 .96877 .97240 .97610 .97988 .98374 .98768 .99171 .99581 1.00000
SPECIAL FUNCTIONS
1
The Beta Function tm−1 (1 − t)m−1 dt
Definition: B(m, n) =
m > 0, n > 0
0
Γ(m)Γ(n) Γ(m + n) B(m, n) = B(n, m) π/2 B(m, n) = 2 0 sin2m−1 θ cos2n−1 θ dθ ∞ tm−1 B(m, n) = 0 (1+t) m+n dt 1 m−1 (1−t)n−1 n B(m, n) = r (r + 1)m 0 t (r+t) dt m+n
Relationship with Gamma function: B(m, n) = Properties:
2 Definition: erf(x) = √
m!(n − 2m)!
(2x)n−2m = cos(n arccos x) = Tn (x)
A-98
(r+t)m+n
The Error Function x 2 −t2 Definition: erf(x) = √ e dt π 0 3 x 1 x5 1 x7 2 x− + − + ··· Series: erf (x) = √ 3 2! 5 3! 7 π Property: erf(x) = − erf(−x)
dt
1 x erf √ 2 2 0 √ To evaluate erf(2.3), one proceeds as follows: For √x2 = 2.3, one finds x = (2.3) ( 2) = 3.25. In the normal probability function table (page A-104), one finds the entry 0.4994 opposite the value 3.25. Thus erf(2.3) = 2(0.4994) = 0.9988. x
Relationship with Normal Probability Function f (t) :
f (t) dt =
2 erfc(z) = 1 − erf(z) = √ π
∞
z
is known as the complementary error function. I: Legendre Name: Legendre Symbol: Pn (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y − 2 xy + n(n + 1)y = 0 y = Pn (x) Explicit Expression: Pn (x) =
1
TeamLRN
e−t dt 2
π
∞
e−t dt 2
z
Orthogonal Polynomials I: Legendre Name: Legendre Symbol: Pn (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y − 2 xy + n(n + 1)y = 0 y = Pn (x) [n/2] 1 2n − 2m n−2m m n Explicit Expression: Pn (x) = n (−1) x 2 m=0 m n Recurrence Relation: (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x) Weight: 1 Standardization: Pn (1)=1 +1 2 Norm: [Pn (x)]2 dx = 2n + 1 −1 (−1)n dn {(1 − x2 )n } Rodrigues’ Formula: Pn (x) = n 2 n! dxn ∞ Generating Function: R−1 = Pn (x)z n ; −1 < x < 1, |z| < 1, √ n=0 R = 1 − 2xz + z 2 Inequality: |Pn (x)| ≤ 1, −1 ≤ x ≤ 1. II: Tschebysheff, First Kind Name: Tschebysheff, First Kind Symbol: Tn (x) Interval:[-1, 1] Differential Equation: (1 − x2 )y − xy + n2 y = 0 y = Tn (x) [n/2] n (n − m − 1)! Explicit Expression: (2x)n−2m = cos(n arccos x) = Tn (x) (−1)m 2 m=0 m!(n − 2m)!
A-98
SPECIAL FUNCTIONS Recurrence Relation: Tn+1 (x) = 2xTn (x) − Tn−1 (x) Weight: (1 − x2 )−1/2 Standardization: Tn (1) = 1 " +1 π/2, n = 0 Norm: −1 (1 − x2 )−1/2 [Tn (x)]2 dx = π, n = 0 √ (−1)n (1 − x2 )1/2 π dn {(1 − x2 )n−(1/2) } = Tn (x) Rodrigues’ Formula: dxn 2n+1 Γ(n + 12 ) ∞ 1 − xz Generating Function: = Tn (x) z n , −1 < x < 1, |z| < 1 1 − 2xz − z 2 n=0 Inequality: |Tn (x)| ≤ 1, −1 ≤ x ≤ 1. III: Tschebysheff, Second Kind Name: Tschebysheff, Second Kind Symbol Un (x) Interval: [-1, 1] Differential Equation: (1 − x2 )y − 3 xy + n(n + 2)y = 0 y = Un (x) [n/2] (m − n)! Explicit Expression: Un (x) = (2x)n−2m (−1)m m!(n − 2m)! m=0 sin[(n + 1)θ] Un (cos θ) = sin θ Recurrence Relation: Un+1 (x) = 2xUn (x) − Un−1 (x) Standardization: Un (1) = n + 1 Weight: (1 − x2 )1/2 +1 π 2 1/2 2 Norm: (1 − x ) [Un (x)] dx = 2 −1 √ (−1)n (n + 1) π dn 2 n+(1/2) {(1 − x ) } Rodrigues’ Formula: Un (x) = (1 − x2 )1/2 2n+1 Γ(n + 32 ) dxn ∞ 1 n Generating Function: = Un (x)z , − 1 < x < 1, |z| < 1 1 − 2xz + z 2 n=0 Inequality: |Un (x)| ≤ n + 1, −1 ≤ x ≤ 1.
IV: Jacobi (α,β) (x) Interval: [-1, 1] Name: Jacobi Symbol: Pn Differential Equation: (1 − x2 )y + [β − α − (α + β + 2)x]y + n(n + α + β + 1)y = 0 (α,β) y = Pn (x) n 1 n + α n + β Explicit Expression: Pn(α,β) (x) = n (x − 1)n−m (x + 1)m 2 m=0 m n−m Recurrence Relation: (α,β)
2(n + 1) (n + α + β + 1) (2n + α + β)Pn+1 (x) = (2n + α + β + 1)[(α2 − β 2 ) + (2n + α + β + 2) × (2n + α + β)x]Pn(α,β) (x) (α,β)
− 2(n + α) (n + β) (2n + α + β + 2)Pn−1 (x) (α,β) Weight: (1 − x)α (1 + x)β ; α, β > 1 Standardization: Pn (x) = n+α n +1 2α+β+1 Γ(n + α + 1)Γ(n + β + 1) Norm: (1 − x)α (1 + x)β [Pn(α,β) (x)]2 dx = (2n + α + β + 1)n!Γ(n + α + β + 1) −1 dn (−1)n (α,β) Rodrigues’ Formula: Pn (x) = n {(1 − x)n+α (1 + x)n+β } α β 2 n!(1 − x) (1 + x) dxn
A-99
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SPECIAL FUNCTIONS ∞ Generating Function: R−1 (1 − z + R)−α (1 + z + R)−β = 2−α−β Pn(α,β) (x)z n , n=0 √ R = 1 − 2xz+ z 2 , |z| < 1 n+q 1 ∼ nq if q = max(α, β) ≥ − n 2 (α,β) Inequality: max |Pn(α,β) (x)| = (x )| ∼ n−1/2 if q < − 12 |Pn −1≤x≤1 x is one of the two maximum points nearest β−α α+β+1
V: Generalized Laguerre (α) Name: Generalized Laguerre Symbol: Ln (x) Interval: [0, ∞] xy + (α + 1 − x)y + ny = 0 Differential Equation: (α) y = Ln (x) n 1 m (α) m n+α (−1) x Explicit Expression: Ln (x) = n − m m! m=0 (α)
(α)
(α)
Recurrence Relation: (n + 1)Ln + 1(x) = [(2n + α + 1) − x]Ln (x) − (n + α)Ln − 1(x) n (α) Weight: xα e−x , α > −1 Standardization: Ln (x) = (−1) xn + · · · n! ∞ Γ(n + α + 1) 2 Norm: xα e−x [L(α) n (x)] dx = n! 0 dn 1 (α) {xn+α e−x } Rodrigues’ Formula: Ln (x) = n!xα e−xdxn∞ Generating Function: (1 − z)−α−1 exp
xz z−1
=
(α)
Ln (x)z n
n=0
Γ(n + α + 1) x/2 x≥0 e ; α>0 n!Γ(α + 1) x≥0 (a) Γ(α+n+1) x/2 |Ln (x)| ≤ 2 − n!Γ(α+1) e ; −1 < α < 0
Inequality: |L(α) n (x) ≤
