Isotopes and Chemical Principles
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
Isotopes and Chemical Principles Peter A. Rock, Editor
A by the Division of Chemical Education, Inc. at the 167th Meeting of the American Chemical Society, Los Angeles, Calif., April 3, 1974
ACS SYMPOSIUM
SERIES
AMERICAN CHEMICAL SOCIETY WASHINGTON
D.
C.
1975
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
11
Data
Library of Congress
Isotopes and chemical principles. (ACS symposium series; 11) Includes bibliographical references and index. 1. Isotopes—Congresses.
2. Chemical reaction, Condi
tions and laws of—Congresses. I. Rock, Peter Α., 1939- ed. II. American Chemical Society.
Division
of Chemical Education.
American Chemical Society. QD466.I86
III. Series:
ACS symposium series; 11.
541'.388
ISBN 0-8412-0225-7
75-2370 ACSmc8 11 1-215
Copyright © 1975 American Chemical Society A l l Rights Reserved PRINTED I N T H E U N I T E D STATES O F
AMERICA
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ACS Symposium Series Robert F. Gould,
Series Editor
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
FOREWORD The ACS SYMPOSIUM
a medium for publishing symposia quickly in book form. The format of the SERIES parallels that of its predecessor, ADVANCES IN CHEMISTRY SERIES, except that in order to save time the papers are not typeset but are reproduced as they are submitted by the authors in camera-ready form. As a further means of saving time, the papers are not edited or reviewed except by the symposium chairman, who becomes editor of the book. Papers published in the ACS SYMPOSIUM SERIES are original contributions not published elsewhere in whole or major part and include reports of research as well as reviews since symposia may embrace both types of presentation.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
PREFACE his volume contains all of the papers presented at the symposium entitled Isotopes and Chemical Principles. The objective of the symposium was to present the major developments, both past and present, in the field of isotope effects in such a way as to make this material available to chemistry teachers who may wish to use isotope effects in their discussions of chemical principles. The order of the paper used at the symposium. The papers fall roughly into two groups. Papers 1-4 deal primarily with theoretical aspects whereas papers 5 - 9 deal primarily with experimental aspects. It is hoped that this symposium volume will serve to convey a fair measure of the utility and power of isotope effects in the continuing development of our understanding of chemical behavior and chemical principles. I would like to thank Bassam Z. Shakhashiri and Jerry A . Bell, the former chairpersons of the Program Committee of the Division of Chemical Education, for their help in bringing about this symposium. PETER A . ROCK
University of California Davis, Calif. January 16, 1975
ix
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
To Marvin J. Stern and T . Ivan Taylor
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
Marvin J. Stern (1935-1974)
T. Ivan Taylor (1909-1973) Xii
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
IN MEMORIUM t the start of this symposium in Los Angeles, Jacob Bigeleisen expressed the deep feeling of sadness of the entire community of isotope scientists at the loss during the past year of two of our esteemed col leagues, Marvin J. Stern and T. Ivan Taylor. This is an attempt to express in a more permanent way our appreciation to them and our sense of loss. Marvin J. Stern Marvin J. Stern was born in New York, Ν. Y., June 1, 1935 and was 38 yrs old at the time of his death on January 29, 1974. During his alltoo-brief career he served on the faculties of Rutgers University, Columbia University, and Yeshiva University. From 1961 to 1969 he was affiliated with the Chemistry Department at Brookhaven National Laboratory as research associate, visiting associate chemist, and research collaborator. During a sabbatical leave from Yeshiva University in 1971-1972 he was a Chaim Weizmann Memorial Fellow at the Weizmann Institute of Science, Rehovot, Israel and a John Simon Guggenheim Memorial Fellow at the University of Kent, Canterbury, England. He was actively involved in stable isotope studies for his entire scientific career, contributing over 30 publications during 1960-1974 on a variety of experimental and theoretical studies of both kinetic and equilibrium isotope effects. His experimental researches contributed greatly to the development of chemical exchange and distillation processes for concentrating the isotopes of nitrogen and oxygen and to the under standing of the relationship between inter- and intra-molecular forces in liquids and vapor pressure isotope effects. At Brookhaven National Laboratory, he participated in some of the first applications of high-speed digital computer calculations to the understanding of isotope effects. Most of his subsequent research career was devoted to such calculations. An initial theoretical study which related the vapor pressure isotope effects in ethylene to force fields in the condensed phase led into an extensive program in which computer calculations on "model molecules" were used to compare "exact" calculations of equilibrium and kinetic isotope effects with various approximation methods. The validity and the applicability xiii
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
limits of a variety of approximation procedures were defined and new approaches were developed to help chemists working with large molecules. Calculations were carried out to study the temperature dependence of isotope effects, and the use of isotope effects in studying quantum mechanical tunnelling in chemical rate processes was investigated. Throughout these studies, Marvin Stern's unusual thoroughness and meticulous attention to detail were uniquely suited to the tasks. The breadth of his interests and of his impact on isotope chemistry can be best appreciated from the fact that five of the participants in this symposium have worked with him. His insights, his tenacity, his thoroughness, and his wit will not be soon forgotten by us. T. Ivan Taylor T. Ivan Taylor was bor was 63 yrs old at the time of his death on July 27, 1973. He received his undergraduate and initial graduate education in Idaho, where he earned B.S. and M.S. degrees at the University of Idaho in Moscow. He continued his graduate education at Columbia University, where he received a Ph.D. degree in 1938. In recognition of his outstanding accomplishments, the University of Idaho conferred an Honorary Doctor of Science degree upon him in 1972. He served on the faculties of the University of Idaho, the University of Minnesota, the University of Iowa, and Columbia University. His most recent tenure at Columbia University extended over 28 yrs, from his appointment as an associate professor in 1945, through his promotion to professor in 1949, until his death this past year. During World War II, he served at the National Bureau of Standards in Washington, D.C., at Oak Ridge, Tenn., and in New York, N.Y. as a member of the research control group of the Manhattan District Atomic Energy Project. In addition to his teaching and research, he found time to serve on visiting and advisory committees for Brookhaven National Laboratory, the National Bureau of Standards, the National Academy of Sciences, and the American Chemical Society. He served a term as chairman of the New York Section of the American Chemical Society and several terms as councilor and as member of the Board of Directors of the New York Section of the American Chemical Society. He also served for many years as a consultant to the Carbide and Carbon Chemicals Corp. at Oak Ridge and as a research collaborator at Argonne National Laboratory and at Brookhaven National Laboratory. Ivan Taylor was an early pioneer in studies of methods for separating stable isotopes and of their use for elucidating kinetics, mechanisms, and the catalysis of reactions. His initial research on isotope separation was in collaboration with Harold C. Urey. He went on to guide personally xiv
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
the studies of numerous post-doctoral associates, as well as those of 22 successful Ph.D. candidates, 14 of whom wrote dissertations either on the separation of isotopes or their use. He and his research group published more than 58 papers dealing with isotopes from 1937 to 1973. One of his major contributions to isotope science was the imaginative development of numerous chemical exchange processes for concentrating a variety of isotopes of the lighter elements. His group explored and developed countercurrent processes using exchanges between gas-liquid, gas—gas, and liquid-solid phases to enrich the rarer isotopes of hydrogen, lithium, carbon, nitrogen, oxygen, and potassium. The nitric acid-nitric oxide exchange process developed in his laboratory for the concentration of nitrogen-15 was the first system to produce this isotope at 99.8% purity. The process has been used during the past 20 yrs to produce laboratory-scale quantities of highly enriched nitrogen-15 and stil production of this useful isotope. Separation plants have been constructed and operated in the United States, United Kingdom, East Germany, Romania, the Soviet Union, and Israel. Innumerable scientific studies using N have been facilitated by Ivan Taylor s work. 1 5
Ivan Taylor was an unusually skilled and creative experimentalist. He never lost the joy of working in the laboratory with his own hands, and he never quite forgot his early days in research when "string and sealing wax" was the rule rather than the exception. In a time of rapidly expanding research costs and increasing dependence on expensive commercial instruments, he was almost fiercely determined to carry out experiments in the most simple and direct manner with equipment of his own design and often his own fabrication. His was a unique mold; we are all the poorer for his loss. WILLIAM
SPINDEL
M A X WOLFSBERG
xv
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1 Quantum Mechanical Foundations of Isotope Chemistry JACOB BIGELEISEN Department of Chemistry, University of Rochester, Rochester, Ν. Y. 14627
Introduction I shall open this symposium with a brief overview of the quantum and s t a t i s t i c a l mechanical foundations of isotope chem i s t r y . A number of monographs already exist dealing with the principles and applications of isotope chemistry i n chemical kinetics, geochemistry, isotope separation, and equilibrium processes. This symposium i s directed, in part, toward estab lishing a bridge from the present habitat of isotope chemistry, graduate students and professional scientists, to the under graduate classroom. For this purpose I find it convenient to develop the fundamental principles of isotope chemistry along h i s t o r i c a l lines. I w i l l retain only those ideas that have stood the test of time and omit all of those dead ends and false turns which are part of the development of any science. Even with this restriction, it i s impossible to cover all of the areas of isotope chemistry. The one area which relates most to all disciplines of isotope chemistry i s the equilibrium i n ideal gases. Other papers in this symposium w i l l start from the ideal gas equilibrium to such topics as tunnelling i n chemical kinetics, isotope separation, reaction mechanisms, condensed phase isotope chemistry and isotope biology. Shortly after the discovery of isotopes Fajans (1)recognized that the thermodynamic properties of solids, which depend on the frequencies of atomic and molecular vibrations, must be d i f f e r ent for isotopes. This idea was put into quantitative form independently by Stern (2) and Lindemann (3,4.). * Much of the research reported in this a r t i c l e was supported by the U.S. Atomic Energy Commission. ** A condensed summary of this a r t i c l e was presented at the Division of Chemical Education Symposium on "Isotopes and Chemical Principles",ACS National Meeting 3 April 1974, Los Angeles. The manuscript was prepared during the tenure of a John Simon Guggenheim Memorial Fellowship. 1
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
Stern-Lindemann Formulation of Vapor Pressure Isotope Effect Stern and Lindemann independently gave the f i r s t quantita tive formulation of an isotope effect. They considered the equilibrium between a Debye solid and an ideal gas composed of monatomic substances. Further they introduced the following assumptions: 1) the oscillations i n a solid l a t t i c e are harmon i c , 2) the potential energy i s isotope invariant, and 3) an oscillator has a zero point energy. Some of these assumptions represented a significant insight into physical processes at the time they were put forward. Anharmonic effects are of major importance i n the equation of state of solids. However, neither experiment nor theory had advanced to the point where anharmonic effects needed to be considered i n the analysis of the constant volume heat capacity, th energies of solids. Th atom shows a small dependence of the potential energy on the mass. The dependence of the Rydberg constant on the mass of the nucleus leads to a small difference i n behavior.of hydrogen and deuterium. The Born-Oppenheimer approximation to the solu tion of the Schrodinger equation for molecules inherently con tains the assumption of an isotope independent potential energy. The small corrections to isotope chemistry from corrections to the Born-Oppenheimer approximation are currently being investi gated by Wolfsberg and are reported in the paper in this sympo sium volume by Wolfsberg and Kleinman. Even before the development of quantum mechanics by Heisenberg and Schrodinger in 1925- 6, in which such concepts as zero point energy follow naturally, d i f f i c u l t i e s with the old quantum theory led scien t i s t s of 1920 to anticipate the existence of a zero point ener gy. It was, therefore, natural for Lindemann to develop the theory of the vapor pressure isotope effect both for the case of the existence of zero point energy and for the non-existence. For the former, he predicted that the isotope effect on vapor pressure would be a small effect, of the order of 0.02 % for P b / P b at 600° Κ (actually Lindemann has an error of a factor of 10 and his result should be reduced to 0.002 %) and that the effect i s a "second order difference". Stern and Lindemann obtained for the monatomic Debye solid under the har monic oscillator, Born-Oppenheimer approximations for a system with zero point energy, Ε = 1/2 h ν » f
206
208
In P7P = (3/40) (Θ/Τ) ο + ...
(1)
2
where P' and Ρ are the vapor pressures of a light and heavy isotope respectively, θ i s the Debye temperature, hv /k, of the heavy isotope and m
Ô - [(H/lO " 1 ]
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1.
BiGELEiSEN
Quantum Mechanical
Foundations
3
Equation (1) i s the f i r s t term of a Taylor expansion valid for (Θ/Τ) < 2rr. For the case of the non-existence of zero point energy, one predicts an isotope effect for the P b / P b vapor pressure ratio two orders of magnitude larger than the predic tion from Eq. (1) at 600° K. In addition the no zero point energy case predicts the P b to have the larger vapor pressure at 600° K. Stern's estimate of the difference i n vapor pressures of Ne and Ne at 24.6° Κ through Eq. (1) led to the f i r s t separ ation of isotopes on a macro scale by Keesom and van Dijk (2). The same theory, without the approximation (Θ/Τ) < 1, was used by Urey, Brickwedde, and Murphy (5) to design a Raleigh d i s t i l lation concentration procedure to enrich HD i n H five fold above the natural abundance level, which was adequate to demon strate the existence o It i s interesting to not van Dijk took their finding that there was a difference i n vapor pressures of the neon isotopes and that the vapor pressure of °Ne i s greater than Ne to be an important experimental proof of the existence of zero point energy. I t i s fortunate that Ur.ey had not yet discovered deuterium at the time of Keesom and van Mjk's work or they might have been misled by the discovery that the vapor pressure of deuterocarbons i s generally larger than hydrocarbons. It i s interesting to speculate what conclu sions would have been drawn from such a finding i n 1931. It i s important to look into the implications of Eq. (1) since the development of the quantum-statistical mechanical theory of isotope chemistry from 1915 u n t i l 1973 centers about the generalization of this equation and the physical interpreta tion of the various terms i n the generalized equations. Accord ing to Eq. (1) the difference i n vapor pressures of isotopes i s a purely quantum mechanical phenomenon. The vapor pressure ^ ratio approaches the classical l i m i t , high temperature, as Τ . The mass dependence of the isotope effect i s 6M/M^ where ôM = M - W . Thus for a unit mass difference i n atomic weights of isotopes of an element, the vapor pressure isotope effect at the same reduced temperature (Θ/Τ) f a l l s off as M~ . Interest ingly the temperature dependence of In P' /P i s T~2 not δλο/Τ where δλ i s the heat of vaporization of the heavy isotope minus that of the light isotope at absolute zero. In fact, i t is the difference between δλ, the difference i n heats of vapor ization at the temperature Τ from δλ. that leads to the Τ law. From Eq. (1) we can write 2 0 6
2 0 8
2 0 8
20
22
2
22
2
β
0
(AG* - AG°' ) RT 0/
,,
m K
1
m
}
(AS° - AS R
P/
)
m
1 (λ - λ') T 2 RT
0/
where (£G° - AG ) (AS° = £S ) , and (λ - λ' ) are respectively the differences i n standard free energies, entropies and heats of vaporization at the temperature T.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1
f 2
4
ISOTOPES AND CHEMICAL PRINCIPLES
Within the approximation of Eq. (1) the entropy change i n the process i s equal to the change i n ô(AG)/T and each are equal to one half the isotope effect on T ~ l times the enthalpy change. The competition between entropy and enthalpy i s independent of temperature and the sign of the difference in free energy change never changes. For the system under consideration, equilibrium between a gas and a solid, this conclusion i s v a l i d even i f one includes the higher terms i n Eq. (1). Later we shall see that the consideration of higher order terms i n Eq. (1) leads to a temperature dependence on the enthalpy-entropy balance, but the difference in free energies of formation of any system from i t s atoms i s always of the sign given by Eq. (2). Although Keesom and van Dijk were led to the successful separation of the neon isotopes through predictions made by Professor Otto Stern base interpreted their measurement Ne/ Ne in terms of a δλ/Τ rather than a δλ/Τ temperature dependence. When I replotted their data in 1956 i n accord with expectations from Eq. (1), I found that a linear extrapolation of the Keesom-Haantjes data predicted a significant difference in vapor pressures of the neon isotopes for the hypothetical solids at i n f i n i t e temperature. There were no other direct con firmations of the T~ law predicted by Eq. (1) and this led to a reinvestigation of the vapor pressures of the neon isotopes. Roth and Bigeleisen (7_ 8) not only confirmed Eq. (1) through their investigation of the vapor pressures of the neon isotopes, cf. Fig. 1, but also showed that the systematic investigation of the vapor pressure isotope effect i n simple substances could give important new information on the l a t t i c e energy, the an harmonic vibrations i n crystals, the melting process and the mean square force between molecules in the liquid state. The latter i s related to the intermolecular potential and the radial distribution function i n the liquid. Systematic investigation of the vapor pressures of the argon (9,10)
22
2
2
9
an
2
2
ln(P'/P) = (1/24)
(3) m m 2 where i s the mean value of the Laplacian of the interaction potential in the^condensed phase. In the harmonic oscillator approximation i s just f i i ^ t h e force constant, ^or a harmonic Debye l a t t i c e (ft / k ) () (m"l) i s just θ . Although Equations (1) and (3) can provide significant i n sight into the behavior of polyatomic molecules, they are inade quate approximationsto account for the role of molecular structure on the isotope chemistry of condensed phases (16). 2
β
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1.
BiGELEiSEN
Quantum Mechanical Foundations
5
Urey-Rittenberg Formulation of Isotope Exchange Equilibria After he had succeeded in enriching deuterium by the Raleigh d i s t i l l a t i o n of liquid hydrogen, Urey undertook both theoretical and experimental investigations of the differences in the chemi s t r y of protium and deuterium compounds. On the theoretical side Urey and Rittenberg (17) u t i l i z e d the methods developed for the calculation of the partition function and the free energy of a diatomic molecule from spectroscopic data. For an ideal gas - (G° - ES)/T = 3/2RlnM + 5/2RlnT + R l n Q
int
~ K(S-T)
(4)
where Ej i s the standard molar internal energy of the molecule at 0° K, Q i s the internal partition function K(S-T) i s the Sackur-îetrode constant ±
t
=
Σ
Ρ
ex
τ P <-E(v,J)/kT) (5) v,J The internal partition function i s a sum of Boltzmann factors, multiplied by their degeneracies, P j , over the vibrational and rotational states of the molecule. In the summation over the rotational states i t i s necessary to consider symmetry selection rules. This leads to a factor S" i n Q ^ <* in Eq. (4) when the rotation i s c l a s s i c a l . In the calculation of the standard free energy change of a chemical reaction from spectroscopic data i t i s necessary to evaluate ΔΕϋ, the energy change at absolute zero. In the general chemical reaction the difference i n electronic binding energies between products and reactants dominates JAE©. The latter are usually not known with sufficient accuracy for the calculation of AG /RT to an absolute value of 10~ or better. Therefore, recourse i s usually made to experimental thermodynamic data for the evaluation A E S (18). The dissociation of the halogens and other diatomic molecules are cases where ΔΕ® has been determined entirely from spectroscopic data to yield values of AG°/RT calculated from Eq. (4) i n quan t i t a t i v e agreement with experiment. For the isotope exchange reaction, e.g., l n t
J
1
T
A N
+
R
L
N
S
e
2
H
2
+ 2DI = D
2
+ 2HI
ΔΕΪ electronic i s identically zero under the Born-Oppenheimer approximation. The only contribution to AE© comes from the molecular vibrations. Ej/Nhc = Σ
±
(η + 1/2)
+ anharmonic corrections
(6)
Thus once the spectroscopic rotational and vibrational constants are known for a pair of isotopic molecules i t i s possible to calculate the partition function ratio of a pair of isotopic molecules
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6
ISOTOPES AND CHEMICAL PRINCIPLES
Q /Q 1
= exp [-(G° - Gp/RT]
2
(7)
from Equations (4) and (6). Q i s the total partition function. The rotational and vibrational spectroscopic constants, moments of inertia and vibrational frequencies, of isotopic molecules are related through atomic masses, bond distances, and force constants (19). The spectroscopic constants used i n the calculât ion of Q ^ / m u s t be consistent with the isotope relations for moments of inertia and vibrational frequencies of the molecules concerned. Frequently the spectroscopic constants for a set of isotopic molecules are best obtained from a least square f i t of a l l available data to the molecular parameters of one isotopic species and the calculation of a l l the other isotopic species through the isotop for an isotope exchang partition functions i n two chemical species. For instance, for the protium-deuterium exchange between hydrogen and hydrogen iodide Κ
=
( Q
D
2
/
Q
H
2
)
/
( Q
D
I
/
Q
H
I
)
2
The experiments of Rittenberg and Urey (20) confirmed their c a l culations on this exchange reaction within the uncertainty of their calculations. Their calculation of the anharmonic cor rection to E|, l i k e many subsequent ones, i s incomplete. In the exchange reaction of protium and deuterium between hydrogen gas and water vapor the deuterium favors the water molecule. At 25° C Κ = (HD0/H 0) / (HD/H ) 2
2
=3.7
In the electrolysis of water deuterium i s depleted by a factor of 3 to 9 compared with the liquid water. The precise difference depends on the electrode materials and the conditions of elec t r o l y s i s . This fractionation i s kinetically controlled. Urey and Greiff (20) extended the treatment given by Urey and Rittenberg for diatomic molecules to polyatomic molecules. In the approximation of harmonic oscillators and r i g i d rotors they obtained
for the partition function ratio of polyatomic molecules. A, B, and C are the principal moments of inertia, M i s the molecular weight and U. = hv /kT. (The factor S /S i n Eq. (8) does not appear i n Urey and Greiff*s Eq. (13) since they 1
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1.
BiGELEiSEN
Quantum Mechanical
Foundations
7
restricted their calculations to isotopic molecules of the same symmetry, S = S ). Whereas, Urey and Rittenberg found values of the total partition function ratio of several diatomic deuterides to the corresponding hydrides corrected for nuclear spin and symmetry of the order of 10 at 298° K, Urey and Greiff found values between 1.25 and 1.35 for the isotopic partition function ratios of a number of compounds of lithium, carbon, oxygen, and sulfur at 298° K. The partition function ratios calculated by them were either independent of temperature, ^ L i / ^ L i , or decreased with temperature. A summary of some of their calculations i s given in Table I. The chemical isotope fractionation factor, α = (Ν./N ) /(N-/N ), , where N. i s the mole fraction of isotope ι ζ a ι ζ b la i i n compound of chemica (Qi/Qo) /(Q-i/Qo)^ molecule ι ζ a ι ζ ο equal number of exchangeable atoms. When a molecule contains more than one equivalent exchangeable atom, e.g., CO^, i t i s sufficient to apply the rule of the mean (22), which states that for isotopic disproportionation reactions, 0
0
f o r
nXY
n-z
Y' = ( n - z ) XY + zXY' ζ η η
AG°/RT = η In (S
/S
x y
in the factor of the ratio,
x y
n
, ) n-z
y
harmonie oscillator approximation. The fractionation for each atom exchanged i s then calculated by the use appropriate power of the reduced partition/function e.g., ( C 0 / C 0 p ' , (C (> ~ /C 0 - ) for the 18
l6
2
1
2
18
lb
1
3
exchange of one oxygen atom. The table of partition function ratios of Urey and Greiff predicts chemical fractionation factors for the isotopes of carbon and oxygen of the order of 1.0 - 1.1 at 25°C. Oxygen exchange between C0 and H ° ( ) predicted to enrich 0 i n CO over H«0 by a factor of 1.054 at 25°C. When such a fractionation i s cascaded i n a countercurrent column, i t provides the basis of a practical economic isotope separation process. The requirements necessary for such a process are reviewed by Spindel i n this monograph. Chemical exchange processes for the enrichment of ^ L i , ^ C , d 1 % were developed by Lewis and MacDonald (23), Hutchison, Stewart and Urey (24), and Thode and Urey (25), respectively. For the development of nuclear energy for military pur poses i n World War II, the Manhattan Project of the U.S. Army Engineers Corps required large quantities of D«0, highly en riched l^B and the fissionable isotope of uranium, u. The most d i f f i c u l t task was the production of kilograms of 90% U from the natural abundance of 0.7%. Many processes were i s
2
2
g
18
a n
2 3 5
2 3
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8
ISOTOPES AND CHEMICAL PRINCIPLES
Table I Calculation of Ratios of Distribution Functions for Light Isotopes H. C. Urey and L o t t i J. Greiff (1935) Ratio 7
Li/ Li
6
7
LiH/ LiH
6
1 8
1 6
c o/c o
18
H
2
l6
2
0/H
16 2
1 8
2
0
1 6
(s o /s o ) 2
l3
13
1 3
2
12
C/ C 12
CO/ CO 1 2
co / co 2
13
2
12
co -/ co = 3
3
3 5
298° Κ
600° Κ
1.2601
1.2601
1.2918
1.2707
1.3200
1.2403
1.3342
1.2400
1.2663
1.2231
1.3018
1.2263
1.3289
1.2033
1.2378
1.1686
1.3437
1.2020
1.3594
1.1987
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1. BiGELEiSEN
Quantum Mechanical
Foundations
9
considered and several different types of production plants were built (26). In the end those processes for the enrichment of D, B , and 235u developed at the SAM Laboratories of Columbia University under the direction of H. C. Urey proved to be the most economical and reliable of a l l the processes developed. Reduced Partition Function Ratio of Bigeleisen and Mayer I joined the SAM Laboratories i n June 1943. I was assigned to a small short term basic research program to investigate the spectra of uranium compounds. The immediate goal was to see whether there were differences in the spectra of 235u and XU
which could lead to a isotopes. A review of the program i n the f a l l of 1943 led to the conclusion that i t was unlikely that a photochemical separ ation process could be developed within the time schedules drawn up by the Manhattan D i s t r i c t . It was decided to phase out the photochemical separation program. We did assemble a signi ficant amount of spectroscopic data, much of which was analyzed by the late Maria Goeppert-Mayer. Under an administrative de cision last in-last out, I was l e f t at the end with closing out the program and the preparation of the f i n a l status report. One of the processes which had been considered early i n the OSRD program for the separation of the uranium isotopes was chemical exchange. It was dropped in favor of the calutron, centrifuges, gaseous and thermal diffusion. By the f a l l of 1943 a l l of these processes were i n trouble and Urey was led to re consider chemical exchange. He asked an assistant and former student, Dr. Isidor Kirschenbaum, to see whether I could calcu late any useful guidelines for chemical exchange separation processes for uranium isotopes from our spectroscopic data. Recall that the chemical isotope fractionation factor α i s the ratio of two partition function ratios. If we use the UreyGreiff approximation to the partition function ratio, Q-^^ Eq. (8), and eliminate the symmetry number factor, 3/2 then 1/2 the chemical exchange fractionation factor i s a product of ratios X α = 3/2 1/2 Î V l V 2 2 2 b A
ZPE ZP:
B
C
)
Exc(a) a
(9) Exc(b)
where the abbreviations ZPE and Exc stand for the zero point energy factor and Boltzmann excitation factors i n Eq. (8),
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
10
ISOTOPES AND CHEMICAL PRINCIPLES
respectively. For compounds of uranium we are dealing with a product of four numbers each of which differs from unity by the order of one per cent or less. The individual factors i n Eq.(9) could be greater or less than unity and the absolute value of lna i s not expected to be larger than 5 χ 10"^. For some of the compounds of interest, i t was not known whether A ^ ^ C ^ A ^ , ^ was unity. Maria Mayer and I had not yet established that UFg, . had the regular octahedral structure. Since some of the factors i n Eq. (9) are larger than unity and some are less than unity, i t was obvious to me that i t would be useful i f some basis could be found to calculate (Q /Q ),or a quantity related to i t , directly rather than the product of the ratio of four numbers. It was immediately obvious that the ZPE and Boltzmann factors i n lniQ^/Q^) could be combined through a 1
2
Taylor series expansion 1
e i-l
1
The following day Maria Mayer v i s i t e d me to find out how I was coming along with the summary of the spectroscopy program. I informed her that I had temporarily set that aside and was using what data we had to estimate chemical exchange factors. Maria Mayer knew a l l the d i f f i c u l t i e s i n the calculation of partition function ratios from incomplete spectroscopic data. She had been through i t i n connection with the heavy water program. She was excited by my approach and said that I could now finish the whole matter by taking out the c l a s s i c a l con tribution to QjQ * added (- ΐ/η,)άη to my Taylor series expansion, we would now have a function related to (Q-,/Q) and useful for the calculation of chemical isotope separation factors. We define I f
w
e
n
o
w
2
±
9
G(u ) - (1/2 1
- - i + - i ) i e ± -1
(10)
u
and a quantity (s/s')f (s/s')f =
±
G(u )Au + ... i
i
For most molecules the classical partition function i s an adequate approximation for the translation and rotation i n
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
(12)
1.
BiGELEisEN
Quantum Mechanical
Foundations
11
ideal gases. These lead to the factors M3/ and (ABC) respectively in Eq. (8). Only the molecular vibration i s quantum mechanical and contributes to ( s / s ) f . We shall show shortly that the reduced partition function, ( s / s ) f , provides an absolute scale for the calculation of isotope separation factors. Within the approximations that (Q ) = (Q ) (Q_ ) = (0 ) , and that the môleëular vibration i s trans'qm ^trans c l harmonic, then the reduced partition function i s 2
7
7
r
v
/
3n-6 (s/s )f= ïï ,
1
U
u. -A i
(u.'- u.)/2 , i % (LlLÊ ~ (1 - e " i ) Λ
1
e
λ
1
}
(13)
U
for ideal gases. The a set of vibrations. Th (Q/Q ) for a set of harmonic oscillators. Development of Eq.(13/ i n a power series in Au. = (u. - u^) gives Eq. (12). We next show that (Q/Q ) ^ ^epenas only on the symmetry number ratio and a mass factor independent of chemical composition. Write the Hamiltonian of the molecule i n Cartesian coordinates. Then the classical partition function 7
7
f
H
Q = (l/h s) J"../ e -
...
d p
(14)
d q
be integrated over the momenta to give 3n 3/2 -V(q)/kT Q = (l/h s)n (2nm kT/h ) J".-J" e V f
r
Z
cl
d
d
1
where η i s the number of atoms i n the molecule. ( Q / Q / )
cl
=
(
e /
e
n
m
/ ) (V i
q
3 n
( 1 5 )
Then
) 3 / 2
( 1 6 )
where m^ and m/ are the masses of the isotopic atoms. We now demonstrate that within the approximation of c l a s s i cal s t a t i s t i c s there i s no chemical isotope fractionation. The symmetry number ratio provides no mechanism for isotope concen tration. Rather i t i s an important s t a t i s t i c a l factor that cor rects for the a p r i o r i probabilities of forming symmetrical vs asymmetrical molecules of the same configuration. Consider the isotopes Y and Y' which form compounds of type A and B. Let there be isotopic exchange of Y and Y between A and B. Define β - (Y/Y ) and γ = (Y/Y ) where (Y/Y ) i s the atom ratio i n the species A. Then the isotope fractionation factor i s 7
7
7
7
A
α = β/γ For systems which obey the rule of the mean (22)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
12
(Q/Q') = (s'/s) g A
(Q/Q' )
A
= (s' / S ) Y
B
b
Then α = (s/s' ) (Q/Q' ) A
A
/((s/s') (Q/Q') B
B
If 3/2 (Q/Q') = (Q/Q') A
(Q/Q') =
Acl
= (s'/s)
(2.)
A
(Q/Q')
B
Then α = 1 Now consider the exchange reaction ΧΥ' + Y = χγ' γ + γ' η n-1 Let the chemical species A and Β be XY Since Y i s an atom, s = s^, «= 1 and
and Y respectively. n
fi
(Q/Q') = (Q/Q^Be! - (*/ιη') B
3/2 γ
Recall (s/s') f = (Q/ 0 A
A
Q
Aqm
= (s/s')
A
/(Q/Q')
(Q/Q')
Âqm
Acl
/
(Q/Q')
Bcl
= β/γ = α Thus the reduced partition function ratio, (s/s')f, i s just the chemical isotope fractionation factor of the chemical species against the gaseous atom. With the convention prime i s the light isotope i t i s easy to prove that In(s/s' )f i s always positive. This follows from the fact that u'^ > u^. We return now to Eq. (10) and examine the G(u) function. I t i s a monotonie positive function. Its value is 1/2 as u goes to i n f i n i t y and i t approaches zero as u goes to zero. At small u i t has the value u/12. For the case of small u, Eq. (11) becomes
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1.
BiGELEiSEN
Quantum Mechanical
ln(s/s')f where δ
2 u ±
* Σ
±
2
=
n àxx /12 ±
±
- u
i
2 .
Foundations
13
2
« Σ δ u /24
(17)
±
2
But E u .
2 = (ft/kT) Σ h
i i
where Σ η ^
i s the trace of the secular equation of the molecular vibra tions. The latter i s independent of the coordinate system. For convenience we choose Cartesian coordinates and obtain Ση,
= Σ — m
a.. i
- Σ — V ±
2
ϋ
m
where a.^ i s the sum of the three orthogonal Cartesian force constants equal to tû . \ o2 Ôy2 ô2 ; i i i operating on the potential energy for the molecular vibrations. We then obtain x
ln(s/s')f ~ 1/24
z
2
"Γ ) V
2
U
(18)
Maria Mayer prepared a summary of the above development in A p r i l 1944 (27) which was reviewed by Edward Teller at the request of H. C. Urey and M. Kilpatrick. Included i n the sum mary paper were some applications of equations (12) and (18) to the possible chemical separation of the uranium isotopes. Edward Teller recognized that in Eq. (18) we had generalized the Herzfeld-Teller theorem to the case of chemical equilibrium in polyatomic molecules. A lucid summary of this development and some of the research i t initiated i s summarized by Clyde A. Hutchison, J r . (29). Late in 1946 Maria Mayer and I were encouraged by W. F. Libby and H. C. Urey to prepare a summary of our work which could be published i n the open l i t erature (29). It was then that Libby called our attention to Waldmann's independent formulation of the reduced partition function ratio and the development of a mathematically equivalent form of Eq. (12) (30). The ln(s/s')f scale i s a positive scale. The value of ln(s/s' )f of a compound i s the maximum isotope fractionation that can be obtained i f the heavy isotope i s to concentrate in that compound. Detailed calculations show that values of ln(s/s' )f for a given isotopic substitution i n a wide range of compounds l i e within a narrow range at a given temperature. This i s one of the reasons why isotope separation i s d i f f i c u l t . In Table II we give some typical order of magnitude values of ln(s/s')f at 300°K. A more detailed systematic discussion of the ln(s/s' )f scale and i t s relationship to isotope fractiona tion factors i s given in London's book on isotope separation(31).
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
Table II Approximate Values of ln(s/s')f at 300° Κ for Isotopes as a Function of Atomic Weight
Isotopic Pair D/H
2.3
13
C/ C
12
80
Se/ Se
2 3 8
ln(s/s')f at 300° Κ
0.1
78
U/
2 3 5
U
0.01 0.002
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1
Quantum Mechanical
BiGELEisEN
Foundations
15
The simplifications and insight that were introduced into isotope chemistry by the reduced partition funcation formulation led H. C. Urey in 1946 to revise and extend the pioneer calcula tions he had made with Rittenberg and Greiff. With the assistance of L. S. Myers, Urey calculated values of f for the isotopes of hydrogen, lithium, boron, carbon, nitrogen, oxygen, chlorine and bromine in a variety of compounds of each of these elements. The results were presented i n the Liversidge lecture late i n 1946 (32). In the course of this work, Urey was led to conceive of the isotopic paleotemperature scale. The latter i s based on the temperature coefficient of the 0 / 0 fractiona tion between carbonate and water. The development of this method over the decades owes much to Epstein and has become an important method i n geochemical science. In his paper Urey makes comparison betwee stants with experimen For the calculation of the reduced partition function ratios of the isotopes of hydrogen, Urey found i t necessary to introduce an anharmonic correction to the zero point energy difference, which in some cases led to corrections to the exchange e q u i l i brium constants of the order of 5 - 10% at 300°K. The most extensive experimental data at the time were the exchange of tritium between hydrogen gas and water which covered the tem perature range 273-600°K (33). For these calculations Urey made use of the extensive calculations of the vibrational f r e quencies and anharmonic corrections in the water molecules carried out by Libby (34). The experimental values are about 2-5% larger than the calculated values. Comparable agreement is found for other exchange reactions involving other elements and compounds. The most important part of the anharmonic correction to isotope exchange equilibria i s in the zero point energy term. The f u l l theory of this correction was developed by Wolfsberg in 1967 (35). We shall outline here the theory for a diatomic molecule 1 8
H = 1/2 Q E/hc = G
2
2
+ 1/2 XQ
+ uu (n + 1/2) -
0
e
Β G
o "
m
α
4
+
12B
24 Β
3 3 r a
α
+
3 / 2
3
4
2
uu (η + 1/2) + ... e
/ α ou \ 2 . eel 1 12B I Β e / e 6 Β
2
) Q + (b/μ ) Q
A
χω e e
(20)
(21)
(22)
3
e
2
e
(19)
2 S.
he ω Β
+ (aAi
1 6
= h/8IÏ I c e
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
(23)
ISOTOPES AND CHEMICAL
16
PRINCIPLES
In Eq.'s (19 - 23), Q i s the normal coordinate and the other terms have the usual spectroscopic significance. The important new term introduced by Wolfsberg i s G . This correction may be as large as the 1/4 χ uu correction i n Eq. (20) and may be of the same or opposite sign. Wolfsberg has extended this method to the water and ammonia molecules. In Fig. II we reproduce the comparison of Bron and Wolfsberg s (36) calculation of exchange equilibrium D
1
^(g)
+
HD
(g)=
+ H
^ ( g ) 2(g)
with experimental data over the temperature range 200-450°K. The agreement between theory and experiment i s of the order of the precision of the experimental data which i 1% bette agreement i s obtained fo The results of Bottinga's (37) calculatio are compared wit the experimental data of Craig, Bottinga,O Neil, Adami, and Truesdale i n Fig. 3. The anharmonic correction i s smaller for oxygen isotope exchange than hydrogen isotope exchange because of the smaller amplitude of vibration. Bottinga's calculation does not include Wolfsberg s G correction. In summary, we see that Eq. (13) when corrected for non-classical rotation and anharmonic effects leads to isotopic exchange equilibrium con stants which agree within the order of one per cent of the experimental value of InK. f
1
Isotope Chemistry and Molecular Forces Much can be learned about the relationship between isotope chemistry and molecular structure and molecular forces by sys tematic calculations from Eq. (13). This can take the form of the study of real molecules or systematic variation i n the properties of real molecules, e.g., bond lengths, force con stants, substituents, etc., and determining the effect of such variations on ln(s/s' )f. This method has been employed to good effect by Wolfsberg and Stern (38) among others (39). An analytical approach to this central problem i n isotope chemistry was initiated by Bigeleisen and Mayer (29) and further developed by Bigeleisen (22, 40-46), most recently i n collaboration with Ishida and Spindel. The analytical approach consists i n the series expansion of In(s/s' f i n powers of x
E
ô
u
2
j
(
o
u
2
J
=
u
2 i
u
2 i
a
n
d
t
h
e
i i i ± ~ ± ) explicit relations between Σάα ^ and atomic masses and molecular force constants. The development over the years has involved extension of the range of v a l i d i t y of Eq. (18) f i r s t through alternate Taylor series expansions (40) and f i n a l l y through the use of f i n i t e orthogonal polynomial expansions (41, 42, 45). The other
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1. BIGELEISEN
Quantum Mechanical
Foundations
17
Journal of Chemical Physics Figure 1. Plot of the logarithm of the reduced partition function of ( Ne/ Ne) solid and liquid derived from vapor pressure data of Bigeleisen and Roth ( 8 ) 22
20
3.0
Journal of Chemical Physics Figure 2. Comparison of theoretical calculations (36) of the equilibrium isotope exchange reaction NH (g) + HD(g) = NH D(g) -j- H (g) with experimental 3
2
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
18
aspect of the development and numerical analysis of expansion of l n ( s / s ) f i n We now summarize the /
has involved physical interpretation the various terms i n the power series terms of ô u ^ J. analytical extensions of Eq. (18).
j+1 BOA . 1-ôu. "4 ln(s/s')f = Σ
23(23) 1
M-l)
* 2 Ou. i = 1<24 ~
i
<
( 2 4 )
^
1
-
> «i
2
< *
]
( 2 5 )
j+ V
a r e
u
+
η
1
[
. 4 . 6 ou. ou. ι , i 28§ô I8Î7440
Σ
'I
2j
0
J
_
1
)
t U i
'
< a > ]
( 2 6 >
t n e
where Β„· " Bernoulli numbers (B = 1/6, B« = 1/30, B = 1/çl, etc.) and W are f i n i t e othrogonal modulating coefficients. For convanience we w i l l use the abbreviation A = (-1)3+1 Β ^(23) (2$) I. The f i n i t e orthogonal modulating 1
coefficients are defined as (41, 42) L
T(n,j,R)
r
= ζα)Τ(η,ά,^ ) + Σ k=l +1
L
-,
k'2j
where P
P
η (-1) C T(n,j,u' ) = Σ —S p-j R p
η /
P
P
(-1) C _S. p=0 R
Σ
p
Here ^
2
- R /k = (u//2 w k ) x
2
(k - 1, 2, ···, L)
and z(j) i s the Riemann zeta function. Any value of the argument u'larger than the actual value w i l l insure convergence of expansion (2). For a polyatomic molecule the modulating coefficient should be chosen i n terms of the largest argument u ' , corresponding to the highest vibrational frequency, to ensure the convergence. The sums of the even
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1. BIGELEISEN
Quantum Mechanical
Foundations
19
powers of are obtained directly from the secular equation for the molecular vibrations: Eu. Σλ
±
2j
2J
- Σ(Λ / k T ) X
3
,
±
(27)
j
- Tr[H ] ,
Λ
(28)
H = F *G ,
(29)
|Η - λΐ J = 0
,
(30)
where F and G are the matrices, respectively discussion to the implications from Σολ. and Σ δ λ ^ , The sums Σδλ^ and Σ δ λ ^ are respectively 3n-6 Σ δλ. - Σ (μ. - μ.) a atoms 7
= 3η-6 Σ δλ./ =
Σ δ
Σ atoms
(μ
8
±
(31)
. ^
(31.)
) a
Σ
2(μ
)μ
3
±~ ~ ^~i i i atoms±'"^ι ι ϋ
( 3 2 )
atoms i *j g
" f
+
+
4
ii
2
f
ii
2
^
e 8
ô
(
g
2
+
«
i j
2
2 ( £
g
\f. i± 3j £
±;}
8 ) = κ1
+
£
Hi^ ±i f
2
ij >
+
)
i k
+ 24 Σ Σ Σ Σ δ(g
where δ ( 8
ôi
Λ
2
%f^±±H^±±%
4
+
£
i i V
,(f -f
8^ " 8^8^·
(32a)
The summations i n Eq. (32a)
run over a l l different values of the distinct subscripts as
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
20
ISOTOPES AND CHEMICAL PRINCIPLES
shown. In Equations (31) and (32), μ' and μ^ are reciprocal atomic masses of the light and heavy isotope respectively, a.. is a Cartesian force constant, g.. and f.. are the elements of the Wilson G and F matrices, respectively. The f i r s t order approximation of ln(s/s' )f through Eq.(26), j = 1, gives values of ln(s/s')f within 10% of the exact solu tion, Eq. (13), even for D/H isotope effects at 300° K. Eq.(26) is an even better approximation to In(s/s')f of heavier elements. Within this limitation, we can now derive the f i r s t order rules of isotope chemistry by substitution of Equations (31) and (31a) into Eq. (26). They are: 1 J
1) Isotope effects, ln(s/s')f, depend only on the masses of the isotopic atoms and the force constants bonding the atom at the sit atoms i n the molecule, 2) Isotope effects between different compounds occur only when there are force constant changes at the site of isotopic substitution. 3) Isotope effects are additive. a) isotopic additivity ,
ln(s/s )f(D
18 2
0/ H
l6 2
0 ) = ln(s/s' ) f ( D
16 2
+ ln(s/s')f(H
0/H
18 2
16 2
0/H
0)
16 2
0)
b) substituent additivity (47) ln(s/s')f(CHDFCl/CH^FCl) = ln(s/s' )f (CI^DF/CR^F) 7
+ In(s/s )f(CH DC1/CH C1) 2
3
- In(s/s' )f(CH D/CH ) 3
4
4) Isotope effects are cumulative ( f i r s t rule of the geometric mean) Inis/s')f(CH D /CH ) = 2 ln(s/s')f(CH D/CH ) 2
2
4
3
4
5) Equivalent isomers have the same isotope chemistry. 7
l n i s / s )f(C H D (0)/C H ) - ln(s/s' )f ( C ^ D ^ / C ^ ) 6
4
2
6
6
- ln(s/s')f(C H D (p)/C H ) For the complete calculation of ln(s/s')f directly from atomic masses and molecular structure, G matrices, and molecular forces, F matrices, i t i s necessary to relate the modulating co efficients, W,, to these quantities. Each W. i s a positive 6
4
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6
6
BIGELEISEN
1.
Quantum Mechanical
Foundations
21
number i n the range 0 < W. < 1. They have been tabulated as a function of u i n the range 0 < u < 25 (48). W. can be deter mined directly from the H = G' F matrix by the row sum-column sum method (42). ThrougK this method i t i s thus possible to calculate ln(s/s')f directly from the F and G matrices to any desired accuracy from Equations (26-28?. In Table III we summarize a correlation of exact values of ln(s/s')f at 300° Κ for C / C substitution in the two homolo gous series (1) CO, OCS, CO (2) Br CO, CI CO, F CO. The cor relation of l n ( s / s ) f with the sum of f i i Çtretch) plus f i x (bend) i s convincing evidence of the relationship between ln(s/s )f and force constants, as predicted from the simple one term expansion of Eq. (26). From Eq. (32) we which enter into the ter 2 2^i. ~ ^ i ^ i i order term 1 1 ^ ' " ^ -.*-î s t a t i s t i c a l mechanical correction term 11 corresponding to the harmonic motion of a mass (m^, πκ) connect ed by a force a^.X. to an i n f i n i t e mass. The second term, ii ι 2ν? Α (μ - l- )Mj j > kinetic energy coupling term which 7
13
12
7
7
W
A
W
A
2
2
a
a
e
/
2
2
2
l
ί
i s
I t :
a
i
,
i s
2
s i m l l a r
f o r m
t 0
t n e
f i r s t
a
i s
a
i
7
accounts for the fact that μ.ι i s not zero and the j th atom shares some of the kinetic energy in the vibration (43). It i s this term which explains why the isotope effect increases as the isotopic atom i s successively bonded to a heavy rather than a light atom, e.g., C / C i n C-C vs C-H for equal force constant. Thus the C-C stretching force constant, 5.09 mdyne A , contrib utes 0.035 to ln(s/s')f i n ethane at 300° Κ while each of the three C-H stretching force constants, 4.70 mdyne A " l , contrib utes but 0.016 to the same quantity. Up to terms in V^A^ou * we can write 13
12
o
7
W |A | - wjc*" 2
7
ln(s/s )f » E W ^ ^ i ^ i i i
1
2
[
(
^>i
)
a
ii
+
2
^ 2μ. . / . ]} 3±
3
(33)
ί
Inasmuch as the term in brackets i n Eq. (33) i s close to unity, Eq. (33) or the analogous one based on Equations (31a) and (32a) may be used to define effective bond force constants. Eq. (33) provides the theoretical basis for, and delimits the range of, applicability of Galimov's method for the calculation of l n ( s / s ) f from empirical bond values (49). Within the framework of the f i r s t order approximation to l n ( s / s ) f , the isotope fractionation factor lna = l n ( s / s ) f In(s/s' ) f always has the same sign, except for small d i f ferences which may arise from W^(A) - W (B) as a function of 7
7
7
A
R
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
22
ISOTOPES AND CHEMICAL PRINCIPLES
Table III 13
12
C / C Reduced Partition Function Ratio at 300° Κ f
Molecule
f
CO
18.5
OCS
16.1, 7.1
0.64(2)
0.1356
co
15.6(2)
0.77(2)
0.1729
2
± i
Stretch
b
±i
Bend
In (s/s' )f 0.0920
Br C0
13.6
ci co
13.7, 3.1(2)
0.94(2), 0.95
0.1391
F C0
15.1, 6.5(2)
1.18(2), 1.52
0.1886
2
2
2
See Reference (46). ^ Number i n parenthesis, e.g., (2), gives the number of equivalent force constants with g'^ φ g ^ .
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1.
BIGELEISEN
Quantum Mechanical
Foundations
23
temperature. Within this approximation isotope fractionation factors do not change sign with temperature. Yet, they do. The latter, which i s known as the cross over, i s principally due to the W A^EÔu * terms. To describe the cross over one must calculate In(s/s )f for each compound at least through the second order terms. A convincing demonstration of the power of the f i n i t e orthogonal polynomial method for the calculation and understanding of the reduced partition function ratio of isotopic molecules was given by the calculation made by Ishida,Spindel, and Bigeleisen (42) of the cross over e q u i l i brium i n the exchange reaction 7
7
16
18
18
16
F ON0 + N0 0 = F ON0 + N0 0 2
2
by a two term expansio the row sum-column sum method and Equations (31a) and (32a) gave Zôu? and Eôu/Î, respectively. A comparison with exact calculations from Eq. (13) i s given i n Fig. 4. The W A Eôu î term i n Eq. (26) allows one to predict the sign and magnituàe of the deviation from the rule of the mean in the harmonic approximation. From this term we can derive the theorem: i n an isotopic disproportionation reaction to form asymmetric molecules from symmetrical ones, the symmetry number corrected equilibrium constant i s always less than unity. Only the term 2(μ^ - μ ^ ) ^ · ^ In Eôu^ leads to deviations from the rule of the mean. ^Quantitative calculation of the deviations from the rule of the mean by Eq. (26) w i l l require at least the term j = 3. The corrections are proportional to μ.a... The largest deviations are expected for the equilibrium /
2
2
0
J
2
J
H
2
+ T
2
= 2 HT
for which the calculated value i s ln(K/4) = - 0.441 at 298° K. The corresponding value for 2 " 0.204. On the other hand the values for H
+
D
=
2 H D
i s
2
H
2°
+
D
2°
=
2 H D 0
are - 0.037 and - 0.038 in the harmonic and f u l l y corrected anharmonic approximations respectively (50). The experimental values of ln(K/4) are some 50 per cent larger than the theoret i c a l values, corresponding to a difference of 2.5 per cent between the experimental and theoretical values of K. This deviation may be real. The f i r s t order correction to In(K/4) for the isotopic disproportionation reaction with two equiv alent atoms i s , in the harmonic oscillator approximation, -W21 1A2' 1 (μ ^ i. - μ ^ i,/ ^)-μj ^i. j. For the H„2 molecule a., i j i s the stretch7
0
0
force constant.
Transformation from Cartesian to valence force
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
24
0 , 5
200
300
400 Τ ( Κ)
500
600
Β
Advances in Chemistry Series Figure 4. Exact and finite orthogonal polynomial ex pansion calculation of the equilibrium (42) F ON0 + N0 0 = F ON0 + NO O 16
18
18
2
2
w
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1.
BIGELEISEN
Quantum Mechanical
Foundations
25
constants shows that (μ' - μ.)μ^.. has a negligible contribu tion from the 0-H stretching force constant i n water. The deviation from the rule of the mean i n the case of water and other polyatomic molecules has i t s origin i n the bending force constants (41, 50). These theorems can be proven in detail from Eq. (32a). Since stretching force constants are approxi mately ten times those of bending force constants, i t i s appar ent that the large deviations from the f i r s t rule of the mean for the isotopic disproportionation of the hydrogen molecules are unique. The development of the f i n i t e orthogonal polynomial method provides the basis for the understanding of the total reduced partition function ratio, ln(s/s')f, of a pair of isotopic molecules and each of the quantum terms, (ft /kT)2l, of order j to ln(s/s')f. To thi have undertaken a systemati In(s/s' )f for isotopes of a number of elements i n a variety of compounds with the molecular force constants in the compounds. It i s of interest to compare the contributions of bending and stretching forces in a homologous series. This comparison i s made i n Table IV, taken from the recent work of Bigeleisen and Ishida (46), where the H/D isotope effects for the molecules CH^O, CH^, Ο^Η^, C H , and C^H^ are intercompared. We can read i l y see the one to one correspondence of the contribution of the C-H stretching force constant to ln(s/s')f with the magni tude of the C-H stretching force constant. In a l l cases the stretching force constant accounts for more than 65 per cent of the isotope effect even at room temperature. There i s also a close parallelism of the sum of the contributions of the bend ing force constants to ln(s/s')f with either the magnitude of the C-H stretching force constant or the sum of the bending force constants in a molecule are correlated. Small deviations from an exact linear relationship between force constants and contribution to In(s/s )f are associated with coupling effects, higher order terms, and geometrical effects i n the case of some bending coordinates. In recent years there has been significant progress i n the quantum mechanical calculation of energies and wave functions of polyatomic molecules. Atomic force constants are related i n a simple manner to molecular electron density functions (51). A significant beginning has been made in the calculation of atom i c force constants (52) and their application to isotope chem istry (53). The development of isotope chemistry has been closely con nected with the development of quantum chemistry and quantum s t a t i s t i c a l mechanics. In many cases isotope chemistry has provided important guides to the development of quantum s t a t i s t i c a l mechanics. In other bases i t has provided important and unique confirmations. We can confidently expect this pattern to continue into the future. 1
2
J
4
7
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1
b
2.353
Number of bending coordinates affected by the isotopic substitution.
2.312
k(p) means planar; (o-p) means out-of-plane; (t) means torsion.
a
2.200
2.321
2.364
Exact
2.309
2.132
Total Σ
2.231
0.789
0.132
0.202
0.355(1)
0.654
0.753
0.289
0.464(2)
(5.03)
1.675
0
C H, 2 4
Σ "bend"
0.631
0.631(3)
(4.70)
1.556
0
C H. 2 6
0.100
0.172
0.352
0.130(1)
(4.92)
1.600
CH. 4
HCX bend (t)
HCX bend (o-p)
HCX bend ( p )
HCH bend (a)
(4.31)
1.478
C-H Stretch
f(mdyne  ^ )
CH 0 2
Coordinate
0
Contributions of Stretching and Bending Forces to ln(s/s' )f(D/H) at 300° Κ
Table IV
2.395
2.470
0.838
0.266
0-572
(5.06)
1.632
C.H. 66
1. BIGELEISEN
Quantum Mechanical
Foundations
27
Literature Cited 1. Fajans, K.,Physik. Z. (1915) 16, 456. 2. Stern, O., (1914), see Keesom, W. H. and Van Dijk, H., Proc. Roy. Acad. Sci. Amsterdam (1931) 34, 42. 3. Lindemann, F. A. and Aston, F. W., Phil. Mag. (1919) 37,523. 4. Lindemann, F. Α., P h i l . Mag. (1919) 38, 173. 5. Urey, H. C., Brickwedde, F.G., and Murphy, G.M., Phys. Rev. (1931) 39, 164; (1932) 40, 1. 6. Keesom, W. H. and Haantjes, J., Physica (1935) 2, 986. 7. Roth, E. and Bigeleisen, J., J . Chem. Phys. (1960) 32, 612. 8. Bigeleisen, J. and Roth, E., J . Chem. Phys. (1961) 35, 68. 9. Lee, M. W., Fuks, S. and Bigeleisen, J., J. Chem. Phys. (1970) 53, 4066 10. P h i l l i p s , J. T., Linderstrom-Lang J. Chem. Phys. (1972 11. Lee, M. W., Eshelman, D. M. and Bigeleisen, J . , J. Chem. Phys. (1972) 56, 4585. 12. Mandel, F., J. Chem. Phys. (1972) 57, 3929. 13. Bigeleisen, J . , Lee, M. W. and Mandel, F., Ann. Rev. Phys. Chem. (1973) 24, 407. 14. Bigeleisen, J., Lee, M. W. and Mandel, F., Accounts of Chemical Research, i n press. 15. Herzfeld, K. F. and Teller, Ε., Phys. Rev. (1938) 54, 912. 16. Bigeleisen, J., J . Chem. Phys. (1961) 34, 1485; see also paper by Van Hook, W. A. i n this monograph. 17. Urey, H. C. and Rittenberg, D., J. Chem. Phys. (1933) 1, 137. 18. Giauque, W. F., J. Am. Chem. Soc. (1930) 52, 4808. 19. Herzberg, G., "Infra-red and Raman Spectra of Polyatomic Molecules", Van Nostrand, New York, (1945). 20. Rittenberg, D. and Urey, H. C., J. Am. Chem. Soc. (1934) 56, 1885. 21. Urey, H. C. and Greiff, L. J., J . Am. Chem. Soc. (1935), 57, 321. 22. Bigeleisen, J., J. Chem. Phys. (1955) 23, 2264. 23. Lewis, G. N. and MacDonald, R. T., J . Am. Chem. Soc. (1936) 58, 2519. 24. Hutchison, C. Α., Jr., Stewart, D. W. and Urey, H. C., J. Chem. Phys. (1940) 8, 532. 25. Thode, H. G. and Urey, H. C., J . Chem. Phys. (1939) 7, 34. 26. Smyth, H. D., "Atomic Energy for Military Purposes", Princeton University Press, Princeton, (1945). 27. Mayer, M. G., S.A.M. Report 4-M-159, April 8, 1944. See also reference No. 28. 28. Hutchison, C. Α., J r . , "Chemical Separation of the Uranium Isotopes", NNES, Div. 3, Vol. 3, Oak Ridge, (1952). 29. Bigeleisen, J . and Mayer, M. G., J. Chem. Phys. (1947) 15, 261.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
28
30. Waldmann, L., Naturwiss. (1943) 31, 205. 31. Bigeleisen, J., i n London, H., "Separation of Isotopes", Geo. Newnes, Ltd., London,(1961). 32. Urey, H. C., J . Chem. Soc. (1947) 562. 33. Black, J. F. and Taylor, H. S., J . Chem. Phys. (1943) 11, 395. 34. Libby, W. F., J . Chem. Phys. (1943) 11, 101. 35. Wolfsberg, Μ., "Advances i n Chemistry",(1969) 89, Am. Chem. Soc. 36. Bron, J. and Wolfsberg, M., J . Chem. Phys. (1972) 57, 2862. 37. Bottinga, Y., J . Phys. Chem. (1968) 72, 800. 38. Stern, M. J. and Wolfsberg, M., J . Chem. Phys. (1966) 45, 2618. 39. Hartshorn, S. R. and Shiner, V J., Jr., J Am Chem Soc (1972) 94, 9002. 40. Bigeleisen, J.,"Proc North Holland Publishing Company, Amsterdam, (1958),121. 41. Bigeleisen, J. and Ishida, T., J. Chem. Phys. (1968), 48, 1311. 42. Ishida, T., Spindel, W. and Bigeleisen, J . , Advan. Chem. Ser. (1969) 89, 192. 43. Bigeleisen, J., Ishida, T. and Spindel, W., Proc. Nat. Acad. Sci. (1970) 67, 113. 44. Bigeleisen, J., Ishida, T. and Spindel, W., J . Chem. Phys. (1971) 55, 5021. 45. Bigeleisen, J. and Ishida, T., J. Am. Chem. Soc. (1973) 95, 6155. 46. Bigeleisen, J . and Ishida, T., J. Chem. Phys., i n press. 47. The substituent additivity rule holds i n the approximation that off diagonal matrix elements, f (i ≠ j), are small. When isotope substitution i s at an end atom, e.g., C H D vs C H , only the interaction of adjacent bending ij
2
5
2
6
coordinates have non-zero values of δg , which reduces the above qualification on substituent additivity. NAPS document 01022, National Auxiliary Publication Service, c/o CCM Information Corp., 909 Third Ave., New York, New York 10022. Galimov, Ε. Α., Geochimica et Cosmochimica, i n press. Wolfsberg, M., Massa, A. A. and Pyper, J. W., J . Chem. Phys. (1970) 53, 3138. Anderson, A. B. and Parr, R. G., J . Chem. Phys. (1970) 53, 3375. Gaughen, R. R. and King, W. T., J . Chem. Phys. (1972) 57, 4530. King, W. T., J . Phys. Chem. (1973) 77, 2770. ij
48.
49. 50. 51. 52. 53.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2 Isotope Effects and Spectroscopy CHARLES P. NASH Department of Chemistry, University of California, Davis, Calif. 95616
Introduction One of the most valuable techniques available to any kind of spectroscopist i s that of isotopic substitution. This paper w i l l discuss a few applications of the method to problems i n the optical spectroscopy of atoms, diatomic and small polyatomic molecules, the vibrations of transition metal complexes, hydrogen bonding, solvation, and optical a c t i v i t y . We shall not treat either magnetic resonance or microwave spectroscopy. Any review which treats optical spectroscopy must acknowledge at its outset the magnificent series of volumes on this subject by Herzberg ( 1 , 2 , 3 , 4 ) . The textbooks by Walker and Straw (5), and King (6) also treat many of the subjects we s h a l l discuss at a somewhat less advanced l e v e l . Atomic Spectra In the domain of atomic spectroscopy, the f i r s t direct observation of deuterium was made in 1932 by Urey, Brickwedde, and Murphy (7), who observed weak satellites of four of the Balmer lines of hydrogen which were shifted to shorter wavelengths by amounts ranging from 1.79 Åfor H at 6536 Åto 1.12 α
for H
δ
at 4102
Å.
Å
Within experimental error the shifts were i n
exact agreement with the predictions of quantum mechanics for the effect of a mass " 2 " nucleus on the reduced mass of the atom. For multi-electron atoms, isotope effects are manifest not only i n the changes i n hyperfine structure arising from nuclear spin changes (1,8), but also i n small shifts i n the energies of s electrons which may be attributed to changes i n the nuclear dimensions (9)· Hindmarsh, Kuhn, and Ramsden (10) have attributed some of the irregularities found i n atomic isotope shifts to the closing of nuclear shells.
29
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
30
ISOTOPES AND CHEMICAL PRINCIPLES
Diatomic Molecule Spectra When two atoms are combined to form a diatomic molecule, isotope effects appear i n the vibrational, rotational, and electronic spectra. To the extent that the Born-Oppenheimer approximation i s v a l i d , the potential function for a given electronic state i s independent of the masses of the nuclei. The harmonic vibration frequencies of two isotopic variants of a diatomic molecule i n the same electronic state are then related by the equation
where ay i s the harmoni variant, and i s the μ = M^/^+Mg) Here
(2)
and Mg are the atomic masses of the atoms comprising the
molecule. One obvious source of differences i n the rotational spectra of isotopically-related diatomic molecules arises from the fact that the energy levels of the r i g i d rotor contain an explicit μ " dependence. Another effect also occurs i f one of the isotopic variants i s a homonuclear molecule and the other i s heteronuclear. 1
For a homonuclear diatomic molecule composed of even (odd) mass-number nuclei, the t o t a l wave function, which we assume to be a product of electronic, vibrational, rotational, and nuclearspin functions, must be symmetric (antisymmetric). I f the electronic wave function i s symmetric, and i f the nuclear spin is zero, as i n the ground state of 0 , only even values of J , the rotational quantum number, are allowed. I f the nuclear spin is not zero, both even and odd values of J (i.e., symmetric and antisymmetric rotational wave functions) are allowed, but with different s t a t i s t i c a l weights. These may be determined from the nuclear-spin part of the wave function. 1 6
2
For a nuclide of spin s the nuclear-spin degeneracy i s n
g =(2s +l). n
n
For the molecule a t o t a l of +
nuclear-spin 2
functions are possible, of which S ( g l ) / are symmetric (ortho), n
n
and g ( g - l ) / 2 are antisymmetric (para). n
n
The ortho (or para)
nuclear spin functions then combine exclusively with either even-J or odd-J rotational states to produce overall wave functions which have the proper symmetry.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2.
NASH
Isotope Effects and
Spectroscopy
31
For H (s =|) the t o t a l wave function must be antisymmetric, the odd-J levels combine with the ortho spin-funetions, and the s t a t i s t i c a l odd-J:even-J ratio i s 3 : 1 . For D or N (s =l) the t o t a l wave function must be symmetric. The even-J levels now combine with the ortho spin-states, and the s t a t i s t i c a l even-J:odd-J ratio i s 6:3=2:1. For heteronuclear diatomic molecules the allowable values of J are not subject to symmetry constraints. 2
n
1 4
2
2
n
In accordance with these considerations, the pure-rotational Raman spectrum (selection rule AJ=±2) of 0 has every second line missing, whereas that of N has a l l lines present, but those arising from even-J states are more intense than those arising from odd-J states (g). Yoshino and Bernstein ( l l ) have observed intensity alternations having s t a t i s t i c a l origins i n both the pure-rotationa rotational fine-structur vibrational band i n the Raman spectra of both H and D . 1 6
2
1 4
2
2
2
If a high-resolution infrared spectrophotometer i s available to them, undergraduates can obtain and analyze i n detail the fundamental vibration-rotation spectra of the four common isotopic variants of hydrogen chloride (H,D, Cl, Cl) ( 12,12.). For these molecules the potential curve i s significantly anharmonic, and a dependence of the rotational constant on the vibrational quantum number i s immediately apparent, since the rotational lines are not evenly spaced. From their data our students have confirmed the constancy of the equilibrium internuclear distances i n these four molecules to within ±0,001 Â , and they have confirmed the v a l i d i t y of Eq. 1 to four decimal places. It must be emphasized that when band-origins rather than harmonic frequencies are used i n Eq. 1, disagreements occur i n the second decimal place. Extensive comparison data are available i n the literature for a l l these molecules ( l 4 , 1 5 , l 6 , 1 7 » l 8 ) . 35
37
It i s of some interest to note here also that Connes et a l . (19) have detected 8 lines of the R-branch of the Δν=2 vibrationrotation band of both H C 1 and H C 1 at about 5700 cm" i n the spectrum of Venus. The agreement between the astronomical data and laboratory data was within ±0.01 cm*" for a l l lines. 3y comparing the relative intensities of the Venusian HC1 and C 0 spectra they estimated a p a r t i a l pressure of HCl^lO" torr, and from the relative intensities of the rotational lines they estimated a rotational temperature of 2^0 K. 35
1
37
1
2
4
The electronic spectra of isotopically varied diatomic molecules reflect the effects of changes i n the vibrational and rotational energy levels of both the ground and the excited electronic states. This subject i s of great h i s t o r i c a l importance, since the isotopes 0 , 0 , C , and N were a l l identified in 1929 on the basis of weak bands i n the electronic spectra of diatomic molecules. 1 8
1 7
13
15
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
32
Giauque and Johnston ( 2 0 , 2 l ) identified a weak band i n the solar spectrum of oxygen near 7595 Â as the (θ,θ) band of the b-»X transition of 0 0 on the basis of an isotopic shift of the bank origin of - 2 . 0 7 cm" vs. the band origin of the (θ,θ) band of 0 , as well as the fact that the band envelope of the homonuclear molecule contained 13 lines whereas that of the heteronuclear molecule contained 26. 1 8
1 6
1
1 6
2
There then followed the identification of c C i n the Swan bands of C by King and Birge ( 2 2 ) : 0 0 i n the solar spectrum by Giauque and Johnston ( 2 £ . 2 4 ) 7 ^ N 0 , N 0 , and N 0 by Naudé ( 2 5 ) ; and C N aiiaP- C 0 by King and Birge ( 2 6 ) . 1 3
1 7
1 2
1 6
2
é
1 3
1 4
5
16
1 4
1 8
1 4
1 7
l6
The Spectra of Small Polyatomic Molecules Isotopic substitutio solution of an enormous of small polyatomic molecules. One such problem i s the structure of the ethane molecule ( 2 7 ) . The eclipsed form of this molecule, having symmetry D , would have three Raman-active vibrationrotation bands, designated v i o v u and v i , for which the selection rule ΔΚ=±1 would predict only one series of Q-branches. The corresponding vibration-rotation bands for the staggered conformation, having symmetry D ^, would obey the selection rules 3n
5
2
3
ΔΚ=±1,±2. Thus two sets of Q-branches would be observed. The CH stretching-region of the C He spectrum contains a complicated admixture of the lines of the νιο fundamental together with those of the v i fundamental and three other combination bands. The νιο fundamental of C D , however, occurs free from any other inter ferences. Two well-defined sets of Q-branches were observed, and hence the staggered structure of ethane was confirmed beyond question. 2
2
6
If one wishes to determine the vibrational force constants for a polyatomic molecule, isotopic substitutions are essential. The general valence force f i e l d for a non-linear polyatomic molecule, expressed i n internal coordinates, contains a t o t a l of -|(3N-6)(3N-5) force constants. Some of these may equal others by the symmetry of the molecule, but a maximum of only (3N-6) vibration frequencies may be observed. The well-known Wilson GF-matrix method (28,29) allows one to break down the (3N-6)x(3N-6T~secular determinant for the molecule into smaller blocks, the number and size of which depend on the symmetry of the molecule i n question. Any n x n block of the block-diagonal secular equation obtained by Wilson's method yields, of i t s e l f , an n x n secular equation of the form |GF - A | = 0
(3)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2. NASH
33
Isotope Effects and Spectroscopy
The G matrix elements are composed only of structural parameters of the molecule (its masses, bond distances, and bond angles), while the F matrix elements, which are called symmetry force constants, are usually linear combinations of the general valence force constants. The roots, λ^, of the secular equation, Eq. 3 , are related to the harmonic vibration frequencies of the η normal modes of that symmetry type by χ
2
±
= ^TT ^.
2
( ) t
1
where the
are expressed i n cm" and c i s the speed of l i g h t .
Except when n=l, the number of independent F matrix elements, §η(η«ϋ), exceeds the number of observable frequencies n Even when n=l the F matrix general force constants If one or more isotopic substitutions are performed on the molecule, new frequencies w i l l be obtained, as well as new G matrix elements. Within the Born-Oppenheimer approximation, however, the F matrix elements w i l l transfer intact to the new molecule. The amount of new information which can be obtained about the elements i n the F matrix i n this way i s , however, limited by several isotope rules which the sets of harmonic frequencies of each symmetry type must obey. One of these, the form of the Teller-Redlich product rule which applies to two isotopic variants having the same molecular symmetry, may be deduced immediately from the secular equation i t s e l f . When the n x n secular determinant i s expanded i n polynomial form, the constant term, which must be equal to the product of the roots, η ΤΤ λ., i s simply the determinantal product |G|·jF|. Thus, i f i=l we designate with superscript j s and k s the properties of two isotopic variants having the same molecular symmetry, we find 1
T
η j ΤΤ λ, i-Ll _ ]£L_ n k , k, ΤΤλ. > i=l
!
TT u>? i=l 1
(5)
G1
1
i=l
In addition to the product rule, there are also sum rules which further r e s t r i c t the number of independent observables when more than two isotopic variants are available. This subject has been discussed i n d e t a i l by Heicklen (^O). One interesting conclusion he reaches i s that for molecules with symmetry the f u l l F matrix can be determined by substituting for a l l but one of the sets of equivalent atoms. In principle, then, CH4 and CH4, C H and C D , or C 6 H and C6H , should suffice to 12
13
12
6
6
6
6
13
6
6
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
34
determine the force-fields for methane or benzene. In practice, however, such a minimum set of data i s rarely enough. For methylamine, H CM , D CMD , D CNH and H G WH have a l l been studied, but almost half the elements of the general forceconstant matrix are s t i l l undetermined (31). Hirakawa, Tsuboi, and Shimanouchi ( j l ) point out that C substitution would be very helpful i n analyzing this particular molecule. 15
3
2
3
2
3
25
3
2
1 3
The vibration frequencies of a molecule are not the only sources of information about the elements of the symmetry forceconstant matrix. In the vibration-rotation spectra of polyatomic molecules, especially those with degenerate vibrational modes, e.g., symmetric tops, there occur certain anomalies i n the spacings of the rotational lines which may be attributed to the Coriolis interaction ( 3 2 ) . This phenomenon, which i s a coupling between the rotational angula i t s vibrational angular coupling constants called zeta constants. The ζ-constants are directly related t o the F and G matrices for the molecule i n question. The ζ-constants for a given molecule must have a sum which may be inferred from the molecular structure, and for a l l symmetric tops, except those which belong to the S 4 and D a point groups, there i s also an isotope rule which they must obey, namely 2
(6)
2
mEcj ( l - f . ) = constant i i 1
Here m i s the mass of the off-axis atom, CD^ i s the harmonic vibration frequency of the i - t h degenerate normal mode, and i s the Coriolis constant for that mode. The 'summation extends over a l l the normal modes belonging to a given degenerate species. The work of Aldous and Mills (33»3^) on the methyl halides provides an excellent example of the determination of force constants. For one of these molecules there are six independent vibrations — t h r e e of class A and three of class E. There are 16 different force constants i n the general valence force f i e l d expressed i n internal coordinates, and 12 force constants i n the symmetrized F matrix, a l l but two of which are linear combinations of the original 16. For CH X and CD X together there are 12 observed frequencies, but the product rule states that within each class only five of the six frequencies are independent observables. For the Ε-vibrations of either CH X or CD X three ζ-constants can be measured. However each molecule must obey a ζ-sum rule, and together they must obey the isotope rule, so that only three of the six ζ-constants are truly independent. ±
3
3
3
3
In addition to the vibration frequencies and the zeta constants, Aldous and Mills included i n their determination of the force f i e l d the data then available on the centrifugal-
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2. NASH
Isotope Effects and Spectroscopy
35
distortion constants, two of which are possible for one of these molecules. These constants are related to derivatives of the moment-of-inertia tensor and the elements of the inverse of the F matrix for both the A and the Ε vibrations (35^,36) · Thus, up To 22 experimental values for each isotopic pair were used by Aldous and Mills to determine the 12 elements of the F matrices for a l l the methyl halides, together with estimates 6? their probable errors. 1
The effects of isotopic substitution on the vibrational spectra of small polyatomic molecules have resulted i n two recent astrophysical observations of considerable interest. The f i r s t extra-terrestrial detection of deuterium was reported i n 1972 by Beer et a l . (37) , who found 11 lines of the P-branch of the 2200 cm" vibration-rotation band of C H Q D i n the spectrum of Jupiter* More recentl intensities of these the atmosphere of Jupiter i s between l / 2 and l / 6 the t e r r e s t r i a l value. 1
In I969 Young (39) published a high-resolution spectrum (which had actually been obtained earlier by Connes' group) showing a very complete vibration-rotation band comprising the 2v transition of ^C^O^O centered at k508 cm" i n the spectrum of Venus. This molecule, unknown i n the laboratory, could be detected i n spite of the abundance of C0 i n the earth's atmosphere, because for a l l the symmetrical isotopes of carbon dioxide (symmetry D , ) this overtone i s symmetry forbidden. On 1
3
2
00
π
the basis of a painstaking analysis of the intensities of the rotational lines Young deduced a .rotational temperature of 2^5+3 K, i n excellent agreement with the value of 2k0 Κ obtained by Connes e_fc a l . (l£) from the spectra of the HC1 species. While i t i s common knowledge that isotopic substitution alters the vibration frequencies of normal modes, i t may be less well known that the intensity of the corresponding infrared absorption band must be affected as well. Crawford (ho) has shown that within the harmonic approximation the integrated intensity of an infrared absorption band, Γ . , i s given by
band Here η i s the concentration of the absorber, I i s the path length, I and I are the incident and transmitted intensities respectively, i s the harmonic frequency of the i ' t h absorption band, N. i s Avogadro's number, c i s the speed of 0
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
36
ISOTOPES AND CHEMICAL PRINCIPLES
l i g h t , *ft i s the dipole moment of the molecule, and i s the i t h normal coordinate. Now μ i s invariant to isotopic sub stitution, but both ω. and are mas s-dependent, and hence the intensity of the band must be mass-dependent as well. f
Crawford (4l) has also shown that the integrated intensities of the infrared absorption bands of isotopically related molecules must obey two sum-rules, one of which, the socalled F-sum rule, may be written Σ Γ . / V · = constant. Here the i summation extends over a l l vibrations of the same symmetry class. Dickson, M i l l s , and Crawford (42) have made an extensive study of the vibrational intensities of the proto and deuteromethyl chlorides, bromides, and iodides. Among other things, they found that the A i vibrations of the methyl bromides and methyl iodides showed excellen but the agreement for th 1
1
In his excellent recent monograph based on his Baker lectures, Herzberg (43) describes the way i n which isotopic studies contributed to the identification of two transient species. In the spectrum of a comet, a band system was found near 4050 a which Herzberg, i n 1942, attributed to the CHg radical. He also found the same bands i n the spectrum of a discharge through methane. In 1949, however, Monfils and Rosen (44) found the identical spectrum from a discharge through CD , and hence the entity responsible for i t could not have been CHg. In 1951 Douglas (45) reported the spectrum of a discharge through an equimolar mixture of ^CE^ and ^CH*. There appeared six bands, and hence the species i n question must have contained three carbon atoms. In 1954 Clusius and Douglas (46) reported the spectrum of a discharge through pure ^CH^. They found an intensity alternation i n the rotational lines having a 3:1 ratio, and they also inferred that i n the spectrum of the ^CH^ discharge every-other rotational line was missing. This kind of behavior, analogous to that cited earlier for diatomic "molecules, is diagnostic for a linear triatomic molecule. Thus the species responsible for the 4050 a band was identified as linear C · 4
3
Herzberg (43_) then describes the actual discovery of the C H 2 radical, which was produced by the flash photolysis of diazomethane. The absorption band, which appeared at 1415 A> shifted when CD^lUg rather than C R 2 N 2 was photolyzed. In sub sequent experiments the photolysis of ^ClfeNg confirmed the presence of a carbon atom i n the radical. Very recently Katayama, Huffman, and 0 Bryan (47) have studied the absorption and photo ionization spectra of several isotopic water molecules i n the vacuum ultraviolet. As part of this investigation they used the spectra of H 2 0 and H 2 0 to establish that the f i r s t electronic excited state of HgO" i s linear. 1
1 6
1 8
4
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2.
NASH
Isotope Effects and
Spectroscopy
37
The Vibrational Spectra of Metal Complexes. In recent years Nakamoto (48) has pioneered i n the use of isotopes of the transition metals i n order to make assignments of the vibrational bands of their complexes. 3y studying the spectra of C r ( a c a c ) and C r ( a c a c ) , Nakamoto, Udovich, and Takemoto (4£) were able to assign a band at 460 cm" to a Cr-0 stretching mode and one at 592 cm" to an out-of-plane ring mode. On the basis of 0 -» 0 isotopic substitution, the 592 cm" band had previously been misassigned as the Cr-0 stretch (50). 50
53
3
3
1
1
1 6
1 8
1
For tetrahedral XY species the t o t a l l y symmetric A i stretching mode involves no motion of the central atom, and hence should y i e l d v i r t u a l l y no isotope shift when that atom i s substituted. The triply-degenerat display an isotope effect able to assign bands i n the Raman spectra of Zn(NE ) atT^O cm" and 410 cm" to the Αχ and F modes respectively. In this instance the isotopes Z n and Z n were used. The ordering of these two levels i n this complex i s somewhat unexpected, since in the vast majority of the X E ^ tetrahalogeno, or X0 species which have been examined the F band occurs at the higher frequency (52). 4
3
1
4
1
2
6 4
6 8
4
2
As a f i n a l example of the use of isotopic substitution i n the study of metal complexes, we cite the use of N0 as a ligand by Collman, Farnham, and Dolcetti (53), who found what they termed "hybridization tautomerism" i n several cobaltn i t r o s y l complexes. From their infrared spectra they inferred a rapid equilibrium between a trigonal-bipyramidal Co(l) species having a linear Co-nitrosy1 geometry, and a square-pyramidal C o ( l l l ) species, i n which the Co-nitrosyl moiety i s bent. l5
Hydrogen Bonding Studies. Much current interest i n the spectroscopy of hydrogen bonded systems attaches to the question of how one might infer the shape of the potential function from the vibrational spectrum of the entity. In this connection Wood and his collaborators have recently made major contributions. They have examined the infrared and Raman spectra of a great number of cations of the form (BFB ) , where Β and B are nitrogen bases or perdeutero-nitrogen bases, and Ρ i s either hydrogen or deuterium. 7
+
7
When Β and B were both trimethy lamine s (54) the NET" stretching band and the ND stretching band were both singlets. The same behavior obtained also when Β and Β were trimethylamine and pyridine (55)· When B=B =pyridine (56), or sub stituted pyridines (57) , and B=H, the NH"" band was s p l i t into a 7
1
+
7
7
1
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
38
ISOTOPES AND CHEMICAL PRINCIPLES
+
broad doublet. The KD stretch, however, was a singlet. Also in these ions the Ν — Ν stretching motion, which was observed i n the infrared spectrum, showed only a small frequency shift when the bridge was deuterated. 7
When Bj^B , but both bases were various pyridines or quinoline, the NIT" band could be characterized as a doublet with an intensity ratio of the components which varied between 1.2 and zero as the pK^ difference of the bases varied between zero and eight (58). The Ν—Ν stretching mode i n these unsymmetrical ions was not substantially more intense than that found when B=B'. Another remarkable feature of the spectra of this set of unsymmetrical ions was the appearance of a new band in the 500-600 cm" region whose frequency increased when the bridge was deuterium substituted 1
1
Wood (59) analyze paper i n which may be found, at least schematically, the spectral consequences (including deuterium isotope effects) to be expected for linear or bent Η-bonded systems having potential functions with single minima, or double minima with either low or high barriers. He interprets those of his (BPB') systems which show a s p l i t t i n g of the ΜΓ " band as having low-barrier double-minimum potential functions. Wood also shows that the anomolous frequency shift of the 550 cm" band on deuteration can be explained on the basis of a well-to-well proton transition occurring i n a low-barrier double-minimum potential which i s markedly asymmetric. +
4
1
Laane (60) has made calculations which show that for doubleminimum, symmetric, but possibly anharmonic, potentials the frequency ratio ω^/ω^ varies depending on the relative sizes of the height of the barrier and the energy of the transition. For a high barrier the frequency ratio i s about l.k. As the barrier height decreases the ratio f a l l s , attaining a minimum value of about 1.2 when the upper level i s near the top of the barrier. With a further decrease i n the barrier the ratio increases again, passing through a maximum value of about 1.6 for a f l a t bottomed well. For a s ingle-minimum harmonic potential the ratio becomes l.k again. On the basis of his calculations Laane has questioned the frequency assignments which Berney ejt a l . . ( 6 l ) have made for acetic acid. These authors have correlated two bands for which ω^/ω^ = 0 . 9 3 . "Very recently Bournay and Maréchal (62) have studied the integrated intensities of the infrared absorption bands of the dimers of acetic acid and acetic a c i d - d For these molecules they find that the intensity of the band attributed to the OH—0 antisymmetric stretching mode i s twice that of the corresponding 0D—0 band, whereas the intensity ratio i n the le
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2. NASH
Isotope Effects and Spectroscopy
39
harmonic approximation should be more nearly /" 2. In their discussion of this anomolous result they conclude that anharmonicity i s not a dominant factor. Rather, they suggest that i n this hydrogen bonded system the Born-Oppenheimer approximation i s invalid; i.e., they propose that the motion of the H atom along the bond induces an electronic transition i n the molecule, and the vibrational spectra have borrowed some intensity from that electronic transition. Ionic Solvation. In recent years isotopic substitution has been used to identify the vibrational modes of cation solvates i n both aqueous and nonaqueous media. Da S i l v e i r a , Marques, and Marques (63 ) obtained the Rama and A l i n both H 0 two depolarized lines i n each case, which they attributed to the Aig, Eg, and F g vibrational modes of octahedral solvates. The 3 +
2
2
frequencies of the A and E vibrations were decreased by a factor of•1.04-1.05 when D 0 was used as the solvent, whereas that of the F vibration decreased by a factor of 1.08. These values are those which would be expected for species having octahedral symmetry. l g
g
2
2 g
In our own laboratory we have obtained the Raman spectra of aqueous solutions of L i C l and L i C l (64). In addition to three depolarized librational bands of water at 720 cm" , 585 cm" , and 462 cm" , we find two other bands i n the low frequency region. One of these i s a polarized band at 420 cm" , independent of the lithium isotope used, which the other i s a depolarized band which occurs at 355 cm" i n the L i C l solutions or 385 cm- i n the L i C l solutions. We interpret these bands as the Αχ and F modes of vibration of L i tetrahedrally solvated by water molecules. Singh and Rock (65) had earlier noted that i f the lithium ion were tetrahedrally solvated, and i f the F vibration frequencies of the solvates were 384 cm" and 358 cm- for Li(0H )J and Li(CE )î respectively, the experimental value of the equilibrium constant of 1.046 for the exchange reaction L i ( s J + LiCl(aq) = L i ( s ) + LiCl(aq) could be explained. 6
7
1
1
1
1
1
1
7
6
+
2
1
2
1
6
7
2
2
7
e
6
7
In an extensive series of papers Popov and his collaborators have studied the solvation of a l k a l i metal cations i n various nonaqueous media, using N a magnetic resonance shifts ( 6 6 ) , and far-infrared spectroscopy. The ion-solvent vibrational bands which they find f a l l i n the 100-500 cm" range. 23
1
In dimethylsulfoxide the bands were identified by using DMS0-d as solvent (6j). In acetone (68) both L i -» L i and acetone-d substitutions were used. In this solvent the frequencies of the "lithium" vibrations showed an anion 7
+
6
6
6
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
+
40
ISOTOPES AND CHEMICAL PRINCIPLES
dependence, indicative of ion pairing, when the mole ratio of acetone to lithium was less than four. In pyridine and substituted pyridines (6°;) lithium isotopes were again used. In these solvents also, lithium halide contact ion pairing was inferred. Concurrently with Popov's work. Roche and Huong (70) reported the cage vibrations of Ca , Mg , and L i i n acetonit r i l e . These authors employed both L i -> L i and CH CN-»CD CN substitutions to identify the absorption bands of the solvates. 2+
7
+
6
+
+
3
3
Optical Activity Induced by Isotopic Substitution. The f i r s t example of optical activity i n a compound of the type R R CHD was reported i n I9A9 by Alexander and Pinkus (71). They measured a specific rotation for 2,3-dideutero-transmenthane of [ a ] = -O.O9 X
2
2 5
the origin of the optical activity, owing to the presence of four asymmetric carbons i n the molecule. In the same year E l i e l (72) reported the synthesis of ethylbenzene with one deuterium i n the a position. The optical activity of this compound ( [ a ] = -O.3O ) clearly derives from 2 5
0
the asymmetry between D and H. Subsequently Streitweiser and Wolfe characterized a number of other benzyl-a-d-derivatives. In I963 S t i r l i n g (jh) reported the preparation of (-)-benzyl-p-tolyl-[ 0i 0]-sulfone, whose rotation ([Œ]^ =-O.I6 ) 16
0
Q
arises from the presence of two different oxygen isotopes bonded to sulfur. A number of other 0 0 sulfones have since been prepared (75,76). 1 6
1 8
Anderson, Colonna, and S t i r l i n g (77) have recently reported (R)-dibenzyl-[ CH CH ]-sulfoxide. In this compound the optical activity ([α] βο =+0.71) originates from having two carbon isotopes bonded to sulfur. 12
13
2
2
2
There have also appeared very recently the studies of Kokke and Oosterhoff ( 7 8 , 7 2 ) . These authors have prepared (lR) - [ 2 - 0 ] -α-fenchocamphoronequinone and (lR) - [ 1 - D ] -afenchocamphoronequinone. In these molecules thes sole source of asymmetry was either an 0 i n the of-diketone function, or a single deuterium atom at a bridgehead. The circular dichroism spectra of these two compounds i n the visible are remarkably different. 1 8
1
8
Literature Cited. 1. 2.
Herzberg, G., "Atomic Spectra and Atomic Structure," Dover Publications, New York, 1944. Herzberg, G., "Spectra of Diatomic Molecules," 2nd ed., Van Nostrand Reinhold, New York, 1950.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2.
NASH
3. 4. 5. 6. 7. 8. 9.
Isotope Effects and Spectroscopy
Herzberg, G., "Infrared and Raman Spectra of Polyatomic Molecules," Van Nostrand Reinhold, New York, 1945. Herzberg, G., "Electronic Spectra and Electronic Structure of Polyatomic Molecules," Van Nostrand Reinhold, New York, 1966. Walker, S. and Straw, Η., "Spectroscopy," Vol. I I , Chapman and Hall Ltd., London, 1962. King, G. W., "Spectroscopy and Molecular Structure," Holt, Rinehart and Winston, Inc., New York, 1964. Urey, H. C., Brickwedde, F. G., and Murphy, G. Μ., Phys. Rev. (1932), 40, 1. Candler, C., "Atomic Spectra," Ch. 19, Van Nostrand Reinhold, New York, 1964. Wilets, L., Hill, D. L., and Ford, K. W., Phys. Rev. (1953), 91,
10. 11. 12.
13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
41
1488.
Hindmarsh, W. R. Phys. Soc. (London) (1954), A67, 478. Yoshino, T. and Bernstein, Η. J., J . Mol. Spectry. (1958), 2, 213. Daniels, F., Williams, J . W., Bender, P., Alberty, R. A., Cornwell, C. D., and Harriman, J. Ε., "Experimental Physical Chemistry," 7 t h ed., pp. 247-256, McGraw-Hill Inc., New York, 1970. Shoemaker, D. P., Garland, C. W., and Steinfeld, J. L., "Experiments i n Physical Chemistry," 3rd ed., pp. 450-459, McGraw-Hill Inc., New York, 1974. Levy, Α., Rossi, I., and Haeusler, C., J . Physique (1966), 27, 526. Levy, Α., Rossi, I., J o f f r i n , C., and Van Thanh, N., J . Chim. Physique (1965), 6 2 , 601. Rank, D. Η., Eastman, D. P., Rao, B. S., and Wiggins, Τ. A., J. Opt. Soc. Am. (1962), 52, 1. Van Horne, Β. Η. and Hause, C. D., J . Chem. Phys. (1956), 25, 56. Pickworth, J . and Thompson, H. W., Proc. Roy. Soc. (London) (1953), 218, 3 7 . Connes, P., Connes, J., Benedict, W. S., and Kaplan, L. D., Astrophys. J . (1967), 147, 1230. Giauque, W. F. and Johnston, H. S., Nature (1929), 123, 318. Giauque, W. F. and Johnston, H. S., J . Amer. Chem. Soc. (1929), 5 1 , 1436. King, A. S. and Birge, R. T., Nature (1929), 124, 127. Giauque, W. F. and Johnston, H. S., Nature (1929), 123, 831. Giauque, W. F. and Johnston, H. S., J . Amer. Chem. Soc. (1929), 51, 3528. Naudé, S. M., Phys. Rev. (1929), 34, 1499. King, A. S. and Birge, R. T., Astrophys. J . (1930), 72, 19. Weber, Α., i n Anderson, A., Ed., "The Raman Effect," Vol. I I , pp. 637-640, Marcel Dekker, Inc., New York, 1973.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
42
28. 29.