VI: Hermite Name: Hermite Symbol:Hn (x) Interval: [−∞, ∞] Differential Equation: y − 2xy + 2ny = 0 [n/2] (−1)m n!(2x)n−2m Explicit Expression: Hn (x) = m!(n − 2m)! m=0 Recurrence Relation:Hn+1 (x) = 2xHn (x) − 2nHn−1 (x) 2 Weight: e−x Standardization: Hn (1) = 2n xn + · · · ∞ 2 √ 2 e−x [Hn (x)] dx = 2n n! π Norm: −∞
2
n
2
−x d ) Rodrigues’ Formula: Hn (x) = (−1)n ex dx n (e ∞ n 2 z Generating Function: e−x +2zx = Hn (x) n! n=0 √ x2 /2 n/2 k2 n! k ≈ 1.086435 Inequality: |Hn (x)|e
H0 = 1 H1 = 2x H2 = 4x2 − 2
x10 = (30240H0 + 75600H2 + 25200H4 + 2520H6 + 90H8 + H10 )/1024 x9 = (15120H1 + 10080H3 + 1512H5 + 72H7 + H9 )/512 x8 = (1680H0 + 3360H2 + 840H4 + 56H6 + H8 )/256
A-100
n! k ≈ 1.086435
Tables of Orthogonal Polynomials H0 = 1 x10 = (30240H0 + 75600H2 + 25200H4 + 2520H6 + 90H8 + H10 )/1024 H1 = 2x x9 = (15120H1 + 10080H3 + 1512H5 + 72H7 + H9 )/512 2 H2 = 4x − 2 x8 = (1680H0 + 3360H2 + 840H4 + 56H6 + H8 )/256 3 H3 = 8x − 12x x7 = (840H1 + 420H3 + 42H5 + H7 )/128 H4 = 16x4 − 48x2 + 12 x6 = (120H0 + 180H2 + 30H4 + H6 )/64 H5 = 32x5 − 160x3 + 120x x5 = (60H1 + 20H3 + H5 )/32 6 4 2 H6 = 64x − 480x + 720x − 120 x4 = (12H0 + 12H2 + H4 )/16 7 5 3 H7 = 128x − 1344x + 3360x − 1680x x3 = (6H1 + H3 )/8 H8 = 256x8 − 3584x6 + 13440x4 − 13440x2 + 1680 x2 = (2H0 + H2 )/4 H9 = 512x9 − 9216x7 + 48384x5 − 80640x3 + 30240x x = (H1 )/2 H10 = 1024x10 − 23040x8 + 161280x6 − 403200x4 + 302400x2 − 30240 1 = H0 L0 L1 L2 L3 L4 L5 L6
=1 x6 = 720L0 − 4320L1 + 10800L2 − 14400L3 + 10800L4 − 4320L5 + 720L6 = −x + 1 x5 = 120L0 − 600L1 + 1200L2 − 1200L3 + 600L4 − 120L5 2 = (x − 4x + 2)/2 x4 = 24L0 − 96L1 + 144L2 − 96L3 + 24L4 3 2 = (−x + 9x − 18x + 6)/6 x3 = 6L0 − 18L1 + 18L2 − 6L3 4 3 2 = (x − 16x + 72x − 96x + 24)/24 x2 = 2L0 − 4L1 + 2L2 = (−x5 + 25x4 − 200x3 + 600x2 − 600x + 120)/120 x = L0 − L1 = (x6 − 36x5 + 450x4 − 2400x3 + 5400x2 − 4320x + 720)/720 1 = L0
P0 = 1 x10 = (4199P0 + 16150P2 + 15504P4 + 7904P6 + 2176P8 + 256P10 )/46189 P1 = x x9 = (3315P1 + 4760P3 + 2992P5 + 960P7 + 128P9 )/12155 P2 = (3x2 − 1)/2 x8 = (715P0 + 2600P2 + 2160P4 + 832P6 + 128P8 )/6435 3 P3 = (5x − 3x)/2 x7 = (143P1 + 182P3 + 88P5 + 16P7 )/429 4 2 P4 = (35x − 30x + 3)/8 x6 = (33P0 + 110P2 + 72P4 + 16P6 )/231 5 3 P5 = (63x − 70x + 15x)/8 x5 = (27P1 + 28P3 + 8P5 )/63 P6 = (231x6 − 315x4 + 105x2 − 5)/16 x4 = (7P0 + 20P2 + 8P4 )/35 7 5 3 P7 = (429x − 693x + 315x − 35x)/16 x3 = (3P1 + 2P3 )/5 8 6 4 2 P8 = (6435x − 12012x + 6930x − 1260x + 35)/128 x2 = (P0 + 2P2 )/3 P9 = (12155x9 − 25740x7 + 18018x5 − 4620x3 + 315x)/128 x = P1 P10 = (46189x10 − 109395x8 + 90090x6 − 30030x4 + 3465x2 − 63)/256 1 = P0 T0 = 1 x10 = (126T0 + 210T2 + 120T4 + 45T6 + 10T8 + T10 )/512 T1 = x x9 = (126T1 + 84T3 + 36T5 + 9T7 + T9 )/256 T2 = 2x2 − 1 x8 = (35T0 + 56T2 + 28T4 + 8T6 + T8 )/128 3 T3 = 4x − 3x x7 = (35T1 + 21T3 + 7T5 + T7 )/64 4 2 T4 = 8x − 8x + 1 x6 = (10T0 + 15T2 + 6T4 + T6 )/32 5 3 T5 = 16x − 20x + 5x x5 = (10T1 + 5T3 + T5 )/16 T6 = 32x6 − 48x4 + 18x2 − 1 x4 = (3T0 + 4T2 + T4 )/8 7 5 3 T7 = 64x − 112x + 56x − 7x x3 = (3T1 + T3 )/4 8 6 4 2 T8 = 128x − 256x + 160x − 32x + 1 x2 = (T0 + T2 )/2 9 7 5 3 T9 = 256x − 576x + 432x − 120x + 9x x = T1 T10 = 512x10 − 1280x8 + 1120x6 − 400x4 + 50x2 − 1 1 = T0 U0 = 1 x10 = (42U0 + 90U2 + 75U4 + 35U6 + 9U8 + U10 )/1024 U1 = 2x x9 = (42U1 + 48U3 + 27U5 + 8U7 + U9 )/512 2 U2 = 4x − 1 x8 = (14U0 + 28U2 + 20U4 + 7U6 + U8 )/256 U3 = 8x3 − 4x x7 = (14U1 + 14U3 + 6U5 + U7 )/128 4 2 U4 = 16x − 12x + 1 x6 = (5U0 + 9U2 + 5U4 + U6 )/64 5 3 U5 = 32x − 32x + 6x x5 = (5U1 + 4U3 + U5 )/32 U6 = 64x6 − 80x4 + 24x2 − 1 x4 = (2U0 + 3U2 + U4 )/16 U7 = 128x7 − 192x5 + 80x3 − 8x x3 = (2U1 + U3 )/8 8 6 4 2 U8 = 256x − 448x + 240x − 40x + 1 x2 = (U0 + U2 )/4 9 7 5 3 U9 = 512x − 1024x + 672x − 160x + 10x x = (U1 )/2 U10 = 1024x10 − 2304x8 + 1792x6 − 560x4 + 60x2 − 1 1 = U0
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Clebsch–Gordan coefficients
j1 m1
×
j2 j m2 m
k
'
= δm,m1 +m2
(j1 + j2 − j)!(j + j1 − j2 )!(j + j2 − j1 )!(2j + 1) (j + j1 + j2 + 1)!
(−1)k (j1 + m1 )!(j1 − m1 )!(j2 + m2 )!(j2 − m2 )!(j + m)!(j − m)! . k!(j1 + j2 − j − k)!(j1 − m1 − k)!(j2 + m2 − k)!(j − j2 + m1 + k)!(j − j1 − m2 + k)!
1. Conditions: (a) Each of {j1 , j2 , j, m1 , m2 , m} may be an integer, or half an integer. Additionally: j > 0, j1 > 0, j2 > 0 and j + j1 + j2 is an integer. (b) j1 + j2 − j ≥ 0. (c) j1 − j2 + j ≥ 0. (d) −j1 + j2 + j ≥ 0. (e) |m1 | ≤ j1 , |m2 | ≤ j2 , |m| ≤ j. 2. Special values: j2 j j1 = 0 if m1 + m2 = m. (a) m m2 m 1 j1 0 j = δj1 ,j δm1 ,m . (b) m1 0 m j 1 j2 j = 0 when j1 + j2 + j is an odd integer. (c) 0 0 0 j1 j j1 = 0 when 2j1 + j is an odd integer. (d) m1 m1 m j1 j2 j : 3. Symmetry relations: all of the following are equal to m1 m2 m j2 j1 j (a) , −m2 −m1 −m j2 j1 j , (b) (−1)j1 +j2 −j m m2 m 1 j1 j2 j , (c) (−1)j1 +j2 −j −m1 −m2 −m j j2 j1 2j+1 j2 +m2 (d) (−1) , 2j1 +1 −m m2 −m1 j j2 j1 2j+1 (−1)j1 −m1 +j−m , (e) 2j1 +1 m −m2 m1 j2 j j1 2j+1 , (−1)j−m+j1 −m1 (f) 2j1 +1 m2 −m −m1 j1 j j2 2j+1 (−1)j1 −m1 , (g) 2j2 +1 m1 −m −m2 j j1 j2 2j+1 (h) (−1)j1 −m1 . 2j2 +1 m −m1 m2 By use of the symmetry relations, Clebsch–Gordan coefficients may be put in the standard form j1 ≤ j2 ≤ j and m ≥ 0.