30. 31. 32. 33. 34. 35. 36. 37· 38. 39. 40. 41. 42. 43.
44. 45. 46. 47. 48. 49. 50. 51. 52. 53· 54. 55. 56.
Wilson, Ε. Β., Decius, J. C., and Cross, P. C., "Molecular Vibrations," McGraw-Hill Inc., New York, 1955. A detailed application of the method to the chloroform molecule has been given by Colthup, Ν. B., Daly, L. H., and Wiberley, S. Ε., "Introduction to Infrared and Raman Spectroscopy," Ch. 14, Academic Press Inc., New York 1964. Heicklen, J., J . Chem. Phys. (1962), 36, 721. Hirakawa, A. Y., Tsuboi, Μ., and Shimanouchi, T., J. Chem. Phys. (1972), 57, 1236. Weber, Α., i n ref. 27, pp. 690-705. Aldous, J . and M i l l s , I. Μ., Spectrochim. Acta (1962), 18, 1073. Aldous, J . and M i l l s , I. M., Spectrochim. Acta (1963), 19, 1567. Kivelson, D. and Wilson 1229. Wilson, Ε. B., J . Chem. Phys. (1957), 27, 986. Beer, R., Farmer, C. B., Norton, R. H., Martonchiek, J. V., and Barnes, T. G., Science (1972), 175, 1360. Beer, R. and Taylor, F. W., Astrophys. J . (1973), 179, 309. Young, L. G., Icarus (1969), 11, 66. Crawford, B., J. Chem. Phys. (1958), 29, 1042. Crawford, B., J . Chem. Phys. (1952), 2 0 , 977. Dickson, A. D., M i l l s , I. Μ., and Crawford, B., J . Chem. Phys. (1957), 27, 445. Herzberg, G., "The Spectra and Structures of Simple Free Radicals," pp. 10-16, Cornell University Press, Ithaca, Ν. Υ., 1971. Monfils, A. and Rosen, Β., Nature (1949), 164, 713. Douglas, A. E. Astrophys. J . (1951), 114, 466. Clusius, K. and Douglas, Α. Ε., Can. J . Phys. (1954), 3 2 , 319. Katayama, D. Η., Huffman, R. E., and O'Bryan, C. L., J . Chem. Phys. (1973), 59, 4309. Nakamoto, K., Angew. Chem. internat. Edit. (1972), 11, 666. Nakamoto, Κ., Udovich, C., and Takemoto, J., J. Amer. Chem. Soc. (1970), 92, 3973. Pinchas, S., Silver, B. L., and Laulicht, I., J . Chem. Phys. (1967), 46, 1506. Takemoto, J . and Nakamoto, Κ., Chem. Commun. (1970), 1017. Nakamoto, Κ., "Infrared Spectra of Inorganic and Coordination Compounds," 2nd ed., pp. 106-112, WileyInterscience, New York, 1970. Collman, J . P. Farnham, P., and Dolcetti, G., J . Amer. Chem. Soc. (1971), 93, 1788. Masri, F. N. and Wood, J . L., J. Mol. Struct. (1972), 14, 217.
Masri, F. N. and Wood, J. L., J. Mol. Struct. (1972), 14, 201. Clements, R. and Wood, J. L., J. Mol. Struct. (1973), 17, 265.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2. NASH
Isotope Effects and Spectroscopy
57·
43
Clements, R. and Wood, J . L., J . Mol. Struct. (1973), 17, 283. 58. Clements, R., Dean, R. L., and Wood, J . L., J . Mol. Struct. (1973), 17, 291. 59· Wood, J . L., J . Mol. Struct. (1973), 17, 307. 60. Laane, J . , J . Chem. Phys. (1971), 55, 2514. 6 1 . Berney, C. V., Redington, R. L., and Lin, K. C., J . Chem. Phys. (1970), 53, 1 7 1 3 . 62. Bournay, J . and Marechal, Y., J . Chem. Phys. (1973), 59, 5077. 63. da S i l v e i r a , Α., Marques, Μ. Α., and Marques, N. M., Mol. Phys. ( 1 9 6 5 ) , 9, 271. 64. Nash, C. P., Donnelly, T. C., and Rock, P. Α., unpublished results. 65. Singh, G. and Rock 57, 66. Greenberg, M. S., chim. Acta (1973), 29A, 1927, and earlier papers cited there. 67. Maxey, B. W. and Popov, A. I., J . Amer. Chem. Soc. (1969), 91, 2 0 . 68. Wong, M. K., McKinney, W. J . , and Popov, A. I., J . Phys. Chem. (1971), 75, 56. 69. Handy, P. R. and Popov, A. I., Spectrochim. Acta (1972), 28A, 1545. 70. Roche, J.-P. and Huong, P. V., B u l l . Soc. Chim. France (1972), 4521. 71. Alexander, E. R. and Pinkus, A. G., J . Amer. Chem. Soc. (1949), 71, 1786. 72. E l i e l , E. L., J . Amer. Chem. Soc. (1949), 71, 3970. 73· Streitweiser, A. and Wolfe, J . R., J . Amer. Chem. Soc. (1959), 8 1 , 4912. 74. S t i r l i n g , C.J.M., J . Chem. Soc. (London) (1963), 5741. 75· Sabol, M. A. and Andersen, K. K., J . Amer. Chem. Soc. (1969), 91, 3603. 76. Annunziata, R., Cinquini, Μ., and Colonna, S., J . Chem. Soc. Perkin Trans. I (1972), 2057. 77. Andersen, Κ. Κ., Colonna, S., and S t i r l i n g , C.J.M., J . Chem. Soc. Sec. D (1973), 645. 78. Kokke, W.C.M.C. and Oosterhoff, L. J . , J . Amer. Chem. Soc. (1972), 9 4 , 7583. 79. Kokke, W.C.M.C. and Oosterhoff, L. J . , J . Amer. Chem. Soc. (1973), 95, 7159.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
3 Isotope Effects and Quantum-Mechanical Tunneling R A L P H E. W E S T O N ,
JR.
Department of Chemistry, Brookhaven National Laboratory, Upton, L. I., Ν. Y.
11973
Introduction The phenomenon of quantum-mechanical tunneling is not observ able i n the macroscopic world which we experience directly. Sup pose that a b a l l i s rolled along the ground towards a small ridge. If i t s kinetic energy i s greater than the potential energy of the b a l l at the top of the ridge it will surmount the barrier; if the kinetic energy i s less than this, the b a l l w i l l not get over the hill. However, a moving particle of atomic or electronic mass does not obey Newtonian mechanics. Instead, it behaves as a wave packet with a wavelength given by the de Broglie expression λ = h/mv
(1)
where h i s Planck's constant, m i s the mass of the particle, and v i t s velocity. For purposes of calibration, it is convenient to remember that a hydrogen atom, moving with thermal velocity at 300 K, has a wavelength of about 1 Å(0.1 nm). When such a particle encounters a barrier, represented by an increase i n potential energy, it does not behave l i k e a macroscopic particle. Instead, there is a f i n i t e probability of "leakage" or "tunneling" through the barrier even if the kinetic energy is less than the potential energy at the barrier summit; conversely, there is a f i n i t e probability of reflection even if the kinetic energy i s greater than this. The extent of tunneling, the transmission probability қ, i s defined by қ= ( A / A ) , where A and A are the wave function amplitudes for the incident and transmitted waves (Fig. 1). 2
t
i
i
t
As one would expect, κ depends on the mass of the particle, its velocity, and the shape and height of the barrier. Two con venient parameters are 44
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
3.
WESTON
Quantum-Mechanical
Tunneling
α = 2frV*/hv*
and
45
ε = E/V*
(2)
where V* is the barrier height and v* i s a "vibrational frequency" defined by 1
1
2
2
v* - (2w)"" {-y" [d V(x)/dx ]}
lâ
(3)
where μ i s the reduced mass appropriate to motion i n the χ direc tion. For a particular form of the potential barrier, V(x), known as the Eckart barrier (see below), the dependence of κ on ε and α is shown i n Fig. 2. The classical limit i s simply a step function at ε = 1, i . e . when the kinetic energy of the particle i s equal to the potential energy at the barrier summit. Large values of a, corresponding to high, or nearly f l a t , barriers lead to a similar form of κ. Low value values of κ with ε muc than unity even when Ε i s greater than V . The physical significance of quantum-mechanical tunneling was recognized very early i n the development of wave mechanics, and there are many examples of physical phenomena i n which tunneling i s important. Here i s a very incomplete l i s t of examples, chosen principally on the basis of h i s t o r i c a l interest: 1. The cold emission of electrons from a metal cathode at a high negative voltage (1). 2. The emission of α-particles from an atomic nucleus (2,3). 3. The effect of the double minimum i n the potential energy for nh3 on vibrational energy levels (Λ). 4. The possibility of tunneling i n a chemical reaction i n volving motion of a proton or a hydrogen atom, which seems to have been f i r s t recognized by R. P. Bell (5). 5. The tunnel diode, a semiconductor device of considerable practical importance. The discovery of this by L. Esaki i n 195658 (6) led to his sharing i n the 1973 Nobel prize for physics (2)· In fact, i t is interesting to note that the other two recipients of the Nobel prize i n physics for that year, B. D. Josephson and I. Giaever, were also honored for discoveries involving tunneling in solids (8,9). The Nobel prize awards should, i n themselves, provide ample proof of the reality and the practical significance of tunnelingI Tunneling i n Chemical Reactions Now that i t has been made evident that tunneling i s predicted by quantum mechanics, and that there are a number of physical manifestations of i t , what about the particular area of chemical reactions?
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
isotopes a n d c h e m i c a l
principles
VU)
Theory of Elementary Gas Reaction Rates Figure 1. Permeability of a barrier to a particle with kinetic energy less than barrier height. The dashed line represents the wave function for a particle of energy, E, moving from the left and interacting with an Eckart barrier (solid line) (29).
0
0.2 0.4
0.6 0.8
1.0
1.2
1.4
1.6
1.8 2.0 2.2
i - E/V* Gas Phase Reaction Rate Theory Figure 2. Transmission probability κ(ε) as a function of reduced energy e(= E/V*) and a(= 2*V*/hv*) for a symmetrical Eckart barrier (30)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
3
Quantum-Mechanical
WESTON
Tunneling
47
The potential energy of interaction for the reactants i n a hypothetical reaction such as A + BC —> AB + C can be represented by a contour map (Fig. 3). Along the reaction path, the potential energy has the form of a one-dimensional barrier, similar to that i n Fig. 1. If the mass of the atom being transferred (B i n this example) i s sufficiently small, and i f the barrier has suitable dimensions, tunneling may become important. I am deliberately evading here some subtle questions, such as: 1. What i s the precise reaction path that should be used to obtain the one-dimensional potential? 2. Is i t correct t when the potential energ These and related questions have been discussed i n d e t a i l by others (10,11). For a one-dimensional barrier of arbitrary shape, numerical methods can be used to calculate the transmission probability κ (12). Before the availability of large computers, this problem was often circumvented by approximating the barrier with an Eckart function (13). The symmetric form of this, which I shall use later, i s 2
V(x) = V*/cosh [(2 V*)\v*x] U
.
(4)
This potential has the great advantage of leading to analytical solutions of the resulting Schrbdinger equation, so that much less computational effort i s required to calculate κ than i s required for an arbitrary barrier shape. It i s ideally suited for testing the qualitative effects of changing properties of the barrier. With an Eckart barrier, the transmission probability for a parti cle with energy Ε is found to be K ( E ) = [cosh(2αε^)-1]/[cosh(2ae^) + δ]
(5)
2
where δ = cosh(4a -π^)^ (a > π/2) or δ = cos|4a -w2p (a < π/2) , and the other quantities have already been defined. 2
In a chemical reaction system at thermal equilibrium, the Boltzmann distribution of molecular energies must be taken into account i n obtaining the average transmission probability. The tunneling factor i s usually defined as the ratio of this averaged transmission probability to that obtained with the classical values K(E)
=0,
Ε < V * ;
κ(Ε)
= 1,
Ε ^ V
American Chemical Society Library
16thPrinciples; St, N.W. In Isotopes and1155 Chemical Rock, P.; Washington, 20036 ACS Symposium Series; American ChemicalD.C. Society: Washington, DC, 1975.
48
ISOTOPES
ANDCHEMICAL
PRINCIPLES
Chemical Kinetics Figure 3. Hypothetical potential energy surface for the collinear reaction A + BC —» AB + C. The solid lines are equipotentials and the dashed line is the reaction path ( 31 ).
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
3.
WESTON
Quantum-Mechanical
Tunneling
49
That i s ,
Γ*(Τ)
(E)exp(-E/RT)dE
IS
=
(6) exp(-E/RT)dE
Γ*(Τ)
exp(V*/RT) J K(E)exp(-E/RT)d(E/RT)
-
.
(7)
This expression must be numerically integrated over the appro priate energy range. Tunneling and Kinetic Isotope Effects The Magnitude of Primary Hydrogen Isotope Effects. As Eq.(l) i l l u s t r a t e s , the wavelength associated with a moving particle i s inversely proportional to the mass. In a comparison of two isotopic species i n a chemical reaction, this mass dependence leads to different values of v* and a. Since a is larger (v* smaller) for the heavier species, the values of κ(Ε) are closer to the classical values, and this shows up i n Γ*(Τ). Since relative differences i n mass are greatest for the isotopes of hydrogen, one might expect important differences i n the tunneling corrections for reactions involving the motion of H, D, or Τ atoms or ions. In fact, almost since the discovery of deuterium, such reactions have been studied i n an attempt to find evidence for tunneling i n a chemical reaction (14). To find evidence for tunneling, one must f i r s t account for kinetic isotope effects that do not depend on tunneling. The most direct method of doing this is by means of activated complex theory, which leads to a formulation of the rate constant i n terms of partition functions of the reactants and the activated complex. Arguments concerning the validity of activated complex theory are easy to provoke and d i f f i c u l t to settle, and I shall not consider this question here. It can then be shown (15,16) that the ratio of rate constants for two isotopic forms of the same reactant (AX| and AX2) is given by k
k
l 2
ν
Γ
ΐ* ΐ* v
2
Γ
2
3 r
"
7
r
i
(
A
X
i*
)
3 r
i-1 Γ (ΑΧ *) ±
2
"
6
r
j-1
j<
A X
) 2
Y V A
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
50
ISOTOPES
A N D CHEMICAL
PRINCIPLES
Here T± = u±e~u±/2/(l-e~u±), u± = hv /kT, and 3r-6 (3r-7) i s the number of real vibrations i n the reactant (activated complex). ±
Given the details of the relevant potential energy surface, or some empirical scheme for predicting the force constants needed to calculate the vibrational frequencies upon which the r^'s depend, one can^evaluate Eq. (8) with or without the tunneling contribution /Γ2 . A comparison with experiment should then reveal whether the tunneling correction i s indeed important. A good example of this approach is provided by the gas-phase reaction systems CF and
3
+ CHD —> CF H or CF D + CD or CHD 3
3
3
3
2
CF + CH D —* 3
2
2
which have been examined i n detail by H. S. Johnston and coworkers (17,18). A comparison of experimental isotope effects and those calculated with and without tunneling i s given i n Fig. 4; evi dently the calculations which include tunneling are i n better agreement with experiment. This approach depends for i t s validity on the accuracy with which Eq. (8) can be evaluated. As potential energy surfaces for larger and larger reaction systems become available, the confi dence with which one can u t i l i z e this method w i l l increase; obviously i t i s only as good as the force constants that go into Eq. (8). However, i n some cases even the magnitude of the experi mental isotope effect is so large that i t seems d i f f i c u l t to ex plain without a contribution from the tunneling correction. For example, the isotopic rate constant ratio k^/k^ for the proton (deuteron) transfer from 4-nitrophenyInitromethane to tetramethy1guanidine has been found to be as high as 45 at 25°C (19). Arrhenius Pre-exponential Factors i n Primary Hydrogen Kinetic Isotope Effects. A second diagnostic test for tunneling i n chemical reactions i s based on the temperature dependence of tunneling. Returning again to Eq. (1), we see that as the momen tum of a particle increases (due to increased thermal energy, for example) i t s wavelength decreases. This, i n turn, leads to a temperature dependence of Γ*, which approaches unity at high temperatures. The way this has been expected to influence an isotopic rate constant ratio has been discussed frequently i n the literature (14), and i s illustrated i n Fig. 5. Kinetic-isotope-effect data can be expressed i n the form =
A (T)e Q
B ( T ) / T
,
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
(9)
3.
WESTON
Quantum-Mechanical
Tunneling
51
Journal of Chemical Physics Figure 4. Rate constant ratio k /k for the abstraction of H or D from CHD, or CH D». The points are experimental values, and the lines are values calculated from different reaction models, (a) No tunneling correction included, (b) Tunneling correction included (18). H
{
n
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
Tunneling | unimportant ρ
0 r
A N D CHEMICAL
PRINCIPLES
Experimental Points
s ?
/
, l n
A
Q,closs
Tunneling Important
1 1
/ ^—In
A
Q
Figure 5. Qualitative portrayal of the temperature dependence of an iso topic rate constant ratio in a reaction where tunneling is important. The Arrhenius intercept which would be obtained at high temperatures (Aç^iaes) and the intercept obtained by extrapolation from low temperatures (A ) are indicated. Q
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
3
Quantum-Mechanical
WESTON
53
Tunneling
where both A Q and Β are written as temperature-dependent, for the sake of generality. If tunneling i s unimportant, i.e., at high temperatures, this expression w i l l behave like the Arrhenius ex pression for individual rate constants, with A Q and Β independent of temperature. Then a plot of l n ( k / k ) vs T - l w i l l be linear, or very nearly so. As the temperature i s decreased and the tun neling component of the isotope effect becomes more important, ln(k^/kj)) w i l l curve upwards, so that B(T) is larger and A Q ( T ) is smaller than they would be without tunneling (Fig. 5 ) . Bell ( 2 0 ) , and more recently Schneider and Stern ( 2 1 ) , have investigated the temperature dependence of AQ for model reactions. The latter workers carried out calculations of k /kj) with Eq. (8) (without the tunneling correction), and by f i t t i n g these values to the form of Eq. ( 9 ) , they obtained A Q as a function of temperature. For several model reaction with larg primar hydroge isotop effects, they attempte adjusting force constants o the mode activated complexes. Ove a temperature range of 2 0 - 1 0 0 0 ° K , values of A Q below 0 . 7 could not be attained. This answers part of the question: In the absence of tunneling, A Q i s restricted to a rather narrow range. The remaining part of the question is the converse: In the presence of tunneling, does AQ l i e outside this range? H
n
H
To answer this, Stern and I ( 2 2 ) used the same model systems previously investigated by Schneider and Stern, but included the tunneling correction i n the isotope effect. Values of v* were those calculated i n their work; barrier heights V* were not needed i n their work, and we have simply assumed values of 1 , 5 , 1 0 , 2 0 , and 3 0 kcal/mole for VJJ . A symmetrical Eckart potential was used. For reasons discussed i n our paper ( 2 2 ) , two sets of calcu lations were made: one with V J J * = Vg* and one with V J J * = V J J * + 1 kcal/mole. Although we examined a l l of the models used by Schneider and Stern, I shall discuss here only the results ob tained from the model for the reactions CF3 + C2H5 or C2H5D — • CF3H or CF3D + C2H5. *
it
F i r s t , the individual tunneling corrections, and , are shown i n Fig. 6 as functions of T~*. These are well-behaved, smooth, functions, although i f V^* is not equal to V^* there i s a crossover temperature below which Γ * < Γ g*. This arises from the term exp(V*/RT) i n Eq. (7); thus, exp[(V *-V *)/RT] is more important than the integral at low temperatures. For the isotopeindependent barrier, the ratio Γ */Γρ* is almost constant at low temperatures because the individual lines are p a r a l l e l . At high temperatures, 1η(Γ||*/Γ])*) approaches zero with a positive curva ture, so that the complete l n ( r * / r * ) vs T" curve i s sigmoid shaped. When the barrier height i s isotope dependent, the shape of l n ( r * / r * ) i s similar at high temperatures, but ultimately the values are decreased because of the exponential term discussed above, and the curve i s almost a straight line with slope Η
H
D
Η
1
H
H
D
D
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
A N DCHEMICAL
PRINCIPLES
Journal of Chemical Physics Figure 6. Arrhenius-type plots of r * , Γ » * , Γ * / Γ * and (k / k) for a model hydrogen-atom abstraction with (a) V * = V * = 10 kcal/mole, (α') V„* = 10, V,,* = 11 kcal/mole (22) w
D
cla8S
Η
0
H
H
D
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
3.
Quantum-Mechanical
WESTON
Tunneling
55
(Vjj*-Vj)*)/R. However, since this term has an exponential temperature dependence, i t does not affect A Q . The temperature dependence of I Ï I A Q (for tunneling alone) can be qualitatively assessed from Fig. 6 by linear extrapolation from a particular temperature region. This was done more precisely by using a least-squares f i t to 1η(Γ|| /Γρ ) or ln(k|j/kn) at four temperatures, and the result ing values of A Q for various barrier heights are plotted i n Fig. 7a. The infinite-temperature limit of l n A Q is zero. At high temperatures, l n  Q w i l l decrease to negative values with decreasing temperature. As the temperature is further decreased, lnÂQ w i l l go through a minimum at the temperature of the i n flection point i n the l n ( r * / r * ) curve, and then increase again as the temperature decreases further. At very low temperatures, IIIAQ ^ is approximately equal to l n r * - l n r * , which i s a very large quantity. As mentione difference between the independent barrier heights. j t u n
> t u n
t u n
H
D
t u n
When the tunneling correction i s combined with the rest of the rate constant ratio, (^H^D^class* resultant form of I ^ A Q is almost identical with that for tunneling only. This i s because ln(k /k ) has almost exactly the Arrhenius form, with ln(AQ class/ v i r t u a l l y independent of temperature (see the curves labelled^NT" i n Figs. 7a and 7a") . t h e
H
D
c l
f
Figs. 7a
1
and 7a" i l l u s t r a t e the following important features:
1. Regardless of the barrier height, there is a f a i r l y small temperature region i n which I I I A Q i s below the limit of -0.69 predicted by Bell's model i n the absence of tunneling (indicated by the lower dashed l i n e ) . The extent of the excursion below this limit increases with increasing barrier height. It is i n this temperature region that most experiments have been done, and earlier calculations of tunneling were made. 2. At lower temperatures, InAQ exceeds the upper limit of 0.35 predicted by B e l l i n the absence of tunneling (indicated by the upper dashed l i n e ) . This i s contrary to what had been expected before our calculations were made, because earlier calculations concentrated on the region to the l e f t of the inflection point i n Fig. 6. 3. The value of AQ does not correlate simply with the^extent of tunneling, which i s indicated by values of ΙηΓ^* for Vg = 10 on the top of Fig. 7. In our work, we also examined other model reactions, and a few other barrier shapes, with similar results.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
56
ISOTOPES
AND
CHEMICAL
PRINCIPLES
Thus, we conclude that experimental A Q values can be used as quantitative indicators of quantum-mechanical tunneling only i n conjunction with model calculations. Although AQ values outside of Bell's range of 0.5-/2 probably cannot be found i n the absence of tunneling, A Q values inside this range do not necessarily indi cate the absence of large tunneling factors. In spite of this somewhat uncertain situation, i t i s interesting to see i f there are experimental values of A Q outside the "non-tunneling" range. Data from reactions i n which tunneling has been invoked are given in Table I. (This does not mean that tunneling i s unimportant i n other reactions, but only that i t has not been specifically searched f o r ) . Without c r i t i c a l l y evaluating the experimental data of Table I, I simply point out that there are several reac tions i n which abnormally low A Q values have been observed. Relative Tritium-Deuteriu isotope experiments are best done with D substitution and others with Τ substitution, there has been for some time an interest i n , and a necessity for, the correlation of the rate constant ratios k / k and k /k . This correlation may be defined as H
D
H
T
ln(k /k ) H
T
Several theoretical investigations of the allowable range i n in the absence of tunneling have been made. Using a simplified model for a hydrogen-transfer reaction, Swain and coworkers (23) ob tained a value of 1.442, considering only zero-point-energy effects. Bigeleisen (24),using a more complete model, proposed a range of 1.33-1.58 for the relative isotope effects, including Wigner tunneling (valid i f κ is close to unity). He stated that exten sive tunneling should lead to abnormally low values of r. More O'Ferrall and Kouba (25) calculated isotope effects for some model proton-transfer reactions involving linear four- and fiveatom transition states. They found deviations of only 2% from the Swain value of r, (1.442), even when tunneling corrections for a parabolic barrier were included. It has also been shown by Lewis and Robinson (26) that tunneling corrections do not necessarily produce significant changes i n the relative isotope effect. Recently, Stern and Vogel (27) investigated relative tritiumdeuterium isotope effects for 180 model reaction systems. They found that, within the temperature range 20-1000°K, was re stricted to the range 1.33 < r^^ 1.58 provided that: 1. ^ H / ^ D f l c t significant force-constant changes between reactant and transition state at the isotopic position(s). A N D
r e
e
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
WESTON
Quantum-Mechanical
Tunneling
Table I. Arrhenius Preexponential Factors and Relative Tritium-Deuterium Isotope Effects Reaction
fa
0.032 1. 4-Nitrophenylnitromethane + tetramethylguanidine (in toluene)
Ref,
-
a
-
a
2. Same reaction i n dich1oromethane
0.45
3. Leucocrystal violet + chloranil
0.04
1.31
b
4. 2-Carbethoxycyclopentanone + F
0.042
1.32
c,d
5. 2-Ni tropropane + 2,4,6-1rime thy1pyridine
0.15
1.39
b
7. 2-Carb ethoxycyclopentanone + chloracetate ion
0.35
1.72
c,d
8. Acetophenone + OH
0.38
1.38
e
9. l-Bromo-2-phenylpropane + ethoxide
0.40
1.48
10. 2-Carb ethoxycy clopentanone + Ό^)
0.44
1.48
f,g c,d
11. Pyridinediphenylborane + H^O
0.94
1.38
h
12. 2,2-Diphenylethylbenzenesulfonate + methoxide
1.1
1.48
i
6. Oxidation of 1-phenyl-2,2,2-tri fluoroethanol
*Ref. 19 b
Ref. 26
C
B e l l , R. P., Fendley, J . Α., and Hulett, J . R., Proc. Roy. Soc. Ser. A (1956), 235, 453.
d
Jones, J . R., Trans. Faraday Soc. (1969), 65, 2430.
e
Jones, J . R., i b i d . (1969), 65, 2138.
f
Shiner, V. J . , J r . and Smith, M. L., J . Amer. Chem. Soc. (1961) , 83, 593. Shiner, V. J . , J r . and Martin, B., Pure Appl. Chem. (1964), 8, 371.
g
^Lewis, E. S. and Grinstein, R. H., J . Amer. Chem. Soc. (1962) , 84, 1158. S f i l l i , Α. V., J . Phys. Chem. (1966), 70, 2705.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
58
ISOTOPES
AND
CHEMICAL
PRINCIPLES
2. kjj/kj. and kjj/kj} are both greater than unity. 11
3. kjj/kj. and k^/k^ exhibit "regular temperature dependences at a l l temperatures, "regular" being defined i n their paper. Stern and I (28) re-examined the model reactions meeting the above three c r i t e r i a , to see what effect the inclusion of tunnel ing would have on the relative isotope effect. The tunneling cor rections were calculated as described i n the preceding sub-section. Although this was done for a large number of model reactions, I s h a l l discuss here only the model for hydrogen-atom abstraction that has already been mentioned i n the preceding sub-section. Again, two variations were considered: i n one, V J J * = = Vj.*; i n the other, V * = V * + 1, V * = V * + 1.45. Figs. 8a and 8b show the ratios of tunnelin Ξ Η*/ Τ* H/T> tw tritium-deuterium tunneling correction D
Γ
Γ
Ξ
T
f
o
R
r t
n
T
R
e
L~ =
lnT
H/T
/lnT
H/D
is indicated. The slightly different shapes of l n T ^ and l n T ^ vs. logT result i n a temperature dependence of t. At high tempera tures , Γ* is given by the Wigner expression, and i t can be shown (24) that R
as Τ l-(v */v *) D
H
T
H
D
—> 0
2
If the barrier height depends on isotopic substitution, there i s a temperature at which is unity and Jt, is zero. At a slightly higher temperature, T ^ becomes unity and t^has a pole (±°°) . R
The relative isotope effect including the tunneling correc tion, which we designate as r , can be expressed i n terms of t and f
LnB
r
=
H/D
st
Γ
n
e r
a
t
lnR
H/D]
e
In this expression, Rg/D ^ constant ratio with no tun neling correction, and R^/n the same ratio including tunneling. It is apparent that the weighting factors multiplying £ and t^are complementary; as a result, plots of the weighted and weighted £ contributions vs. logT w i l l be, approximately, displaced "mirror images" (cf. Figs. 8a and 8b ). Because the two contributions are not exactly out of phase, a slight temperature dependence of r^ remains. Fig. 9 illustrates the effect on t^and on of f
f
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
3.
WESTON
Quantum-Mechanical
Tunneling
* Η( Η η Γ
147.6
32.6
4.40
0.39
147.6
32.6
ν
59
= Ι 0 )
4.40
Q39
147.6
32.6
4.40
0.39
Figure 7. Arrhenius pre-exponential factors as functions of temperature for a model hydrogen-atom abstraction with various barrier heights: (a), tunneling cor rection only, V * = V *; (α'), complete isotope effect, V * = V *; (a")> com plete isotope effect, V^* = V * + 1. The curves are labeled with V * in kcal/ mole. (NT indicates that no tunneling correction is included.) Numbers at the top of each graph are ΙηΓ * for V * =10, as a function of T . Dashed lines are at A = Vz andV~2 (22). D
H
D
H
H
H
Η
H
ar!l
Q
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
60
ISOTOPES
147.6 32.6 1 1 1 1 11II
2.0
4.40
1
A N DCHEMICAL
PRINCIPLES
0.39
! 1 1 1 1 III ία)
1.8 - 5
1
1.6
10
1.4 _
2C ' * 3 0
1.2
1 ! 1 1 11II
1
1 1 llllll
Journal of Chemical Physics Figure 9. Relative tritium-deuterium kinetic isotope effects as functions of tem perature with various barrier heights: (a),(b), tunneling only; (a'),(b') complete isotope effect. Unprimed letters are for isotope-independent barrier heights, and primed letters are for V * = V,/* + 1, V * = W + ΙΛ5 kcal/mole. The curves are labeled with V * in kcal/mole. Numbers at the top of the graphs are ΙηΓ * for V * = 10 kcal/mole, as a function of log T. Dashed horizontal lines are the nontunneling limits," 1.33 and 1.58. The infinite-temperature limit of the relative isotope effect is indicated by "n" at right border (28). f
D
T
H
Η
H
<e
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
3.
Quantum-Mechanical
WESTON
61
Tunneling
f
changes i n V*; note that deviations of £ vs logT from a smooth, monotonie curve increase as the barrier height increases. The important point, however, i s that even when tunneling i s very significant (cf. legend at top of Fig. 9), rj does not greatly exceed the "nontunneling" range of 1.33-1.58. Furthermore, contrary to the assumptions of earlier authors, rj drops below the lower limit only for the highest barrier tested, and over a narrow temperature range. Although tunneling i s important at these temperatures, the tunneling correction i s even larger at lower temperatures where rj exceeds the upper range of 1.58. Except for the lowest barrier, where relative differences between, H*» D*> * T * S » the conclusions for the case when the barrier height i s isotope-dependent are very similar. Note that the abnormal behavior weighting factor to giv The only cases for which the discontinuity i n r j i s retained are physically unrealistic models i n which secondary isotope effects are combined with an isotope-dependent barrier height. V
V
an<
V
a r e
l a r
e
The large number 60) of other model reaction systems we investigated a l l gave results qualitatively similar to those discussed above, but with subtle differences dependent on the particular model. We conclude from this work that: 1. Even when quantum-mechanical tunneling i s important, the limits 1.33 ^ r, ^ 1.58 are not l i k e l y to be exceeded, i f they would not be exceeded i n the absence of tunneling. 2. There i s no general correlation between the magnitude of £ and the importance of tunneling. 3. The temperature dependence of £ i s qualitatively the same for a wide range of barrier heights and reaction models. Experimental data for some systems i n which both deuterium and tritium isotope effects have been measured are given i n Table I. Again, the warning must be issued that other reactions undoubtedly exist i n which tunneling i s important but has not been searched for by this method. Nevertheless, there i s only one reaction l i s t e d (No. 7) where the experimental value of £ is significantly outside the range of 1.33-1.58. Interestingly, the temperatures at which this reaction was studied are close to the temperature at which a minimum value of i:' i s calculated from the model discussed above. Conclusion:
What About Chemical Principles?
I hope that the details of the preceding sections have not diverted the reader's attention from the general subject of this
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
62
ISOTOPES
A N D C H E M I C A L PRINCIPLES
symposium: "Isotopes and Chemical Principles". the principle that underlies these details:
Let me reiterate
Classical méchantes cannot be relied upon to provide an adequate and accurate description of the motion of light atoms (particularly isotopes of hydrogen) during a chemical reaction. The wave nature of the moving particle must be taken into account because it permits "tunneling" through the potential energy barrier separating reaetants from products. Because the wavelength (and hence the extent of tunneling) of the moving atom is mass-dependenty kinetic isotope effects will reflect this wavemechanical property.
unfortunately, the diagnostic methods that have been used to search for tunneling i n actual experiment hav t led t equivocal proof. In effect structure that has been b u i l t o t s somewhat rickety. Antici pated developments that w i l l improve the sturdiness of this structure over the next few years are: 1 . The calculation of accurate potential energy surfaces for reacting systems consisting of a few atoms. 2. Complete wavemechanical calculations, with these potential energy surfaces, of reactive cross sections and other details of reaction dynamics. 3. Precise measurements of cross sections for these reaction systems, with reaetants i n well-defined s tates. Acknowledg ement The tragic death of Marvin Stern brought to an untimely end the promising career of a good friend and respected colleague. His contributions to our calculations described here and i n more detail elsewhere ( 2 2 , 2 8 ) cannot be overestimated. To this work he brought his insight into the significant features of the problem and his uniquely high standards of meticulous attention to d e t a i l . This paper i s based on research supported by the U. S. Atomic Energy Commission.
Literature Cited 1.
Fowler, R. H. and Nordheim, L., Proc. Roy. Soc. Ser. A 119, 173.
2.
Gamow, G., Z. Phys.
(1928),
51,
204.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
(1928),
3. WESTON
Quantum-Mechanical Tunneling
63
3. Gurney, R. W. and Condon, Ε. U., Nature (1928), 122, 439. 4. Morse, P. M. and Stueckelberg, E. C. G., Helv. Phys. Acta (1931), 4, 335. 5. B e l l , R. P., Proc. Roy. Soc. Ser. A (1933), 139, 466. 6. Esaki, L., Phys. Rev. (1958), 109, 603. 7. Esaki, L., Science (1974), 183, 1149. 8. Josephson, B. D., i b i d . (1974), 184, 527. 9. Giaever, J., i b i d . (1974), 183, 1253. 10. Truhlar, D. G. and Kuppermann, Α., J . Am. Chem. Soc. (1971), 93, 1840. 11. Truhlar, D. G. and Kuppermann, Α., J . Chem. Phys. (1972), 56, 2232. 12. Le Roy, R. J . , Quickert, Κ. Α., and Le Roy, D. J., Trans. Faraday Soc. (1970), 66, 2997. 13. Eckart, C., Phys. Rev (1930) 35 1303 14. For a review, cf. 15. Bigeleisen, J . , J 16. Weston, R. E. and Schwarz, Η. Α., "Chemical Kinetics", p. 109, Prentice-Hall, Englewood C l i f f s , N. J., 1972. 17. Johnston, H. S. and Tschuikow-Roux, E., J . Chem. Phys. (1962), 36, 463. 18. Sharp, T. E. and Johnston, H. S., J . Chem. Phys. (1962), 37, 1541. 19. Caldin, E. F. and Mateo, S., J . Chem. Soc. Chem. Comm. (1973), 854. 20.
Bell,
R.
P.,
"The
Proton
in
Chemistry",
Chap
XI,
Cornell
Press, Ithaca, Ν. Y., 1959.
Univ.
21. Schneider, M. E. and Stern, M. J . , J . Am. Chem. Soc. (1972), 94, 1517. 22. Stern, M. J . and Weston, R. E., J r . , J . Chem. Phys. (1974), 60, 2808. 23. Swain, C. G., Stivers, E. C., Reuwer, J . F., J r . , and Schaad, L. J., J . Amer. Chem. Soc. (1958), 80, 5885. 24. Bigeleisen, J . , "Tritium i n the Physical and Biological Sciences", International Atomic Energy Agency, Vienna, 1962, vol. 1, p. 161. 25. More O'Ferrall, R. Α., and Kouba, J . , J . Chem. Soc. Β (1967), 985. 26. Lewis, E. S. and Robinson, J . K., J . Amer. Chem. Soc. (1968), 90, 4337. 27. Stern, M. J . and Vogel, P. C., i b i d . (1971), 93, 4664. 28. Stern, M. J . and Weston, R. E., J r . , J . Chem. Phys. (1974), 60, 2815. 29. Bunker, D. L., "Theory of Elementary Gas Reaction Rates", i n "The International Encyclopedia of Physical Chemistry and Chemical Physics", Topic 19, v o l . 1, p. 30, Pergamon Press, Oxford, 1966. 30. Johnston, H. S., "Gas Phase Reaction Rate Theory", p. 45, The Ronald Press Co., New York, 1966. 31. Ref. 16, p. 36.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
4 Corrections to the Born-Oppenheimer Approximation in the Calculation of Isotope Effects on Equilibrium Constants MAX WOLFSBERG
and L A W R E N C E I. K L E I N M A N
Department of Chemistry, University of California, Irvine, Calif. 92664
Introduction The Born-Oppenheimer (B.O.) approximation is the cornerstone of most theories dealing with the effect of isotopic substitution on molecular properties. Within the framework of this approximation, the potential energy surface for the vibrational-rotational motions of a molecular system depends on the nuclear charges of the substituent atoms and on the number of electrons in the system but is independent of the masses of the nuclei. Thus isotope effects arise from the fact that nuclei of different mass move differently on the same potential surface. Among the theories into which the B.O. approximation has been incorporated are the theory of isotope effects on the rotational-vibrational energy levels of a molecule, the theory of isotope effects on chemical equilibria, and various theories of isotope effects on chemical rates. The relative success of these theories in the interpretation of experimental results suggests that the B.O. approximation must be a relatively "good" approximation. The first order correction to the B.O. approximation has been formulated by Van Vleck (1) for diatomic molecules. Subsequent quantitative calculations have been largely restricted to one and two electron molecules. When corrections to the B.O. approximation are introduced, the energy of the nonrotating non-vibrating molecule will generally depend on isotopic substitution. Thus the energy difference between the non-rotating non-vibrating products and the corresponding reaetants in the gas phase isotopic exchange reaction HX + H D = DX + H
(1)
2
64
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
4.
WOLFSBERG
AND
KLEiNMAN
Isotope Effects on Equilibrium
Constants
65
need no longer vanish. This energy difference will be desig nated here as A E L E C . If A E L E C is not equal to zero, the value of the equilibrium constant calculated by the usual methods within the framework of the B.O. approximation must be multiplied by a factor differing from unity which is called here K(BOELE). K(BOELE) will be referred to as the elec tronic isotope effect although it is actually the deviation of this quantity from unity which is important. It follows, however, from the work of Van Vleck and others that, to first order, this energy difference is still zero for the reaction 2HD == H + E>2. On the other hand, it also follows from Van Vleck s calculations, that the failure of the B.O. approximation favors the reaetants of the reaction H + HD = D+ + H by 29 c m " (i.e. ΔΕΙΙ,ΕΟ = 29 cm" value for this reaction is about 0.87. f
2
+
1
We decided to investigate corrections to the B.O. approx imation for a number of diatomic hydrides and deuterides in order to obtain information about electronic isotope effects on the gas phase isotopic exchange reactions (1). Such calculations require a knowledge of electronic wave functions for molecules and the evaluation of a number of integrals. While such calculations would at one time have been very difficult, the availability of a large digital computer makes such calculations feasible now. The results of some of these calculations will be reported in the following sections; they are reported in more detail elsewhere (2,3,4). Theory The theory here follows the development of Van Vleck (1) and Born (5). In atomic units (h=l, mass of electron=l), the nonrelativistic Schroedinger equation for a diatomic molecule with η electrons and nuclei of masses M and is a
[ - έ ζ ν 1 - ^ 4 | ν
! 1
+
ν]Φ =Ε Φ τ
τ
τ
(2)
where all coordinates are relative to an arbitrary but fixed laboratory origin and V is the Laplacian operator. The potential V consists of coulomb interactions among the charged particles (and possibly may contain spin terms) but it does not depend on nuclear masses. After the motion of the center of mass is separated out and the electronic coordinates are 2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
66
ISOTOPES
AND CHEMICAL
PRINCIPLES
transformed to the center of mass of the nuclei (CMN) system, the Schroedinger equation becomes 1 Î ' J Ï I
^
1 R
V
1
n
" 2 Σ
i=l
V·
"
2
(M
n
+MJ
a
v
b
7
.Σ,
ι, j=l
Yi
' Yj
+
J
V(R,r.)] Φ = ΕΦ
(3)
where R is the relative position vector of the two nuclei, the are electronic coordinates relative to axes whose origin is at the CMN and μ = Μ Μ^/(Μ +Μ^). The rj coordinates are taken to be molecule fixed here but the differentiations must be carried out with the coordinate with respect to space fixed CMN axes. The B.O. electronic wavefunctions and energies are defined as the normalized eigenfunctions and eigenvalues of the following Schroedinger equation: Ε
c
"I
p
V Î
t
+v
α
]
*
( r
£i
; R ) =
E
r
( R )
(
* r ^i
;
R
)
'
( 4 )
This is the usual electronic Schroedinger equation for a diatomic molecule used by quantum chemists. Neither E nor Φ (except insofar as the origin of the electronic coordinate system here reflects the nuclear masses) will depend on the masses of the nuclei. Since the B.O. wavefunctions form a complete set in the space of the electronic coordinates, it is possible to expand the total wavefunction in terms of the B.O. states: e
φ
(£i>5) = Σ
F
r Φ * r ïi (
; R )
'
( 5 )
Substitution erf this expansion into (3), left multiplication by a particular Ψ and integration over all electronic coordinates yields the coupled equations for Fp (R) Γ
V
2
R
+E£(R)-E] F ( R ) Ç c r
+
r r
,F
r f
(R)=Q
with
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
(6)
4.
WOLFSBERG
AND
KLEiNMAN
° Γ Γ ' TJI =
C
2
Isotope Effects on Equilibrium
J* * r YR
- 2 ^ y l * ;
X
1D
* r £ ' YR
+
Constants
V
J" * r R
V
D
£
67
]
Ei'Sj
= 1
(7)
The zeroeth-order approximation consists of neglecting all C terms so that ί-ΎΪ
V^ E +
E R
(8)
(R)]F (R) =EF (R). R
R
This is the Born-Oppenheime wavefunction for rotational-vibrational motion. The potential energy in the Schroedinger equation (8) for rotational-vibra tional motion is the electronic energy as a function of R, E|,(R), which is independent of the masses of the nuclei. The first-order approximation to eq. (6), the so-called adiabatic approximation, consists of neglecting the off-diagonal matrix elements of C (i.e. Cpp τ= 0 when Γ Φ Γ ) . It can be shown (6,7) that, for the ground state of H , the adiabatic correction accounts for 99.8% of the total correction to the B.O. approximation. A similar statement is correct (8,9) for the ground state of H . We expect that the adiabatic correction is sufficient for all the systems considered here and subsequent discussion will be restricted to the adiabatic correction. ?
2
+
2
In order to simplify the notation, the subscripts on E (R), F(R), % and C are henceforth omitted. Then, in the adiabatic approximation, E
ί - γ £ V R + E ( R ) + C ] F(R) = EF(R) . G
(9)
It should be noted that C is in general a function of R since Φ is a function of R. F(R) is the rotational-vibrational wavefunction of the molecule and eq. (9) is the Schroedinger equation for this wavefunction. The potential energy term in eq. (9) is the sum of E (R), the B.O. electronic energy as a function of R, which is independent of nuclear masses, and the adiabatic correction C, which does depend on nuclear masses. It is clear that the subsequent interest of the present discussion centers on C, which is sometimes referred to as the diagonal nuclear motion E
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
68
ISOTOPES
AND CHEMICAL
PRINCIPLES
correction. F(R) may be approximated in the usual manner (10) by the producFof a vibrational and a rotational wavefunction
F(R) = ^ f ( R )
Y j M
(0,0).
f (R) is then found to satisfy the vibrational wave equation -Ê?
+
2 ^
+ e 6 ( R )
+ C] f(R) = Ε f(R)
(10)
The term involving J i The R dependence of this term gives rise to rotational-vibra tional interaction. The interest here will be restricted to *Σ states. It can be shown that the first term in eq. (7) for C p must vanish. The differentiation V R * in the second term must be performed with the coordinates of the electrons held constant with respect to space fixed axes whose origin is at the CMN. This term can be further simplified as is described elsewhere (10,2), r
In all the calculations discussed here, the electronic wavefunctions of the *Σ states are filled shell single determinant LCAO-MO wavefunctions. The integrals which occur in C can then be expressed first as integrals over molecular orbitals and subsequently as integrals over atomic orbitals. As a first approximation, one may assume that C is independent of R. C is calculated at the experimental equili brium internuclear distance and is then the adiabatic correc tion to the zero of vibrational energy. In this approximation, the only contribution of the corrections to the B.O. approxima tion to the equilibrium constant of exchange reaction (1) will result from the quantity designated as A E L E C in the Introduc tion (See also eqs. (12) and (13)). In our later calculations (4), which will not be discussed in detail here, we have calculated C at several internuclear distances. We have found that C then will contribute not only to the zero of vibrational energy but will also lead to an isotope dependent shift of the equilibrium internuclear distance and of the various vibrational constants. However, the distance dependence of C is such in our calcula tions that the correction to the zero of vibrational energy so calculated at the new equilibrium internuclear distance is,
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
4.
WOLFSBERG A N D
KLEiNMAN
Isotope Effects on Equilibrium
Constants
69
within the significance of the calculations, equal to the value of C calculated at the original equilibrium internuclear distance. Moreover, the isotope dependent shifts of equilibrium inter nuclear distance and of vibrational constants are sufficiently small so that the contribution of these shifts to the equilibrium constant of (1) appears negligible. The emphasis in the following section will therefore be on the evaluation of C at the experimental equilibrium internuclear distance of a molecule and the subsequent evaluation of aELEC and the electronic isotope effect K(BOELE) on the equilibrium constant. Calculations Calculations wer *Σ states of H , LiH, BH, NH, and H F and the corresponding deuterated molecules with the LCAO-MO wavefunction of Coulson for H (11) and the LCAO-MO-SCF functions of Ransil (12) for the other molecules. These wavefunctions contain a minimum basis set of inner and valence shell Slater-type orbitals with the orbital exponents optimized at the experi mental equilibrium internuclear distance. The *Σ states are the ground states of the respective molecules except in the case of NH. The normalized molecular orbitals ψ . are expressed in terms of atomic orbitals 04, 2
2
1
(11) The C j i coefficients are available in the published literature only at the experimental internuclear distance of the respective molecules. Thus, in the calculation of C at the equilibrium internuclear distance, contributions which arise from the variations of the coefficients with distance were ignored except for the contribution which arises from the fact that must remain normalized as a function of distance. We will designate as OL the contributions to C from the variations of such coefficients; CL was thus evaluated incompletely in these calculations. With the minimum basis set wavefunctions, we calculated
AC = C(HX) - C(DX) and
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
(12)
70
ISOTOPES
AND CHEMICAL
PRINCIPLES
Δ AC = - U E L E C = C(HX) + C(HD) - C(DX) - C(H ) 2
= AC(BX) - AC(H ).
(13)
2
The electronic isotope effect on the equilibrium constant of equilibrium (1) is then calculated as K(BOELE) = exp {AAC/kT).
(14)
In these calculations, we found (2) varying effects with a maximum effect, K(BOELE) =1.10, in the case of X = B. It was decided to improve these calculations by using better electronic wavefunction orbital wavefunctions were still used. However, the mole cular orbitals were expressed in terms of a so-called extended basis set of gaussian atomic orbitals (for details see reference (3)). The Hartree-Fock-self-consistent-field (HFSCF) pro cedure was carried out with the digital computer program POLYATOM. The quality of the wavefunctions is not quite what would be called Hartree-Fock limit wavefunctions. Calculations were carried out at several internuclear distances and C was calculated with the inclusion of the factor CL correctly calcu lated. The calculations were extended to include the Σ ground states of several ions and also erf HC1. ί
Cade and Huo (13) have calculated near Hartree-Fock limit wavefunctions for LiH, BH, HF and HC1. The molecular orbital coefficients c^j are available in the literature again at only the equilibrium internuclear distance. Thus again CL values can not be completely calculated. In Table I, the C values at the experimental equilibrium internuclear distances calculated for LiH, BH, HF, and HC1 with (1) minimum basis set wavefunc tions, with (2) our extended basis set wavefunctions and with (3) the near Hartree-Fock limit wavefunctions are compared. In order to assess the quality of the various wavefunctions, the respective electronic energies are compared with those of the corresponding near Hartree-Fock limit wavefunctions. For the minimum basis set and the near Hartree-Fock limit calculations, the correct CL values of the extended basis set calculations were employed to calculate C . It is seen that both the C values and the AC values for the extended basis set calculations approach closely those of the near Hartree-Fock limit calcula tions. For H , we carried out calculations not only for an extended basis set but also for a large extended basis set which 2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
β
0.001
Extended
0.92
1.27
HF
HC1 Near H. F.
Extended
0.023
0.007
Extended Near H. F.
0.534
Minimum
e
—
5
7
534
—
8
0.002
Extended Near H. F.
226
0.057
Minimum
—
11
218
_1
1349.81
1346.96
23.59
22.07
16.46
17.55
600.82 598.54
35.47
48.30
360.06 646.17
47.68
49.74
30.81
31.24
31.56
1
ACicm- )***
359.05
364.63
188.27
189,06
189.55
C(cm )
e
e
e
* ΔΕ (hartree units) = E - E (Near H. F.) Here E is the electronic energy of the molecule calculated with the given wavefunction and E (Near H. F.) is the electronic energy of the molecule calculated with the near Hartree-Fock limit wavefunction. * ΔΕ/Ε = ΔΕ/Ε (Near H. F.) ** AC = C(HX) - C(DX)
1.24
BH
Near H. F.
0.017
Minimum
1.60
LiH
7
Wavefunction
R(Â)
Molecule
ΔΕ*
Adiabatic Corrections with Different Wavefunctions
Table I
72
ISOTOPES
AND CHEMICAL
PRINCIPLES
yields results of near Hartree-Fock limit quality. In the case of H , the results of the calculation of C and AC with the almost exact wavefunctions of Kolos and Wolniewicz (8) are also available. The latter type of wavefunction can be reproduced in a molecular orbital type of calculation only if configuration interaction is included. Table II compares the results of the various calculations. For the minimum basis set calculation, the Û. value calculated with the minimum basis set was employed since the relative values in the molecular orbital are determined by symmetry. This procedure may be termed somewhat arbitrary (3). The extended basis set and the large extended basis set yield very similar results (as does even the minimum basis set). Table Π does emphasize that configuration interaction may be significan Configuration interaction has not been employed in the subse quent calculations here and this neglect may be a source of error. Hopefully, the consistent neglect of configuration inter action leads to cancellation. We intend to undertake configura tion interaction calculations at a later time. 2
Table II Adiabatic Corrections for H at R = 0.74Â with Different Wavefunctions 2
Cicm" )
AC* (cm )
Minimum
99.07
24.75
Extended
100.94
25.22
Large Extended
101.31
25.31
Kolos and W.
114.59
28.63
Wavefunction
1
-1
* AC = C(H ) -C(HD) 2
Table ΠΙ summarizes the results calculated with the extended basis set wavefunctions (except for H where the large extended basis set was employed). The calculations are at the respective equilibrium internuclear distances. As a matter of interest, CL values are listed to show that they need not be negligible. For the X atom in HX the nuclear mass of the most abundant isotope was employed unless otherwise stated. The 2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
4.
WOLFSBERG AND K L E I N M A N
Isotope Effects on Equilibrium
Constants
Table ΠΙ Adiabatic Corrections (in cm" ) 1
Molecule H
2
R(A)
a
C
0.74
7.63
101.31
5.72
76.00
1.49
210.12
HD 6
LiH
e
LiD
0.69
178.89
7
LiH
1.50
189.06
7
LiD
0.71
1.60
BeH
+
BeD
+
BH
1.31
1.24
BD CH
+
CD
+
NH
1.13
1.05
ND HF
0.92
DF HC1 DC1
1.27
3.40
245.74
1.78
217.61
8.60
359.05
4.14
311.37
4.76
454.12
2.40
409.74
4.07
515.78
2.27
478.07
6.41
600.82
3.46
583.27
4.76
1346.96
2.37
1324.89
Δ
25.31
31.24
28.12
47.68
44.37
37.71
17.55
22.07
* A C = C (HX) - C (DX)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
73
74
ISOTOPES
AND CHEMICAL
PRINCIPLES
lithium hydride calculations demonstrate that, while C does depend on the nuclear mass of X, ÙC is independent of the X mass. It is now easy to evaluate the electronic isotope effects on the equilibria (1). The results at 300 Κ are shown in Table IV. The electronic isotope effects on the various equilibrium constants vary from 0.96 to 1.11. Table IV The Electronic Isotope Effect K(BOELE) at 300 Κ Reaction (X)
Δ A C * (cm" )
K(BOELE)* *
D
0.0
1.00
Li
5.92
1.029
2.81
1.014
22.36
1.113
19.06
1.096
Ν
12.40
1.061
F
-7.77
0.963
Cl
-3.25
0.985
Be
+
Β C
+
* A A C = [ C(HX) - C(DX) ] - [ C(H ) - C(HD) ] 2
* * K(BOELE) = exp (A A C A T ) We have evaluated the electronic isotope effect for some gas phase heavy atom exchange reactions only with minimum basis set wavefunctions, e
L i + LiH = L i L i + LiH 2
14
N
2
7
6
7
6
+ NH = N N + NH . 15
14
15
14
We find K(BOELE) in these two cases at 300 Κ to be 1.001 and 0.999 respectively. For self-exchange equilibria like 'Lit + L i 7
2
= 2 Li Li , 6
7
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
4.
WOLFSBERG
AND
KLEiNMAN
Isotope Effects on Equilibrium
Constants
75
one finds again that K(BOELE) is unity. The deviation of K(BOELE) from unity in heavy atom isotope effects is quite small as expected. We have also calculated the isotope dependent changes in the equilibrium internuclear distances and the isotope dependent changes in the vibrational constants due to the corrections to the B . O . approximation. In Table V, we list some £ R values and also some Δω values (the changes in the harmonic vibra tional frequencies). These changes are expected to lead only to very small effects on the exchange equilibrium constants. They are discussed in more detail elsewhere (4). e
Adiabatic Corrections to R and co e Molecule
Δ*α· (cm )
A*R (Â)
-1
e
2.2 χ ΙΟ"
4
-2.1
HD
1.6 χ ΙΟ"
4
-1.4
Da
1.1 χ ΙΟ"
4
-0.7
LiH
3.8 χ 10~
4
-0.5
LiD
2.2 χ 1 0
-4
-0.2
H
2
* Δ = Adiabatic - Born Oppenheimer Conclusions We have found that corrections to the B.O. approximation may lead to measurable effects on isotopic exchange equilibrium constants involving hydrogen and deuterium. While our calcu lations do not yet include the effect of configuration interaction, there i s no reason to expect agreement between theoretical and experimental isotope effects on equilibrium constants to be better than a few percent unless corrections to the B.O. approx imation are included in the theoretical calculation. We hope to be able to report in the very near future on the comparison between a very accurately experimentally determined equili brium constant involving diatomic hydrides and deuterides and a corresponding theoretical calculation including the corrections
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
76
ISOTOPES
AND CHEMICAL
PRINCIPLES
to the B.O. approximation. Acknowledgements We are indebted to Professor H. F. Schaefer III for making available to us his version of the program POLYATOM. This work was supported by the U.S. Atomic Energy Commission under Contract No. AT(04-3)-34, Project Agreement No. 188. Literature Cited 1. Van Vleck, J.H., J. Chem. Phys. (1936) 4, 327. 2. Kleinman, L.I., and Wolfsberg, Μ., J. Chem. Phys. (1973) 59, 2043. 3. Kleinman, L.I., and Wolfsberg, M . , J. Chem. Phys. (1974) 60, 4740. 4. Kleinman, L.I., and Wolfsberg, M . , J. Chem. Phys. (1974) 60, 4749. 5. Born, M . , and Huang, Κ., "Dynamical Theory of Crystal Lattices" pp. 406-407 (Oxford University Press, New York 1954). 6. Hunter, G . , and Pritchard, H.O., J. Chem. Phys. (1967) 46, 2146, 2153. 7. Kolos, W., and Wolniewicz, L . , J. Chem. Phys. (1965) 43, 2429. 8. Kolos, W., and Wolniewicz, L . , J. Chem. Phys. (1964) 41, 3663, 3674. 9. Poll, J . D . , and Karl, G . , Can. J. Phys. (1966) 44, 1467. 10. Kronig, R. de L . , "Band Spectra and Molecular Structure" pp. 6-16 (Cambridge University Press, London 1930). 11. Coulson, C . A . , Trans. Faraday Soc. (1937) 33, 1479. 12. Ransil, B.J., Rev. Mod. Phys. (1960) 32, 245. 13. Cade, P . E . , and Huo, W.H., J. Chem. Phys. (1967) 47, 614, 649.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
5 Isotope Separation Processes WILLIAM SPINDEL Department of Chemistry, Yeshiva University, New York, Ν. Y. 10033
Stable isotopes ca b few of the methods whic t i n g isotopes are listed exhaustive.
d b
variet
f methods
,
complet
An a p p r e c i a t i o n of the v a r i e t y of conceivable s e p a r a t i o n pro cesses f o r a p a r t i c u l a r task can be gleaned from the f a c t that a survey (9) c a r r i e d out i n 1953 by a group at the Esso Research and Engineering Company f o r the U.S. Atomic Energy Commission, of pos s i b l e methods f o r the production of heavy water (D O), examined 98 p o t e n t i a l processes. In a s i m i l a r v e i n , an advisory committee of the AEC i n 1971 examining the p o t e n t i a l merits of known pro cesses f o r the s e p a r a t i o n of uranium i s o t o p e s , evaluated at l e a s t 25 processes other than gaseous d i f f u s i o n and gas c e n t r i f u g e meth ods (10). Almost all of the processes l i s t e d i n the t a b l e are probably u s e f u l in varying degrees f o r separating isotopes of any of the elements i n the p e r i o d i c t a b l e . None of the processes listed i s c l e a r l y s u p e r i o r to all the others f o r every isotope separating purpose. 2
Although
the s i n g l e - s t a g e s e p a r a t i o n f a c t o r is the best s i n g l e
77
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
78
ISOTOPES
AND
CHEMICAL
PRINCIPLES
measure of the r e l a t i v e ease of s e p a r a t i o n , i t alone does not determine the optimum process f o r a s p e c i f i c separation task. The best method of separation depends on the p r o p e r t i e s of the element, the degree of s e p a r a t i o n d e s i r e d , and the s c a l e of the operation. Even f o r a given i s o t o p e , there i s not one best method. Some general conclusions regarding isotope separation methods drawn by Manson Benedict (11) almost twenty years ago, have not been i n v a l i d a t e d to date by any d i r e c t experimental demonstration. 1. The most v e r s a t i l e means f o r the production of research q u a n t i t i e s of isotopes i s the electromagnetic method. 2. The simplest and most inexpensive means f o r s m a l l - s c a l e separation of many isotopes i s the C l u s i u s T h e r m a l - d i f f u s i o n column. 3. D i s t i l l a t i o n an c a l methods f o r the l a r g e - s c a l e separation of the l i g h t e r elements. 4. Gaseous d i f f u s i o n and the gas c e n t r i f u g e are most economi c a l f o r the l a r g e - s c a l e separation of the h e a v i e s t elements. In t h i s paper a number of isotope separating processes w i l l be examined, p a r t i c u l a r l y those u t i l i z e d on a l a r g e i n d u s t r i a l s c a l e , and the bases f o r the above conclusions w i l l be presented. F i n a l l y , the fundamental p r i n c i p l e s of a photochemical method of isotope separation based upon e x c i t a t i o n by l a s e r l i g h t , which has e x c i t e d a great deal of current i n t e r e s t , w i l l be o u t l i n e d . Electromagnetic
Separators
Electromagnetic separators, r e a l l y l a r g e - s c a l e mass spectrometers , c a l l e d Calutrons because of t h e i r e a r l y development at the U n i v e r s i t y of C a l i f o r n i a C y c l o t r o n Laboratory, were o r i g i n a l l y constructed f o r the Manhattan D i s t r i c t during the second world war i n order to separate 235y ^ kilogram q u a n t i t i e s . At the height of t h i s e f f o r t some 1100 u n i t s were i n o p e r a t i o n , i n two s i z e s , a 48-inch radius u n i t c a l l e d an alpha Calutron and a 24-inch radius u n i t , the beta Calutron. Figure 1 shows a schematic view of a beta separator. In 1945 production of U by t h i s method was discontinued because the gaseous d i f f u s i o n process could be operated at much lower cost and most of the Calutrons were d i s mantled. Only two of the alpha u n i t s and 72 of the beta u n i t s remain i n operation today at the Oak Ridge N a t i o n a l Laboratory. These have been devoted f o r the past 25-30 years to the separation of a most amazing v a r i e t y of i s o t o p e s , both s t a b l e and r a d i o a c t i f . Some 200 kilograms of enriched isotopes i n c l u d i n g over 250 n u c l i d i c species have been produced by the f a c i l i t y . A recent r e t r o s p e c t i v e paper by W. 0. Love (12) o u t l i n e d the accomplishments of the Oak Ridge Electromagnetic Separations Department over n
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
5.
SPINDEL
Isotope Separation Processes
79
the past three decades. Figure 2 and Table 2 summarize the production of separated isotopes by this f a c i l i t y , and indicate the levels of isotopic purities obtainable by the electromagnetic method. The highly purified samples listed i n Table 2. required two or at most three passes through the separator. These results elegantly demonstrate the extreme v e r s a t i l i t y of the electromagnetic method. Table 2. Uranium and plutonium isotope separations. (12)
Mass
Weight Collected
Assay range (%)
233 234 235 236
Uraniu 220 16 530 280
99.99-99.9986 30.86-94.49 98.64-99.9988 85.07-99.996
238 239 240 241 242 244
Plutonium 5 232 238 120 360 3
99.48-99.9988 95.4-99.999 97.03-99.993 83.31-99.997 81.45-99.987 0.61-99.06 Science
Thermal Diffusion The thermal diffusion effect, namely that i n a gaseous mixture subjected to a temperature gradient, as for example i n a vessel with walls at different temperatures, a concentration gradient is established, was f i r s t predicted theoretically by Enskog (13) and by Chapman (14) and confirmed experimentally by Chapman and Dootson (15). It remained for Clusius and Dickel (16) i n 1938 to transform the thermal diffusion effect from a laboratory curiosity into a useful and simple method for separating gaseous, l i q u i d , and isotopic mixtures by devising the thermal-diffusion (or thermo-gravitational) column whose operational principle i s illustrated i n Figure 3. Heating the inner wire (or tube) and cooling the outer wall of a column produces a convective flow pattern, as shown, descending along the cold wall and rising along the heated wire. This convective flow i s super-imposed upon the radial concentration gradient produced by the thermal diffusion effect. The
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
Figure 1.
Η· 2989 35.9
Να
K
Be
M f l
8200 45 A
O.O03 ,.3 2ββ9 47.7
Co
Β
«3.6 659 45.2
Ag 12.4 Ct
C
d
Bo
Au
Hg "
Fr
Ro
8985 78Λ 2,63 30.5 «963 40.9 648 422
6
0
C
N
ÎE°3 3757 ,46
8 1
Se
18.2 Sr
63 9.5
Al
Cu Rb
PRINCIPLES
Diagram of a beta calutron separator ( 12 )
H U
AND CHEMICAL
ΤΙ
7,3 ,4.6
6
,663 22.,
β
Y
Zr
In
Sn
L
°
ΤΙ
33, 7.» 294 8.6 2539 ,7.0
256, 4,.5
Mf Pb
23,4 59.7 4932 ,09 602 24.5 5280 ,05
4, 2.5
P
0
3447 ,27
S
,86 5.8
V
A»
C
0.022
f
s..
4088 35.3 7 8 4
3,0 No
S
b
To
Β
'
^9603 ,50 384 5.4 ,427 28.7 5, 0.6
Τ
·
W Po
227, 27.9 9576 92.9
F
C l
Ν·
43, 26.,
Ar
Μη
* Te
IPar
r «ir
ir
•
I
"*
Ι*»HT 572 22.8
370 6.7
At
Ac
r
v.
0
I· all" nil" al
39.5 Pu
,506 24.3
_ ^ T O T A L ESTIMATED WEIGHT (grwna)
""1,11 "». """^ THOUSANDS e
OF TANK-HOURS
Science Figure 2. Summary of isotope separations by Oak Ridge Electromagnetic Separations Department through 1972 (12)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
5.
SPINDEL
81
Isotope Separation Processes
counter-current flow multiplies the single-stage separation effect and concentrates the component which diffuses preferentially toward the hot wall at the top of the column, while the component which diffuses preferentially toward the cold wall concentrates at the bottom. This multi-stage separation i s essential because the concentration difference between hot and cold walls i s quite small (^10""2) for isotopic mixtures. Usually rather than separating isotopes i n a single long column, a series or series-parallel arrangement of columns (a cascade) i s constructed with individual units perhaps 3-5 meters in length, and appropriate interconnections to permit flow from the top of one unit to the bottom of a succeeding unit. A typical cascade of eleven thermal diffusion columns used at the Mound Laboratory of the U.S.A. (17) i s illustrated i the rate of production of enriched material i s indicated. It i s worth noting that 38ΑΓ concentrates i n the middle of the cascade. This i s typical behavior for an isotope intermediate in mass between two others. The v e r s a t i l i t y of the thermal diffusion method and the separations achieved with simple laboratory cascades by Clusius laboratory for various isotopes i s elegantly demonstrated by the data i n Table 3. 1
Table 3.
Isotopes Separated by K. Clusius by Thermal Diffusion (8)
Year
Isotopes
1939 1939 1942 1942 1950 1950 1953 1955 1956 1959 1959 1960 1962
35C1
371 C
Kr Kr <>Ne »N 13C
8 4 8 e 2
2
*Ne 0» Ar Ne Ar
1 8 3 8 22 3 6
Natural Abundance
Separation Factor
Final Purity
75.7 24.3 57.1 17.5 90.5 0.37 1.09 8.9 0.275 0.204 0.064 9.21 0.37
53 775 45 940 210 135,000 45,000 810 96,500 200,000 9,750,000 12,500 3,300,000
99.4 99.6 98.3 99.5 99.95 99.8 99.8 99.0 99.6 99.75 99.984 99.92 99.991
Adv. Chem,
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
A N DCHEMICAL
Countercurrent Separation Processes Figure 3.
Thermal diffusion column ( 7 )
.0.5ml/hr (Impurities) -2.0ml/hr (Product)
Concentra tion.% 36 38 40 99.8 0.2 0
92.8 7.0 0.2
8.5 30.3 61.2
2.0
1.3 97.7
L^S LnS ί-S LnS L ^ u Î
Feed
1
1
3
1
3
3
3
1
2
0.27 0.06 99.66 500 ml/hr
Gaseous Isotope Separation at Mound Lab. Figure 4.
Hot wire thermal diffusion cascade for separating argon isotopes (17)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
PRINCIPLES
5.
SPINDEL
Isotope Separation Processes
83
The foregoing d i s c u s s i o n f a i r l y w e l l demonstrates that on a small s c a l e any d e s i r e d isotope can be separated e i t h e r by the electromagnetic or the thermal d i f f u s i o n method. In contrast to these l a b o r a t o r y - s c a l e processes, the separations o f the heavier isotope D, of the l i g h t e s t element, hydrogen, and o f the l i g h t e r isotope 235u, o f the h e a v i e s t n a t u r a l element uranium, are c a r r i e d out on a l i t e r a l l y enormous i n d u s t r i a l s c a l e . Uranium Enrichment by Gaseous D i f f u s i o n The United States has three gaseous d i f f u s i o n p l a n t s (18) f o r separating 235u l o c a t e d a t Oak Ridge, Tenn., Paducah, Kentucky; and Portsmouth, Ohio. The t o t a l cost o f these p l a n t s i s about 2.4 b i l l i o n d o l l a r s and the annual operating costs (at f u l l capacity) i s abou consumption i s about 600 annual e l e c t r i c a l energy cost o f over 300 m i l l i o n d o l l a r s . The production capacity o f these p l a n t s i s the equivalent o f 75000 kg. (75 m e t r i c tons) of 90% 5 u per year. Further, the AEC has estimated that even i f the e x i s t i n g p l a n t s are upgraded with newer technology and operated at a higher power l e v e l , to produce an a d d i t i o n a l 60% of enriched uranium, by 1980 a d d i t i o n a l e n r i c h ing capacity w i l l be required i f the U.S. i s t o supply most o f the free-world needs f o r uranium enrichment. A new p l a n t i s p r o j e c t e d a t a cost of 1-1.2 b i l l i o n d o l l a r s with an annual pro duction capacity equivalent to 38.5 m e t r i c tons o f 90% 235TJ. 2 3
Figure 5 shows an a e r i a l view of the Portsmouth, Ohio p l a n t (the newest one), t y p i c a l o f the three i n s i z e . The b u i l d i n g s cover 93 acres of ground but have a f l o o r area about three times as great. The l a r g e r two b u i l d i n g s (χ-330 and x-333) are each about a h a l f mile long by 550 f e e t wide! The operating p r i n c i p l e s o f multi-stage isotope separating processes, such as the gaseous d i f f u s i o n process, were f i r s t developed by K a r l Cohen (19), and f u r t h e r exposed i n a form more d i r e c t l y u s e f u l t o chemical engineers by Benedict and P i g f o r d (1). B i g e l e i s e n has presented a concise summary o f the theory i n h i s review (8). At the heart of an isotope separation cascade i s the elemen tary separating u n i t , or stage which separates a feed stream c a r r y i n g F moles/time of an i s o t o p i c mixture c o n t a i n i n g a mole f r a c t i o n x f o f the d e s i r e d i s o t o p e , i n t o a product (enriched) stream, Ρ at mole f r a c t i o n Xp, and a waste (depleted) stream, W at mole f r a c t i o n x^. For that matter, the same separation process i s c a r r i e d out by the e n t i r e cascade. Two m a t e r i a l balance equations govern the operation o f a separating u n i t o r an e n t i r e cascade.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
84
Total: Isotopic:
AND
CHEMICAL
PRINCIPLES
F = Ρ + W
(l)
Fx- = Px + Wx f ρ w
(2)
The elementary separation factor for a single-separating unit on a two component mixture i s defined as -ι it Elementary factor:
_ (x/l-x)enriched α = ; , —r—:—, ^ , (x/l-x) depleted
13;
Λ
For the separation of uranium isotopes by gaseous diffusion of UF^, the theoretical limiting value of a i s given by Graham's law
m
/ U S
The stages of a cascade are usually combined as shown i n Figure 6. The feed to each stage i s made-up by combining the de pleted stream from the stage above with the enriched stream from the stage below. A material balance anywhere inside the cascade shows that the depleted flow from the (n + l ) t h stage i s just equal to the enriched flow from the η th stage minus the product withdrawn at the top of the cascade. From the material balance equations 1 and 2, and material balances within the cascade, the operating parameters for an isotope separating cascade can be determined (1,8,19). The minimum number of separating stages, N, required to achieve a given overall separation at total reflux (no enriched product withdrawn) i s (x /1-x ) Separation, S = , / ^ ) = w w Ρ
Ρ
1
N
min
=
l n
S / l n
χ
α
N
α
,
(4)
( 5 )
The minimum reflux ratio required, at the feed point, to produce a given product rate Ρ of material at isotopic composition Xp, in systems where α i s close to 1, is <
L/P
> mm
(S"i)? (i-x ) f
( 6 )
£
This minimum feed/product rate would require an i n f i n i t e number of separating stages in the cascade. Neither of these limiting cascade parameters are suitable for an actual isotope production task; the minimum stage cascade
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
5.
SPINDEL
85
Isotope Separation Processes
AEC Gaseous Diffusion Plant Operations Figure 5.
Aerial view
Product stream Ρ mole/sec molt fraction y
p
Top stage
Stage η+ 2
*«+2 Stage η + 1
L -P n
Stage η
Enriching section
Feed stage
Feed stream F mole/sec mole fraction x
f
Bottom stage Waste stream W mole/sec mole fraction *
Stripping section
w
Encyclopedia of Chemical Technology Figure 6.
Separation stages arranged to form a simple cascade (6)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
86
ISOTOPES
AND
CHEMICAL
PRINCIPLES
produces no material at the desired concentration, the minimum flow cascade requires an i n f i n i t e number of stages. The ideal cascade i s one in which the separation achieved per stage i s just sufficient to make the composition of the depleted stream from the n+1 stage exactly equal to that of the enriched stream from the n-1 stage. Thus no entropy of remixing i s produced when these streams are combined to form the feed of the η th stage. This cascade i s often called the no-remixing cascade. In the ideal cascade, ff
χ
lf
n+1 n-1 η = χ = x w ρ f £
N
ideal
= 2 N
min
(7)
Since the reflux ratio (flow/product) required at each stage in a cascade (eqs. 6 and 7) i s related to the isotopic concentra tion at that point, the size of the separating units are tapered from the feed point towards the ends of the cascade in order to minimize the size and cost of the plant, the pumping energy con sumed, the hold-up of enriched material, and the time required to reach steady-state enrichment. The characteristic shape of an ideal cascade i s indicated i n Figure 7 which diagvomatioa1Vy depicts the parameters for an ideal gaseous diffusion cascade to produce 1 kg of 90% 235TJ from natural U F 5 containing 0.711% 235TJ and discarding waste UF6 at 0.200% 235TJ. The v e r t i c a l height at any point in the figure i s proportional to the stage number measured from the waste end of the cascade, the width i s propor tional to the total interstage flow at that stage. The quantities of uranium feed required and waste produced, the numbers of stages in the enriching and stripping sections (above and below the feed point), and the inter-stage flow at any stage are a l l calculated by eqs. 1-7 and the elementary factor 1.0043. Thus, from eq. 7, and the product and feed isotope concentrations, the interstage flow at the feed point i s calculated α/Ρ) . 0Q711) ^'^ideal · (.0043) (.00711) (1-.00711) 2
m
5 8
Ζ
D O
8 2 9
>
(8)
o z y
Almost 60,000 moles of natural abundance uranium flow through the cascade, at the feed stage for each mole of 90% 235TJ withdrawn. The total interstage flow,J, in the cascade, i.e. the total area enclosed in Figure 7, i s (20) J
=
"(odl
2
p
t -V(*p) + W-V(x ) - F-V(x )] w
f
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
(9)
5.
SPINDEL
Isotope Separation Processes
87
where V(x) i s the function (2x-l) In x$.-x), often called the value function or separation potential. The separation potential is a function of composition only and i s dimensionless. It has a value of zero at χ = 0.5 and i s positive for a l l other values of x, symmetrically about the minimum at χ = 0.5. Eq. 9 i s of major importance for estimating the size and cost of an isotope separation plant. It indicates that the total flow i s a product of two factors; the f i r s t of these, proportion al to 1/(a-1)2 i s a function only of the elementary separation factor which i s determined by the separation process used. The second factor [in square brackets], which i s usually called the separative duty or separative work units (S.W.U.) i s a function only of quantities and concentrations of feed, product, and waste. It has the sam t i t i e s of material, an used to accomplish the separation task. The significance of the magnitude of the elementary factor i s immediately apparent; a two-fold reduction i n (a-1) requires an increase i n the total flow by a factor of 4. Since for a gaseous diffusion process, the total flow rate i s closely related to the total area of porous barriers, the total pumping capacity and the total power consumption required, a l l the associated costs vary proportion ately. The S.W.U. provides a quantitative measure of the isotope separation task for any conceivable process. For the task des cribed i n Figure 7, 227.3 kg S.W.U. are required per kg of 90% 235TJ. For gaseous diffusion, with (a-1) = .0043, 98.2 million kg U must be pumped through the cascade i n order to produce 1 kg of product. It i s most interesting to note from the shape of the ideal cascade i n Figure 7, and from eq. 9, that most of the area in the figure, and therefore most of the volume of the cascade, and the energy which must be supplied, i s associated with enriching the isotope from natural abundance by the f i r s t factor of ten, say from 0.711% to 7% 235u. To prepare 1 kg of 235TJ t 7% con centration requires 77% of the S.W.U. required to produce a kg of 235TJ at 90% enrichment. The cost of enrichment by any combina tion of processes i s essentially determined by the process used at the base of the cascade. a
To give some feeling for the magnitude of cost of enriching uranium, the A.E.C. estimates (18) that in the most efficient projected gaseous diffusion plants, 0.266 W of continuous elec t r i c a l energy w i l l be consumed per S.W.U.»yr. This corresponds to a consumption of 530,000 kW-hr per kg of 90% 35TJ, At 5.5 mills per kw*hr the cost of e l e c t r i c a l energy i s $2900 per kg of 9Œ 235u. 2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
88
ISOTOPES
AND CHEMICAL
PRINCIPLES
The essential elements of a gaseous diffusion cascade for 235TJ enrichment, their physical arrangement and inter-connections are indicated i n Figure 8. The inter-relationships between the motors, compressors, heat exchangers, diffusion membranes, and stage control valves are a l l shown. The largest size diffusion stages used in one of the AEC plants are seen i n the photograph in Figure 9, which shows the equipment size i n comparison to a man. Deuterium Enrichment by Exchange and D i s t i l l a t i o n Now l e t us examine deuterium enrichment—first the scale of the current effort, and a projection of the deuterium requirements in the future. The united States has constructed two plants, each capable of producin 500 plants i s dismantled, an at about one third of i t s capacity, producing 180 tons of D2O per year. In Canada, two plants with a combined output of 1600 tons D2O per year have recently been completed (21). D2O i s used as a neutron moderator and heat transfer medium for power reactors. The D 0 requirement i s about one ton D2O per megawatt of e l e c t r i c a l capacity. In the United States the development of power reactors has moved toward reactors fuelled with enriched 235u, using light water as moderator and either gas or liquid metals as coolants. In Canada the direction has been toward the use of natural abundance uranium (0.7% 235TJ) as fuel, with D2O as moderator. 2
It i s of interest to project the deuterium requirements for use as a fusion fuel i n controlled thermonuclear reactors (CTR). Robert Hirsch, Director of the U.S.A.E.C. Division of Controlled Thermonuclear Research has predicted the production of s i g n i f i cant amounts of fusion energy by 1980, and fusion power commer cialization before the turn of the century (22). A report pre pared by a group at Brookhaven National Laboratory (23) can be used for estimating energy and deuterium requirements i n a future fusion-based energy regime, i n which a l l f o s s i l fuels (except for ship and petrochemical feed requirements) are replaced with synthetic fuels produced by D-D fusion reactors, and a l l electrici ty production i s by CTR. They estimate that by the year 2020, 500 χ 1015 BTU of fusion energy would be required to supply a l l U.S. energy needs (with exceptions noted above). The total amount of deuterium needed to produce this energy (assuming 100% e f f i ciency for the D-D fusion reaction) would be only 1000 tons D2 per year (5000 tons D 0). If the technical problems can be over come, the fusion process promises to be astonishingly efficient from the standpoint of resource consumption! 2
Four, of the many processes proposed and used for deuterium
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
5.
SPINDEL
Isotope Separation Processes
174.7 kg WASTE x =0.0020 w
HEADS FLO α = 1.0043
Figure 7. Parameters of an ideal separation cascade f uranium isotope separation or
AEC Gaseous Diffusion Plant Operations Figure 8.
Arrangement of gaseous diffusion stages ( 18 )
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
89
90
ISOTOPES
AND CHEMICAL
PRINCIPLES
enrichment, w i l l now be examined. Two are d i s t i l l a t i o n process es and the remaining two are chemical exchange processes. The basic operational parameters of the d i s t i l l a t i o n processes, water d i s t i l l a t i o n and hydrogen d i s t i l l a t i o n , are compared in Table 4. The water d i s t i l l a t i o n system was actually used i n early plants
Table 4. D i s t i l l a t i o n process requirements to produce deuterium at 99.8% D from natural feed at 0.0149% D. (4,24) Water Distillation effective α temperature pressure min.no.stages min.reflux ratio percent recovery moles feed/moles product Operating Cost ($/lb D 0) 2
Hydrogen Distillation
1.05 50°C
1.52 23 Κ
141,000 5 133,940 176
19,600 90 7442 16
built and operated in the U.S. between 1943 and 1945. These plants were shut down in 1945 because of the high operating costs. The hydrogen d i s t i l l a t i o n process appears extremely favorable be cause of the large effective fractionation factor, and the total energy cost for producing deuterium by this process has been estimated (24) to be about 1700 kw · hr per pound of D 0. This corresponds to an energy cost of $9.30 per pound of D£0 (at 5.5. mills/kW · hr) ; the energy consumption i s probably the lowest of any of the deuterium separating processes. Several plant designs for producing deuterium by d i s t i l l a t i o n of liquid hydrogen were actually carried out during the decade 1941-1951 (25-27). Be cause of the technical problems of handling large amounts of liquid hydrogen at cryogenic temperatures of ^20 Κ , and because of the unavailability of sufficiently large quantities of hydro gen gas to serve as feed for a large-scale deuterium enriching plant, no hydrogen d i s t i l l a t i o n plant has ever been constructed in the U.S. Several relatively small hydrogen d i s t i l l a t i o n plants have been constructed and operated in the past 15 years in France, Germany, India and the Soviet Union. The reported magnitude of D2O production i n these plants i s in the range 3-14 tons D2O per year. 2
It seems of interest to re-examine the hydrogen d i s t i l l a tion system, now, in light of current liquid hydrogen technology, and of projected hydrogen production as a synthetic fuel. At present, the U.S. production capacity for liquid hydrogen, mainly for use in the space program, i s about 150 tons of liquid hydro gen per day. This, by i t s e l f , would permit the extraction of 15
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
5.
SPINDEL
91
Isotope Separation Processes
tons deuterium (75 tons D2O) per year. More to the point i s a developing interest i n the use of liquid hydrogen as an aircraft fuel, particularly for hypersonic a i r c r a f t . Design studies have shown that the flying range of a large transport could be i n creased 30-40% using liquid hydrogen as fuel (28). Projections regarding liquid hydrogen consumption by aircraft (29) indicate that i f only ^3.5% of a i r transport i n the U.S. in the year 2000 were hydrogen fuelled, liquid hydrogen production would be 4.25 million tons per year, and the extractable deuterium, by d i s t i l l ation, would be 1100 tons per year. It should be noted that the energy consumption for the extraction of deuterium from liquid hydrogen, by a d i s t i l l a t i o n process operating parasitioally on a liquid hydrogen plant, would probably be lower by one or two orders of magnitude than the energy consumption previously e s t i mated for hydrogen d i s t i l l a t i o systems Previou estimate based on using gaseou energy would be supplied fo the purpose o isotope extraction; in parasitic operation on a liquid hydrogen f a c i l i t y only the heat losses from the d i s t i l l a t i o n columns would correspond to energy required for the deuterium extraction. Two significant chemical exchange reactions useful for the concentration of deuterium are HDS(gas) + H 0(liq) = H S(gas) + HDO(liq) o
2
0
2
HD(gas) + NH (liq) = H^gas) + NH^Diliq) 3
(10) .
A simple system for u t i l i z i n g these exchange reactions for deu terium concentration i s illustrated i n Figure 10. In both the exchange reactions l i s t e d above, the desired isotope, deuterium concentrates i n the liquid phase, so the phase conversion unit i n Figure 10 consists of a chemical reactor which converts the en riched compound (HDO or NH2D respectively for the reactions listed) into the isotopically depleted compound (hydrogen gas or hydrogen sulfide gas). The exchange tower must contain sufficient separating stages to multiply the single stage enriching factor, α to produce the desired degree of overall separation. As indicated earlier, interstage flows are very large i n isotope separation processes, and the quantities of processing chemicals required for reflux reactions (phase conversion) repre sent a major expense in any chemical exchange separation system, unless the process i s parasitic on a chemical manufacturing process and/or the by-products of the reflux reactions upgrade the value of the reflux chemicals. An elegant way to avoid the need for reflux chemicals i s provided by using a dual-temperature exchange system of the type f i r s t proposed, independently by Spevack (30) and by Geib (31). Such a system uses the variation of the exchange equilibrium constant with temperature, to substi-
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
92
ISOTOPES
A N DC H E M I C A L
PRINCIPLES
AEC Gaseous Diffusion Plant Operations Figure 9. Close-up view of actual equipment (diffusers and compressor) in a gaseous diffusion plant (18)
LIQUID FEED "
WASTE GAS"*"
EXCHANGE TOWER
-PRODUCT
PHASE CONVERSION Figure 10. Simple chemical exchange system for isotope separation for an isotope concentrating in the liquid phase
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
5.