A-102
m2
m
j1
j
− 12
0
1 2
1
0
1 2
1 2
1
1 2
0
1 2
1
1 2
1 2
1 2
1
1
1 2
1 2
j1 m1
√
1
m2
m
j1
j
−1
0
1
1
−1
0
1
2
− 12
0
1 2
3 2
− 12
1 2
1
1
− 12
1 2
1
2
0
0
1
2
0
0
1 2
3 2
0
1 2
1 2
3 2
0
1 2
1
1
0
1 2
1
2
0
1
1
1
2 2 √ 3 2 √ 2 2 √ 3 2
1 2 m2
j m
m2
m
j1
j
≈ 0.707107
0
1
1
2
≈ 0.866025
1 2
0
1 2
3 2
≈ 0.707107
1 2
1 2
1
1
≈ 0.866025
1 2
1 2
1
2
1 2
1
1 2
3 2
1 2
3 2
1
2
1
0
1
1
√
1 ≈ 1.000000 j1 1 j m1 m2 m √ 2 2 √ 6 6 √ 2 2 3 4 √ 5 4 √ 6 3 √ 3 2 √ 6 3 √ 2 4 √ 10 4 √ 2 2
j1 m1
≈ 0.707107 ≈ 0.408248 ≈ 0.707107 ≈ 0.750000 ≈ 0.559017 ≈ 0.816496 ≈ 0.866025 ≈ 0.8164967 ≈ 0.353553 ≈ 0.790569 ≈ 0.707107
1
0
1
2
1
1 2
1 2
3 2
2 2 √ 2 2 √ − 42 √ 10 4 √ 30 6 √ 105 12 √ − 22 √ 6 6 √ 3 3
1
1 2
1
1
− 34
1
1 2
1
2
1
1
1 2
3 2
1
1
1
1
√
≈ 0.707107 ≈ 0.707107 ≈ −0.353553 ≈ 0.790569 ≈ 0.912871 ≈ 0.853913 ≈ −0.707107 ≈ 0.408248 ≈ 0.577350 ≈ −0.750000 ≈ 0.559017
10 4 √ − 22 √ 2 2
≈ 0.790569
1
≈ 1.000000
√
1
1
2
1
3 2
1 2
3 2
1
3 2
1
2
√ 105 12
1
2
1
2
1
TeamLRN
5 4
1
A-103
1 j m2 m
≈ −0.707107 ≈ 0.707107
≈ 0.853913 ≈ 1.000000
NORMAL PROBABILITY FUNCTION Table of the normal distribution For a standard normal random variable (Φ(z) is the area under the Standard Normal Curve from −∞ to z). Proportion of the total area (%) 68.27 90 95 95.45 99.0 99.73 99.8 99.9
Limits µ − λσ µ−σ µ − 1.65σ µ − 1.96σ µ − 2σ µ − 2.58σ µ − 3σ µ − 3.09σ µ − 3.29σ x Φ(x) 2[1 − Φ(x)] x 1 − Φ(x)
3.09 10−3
µ + λσ µ+σ µ + 1.65σ µ + 1.96σ µ + 2σ µ + 2.58σ µ + 3σ µ + 3.09σ µ + 3.29σ 1.282 0.90 0.20 3.72 10−4
1.645 0.95 0.10 4.26 10−5
1.960 0.975 0.05 4.75 10−6
Remaining area (%) 31.73 10 5 4.55 0.99 0.27 0.2 0.1
2.326 0.99 0.02 5.20 10−7
2.576 0.995 0.01 5.61 10−8
3.090 0.999 0.002 6.00 10−9
6.36 10−10
Areas under the Standard Normal Curve from 0 to z z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9
0 .0000 .0398 .0793 .1179 .1554 .1915 .2258 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 4990 4993 4995 4997 4998 4998 4999 4999 5000
1 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2612 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .4991 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .5000
2 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2652 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .4991 .4994 .4995 .4997 .4998 .4999 .4999 .4999 .5000
3 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .4991 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000
4 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2996 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000
A-104
5 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000
6 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000
7 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .4992 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000
8 .0319 .0714 .1103 .1480 .1844 .2190 .2518 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .4993 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000
9 .0359 .0754 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .4999 .5000
Common sample size calculations Parameter
Estimate
Sample size
µ
x ¯
n=
p
pˆ
n=
µ2 − µ2
x ¯1 − x ¯2
n1 = n2 =
(zα/2 )2 (σ12 + σ22 ) E2
p1 − p2
pˆ1 − pˆ2
n1 = n2 =
(zα/2 )2 (p1 q1 + p2 q2 ) E2
z
α/2
· σ 2
E (zα/2 )2 · pq E2
Common one sample confidence intervals Parameter
Assumptions
100(1 − α)% Confidence interval
µ
n large, σ 2 known, or normality, σ 2 known
σ x ¯ ± zα/2 · √ n
µ
normality, σ 2 unknown
σ2
normality
p
binomial experiment, n large
s x ¯ ± tα/2,n−1 · √ n (n − 1)s2 (n − 1)s2 , χ2α/2,n−1 χ21−α/2,n−1
pˆ(1 − pˆ) pˆ ± zα/2 · n
Common two sample confidence intervals 100(1 − α)% Confidence interval
Parameter
Assumptions
µ1 − µ2
normality, independence, σ12 , σ22 known or n1 , n2 large, independence, σ12 , σ22 known
(¯ x1 − x ¯2 ) ± zα/2 · (¯ x1 − x ¯2 ) ±
σ12
=
σ22
µ1 − µ2
normality, independence, unknown
µ1 − µ2
normality, independence, σ12 = σ22 unknown
σ2 σ12 + 2 n1 n2
1 1 + n1 n2 (n1 − 1)s21 + (n2 − 1)s22 s2p = n1 + n2 −2 s2 s21 (¯ x1 − x ¯2 ) ± tα/2,ν · + 2 n n 1 2
t α2 ,n1 +n2 −2 · sp
ν≈
s2 1 n1
2 (s2 1 /n1 ) n1 −1
+ +
s2 2 n2
2
2 (s2 2 /n2 ) n2 −1
µ1 − µ2
normality, n pairs, dependence
sd d¯ ± tα/2,n−1 · √ n
p1 − p2
binomial experiments, n1 , n2 large, independence
(ˆ p1 − pˆ2 )±
pˆ2 (1 − pˆ2 ) pˆ1 (1 − pˆ1 ) zα/2 · + n1 n2
A-105
TeamLRN
PERCENTAGE POINTS, STUDENT’S t-DISTRIBUTION This table gives values of t such that F (t) =
Γ n+1 x2 n+1 2 1 + dx − √ n n 2 nπΓ 2 −∞ for n, the number of degrees of freedom, equal to 1, 2, . . . , 30, 40, 60, 120, ∞; and for F (t) = 0.60, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, and 0.9995. The t-distribution is symmetrical, so that F (−t) = 1 − F (t) n/F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
.60 .325 .289 .277 .271 .267 .265 .263 .262 .261 .260 .260 .259 .259 .258 .258 .258 .257 .257 .257 .257 .257 .256 .256 .256 .256 .256 .256 .256 .256 .256 .255 .254 .254 .253
.75 1.000 .816 .765 .741 .727 .718 .711 .706 .703 .700 .697 .695 .694 .692 .691 .690 .689 .688 .688 .687 .686 .686 .685 .685 .684 .684 .684 .683 .683 .683 .681 .679 .677 .674
t
.90 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.296 1.289 1.282
.95 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.671 1.658 1.645
.975 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000 1.980 1.960
.99 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.390 2.358 2.326
.995 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660 2.617 2.576
.9995 636.619 31.598 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.767 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.460 3.373 3.291
*This table is abridged from the “Statistical Tables” of R. A. Fisher and Frank Yates published by Oliver & Boyd. Ltd., Edinburgh and London, 1938. It is here published with the kind permission of the authors and their publishers.
χ2 F (χ)2 =
χ2
1
0
x!
A-106
n/F ∞
PERCENTAGE POINTS, CHI-SQUARE DISTRIBUTION This table gives values of χ2 such that
χ2
1 x(n−2)/2 e−x/2 dx 2n/2 Γ n2 for n, the number of degrees √ of freedom, √ equal to 1, 2, . . ., 30. For n > 30, a normal approximation is quite accurate. The expression 2x2 − 2n − 1 is approximately normally distributed as the standard normal distribution. Thus χ2α , the α-point of the distribution, may be computed by the formula √ 1 χ2α = [xα + 2n − 1]2 , 2 F (χ)2 =
0
where xα is the α-point of the cumulative normal distribution. For even values of n, F (χ2 ) can be written as 1 − F (χ2 ) =
x −1
x=0 1 2 χ 2
e−λ λx x!
1 n. 2
with λ = and x = Thus the cumulative Chi-Square distribution is related to the cumulative Poisson distribution. Another approximate formula for large n
3 2 2 2 χα = n 1 − + zα 9n 9n n = degrees of freedom zα = the normal deviate (the value of x for which F (x) = the desired percentile). x 1.282 1.645 1.960 2.326 2.576 3.090 F (x) .90 .95 .975 .99 .995 .999 χ2.99 = 60[1 − 0.00370 + 2.326(0.06086)]3 = 88.4 is the 99th percentile for 60 degrees of freedom.
χ2
1 xn−2/2 e−x/2 dx 2n/2 Γ n2
F (χ2 ) = 0
( n F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
.005 .0000393 .0100 .0717 .207 .412 .676 .989 1.34 1.73 2.16 2.60 3.07 3.57 4.07 4.60 5.14 5.70 6.26 6.84 7.43 8.03 8.64 9.26 9.89 10.5 11.2 11.8 12.5 13.1 13.8
.010 .000157 .0201 .115 .297 .554 .872 1.24 1.65 2.09 2.56 3.05 3.57 4.11 4.66 5.23 5.81 6.41 7.01 7.63 8.26 8.90 9.54 10.2 10.9 11.5 12.2 12.9 12.6 14.3 15.0
.025 .000982 .0506 .216 .484 .831 1.24 1.69 2.18 2.70 3.25 3.82 4.40 5.01 5.63 6.26 6.91 7.56 8.23 8.91 9.59 10.3 11.0 11.7 12.4 13.1 13.8 14.6 15.3 16.0 16.8
.050 .00393 .103 .352 .711 1.15 1.64 2.17 2.73 3.33 3.94 4.57 5.23 5.89 6.57 7.26 7.96 8.67 9.39 10.1 10.9 11.6 12.3 13.1 13.8 14.6 15.4 16.2 16.9 17.7 18.5
.100 .0158 .211 .584 1.06 1.61 2.20 2.83 3.49 4.17 4.87 5.58 6.30 7.04 7.79 8.55 9.31 10.1 10.9 11.7 12.4 13.2 14.0 14.8 15.7 16.5 17.3 18.1 18.9 19.8 20.6
.250 .102 .575 1.21 1.92 2.67 3.45 4.25 5.07 5.90 6.74 7.58 8.44 9.30 10.2 11.0 11.9 12.8 13.7 14.6 15.5 16.3 17.2 18.1 19.0 19.9 20.8 21.7 22.7 23.6 24.5
A-106 TeamLRN
.500 .455 1.39 2.37 3.36 4.35 5.35 6.35 7.34 8.34 9.34 10.3 11.3 12.3 13.3 14.3 15.3 16.3 17.3 18.3 19.3 20.3 21.3 22.3 23.3 24.3 25.3 26.3 27.3 28.3 29.3
.750 1.32 2.77 4.11 5.39 6.63 7.84 9.04 10.2 11.4 12.5 13.7 14.8 16.0 17.1 18.2 19.4 20.5 21.6 22.7 23.8 24.9 26.0 27.1 28.2 29.3 30.4 31.5 32.6 33.7 34.8
.900 2.71 4.61 6.25 7.78 9.24 10.6 12.0 13.4 14.7 16.0 17.3 18.5 19.8 21.1 22.3 23.5 24.8 26.0 27.2 28.4 29.6 30.8 32.0 33.2 34.4 35.6 36.7 37.9 39.1 40.3
.950 3.84 5.99 7.81 9.49 11.1 12.6 14.1 15.5 16.9 18.3 19.7 21.0 22.4 23.7 25.0 26.3 27.6 28.9 30.1 31.4 32.7 33.9 35.2 36.4 37.7 38.9 40.1 41.3 42.6 43.8
.975 5.02 7.38 9.35 11.1 12.8 14.4 16.0 17.5 19.0 20.5 21.9 23.3 24.7 26.1 27.5 28.8 30.2 31.5 32.9 34.2 35.5 36.8 38.1 39.4 40.6 41.9 43.2 44.5 45.7 47.0
.990 6.63 9.21 11.3 13.3 15.1 16.8 18.5 20.1 21.7 23.2 24.7 26.2 27.7 29.1 30.6 32.0 33.4 34.8 36.2 37.6 38.9 40.3 41.6 43.0 44.3 45.6 47.0 48.3 49.6 50.9
.995 7.88 10.6 12.8 14.9 16.7 18.5 20.3 22.0 23.6 25.2 26.8 28.3 29.8 31.3 32.8 34.3 35.7 37.2 38.6 40.0 41.4 42.8 44.2 45.6 46.9 48.3 49.6 51.0 52.3 53.7
PERCENTAGE POINTS, F -DISTRIBUTION This table gives values of F such that F Γ m+n m 2 n mm/2 nn/2 xm−2/2 (n + mx)−(m+n)/2 dx F (F ) = Γ 2 Γ 2 0 for selected values of m, the number of degrees of freedom of the numerator of F ; and for selected values of n, the number of degrees freedom of the denominator of F . The table also provides values corresponding to F (F )=.10,.05,.025,.01,.005,.001 since F1−α for m and n degrees of freedom is the reciprocal of Fα for n and m degrees of freedom. Thus 1 1 = = .164 F.05 (4, 7) = F.95 (7, 4) 6.09
A-107
F
F (F ) = 0 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
F =
s2 1 s2 2
Γ m+n 2 n mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .90 Γ m Γ 2 2
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
39.86 8.53 5.54 4.54 4.06 3.78 3.59 3.46 3.36 3.29 3.23 3.18 3.14 3.10 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94 2.93 2.92 2.91 2.90 2.89 2.89 2.88 2.84 2.79 2.75 2.71
49.50 9.00 5.46 4.32 3.78 3.46 3.26 3.11 3.01 2.92 2.86 2.81 2.76 2.73 2.70 2.67 2.64 2.62 2.61 2.59 2.57 2.56 2.55 2.54 2.53 2.52 2.51 2.50 2.50 2.49 2.44 2.39 2.35 2.30
53.59 9.16 5.39 4.19 3.62 3.29 3.07 2.92 2.81 2.73 2.66 2.61 2.56 2.52 2.49 2.46 2.44 2.42 2.40 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.28 2.23 2.18 2.13 2.08
55.83 9.24 5.34 4.11 3.52 3.18 2.96 2.81 2.69 2.61 2.54 2.48 2.43 2.39 2.36 2.33 2.31 2.29 2.27 2.25 2.23 2.22 2.21 2.19 2.18 2.17 2.17 2.16 2.15 2.14 2.09 2.04 1.99 1.94
57.24 9.29 5.31 4.05 3.45 3.11 2.88 2.73 2.61 2.52 2.45 2.39 2.35 2.31 2.27 2.24 2.22 2.20 2.18 2.16 2.14 2.13 2.11 2.10 2.09 2.08 2.07 2.06 2.06 2.05 2.00 1.95 1.90 1.85
58.20 9.33 5.28 4.01 3.40 3.05 2.83 2.67 2.55 2.46 2.39 2.33 2.28 2.24 2.21 2.18 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.04 2.02 2.01 2.00 2.00 1.99 1.98 1.93 1.87 1.82 1.77
58.91 9.35 5.27 3.98 3.37 3.01 2.78 2.62 2.51 2.41 2.34 2.28 2.23 2.19 2.16 2.13 2.10 2.08 2.06 2.04 2.02 2.01 1.99 1.98 1.97 1.96 1.95 1.94 1.93 1.93 1.87 1.82 1.77 1.72
59.44 9.37 5.25 3.95 3.34 2.98 2.75 2.59 2.47 2.38 2.30 2.24 2.20 2.15 2.12 2.09 2.06 2.04 2.02 2.00 1.98 1.97 1.95 1.94 1.93 1.92 1.91 1.90 1.89 1.88 1.83 1.77 1.72 1.67
59.86 9.38 5.24 3.94 3.32 2.96 2.72 2.56 2.44 2.35 2.27 2.21 2.16 2.12 2.09 2.06 2.03 2.00 1.98 1.96 1.95 1.93 1.92 1.91 1.89 1.88 1.87 1.87 1.86 1.85 1.79 1.74 1.68 1.63
60.19 9.39 5.23 3.92 3.30 2.94 2.70 2.54 2.42 2.32 2.25 2.19 2.14 2.10 2.06 2.03 2.00 1.98 1.96 1.94 1.92 1.90 1.89 1.88 1.87 1.86 1.85 1.84 1.83 1.82 1.76 1.71 1.65 1.60
60.71 9.41 5.22 3.90 3.27 2.90 2.67 2.50 2.38 2.28 2.21 2.15 2.10 2.05 2.02 1.99 1.96 1.93 1.91 1.89 1.87 1.86 1.84 1.83 1.82 1.81 1.80 1.79 1.78 1.77 1.71 1.66 1.60 1.55
61.22 9.42 5.20 3.87 3.24 2.87 2.63 2.46 2.34 2.24 2.17 2.10 2.05 2.01 1.97 1.94 1.91 1.89 1.86 1.84 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.72 1.66 1.60 1.55 1.49
61.74 9.44 5.18 3.84 3.21 2.84 2.59 2.42 2.30 2.20 2.12 2.06 2.01 1.96 1.92 1.89 1.86 1.84 1.81 1.79 1.78 1.76 1.74 1.73 1.72 1.71 1.70 1.69 1.68 1.67 1.61 1.54 1.48 1.42
62.00 9.45 5.18 3.83 3.19 2.82 2.58 2.40 2.28 2.18 2.10 2.04 1.98 1.94 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.72 1.70 1.69 1.68 1.67 1.66 1.65 1.64 1.57 1.51 1.45 1.38
62.26 9.46 5.17 3.82 3.17 2.80 2.56 2.38 2.25 2.16 2.08 2.01 1.96 1.91 1.87 1.84 1.81 1.78 1.76 1.74 1.72 1.70 1.69 1.67 1.66 1.65 1.64 1.63 1.62 1.61 1.54 1.48 1.41 1.34
62.53 9.47 5.16 3.80 3.16 2.78 2.54 2.36 2.23 2.13 2.05 1.99 1.93 1.89 1.85 1.81 1.78 1.75 1.73 1.71 1.69 1.67 1.66 1.64 1.63 1.61 1.60 1.59 1.58 1.57 1.51 1.44 1.37 1.30