Isotope Separation Processes
SPINDEL
93
tute thermal r e f l u x i n s t e a d of chemicals f o r r e t u r n i n g the des i r e d isotope from the enriched compound to the i s o t o p i c a l l y dep l e t e d compound. The magnitude of the exchange constants f o r deuterium exchange, and t h e i r v a r i a t i o n with temperature, make a dual-temperature exchange process p a r t i c u l a r l y s u i t a b l e f o r conc e n t r a t i n g isotopes of hydrogen. Figure 11 i l l u s t r a t e s a s i m p l i f i e d arrangement of components and the magnitude of operating parameters f o r the concentration of deuterium by dual-temperature exchange between hydrogen s u l f i d e gas and l i q u i d water. i s c i r c u l a t e d i n a closed loop countercurrent to a descending stream of water. In the c o l d e r column deuterium concentrates i n the l i q u i d phase; the e q u i l i b r i u m constant f o r the exchange at the c o l d temperature, 30°C, i s 2.20. In the h o t t e r column, 130°C i s 1.69, and deuteriu gaseous phase. Thus, dual-temperatur operatio f o r a chemical r e a c t i o n to return the d e s i r e d isotope from the phase i n which i t enriches to the phase i n which i t i s depleted. In operation, feed water at the n a t u r a l deuterium abundance of 145 ppm i s fed to the top of the c o l d column, depleted water at 120 ppm deuterium i s discarded from the bottom of the hot column, and enriched D2O i s withdrawn between the c o l d and hot columns. The e f f e c t i v e s i n g l e - s t a g e enrichment f a c t o r i s simply the r a t i o °f c o l d / h o t 1·26, f o r t h i s process at the i n d i c a t e d operating temperatures. This e f f e c t i v e s i n g l e - s t a g e enrichment f a c t o r determines the number of separating stages required i n the hot and c o l d columns r e s p e c t i v e l y , to achieve the d e s i r e d degree of enrichment. The maximum f r a c t i o n of the isotope which i s e x t r a c t able from the feed i n t h i s operating mode, i s simply the d i f f e r ence i n the a s at the two operating temperatures d i v i d e d by the l a r g e r of the two values. These r e l a t i o n s h i p s e f f e c t i v e l y l i m i t the use of a dual temperature system to the separation of hydrogen isotopes. The operating temperatures shown i n Figure 11 f o r the H2S-water exchange system are determined by the p r a c t i c a l considerations that I^S-hydrate p r e c i p i t a t e s i n the low temperature column below 30°C, and at the operating pressure of 300 p s i , the mole f r a c t i o n of water i n the gaseous phase becomes excessive above 130°C. a
a
=
T
A dual-temperature system requires twice the number of separating stages (hot and c o l d columns) needed f o r a s i n g l e temperature system with the same e f f e c t i v e a, but the need f o r r e f l u x i n g chemicals i s eliminated, and the heat energy required to maintain the temperature difference can be g r e a t l y reduced by the use of heat exchangers at appropriate points to pre-heat the gas and l i q u i d streams entering the hot column, and to p r e - c o o l the gas entering the c o l d column. A l l of the major plants c u r r e n t l y producing
deuterium i n
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ISOTOPES
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quantities >20 tons per year u t i l i z e the I^S-water dual-tem perature exchange system. Figure 12 shows schematically the cas cade arrangement used at the Savannah River plant for the produc tion of 500 tons of D2O per year (32). Concentration of deuterium from the 15% level produced by exchange, to the f i n a l product concentration of 99.8% D i s accomplished by vacuum d i s t i l l a t i o n of water. It should be noted however, that practically a l l of the separative work i s expended i n getting from the feed concentration of 145 ppm to 15% D. This task requires 7383 SWU; about 0.7% additional SWU are needed to produce the f i n a l product. The economics of the process i s determined at the base of the cascade! Fundamental parameters for the ammonia-hydrogen exchange reaction which i s listed in eq. 10, are much more favorable than the equivalent factors fo I^S-wate by Claeys, Dayton and Wilmart efficient homogeneous catalyst for the exchange immediately stimulated a group at Brookhaven National Laboratory, led by Bigeleisen to carry out extensive experimental and theoretical studies of this system (34,35). They determined that a dualtemperature system operating with hot column at 70°C (a = 2.9) and a cold column at -40°C (a = 5.9) would have an effective α = 2.0, and would permit extraction of 50% of the deuterium from the ammonia feed. They also showed that the exchange, when catalyzed by NaNH2 dissolved i n the liquid, was sufficiently rapid even at the lower temperature to reach equilibrium i n reasonably sized exchange columns. To date, this system i s being used i n a single-temperature plant in France producing about 20 tons D2O per year, about which only few details have been published. The major limitation on the use of this process has been the a v a i l a b i l i t y of sufficient quantities of ammonia for plant feed. Even a plant producing 1000 tons ammonia per day would only provide sufficient feed to permit production of 60-70 tons D2O per year. Again, one sees the interplay between science and technology, and one i s faced with the fact that technological rather than s c i e n t i f i c considerations are often the over-riding determinants for large seale isotope separation. Photochemical Isotope Separation
Processes
For many years attempts have been made to use photochemical processes for the separation of isotopes. The basic idea i s to u t i l i z e the difference in the absorption spectra of different isotopic species, and by use of sufficiently monochromatic light of an appropriate wavelength to excite only one of the species to an upper energy level. The excited species may then be separated by chemical or physical means from i t s isotopic partners; the separating process need not have any inherent isotopic selectivity. A particularly successful application of the method was to the separation of Hg isotopes by Gunning et a l . (36,37). For example,
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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Isotope Separation Processes
Fraction of isotope extracted
95
a cold — ahot
WATER 145 ppm
I COLD » 2.20 H S 2
300 pel PRODUCT 15 % D 0 2
WATER
J_
Figure 11. Simplified flow sheet for a dual-temperature exchange system for concentrating deuterium
Feed 0.0145 % D
Product 15 % D
Waste 0.012 %D' Stage I
Stage Π Chemical Engineering Progress
Figure 12. Simplified schematic of Savannah River exchange unit showing principal towers and liquid flow paths. There are 24 such units (500 tons D 0/year). Stage 1: hot towers, 12' dia. X 70 trays; cold towers, 1Γ dia. χ 70 trays. Stage 2: hot towers, 2 X 6.5' dia. X 70 trays ea.; cold towers, 2 X 6.5' dia. X 85 trays ea. (32). 2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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ISOTOPES AND CHEMICAL PRINCIPLES
a monoisotopic 202Hg resonance lamp was used to i l l u m i n a t e mercu ry vapor containing the n a t u r a l abundance mixture of Hg isotopes (196, 198-202, 204). E s s e n t i a l l y only H g isotopes were e x c i t e d to an upper e l e c t r o n i c s t a t e , i n which they reacted with an added gas such as H2O, to form HgO which could then be sepa rated e a s i l y from unreacted Hg. Values o f α as high as 3 were obtained f o r 202Hg. 2 0 2
The e s s e n t i a l requirements f o r any photochemical isotope separation scheme f o l l o w : 1. Absorption spectrum with a w e l l resolved isotope s h i f t ( v i b r a t i o n a l - r o t a t i o n a l or v i b r o n i c f o r molecules, e l e c t r o n i c f o r atoms) 2. A l i g h t source e x c i t e absorption by on intense to e x c i t e a s u b s t a n t i a l f r a c t i o n of the i s o t o p i c species i n the gas mixture. 3. D e a c t i v a t i o n o f e x c i t e d species by c o l l i s i o n s with other species must be minimized; energy t r a n s f e r from one i s o t o p i c species to another must not occur before step 4. 4. A chemical or p h y s i c a l process which separates e x c i t e d species from the others. The f i r s t and l a s t of these requirements are dependent on the s p e c i f i c i s o t o p e s , molecular species and chemical or p h y s i c a l processes. They cannot be changed, but the appropriate s e l e c t i o n depends on the ingenuity o f the i n v e s t i g a t o r . Requirements two and three, on the other hand, have been v i r t u a l l y unobtainable u n t i l the development of l a s e r s , which produce very l a r g e outputs of r a d i a n t energy with an extremely narrow band width. A l s o , the a b i l i t y o f a l a s e r t o emit t h i s l a r g e amount o f monochromatic radiant energy w i t h i n a time i n t e r v a l as short as s e v e r a l nano seconds i s important i n meeting the energy t r a n s f e r requirements. Two recent papers by Letokhov (38) and by Moore (39) contain e x c e l l e n t and d e t a i l e d d i s c u s s i o n s o f the a p p l i c a t i o n o f l a s e r s to i s o t o p e s e p a r a t i o n . The approaches f a l l i n t o two broad cate gories which may be c h a r a c t e r i z e d as one-step and two-step pro cesses. The one-step process i s p a r t i c u l a r l y simple conceptually but not as g e n e r a l l y a p p l i c a b l e . I t i n v o l v e s s e l e c t i v e e x c i t a t i o n o f a s u i t a b l e molecule to an upper p r e - d i s s o c i a t i v e s t a t e . This upper s t a t e i s a n o n - d i s s o c i a t i v e one whose p o t e n t i a l energy surface i n t e r s e c t s another surface corresponding to a d i s s o c i a t i v e s t a t e . Such a system i s i l l u s t r a t e d i n Figure 13. I f the d i s s o c i a t i v e l i f e t i m e i s s h o r t e r than the r a d i a t i v e l i f e t i m e , then s e l e c t i v e p h o t o - e x c i t a t i o n can produce i s o t o p i c a l l y enriched
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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dissociation products. Yeung and Moore (40) irradiated an equimolar mixture of D2CO and H2CO using a frequency-doubled Ruby laser (3472 Â ) . The hydrogen gas formed by photo-dissociation of the formaldehyde showed a 6:1 D to H ratio! To maximize isotopic separation, the compound should be irradiated with a narrow line in a spectral region where the undesired isotopic species i s r e l atively transparent. Moore (41) has observed the absorption spectra of the isotopic (^C and C) formaldehydes and concluded that irradiation with light of an appropriate wavelength should yield l^CO preferentially as a dissociation product. 12
Selective two-step processes are more generally applicable to isotope separation. Several approaches are indicated i n Figure 14. The diagram on the l e f t (a) shows approaches to selective photo-ionization. A atoms to the upper state cient energy to ionize atoms in state 2, but insufficient to ion ize those i n the ground state, then ionizes the excited atoms. It should be noted that the second photon, V2 need not be highly monochromatic. Alternatively, the frequencies V-^ and V2 may be identical, as indicated by V3. The V3 frequency must be reso nant with a transition of one of the isotopic species, and of sufficiently high frequency to ionize the excited atom. The dia gram on the right of Figure 14, illustrates a two-step, photo-dis sociation process. The selective photon, excites a vibration a l transition. The level excited should be sufficiently high so that i t s thermal population i s negligible, and i t s absorption co efficient for i>2 is appreciable, while the corresponding c o e f f i cient from the thermally populated state i s negligible. If the A and Β fragments are chemically stable, one need only separate isotopically enriched A or Β species from the remaining AB mole cules by a simple chemical separation. If the fragments are re active, a chemical trapping scheme which does not induce isotopic scrambling must be devised. Notice that the two-step processes are really limited to laser light sources, because the two light pulses must be closely synchronized to avoid energy transfer. One can only estimate very crudely at this time the energy requirements for laser isotope separation. Moore has pointed out (39) that a process yielding one separated atom for each 3300 Â photon absorbed, requires 0.1 kw. hr. of light energy per mole of separated isotope. Assuming a laser efficiency of 10"* this corresponds to an energy consumption of 1000 kw. hr. per mole. Naturally, these calculations essentially represent a theoretical minimum goal to be approached. To date, the experimentally demonstrated f e a s i b i l i t y of uranium isotope separation has been l i m i t ed to the preparation of about 10^ 235jj atoms per second (42), but the magnitude of the industrial enterprise involved i n isotope separation certainly appears to j u s t i f y an expanded level of fundamental research i n this direction.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
A N DC H E M I C A L
Figure 14. Schematic diagram of selective turnstep processes: (a) photo-ionization, (b) photodissociation (39)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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Acknowledgement The author i s most pleased to acknowledge the assistance of his colleagues, George W. Flynn and Ralph E. Weston through numer ous discussions of isotopic separation "by laser excitation. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Benedict, Μ., and Pigford, Τ. Η., "Nuclear Chemical Engineer ing", McGraw-Hill, New York, 1957. Kistemaker, J . , Bigeleisen, J . , Nier, A.O.C., (Eds.), "Pro ceedings of the International Symposium on Isotope Separa tion", North-Holland Publ. Co., Amsterdam, 1958. Koch, J., (Ed.), "Electromagnetic Isotope Separators and Ap plications of Electromagneticall Holland Publ. Co. London, H., (ed.), "Separation of Isotopes", George Newnes, Ltd., London, 1961. J . Chim. Phys., (1963), 6 0 , (1 and 2), Symposium on "Physical Chemistry of Isotope Separation". Schacter, J . , Von Halle, Ε., Hoglund, R. L., "Encyclopedia of Chemical Technology", Standen, Α., (Ed.), 7, 9 1 , Wiley and Sons, New York, 1965. Pratt, H. R. C., "Countercurrent Separation Processes", Elsevier Publ. Co., Amsterdam, 1967. Bigeleisen, J . , i n "Isotope Effects i n Chemical Processes", Advances i n Chemistry Series, 89, Am. Chem. Soc., Washington, D. C., 1969, Chapter 1. Barr, F. T., Drews, W. P., Chemical Engineering Progress, (1960), 56, 49. Benedict, M., Berman, A. S., Bigeleisen, J . , Powell, J . Ε., Shacter, J . , Vanstrum, P. R., "Report of Uranium Isotope Separation Review Ad Hoc Committee", ORO-694, Oak Ridge, Tenn., June 1972. Ref. 1, p. 516. Love, L. O., Science, (1973), 182, 343. Enskog, D., Z. Physik, (1911), 12, 56, 533. Chapman, S., P h i l . Trans, Roy. Soc. London Ser. A, (1916), 216, 279. Chapman, S., Dootson, F. W., P h i l . Mag., (1917), 33, 248. Clusius, K., Dickel, G., Naturwiss., (1938), 2 6 , 546. Haubach, W. J., Eck. C. F., Rutherford, W. M., Taylor, W. L., "Gaseous Isotope Separation at Mound Laboratory-1963", MLM1239, Miamisburg, Ohio, June 1965. "AEC Gaseous Diffusion Plant Operations", ORO-684, Washington, D. C., January 1972. Cohen, K., "The Theory of Isotope Separation", McGraw-Hill, New York, 1951. Ref. 1, p. 396.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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21. 22. 23.
24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
AND CHEMICAL
PRINCIPLES
Bancroft, A. R., "The Canadian Approach to Cheaper Heavy Water", AECL-3044, Chalk River, Ontario, February, 1968. New York Times, February 2 8 , 1974. Powell, J . , Salzano, F., Sevian, W., Hoffman, Κ., Bezler, P., "An Evaluation of the Technical, Economic and Environmental Features of a Synthetic Fuels Economy Based on Fusion Reactors", ENL-18430, Upton, Ν. Y., November 1973. (Figure 3.5). Benedict, Μ., "Survey of Heavy Water Production Processes", Proc. Int'l Conf. on Peaceful Uses of Atomic Energy, (1956), 8, 377, United Nations, Ν. Y. Clusius, Κ., Starke, Κ., Z. Naturforschung, (1949), 4A, 549. Hydrocarbon Research, Inc., "Low Temperature Heavy Water Plant", ΝΥO-889, Ν Υ. Ν Υ. March 1951 Murphy, G. M., (Ed.) III, 4F, McGraw-Hill TID-26136, "Hydrogen and Other Synthetic Fuels--A Summary of the Work of the Synthetic Fuels Panel", September, 1972, Superintendent of Documents, Washington, D. C. AET-8, "Reference Energy Systems and Resource Data for use in The Assessment of Energy Technologies", A p r i l , 1972, Assoc. Univ. Inc. Upton, New York. Spevack, J . , Report MDDC-891(1947). Clusius, Κ., et al., FIAT Review of German Science (19391946), Physcial Chemistry, p. 19ff. Bebbington, W. P., Thayer, V. R., Chem. Eng. Progress, (1959) 55, 70. Claeys, Y., Dayton, J . C., Wilmarth, W. K., J . Chem. Phys., (1950), 18, 759. Perlman, M., Bigeleisen, J . , Elliot, N., J . Chem. Phys., (1953), 2 1 , 70. Bigeleisen, J . , Ref. 2., p. 121. Pertel, R., Gunning, Η. Ε., Can. J . Chem., (1959), 37 35. Gunning, Η. Ε., Strausz, O. P., Adv. i n Photochem., (1963), 1, 2 0 9 . Letokhov, V. S., Science, (1973), 180, 451. Moore, C. B., Accounts of Chem. Res., (1973), 6 , 323. Yeung, E. S., Moore, C. B., Appl. Phys. Lett., (1972), 2 1 , 109.
4 1 . Moore, C. B., private communication. 42. Chem. and Eng. News, (1974), 52, No. 27, p. 24 (July 8). *Research supported by U.S.A.E.C. under Contract AT(11-1)-3581. †Present address:
Division of Chemistry and Chemical Technology National Academy of Sciences-National Research Council, 2101 Constitution Avenue, Washington, D. C. 20418
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6 Condensed Phase Isotope Effects, Especially Vapor Pressure Isotope Effects: Aqueous Solutions W. A L E X A N D E R V A N
HOOK
Chemistry Department, University of Tennessee, Knoxville, Tenn. 37916
Abstract The theory of isotope effects i n condensed phase systems, especially vapor pressure isotope effects (VPIE) is b r i e f l y reviewed. It i s pointed out that the VPIE can be employed as one measure of the effect of intermolecular forces on the motions of molecules i n condensed phases. This i s illustrated with a number of examples from the recent literature and from our own laboratory. A more detailed description of our recent work on thermodynamic solvent isotope effects i n aqueous systems i s presented. Experiments on vapor pressures, freezing points, and heats of solution and dilution of solutions of electrolytes i n HOH and DOD are described. Implications are discussed with respect to the aqueous solvent structure problem. A.
Introduction
This symposium has revealed a number of different aspects of the physical and theoretical bases of isotope effects. Generally, we are interested in the effect of isotopic substitution on some process. In the direct approach one formulates a description of the process i n terms of a set of energy levels describing the "before" and "after" states of each isotopic isomer which enters. With those energy states as input information one then calculates population distributions, and isotope effects on population distributions, evaluates partition functions, takes ratios, and thereby determines the isotope effects (IEs). More commonly, detailed information concerning energy level distributions i s not available, so one i s forced to the alternate point of view that the measured bulk property IEs serve as probes to investigate IEs on the energy level distributions. Further, we note that energy level distributions can often be theoretically calculated to sufficient precision from appropriate potential energy functions. These functions relate the potential energy to interparticle geometry. It i s well established that properly evaluated potential 101
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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functions are independent of isotopic substitution to a good approximation (but see the paper by Wolfsberg ÇL)). It follows that measured isotope effects can serve as probes i n investigations of some details of the nature of a given potential energy surface. This i s the basic reason for s c i e n t i f i c interest in isotope effects; thus, kineticists make certain deductions about the nature of transition states by reasoning from observed kinetic isotope effects (2); i n a similar vein we w i l l interpret measured isotope effects on condensed phase properties i n terms of the intermolecular forces on the motions of molecules i n condensed systems. Most of the material i n the present talk w i l l be concerned with the Vapor Pressure Isotope Effect (VPIE). This i s for two good reasons. F i r s t , the VPIE i s the most throughly studied of a l l condensed phase effect mation i s available to exemplif Second, the methods used to formulate a description of the VPIE can to a large extent be carried over and applied to other kinds of condensed phase effects, so a certain generality i s present. More superficial, but equally useful i s the fact that i t i s only with the VPIE that one i s naturally led to employ the attenuated vapor as the reference standard state. This simplifies things and also has a good deal of appeal to anyone brought up on Lewis and Randall (3). A number of recent reviews containing s i g n i f i cant amounts of material on the VPIE are available. Among these, the recent ones by Bigeleisen, Lee and Mandel (4) and by Jancso and Van Hook (5), afford convenient access to the literature. The effect of isotopic substitution on the vapor pressure i s an old problem — the f i r s t theoretical calculations were carried out by Lindeman (6) for monatomic Debye solids. On the basis of this work i t was suggested that experimental investigation of the VPIE might prove a powerful tool i n settling the then controvers i a l question of the existence of the zero point energy. The f i r s t experiments were carried out i n 1931 by Keesom and van Dijk ( 2 ) who succeeded i n obtaining a p a r t i a l separation of the isotopes of neon i n a d i s t i l l a t i o n experiment. Very soon thereafter other separations using the VPIE were successful, and i n the middle and late t h i r t i e s a number of authors including Scott, Brickwedde, Urey and Wahl ( 8 ) and Topley, Bailey and Eyring (9) applied the concepts of s t a t i s t i c a l thermodynamics to the problem of the VPIE of structured molecules. They pointed out that there were a number of factors which could contribute to the effect. Herzfeld and Teller (10) i n an important paper which appeared in 1938, showed that at low temperatures the VPIE i s principally determined by the isotopic zero point energy differences associated with the external degrees of freedom. At high temperatures, on the other hand, a classical treatment i s j u s t i f i e d and there i s no isotope effect. The authors nicely showed that at the intermediate temperatures commonly encountered care must be taken to properly account for excitation into higher quantum levels. We
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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have seen earlier (11) that this can be accounted by a number of different but equivalent approaches. The original authors developed the VPIE i n terms of an expansion about the c l a s s i c a l limit in powers of successively higher quantum corrections. The method is therefore limited to "almost c l a s s i c a l " systems. In the case of the intermolecular part of the potential the quantum correction i s larger i n the condensed than i n the vapor phase because the mean square force i n the (ideal) vapor i s zero. It follows that the lighter isotope exhibits the higher vapor pressure. The inverse effect often observed for molecules with structure was attributed to the changes i n the intramolecular potential which occur on condensation. In spite of this rather clear and early statement of the theoretical basis for the VPIE, confusion persisted u n t i l a reformulation was given almost twenty-five years later by Bigeleisen (12) terms of reduced partitio giving a demonstration of the important connection between the VPIE and details of the molecular structure. A good deal of the present paper i s concerned with this point. We w i l l return to a quantitative formulation of the Bigeleisen approach after a brief consideration of the p i c t o r i a l model described below. B.
A Physical Picture
In examining the VPIE, we are concerned with the equilibrium between the condensed and vapor phases. The latter may be approximated as an ideal gas ( i f this i s not adequate methodologies exist for correcting to the ideal gas reference state). In the gas the freely rotating and translating molecule may be regarded as a set of 3n-6 coupled oscillators (n i s the number of atoms, i f the molecules are linear there are 3n-5 oscillators, i f they are monatomic there are no oscillators and rotation i s not defined) . Two gross effects must be considered on condensation. First intermolecular forces i n the condensed phase cause perturbations of the energy levels characterizing the internal oscillator energies. A given energy level may either be raised or lowered with respect to i t s gas phase value (blue or red shifted). Det a i l s w i l l depend on the specific natures of the intermolecular force and the type of oscillator (for example i t i s well established that stretching motions of non-polar molecules are red shifted on condensation by the dispersion interaction). The second effect arises because the six (five for linear, three for monatomic molecules) external motions of the molecule, originally free (zero frequency) i n the attenuated vapor, become hindered (of real frequency) as the molecule condenses into the attractive well defined by the intermolecular potential. These six external condensed phase modes are necessarily blue shifted from the (zero) gas phase values. Notice that for both internal and external modes the reduced masses of the particular motions are such that the more lightly substituted molecule undergoes the larger s h i f t .
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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104
AND
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PRINCIPLES
Since the intramolecular motions can shift either to the blue or red the net isotopic difference of the shifts can be either positive or negative, and can therefore lead to positive (normal) or negative (inverse) VPIEs. In general, excitation into upper v i brational levels must be considered and these contributions can lead to complicated temperature dependencies (14). Some of the ideas outlined above are presented i n a conventional schematic fashion i n Figure 1. Diagrams are given for both internal and external modes. C.
Functional Forms
The ideas that we have crudely expressed above and in Figure 1 have been quantified by Bigeleisen (12) who derived an expression for the VPIE. In thi In 1^- = In ~ Ρ s
f
f c
- In
f s'g
+ (RT)" (P V
2
1
+ (Β Ρ + ^ C P ) - (RT)" / ο ζ ο ν
V ?
- PV) '
x
(BP+^CP )' ο 2 ο
P'dV
T
(1)
f
f
lighter isotope, P /P i s the VPIE, and ( s / s ) f and ( s / s ) f are Reduced Partition Function Ratios (RPFRs) for 8he condensed and gas phases as introduced by Bigeleisen and Mayer (13). The (s/s')f terms i n the equation represent the differences i n quan tum effects between the condensed and gaseous states. The cor rection terms account for the effect due to the difference be tween the Gibbs and Helmholtz free energies and from gas imper fection. They are necessary because the (separated) samples are being compared at the same temperature but at different pressures. If a comparison at the same condensed phase molar volume i s de rived then the additional correction, (RT)-l / P dv, i s re quired. The correction terms are usually small and may often be neglected. They have been evaluated for a number of different systems by various authors (15-18). A simplification to equation (1) obtains i f ln(P'/P) i s i t self small. With Β « B and V ~ V and negelecting (RT)" / P'dv and terms of order G P then f ° l n ^ = l n f [l P ( B - | ] (2) g
v ?
f
v
f
1
v
v
2
+
o
f )
Equations (1) and (2) refer to measurements on separated isotopes. It i s also important to develop a relation between the reduced partition function ratio (RPFR) and measured separation factors, a, referring to single stage isotopic enrichments (on d i s t i l l a tion) for dilute nonideal solutions. In the case of i n f i n i t e d i lution the molar volumes of the two isomers are identical (they are determined by environment) and one obtains (19) ι ι ,N\ Ina = l n ( — )
v a p o r
,,N\ /(—)
f
l i q u ± d
- c (V* - V ) = In j- , , 2 β
ν
2
κ τ
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
Ο)
6.
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Condensed Phase Isotope Effects
105
Here β i s the isothermal compressibility and V the molar volume of the condensed phase. The last term i s usually very small and i s often neglected. This accounts for the common and mistaken idea that RPFRs are directly measured i n d i s t i l l a t i o n experiments. Equations (1), (2) and (3) relate the RPFRs to the measured VPIEs. The next step i s to develop a methodology relating RPFRs to the changes i n the 3nN dimensional potential function which occur on condensation. The alternative, to directly measure the isotope effects on each and every energy level that contributes to RPFR, i s not only tedious and impossible, but also inelegant. Therefore, a "standard manual" for calculating isotope effects would now plunge into a detailed exposition of the methodology of calculating partition functions and reduced partition functions for condensed and vapor phase molecules given the molecular struc ture and some informatio The commonly employed procedure terms of the isolated molecule gas phase approximation. Conven tionally, electron-nuclear coupling effects are ignored (that i s the Born-Oppenheimer approximation i s assumed, we have earlier seen some indications of the inadequacy of this approximation (1)), and then the molecular RPFRs are further approximated as a product of rotational, translational and internal vibrational parts. The rotation and translation contributions to RPFR are zero in the ideal gas reference because the potential for these modes i s zero. In the condensed phase they may be expressed in terms of the mean square forces exerted on the translational motions and the mean square torques on the rotational ones. Next the intramolecular degrees of freedom are often approximated with harmonic oscillator potentials. (This i s normally reasonable because our interest l i e s in accurate assessments of isotopic d i f ferences in the changes the frequencies on condensation, not i n the absolute value of the frequencies themselves. Therefore, we expect that many of the d i f f i c u l t i e s connected with small anharmonicities might cancel when we take ratios to determine the VPIE. S t i l l , i t i s to be hoped that refinements w i l l soon permit an ac curate assessment of the anharmonic corrections. Wolfsberg (20) and Hulston (21) have cleared up apparent d i f f i c u l t i e s i n the methodology commonly employed to evaluate anharmonic corrections for polyatomic systems.) The isotope effects on the intramolec ular energy level distributions can now be calculated straight forwardly from the intramolecular potential. This potential i s isotope independent within the frame of reference defined by the validity of the (usually very good) Born-Oppenheimer approxima tion. We note that the procedure has been subjected to many and varied experimental verifications, principally by spectroscopic techniques (22). It i s well established. The problem i s more complicated for the condensed phase. Here i t becomes necessary to include the mutual interactions be tween many molecules. We have already seen schematically (Figure 1) that the interaction may be thought of i n f i r s t approximation
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
106
ISOTOPES
AND CHEMICAL
PRINCIPLES
as consisting of two parts. F i r s t , the c l a s s i c a l translations and rotations of the gas phase now become quantized as the molecules " f a l l into" the attractive well. Second, the same attractive i n termolecular forces which account for that well perturb the intra molecular energy levels from the gas phase values. Factorization of these two effects, one from the other, i s clearly not exact. Each i s a manifestation of one and the same intermolecular force f i e l d and the two must be intimately coupled. In other terms the factorization of the canonical partition function per mole of ma t e r i a l into η single particle microcanonical partition functions Q = q , which was exact (within the present context) i n the atten uated gas reference state, i s now conceptually inadequate. Even so, most calculations have proceeded by considering the motions of the "average condensed phase molecule (and i t s isotopic isomer)" in terms of their microcanonica culations are i n fact s t r i c t l culations outlined above. As a f i r s t step factorization into internal, rotational and translational parts i s assumed, and the three different kinds of motion are presumed to be describable with appropriate isotope independent potential functions. Stern, Van Hook and Wolfsberg (23) have described a model calculation based on the Bigeleisen theory. In the BSVHW approach, the har monic approximation i s applied to a l l 3n modes of the "average molecule". In this approximation, equation (1) becomes after ne glect of the correction terms n
f
c g
3n-6 a-6
e x
( V ^ c
f r e q v
Χ
u
±
[
i n tlrnalW^ ternal
u
P( i"- ) /
^ ^ - i ) π
g
/ 2 r
u
Π — - u external
T
2
(l-«p(-u') )/(l-exp)-u ) )
c
c
]
±
^l-exp^upg/l-exp^)^ /U -Uxιr 1 - e x p ( - u ) f
[exp (—ζ r
T
) J [ -ζ 7—TJ 2 1-exp(-u) c
Ί
,,χ
c
1 W
r
The result has been called the "complete" equation. The frequen cies which enter may be calculated from mass and geometry consid erations using standard techniques. The 3n dimensional PE matrix is taken as isotope independent. Computer programs are available for this purpose. The potential and kinetic energy parameters describing the problem are expressed as matrix elements i n the 3n dimensional space. Off diagonal elements couple the different modes, and rotation-translation and internal-external interactions thereby enter i n a natural way. These effects are important i n VPIE considerations and the model has been used widely (4,5) · high temperatures, certain single modes, or ultimately a l l of the modes, may become excited and then can be treated i n terms of se ries expansions i n powers of quantum corrections. Such methodol ogies have been developed by Bigeleisen and co-workers (24) and by Jancso (25) i n d e t a i l . Alternatively at low enough temperatures certain other modes may be treated i n the zero-point energy ap proximation. In especially favorable cases the different motions A t
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6.
VAN
HOOK
Condensed Phase Isotope Effects
107
can be separated into a high lying set (generally the internal v i brations) which i s treated i n the zero point energy approximation, and a low lying one (the l a t t i c e modes) handled by the f i r s t order quantum correction. In this fashion one obtains the widely quoted approximate relation Ί
In
P —
f
f
« . c In — g
=
A
Β - ψ
-ζ
τ
... (5)
ζ
where the f i r s t term i s interpreted as the f i r s t order quantum correction A
and the Β term account high lying modes. .. , int = r "Σν· 8
nV-#]
h(r)
2
-
int int Σν· > - d ν,
(6)
-
int Σ v. )]
(7)
Some discussion of the BSVHW model i s i n order. Two impor tant criticisms come rapidly to mind. F i r s t the model i s severely limited by virtue of i t s adoption of the average molecule approxi mation. This i s not physically reasonable. The motions of neigh boring molecules must be intimately coupled through the intermo lecular force f i e l d and this i s only crudely recognized. The second criticism i s concerned with the problem of anharmonicity. Clearly the harmonic approximation i s more suitable for internal than i t i s for external modes. While i t may be possible to intro duce anharmonic corrections for these modes, i t would appear that i f this i s to be done in a r e a l i s t i c fashion a number of higher order terms should be included (the motions are highly anharmon ic) . The concomitant higher order interactions introduced i n such a treatment may well serve to destroy the chief appeal of the BSVHW approach which l i e s i n i t s simplicity and i n the straight forward method by which i t handles internal-external interactions. Gordon (26) and others abandon this latter advantage by assuming separability of the internal and external motions. The external motions are then described i n terms of the mean square forces and torques exerted on the average condensed phase molecule. This i s certainly more r e a l i s t i c for these modes than i s the introduction of the harmonic approximation, but i t i s achieved at the high cost of neglecting the consequences of internal-external coupling. A different point of view has been taken by Fang and Van Hook in a qualitative discussion of the VPIE (27) . These authors ob tained data on the isotope effects on the vapor phase second v i r i a l coefficients of the deuterated methanes. The data were i n good agreement with those of earlier workers (28). In the inter pretation, separability of the internal vibrations form the exter nal motions of the molecules was assumed. This approximation i s commonly used i n considering vapor phase intermolecular potentials.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
isotopes a n d c h e m i c a l p r i n c i p l e s
108
The interaction i s considered as equivalent to that between two nonvibrating but polarizable molecules. This amounts to separating out and averaging over the intramolecular contributions before introducing consideration of the intermolecular modes. In terms of this model i t i s readily demonstrated that the v i r i a l c o e f f i cient isotope effect can only be interpreted i n terms of isotope dependent parameters for the Lennard-Jones force f i e l d . The v a l ues extracted from the VCIE are within experimental error of the measured isotope effects on the polarizabilities (29). These i n turn were shown to be consistent with the IEs on molecular size as determined by electron diffraction (30) or molar volume measurements (31). The authors note that the IE on molecular size is well understood i n terms of the vibrational properties of the isolated molecules. An isotope independent intramolecular potent i a l function may be employe mean square amplitudes o found to be proportional to the molecular size and polarizability isotope effects. This correlation indicates that the isotope dependence of the effective intermolecular potential arises from the approximate procedure of separating the internal and external motions and the neglect of certain details of the vibrational properties of the interacting molecules. The authors emphasize this point. The isotope dependence i n the intermolecular potential should not be regarded as an apparent contradiction to standard approaches based on the Born-Oppenheimer approximation. Rather i t only underscores the crudeness of the standard method of separating out and averaging over the internal modes before considering the motions on the external potential surface(s). They (27) go on to indicate that an analagous approach may be useful in handling the external contribution to the VPIE. This would amount to substituting an isotope dependent external force f i e l d i n the average molecule approximation i n place of a detailed consideration of numerous higher order internal-external coupling and anharmonicity effects. Such isotope dependent force constants are to be regarded principally as a convenience i n the sense that i t i s easier to approximate the real situation with an isotope dependent effective (perhaps harmonic) f i e l d than i t i s to attempt to solve the complicated many molecule problem exactly. The approach which i s suggested has some similarities to earlier applications of DeBoer s modification of the law of corresponding states (32) to the VPIE of polyatomic systems following Steele (33). It i s to be presumed that this suggestion w i l l prove equally controversial. The magnitude of the expected isotope dependencies i s small. For the translation modes of CH* and CD , an extreme case, a ten percent difference i n the translation force constant was estimated as an upper l i m i t . This would corrospond to about a 5% shift i n the associated frequency, but this i s not directly detectable because of the broad IR band associated with the l a t t i c e modes i n the l i q u i d . Normally corrections of this type are expected to be quite small. For practical purposes, they can be ignored when the f
4
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6.
VAN
Condensed Phase Isotope Effects
HOOK
109
BSVHW model i s employed to correlate isotope effects among a series of molecular isotopic isomers. D.
Specific Examples of VPIE Studies
While this paper i s not intended to serve as a review, a few examples from the literature (including some work from our own laboratory) w i l l serve to i l l u s t r a t e points of interest concerning the theory. 1. The Monatomic Gases. Extensive data on rare gas VPIEs i s available, most recently from Bigeleisen and co-workers (34). Jancso and Van Hook have reviewed the situation as of 1973 (5). The results on monatomic liquids are of particular interest be cause r e a l i s t i c comparison tions are possible. Th only degrees of freedom are the translational ones. In the data analysis the mean value of the Laplacian of the positional energy in the condensed phase (the mean square force) i s obtained and can be compared with results from calculated or experimental radial distribution functions. Specific intermolecular potential functions can be tested i n this fashion. In the context of the present paper i t i s worth noting that application of c e l l model calculations, i n both the harmonic and various anharmonic approxi mations, demonstrate these to be inadequate. The vibrational cou pling between atoms i s completely ignored. However, good agree ment was found with calculations based on the improved self con sistent phonon theory of an anharmonic l a t t i c e using an LJ 13-6 potential. Such results reenforce the earlier criticism of the BSVHW model for the VPIE of structured molecules (Section C). In another calculation Weeks-Chandler-Anderson perturbation theory was used to calculate the mean Laplacian from the LJ 6-12 poten t i a l by Mandel (35). The calculated 's were used to obtain isotope separation factors which were then compared with experi ment. The comparison i s given in Figure 2 and is really quite good 2
2
2. Diatomic and Linear Molecules: The Effect of Rotation and the Rotation-Translation Interaction. The possible importance of a rotational contribution to the VPIE i s nicely demonstrated by the experimental results for NNO isotopic isomers. The results are old ones due to Bigeleisen and Ribnikar (36). Here the pertinent ob servation i s that the VP difference between the isomers N N Ο (M - 45), and N N Ο (M = 45) i s as large as that between n i5 i6 i f i f i6 ) the mass ratio i s unity, but there i s an isotope effect on the moments of inertia; i n the second there i s an isotope effect on mass but the moments of inertia are equal within 7 parts i n 10 . Other evidence indicates that the change i n intramolecular force constants on condensation i s negligible. It was concluded that a sizable barrier to rotation exists i n this condensed phase. 15
1 6
N
llf
N
0
( M
=
4 5 )
a n d
15
N
1 6
N
0
( M
=
4 4
m
I n
t h e
f i r s t
p
a
i
r
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
1If
ISOTOPES
ANDCHEMICAL
PRINCIPLES
Figure 1. Potential energy diagrams, a. external transhtion (gas phase); b. external translation (condensed phase); c. internal vibration (gas phase); d. internal vibration (condensed phase). In a, r denotes the average intermolecular distance in the gas phase, in b, r denotes the value of the intermolecular distance evaluated at the minimum, and in c and d, r denotes the value of the coordinate describing the molecular distortion evaluated at the minimum. Notice for the exter nal motions the zero point energy change on condensation, (E' — E ) — E' — E ) > 0, because E' E ^ 0, but for the internal mo tions it may be positive, negative, or zero depending on the effect of the intermolecular forces on the specific motion under consideration. 0
0
0
c
v
v
v
c
v
50
80
90
100
110
120
130
W<>
T(°K) Journal of Chemical Physics Figure 2. T ln(f/fg) as a function of temperature for argon. The solid line is the difference between the liquid and the gas values represented by the dashed lines. *o experimental results on (34). · vapor pressure, ο separation factor. The lines are calculated (35). 2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6.
VAN
Condensed Phase Isotope Effects
HOOK
111
It i s interesting that this prediction was confirmed by infrared measurements 11 years later. Effects which couple rotational and translational motions sometimes occur when centers of mass and geometry i n a molecule do not coincide. Friedman (37) developed an expression describing this situation i n the limit of small quantum corrections. He considered relations between the center of e l e c t r i c a l charge (or i n teraction) (CI) , the center of mass (CM) , and the geometric center (CG). The geometric relationships between the centers are expressed by Equation 8. R(nL - m )
Here m-^ and m^ are th and d = |CG - CI|. Fo same element, d = 0. Friedman obtained an expression for the f i r s t quantum correction to the configurâtional part of the p a r t i tion function, %1
eff
in terms of an effective mass M ff and the mean squared force, , exerted on one molecule by a l l of the others. The effective mass takes the rotational contribution into account. e
M
jpr;
1 M
,2 3
2
Ma I
eff The approach i s tested by considering the relative vapor pressures of three isotopic molecules. One finds for the isotopic nitrogens H" N * , N " N and N N . 1
Ï s
1
5
R - In
L
ι
S
χ
Ι *- 'Ζ"~"
= 0.495
* 14-1415-15 The measured value i s 0.494 + 0.002. In the absence of transla tion-rotation coupling R = 0.5. The theoretical ratio i s inde pendent of temperature and this, too, i s i n agreement with experi ment. Gordon (26) gave a somewhat similar approach i n his discus sion of the isotopic CO molecules. Rather than introduce an ef fective mass he considered the VPIE as arising from four terms; (i) the mean square force (translational contribution), ( i i ) the mean square torque (rotational contribution), ( i i i ) the change i n torque on isotopic substitution, and (iv) the intramolecular fre quency shifts on condensation. Effects ( i i ) and (iv) can both be obtained from moment analysis of IR or Raman vibrational bands. Therefore by combining VPIE and spectroscopic data, values for both the mean square force and mean square torque can be obtained.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
112
AND
CHEMICAL
PRINCIPLES
The method was illustrated by application to the isotopic COs (38) (Table 1). Friedman and Kimel (39) also evaluated mean square torques for CO in the liquid phase but their calculated values are much smaller than those derived by Gordon from measurements of the third and fourth moments of IR and Raman spectra. The most l i k e l y reason for the discrepancy i s that Friedman's assumption of a spherically symmetric external force f i e l d for CO i s not j u s t i fied. 3. Polyatomic Molecules: Hydrogen-Deuterium Effects. A rep resentative sampling of measured HD VPIE effects i s shown i n Fig ure 3. The information which i t contains i s far from exhaustive but does include a number of different kinds of effects. At the moment we are only concerned with gross features. These exhibit a wide difference, varyin a l l th fro approximatel 10% D normal for HOH/DOD at inverse for substitutio nonpola hydrocarbons. the large normal effects are connected with molecules which strongly associate in the condensed phase, and associate through the position of isotopic substitution. The inverse effects often display minima in the VPIE temperature curves as expected from the earlier discussion of applicable theory (Equations 4 and 5). One interesting exception to the behavior (associated compounds=normal effects; nonassociated compounds=inverse effects) i s that of the carboxylic acids. Some examples are shown in Figure 4 (40). The results indicate that deuteration at methyl or methy lene, and also at carboxyl, increases the vapor pressure. The un expected sign of the carboxyl effect i s attributed to the fact that organic acids are highly associated, not only in the l i q u i d , but also in the vapor, and this must be properly taken into ac count. More commonly, association in the vapor phase i s not im portant and then normal effects are readily attributable to unus ually large values for the intermolecular condensed phase frequen cies, these can outweigh the normal red shifts in internal fre quencies usually observed on condensation. For example, i n the condensation of water vapor to the liquid at 313 Κ there i s a net red shift of about 283 cm"* in the internal frequencies but this is more than compensated by the appearance of three librational frequencies around 500 cm" and three hindered translations around 160 cm" . The VPIE i s therefore normal and large. We shall discuss the water problem in more detail below. It i s evi dent from Figure 3 that a similar situation obtains for deutera tion on the NH group of methyl amine and to a lesser extent for substitution at acetylenic hydrogen positions which are also known to associate, but clearly the behavior of the compounds substitut ed at the non-polar methyl and methylene positions i s different. 1
1
1
2
4. Applications of the BSVHW Model - i (methane). The prin cipal successes of the BSVHW model have been in application to hydrocarbon VPIEs. As a f i r s t example we consider the calcula-
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6.
V A N HOOK
Table 1 .
Condensed Phase Isotope Effects
Predicted VPIE and Changes i n Mean-Squared Torques of the Isotopes of Liquid CO at 77 K. (26) f
(P-P )V RT Isotope i2 i6 G
1 2
C 0 12 18 C
0
13 16 C
0
13 17
0
C
1 3
1 4
o
0
1 7
113
f
2
2
<(0 U) > - <(0U) > <(0U) > % Z
0
a
0.00285
3.7
0.0056
7.5
0.0107
-1.1
c
1 8
o
0.0132
2.3
c
1 6
o
0.0147
-8.1
*c o 14 ie
0.0199
-5.1
0.0202
-2.1
l
1 7
G
0
5
Measured values. (38)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES A N D C H E M I C A L
114
oT^~2 λ
3
** 5 β
7
β
9
fl
.α, .«Ο -180
f? 0~~ *
-aoo Chemical Reviews
Figure 3.
Hydrogen-deuterium vapor pressure isotope effects for some compounds ( 5 )
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
PRINCIPLES
6.
Condensed Phase Isotope Effects
V A N HOOK
115
Table 2. F Matrix Elements for Isotopic Methanes
Gas
F -F - . , F -F . . , F -F gas solid gas liquid gas ads A
5.495 F ,CH Stretch (mdyn/A)
0.063
0.043
0.051
0.568
0.000
0.003
0.002
, (mdyn)
0.000
-0.010
0.000
g
F^HCH bend (mdyn*A)
0.165 F
C U
0.124
f
, (mdyn/A)
0.00
0.019
f
mdyn»A
0.000
0.000
0.000
0
F ,CH* Translation * (mdyn/A)
-0.096
-0.057
-0.063
F ,CH Rotation (mdyn«A)
-0.008
-0.010
-0.021
Reference
(41)BCJ
(41)BCJ
(42)VH
m9
0
4
Γ
12
1
Calculated Frequency Differences of C Hft (cm ) , [ 3143.71(A)
CH stretch
1574.2 (E)
HCH bend
3154.1a(F)
CH stretch
1357.4*(F)
HCH bend
2
trans(F) Rot(F)
v g
~
v c o n d
3
16.9
11.5
12.8
0.0
3.9
2.3
18.7
13.9
15.5
0.0
4.8
2.3
100.5
77.7
81.5
63.5
72.3
106.0
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
116
AND
CHEMICAL
PRINCIPLES
tions of Bigeleisen, Cragg and Jeevanandam (41) on methane. The authors successfully used a single isotope independent force f i e l d , Table _2, to rationalize the available VPIE data (from their own and another laboratory). A similar calculation on methane (vapor) = methane (adsorbed) effects was independently carried out by Van Hook (42) . In either calculation the point of primary i n terest was the rotational contribution. The discussion i s most conveniently made i n terms of the approximate relation, Equation 5. The contribution of the internal modes i s found primarily in the Β term because these frequencies are large. The l a t t i c e con tribution (A term) i s from both translation and rotation. The ef fects may be sorted out, one from the other, by considering the behavior of molecules of the same total mass but different moments of inertia, e.g., CH D/C H , CH D /C Hi»/CH T, etc. In other words data from a single isotopi pair h CHx»/CH D onl fixes the total l a t t i c e contribution sistent with a large numbe tional contributions. The experimental determination of A's (Table 3) for the different species unequivocally shows that rota tion must contribute i n the solid and liquid (41) and adsorbed (42) phases, and therefore must be hindered. It allows the ratio of librational to translational force constants to be fixed. 13
3
1A
4
2
2
3
Table 3.
Relative A Values for Isotopic Methanes Experimental Calculated for Solid (41) Liquid (41) Adsorbed Free Rotor BCJ BCJ VH CH D/ CH* 0.8 4.3 3.1 2.9 CH C / *CH 0.9 2.3 0.7 2.2 CH T/ CH*
(42)
13
3
X/
2
2
2
14
3
The detailed force fields used in the complete harmonic cal culation are shown in Table 2 where the frequency shifts on phase change (which give rise to the isotope effects) are entered at the bottom. The agreement between the observed (spectroscopic) and calculated shifts i s within the experimental precision with which the latter have been determined. The agreement between calculated and observed VPIEs i s shown in Figure 5 where the calculated ef fects are plotted as the long dashes. The agreement i s good in most details especially considering that one (isotope and tempera ture independent) approximate force f i e l d has been applied to c a l culate effects for seven different isotopic isomers. The agree ment extends to the heats of fusion and vaporization as calculated from the slopes which are in reasonable agreement with the c a l o r i metrically determined values; differences are on the order of a few calories per mole (5). S t i l l , i t i s clear that the model i s subject to refinement. It does not f i t even a l l of the vapor pressure data simultaneously to within the experimental precision. This i s particularly noticeable in the case of C substitution. The authors (41) suggest that a refined calculation in which due 13
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6.
VAN HOOK
111
Condensed Phase Isotope Effects
25
2,6
2,7
2,8
23
3.0
31
3.2
33
3.4
60
50
40
30
20
;
3,5
-O07J 120 110 100 90
80
70
fC
10
Chemical Reviews Figure 4. Vapor pressure isotope effects for organic acids deuterated at the carboxyl position (5, 40). X = CH.COOH, Ο = CH CH CH COOH, • = (CH ) CHCOOH, Δ = (CH ),CHCH COOH. 3
d 2
8
3
9
Î0
Ϊ1
12
Ϊ5
2
2
%~
10/T 5
Chemical Reviews Figure (41 ).
5.
Vapor pressures of the isotopic methanes fit to data; calculated from Table 2 (5).
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2
ISOTOPES
118
AND
CHEMICAL
PRINCIPLES
account i s taken of stretch-bend interactions and of the contribu tion of the anharmonicity in the external motions would improve the agreement. It i s interesting, perhaps disturbing, to note that in the model the librational frequencies are found to be higher i n the liquid than in the solid. We note here with respect to the intermolecular portion of the methane isotope effects that an alternative approach has been suggested by Steele (33) who ad vocates an application of the theory of corresponding states. A similar approach has been qualitatively discussed by Fang and Van Hook (27). i i The deuterated ethylenes. The VPIEs of this series of compounds form an excellent example of the importance of coupling effects between the internal and external degrees of freedom. Ex tensive data are availabl culations have been performe (23) by Bigeleisen and Ishida (44). In these calculations the c e l l model was solved giving 18 harmonic oscillator frequencies for each isotopic isomer. F u l l account was taken of the contribu tions of the coupling terms in the F and G matrices, including coupling between the different external modes (44). The vapor pressure differences between the c i s , trans, and gem dideutero isomers i s of particular interest. These are principally due to hindered rotation in the liquid (i.e., the principal moments of inertia are different for these three molecules), but in addition, superimposed on this effect, there i s a ZPE effect due to coupling of the hindered rotation with certain internal vibrations. The specific internal frequencies which couple with the rotation are symmetry dependent. The same effect i s manifested by the shifts in the internal frequencies themselves on condensation (i.e., in the Β as well as the A term) . In the absence of the effect the law of the geometric mean would be obeyed, and B(d-2)/B(d-l) = 2.00. The symmetry dependent perturbations may theoretically be shown to necessarily lower this value. The calculated and experi mental values are compared below: (B(d-2)/B(d-1)) trans cis gem
1.83 1.89 1.92
theor
(B(d-2)/B(d-l))
expt
1.88 1.93 1.98
Agreement in the ordering trans-cis-gem i s cited as particularly strong evidence in favor of the correctness of the approach. Note that the order i s symmetry, not model, dependent. The magnitude of the shift does depend on the parameters inserted into the cal culations. In the refined calculation (44) the authors indicate that anharmonicity i n the rotational potential of the liquid must be taken into account to f i t the data accurately. Later (45) the same force f i e l d was used to correlate the molecular volume iso-
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6.
VAN
Condensed Phase Isotope Effects
HOOK
tope effects for the deuterated
119
ethylenes.
l i i Other examples. Detailed model calculations have also been performed for a number of other systems. These include studies on the isotopic isomers of ethane (46), proplyene (47), acetylene (48), methyl acetylene (49), alcohols (50), hydrogen sulfide (51), water (52,53), and numerous other compounds (5). Generally speaking,the theory has managed to successfully correlate the IEs of complete sets of isotopic isomers using isotope independent force fields consistent with the available spectroscopic data. I t is to be emphasized, however, that the spectroscopic information is generally not complete enough to define a unique force f i e l d and i t has proved necessary to appeal in part to the isotopic data i t s e l f i n completing the parametrization. For example, in references (23), (46), and was used to f i x the rati (within, of course, the bounds set by the spectroscopic informa tion). The rest of the experimental data, including the information on the temperature coefficients, serve to test the theory. Similarly we have seen for the isotopic methanes that data on one pair of ratios comparing isomers of equivalent mass (i.e., CH*CH D and C H -C H*) was used to f i x the relative contributions of the translational and rotational modes. Again the approach i s tested by the data for other sets of ratios ( C H 4 - C H 2 D 2 and CH*»CH T and C H -C H , etc.). Within this context i t seems f a i r to say that the Bigeleisen formulation of the VPIE, as parametrized through the BSVHW model, i s reasonably successful i n interpreting VPIEs of polyatomic molecules. I t i s particularly useful i n correlating effects in series of isotopic isomers (i.e. C 2 H 6 - C 2 H 5 D . . ..C D , or H0H-H0D-D0D....TOT, etc.). I t i s therefore reasonable to conclude that i t i s experimentally and theoretically established that the VPIE and related isotope effects can usefully serve as probes i n Investigations of intermolecular forces i n condensed phases. 12
3
13
4
12
3
2
E.
14
4
4
6
Recent Work In Our Laboratory: Aqueous Solvent Effects
1. Introduction. Our recent work has been directed to studies on the equilibrium solvent isotope effects of aqueous solutions. The goal of this program i s to contribute new and hopef u l l y useful information to current discussions of the nature of the structure of water and of aqueous solutions. We have now accumulated a good deal of information about isotope effects on the properties of electrolyte solutions including vapor pressure, c a l orimetric, and freezing point studies. In addition, we have made and are continuing studies on nonelectrolyte aqueous solutions especially on systems expected to show hydrophobic bonding. However, the present report w i l l be limited to a discussion of the work on electrolyte solutions. The experiments are designed to compare solvent isotope effects i n the solutions with the isotope
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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effects of the pure solvents. Calculations due to Van Hook (52) have demonstrated that the BSVHW model i s applicable to the iso topic waters and to the mixed solvents. In the latter case the assumption of ideality of H0H-H0D-D0D mixtures has been shown to be consistent with the model calculations on the pure isotopic isomers. Since the calculated effects are dependent on the force constant inputs to the model, the rationale of our present experi ments i s that changes in the isotope effects with increasing salt concentration can be correlated with changes in the solvent model, in other words with structural considerations. The experiments are l i k e l y to prove useful even though i t i s recognized that pres ent models for water i t s e l f may be naive, certainly this i s so for the average molecule c e l l model. On the other hand, i t i s our hope that the trends which should become evident upon the gradual accumulation of data for a number of systems w i l l help point the way to proper refinement In any discussion o thermodynamic the standard i s an important one and must be constantly kept i n mind. Our standard state, the conventional one, i s the hypothet i c a l solution of unit aquamolal concentration, taken to the Henry's law i n f i n i t e dilution l i m i t . (A one aquamolal solution i s one containing 1 mole of solute per 55.508 moles of solvent f o r H 0 this i s 1000 grams. Our choice of the aquamolal concentration scale insures that a l l isotope effects are compared at equivalent mole fractions.) The properties of the solution are known in a thermodynamic sense when the standard and excess molal free ener gies, μΪ and yjj, y f and p f and their appropriate température and pressure derivatives, H ? and H§, H? and S f , S| and S S , S ? and S f , V? and V?, vf* and v f , and Cp? and CpS, Cp? and Off, etc. , and the solvent isotope effects on a l l of these properties are known as functions of temperature and concentration. In the present paper we focus attention on the isotope effects. The standard state transfer properties of the solvent Δμ? = μ?(Η) - μΐ(ϋ), ΔΗ? - H?(Η) - HÎ(D), etc. are defined i n terms of the properties of the pure solvents because of our selection of the standard state. Thus Δμ? = -RT In P°/P°, etc. These values are well es tablished. The standard state transfer properties for the solute are not known. They correspond to the isotope effect per mole of salt on the process of transferring an infinitesimal quantity of salt from i t s standard state (crystal, latm, the temperature of the experiment) to the pure solvents. The solution which i s formed i s at i n f i n i t e dilution. The excess transfer isotope effects refer to the difference in the IE per mole of salt on the transfer be tween isotopic solutions (cone. M); and that for the standard state transfer (cone. 0). The experiments discussed in the pres ent a r t i c l e are designed to measure the excess isotope effects on the p a r t i a l molal free energies and enthalpies of the solvent and the electrolyte solute. Three approaches w i l l be described. In the f i r s t , precise measurements of the VPIEs over solutions at equal aqualmolal con2
x
x
x
x
x
x
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
x
6.
VAN HOOK
Condensed Phase Isotope Effects
121
centrations give the isotope effects on the solvent activity (ex pressed i n terms of the osmotic coefficients) and hence the s o l vent excess free energy of transfer as a function of temperature and concentration. The solute excess free energies can be obtain ed v i a a Gibbs-Duhem-Bj errum integration i f sufficient data are available to permit reliable extrapolation to i n f i n i t e dilution. The excess p a r t i a l molal enthalpies and entropies can then be ob tained by differentiation with respect to the temperature. In the second series of experiments, we describe freezing point measure ments on solutions of NaCl in HOH and DOD carried out as a func tion of concentration to low concentration. These measurements are designed to evaluate the assumptions earlier employed for the Gibbs-Duhem integrations of the VPIE data. In the third approach, we describe direct calorimetric measurements of isotope effects on the excess and standard heats of solution The goal of this pro gram i s to ultimately emplo model calculations of th Unfortunately cannot yet report on t h i s . Instead we w i l l phenomenologically describe some of our experimental results and then discuss them in terms of current qualitative ideas of solution structure. 2. Experimental. The vapor pressure isotope effects were measured on either of two systems. Each was based on the same principle and we b r i e f l y describe the later and improved appara tus. The earlier equipment has been previously described (54). In our new apparatus two sets of two sample pyrex vessels are con tained i n an aluminum block mounted i n a "Tronac" temperature bath which can be thermostated at any desired temperature between 0 and 80°C. The control i s good to about +3 χ 10~4°c i n the most favor able region (around room temperature), not that good at elevated temperatures. We estimate that the temperature differences within the block (and accordingly between the samples) as significantly better than +1 x 10"4°. The vessels contain glass encased magnet i c bars, M, and are stirred from below. The c e l l s are of approxi mately 20ml volume and are constructed as diagrammed below.
Sample Cell
Degassing C e l l
Connecting tube A i s made from 6mm OD pyrex and after exiting through the bath l i d i s joined by means of a glass-metal seal,
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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ISOTOPES
AND CHEMICAL
PRINCIPLES
through a valving manifold, to the d i f f e r e n t i a l capacitance trans ducer. The entire valving manifold, transducers, etc. i s thermostated around 90°C to +0.01 degrees. F i l l tube Β i s made from 3mm pyrex and i s connected by means of a second valving manifold to our degassing f a c i l i t i e s . The degassed solutions can be trans ferred under vacuum. A l l valves i n both manifolds are high vacuum bellows type. The vacuum integrity of the system i s such that the leak rate (system isolated from pumps) i s less than 3 microns per normal working day. Proper degassing of the samples i s c r i t i c a l . We employ numerous freeze-pump-thaw cycles and several sublima tions i n a specially designed pyrex vessel which i s connected to the high vacuum manifold through a glass metal seal, a stainless steel bellows, and a high vacuum bellows valves. Both the degas sing and the measurements are therefore carried out exclusively i n pyrex. I t i s only momentaril samples are exposed to metal In an experiment two of the sample vessels contain the two different isotopic solutions. The pressure difference between them i s monitored to approximately +0.03% or +0.001 mm, whichever is larger. The other two vessels contain a solution of the normal isomer being compared against a standard solution, pure solvent, or vacuum. One run over the temperature range therefore furnishes us with a set of data on the total pressure and the isotopic pres sure ratio as a function of temperature. The calorimetric measurements were carried out either on an isoperibol solution calorimeter built by us following conventional (55) design and employing thermistor sensors and e l e c t r i c a l heat ing and Peltier cooling, or (later) on a "Tronac 880" isothermal solution calorimeter. The freezing point measurements employed a modification of an old technique (56) similar to the Beckman method. I t was suggest ed from our experience i n adiabatic calorimetry. In the experi ments a thermistor forming one arm of a Wheatstone bridge i s con tained i n a well stirred mixture of ice and water (or heavy ice and heavy water). Its resistance i s monitored and defines a tem perature (the scale i s set i n separate calibration experiments by resistance thermometry). After a stable base line i s established (corresponding to ±2 or 3 χ 10~5 deg.) salt or concentrated solu tion i s added, the temperature drops, and a new base line i s es tablished. A sample i s taken and the temperature drop recorded. The chemical analysis for concentration i s c r i t i c a l . We have de veloped a technique based on a suggestion of Richards (57) which is good to +0.02%. In this method, the chloride i s precipitated with s i l v e r delivered from a weighing burette. The t i t r a t i o n i s followed potentiometrically and the end point deliberately exceed ed by one or two drops. After curing and f i l t r a t i o n , excess s i l ver i s determined by AA spectrophotometry. Our data i n Η0Η as solvent i s i n quantitative agreement with Scatchard and Prentice (56) and defines the osmotic coefficients to +0.0005 unit i n either solvent.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6.
VAN
Condensed Phase Isotope Effects
HOOK
123
3. Discussion. First we consider the VPIE data. The changes in the isotopic pressure ratios are readily related to the isotope effects on the osmotic coefficients (58). AinR =
mΡ·/Ρ· - m
- m
(ψ - « y
=
Η
(9)
In this equation m refers to the aquamolal concentration, the °s to the pure solvent effects, a is the solvent activity and φ the osmotic coefficient. Data for a number of different electrolyte solutions at one selected temperature are shown in Figure 6. The solutions a l l show a smaller VPIE than do the pure solvents. In the experiments AlnR i s measured as a function of temperature and concentration, and we have phenomonologically f i t the data (58,59) using the extended Debye-Huckel theory. φ - 1 --|f
1
[K^ ) Κ = 1+ aI
X / Z
(10)
In the equation I i s the ionic strength, S i s the DH limiting slope, and a i s a parameter. We have presented arguments else where (58) that the term in S/a I i s isotope independent. In that event 3
Δφ -
5
5
^°
8
AlnR = (B -B ) I + ( C ^ ) H
D
2
I + ....
(11)
The ΔΒ(Τ), AC(Τ), etc. functions are derived from the data by least squares. They are tabulated for the different salts, which have been investigated (j>8,59). The solvent excess free energies and excess enthalpies are readily obtained as functions of temper ature and concentration. The data i n Figure 6 shows the 25 degree isotherms. The effects are small for 1:1 electrolytes. Thus even at 10m, corrosponding to a water/ion ratio of only 2.8, the VPIE of KF (for example) i s depressed by only 4.4% (63 parts i n 1420, ln(P|/Pg) - 0.142 at 25°), that of NaBr by 14.6%, etc. The effects are about one order of magnitude larger for the CaCl solutions. This i s just what i s expected from the ionic strength relation given i n the extended Debye-Hîickel theory, equation (10). (Note AlnR = ΔΒ·Ι + for a 1:1 electrolyte I = m, but for a 2:1 salt I = 3M so AlnR = 9 ΔΒ·πι ) . The values of AlnR may be reexpressed in terms of the "structural temperature" concept original ly introduced by Bernai and Fowler (60) . This parameter i s defined as that temperature increase for the pure solvent which gives a change in the VPIE equivalent to that observed upon addition of the s a l t . They are shown on the right hand side of the graphs. In this framework i t i s clear that the electrolytes are structure breakers. At high concentrations the effects are considerable, reaching 13° for 15m L i C l , and about 100° for 10m CaCl . It i s interesting to note that the differences between NaCl, NaBr, Nal are not as marked as those between L i C l , NaCl, NaBr, Nal. -This i s 2
2
2
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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PRINCIPLES
consistent with the trends observed i n the values of the osmotic coefficients themselves (61). The data on the temperature dependencies are important. At a given ionic strength, minima are indicated in a l l cases and there i s a rough correlation between the temperature of the mini mum and the structure breaking character. A Gibbs-Duhem extra polation and integration followed by application of the GibbsHelmholtz equation allow the temperature coefficients to be com pared with direct calorimetric measurements of the isotope effects on the excess solute enthalpies. The comparison at 25°C with the available data i s shown in Table 4-, A significantly large and negative excess heat capacity i s indicated and this has been cor roborated by Craft and Van Hook (62) with direct calorimetric mea surements at 10, 25, 50 and 75°C. These results are reviewed i n Figure They are quantitatively consistent with the hypothesis that the aqueous structur This reassuring result i ideas. Table 4 ex Comparision of Excess Enthalpies of Transfer ΔΗ (H 0 -> D 0) at 25°C and Selected Concentrations 2
2
Salt NaCl KC1 LiCl NaBr Nal CaCl
/ , , i n eal/mole.
Concentration
2
2 2 4 2 2 5.3
Present Work 34+6 15+24 -4+6 37+7 43+7 243+10
Calorimetric (and Ref.) 39(67), 34(68), 32(62), 36(58) 39(67), 40(68), 33(62) 12(67), 47(68) 36(67), 36(68) 31+6(68) 226+20(59)
In Figure <8, preliminary data of Craft and Van Hook (62) on freezing points on NaCl solutions i n HOH and DOD are shown. This experimental program was instituted following a suggestion of H.S Frank (63) that the earlier extrapolation procedures to i n f i n i t e dilution (58,59) deserved c r i t i c a l examination. The f i t to the data i s given by 2
2λ m ψ- = Η *H Θ
+ (Κχ + K m
| i =(0.0201 + 0.005) Η ~ + (2.63 + 0.41 χ 1θ"" j-) - (1.08 + 0.20 χ 1θ" ) | - (11) 2
+ ....)
Λ
Ζ
«
Θ
9
3
Ζ
The intercept corresponds to a predicted isotope effect on the heats of fusion of the pure solvent of 7 1 + 2 cal/m, i n good agreement with the best calorimetric value 6 5 + 4 cal/m (64) tak ing Tf(H) - Tf(D) = -3.82. The present result i s somewhat more precise than the old calorimetric data. The limiting slope i s in good agreement with the value predicted from the extrapolated
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6.
VAN
HOOK
Condensed Phase Isotope Effects
f
I
NaC/
Jm
Ï NaCl 5m SKCi
3m
ΔΗ: 2*
I
5
2S
SO
7S
Figure 7. Solvent isotope effects on excess heats of solution at different temperatures
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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temperature coefficients of the VPIE studies (58) and sub stantiates the validity of our earlier Gibbs-Duhem extrapolation technique. The curvature in the isotopic freezing point vs. mo l a l i t y data i s significantly more than was expected from the VPIE experiments (58,59). In these experiments the available data was f i t over a broad temperature range to the simplest s t a t i s t i c a l l y significant form; Φ
Η
" Φ
P
Β
= KY (T)m
The freezing point data require at least one higher-order term. Φ
Η
- Φ
= KÎ (T)m + K2 (T)m P
Β
P
2
The observation above i polated to 0°C i t is in since κΥ^(Τ) i s obtained from concentration data between 1.1 and 5m while KfP is the limiting slope at much lower concentrations we must conclude that the temperature dependence of KfP(T) i s such that i t s contribution has essentially vanished by room temperature (or thereabouts). Thus we have qualitatively predicted the be havior of the isotope effect on the excess heat capacity in agree ment with the observed behavior (Figure 7). This i s equivalent to the statement that the longer range ordering terms in the extended theory are relatively more important at the lower temperatures and have larger temperature coefficients there. 5. Conclusions. We do not have any definite conclusions con cerning the structure of water or aqueous solutions. These must wait for more complete interpretive calculations. However, i t i s clear that the effects which we have discussed above, although small, are self-consistent (that i s from VPIE, to freezing point, to calorimetric data). Also they are sensitive to structural con siderations. The general trends exhibited for a l l the salts are especially evident on examination of the CaCl data as shown in Figure 9_. The effects for this salt are the largest of a l l those studied and for this reason the relative errors are smallest. The general trend to smaller excess isotope effects at either high concentrations or high temperatures i s apparent and i s consistent with the idea that aqueous structure i s considerably decreased under these conditions. We conclude that systematic investiga tions of these aqueous solvent isotope effects w i l l i n the end make a significant contribution to the analysis and eventual understanding of the aqueous solvent structure problem. Quantita tively speaking this probably awaits further systematic accumula tion of data. We are presently engaged in this enterprise and are currently examining a number of hydrophobic electrolyte and nonelectrolyte molecules as solutes. 2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
6.
VAN
HOOK
Condensed Phase Isotope Effects
Figure 8.
Figure 9.
Solvent isotope effects on the freezing points of sodium chloride solutions
Isotope effects on the solute excess thermodynamic properties of calcium chloride solutions ( 59 ).
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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ANDCHEMICAL
PRINCIPLES
Acknowledgement This research has been supported by the National Science Foundation under grant GP-25728 and by the National Institutes of Health, GMS-19991-02. Literature Cited. 1· Wolfsberg, Μ., This Symposium. 2. "Isotope Effects i n Chemical Reactions", ACS Monograph 167, C. J . Collins and N. S. Bowman, Eds., Van Nostrand-Rheinhold, New York, 1971. 3. Lewis, G. N. and Randall, M., Revised by K. S. Pitzer and L. Brewer, "Thermodynamics" 1961. 4. Bigeleisen, J . , Lee, M. W. and Mandel, F., Ann. Rev. Phys. Chem., (1973), 24, 407 . 5. Jancso, G. and Van Hook, W. Α., Chemical Reviews (in press). 6. Lindemann, F. Α., P h i l . Mag., (1919) 38, 17 ; Lindemann F. Α., and Aston, F. W., P h i l . Mag. (1919) 37, 523 . 7. Keesom, W. H. and van Dijk, H., Proc. Acad. Amsterdam, (1931) 34, 42. 8. Scott, R. B., Brickwedde, F. C., Urey, H. C. and Wahl, M. H., J . Chem. Phys., (1934) 2, 454 . 9. Topley, B. and Eyring, H., J . Chem. Phys., (1934) 2, 217; Bailey, C. R. and Topley, B., J . Chem. Soc., (1936), 921. 10. Herzfeld, K. F. and Teller, Ε., Phys. Rev., (1938) 54, 912 . 11. Bigeleisen, J . , This Symposium. 12. Bigeleisen, J . , J . Chem. Phys., (1961) 34, 1485 . 13. Bigeleisen, J . and Mayer, M. G., J . Chem. Phys., (1947) 15, 261 . 14. Stern, M. J . , Spindel, W. and Monse, E. U., J . Chem. Phys., (1968) 48, 2908 ; Monse, E. U., Spindel, W. and Stern, M. J., Advances i n Chemistry, (1969) 89, 148. 15. Van Hook, W. Α., J . Chem. Phys., (1967) 46, 1907 . 16. Kiss, I., J a k l i , G. and Illy, H., Acta Chim. Hung., (1972) 71, 59 . 17. Kiss, I., J a k l i , G., Jancso, G. and Illy, H., Acta Chim. Hung., (1967) 51, 65 . 18. Gellai, B. and Jancso, G., Ber. Bunsenges. phys. Chem., (1971) 75, 156 . 19. Bigeleisen, J . , J . Chem. Phys., (1963) 39, 769 . 20. Wolfsberg, M., J . Chem. Phys., (1969) 50, 1484 ; Advances i n Chemistry, (1969) 89, 185 . 21. Hulston, J . R., J . Chem. Phys., (1969) 50, 1483 . 22. Nash, C. P., This Symposium. 23. Stern, M. J . , Van Hook, W. Α., and Wolfsberg, M., J . Chem. Phys., (1963) 39, 3179 .
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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V A N HOOK
24.
25. 26. 27. 28.
29.
30. 31.
32. 33.
34.
35. 36. 37. 38.
39. 40. 41. 42. 43.
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129
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56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68.
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PRINCIPLES
Ishida, T. and Bigeleisen, J . , J . Chem. Phys., (1968) 49, 5498. Menes, F., Dorfmüller, T. and Bigeleisen, J . , J . Chem. Phys., (1970) 53, 2869. Van Hook, W. Α., J . Chem. Phys., (1966) 44, 234. McDaniel, R. L. and Van Hook, W. Α., J . Chem. Phys., (1970) 52, 4027. P h i l l i p s , J . T. and Van Hook, W. Α., J . Chem. Phys., (1970) 52, 495. Van Hook, W. Α., J . Chem. Phys., (1967) 46, 1907. Borowitz, J . L. and Klein, F. S., J . Phys. Chem., (1971) 75, 1815; Kiss, I., J a k l i , G., Jancso, G. and Illy, Η., J . Chem. Phys., (1967) 47, 4851; Linderstrom-Lang, C. U. and Vaslow, F., J . Phys. Chem., (1968) 72, 2645; Kiss, I., J a k l i , G., Jancso, G. and Illy, Gellai, B. and Jancso (1971) 75, 156. Van Hook, W. Α., J . Phys. Chem., (1968) 72, 1234; (1972) 76, 3040. Majoube, Μ., J . chim. Phys., (1971) 68, 625; Wolff, H., "Phys. Ice, Proc. Int. Symp. 3rd.", 1968, p. 305. Van Hook, W. Α., Isotopenpraxis, (1968) 4, 161. Arnett, Ε. M. and McKelvey, D. R., "Solute Solvent Inter actions", Coetzee, J . F. and Ritchie, C. D., Eds., Interscience, New York, 1969. Scatchard, G. and Prentice, S. S., J . Amer. Chem. Soc., (1933) 55, 4355. Richards, T. W., as quoted by J . R. Partington, "Textbook of Inorganic Chemistry", Macmillan, London, 1950, p. 91. Pupezin, J . , J a k l i , G., Jancso, G. and Van Hook, W. Α., J . Phys. Chem., (1972) 76, 743. J a k l i , G. Chan. T. C. and Van Hook, W. Α., J . Soln. Chem., (in press). Bernal, J . D. and Fowler, R., J . Chem. Phys., (1933) 1, 515. Robinson, R. A. and Stokes, R. Η., "Electrolyte Solutions", 2nd Ed., Butterworths, 1962. Craft, Q. C. and Van Hook, W. Α., (in preparation). Frank, H. S., Personal Communication. Rossini, F. D., Knowlton, J . W. and Johnston, H. L., J . Res. Natl. Bur. Stand., (1940) 24, 369. Wood, R. H., Rooney, R. A. and Braddock, J . N., J . Phys. Chem., (1969) 73, 1673. Desnoyers, J . E., Francescon, R., Picker, P. and Jolicoeur, C., Can. J . Chem., (1971) 49, 3460. Wu, Y. C. and Friedman, H. L., J . Phys. Chem., (1966) 70, 166. Greyson, J . and Snell, H., J . Phys. Chem., (1969) 73, 4423.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7 The Electrochemical Determination of Equilibrium Constants for Isotope-Exchange Reactions PETER A. ROCK Department of Chemistry, University of California, Davis, Calif. 95616
I.
Introduction The basic idea involve to study the thermodynamic isotope-exchang set up the appropriate electrochemical c e l l s i n duplicate, one with one isotope of the particular element of interest, and the other with the other isotope. The thermodynamic quantities for the isotope-exchange reaction are then obtained as a difference in the measured values for the two separate isotope c e l l s . The difference method, which was f i r s t used by Abel, Bratu, and Redlich (1) (1935) and by Korman and La Mer (2) (1936) i n their pioneering studies of hydrogen isotope effects, i s outlined below for a particular hydrogen-isotope-exchange reaction: hydrogen c e l l H (g,Pt)|HCl(H O)|AgCl(s)|Ag(s) 2
2
H (g) + 2AgCl(s) = 2Ag(s) + 2HCl(H O) 2
2
(1)
deuterium c e l l D (g,Pt)|DCl(D O)|AgCl(s)|Ag(s) 2
2
D (g) + 2AgCl(s) = 2Ag(s) + 2DCl(D O) 2
2
(2)
The isotope-exchange reaction i s D (g) + 2HCl(H O) = H (g) + 2DCl(D O) 2
2
2
2
(3)
At and the ε° value for reaction (3) i s given by 298.15 Κ the reported value(3,4,5)of i s - 4 . 3 4 mV (mole-fraction composition scale), which yields an equilibrium constant of K = 0.713 (298.15 K). 3
The difference procedure outlined above i s not useful, as such,for elements other than hydrogen, because the difference between the two ε° values i s then usually smaller than the sum of the absolute errors in the ε° values for the two isotope c e l l s . However, if we use electrochemical double c e l l s without l i q u i d 131
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
132
AND CHEMICAL
PRINCIPLES
junction in which the two isotope c e l l s are connected i n series opposition to one another, then we can measure directly the & value for the isotope-exchange reaction. For the reaction consid ered above, an appropriate double c e l l i s D (g,Pt)|DCl(D 0)|AgCl(s)|Ag(s)|AgCl(s)|HCl(H 0)|E2(g,Pt) (k) 2
2
2
the overall c e l l reaction for which i s reaction ( 3 ) above. Experimental Results f o r Lithium-Isotope-Exchange Reactions. The c e l l diagram for a double-cell used for the study of lithium-isotope-exchange reactions i s as follows II.a.
7
Li(s)| LiBr(solnO|TlBr^^
(5)
7
This double c e l l (Figur l ) involve following electrode reaction l y (left-to-right) i n th 7
7
fou
electrode
t which th
+
L i ( s ) = L i ( s o l n ' ) + e"
TlBr(s) + e" = H(Hg) + Br" (soin') EL(Rg) + Br" ( s o i n ) = ELBr(s) + e~ 6
+
6
L i ( s o l n ) + e-
= Li(s)
The overall c e l l reaction i s given by the sum of these four reactions, namely 7
6
6
7
L i ( s ) + LiBr(soln) = L i ( s ) + LiBr(soln')
(6)
Notice that Tl(Hg) and HBr(s ) do not appear i n the net c e l l reaction because the EL(Hg)| TlBr(s)| Br" electrode functions as the cathode of one side and the anode of the other side of the double c e l l . The solvent need not be the same i n the two halves o f the double c e l l (see c e l l (h)) although i n most cases of interest the solvent i s the same (6). 9
Application to the Nernst equation to reaction (6) yields ε
ο
=
e
+
RT ^ 4
a7
LiBr(soln-) a6
{ j )
LiBr(soln)
The application of equation (7) to a specific solvent case i n volves assumptions as to the nature of the dissolved LiBr. There are two limiting cases i n the thermodynamic analysis: (1) i f the dissolved LiBr can be treated as a non-electrolyte, then ^iBr
= m
Y
LiBr LiBr
( 8 )
for each isotopic species; whereas (2)
i f the dissolved LiBr can be treated as a strong (i.e., com-
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
ROCK
Equilibrium
Constants for Isotope Exchange Reactions
Journal of Chemical Physics Figure 1. Schematic of the double cell. i » t ^ ^ I LiBr(soln )\TlBr(s)\Tl(Hg)\TlBr(s)\ LiB^ Li(s). A,A are the outer cell compartments. The top is a standard taper 14/20 joint. B,B are the inner cell compartments containing Tl(Hg)\TlBr. C,C are droppers for sprinkling TlBr over Tl(Hg) in vacuum. D,D are the tubes con necting the cell to the LiBr and LiBr solutions in PC. E,E are connections to the vacuum line. F,F are the alligator clamps, and G,G are the Li and Li metal electrodes. H is the capillary connecting the cell to the Tl(Hg) reservoir. Ρ is the platinum wire central electrode for measur ing side potentials. The scale is 1:2.2 (8). 7
7
f
6
6
7
7
6
6
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
133
134
ISOTOPES
ANDCHEMICAL
PRINCIPLES
pletely dissociated) electrolyte then ^ i C l = *ϊ,1
+X
V -
=
\i+
X
Χ V -
Y
( 9 )
?(LiBr)
for each isotopic species. In equation (9) Y i s the mean ionic a c t i v i t y coefficient. The isotope effect on "the activity co efficient i s negligible ( i . e . at the same concentration γ = Y ) for a l l isotopes except those of hydrogen where the effect i s about (3) 2$ i n log Y at 0.05 molal HCl(aq). The decision as to how to treat the dissolved electrolyte i s based i n part on independent studies of the LiBr solutions. For example, emf measurements on the separate side c e l l s can be carried out i n order to obtain the necessary activity coefficient data. In the case of weak electrolyte solutes, equation (7) becomes +
7
6
±
e
whereas for the strong electrolyte case we have
°= fHSi
e
£+
ta)
There are two c r i t i c a l tests of the c e l l data that must be carried out i n order to establish the v a l i d i t y of the postulated c e l l reaction. The f i r s t of these i s the voltage-current revers i b i l i t y check which i s based on the fact that around an equilib rium point the restoring force that acts to bring the system back into equilibrium i s directly proportional to the displacement for small displacements. The r e v e r s i b i l i t y check for an electro chemical c e l l consists of a plot o f measured c e l l emf vs meterscale deflection on the null-detector. I f the resulting plot i s linear and does not show any hysteresis loop, then that particular c e l l can be safely regarded as behaving reversibly. The second c r i t i c a l test of the c e l l reaction i s provided by the constancy of the E° values calculated from several c e l l s with different concentrations of the isotopic solute species. The variations in the concentration ratios should be such as to produce both positive and negative values of the measured c e l l voltages. The results ( j , 8 ) of measurements on double c e l l s of type (5) for the solvents diglyme ( = 7 . 2 ) and propylene carbonate CH CHCH (e = 6 5 . 5 ) are given i n Table I. ^A e
3
2
In both cases the c e l l reaction i s 7
6
6
7
Li(s)·+ LiBr(soln) = L i ( s ) + LiBr(soln)
(12 )
however, in diglyme LiBr i s a weak electrolyte, whereas, i n pro-
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
Equilibrium
ROCK
Constants for Isotope Exchange Reactions
135
TABLE I. Results of measurements on the c e l l : 7
7
6
6
L i ( s ) I L i B r (digl,m )| TlBr(s ) | TL (Hg ) | T3.Br (s)| L i B r (digl ι%)1 Li(s). 7
3
(τ = 296.6 ± 1.0 κ)
e
ε° a
m (mol -kg" )
(mol-kg" )
obs (mV)
(mv)
0.03779
0.03099
- 4 . I 5 ± 0.20
Ο.92
o .03U5
Ο.Ο5923
14.82 ± 0.20
0.74
0.03524
0.03524
0.97 ± 0.10
0.97
0.05874
0.0603
7
1
1
.8
C a l c u l a t e d using Equation (lo) with Y
7
= Y. 6
Results of measurements on the c e l l : 7
7
Li(s)| LiBr(PC,m )|TlB^ 7
(τ = 296.6 ± 1.0 κ)
e° a (mv)
m (mol -kg" )
me (mol «kg" )
0.05075
0.05080
0.03372
0.04532
13.74 ± 0.05
0.66
0.04381
0.03775
-5.62 ± O.O9
Ο.95
0.04720
0.03903
- 7 . 8 3 ± 0.04
0.53
0.03180
0.03374
3.49 ± 0.08
7
1
e
obs (mv)
1
0.86
± 0.06
0.82
0.8k
0.13
C a l c u l a t e d u s i n g Equation ( l l ) w i t h In Y
mv)
= - I.6993 m / 2
±
(l+m ) - 0 Λ 5 Ι m. 2
pylene carbonate LiBr can be treated as completely dissociated. The emf results confirm the postulated c e l l reaction i n both cases. The investigation of lithium-isotope-exchange reactions involving the isotopic metals and the isotopic ions i n aqueous solution presents special problems because of the reaction of lithium metal with water. The clue to the resolution of this
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
136
AND CHEMICAL
PRINCIPLES
problem was provided by the method used by G. N. Lewis to determine the standard potentials of the a l k a l i metals i n aqueous media. Lewis and coworkers measured separately the voltages of the c e l l s Na (s )| Nal (amine )|Na(Hg) Na (Hg ) I NaCl (aq) | RfeClg (s ) | Hg U) where the sodium amalgam i s the same i n both c e l l s . The sum of the measured voltages for the two c e l l s yields the voltage of the hypothetical c e l l Na (s ) I NaCl (aq) | H g C l (s ) | Hg U) 2
2
The c e l l that we have used (g) to study exchange between the isotopic lithium metal ruple c e l l , an eight-electrod quadruple c e l l (Figure 2 ) i s as follows (pc - propylene carbonate) 7
7
7
7
L i ( s ) I LiBr(pc)1 Li(Hg)| LiCl(aq)|Hg Cl (s)|HgU)2
6
6
2
6
(13)
6
-|Hg Cl (s)| LiCl(aq)| Li(Hg)| LiBr(pc)| Li(s) 2
2
the postulated electrode reactions of the quadruple c e l l are 7
7
7
+
L i ( p c ) + e" 7
+
L i ( s ) = L i ( p c ) + e" 7
= Li(Hg) 7
+
Li(Hg) = Li (aq,m ) + e" 7
i H g C l ( s ) + e" = HgCt) + Cl"(aq,m ) 2
2
7
HjgCt) + Cl-(aq m ) = £ H g C l ( s ) + e* 3
6
+
Li (aq
2
2
6
5 m 6
6
6
6
) + e~ = Li(Hg) 6
+
Li(Hg) = L i ( p c ) + e"
+
6
L i ( p c ) + e" = L i ( s )
The sum of the above eight electrode reactions yields 7
6
7
L i ( s ) + ^iCliaq^me) = L i ( s ) + LiCl(aq,m )
(ik)
7
Note that both lithium amalgams as well as the central Hg(t) and H g C l ( s ) phases do not appear i n the net c e l l reaction. Consequently we do not need to know the concentrations of L i and L i in the amalgams. The key to the success of the above quadruple c e l l i s the combination of a high overvoltage for hydrogen evolution on a mercury surface (ca. IV ), together with a very low concentration of lithium i n the amalgams ( X ^ ~ 10" ). The 2
2
7
6
5
results of emf measurements on the quadruple c e l l show that reaction (lh) i s indeed the c e l l reaction of the quadruple c e l l .
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
Equilibrium
ROCK
Constants for Isotope Exchange Reactions
137
Journal of Chemical Physics Figure 2. Schematic of the quadruple cell. Li(s ) \ LiBr( pc ) 1 Li( H g ) \ LiC l(aq)\Hg Cl (s)\Hg(l) IHg Ck(s)1 LiCl(aq)1 Li(Hg)\ LiBr(pc)\ Li(s). A,A are the outer cell compartments containing isotopic lithium electrodes in isotopic lithium bromides in PC; C , C and C are the capillary tubes connected to Li(Hg) Li(Hg), and Hg(l) reservoirs, respectively: G , G , H and H are the delivery tubes connected to the LiBr(PC), LiBr(PC), LiCl(aq), and LiCl(aq) solutions, respectively. E,E and F,F are connections to the vacuum line manifold; S,S are droppers for preparing the calomel electrodes. P , P , and P are connection points for electrode leads that permit voltage readings of various sections of the complete cell ( 9 ). 7
6
2
6
6
7
7
7
z
z
6
7
6)
7
m
7
6
6
7
>
6
6
7
6
7
6
m
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7
ISOTOPES
138
The measured value β° i s 1.16 equilibrium constant of 1 . 0 4 6
AND
CHEMICAL
PRINCIPLES
± O.3O mV, which corresponds to an ± 0 . 0 1 3 ( 2 9 6 . 6 K).
Lithium-isotope-exchange reactions involving the metals and pure solid salt phases can also be studied i n electrochemical c e l l s . Such reactions are particularly simple to analyze electrochemic a l l y because ε = £°. An example i s the c e l l (lo) 7
T
6
(15 )
6
Li(s)| LiF(s)|KBF (pc)| LiF(s)| Li(s) 4
Lithium fluoride i s insoluble in propylene carbonate, and the KBF acts as a source and sink for fluoride ions while keeping the fluoride ion concentration low and thereby preventing the attack by fluoride ion on the solvent. (Tetraalkylammonium fluorides decompose propylene carbonate, presumably by promoting ring opening followed b reactions for c e l l (15) 4
7
7
L i ( s ) + BF "(pc) = L i F ( s ) + BF (pc) + e" 4
6
3
6
L i F ( s ) + EF (pc) + e" = BF ~(pc) + L i ( s ) 3
4
and the net c e l l reaction i s 7
6
6
(l6)
7
Li(s) + LiF(s) = L i ( s ) + LiF(s)
Measurements (lo) on c e l l (15) have established reaction (16) as the c e l l reaction; the measured value of ε° i s 2 Λ 9 ± O.I5 mV. Reaction (16) exhibits the largest lithium isotope effect that has been found experimentally. The measured equilibrium constants for the four different types of lithium-isotope-exchange reactions are summarized i n Table II. TABLE II.
Lithium-Isotope-Exchange Reactions Κ (296.6
Reaction
(ε° 7
L i ( s ) + LiBr(digl) = L i ( s ) + LiBr(digl)
6
6
7
7
L i ( s ) + L i Br" (pc J = L i ( s ) + L i B r " (pc )
6
6
7
7
L i ( s ) + L i Cl"(aq) = L i ( s ) + L i C l " (aq)
6
6
7
7
Li(s) + LiF(s) = L i ( s ) + LlF(s)
+
6
7
κ)
mv)
I.O35 ± 0 . 0 0 7 (8) ( 0 . 8 7 ± 0.18) I . 0 3 0 ± 0 . 0 0 5 (8) (Ο.76 ± 0 . 1 3 ) 1 . 0 4 6 ± 0 . 0 1 3 (3) (1.16
6
in
1.10 (2.49
±0.30)
± 0.01 ± 0.15
(lo) )
II.b. Experimental Results for Hydrogen-Isotope-Exchange Reactions. Equilibrium constants for hydrogen-isotope-exchange reactions can be obtained either by the difference method or by the doublec e l l method. The difference method for hydrogen-isotope-exchange
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
ROCK
Equilibrium
Constants for
Isotope Exchange Reactions
139
reactions i s outlined i n the introduction. The general c e l l diagram of a double c e l l that has been used ( l l ) to study hydrogen isotope-exchange reaction i s (17)
D (g,Pt)|DCl(soln)|^ 2
The postulated electrode reactions for the double c e l l are as follows D (g) 2
+
= 2D (soln) + 2e"
2TlCl(s) + 2e" = 2Cl"(soln) + 2Tl(ng) 2Tl(Hg) + 2Cl-(soln') = 2TlCl(s) + 2e" +
2H (soln') + 2e" = The net c e l l reaction i trode reactions
give
Hg(g) b
th
f th
abov
fou
D (g) + 2HCl(soln') = Hg(g) + 2DCl(soln) 2
elec (18)
If the HCl(soln) can be treated as a strong electrolyte, then (mY±)
ε° = ε + *
"
"
r
"
whereas, i f HCl(soln) i s a weak electrolyte, then
V
= ε +
W te) w ITTPJ ωto
1^1
++ |
^ |
( m Y )
to
DCl
(mY
)
(20)
HCl
The dual requirements of non-exchangeable solvent hydrogen atoms and electrode r e v e r s i b i l i t y severly limits the number of cases in which the solvent can be made the same on both halves of the double c e l l . One such case i s provided by the solvent N,Ndimethylformamide (DMF ) CH
3
.0
CH
3
H
The hydrogen atoms of DMF (e = 3 7 ) do not exchange at a measureable rate with DC1 under anhydrous conditions ( l l ) , and both the hydrogen electrode and the Tl(Hg)|TlCl(s)|Cl"(DMF) electrodes are reversible i n D M F . The overall c e l l reaction i s D ( g ) + 2HC1(DMF) = Ife(g) + 2DC1(DMF) 2
(21 )
In this case i t proved possible to obtain ε° values for reaction (2l) involving both the undissociated H C I ( D M F ) and the completely dissociated H C I ( D M F ) standard states. Even though H C I ( D M F ) i s a weak electrolyte ( i ^ =2-7 χ 10" at 25°c), Petrov and Utaanskii (l2 ) were able to establish a strong electrolyte standard state by 4
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
140
AND CHEMICAL
PRINCIPLES
going to very low H C I ( D M F ) (<10~ molal) concentrations; conse quently, Y values for H C I ( D M F ) were available. The calculated 6° values based on strong and weak electrolyte standard states are 7.64 ± Ο . 3 8 and 8.12 ± Ο . 5 4 mV, respectively ( 2 9 6 . 6 K). The corresponding equilibrium constants for the reactions 5
±
D (g ) + 21^01" (DMF ) = H (g ) + 2D C1~ (DMF )
(22 )
+
2
2
D (g) 2
+ 2HC1(DMF) = Hg(g)
are K. = 1 . 8 2 ± 0 . 0 5 and Κ . = 1.89 ion neut values y i e l d a value of y
K
(
= ^ « Λ ο η
)
4
1
= ·
0
±
0.08.
* °-
2
(23)
+ 2DCl(DMF)
fe
0k
These two Κ
9
6 6
K
)
for the reaction D ( D M F ) + HC1 (DMF ) = IT" (DMF ) + D C l ( D M F )
(24)
1
+
The value of Κ for reaction ( 2 4 ) was also determined inde pendently by a conductometric method (13 ) i n which Κ was deter mined as the ratio of Κ values for the reactions = EHDMF) + CI"(DMF)
(25)
D C 1 (DMF ) = D (DMF ) + C I " (DMF )
HCI(DMF)
(26 )
+
The value of Κ obtained was (25°c) _ K
( 2 . 6 8 ± 0,05) X 1 0 ~ ~ (2.64 ± 0 . 0 2 ) x l 0 -
— 1
4
4
04
OP + 0 - ' " 1
0
2
An example o f a double-cell i n which the solvents are not the same i s the c e l l (14,18) (ROH Ξ CH (CH ) CH 0H and ROD = CH3(CHg, ) CH 0D) 3
4
2
4
2
2
(27)
D (g,Pt)|DCl(ROD)|llCl(s)^ 2
for which the net c e l l reaction i s D (g) + 2HC1(R0H) = H (g) + 2DC1(R0D) 2
2
(28)
The results of measurements on c e l l ( 2 7 ) y i e l d an £° value for reaction ( 2 8 ) , calculated on the basis of strong electrolyte standard states for H C I ( R O H ) and D C I ( H O D ) , of 3 . 5 3 ± Ο . 9 5 mV at 296.Ο K; the value of the equilibrium constant i s I . 3 2 ± 0.10. The equilibrium constants for several hydrogen-isotopeexchange reactions involving elemental hydrogen are given i n Table I I I .