62.79 9.47 5.15 3.79 3.14 2.76 2.51 2.34 2.21 2.11 2.03 1.96 1.90 1.86 1.82 1.78 1.75 1.72 1.70 1.68 1.66 1.64 1.62 1.61 1.59 1.58 1.57 1.56 1.55 1.54 1.47 1.40 1.32 1.24
63.06 9.48 5.14 3.78 3.12 2.74 2.49 2.32 2.18 2.08 2.00 1.93 1.88 1.83 1.79 1.75 1.72 1.69 1.67 1.64 1.62 1.60 1.59 1.57 1.56 1.54 1.53 1.52 1.51 1.50 1.42 1.35 1.26 1.17
63.33 9.49 5.13 3.76 3.10 2.72 2.47 2.29 2.16 2.06 1.97 1.90 1.85 1.80 1.76 1.72 1.69 1.66 1.63 1.61 1.59 1.57 1.55 1.53 1.52 1.50 1.49 1.48 1.47 1.46 1.38 1.29 1.19 1.00
=
S1 S2 /n, m
where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common
2
variance σ and based on mandndegrees of freedom, respectively. F Γ m+n 2 n mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .95 F (F ) = Γ 2 Γ m 0 2 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
F =
s2 1 s2 2
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
161.4 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.32 4.30 4.28 4.26 4.24 4.23 4.21 4.20 4.18 4.17 4.08 4.00 3.92 3.84
199.5 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.47 3.44 3.42 3.40 3.39 3.37 3.35 3.34 3.33 3.32 3.23 3.15 3.07 3.00
215.7 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 3.07 3.05 3.03 3.01 2.99 2.98 2.96 2.95 2.93 2.92 2.84 2.76 2.68 2.60
224.6 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.73 2.71 2.70 2.69 2.61 2.53 2.45 2.37
230.2 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.59 2.57 2.56 2.55 2.53 2.45 2.37 2.29 2.21
234.0 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.47 2.46 2.45 2.43 2.42 2.34 2.25 2.17 2.10
236.8 19.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.49 2.46 2.44 2.42 2.40 2.39 2.37 2.36 2.35 2.33 2.25 2.17 2.09 2.01
238.9 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.18 2.10 2.02. 1.94
240.5 19.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.12 2.04 1.96 1.88
241.9 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.22 2.20 2.19 2.18 2.16 2.08 1.99 1.91 1.83
243.9 19.41 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.00 1.92 1.83 1.75
245.9 19.43 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.18 2.15 2.13 2.11 2.09 2.07 2.06 2.04 2.03 2.01 1.92 1.84 1.75 1.67
248.0 19.45 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.10 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.93 1.84 1.75 1.66 1.57
249.1 19.45 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.51 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.05 2.03 2.01 1.98 1.96 1.95 1.93 1.91 1.90 1.89 1.79 1.70 1.61 1.52
250.1 19.46 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.57 2.47 2.38 2.31 2.25 2.19 2.15 2.11 2.07 2.04 2.01 1.98 1.96 1.94 1.92 1.90 1.88 1.87 1.85 1.84 1.74 1.65 1.55 1.46
251.1 19.47 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.96 1.94 1.91 1.89 1.87 1.85 1.84 1.82 1.81 1.79 1.69 1.59 1.50 1.39
252.2 19.48 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 2.49 2.38 2.30 2.22 2.16 2.11 2.06 2.02 1.98 1.95 1.92 1.89 1.86 1.84 1.82 1.80 1.79 1.77 1.75 1.74 1.64 1.53 1.43 1.32
253.3 19.49 8.55 5.66 4.40 3.70 3.27 2.97 2.75 2.58 2.45 2.34 2.25 2.18 2.11 2.06 2.01 1.97 1.93 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.71 1.70 1.68 1.58 1.47 1.35 1.22
254.3 19.50 8.53 5.63 4.36 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71 1.69 1.67 1.65 1.64 1.62 1.51 139 1.25 1.00
=
S1 S2 /n, m
where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common
2
variance σ and based on m and n degrees of freedom, respectively.
A-108
TeamLRN
F
F (F ) = 0 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
F =
s2 1 s2 2
Γ m+n 2 n mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .975 Γ m Γ 2 2
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
647.8 38.51 17.44 12.22 10.01 8.81 8.07 7.57 7.21 6.94 6.72 6.55 6.41 6.30 6.20 6.12 6.04 5.98 5.92 5.87 5.83 5.79 5.75 5.72 5.69 5.66 5.63 5.61 5.59 5.57 5.42 5.29 5.15 5.02
799.5 39.00 16.04 10.65 8.43 7.26 6.54 6.06 5.71 5.46 5.26 5.10 4.97 4.86 4.77 4.69 4.62 4.56 4.51 4.46 4.42 4.38 4.35 4.32 4.29 4.27 4.24 4.22 4.20 4.18 4.05 3.93 3.80 3.69
864.2 39.17 15.44 9.98 7.76 6.60 5.89 5.42 5.08 4.83 4.63 4.47 4.35 4.24 4.15 4.08 4.01 3.95 3.90 3.86 3.82 3.78 3.75 3.72 3.69 3.67 3.65 3.63 3.61 3.59 3.46 3.34 3.23 3.12
899.6 39.25 15.10 9.60 7.39 6.23 5.52 5.05 4.72 4.47 4.28 4.12 4.00 3.89 3.80 3.73 3.66 3.61 3.56 3.51 3.48 3.44 3.41 3.38 3.35 3.33 3.31 3.29 3.27 3.25 3.13 3.01 2.89 2.79
921.8 39.30 14.88 9.36 7.15 5.99 5.29 4.82 4.48 4.24 4.04 3.89 3.77 3.66 3.58 3.50 3.44 3.38 3.33 3.29 3.25 3.22 3.18 3.15 3.13 3.10 3.08 3.06 3.04 3.03 2.90 2.79 2.67 2.57
937.1 39.33 14.73 9.20 6.98 5.82 5.12 4.65 4.32 4.07 3.88 3.73 3.60 3.50 3.41 3.34 3.28 3.22 3.17 3.13 3.09 3.05 3.02 2.99 2.97 2.94 2.92 2.90 2.88 2.87 2.74 2.63 2.52 2.41
948.2 39.36 14.62 9.07 6.85 5.70 4.99 4.53 4.20 3.95 3.76 3.61 3.48 3.38 3.29 3.22 3.16 3.10 3.05 3.01 2.97 2.93 2.90 2.87 2.85 2.82 2.80 2.78 2.76 2.75 2.62 2.51 2.39 2.29
956.7 39.37 14.54 8.98 6.76 5.60 4.90 4.43 4.10 3.85 3.66 3.51 3.39 3.29 3.20 3.12 3.06 3.01 2.96 2.91 2.87 2.84 2.81 2.78 2.75 2.73 2.71 2.69 2.67 2.65 2.53 2.41 2.30 2.19
963.3 39.39 14.47 8.90 6.68 5.52 4.82 4.36 4.03 3.78 3.59 3.44 3.31 3.21 3.12 3.05 2.98 2.93 2.88 2.84 2.80 2.76 2.73 2.70 2.68 2.65 2.63 2.61 2.59 2.57 2.45 2.33 2.22 2.11
968.6 39.40 14.42 8.84 6.62 5.46 4.76 4.30 3.96 3.72 3.53 3.37 3.25 3.15 3.06 2.99 2.92 2.87 2.82 2.77 2.73 2.70 2.67 2.64 2.61 2.59 2.57 2.55 2.53 2.51 2.39 2.27 2.16 2.05
976.7 39.41 14.34 8.75 6.52 5.37 4.67 4.20 3.87 3.62 3.43 3.28 3.15 3.05 2.96 2.89 2.82 2.77 2.72 2.68 2.64 2.60 2.57 2.54 2.51 2.49 2.47 2.45 2.43 2.41 2.29 2.17 2.05 1.94
984.9 39.43 14.25 8.66 6.43 5.27 4.57 4.10 3.77 3.52 3.33 3.18 3.05 2.95 2.86 2.79 2.72 2.67 2.62 2.57 2.53 2.50 2.47 2.44 2.41 2.39 2.36 2.34 2.32 2.31 2.18 2.06 1.94 1.83
993.1 39.45 14.17 8.56 6.33 5.17 4.47 4.00 3.67 3.42 3.23 3.07 2.95 2.84 2.76 2.68 2.62 2.56 2.51 2.46 2.42 2.39 2.36 2.33 2.30 2.28 2.25 2.23 2.21 2.20 2.07 1.94 1.82 1.71
997.2 39.46 14.12 8.51 6.28 5.12 4.42 3.95 3.61 3.37 3.17 3.02 2.89 2.79 2.70 2.63 2.56 2.50 2.45 2.41 2.37 2.33 2.30 2.27 2.24 2.22 2.19 2.17 2.15 2.14 2.01 1.88 1.76 1.64
1001 39.46 14.08 8.46 6.23 5.07 4.36 3.89 3.56 3.31 3.12 2.96 2.84 2.73 2.64 2.57 2.50 2.44 2.39 2.35 2.31 2.27 2.24 2.21 2.18 2.16 2.13 2.11 2.09 2.07 1.94 1.82 1.69 1.57
1006 39.47 14.04 8.41 6.18 5.01 4.31 3.84 3.51 3.26 3.06 2.91 2.78 2.67 2.59 2.51 2.44 2.38 2.33 2.29 2.25 2.21 2.18 2.15 2.12 2.09 2.03 2.05 2.03 2.01 1.88 1.74 1.61 1.48
1010 39.48 13.99 8.36 6.12 4.96 4.25 3.78 3.45 3.20 3.00 2.85 2.72 2.61 2.52 2.45 2.38 2.32 2.27 2.22 2.18 2.14 2.11 2.08 2.05 2.03 2.00 1.98 1.96 1.94 1.80 1.67 1.53 1.39
1014 39.49 13.95 8.31 6.07 4.90 4.20 3.73 3.39 3.14 2.94 2.79 2.66 2.55 2.46 2.38 2.32 2.26 2.20 2.16 2.11 2.08 2.04 2.01 1.98 1.95 1.93 1.91 1.89 1.87 1.72 1.58 1.43 1.27
1018 39.50 13.90 8.26 6.02 4.85 4.14 3.67 3.33 3.08 2.88 2.72 2.60 2.49 2.40 2.32 2.25 2.19 2.13 2.09 2.04 2.00 1.97 1.94 1.91 1.88 1.85 1.83 1.81 1.79 1.64 1.48 1.31 1.00
=
S1 S2 /n, m
where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common
2
variance σ and based on mandndegrees of freedom, respectively. F Γ m+n 2 n mm/2 nn/2 x(m/2)−1 (n + mx)−(m+n)/2 dx = .99 F (F ) = Γ 2 Γ m 0 2 ( n m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞
F =
s2 1 s2 2
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
4052 98.50 34.12 21.20 16.26 13.75 12.25 11.26 10.56 10.04 9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.18 8.10 8.02 7.95 7.88 7.82 7.77 7.72 7.68 7.64 7.60 7.56 7.31 7.08 6.85 6.63
4999.5 99.00 30.82 18.00 13.27 10.92 9.55 8.65 8.02 7.56 7.21 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.78 5.72 5.66 5.61 5.57 5.53 5.49 5.45 5.42 5.39 5.18 4.98 4.79 4.61
5403 99.17 29.46 16.69 12.06 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.87 4.82 4.76 4.72 4.68 4.64 4.60 4.57 4.54 4.51 4.31 4.13 3.95 3.78
5625 99.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.37 4.31 4.26 4.22 4.18 4.14 4.11 4.07 4.04 4.02 3.83 3.65 3.48 3.32
5764 99.30 28.24 15.52 10.97 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.69 4.56 4.44 4.34 4.25 4.17 4.10 4.04 3.99 3.94 3.90 3.85 3.82 3.78 3.75 3.73 3.70 3.51 3.34 3.17 3.02
5859 99.33 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.81 3.76 3.71 3.67 3.63 3.59 3.56 3.53 3.50 3.47 3.29 3.12 2.96 2.80
5928 99.36 27.67 14.98 10.46 8.26 6.99 6.18 5.61 5.20 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.77 3.70 3.64 3.59 3.54 3.50 3.46 3.42 3.39 3.36 3.33 3.30 3.12 2.95 2.79 2.64
5982 99.37 27.49 14.80 10.29 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 3.71 3.63 3.56 3.51 3.45 3.41 3.36 3.32 3.29 3.26 3.23 3.20 3.17 2.99 2.82 2.66 2.51
6022 99.39 27.35 14.66 10.16 7.98 6.72 5.91 5.35 4.94 4.63 4.39 4.19 4.03 3.89 3.78 3.68 3.60 3.52 3.46 3.40 3.35 3.30 3.26 3.22 3.18 3.15 3.12 3.09 3.07 2.89 2.72 2.56 2.41
6056 99.40 27.23 14.55 10.05 7.87 6.62 5.81 5.26 4.85 4.54 4.30 4.10 3.94 3.80 3.69 3.59 3.51 3.43 3.37 3.31 3.26 3.21 3.17 3.13 3.09 3.06 3.03 3.00 2.98 2.80 2.63 2.47 2.32
6106 99.42 27.05 14.37 9.89 7.72 6.47 5.67 5.11 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.17 3.12 3.07 3.03 2.99 2.96 2.93 2.90 2.87 2.84 2.66 2.50 2.34 2.18
6157 99.43 26.87 14.20 9.72 7.56 6.31 5.52 4.96 4.56 4.25 4.01 3.82 3.66 3.52 3.41 3.31 3.23 3.15 3.09 3.03 2.98 2.93 2.89 2.85 2.81 2.78 2.75 2.73 2.70 2.52 2.35 2.19 2.04
6209 99.45 26.69 14.02 9.55 7.40 6.16 5.36 4.81 4.41 4.10 3.86 3.66 3.51 3.37 3.26 3.16 3.08 3.00 2.94 2.88 2.83 2.78 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.37 2.20 2.03 1.88
6235 99.46 26.60 13.93 9.47 7.31 6.07 5.28 4.73 4.33 4.02 3.78 3.59 3.43 3.29 3.18 3.08 3.00 2.92 2.86 2.80 2.75 2.70 2.66 2.62 2.58 2.55 2.52 2.49 2.47 2.29 2.12 1.95 1.79
6261 99.47 26.50 13.84 9.38 7.23 5.99 5.20 4.65 4.25 3.94 3.70 3.51 3.35 3.21 3.10 3.00 2.92 2.84 2.78 2.72 2.67 2.62 2.58 2.54 2.50 2.47 2.44 2.41 2.39 2.20 2.03 1.86 1.70
6287 99.47 26.41 13.75 9.29 7.14 5.91 5.12 4.57 4.17 3.86 3.62 3.43 3.27 3.13 3.02 2.92 2.84 2.76 2.69 2.64 2.58 2.54 2.49 2.45 2.42 2.38 2.35 2.33 2.30 2.11 1.94 1.76 1.59
6313 99.48 26.32 13.65 9.20 7.06 5.82 5.03 4.48 4.08 3.78 3.54 3.34 3.18 3.05 2.93 2.83 2.75 2.67 2.61 2.55 2.50 2.45 2.40 2.36 2.33 2.29 2.26 2.23 2.21 2.02 1.84 1.66 1.47
6339 99.49 26.22 13.56 9.11 6.97 5.74 4.95 4.40 4.00 3.69 3.45 3.25 3.09 2.96 2.84 2.75 2.66 2.58 2.52 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.17 2.14 2.11 1.92 1.73 1.53 1.32
6366 99.50 26.13 13.46 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.17 3.00 2.87 2.75 2.65 2.57 2.49 2.42 2.36 2.31 2.26 2.21 2.17 2.13 2.10 2.06 2.03 2.01 1.80 1.60 1.38 1.00
=
S1 S2 /n, m
where s21 = S1 /m and s22 = S2 /n are independent mean squares estimating a common
2
variance σ and based on m and n degrees of freedom, respectively.
A-109
APPENDIX B: SOURCES OF PHYSICAL AND CHEMICAL DATA
TeamLRN
SOURCES OF PHYSICAL AND CHEMICAL DATA In addition to the primary research journals, there are many useful sources of property data of the type contained in the CRC Handbook of Chemistry and Physics. A selected list of these is presented here, with emphasis on print and electronic sources whose contents have been subject to a reasonable level of quality control.