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
ROCK
Equilibrium
Constants for Isotope Exchange Reactions
TABLE III.
141
Hydrogen-Isotope-Exchange Reactions Κ ( 2 9 6 . 6 Κ)
Reaction
(β° i n my)
D (g) + 2HCl(g) = H (g 2
2
D (g) + 2HC1(DMF) = H (g 2
2
+ 2DCl(DMF)
+
Ife(g
+ 2D Cl'(DMF)
+
Ife(g
+ 2D+C1"(ROD)
H (g
+ 2D C1~(D 0)
D (g) + 2H C1~ (DMF ) 2
D (g) + 2H C1~(R0H) 2
+
D (g) + 2H C1~(H 0) 2
2
2
1.99* (8.87)* I.89 ± 0.08 (8.12 ± 0.54) 1.82 ± 0.05 (7-64 ± Ο.38) 1.32 ± 0.10 (3.53 ±0.95) 0.713+
+ 2DCl(g)
+
+
2
(-4.34 mv) ^Calculated value, se Phys. 1, I39 (I933) and reference (3JL). +References ( l ) , (3), (4), (5) In an isotope-exchange reaction involving solvent asymmetry, for example, reactions ( 3 j and (28), the numerical values of £° and of Κ depend on the concentration scale chosen for the scOutâonphase species. For example, application of the Nernst equation to the reaction D (g) + 2HCl(soln-h) = Ife(g) + 2DCl(soln-d) 2
(29)
yields
ε = ε°
I V cia
^
\%)
*
(50)
where both ε° and the a c t i v i t i e s depend on the choice of concen tration scales. The c e l l voltage, ε, i s a directly measureable quantity and consequently the value of ε must be independent of the concentration scale chosen, therefore we can write (using equation (30)) [c = molarity (a. = c.y^); m = molality (a^ =
If we take the l i m i t of both sides of these equations as the con centrations go to zero, and we note that i n the l i m i t c/m = ^and
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
142
AND CHEMICAL
PRINCIPLES
x/m = M /lOOO (where ρ i s the density of the pure solvent and s s M i s the molecular weight of the solvent), then we obtain s
β)
aê-Ç-^f^l =
(35)
&a\
2RT
Combination of equations ( 3 3 ) and ( 3 4 ) with the relation ε° = ^-p£nK yields the results ?
Κ
For
χ
=
κ(^Υ m
VW
= κ / ^ Ϊ
and Κ c
m
\ sh/
(35)
P
reaction ( 3 ) we have 25°C &
= - 9 - 7 8 mv
K
m
= 0Λ67
S° = - 4 . 5 3 mv
K
c
= O.703
e°
Κ
m
χ
= - 4 . 3 4 mV
II.c. Gas-Phase Reactions. reactions, for example
χ
=0.713
Gas-phase,
hydrogen-isotope-exchange
D (g) + 2HCl(g) = H (g) + 2DCl(g) 2
2
(36)
can also be studied i n electrochemical c e l l s . The determination of accurate (< ± 1$ error) values of equilibrium constants for gas-phase isotope-exchange reactions can provide the necessary data for a c r i t i c a l test of the theory of isotope-exchange reac tions. The need for such an experimental test was f i r s t brought to our attention by Professor Max Wolfsberg. Kleinman and Wolfsberg (l^) have calculated that there i s a Born-Oppenheimerapproximation failure in the range of 3 to 10fo at 300 Κ i n the equilibrium constants for hydrogen-isotope-exchange reactions of the type HX(g) + HD(g) = DX(g) + Hg(g)
(37)
where X i s L i ( 2 . 9 $ ) , Β, N, or F (lO.l/o). Prior to the work of Kleinman and Wolfsberg (see paper i n this volume) i t was generally believed that the Born-Oppenheimer approximation contributed at
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
ROCK
Equilibrium
Constants for Isotope Exchange Reactions
143
most an error of about 0.1$ to the equilibrium constant of an isotope-exchange reaction. Reactions of the type ( 3 7 ) are not suitable for study i n electrochemical c e l l s ( 1 6 ) because they involve the isotopicallymixed species HD. However, a reaction like ( 3 6 ) can be investi gated directly i n an electrochemical c e l l . The basic idea i s to compare an experimental value of Κ for reaction (36) with two calculated (by Wolfsberg, et al.) values of K, one obtained invoking the Born-Oppenheimer approximation, and the other obtain ed without invoking the Born-Oppenheimer approximation. In this way the r e l i a b i l i t y of the Born-Oppenheimer approximation for hydrogen-isotope-exchange reaction can be tested experimentally. Reaction ( 3 6 ) i s being studied i n cells of the type D (g,Pt)|DCl(D 0)|H^Cl It 2
2
It HCl(g)
DCl(g)
where the DCl(D o) solution i s i n equilibrium with a gas phase containing DCl, D , and D 0, and the HClÎH^o) solution i s i n equilibrium with a gas phase containing HD1, H , and HgO. The basic idea for the design of the respective halves of the double c e l l ( 3 8 ) stems from the work of Aston and G i t t l e r ( l j ) on the cell 2
2
2
2
H (g)|HCl(soln)|AgCl(s)|Ag(s). tl HCl(g) 2
The net c e l l reaction for c e l l ( 3 8 ) i s D (g ) + 2HG1 (H 0 ) = Hs (g ) + 2DC1 (D 0 ) 2
2
2
(39)
Also, for the conditions prevailing i n the c e l l 2H3l(g) =2HCl(H 0)
ÙG = 0
(40)
2DC1(D 0) = 2DCl(g)
AG = 0
(41)
2
2
Addition of equations ( 3 9 ) , ( 4 θ ) , and (41 ) yields D (g) + 2HCl(g) = H (g) + 2DCl(g) 2
2
and therefore AG
42
= Δ&39 = AG
42
+ RTto a
HCl(g)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
(42)
ISOTOPES
144
AND CHEMICAL
PRINCIPLES
Also Δ&39 = where 839 i s the measured voltage of the c e l l . Thus (assuming ideal-gas behavior at pressures less than 100 Torr, for the purposes of discussion) we can write
-42
P
D ] 2
A measured value of 639, together with the corresponding measured values of the pressures Ρ _ , , , P__ , and P^ (manometers), ηΊ
UGJL
liLJ-
lig
Dg
yields a value of ε , no extrapolations are necessary and measurements can be mad HCl(aq), Ρ___, ~ kl Torr and P „ = k Torr, whereas for m 4 2
Ί
TT
MUX
—
JtigU
H C l i a q J j . P ^ ~ 120 Torr and Ρ = 3 Torr. Our preliminary measurements (18) on c e l l (38) show that the cells are remarkably stable (emf fluctuation less than 12 μν over k8 hours) and reversible. The major experimental problem concerns the determination of the composition of the gas phase in e q u i l i brium with the c e l l electrolyte. Experimental pressure measure ments give the sums Ρ + Ρ„ ^ and Ρ _, + ρ + Ρ ; the gas β
Q
τι/Ί1
nl/JL
ττ
*%U
Ί
ribJ-
Jig
Ά%>Ό
phases i n the two halves of the double c e l l must be analyzed i n order to obtain P _ and Ρ__, . If the measured pressure ratios are good to 0 . 3 $ (which corresponds to 0 . 1 Torr i n the pressures for roughly equal pressures), then Κ can be obtained to with an error of less than 0 . 5 $ . An example of some preliminary data for c e l l (38) i s the following (20°c) 6 = 4-596 ± 0.011 mV, P + rr
(1
Ί
H C l
Rs>0 Ηβ 7Ο6.3Ο Torr, P + Ρ = 70.65 Ίοντ, and P = 697.6Ο Torr. If we assume that ( P Ep V (P + P q ) ~ ^ c i / ^ H C l because ' was we had not perfected our gas-phase analysis method), then we compute that Κ = 1 . 9 9 . Further measurements on c e l l (38) are i n progress. P
=
6
0
,
7
2
T
o
r
r
?
Ρ
=
DCl
D
HC1
+
DC1
H
a
fc
t
n
e
t
i
m
e
t
n
i
0
s
c
e
l
1
T V X l
II.d.
Calcium-Isotope-Exchange Reactions. There are several stable isotopes of calcium (40, 4 2 , 4 4 , 46, and 48). The 20$ mass difference between ° C a and C a i s larger than the 17$ mass difference between L i and L i . The 4
6
4 8
7
square root of the mass ratio i s (48/40) = I.O95 for calcium and 2
( 7 / 6 F = I.080 for lithium. The electrochemical determination of equilibrium constants for isotope-exchange reactions of the type (12)
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
Equilibrium
ROCK
48
Constants for
Isotope Exchange Reactions
145
Ca(s) + CaX (soln) = Ca(s) + CaX (soln)
(44)
48
(45)
40
40
48
2
2
and 4
4
48
C a ( s ) + °Cax(soln) = °Ca(s) + Cax(soln)
has not proved possible to date, because the conditions under which calcium metal electrodes behave reversibly have not been discovered. However, calcium-isotope-exchange reactions of the type 48
4 0
2 +
4O
4 8
2 +
CaC0 (s) + C a ( a q ) = CaC0 (s) + C a ( a q ) 3
3
as well as analogous reactions involving hydroxyapatite apatite and other calciu interest, can be conveniently c e l l s involving electrodes of the t h i r d kind.
(46) fluor
An electrode of the f i r s t kind involves a metal .lie phase i n contact with an electrolyte phase containing the ions of the metal, for example 2+
Fb(s)|Pb (aq) Pb(s) τ± Fb (aq) + 2e"
(4τ)
2+
An electrode of the second kind involves a metallic phase i n contact with an electrolyte solution i n which the a c t i v i t y o f the metal ions i n equilibrium with the metallic phase i s controlled by an anionic, solution-phase species that forms an insoluble salt with the metal ion, for example Fb(s)|PbS0 (s)|SOf'(aq) 4
(48)
Pb(s) + SOf-(aq) 5± FbS0 (s) + 2e~ 4
An electrode of the third kind involves a metallic phase i n contact with an electrolyte solution i n which the a c t i v i t y of the metal ions i n equilibrium with the metallic phase i s controlled by a different cat ionic, solution-phase species through the presence of two insoluble salts of the respective cations, for example Fb(s)| FbC0 (s ) ,CaC0 (s)| G a 3
3
2+
(aq)
2+
Pb(s) + CaC0 (s) ^FbG0 (s) + Ca (aq) + 2e" 3
3
(49)
Electrodes of the third kind were f i r s t reported by Le blanc and Harnapp (gO ), and the particular type of third-kind electrode outlined above was used by Jakuszewski and Tanieuska-Osinka (2l) to determine the Gibbs energies of formation of Ca (aq). 2+
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
146
AND CHEMICAL
PRINCIPLES
The c e l l diagram for a double c e l l involving electrodes of the third kind, for which the net c e l l reaction i s equation (46) i s (22) Fb (Hg ) I FbC0 (s ) , CaC0 (s ) 1 CaCl (aq) | H g C l (s )| HgU)2-phase 48
48
3
3
2
2
2
- H g U ) | H g C l (s ) 1 CaCl (aq) | CaC0 (s),FbC0 (s ) | Fb (Hg ) 2-phase 40
2
2
4O
2
3
3
(50 )
where we have employed a two-phase lead amalgam rather than Pb(s) because the former can be prepared i n a reproducible, strain-free state. We are presently using c e l l s of the type (50) (both with aqueous and nonaqueous c e l l electrolytes) to investigate the thermodynamics of calcium-isotope-exchange reactions The data from such c e l l s should provid methods (such as mass-spectrometri phases), which suggest that there i s no isotopic fractionation i n the precipitation of calcium salts from aqueous media (23,24 ) involving Ca (aq). The results of Heumann (24) on isotope exchange between Ca (aq) and calcium ions chelated on Dowex A l resin suggest that i f c e l l (50) i s run with chelated calcium ion in solution (e.g. CaEDTA , or calcium chelated by biologicallyimportant ligands), then isotopic fractionation might be observed. Fractionation of calcium isotopes between inorganic salts and chelated calcium ion i s a process of obvious potential interest as a probe into biological processes. 2+
2+
2+
III.a. Isotope-Exchange Reactions as a Probe for the Study of Ion Solvation. In thermodynamics we are no longer content to know the value of an equilibrium constant for a chemical reaction; we also want to know why the equilibrium constant has that particular value. In other words, we would l i k e to be able to calculate the value of the equilibrium constant from microscopic properties (atomic masses, bond lengths, bond angles, force constants, dissociation energies, etc.) together with quantum-mechanical and statisticàlthermodynamic theory. Isotope-exchange reactions are the simplest class of chemical reactions, and consequently isotope-exchange reactions have played a central role i n the development of theories of chemical reactions. The development of the theory of isotope-exchange reactions i s outlined by Bigeleisen i n the opening paper of this symposium. The Bigeleisen-Mayer theory o f isotope-exchange reactions (2^,26) has proved remarkably successful i n predicting equilibrium constants for gas-phase reactions and for reactions involving gases and pure condensed phases. Although some major questions remain to be answered, especially .in connection with the accuracy of the Born-Oppenheimer approximation for hydrogenisotope-exchange reactions (see paper by Wolfsberg and Kleinman
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
Equilibrium
ROCK
Constants for Isotope Exchange Reactions
147
in this volume), the overall status of the theory for such reac tions appears to be excellent. The accuracy with which equilib rium constants can be calculated for reactions involving only gases and/or pure condensed phases r i v a l s , and i n some cases exceeds, the accuracy with which the equilibrium constants can be measured. The picture i s not so clear for isotope-exchange re actions involving solution-phase species, because of major un certainties associated with the nature and structure of solutionphase species; this problem i s especially acute i n the case of solvated ions. The determination of equilibrium constants for heterogeneous, isotope-exchange reactions involving isotope-exchange between pure phases and solution phases yields partition-function ratios for the isotopic, solution-phas species I favorabl comparison of calculate ratio with the experimenta distinguis between possible solute models. The statistical-thermodynamic constant of reaction (51) 7
6
6
(51)
7
L i ( s ) + L i x ( s o l n ) - ^ L * L i ( s ) +_ Lix(soln) /
AG
1 7
expression for the equilibrium
constant \ Ρ
^temperature/
6
L i ( s ) + Lix(g)
^52
— 2 2 * 6
L
i
(
s
)
T
+
L
i
x
(
g
)
(
5 2
)
is Κ
feiiaA
=
/^ix(soln)\
θ χ ρ (
.
ρ Δ ν / Ε τ )
(
,
\ ^ L i ( ) / V^LixUola)/ s
where q^ terms are the partition functions (including the zeropoint energy factors) of the various species evaluated i n the respective standard states. The volume change, Δ¥, for a symmetric lithium- isotope-exehange_reaetion jsuch as reaction (51) i s essentially zero (27), because ^ 7 7 ^ ( 3 ) = 6 J L ( ) v
7
^ LiX
=
6
^ L i X ^°
w i > t h i n
±
°·2$·
L
S
A N C I
Consequently, equation (53) can
be written as
W i f e ) /
V^LUcCsoln)/
We shall adopt the following standard states i n which the various partition-function ratios w i l l be evaluated: (a) solids and liquids - the pure, unstrained material at one standard atmosphere
American Chemical Society Library 1155 16th St., N.W.
In Isotopes and Chemical Principles; Rock, P.; Washington, D.C. 2003S ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
148
AND CHEMICAL
PRINCIPLES
pressure (I.OI3 χ 10 Ν·ια" ) and the temperature of interest ; (b) gases - the hypothetical, ideal gas at one standard atmosphere and the temperature of interest; and (c) solutes - the hypo thetical, ideal solution containing unit-mole-fraction of the solute. 5
2
For the thermodynamic cycle given above - equations ( 5 1 ) and (52) - we have AG*
X
= AG
5 2
+ AG ° + AGp
(55)
and therefore (using ù&° = -RT £n K) *51
= K
52 ^
(56)
Kp
but as =
LiJC(g)
a6
m
Lix(soln)
LUc(soln) a7
L:LX(g) _ a
LiLx(g)
j
(
7
7
* L:Lx(g) ~ ^ L i X
a
where |(g) i s the Henry's law constant for the solute i i n the Combination of equations ( 5 6 ) and ( 5 7 ) yields
particular solvent.
K51
=Ks
2
(
( I W * W
5
8
)
In other words the treatment of solution-phase species (i.e. taking K = ) as gas-phase species i s equivalent to the assumption that there i s no isotope effect on the Henry s law constant for the isotopic solutes. 51
1
An experimental determination of Κ for a reaction of the "tyP (51 ) can be combined with a calculated value for the p a r t i tion-function ratio of the isotopic solids, Q 6 j | ( ) / l 7 T i ( ) obtain an "experimental" value for the partition-function ratio of the solute species, ^ ( s o l n / ^ L U c t s o l n ) ' e
c
j:
<
s
,
i
J
t
o
s
IH.b. Evaluation of Part i t ion-Function Ratios for Isotopic Solids. The quantitative evaluation of the partition-function ratio K
(
s = ^Li(s/^Li(s)
5
9
)
for the isotopic lithium metals requires a choice of model for the solids. The choice of a particular model for the solid phase determines the form of the frequency distribution function for the l a t t i c e , which i n turn determines the partition-function ratb. A mole of a monatomic solid has 3N - 6 2? 3 · (where N i s Avogardo s number) vibrational frequencies (normal modes). The Ν
0
0
1
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
Equilibrium
ROCK
Constants for Isotope Exchange Reactions
149
partition function for a one-dimensional, harmonic o s c i l l a t o r of frequency i s given by v
oj. = exp ((D -hv/2ykT) ^L-expC-iiv/kT)^
(60)
1
e
where D i s the dissociation energy measured from the "bottom of the potential well. For isotopic species the D values are equal (Born-Oppenheimer approximation). I f we assume that the solid consists of 3N harmonic oscillators, then Κ can be written s Q
0
•exp(-hv /kT)| i7
κ
Τ
=
ι •exp(-hv /kT}(
exp{-h(v
l6
•v )/2klj i7
(61)
i6
i=l
'
An Einstein solid i s one i n which a l l of the atoms are treated as three-dimensional harmonic oscillators with frequency v^. Thus for monatomic Einstein solids, equation (61 ) becomes ( 1-exp(-hv/kT)) K
3
=
s (l-exp(-hv^kT)|
, e x
Λ
P{-3^VV
) / 2 k T
}
( 6 2 )
The Einstein characteristic temperature i s defined by the relation hv =k9 E
(63)
E
and thus equation ( 6 2 ) can also "be written as il-expi-e^ /τη /T)f (i-exp(-e_
s
,
Κ s The Einstein frequent i s inversely proportional to the atomic mass of the l a t t i c e atoms and thus θ.
=θ
Thus the calculation of K or θ-
Q
1
(65)
2
(m /me) 7
requires an experimental value of θ_
, which can be obtained from heat capacity measurements.
It i s usually θ
β
(=hv^/k), the Debye characteristic temperature
which i s reported, but θ„ can be calculated from θ ill
η
using the
JJ
relation (j,8)
θ, = 3Θ-Λ
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
(66)
150
ISOTOPES
AND CHEMICAL
= 353K ( 2 8 ) , then we compute that Κ
If we take JJy
——
PRINCIPLES
= 0 . 7 8 1 1 at S
25°C for Einstein solids.
The atoms i n a Debye solid are treated as a system of weakly coupled harmonic o s c i l l a t o r s . Normal modes with wavelengths that are large compared to the atomic spacing do not depend on the discrete nature of the crystal l a t t i c e , and consequently these normal modes can be obtained by treating the crystal as an iso tropic elastic continuum. In the Debye treatment of a solid a l l of the normal modes are treated as elastic waves. The partition function for a Debye solid cannot be obtained i n closed form, but the thermodynamic functions for a Debye solid have been tabulated as a function of θ^/τ. For the pair of isotopic metals L i ( s ) 6
7
and L i ( s ) we have (Δν = Δ0° = ΔΑ
+ ΡΔν
= ΔΑ°
and 0
Κ = exp(-M°/RT) = exp|-A(A -E )/RT^exp(-AE /RT) 0
0
(67)
The function ( A ° - E ) / R T i s tabulated ( 2 2 2 θ ) as a function of Θ^/Τ, and the zero-point-energy term i s given by 0
?
ΔΕο/RT = 9 ( θ
where θ have
= θ
D e
D
2
(π^/ιι^ ) .
η
-θ-
(68)
)/8T
Thus, for the isotopic Debye solids we
7
^ j ^ H ^ h P ^ Î = 353 Κ, then we compute that Κ
If we take Dy
(69
'
= Ο.78Ο9 at 25°C. S
The most sophisticated model for the solid phases i s that developed by Born and von Karman (32 ). We have carried out a Born-von Karman (BVK) l a t t i c e calculation for the isotopic lithium metals. In our calculations we have followed closely the methods described by de Launay (jg). In our model for the atomic inter actions we chose to consider only central forces, and to take only the forces between nearest and next-nearest neighbors as signif icant. This choice was dictated mainly by the fact that the secular equation for a bcc (as i s lithium; crystal with this type of force interaction has been treated (^k). In our calculations we used the root-sampling method and solved the _secular equation for 3 3 I I values of the wave propagation vector, k. These values of k were obtained from the coordinates of a cubic grid of 3 3 I I points i n the irreducible element of the f i r s t B r i l l o u i n zone. The nearest- and next-nearest-neighbor force constants, Ο Ί and a ? 2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
Equilibrium
ROCK
Constants for Isotope Exchange Reactions
151
were calculated from the macroscopic elastic constants C , C and C , using the relations given by Bauer ( 3 ^ ) , v i z . , a = 1X
44
1 2
x
— a C , and a 44
2
= 2 a ( c - C ) , where a i s the lithium bcc l a t t i c e 1 1
1 2
2
parameter, 2a =3.50 A. The values of the l a t t i c e constants used were those reported by Nash and Smith (36). In the harmonic approximation the l a t t i c e constants should be independent of temperature. This i s not confirmed experiment a l l y . We have there fore used the l a t t i c e constants reported for the lowest temper ature that was investigated experimentally (77 κ), because at this temperature the l a t t i c e vibrations w i l l be more nearly harmonic. The calculated l a t t i c e spectrum i s approximated as a histo gram of the number of (Figur 3 a ) partition function ratio lithium metals can be calculated from our histogram of the lattice spectrum using the equation J s
3
S
3
g
jexp(-3hv /2kT)[l-exp(-hv^/kT)]" } ie i6
ΤΓ | e x p ( - 3 h v / 2 k T ) [ l - e x p ( - h v / k T ) ] - } i 7 i7
i7
where g^ i s the fraction of o s c i l l a t o r s having the frequency 2jg^ = 1.
Substitution of the g^ values obtained from the BVK
calculations into equation (70) yields Κ
= 0.7866 at 298.15 K. s The results obtained for Κ show that the value of Κ i s not s s strongly dependent on the detailed nature of the model chosen for the isotopic solids; the t o t a l variation i n Κ i s less than 1$. However, when considered from the point of view of lithiumisotope-exchange reactions, a 1$ variation of Κ may be as much as 30^ of the t o t a l isotope effect. BVK-lattice calculations can also be carried out for isotopic lithium salts, and we have carried such calculations for the isotopic pair of salts L i F ( s ) and L i F ( s ) (32 ). Within the harmonic-oscillator approximation the partition-function ratio for the isotopic lithium fluorides can be written as 1/N (^/l-exp^hv^/kTA /-h(v -v )U s
7
6
i 7
^LlKs)
=
{ Jl
\l>exp(-hv /kTj; ^ i 7
\
i 6
2 k T — ])
+
( 7 1 )
where Ν i s the number of L i or F" ions i n the crystal. The evalu ation of equation (71) requires a knowledge of the frequency dis tribution function for the LiF(s) crystal l a t t i c e . Lithium fluo ride i s presumed to be an ionic crystal having a NaCl-type
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
ANDCHEMICAL
À
Figure 3a. Born-von calculated using elastic constants determined at 77°K. The vertical axis is the fraction of oscillators in a frequency range of 0.05. The horizontal axis is the frequency in reduced, dimensionless units, (3Μπ /2α,) (8). 2
1/2
1400
1200 1000
! ! :' · » · ι .ι
800
»!
N(v) 600
t
400
I
I
Λ \
200
0.0
0.5
1.0
2.0
1.5
2.5
3.0
13 ν χ 10'
Figure 3b. Born-von Karman vibrational frequency distribution for the LiF lattice (- · ·) and for the LiF lattice (—) calcu lated with 0°K input data. The line occurs in the region where the two distributions are indistinguishable ( 10 ). 7
6
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
PRINCIPLES
7.
Equilibrium
BOCK
Constants for Isotope Exchange Reactions
153
structure. The -vibrational spectrum for a NaCl-type of l a t t i c e was f i r s t calculated by KeUermann (38 , . 2 2 ) ; the calculational methods developed by KeUermann were refined by Sayre and Beaver (ho) and these methods have been applied to normal-isotopicabundance LiF(s) by Karo (41 ) . The KeUermann treatment involves the separation of the interionic forces into short-range repulsive forces between nearest neighbors only, and long-range Coulombic forces that extend over the entire crystal l a t t i c e . 3
The evaluation of the Coulombic force constants requires the summation of the Coulombic interactions over the whole of the crystal l a t t i c e for each wave vector considered. KeUermann, and Sayre and Beaver have carried out this summation for k& evenlyspaced wave vectors i n the irreducible element of the f i r s t Brillouin zone of NaCl(s) An exact evaluation of equation (71) would require the solutio evenly-spaced wave vector root-sampling method these N/48 wave vectors are approximated by an evenly-distributed grid of points i n the irreducible element. The 48 points i n the irreducible element for which Coulombic force constants have been calculated correspond to 1000 wave vectors i n the f i r s t Brillouin zone. For a given wave vector six equations of motion (assuming only central forces) are necessary to describe the three normal components of the displacent for the two types of atoms. In order for wave re-enforcement to occur these s i x equations must be simultaneously satisfied. Thus for a given wave vector six normal modes w i l l be obtained; 1000 wave vectors y i e l d 6000 of the 6N normal modes of L i F ( s ) . Because the fre quencies are very densely packed i t i s assumed that the normali zation to 6N normal modes does not affect the form of the fre quency distribution. The numerical evaluation of the frequency distribution of L i F requires the ionic masses, the equilibrium internuclear distances, and the compressibility as input param eters. The calculated frequency distribution functions for L i F ( s ) and L i F ( s ) are shown i n figure 3b. As can be seen i n the figure the isotopic shift i n the frequency distribution be comes much more pronounced as the frequency increases. 7
6
Equation (7I) can be rewritten i n the root-sampling approxi mation as 288 "/l-exp(-hv. /kT)\ -jT^l-exp(-hv /kT)/
s.j/1000 /-h(v -v )\"
6
7
^ LiF(s )
=
i7
*>L1F(B)
^
L
i7
e
x
p
^
2kT
i6
/
(72)
.
where s^ i n equation (72) i s the number of times that the i t h wave vector i s reproduced by the symmetry operations of the Brillouin zone ( Σ ξ ^ s^ = 6000 ).
The repeat product i n equation
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
154
ISOTOPES
AND
CHEMICAL
PRINCIPLES
(72) i s over the 288 (= 6 χ 48) eigenvalues which result from the solution of the equations of motion for the 48 points consi dered. The value of ( ) / ( s ) " ^ 298.15 Κ c a
L c u l a
b e ( i
a
s
(using 0 Κ values for the input parameters) i s I . 3 7 6 , whereas with room temperature values for the input parameters the cal culated value i s I . 3 6 7 . Combination of the BVK results for the solid fluorides with the BVK results for the isotopic lithium metals yields a calcu lated value of Κ = 1.082 for the reaction 7
6
6
7
Li(s) + LiF(s) = L i ( s ) + LiF(s)
(73 )
compared to an experimenta III.c. Evaluation of Partition-Function Ratios for Isotopic Solute Species. The Symmetric Solvent Case. The simplest possible model for the partition-function ratio for isotopic species i n a common solvent involves the assumption that the solute-phase species can be treated as gas-phase species. As noted e a r l i e r , as far as isotope-exchange reactions are con cerned, this assumption i s equivalent to the assumption that there i s no vapor pressure isotope effect. For the reaction 7
6
6
7
L i ( s ) + L i B r ( s o l n ) = L i ( s ) + LiBr(soln)
(12 )
the gas-phase model assumes that q7
LiBr(soin) ~
^LiBrisoln)
q7
LiBr(g)
(jk)
^LiBrig)
The principal condition for the applicability of equation (74) i s that the dissolved LiBr exists i n solution as undissociated dia tomic molecules and not as ions. There must, of course, be a significant amount of cancellation of terms i n the partitionfunction ratio for the solute species. In particular a l l purely electrostatic interactions (e.g., ion-dipole, dipole-dipole, etc.) which are isotope independent, should cancel out. Dissolution of LiBr would be expected to lower the stretching frequency for the LiBr bond (42). The decrease i n ω decreases the ratio of —• jjiBr vibrational p a r t i t i o n functions, primarily through the decrease in the vibrational zero-point energy. However, this effect would be partial1 y offset by an increase i n the zero-point energy aris ing from restricted rotation i n the solution phase. Lastly, placing LiBr i n a solvent cage w i l l increase the ratio of trans l a t i o n a l p a r t i t i o n functions. Presumably, these opposing effects for the most part cancel out, primarily because the frequency
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
ROCK
Equilibrium
Constants for Isotope Exchange Reactions
155
shift on dissolution amounts at most to a few percent The ratio of partition functions for the gaseous isotopic bromides is given i n the harmonic-oscillator, rigid-rotor approximation by the expression (kk)
q7
LiBr(g)
^LiBrCg)
/
m y + I
V\
3 /
k ^
Y^Br)
j i-exp (-hca) /kT)| R
fiyp
j-hc (α) -ω )/ , Λ
ω
}^^^ τ/^))
7
}
2kT
β
\
where the μ^ values are the reduced masses,
H
= m
i Br/ V Br m
(
m
)
( τ 6 )
The (harmonic o s c i l l a t o r ) vibrational frequencie expression 1
u> = ω ( μ / μ ) / 6
τ
7
related b (77)
2
6
The use of equations (75), (76), and (77) together with the value of u) = 576.2 cur (V?), and K = Ο.7865 yields a value of Κ = 1
7
1.035
g
(at 296.I5 K) for the reaction 7
6
6
7
L i ( s ) + LiBr(g) = L i ( s ) + LiBr(g)
(78)
When anharmonic corrections are included (anharmonic o s c i l l a t o r , nonrigid rotor, vibration-rotation coupling, and the Wolfsberg G term), the calculated value of Κ remains unchanged. The ex perimental value of Κ for reaction (12) with diglyme as solvent i s I.O35 ± 0 . 0 0 7 ; the good agreement between the experimental and calculated values of Κ suggests that approximation (jk ) is a satis factory approximation. 0
The approximation of the partition-function ratio for lithium ions i n solution by the p a r t i t ion-function ratio for the uisoliated, gaseous lithium ions i s not a good approximation. The calculated Κ value for the reaction 7
6
6
7
+
L i ( s ) + Li+(g) = L i ( s ) + L i ( g )
(79)
i s O . 9 9 I , whereas the experimental value of Κ for the reaction 7
6
+
6
7
+
Li(s) + Li (pc) = Li(s) + Li (pc)
(80)
i s I.030 ± O.OO5. Because propylene carbonate (pc) i s usually only weakly coordinated to metal ions, a solute model involving a specific solvated-ion structure does not seem appropriate. We have treated the isotopic lithium ions i n propylene carbonate as particles in cubical boxes (cavity model (8))
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
156
7
6
6
AND
CHEMICAL
PRINCIPLES
7
L i ( s ) + Li+(eav) = L i ( s ) + Li+(cav)
(81 )
The partition-function ratio for the isotopic, caged ions i s then
^Li+( av)
f n=l
C
e
x
p
(
.
n 2 h 2 / 8 m e b 2 k T
)
where b i s the cavity dimension.. The value of b = 1-377 Â makes the partition function ratio equal to I.3IO (note that I.O30/ Ο.7865 = I . 3 I 0 ) ; this i s the value of the partition-function ratio for the ions that yield (80). The diameter of about 2.58Â i n this model. The predicted η = 1 to η = 2 transla tional transition energy i s 37 Λ cm ; such translational trans itions have been observed for neutral species in solution (45 ). o
-1
Isotope-exchange reactions involving strong, specific solva tion of the solute apparently require the explicit incorporation of the solvated ions into the model. An example i s provided by the reaction 7
6
+
6
7
(83)
+
Li(s) + Li (aq) = Li(s) + Li (aq)
The aquated lithium ion has a four-coordinate, tetrahedral structure (46); the model for reaction (83) then becomes (3) 7
6
6
7
L i ( s ) + Li(OH )t(g) = L i ( s ) + Li(OHfe)î(g)
(84)
2
The Lifoïïg)^ species i s a 13-atom system with 33 normal vibrational modes. The number of normal vibrations can be decreased to 9 by treating the H 0 molecule as an 0 atom (pseudoatom approximation). Pseudoatom approximations have proved successful in the analysis of vibrational spectra f<j>r complex molecules (4j). Of the nine normal modes of the L i ^ 0 unit, there i s only one triple-degenerate, genuine vibrational mode that involves movement of the lithium atom. Consequently, only this vibrational mode w i l l change frequency on the substitution of L i for L i . The ratio of partition functions for the aquated ions i s given by 1 8
2
1
4
7
^Li(0 )t(g) _ K ^ O J %
3
/
2
fl-exp(-hc^/kT)j
3 e x p
6
feieK-cQJ
Because the isotopic substitution takes place at the center of mass, the ratio of rotational partition functions i s unity. The
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
Equilibrium Constants for Isotope Exchange Reactions
ROCK
157
frequencies <% and U) are related through the Teller-Redlich product rule by the expression T
ω
6
= o)
7
I (my/me ) ( m ^ r n ^ )/
(86 )
(m +k^ T
0
A l l that i s needed to compute Κ for reaction (84) i s a value for either u> or m . The value of ω that yields a value of Κ = 1.046 (the experimental value of Κ for reaction (83) ) i s (% = 384 cm" ; the corresponding calculated value of ® i s 358 cm- . Subsequent to the prediction of these values for <% and ω , laser Raman studies (56 ) of concentrated L i C l ( a q ) and L i C l ( a q ) solutions have led to the detection of (% at 380 cm" and ω at 355 cm" . 7
6
β
1
1
T
7
6
7
1
1
7
Another example of strong specific solvation of isotopic ions i s found i n the (87)
D (g) + 2H (DMF) = H (g) + 2D (DMF) 2
2
The a v a i l a b i l i t y of the amine nitrogen of DMF as a proton accep tor suggests that a reasonable model for reaction (87) i s OH
Cj D
D (g) + 2 H-!J-ïï[-CH3(g) = H (g) + 2 H-C-N -CH (g) +
+
2
2
3
CH3
(88)
CH3
Unfortunately, the frequencies of the protonated DMF that are necessary to compute Κ for reaction (88) have not been obtained experimentally. However, an estimate of Κ for reaction (88) can be obtained i f i t i s assumed that there are only two normal vibra tional modes of protonated DMF that exhibit an appreciable H / D isotope effect. The two normal modes are: (a) the mode that i s predominantly an N-H stretching frequency; and (b) the norma! mode that i s predominantly an I-H bending frequency. The reported values of the N-H stretching frequencies of amine salts l i e i n the range 2700-2250 cm" , and N-H bending frequencies from amides usually occur around 1250 cm" . If we assume that the N-H stretching frequency i s 2250 cm" , then the value of the N-H bending frequency that i s required to make the calculated value of Κ for reaction (88) equal to the experimental value of Κ = 1.82 for reaction (87) i s 1180 cm" ( l l ) . This model calculation i s admittedly crude, but i t i s encouraging that the values of the frequencies required to give agreement with experiment are reason ably close to representative experimental values for these quantities. 1
1
1
1
I l l . d . Evaluation of Part i t ion-Function Ratios for Isotopic Solute Species. The Asymmetric Solvent Case. The calculation of equilibrium constants for isotope-exchange reactions involving solvent asymmetry, such as the reactions
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
158
AND
CHEMICAL
PRINCIPLES
(89)
D (g) + 2HC1[H 0] = H (g) + 2DC1[D 0] 2
2
2
2
and D (g) + 2HC1[R0H] = H (g) + 2DCl[R0D] 2
2
where R i s n-hexyl and the species i n square brackets i s the solvent, requires a detailed consideration of transfer (or medium) effects as well as exchange effects (48, 4g, 5 0 , ^2, £2, ^4). The equilibrium constant for reaction (89), based on molefraction-compos i t ion, strong-electrolyte standard states for HC1[H 0] and DCl[Dp0]. i s 0.713 at 298.15 K. Because this value of Κ for reaction 189) refers to strong electrolytes^we can re write equation (89) as 2
+
D (g) + 2H [H 0] + 2C1~[H 2
2
Reaction (90) can be written i n even more detail i n order to show e x p l i c i t l y the stripping of the solvent from the solvated hydronium ions that accompanies the exchange reaction Κ D (g) + 2HQ0 (0H ) [Hg0] + 2C1"[H 0] + 2(n+l) D 0U) = +
2
2
n
2
2
3
2
n
2
(91)
E 0(l)
+
Hs(g) + 2D 0 (0D ) [D 0] + 2Cl"[D 0] + 2(n+l)
z
2
where n, the solvation number of the hydronium ion, i s assumed to be the same i n H 0 and D 0. 2
2
(91) can be broken down into the following sequence
Reaction of steps: (i)
the gas-phase, isotope-exchange reaction K_ D (g) + 2H 0+(g) + 2D 0(g) =S H (g) + 2D 0 (g) + 2H 0(g) (92) +
2
3
2
2
3
2
+
+
( i i ) The solvent-stripping reaction, i n which H 0 and D 0 are stripped of their solvation sheaths and are placed i n hypothetical, cavities i n the respective liquids which have dielectric con stants equal to the bulk values Κ 2H 0 (0H ) [H 0] + 2D 0 (cav) + 2nD 0 U) = 3
+
3
+
3
2
n
2
3
2
2D 0 (0D ) [D 0 ] + 2H 0 (cav ) + 2nH 0 U) +
3
+
2
n
2
3
2
(93 )
( i i i ) The removal to the gas phase of the hydronium ions from their respective solvent cavities, and the transfer of CI" from H 0 to D 0 2
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
Equilibrium
ROCK
Constants for Isotope Exchange Reactions
2D 0 (g) + 2H 0 (cav) + 2Cl"[HgO] +
+
3
159
=
3
2H 0 (g) + 2D 0 (cav) + 2C1"[D 0] +
(9A)
+
3
3
2
(iv) The vaporization of two moles of D oCc) and the condensa tion of two moles of H 0(g) 2
2
Κ 2D 0 U) + 2H 0 (g) ==== 2H 0 il) 2
2
2
+ 2D 0 (g)
(95 )
2
Reaction (91) i s equal to the sum of reactions (92), (93)5 (94), and (95), and therefore
where K = Ο.713 at 298.15 K. The value of K p i s given by the x
va
square of the ratio of vapor pressures of pure O o(t) and pure H 0Ct); K = (p° /p° ) = 0.756 at 298.I5 K. The value o f 2
2
2
y a p
0
Q
can be calculated by standard s t a t i s t i c a l thermodynamic methods (25 ) to be Κ^ =1.11 ± 0 . 2 0
(18), where the uncertainty i n K
arises
g
from uncertainties i n the vibrational frequencies for the hydro nium ion. The value of Κ can be estimated from the Born equat ion as ^
=
-
p (
B
j
-
^ R
(9T)
^ \ D o
T
E
2
€
H OAH O 2
+
3
r
ci-/|
from which we compute (* Q+ ~ c i " = !-82 Â, j) q ~ 77-936, r
e
Η
€
H 0
=
7
^ * 3 ° 3 ) that Kg = Ο . 9 6 Ϊ .
Combination of these results
yields a value of Κ = 0.88 ± 0 . 2 0 . The closeness of Κ to unity ss ss suggests that the water molecules bonded to the hydronium ion are not significantly more strongly bonded than the water molecules that are bonded to one another i n bulk water, because otherwise there would be an appreciable isotope effect on Κ . These ss results suggest, i n aggrement with the conclusions reached by Heinzinger and Weston (42), that the hydronium ion, H 0 , i s an adequate representation of the proton i n water, and that i t i s not necessary to invoke (55 ) the species Εφ% to understand hydrogen-isotope-exchange reactions i n water. 3
+
Acknowledgments : This work would not have been accomplished with out the special efforts of my students: John C. Hall (lithiumisotope-exchange i n diglyme and the BVK calculations);
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
160
Leonard F. Silvester (hydrogen-isotope-exchange i n DMF, the preliminary work on the experimental test of the Born-Oppenheimer approximation, and the experiments on exchange between the isotopic metals and the isotopic fluorides); Gulzar Singh (lithium-isotope-exchange reactions i n propylene carbonate and the quadruple c e l l for the exchange reaction between the isotopic lithium metals and the aqueous ions); Carlos H. Contreras-Ortega (hydrogen isotope exchange i n n-hexanol); and Arnold Z. Gordon (calcium-isotope-exchange reactions). Numerous helpful and enjoyable discussions with C. P. Nash are also acknowledged. This research was supported by the AEC under contract AT(cA3)-5^ Ρ·Α. I69, by the NSF under Grant GP 53264, and i n the early stages by the Committee on Research, UCD. Literature Cited 1.
Abel, Ε., Bratu, Ε., and Redlich, O., Z. physik. Chem. (1934) A170, 153; (1935) Α173, 353. 2. Korman, S. and La Mer, V.K., J . Amer. Chem. Soc. (1936) 58, 1396. 3. Gary, R., Bates, R.G., and Robinson, R.A., J . Phys. Chem. (1969) 68, 1186. 4. Goldblatt, M. and Jones, W.M., J. Chem. Phys. (1969) 51, 1881. 5. Noonan, E. and La Mer, V.K., J . Phys. Chem. (1939) 43, 247. 6 . Electrochemical double c e l l s are ideally suited for the study of Gibbs energies of transfer. For example, cell (5) could be set up with normal-isotopic-abundance lithium on both sides, but dissolved in different solvents. The cell reaction i s then LiCl(soln) = LiCl(soln'). 7· Hall, J.C., Murray, J r . , R.C., and Rock, P.Α., J . Chem. Phys. (1969) 51, 1145. 8. Singh, G., Hall, J.C. and Rock, P.A., J . Chem. Phys. (1972) 56, 1856. 9. Singh, G. and Rock, P.Α., J . Chem. Phys. (1972) 57, 5556. 10. Hall, J.C., Silvester, L.F., Singh, G., and Rock, P.Α., J . Chem. Phys. (1973) 59, 6358. 11. Silvester, L.F., Kim, J.J., and Rock, P.A., J . Chem. Phys. (1972) 56, 1863. 12. Petrov, S.M. and Umanskii, Yu. I., Russ. J . Phys. Chem. (1967) 41, 449. 13. Silvester, L.F. and Rock, P.A., J . Chem. Eng. Data (1974) 19, 98 14. Contreras-Ortega, C.H. and Rock, P.A., J . Electrochem. Soc. (1974) 121, 1048. 15. Kleinman, L.I., and Wolfsberg, M., J . Chem. Phys. (1973) 59, 2043.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
7.
ROCK
Equilibrium
Constants for Isotope Exchange Reactions
16.