A. Data Journals 1. Journal of Physical and Chemical Reference Data – Published jointly by the National Institute of Standards and Technology and the American Institute of Physics, this quarterly journal contains compilations of evaluated data in chemistry, physics, and materials science. It is available in print and on the Internet. [ojps.aip.org/jpcrd/] 2. Journal of Chemical and Engineering Data – This bimonthly journal of the American Chemical Society publishes articles reporting original experimental measurements carried out under carefully controlled conditions. The main emphasis is on thermochemical and thermophysical properties. Review articles with evaluated data from the literature are also published. [pubs.acs.org/journals/jceaax/index.html] 3. Journal of Chemical Thermodynamics – This journal publishes original research papers that include highly accurate measurements of thermodynamic and thermophysical properties. [http://www.sciencedirect.com] 4. Atomic Data and Nuclear Data Tables – This is a bimonthly journal containing compilations of data in atomic physics, nuclear physics, and related fields. [www.sciencedirect.com] 5. Journal of Phase Equilibria and Diffusion – This journal presents critically evaluated phase diagrams and related data on alloy systems. It is published by ASM International and is the successor to the previous ASM periodical Bulletin Of Alloy Phase Diagrams. [www.asm-intl.org.] 6. Journal of Chemical Information and Computer Sciences – Although not a true data journal, it contains many papers on the prediction of physical property data from molecular structure. It is published by the American Chemical Society. [pubs.acs.org/journals/jcisd8/index.html]
B. Data Centers This section lists selected organizations that perform a continuing function of compiling and critically evaluating data in specific fields of science. 1. National Institute of Standards and Technology – Under its Standard Reference Data program, NIST supports a number of data centers in chemistry, physics, and materials science. Topics covered include thermodynamics, fluid properties, chemical kinetics, mass spectroscopy, atomic spectroscopy, fundamental physical constants, ceramics, and crystallography. Address: Office of Standard Reference Data, National Institute of Standards and Technology, Gaithersburg, MD 20899 [www.nist.gov/srd/]. 2. Thermodynamics Research Center – Now located at the National Institute of Standards and Technology, TRC maintains an extensive archive of data covering thermodynamic, thermochemical, and transport properties of organic compounds and mixtures. Data are distributed in both print and electronic form. Address: Mailcode 838.00, 325 Broadway, Boulder, CO 80305-3328 [www.trc.nist.gov] . 3. Design Institute for Physical Property Data – Under the auspices of the American Institute of Chemical Engineers [www.aiche.org/dippr/], DIPPR offers evaluated data on industrially-important chemical compounds. The largest project deals with physical, thermodynamic, and transport properties of pure compounds. Address: Brigham Young University, Provo, UT 84602 [dippr.byu.edu] . 4. Dortmund Data Bank – Maintains extensive databases on thermodynamic and transport properties of pure compounds and mixtures of industrial interest. The data are distributed through DECHEMA, FIZ CHEMIE, and other outlets. An abbreviated database system is also available for educational use. Address: DDBST GmbH, Industriestr. 1, 26121 Oldenburg, Germany [www.ddbst.de]. 5. Cambridge Crystallographic Data Centre – Maintains the Cambridge Structural Database of over 250,000 organic compounds. The data files and manipulation software are distributed in several ways. Address: 12 Union Rd., Cambridge CB2 1EZ, UK [www.ccdc.cam.ac.uk]. 6. FIZ Karlsruhe – In addition to many bibliographic databases, FIZ Karlsruhe maintains the Inorganic Crystal Structure Database in collaboration with the National Institute of Standards and Technology. The ICSD contains the atomic coordinates and related data on over 50,000 inorganic crystals. Address: Fachinformationszentrum (FIZ) Karlsruhe, Hermann-von-Helmholtz-Platz 1, D-76344 EggensteinLeopoldshafen, Germany [crystal.fiz-karlsruhe.de]. 7. International Centre for Diffraction Data – Maintains and distributes the Powder Diffraction File (PDF), a file of x-ray powder diffraction patterns used for identification of crystalline materials. The ICDD also distributes the NIST Crystal Data file, which contains lattice parameters for over 235,000 inorganic and organic crystalline materials. Address: 12 Campus Blvd., Newton Square, PA 19073-3273 [icdd.com]. 8. Research Collaboratory for Structural Bioinformatics – Maintains the Protein Data Bank (PDB), a file of 3-dimensional structures of proteins and other biological macromolecules. Address: Department of Chemistry and Chemical Biology, Rutgers University, 610 Taylor Road, Piscataway, NJ 08854-8087 [www.rcsb.org]. 9. Toth Information Systems – Maintains the Metals Crystallographic Data File (CRYSTMET). Address: 2045 Quincy Ave., Gloucester, ON, Canada K1J 6B2 [www.tothcanada.com]. 10. Atomic Mass Data Center – Collects and evaluates high-precision data on masses of individual isotopes and maintains a comprehensive database. Address: C.S.N.S.M (IN2P3-CNRS), Batiment 108, F-91405 Orsay Campus, France [csnwww.in2p3.fr/amdc/]. 11. Particle Data Group – International center for data of high-energy physics; maintains database of properties of fundamental particles, which is published in both print and electronic form. Address: MS 50-308, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 [pdg.lbl.gov]. 12. National Nuclear Data Center – Maintains databases on nuclear structure and reactions, including neutron cross sections. The NNDC is the U. S. node in an international network of nuclear data centers. Address: Brookhaven National Laboratory, Upton, NY 11973-5000 [www.nndc.bnl.gov].
B-1
SOURCES OF PHYSICAL AND CHEMICAL DATA (continued) 13. International Union of Pure and Applied Chemistry – Address: PO Box 13757, Research Triangle Park, NC 27709-3757 [www.iupac.org]. IUPAC supports a number of long-term data projects, including these examples: a. Solubility Data Project – Carries out evaluation of all types of solubility data. The results are published in the Solubility Data Series, whose current outlet is the Journal of Physical and Chemical Reference Data. [www.unileoben.ac.at/~eschedor/] b. Kinetic Data for Atmospheric Chemistry – Maintains a comprehensive database on the kinetics of reactions important in the chemistry of the atmosphere. [www.iupac-kinetic.ch.cam.ac.uk/] c. International Thermodynamic Tables for the Fluid State – Prepares definitive tables of the thermodynamic properties of industrially important fluids. Thirteen volumes have been published by IUPAC. [http://www.iupac.org/publications/books/seriestitles/]
C. Major Multi-Volume Handbook Series 1. Chapman & Hall/CRC Chemical Dictionaries – These originally appeared in print form as the Dictionary of Organic Compounds, Dictionary of Natural Products, etc. They are now published in electronic form and are available in CDROM format [www.crcpress.com] and on the Internet [www.chemnetbase.com]. The consolidated version, called the Combined Chemical Dictionary, has data on more than 450,000 compounds spanning all branches of chemistry. The coverage includes physical properties, biological sources, hazard information, uses, and literature references. 2. Properties of Organic Compounds – Originally published in three editions as the Handbook of Data on Organic Compounds, it is now in electronic form as Properties of Organic Compounds. The database includes about 30,000 compounds; physical properties and spectral data (mass, infrared, Raman, ultraviolet, and NMR) are covered. It is offered as CDROM [www.crcpress.com] and web access [www.chemnetbase.com]. 3. Beilstein Handbook of Organic Chemistry – The classic source of data on organic compounds, dating from the 18th century, Beilstein was converted to electronic form in the last decade of the 20th century. Over 8 million compounds and 5 million chemical reactions are now covered, with a broad range of physical properties as well as synthetic methods and ecological data. The database is accessed by the CrossFire software [www.mdli.com]. 4. Gmelin Handbook of Inorganic and Organometallic Chemistry – A subset of the information in the print series has been converted to electronic form and is now distributed in the same manner as Beilstein. In addition to the standard physical properties, the coverage includes a wide range of optical, magnetic, spectroscopic, thermal, and transport properties for about 1.4 million compounds [www.mdli.com]. 5. DECHEMA Chemical Data Series – DECHEMA distributes the DETHERM database, which emphasizes data used in process design in the chemical industry, including thermodynamic and transport properties of about 20,000 pure compounds and 90,000 mixtures. Access is available through in-house databases and via the Internet. [www.dechema.de]. 6. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology - Landolt-Börnstein covers a very broad range of data in physics, chemistry, crystallography, materials science, biophysics, astronomy, and geophysics. Hard-copy volumes in the New Series (started in 1961) are still being published, and the entire New Series is now accessible on the Internet [www.landolt-boernstein.com].
D. Selected Single-Volume Handbooks The following handbooks offer broad coverage of high-quality data in a single volume. This list is only representative; an extensive listing of handbooks in all fields of science may be found in Handbooks and Tables in Science and Technology, Third Edition (Russell H. Powell, ed., Oryx Press, Westport, CT, 1994). 1. American Institute of Physics Handbook – Although an old book, it contains much data that is still useful, especially in acoustics, mechanics, optics, and solid state physics. (Dwight E. Gray, ed., McGraw-Hill, New York, 1972) 2. Constants of Inorganic Substances - This book presents physical constants, thermodynamic data, solubility, reactivity, and other information on over 3000 inorganic compounds. Since it draws heavily on Russian literature, it contains a great deal of data that does not make its way into most U. S. handbooks. (R. A. Lidin, L. L. Andreeva, and V. A. Molochko, Begell House, New York, 1995) 3. Handbook of Chemistry and Physics – Now in the 84th Edition, the CRC Handbook covers data from most branches of chemistry and physics. The annual revisions permit regular updating of the information. Also available on CDROM [www.crcpress.com] and the web [hbcpnetbase.com]. (David R. Lide, ed., CRC Press, Boca Raton, FL, 2002) 4. Handbook of Inorganic Compounds – This book covers physical constants and solubility for about 3300 inorganic compounds. Also available on CDROM [www.crcpress.com]. (Dale L. Perry and Sidney L. Phillips, eds., CRC Press, Boca Raton, FL, 1995) 5. Handbook of Physical Properties of Liquids and Gases – This is a valuable source of data on all types of fluids, ranging from liquid and gaseous hydrocarbons to molten metals and ionized gases. Detailed tables of physical, thermodynamic, and transport properties are given for temperatures from the cryogenic region to 6000 K. Both Western and Russian literature is covered. (N. B. Vargaftik, Y. K. Vinogradov, and V. S. Yargin, Begell House, New York, 1996) 6. Handbook of Physical Quantities – The range of coverage is somewhat similar to the CRC Handbook of Chemistry and Physics, but with a stronger emphasis on physics than on chemistry. Solid state physics, lasers, nuclear physics, geophysics, and astronomy receive considerable attention. (Igor S. Grigoriev and Evgenii Z. Meilikhov, eds., CRC Press, Boca Raton, FL, 1997) 7. Kaye & Laby Tables of Physical and Chemical Constants – Kaye & Laby dates from 1911, and the 16th Edition was prepared in 1995 by a committee of experts. The coverage extends to almost every field of physics and chemistry; data on a limited number of representative substances or materials are given for each topic. (Longman Group Limited, Harlow, Essex, UK, 1995)
B-2
TeamLRN
SOURCES OF PHYSICAL AND CHEMICAL DATA (continued) 8. Lange’s Handbook of Chemistry – Provides broad coverage of chemical data; last updated in 1998. Also available on the web [www.knovel.com]. (John A. Dean, ed., McGraw-Hill, New York, 1998) 9. Recommended Reference Materials for the Realization of Physicochemical Properties – This IUPAC book emphasizes highly accurate data on substances and materials that can be used as calibration standards. It covers physical, thermal, optical, and electrical properties. (K. N. Marsh, ed., Blackwell Scientific Publications, Oxford, 1987) 9. The Merck Index – Now in its 13th Edition (published in 2001), The Merck Index is a widely used source of data on over 10,000 compounds, chosen particularly for their importance in biology, medicine, and ecology. A short monograph on each compound gives information on the synthesis and uses as well as physical and toxicological properties. Also available on CDROM [www.camsoft.com]. (Maryadele J. O’Neil, ed., Merck & Co., Whitehouse Station, NJ, 2001)
E. Summary of Useful Web Sites for Physical and Chemical Properties Most of the web sites in the following list provide direct access to factual data on physical and chemical properties. However, the list also includes portals that link to different property databases or describe the procedure for gaining access to electronic sources of property data. There are also a few chemical directory sites, which are useful for obtaining formulas, synonyms, and registry numbers for substances of interest. Web Site
Address
Comments
Acronyms and Symbols Advanced Chemistry Development
www3.interscience.wiley.com/stasa/ www.acdlabs.com
Alloy Center
www.asminternational.org/ alloycenter/ www.geo.arizona.edu/AMS/ amcsd.php csnwww.in2p3.fr/amdc/ www.mdli.com www.ccdc.cam.ac.uk www.chemnetbase.com/scripts/ ccdweb.exe www.chemfinder.com www.oscar.chem.indiana.edu/cfdocs/ libchem/acronyms/ acronymsearch.html chem.sis.nlm.nih.gov/chemidplus/ www.chemindustry.com/chemicals/ www.chemnetbase.com
American Mineralogist Crystal Structure Database Atomic Mass Data Center Beilstein Cambridge Structural Database Chapman & Hall/CRC Combined Chemical Dictionary Chemfinder Chemical Acronyms Database
ChemIDplus ChemIndustry CHEMnetBASE ChemWeb Databases Coblentz Infrared Spectra CODATA Home Page Crystallography Open Database (COD) DECHEMA (DETHERM) DIPPR Pure Compound Database Dortmund Data Bank Enzyme Nomenclature Database FDM Reference Spectra Databases FIZ Chemie Berlin FIZ Karlsruhe - ICSD Fundamental Physical Constants Gmelin Handbook of Chemistry and Physics Hazardous Substances Data Bank IUPAC Home Page IUPAC Kinetics Data IUPAC Nomenclature Rules IUPAC Solubility Data Project Knovel.com
www.chemweb.com/databases/ www.galactic.com/coblentz/ www.codata.org www.crystallography.net www.dechema.de dippr.byu.edu www.ddbst.de www.expasy.ch/enzyme/ www.fdmspectra.com/ www.fiz-chemie.de crystal.fiz-karlsruhe.de physics.nist.gov/constants www.mdli.com hbcpnetbase.com toxnet.nlm.nih.gov/cgi-bin/sis/ htmlgen?HSDB www.iupac.org www.iupac-kinetic.ch.cam.ac.uk/ www.chem.qmul.ac.uk/iupac/ www.unileoben.ac.at/~eschedor/ www.knovel.com
B-3
Free servcie; useful for indentifying acronyms for chemicals Chemical directory, with programs for estimating physical and spectral properties Physical, electrical, thermal, and mechanical properties of alloys Lattice constants of minerals See B.10 See C.3 See B.5 See C.1 Chemical directory, with links to several property databases Useful for associating chemical names and acronyms
Chemical directory Chemical directory Portal to C&H/CRC Chemical Dictionaries, Handbook of Chemistry and Physics, Properties of Organic Compounds, etc. Portal to many databases IR spectra on CDROM Thermodynamic key values and fundamental constants Crystal data on 12,000 compounds See C.5 See B.3 See B.4 IUBMB nomenclature for enzymes Infrared, Raman, and mass spectra Portal to DETHERM (C.5) and Dortmund Data Bank (B.4) See B.6 CODATA fundamental constants See C.4 Web version of CRC Handbook Physical and toxicological properties of chemicals of health or environmental importance See B.13 See B.13.b Useful site for organic and biochemical nomenclature See B.13.a Portal to Lange’s Handbook, Perry’s Chemical Engineers’ Handbook, etc.
SOURCES OF PHYSICAL AND CHEMICAL DATA (continued) Web Site Landolt-Börnstein MatWeb
Address www.landolt-boernstein.com www.matweb.com
Metals Crystallographic Data File NASA Chemical Kinetics Data National Center for Biotechnology Information National Nuclear Data Center National Toxicology Program NIST Atomic Spectra Database
www.tothcanada.com jpldataeval.jpl.nasa.gov www.ncbi.nlm.nih.gov
NIST Ceramics Webbook NIST Chemistry Webbook NIST Data Gateway NIST Physical Reference Data NLM Gateway NMR Shift DB Particle Data Group Polymers — A Property Database Powder Diffraction File Properties of Organic Compounds Protein Data Bank SpecInfo Spectra Online STN Easy STN Easy-Europe STN Easy-Japan Syracuse Research Corporation Table of Isotopes Thermodynamics Research Center TOXNET Wiley Interscience
Comments See C.6 Thermal, electrical, and mechanical properties of engineering materials See B.9 Kinetic and photochemical data for stratospheric modeling Portal to GenBank and other sequence databases
www.nndc.bnl.gov ntp-server.niehs.nih.gov physics.nist.gov/cgi-bin/AtData/ main_asd www.ceramics.nist.gov/webbook/ webbook.htm webbook.nist.gov srdata.nist.gov/gateway/ physics.nist.gov/PhysRefData/ gateway.nlm.nih.gov/gw/Cmd www.nmrshiftdb.org pdg.lbl.gov www.polymersdatabase.com/ icdd.com www.chemnetbase.com/scripts/ pocweb.exe www.rcsb.org www.chemicalconcepts.com spectra.galactic.com/SpectraOnline/ stneasy.cas.org stneasy.fiz-karlsruhe.de stneasy-japan.cas.org esc.syrres.com/interkow/database.htm ie.lbl.gov/education/isotopes.htm www.trc.nist.gov toxnet.nlm.nih.gov www3.interscience.wiley.com/ reference.html
See B.12 Chemical health and safety data Energy levels, wavelengths, and transition probabilities of atoms and atomic ions See B.1 Broad range of physical, thermal, and spectral properties Portal to all NIST data systems; see B.1 Atomic and molecular spectra, cross sections, x-ray attenuation, and dosimetry data Portal to all National Library of Medicine databases NMR data submitted by users See B.11 Properties of commercial polymers See B.7 See C.2 See B.8 IR, NMR, and mass spectra IR, UV, NMR, Raman, and mass spectra (unreviewed) Chemical directory (and access to Chemical Abstracts)
Properties of environmental interest Nuclear energy levels, moments, and other properties See B.2 Portal to HSDB and other databases on hazardous chemicals Portal to Kirk-Othmer Encyclopedia of Chemical Technology, Ullmann’s Encyclopedia of Industrial Chemistry, Encyclopedia of Reagents for Organic Synthesis, etc.
B-4
TeamLRN