161
The reason i s that i n the c e l l the isotopically-mixed species undergoes an isotopic disproportionation reaction, e.g., 2HD = H + D . 17. Aston, J.G. and G i t t l e r , F.L., J . Amer. Chem. Soc. (1955) 77, 3173. 18. Contreras-Ortega, C.H., Silvester, L.F. and Rock, P.A. unpublished results. 19. It i s o f interest to note that Ca (g) i s stable with respect to disproportionation into Ca(s) and Ca (g). Consequently, in weakly solvating media Ca (soln) may be stable relative to disproportionation. 2 0 . Le blanc, M. and Harnapp, O., Ζ. physik. Chem. (1933) A166, 321. 21. Jakuszewski, B. and Taniewska-Osinska S Roczniki Chem (1962) 36, 329. 2 2 . Gordon, A.Z. and , unpublishe 2 3 . Stahl, W. and Wendt, I., Earth Plant. S c i . Letters (1968) 5, 184. 24. Heumann, K.G. and Lieser, K.H., Z. Naturforsch (1972) 27b, 126. Heumann, K.G., Z. Naturforsch (1972) 27b, 492. 25· Bigeleisen, J . and Mayer, M.G., J . Chem. Phys. (1947) 15, 2 6 1 . 26. Bigeleisen, J., "Proceedings of the International Symposium on Isotope Separation" (North-Holland Pub. Co., Amsterdam, 1958) pp. 121-157. 27. Snyder, D.D. and Montgomery, D.J., J . Chem. Phys. (1957) 27, 1033 2 8 . Martin, D.L., Physica (1959) 25, 1193. 2 9 . Gray, D.E., "American Institute of Physics Handbook" (McGrawHill Book Co., New York, N.Y., 1963) 2nd. ed. pp. 4/S2. 3 0 . Lewis, G.N. and Randall, M. "Thermodynamics" Second Revised Edition (McGraw-Hill Book Co. Inc., New York, N.Y. 1961) Revised by Pitzer, K.S. and Brewer, L. p. 6 5 9 . 31. H i l l , T.L., "Introduction to S t a t i s t i c a l Thermodynamics" (Addison-Wesley, Pub. Co., Reading, MA., 1960) p. 1 0 0 . 32. Born, M. and Huang, K. "Dynamical Theory of Crystal Lattices" (Oxford U.P., London, 1940) 3 3 . de Launay, J . , Solid State Phys. (1957) 2 , 2 2 0 . 34. Fine, P.C., Phys. Rev. (1939) 5 6 , 355. 35. Bauer, Ε., Phys. Rev. (1953) 9 2 , 5 8 . 36. Nash, H.C. and Smith, C.S., Phys. Chem. Solids (1959) 9, 113. 37. Hall, J.D., Silvester, L.F., Singh, G., and Rock, P.A., J. Chem. Phys. (1973) 5 9 , 6358. 38. Kellermann, E.W., Philos, Trans. R. Soc.Lond. (1940) A238, 513. 39. Kellermann, E.W., Proc. R. Soc. (1941) A178, 17. 40. Sayre, E.V. and Beaver, J.J., J . Chem. Phys. (1950) 18, 585. 41. Karo, A.M., J . Chem. Phys. (1959) 31, 1489. 42. Herzberg, G., "Molecular Spectra and Molecular Structure I I . Infrared and Raman Spectra of Polyatomic Molecules", (Van Nostrand Co. New York, 1964) p. 534. 2
2
+
2+
+
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES AND CHEMICAL PRINCIPLES
162
43. 44. 45. 46. 47· 48. 49. 50. 51. 52. 53.
Jones, W.J. i n "Infrared Spectroscopy and Molecular Structure," Davies, M. (Ed.) (Elsevier, New York, 1963) p. 160. Bigeleisen, J . , "Proceedings of the International Symposium on Isotope Separation" (North-Holland Pub. Co., Amsterdam, 1958) pp. 121-157. Burgiel, J.C., Meyer, H., and Richards, P.L., J . Chem. Phys. (1965) 43, 4391. Narten, A.H., Vaslow, F. and Levy, H.A., J . Chem. Phys. (1973) 5 8 , 5017. Herzberg, G., "Molecular Spectra and Molecular Structure I I . Infrared and Raman Spectra o f Polyatomic Molecules" (Van Nostrand, New York, 1964) p. 235. Halevi, E.A., Israel J . Chem. (1971) 9, 385 Heinzinger, K. and Weston, R.E., J . Phys. Chem. (1964) 6 8 , 744. Heinzinger, K. and 2179. Kingerley, R.W. and La Mer, V.K., J . Amer. Chem. Soc. (1941)
63, 3256.
Swain, C.G. and Bader, R.F.W., Tetrahedron (1960) 1 0 , 182. O'Ferrall, R.A.M., Koeppl, G.W., and Kresge, A.J., J . Amer. Chem. Soc. (1971) 93, 1 , 9 . 54. Van Hook, W.A., J . Phys. Chem. (1968) 72, 1234. 55. Eigen, M. and DeMaeyer, L., Proc. R. Soc. London (1958) Ser A, 247,505. 56. Nash, C.P., Donnelly, T. and Rock, P.A., unpublished results.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8 Isotope Effects and Reaction Mechanisms V. J. S H I N E R , JR. Department of Chemistry, Indiana University, Bloomington, Ind.
47401
Introduction T h e subject of t h i s w e l l - d e v e l o p e d that one c a n only hope i n the s p a c e a v a i l a b l e h e r e to g i v e the n o n - s p e c i a l i s t s o m e g e n e r a l knowledge of the u t i l i t y , the s c o p e and the l i m i t a t i o n s of the u s e of i s o t o p e r a t e effects i n the study of r e a c t i o n m e c h a n i s m s and an i n t r o d u c t i o n to the m o r e d e t a i l e d l i t e r a t u r e f o r those who m a y w i s h to d e l v e d e e p e r . It i s v e r y m u c h to be hoped that this t e c h n i q u e w i l l c o m e into o c c a s i o n a l , i f not r e g u l a r , u s e by a l l who u n d e r t a k e the i n v e s t i g a t i o n of r e a c t i o n m e c h a n i s m s and w i l l not be c o n s i d e r e d to lie m a i n l y i n the p r o v i n c e of those who m a k e i t a s p e c i a l t y . P r o b a b l y the m a i n barrier to m o r e g e n e r a l u s e of t h i s technique i s the s o m e what r e f i n e d a c c u r a c y g e n e r a l l y r e q u i r e d i n the m e a s u r e m e n t of r e a c t i o n r a t e s or i n the m e a s u r e m e n t of i s o t o p e r a t i o s if the competitive technique i s used. T h e s e experimental r e q u i r e m e n t s c a n be m a s t e r e d r e a s o n a b l y r e a d i l y w i t h the m o r e s o p h i s t i c a t e d c o m m e r c i a l i n s t r u m e n t s w h i c h a r e now g e n e r a l l y a v a i l able. I do not intend h e r e to d i s c u s s e x p e r i m e n t a l t e c h n i q u e s but r a t h e r the g e n e r a l f r a m e w o r k for the i n t e r p r e t a t i o n of r e s u l t s w h i c h has b e e n b u i l t up by m a n y i n v e s t i g a t o r s o v e r the c o u r s e of the l a s t t w e n t y - f i v e or so y e a r s . (1) T h e e a r l i e r c o n t r i b u t o r s to this s y m p o s i u m have outlined the b a s i c t h e o r y of the i n t e r p r e t a t i o n of the e f f e c t s of i s o t o p i c s u b s t i tution on r e a c t i o n r a t e s o r i g i n a l l y d e v e l o p e d i n d e p e n d e n t l y by B i g e l e i s e n and M a y e r (2) and M e l a n d e r . (3) T h e y showed that i t i s the g e o m e t r i c a l s t r u c t u r e , n u c l e a r m a s s e s and, m o s t i m p o r tantly, the v i b r a t i o n a l f o r c e f i e l d s of i n i t i a l and t r a n s i t i o n states that d e t e r m i n e the m a g n i t u d e of i s o t o p e effects on r e a c t i o n r a t e s . S i n c e t h e s e p r o p e r t i e s of the i n i t i a l state r e a e t a n t s a r e subj e c t to r e a s o n a b l y d i r e c t o b s e r v a t i o n or d e r i v a t i o n , the r e a c t i o n m e c h a n i s m s c h e m i s t uses experimentally m e a s u r e d isotope effects on r e a c t i o n r a t e s p r i n c i p a l l y as a p r o b e f o r f e a t u r e s of the t r a n s i t i o n state v i b r a t i o n a l f o r c e f i e l d . P r i m a r i l y t h r o u g h
163
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
164
AND
CHEMICAL
PRINCIPLES
the e f f o r t s of W o l f s b e r g and S t e r n (4,5,6) c o m p u t e r p r o g r a m s a r e a v a i l a b l e w h i c h a l l o w the c a l c u l a t i o n of expected i s o t o p e r a t e effects f r o m c o m p l e t e l y s p e c i f i e d s t r u c t u r e s and f o r c e f i e l d s of r e a c t a n t and t r a n s i t i o n states c o n t a i n i n g up to 30 a t o m s . This k i n d of d e t a i l e d m o d e l l i n g has been s e r i o u s l y i n v e s t i g a t e d f o r only a r e l a t i v e l y s m a l l n u m b e r of r e a l r e a c t i o n s . (7,8,9,10) S u c h a n a l y s e s a r e c r u c i a l to a g r e a t e r u n d e r s t a n d i n g of tEe~prec i s e d e t a i l s of r e a c t i o n s but p r e s e n t l y this a s p e c t of the subject i s s t i l l i n i t s i n f a n c y and v e r y m u c h i n the p r o v i n c e of s p e c i a l i s t s . R a t h e r than r e v i e w t h e s e quantitative r e s u l t s I w i l l h e r e t r y to i l l u s t r a t e s o m e of the g e n e r a l q u a l i t a t i v e f e a t u r e s of the t h e o r e t i c a l a n a l y s i s that can be u s e d by r e a c t i o n m e c h a n i s m s c h e m i s t s a l o n g w i t h other e l e m e n t s of t h e i r t r a d i t i o n a l a r m a m e n t a r i u m to d i s t i n g u i s h a m o n g a p r i o i m e c h a n i s t i c p o s s i b i l i t i e s . The traditional reactio of a d i s a d v a n t a g e t h i n k i n a n d / o r v i b r a t i o n f r e q u e n c i e s b e c a u s e m o s t other t e c h n i q u e s f a m i l i a r to h i m do not r e l a t e d i r e c t l y to f o r c e f i e l d s but to s u c h " e l e c t r o n i c " p r o p e r t i e s as total e n e r g y , e l e c t r o n d i s t r i b u t i o n , d i p o l e m o m e n t or g e o m e t r i c s t r u c t u r e . Of c o u r s e a l l of t h e s e things a r e r e l a t e d t h r o u g h f u n d a m e n t a l t h e o r y , but so f a r quan t u m m e c h a n i c a l c a l c u l a t i o n s have not y i e l d e d v e r y s u c c e s s f u l explanations of s u c h things as the effects of substituents on r e a c tion r a t e s of o r g a n i c c o m p o u n d s . N e i t h e r h a v e they yet p r o v i d e d v e r y a c c u r a t e f o r c e constants for h y p o t h e t i c a l t r a n s i t i o n states, but I think i t i s r e a s o n a b l e to hope f o r p r o g r e s s t o w a r d this g o a l and I would e n c o u r a g e t h e o r e t i c i a n s to give m o r e thought and effort to the c a l c u l a t i o n of m o l e c u l a r f o r c e constants. Success i n t h e s e e f f o r t s would g r e a t l y a s s i s t i n the m e c h a n i s t i c i n t e r p r e t a t i o n of i s o t o p e e f f e c t s . P r i m a r y Isotope E f f e c t s If we c o n s i d e r the d i s s o c i a t i o n of a d i a t o m i c m o l e c u l e , the e n e r g e t i c s of the s y s t e m can be q u a l i t a t i v e l y r e p r e s e n t e d as i n f i g u r e 1, w h e r e i n i t i s a s s u m e d for p u r p o s e s of s i m p l i c i t y that the t e m p e r a t u r e i s low enough so that a l l m o l e c u l e s a r e i n t h e i r l o w e s t v i b r a t i o n a l state and t h e r e f o r e contain the z e r o point v i b r a t i o n a l energy. S i n c e , by the B o r n - O p p e n h e i m e r a p p r o x i m a t i o n the m o l e c u l a r e n e r g y and t h e r e f o r e the i n t e r a t o m i c r e s t o r i n g f o r c e s depend only on n u c l e a r and e l e c t r o n i c c h a r g e s and not on the n u c l e a r m a s s e s , the v i b r a t i o n a l f o r c e constants w i l l be the s a m e f o r both i s o t o p i c m o l e c u l e s . H o w e v e r , the h e a v i e r m o l e c u l e , A - Β w i l l v i b r a t e m o r e slowly, as expected f r o m Hook's Law, than the l i g h t e r m o l e c u l e *A-B. S i n c e the z e r o point e n e r gy i s p r o p o r t i o n a l to the v i b r a t i o n f r e q u e n c y , A - B r e s t s l o w e r i n the p o t e n t i a l w e l l than *A-B and r e q u i r e s a h i g h e r " a c t i v a t i o n e n e r g y " to r a i s e i t to the d i s s o c i a t e d state. It t h e r e f o r e should on the a v e r a g e a t t a i n that state at a s l o w e r r a t e than the l i g h t e r m o l e c u l e and we expect a n o r m a l i s o t o p e r a t e effect, i . e. , the lighter molecule reacts faster. 2
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8.
SHINER
Isotope Effects and Reaction
Mechanisms
165
Of course, translational and rotational partition functions are usually different for molecules differing in isotopic substitu tion and,in general,will change on activation,leading to effects on reaction rates. Kinetic isotope effects from these sources are, however, generally dominated by the vibrational zero-point ener gy effects which are the main focus of all general, qualitative discussions. This simply explains the occurrence of a primary isotope rate effect, "primary" referring to the fact that the rate deter mining step involves a bond to the isotopically substituted atom. However, many reactions which show large primary isotope ef fects do not involve dissociation into free particles but rather are displacement reactions wherein the isotopic atom is abstrac ted by an attacking agent Since the isotopic atom is bound both in reactant and produc how can the occurrenc ses be explained? This can be seen with the aid of figure 2 wherein the zero point energy levels are depicted qualitatively for diatomic initial and final states and for the triatomic "abstraction" transition state. (The inclusion of bending modes for the transition state is necessary, of course, in a rigorous treatment but will be ignor ed for simplicity in the qualitative argument. ) The stretching motions of the triatomic transition state can be visualized in the same way that one would visualize the stretching modes for a stable molecule, except that, from tran sition state theory, we expect one of these modes to have no re storing force but rather to be replaced by a translation in one direction to give products and in the other direction to give re aetants. The arrows in the center of the figure represent the atomic movements expected for triatomic molecular stretching motions. The symmetrical stretch, ν , has a restoring force and is a normal vibrational mode of the transition state. One i n tuitively knows this because if there were no restoring force for this motion it would continue indefinitely with the resulting dis sociation of all three atoms giving a reaction which is not the ab straction process. The asymmetrical stretching mode expected for a normal molecule i s , for the triatomic abstraction transi tion sta^te, the reaction coordinate; in this motion in one direc tion, v^, the C A bond contracts while the A B bond stretches, there is no restoring force and the continuation of t h i | transla tion produces products; the reverse of this motion, ν , produces reaetants. It is important to note that i f the transition state is symmetrical with the force of attraction between A and Β equal to that between A and C and if the masses of Β and C are equal, the frequency of the symmetric stretching mode will be the same for both isotopes of A, since A will not move in this vibration. Even if the transition state is not quite symmetrical the motion of A in the symmetrical stretching vibration will be relatively ς
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
166
Figure 2.
A N DCHEMICAL
PRINCIPLES
Abstraction reactions. C + *A — Β
—» C — *A + Β.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8.
SHINER
Isotope Effects and
Reaction Mechanisms
167
unimportant and the zero point energy difference between the isotopic molecules will be small. Thus,for the symmetrical iso topic transition states in the abstraction reaction, there is no difference in zero point energies and one expects a large p r i mary isotope effect because the difference in zero point energies in the initial state contributes to make the activation energy for the lighter isotope smaller than that for the heavier isotope. This analysis has always appealed to me as one of the nicest ex perimental demonstrations of the major hypothesis of transition state theory. The primary isotope effect shows that the transi tion state has a translational degree of freedom which replaces a vibrational one expected in a normal molecule. Thus, whether the reaction involves a dissociation or a trans fer, a simple diatomic molecular model can give useful semi quantitative results indicativ tope effects which migh Some values calculated in this way are shown in Table 1. (11) In most cases primary isotope effects about this size have been observed, and results of this magnitude are taken as prima facie evidence that the isotopically substituted bond is being broken in the rate-determining step. There is another obvious qualitative problem that immediate ly arises in examining any reasonably large collection of p r i mary isotope effect data, particularly results on isotope effects in hydrogen transfer reactions. This is that many isotope ef fects in abstraction reactions are much smaller than the expec ted maxima. One explanation, due to Westheimer (12) and to Melander, (13) as to how this can arise is illustrated qualitative ly in figure 3. In a very exergonic reaction, the reaction coordi nate involves a stretching type motion in which A and Β together move toward C. The stretching motion,ν , is perpendicular to the reaction coordinate, has a restoring ft>rce, and involves the motion of the isotopic atom; hence there is a zero point energy difference in the stretching mode in the transition state between the two isotopic molecules and an activation energy difference that is less than the maximum. In the extreme, one could look at this as a diffusion controlled reaction with the approach of C to the A-B unit as the reaction coordinate and the internal vibra tion of A-B in the transition state being unaffected by C. This analysis suggests that p r i m a r y isotope effects in a series of re lated reactions will vary with reactivity; the maximum effect being found for nearly thermoneutral reactions having transition states which are nearly symmetrical and increasingly smaller effects being exhibited by processes either increasingly endergonic or increasingly exergonic. Such behavior has apparently been found for both proton abstractions and for hydrogen atom abstractions. This would seemingly limit the utility of the p r i m a r y isotope effect as a criterion of mechanism, but in practice one usually knows when to expect the occurrence of such asym-
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
168
AND
CHEMICAL
PRINCIPLES
a b T a b l e I.
Isotopic m o l e c u l e s 1
v
V 2
for Diatomic M o l e c u l a r
V
il/ 2L
^Α (25°) 2
VM200'
2
C-*H
2
C- H
3015
2212
1.36
6.93
3. 39
c-
1 2
c
c-
1 3
c
1129
1107
1. 020
1. 055
1.036
c-
1 2
c
c-
1 4
c
1129
1088
1.038
1. 109
1.069
1113
1086
1. 025
1. 068
1. 045
915
908
1. 008
1. 018
1.013
804
798
1. 007
1. 014
1.010
14
C- N c-
1 6
o
3 5
1 8
c-
3 2
c-
113
15
C- N
C- S
a
1
C a l c u l a t e d Isotope E f f e c t s Dissociation Model
o
C-^S
ci
c-
3 7
ci
T h e s e results have been calculated using the equation (11) * ^
k /k = s in h x
s
2
„, •„ . Α. hCv Tin
=
A
>
Τ
Έ
Γ
2~Pr
n n
'
^ measured in c m s
-1
and
0. 71929V T
T h e s e c a l c u l a t i o n s a r e e s p e c i a l l y e a s y to p e r f o r m with one of the new " e l e c t r o n i c s l i d e r u l e " pocket c a l c u l a t o r s i f it has a " h y p e r b o l i c s i n e " key. F o r s u c h an a p p r o x i m a t i o n one only needs to know the i s o t o p i c m a s s e s and the a p p r o x i m a t e v i b r a t i o n f r e q u e n c y for one of the i s o t o p i c m o l e c u l e s ; the f r e q u e n c y f o r the second i s o t o p i c m o l e c u l e can be c a l c u l a t e d f r o m that for the f i r s t and the r e d u c e d m a s s r e l a t i o n s h i p , e. g. :
*CD - - C H
f(m , ^(m
C
c
+m ) / m , +m ) / m D
H
w
" m C
\ 1/2
5
c
· rr^ /
"
/14. l 2 « l \ l / 2 777T:) = v U2-2-13/ 1 / 2
H
« 0. 73380
S i n c e the r e d u c e d m a s s r e l a t i o n s h i p a s s u r e s that the r a t i o of f r e q u e n c i e s f o r heavy and light m o l e c u l e s w i l l be v e r y n e a r l y c o r r e c t , any e r r o r i n the e s t i m a t e d f r e q u e n c y w i l l i n g e n e r a l c a u s e only an a p p r o x i m a t e l y p r o p o r t i o n a l e r r o r i n the l o g a r i t h m of the c a l c u l a t e d i s o t o p e effect so that the f r e q u e n c y does not have to be e s t i m a t e d with high p r e c i s i o n to get r e a s o n a b l e ans wers. T h e e x a m p l e s c i t e d a s s u m e that the a t o m s a r e held together by a s i n g l e bond; c a l c u l a t i o n s for e x a m p l e s with m u l t i p l e bonds could be c a r r i e d out u s i n g the a p p r o p r i a t e f r e q u e n c i e s .
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8.
Isotope Effects and
SHINER
Reaction Mechanisms
169
m e t r i c a l t r a n s i t i o n states as, for example, i n the t r a n s f e r of a p r o t o n between b a s e s of s t r o n g l y u n e q u a l b a s i c i t y . In addition, this effect c o u l d be u s e d , f o r e x a m p l e , to d e t e r m i n e the e f f e c t i v e b a s i c i t y of a p r o t o n a b s t r a c t i n g site i n a r e a c t i o n and to d i s t i n g u i s h t h e r e b y between a l t e r n a t i v e t r a n s i t i o n state m o d e l s . B e l l (14) has pointed out that a plot of h y d r o g e n k i n e t i c i s o t o p e effects for a v a r i e t y of p r o t o n t r a n s f e r r e a c t i o n s vs. the d i f f e r ence i n pK between the two b a s e s shows a c e n t r a l m a x i m u m n e a r ΔρΚ = 0. 0 and d e c r e a s i n g l i m b s f o r ΔρΚ < 0 or > 0. An e x a m p l e o f this i s shown i n f i g u r e 4 f r o m B e l l and Cox (14). P r y o r and K n e i f f (15) have shown that a s i m i l a r r e l a t i o n s h i p holds f o r a s e r i e s of r a d i c a l s r e a c t i n g with t h i o l s i f the i s o t o p e effects a r e plotted vs. the heats of r e a c t i o n s . S i m s and c o w o r k e r s (16) have r e c e n t l y pointed out that c a r bon t r a n s f e r p r o c e s s e s s i m i l a r i s o t o p e effec c a r b o n i s o t o p e effects i n S^2 d i s p l a c e m e n t s on c a r b o n m i g h t be s m a l l b e c a u s e bonding about c a r b o n i s q u a l i t a t i v e l y " p r e s e r v e d " i n the t r a n s i t i o n state. T h e e s s e n c e of t h i s , as of any s y n c h o nous d i s p l a c e m e n t p r o c e s s , i s that the e n e r g y b i l l to be paid for b r e a k i n g the " o l d " bond i s p a r t i a l l y r e d u c e d by the e n e r g y gained i n f o r m i n g the "new" bond. H o w e v e r , the a n a l y s i s w h i c h we have m a d e h e r e s u g g e s t s that, for n e a r l y s y m m e t r i c a l t r a n s i t i o n states, c a r b o n m o t i o n should be s t r o n g l y i n v o l v e d a l o n g the r e a c t i o n c o o r d i n a t e and c a r b o n i s o t o p e r a t e effects should be l a r g e . T h e m o d e l c a l c u l a t i o n s of S i m s et a l . (16) i n d i c a t e how the effect should v a r y as the t r a n s i t i o n state bonding v a r i e s . T h i s i s shown i n f i g u r e 5 f o r the i n c o m i n g n u c l e o p h i l i c a t o m s oxygen, c h l o r i n e or s u l f u r d i s p l a c i n g c h l o r i d e i o n f r o m b e n z y l c h l o r i d e . T h e u p p e r c u r v e s d i s p l a y the C / C effect and the l o w e r c u r v e the C l / C l l e a v i n g g r o u p effects as a function of bond o r d e r (n ) of the f o r m i n g bond, a s s u m i n g that the b r e a k i n g bond has bond order, n e q u a l to ( l - n ) . A l t h o u g h this g e n e r a l kind of b e h a v i o r i s c e r t a i n l y to be expected, no a v a i l a b l e e x p e r i m e n t a l e v i d e n c e s u p p o r t s the e x i s t e n c e of a c u r v e w i t h a m i x i m u m . M a n y c o m m o n S 2 r e a c t i o n s a r e p r o b a b l y not m a n y k i l o c a l o r i e s p e r m o l e away i r o m t h e r m o n e u t r a l i t y s ο the c o m m o n o b s e r v a t i o n of C / C effects i n the r a n g e of 8-12% ( l 7) s e e m s to c o n f i r m the e x i s t e n c e of n e a r m a x i m a l i s o t o p e effects i n the c e n t r a l r e g i o n of the plot, but t h e r e i s as yet no e v i d e n c e s u p p o r t i n g the s y s t e m a t i c f a l l - o f f w i t h n > 0. 5 or < 0. 5. One of the p r o b l e m s i n i n t e r p r e t i n g a wide v a r i e t y of data on this c l a s s of r e a c t i o n s i s that b i g c h a n g e s i n n u c l e o p h i l i c i t y u s u a l l y a r e a c c o m p a n i e d by c h a n g e s i n a t t a c k i n g atom. T h i s u s u a l l y c a u s e s b i g changes both i n bonding strength, as r e f l e c t e d i n the f o r c e constants f o r n o r m a l s i n g l e bonds, and i n m a s s , w h i c h of c o u r s e has i t s own effect on t r a n s i t i o n state s y m m e t r y independent of the f o r c e constant s i t uation. T h u s one needs a c a r e f u l l y d e s i g n e d set of e x p e r i m e n t s and a c o m p a n i o n set of t h e o r e t i c a l c a l c u l a t i o n s to a p p r o p r i a t e l y i n v e s t i g a t e the phenomenon. No doubt we w i l l see r e s u l t s of e f f o r t s t o w a r d t h i s end i n the next few y e a r s . 1 2
3 5
1 4
3 7
2
l9
2
N
1 2
1 4
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
Figure 3.
AND CHEMICAL
PRINCIPLES
Very exergonic abstraction reaction.
C + *A - Β -> C - Ά + B.
1.08
X OXYGEN IE CHLORINE m SULFUR
1.07
1.06
1.05
1.04
1.03
1.02
1.01
ι
00 ' υ υ
1
00
0.2
Q4
1
1
OS
06
I - n, « n
1 1.0
2
Chemical Reviews Figure 4. Relations between the primary iso tope effect (k /k ) and the pK difference be tween the reacting bases in proton transfer H
D
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8.
SHINER
Isotope Effects and Reaction Mechanisms
i f o'
10r
171
.·5
ν % ν 4·\
.ο ·7 pQ
•3
08 \
\ \
06
2· Γ
-15
-10
5
- ΔρΚ
0
Journal of the Chemical Society, Part Β Figure 5. Calculated carbon-14 (upper curves) and chlorine-37 (lower curves) iso tope effects for displacement on benzyl chlorine by oxygen, chlorine, and sulfur as a function of bond order
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
172
AND
CHEMICAL
PRINCIPLES
Secondary Isotope Effects Having, thus far, given a very brief survey of the general shape of the theory and results on p r i m a r y isotope effects in thermal reactions, I would like next to turn to secondary isotope effects. Here,for the sake of simplicity and brevity,I will con fine my examples to the isotopes of hydrogen,since secondary isotope effects are usually so small that their reliable observa tion is difficult and they have been reported for isotopes other than those of hydrogen in only a few instances. It is convenient to divide secondary isotope effects into at least two kinds (one of which could be further subdivided): 1* a-effects. The isotopic atom is bound to an atom under going bond rupture and/or formation through its other valences. One can readily imagine several possible "sources" of an a. The breakin constant to the a-substituted isotopic atom. b. A forming bond at the reaction center introduces a new bending force constant to the a-substituted isotopic atom. c. Rehybridization of the reaction center causes a change in bond strength and therefore a change in bond force constant to the α-substituted isotopic atom. d. Changes in electron density at the reaction center change the bond strength and bond force constant at the α-substituted isotopic center. e. A change in steric hindrance at the reaction center changes the force constants at the isotopic atom. 2. The second general kind of secondary isotope effects are those caused by isotopic substitution at positions in the molecule more remote from the reaction center. There are two general possible causes for such effects. a. Electronic or steric intramolecular interactions be tween isotopic center and reaction center cause force constant differences between initial and transition states. b. "No force constant change" effects could be caused by changes in moments of inertia, or masses or in me chanical coupling of vibrations. Wolfsberg and Stern (4) have shown that these effects generally are expect ed to amount to no more than a few percent per D atom (replacing H) for example. Many isotope effects due to substitution of deuterium for hy drogen at positions β- or more remote from the reaction center have been observed. (18) There is only space here to refer the reader to more detailed"summaries or to the original literature and to comment that each of the "big three" mechanisms of intra molecular interaction between reaction and isotopic centers has been shown in one case or another to cause isotope effects: 1) Hyperconjugative electron release from a C-H bond causes normal
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8.
Isotope Effects and Reaction
SHINER
Mechanisms
173
deuterium isotope effects of the order of krr/k-Q =1.30 or less per D atom and the effect is qualitatively proportional to the ex pected change in hyperconjugative demand. 2) Inductive electron acceptance by a C-H bond causes small normal isotope effects (D compound reacting slower than H compound) of a few percent per D or less depending on demand. (19) 3) Increased crowding causes inverse isotope effects, the lar~gest so far observed being around 1 0 % per D. (20) Of course,effects in directions opposite to those enumeratecTabove give the opposite kind of isotope effect i . e. , inductive electron release by C-H causes an inverse iso tope effect. The interaction effects referred to in each case are those in the transition states relative to the respective initial states. I would like to take the rest of the space available to discuss some general qualitativ results obtained from philic solvolytic substitution reactions. an
We can represent the dissociation reaction of a bond having α-d substituent as follows:
H
H \ a
a
\
C—X
—X
4
c
Λ
+
X
Ct-Effect: Dissociation Reaction The example involves a reacting carbon atom with the attach ments to the two other valences unspecified; qualitatively this makes no difference in principle and the same general arguments should hold for other kinds of reacting atoms. A s Wolfsberg and Stern (4) have argued, the partial breaking of the C-X bond in the reaction transition state should cause the bending force constant, *HCX' * ^ t ^ to be less than that of the initial state, fjjçx* Thus the zero point energy differences between H and D compounds should be smaller in the transition state (assuming no big changes in the other force constants constraining the H) and the activation energy for the reaction of the H compound should be smaller than that for the reaction of the D compound and a normal isotope effect should result. If we consider a series of related transition states having greater and greater degrees of C-X bond extension we would expect the isotope rate effect to increase with bond extension but to a smaller and smaller degree until a point is reached where the X group no longer has any influence on the transition state vibrations of the a-H. Thus for reactions of this type the secondary isotope effect nt
ie
r
a
n
s
o
n
s t a t e
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
174
A N D CHEMICAL
PRINCIPLES
s h o u l d be a m e a s u r e of the extent of C - X bond b r e a k i n g i n the t r a n s i t i o n state, a s y m p t o t i c a l l y a p p r o a c h i n g s o m e m a x i m u m v a l u e c o r r e s p o n d i n g to c o m p l e t e bond c l e a v a g e . O f c o u r se,the m a x i m u m effect o b t a i n a b l e would depend on the n a t u r e of X and the s t r e n g t h of i t s bond to C, p a r t i c u l a r l y as r e f l e c t e d i n the i n i t i a l state b e n d i n g f o r c e constant. F o r a c o n c e r t e d d i s p l a c e m e n t the s i t u a t i o n i s s i m i l a r except that a new bond to the e n t e r i n g group,Y, has been p a r t i a l l y f o r m ed i n the t r a n s i t i o n state,
H
Ήα Y
+
a
\
X-X
---C
\
a -Effect: Concerted Displacement Reaction In the t r a n s i t i o n state the f o r c e constant, f j ^ c y , f o r the new bond p a r t i a l l y , at l e a s t , c o m p e n s a t e s f o r the r e d u c t i o n i n the i n i t i a l state f o r c e constant, ^jq-^* that one expects the α-d i s o t o p e effect to be s i g n i f i c a n u y l o w e r than that f o r a s i m p l e d i s sociation. s
o
In the w o r k f r o m m y own l a b o r a t o r y o v e r the l a s t ten y e a r s or s o we have u s e d a c o m b i n a t i o n of e x p e r i m e n t a l m e a s u r e m e n t s on s e l e c t e d r e a e t a n t s and m o d e l c a l c u l a t i o n s r e l a t i n g to α-deu t e r i u m effects on the r a t e s of s o l v o l y t i c n u c l e o p h i l i c d i s p l a c e m e n t s on c a r b o n to t r y to c o r r e l a t e a l l known r e s u l t s on s u c h r e a c t i o n s i n t e r m s of S c h e m e I. T h i s i s a g e n e r a l i z e d s c h e m e i n c o r p o r a t i n g the S-,1 and S-,2 m e c h a n i s m s of H u g h e s and Ingold w i t h the i d e a s of "Winstein and c o w o r k e r s and o t h e r s on the e x i s t e n c e of i o n p a i r i n t e r m e d i a t e s in r e a c t i o n s h a v i n g c a r b o n i u m i o n c h a r a c t e r . T h e c o v a l e n t sub s t r a t e h a v i n g a bond between the e l e c t r o n e g a t i v e l e a v i n g g r o u p X and c a r b o n i s r e p r e s e n t e d b y RX; the c o m p o u n d having the op p o s i t e c o n f i g u r a t i o n at c a r b o n i s r e p r e s e n t e d b y X R . T h e S.^2 r e a c t i o n with a p r o t i c s o l v e n t n u c l e o p h i l e , S H , i s r e p r e s e n t e d b y the a r r o w i n d e n t i f i e d w i t h r a t e constant 1^. T h e other p r o d u c t s ^ Η and X a r e o m i t t e d f o r s i m p l i c i t y but a r e u n d e r s t o o d to be f o r m e d i n this p r o c e s s as w e l l as b y the p r o c e s s e s w i t h r a t e c o n s t a n t s kg, k and k T h e s o l v o l y t i c s u b s t i t u t i o n p r o d u c t SR i s r e p r e s e n t e d a s b e i n g f o r m e d i n the S 2 r e a c t i o n f r o m r e a c t ?
6
r
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8.
SHINER
Isotope Effects and Reaction Mechanisms
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
175
ISOTOPES A N D
176
CHEMICAL
PRINCIPLES
ant of the o p p o s i t e c o n f i g u r a t i o n R X . R X c a n a l s o i o n i z e w i t h a f i r s t o r d e r r a t e constant k to p r o d u c e the tight i o n p a i r , R X", v i s u a l i z e d a s two i o n s i n d i r e c t contact- T h i s i n t e r m e d i a t e c a n (1) " r e t u r n " with a f i r s t o r d e r r a t e constant k_j to c o v a l e n t R X or (2) f u r t h e r s e p a r a t e w i t h a ra|e constant k to the s e c o n d c a r b o n i u m i o n type i n t e r m e d i a t e , R βχ~, the s o l v e n t separated i o n p a i r , where* // r e p r e s e n t s a s o l v e n t m o l e c u l e between R and X ~ or (3) b y R r o t a t i n g r e l a t i v e to X " p r o d u c e X " R , the tight i o n p a i r of c o n f i g u r a t i o n o p p o s i t e the o r i g i n a l tight i o n p a i r or (4) u n d e r g o n u c l e o p h i l i c a t t a c k b y s o l v e n t to p r o d u c e s u b s t i t u t i o n p r o d u c t expected to b e of o p p o s i t e c o n f i g u r a t i o n to R X * b e c a u s e the f r o n t s i d e i s s t i l l " p r o t e c t e d " f r o m n u c l e o p h i l i c attac^. by the c l o s e p r o x i m i t y of^C". T h e s o l v e n t s e p a r a t e d i o n pair,R / ^ " , c a n (1) r e t u r n to R X", (2) i n v e r t i t s con figuration^ to X " / R , (3) dissociate completely t collapse by nucleophili both SR ar^d RS, p r o b a b l y unequa amounts. e free carbon i u m i o n R c a n ( l ) r e t u r n to R / X ~ o r X ~ / R w i t h equal f a c i l i t y by a s s o c i a t i n g w i t h a k i n e t i c a l l y f r e e a n i o n X " o r (2) f o r m SR and R S i n equal a m o u n t s by n u c l e o p h i l i c a t t a c k b y s o l v e n t . In this s c h e m e i t i s a s s u m e d that a s o l v e n t m o l e c u l e i s always r e a s o n a b l y c l o s e l y p o s i t i o n e d to R on f r o n t o r b a c k o r both s i d e s i f X " does not o c c u p y one o r the other of t h e s e s i t e s . T h e effects of added s a l t s , o r added n o n s o l v e n t n u c l e o p h i l i e s , the i n c u r s i o n of e l i m i n a t i o n o r r e a r r a n g e m e n t a r e not i n c l u d e d but c o u l d r e a d i l y b e s o b y obvious e l a b o r a t i o n s of the s c h e m e . x
2
In c l a s s i f y i n g a m e c h a n i s m a c c o r d i n g to this g e n e r a l s c h e m e it i s f i r s t i m p o r t a n t to e s t a b l i s h w h i c h of the s e v e r a l p o s s i b l e steps i s the r a t e d e t e r m i n i n g one and i n w h i c h step the c o v a l e n t bond to the i n c o m i n g n u c l e o p h i l e i s f o r m e d ("product f o r m i n g step"). F o r e x a m p l e ,if the Reaction goes i n the s e q u e n c e of steps through the tight i o n p a i r (R X " ) to the solverjt s e p a r a t e d i o n p a i r (R / X " ) and then b y n u c l e o p h i l i c attack on R to p r o d u c t , the r a t e d e t e r m i n i n g step c o u l d be a n y of the t h r e e l a b e l e d with r a t e constant k ^ k o r k . T h e step with r a t e constant kj w i l l b e r a t e d e t e r m i n i n g i f k 2> k . T h e step w i t h r a t e constant k w i l l be r a t e d e t e r m i n i n g i f k « k , and 1^ :» k_ . T h e step with r a t e constant w i l l be r a t e d e t e r m i n i n g i f k » k and k_ » k^. 2
6
2
e l
2
2
e l
2
-]L
2
2
F r o m the point of v i e w of the α - i s o t o p e effect, one c a n i d e n tify t h r e e g e n e r a l c l a s s e s into w h i c h a l l r e a c t i o n s of this m e c h a n i s t i c s c h e m e w i l l f a l l : ( l ) the one r e a c t i o n i n w h i c h the t r a n s i t i o n state has two p a r t i a l c o v a l e n t bonds to c a r b o n , i . e. , the S^2 r e a c t i o n l a b e l e d with r a t e constant 1%. T h e α - e f f e c t i n this type of r e a c t i o n s h o u l d be l o w (see above). (2) R e a c t i o n s i n w h i c h the t r a n s i t i o n state has one p a r t i a l c o v a l e n t bond to carbon. T h e s e i n c l u d e those r e a c t i o n s with r a t e d e t e r m i n i n g steps l a b e l ed w i t h k kg, k£ and k . T h e α - i s o t o p e effect i n t h e s e types of r e a c t i o n s should be l a r g e but not at the m a x i m u m . (3) R e a c t i o n s i n w h i c h the t r a n s i t i o n state has no p a r t i a l c o v a l e n t bonds to c a r 1 }
7
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8.
Isotope Effects and
SHINER
Reaction Mechanisms
177
bon. T h e s e i n c l u d e the r e a c t i o n s w i t h r a t e d e t e r m i n i n g steps l a b e l e d k , k , and k^. T h e i s o t o p e effects i n a l l of these type r e a c t i o n s should be at the m a x i m u m . 2
8
It i s f u r t h e r i m p o r t a n t to r e a l i z e that the m a x i m u m i s o t o p e effect w i l l depend on the n a t u r e of the l e a v i n g g r o u p X and that the r e a c t i o n s i n c l a s s e s 1 and 2 above w i l l show i s o t o p e effects that w i l l , to s o m e extent, depend on r e a c t i v i t y and (except f o r c a s e s w h e r e the step l a b e l e d kj i s r a t e d e t e r m i n i n g ) on the n a t u r e of the i n c o m i n g n u c l e o p h i l e . Of c o u r s e , it i s to be expected that m a n y r e a c t i o n s w i l l not c l e a r l y have one step as c o m p l e t e l y r a t e d e t e r m i n i n g but might, f o r e x a m p l e , i f k_j — k have k and k both p a r t l y r a t e - d e t e r m i n i n g . 2
x
2
In our e f f o r t s to c l a s s i f s c h e m e and to d e t e r m i n ferent m e c h a n i s m s w (18) e x a m p l e g ple a l k y l , α - p h e n y l e t h y l , b e n z y l and"~propargyl c o m p o u n d s hav i n g h a l i d e and sulfonate l e a v i n g g r o u p s i n s o l v e n t s ethanol-water, t r i f l u o r o e t h a n o l - w a t e r and m o r e r e c e n t l y t r i f l u o r o a c e t i c a c i d water. In T a b l e 2 a r e shown the m a x i m u m v a l u e s f o r the α-d e f f e c t s w h i c h we b e l i e v e a p p l y f o r the d i f f e r e n t l e a v i n g g r o u p s studied. F o r e a c h of t h e s e l e a v i n g g r o u p s , e f f e c t s of the s i z e i n d i c a t e d have been o b s e r v e d i n one or m o r e r e a c t i o n s w h i c h show a l l of the c h a r a c t e r i s t i c s of the Ingold S 1 or the W i n s t e i n L i m c l a s sification. M
T a b l e 2. A p p r o x i m a t e M a x i m u m α - d e u t e r i u m B a t e E f f e c t s f o r D i f f e r e n t L e a v i n g G r o u p s A t t a c h e d to Saturated Carbon.
Leaving Groups
V oD k
-OS0 R
-CI
-Br
1. 23
1. 16
1. 12
2
-I 1. 09
A l l of the r e a c t i o n s so f a r studied w h i c h show t h e s e m a x i m a have been c l a s s i f i e d as i n v o l v i n g r a t e d e t e r m i n i n g f o r m a t i o n of the solvent s e p a r a t e d i o n p a i r f o l l o w e d by n u c l e o p h i l i c attack. R e a c t i o n s going v i a f r e e c a r b o n i u m i o n s a r e m u c h r a r e r , and although they undoubtedly e x i s t , we have not m e a s u r e d α-d ef fects f o r any of them. We have a l s o studied r e a c t i o n s c l a s s i f i e d as h a v i n g the step labeled with h or the step l a b e l e d w i t h kc, r a t e d e t e r m i n i n g . T h e s e r e a c t i o n s show α-d effects about 2/3 r d s or 3/4 ths of the l i s t e d m a x i m u m v a l u e s . In a d d i t i o n , a n u m b e r of r e a c t i o n s w h i c h we have e x a m i n e d s e e m to c l e a r l y i n v o l v e S^2 attack by solvent, , and to show i s o t o p e effects between 0. 96 and 1. 06 w i t h r e a c tions i n v o l v i n g the i o d i d e l e a v i n g g r o u p f a l l i n g i n the l o w e r end of the r a n g e and those w i t h a sulfonate g r o u p i n the u p p e r end. lf
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
178
ANDCHEMICAL
PRINCIPLES
Classically, on of the most interesting problems in solvolysis mechanisms relates to the reactions of secondary alkyl sul fonates. Therefore, I will choose three examples from this group of compounds to illustrate the kinds of conclusions which we derive from the study of α-deuterium isotope rate effects. The esters are those of the isopropyl, pinacolyl (3, 3-dimethyl2-butyl) and 2-adamantyl groups; to obtain convenient reaction rates in the range of solvents involved we have used p-toluenesolfonate, p- br omob en ζ en e sulfonate and 2, 2 , 2-trifluoroethanesulfonate esters. We have shown, in at least one of the solvents involved, that a change among these three leaving groups causes only a few tenths of a percent of less, change in the isotope effect. CH* I CH* —CH—CH* CHa-C-*—CH-CH Ο CH S 0 - C H4~ C H isopropyl p-toluenesulfonate 2
6
3
SO - C H — Br pinacolyl p-bromobenzenesulfonate z
6
4
Η
0-S0 -CH -CF 2
2
3
2-adamantyl trifluoroethanesulfonate The rates of solvolysis of one or more of the three sulfonate esters of each of these three alkyl groups have been measured in ethanol-water, 2, 2, 2-trifluoroethanol-water and trifluoacetic acid-water solvents. In addition, the corresponding a-deuteroesters have been synthesized and their solvolysis rates mea sured under identical conditions. F o r ethanol-water and t r i fluoroethanol water mixtures the rates were measured conductimetrically and the first order rate constants obtained with an electronic computer using a doubly weighted non-linear least squares routine. (21) The reactions were followed for about two half-lives starting at about ΙΟ- M initial concentration and tak ing a total of about 150 readings for each reaction. The stand ard e r r o r s in the rate constants and the reproducibility are both of the order of 0. 1%. The rates of solvolysis in trifluoroacetic acid were measured using a C a r y 118 ultraviolet spectrophoto meter and following the reduction in the ester absorbance around 270 nm. from about 1. 2 o. d. to about 0. 7 o. d. at an initial con centration about 3 χ 10" molar. (22) A computer program s i m i lar to that mentioned above was useH to obtain the first order rate constants from around 200 readings per reaction. Standard e r r o r s and reproducibility in trifluoroacetic acid are only around 1%. apparently because the reactions are not precisely first order. In Table 3 are given the isotope effects expressed as the ratio of the first order rate constants for hydrogen and deuterium 3
3
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8.
SHINER
Isotope Effects and Reaction Mechanisms
179
Table 3.
α-d Effects and Relative Rates of Solvolyses at 25°
Solvent
Isopropyl Pinacolyl Sulfonates , Sulfonates k / k , k/k . d k/k , H ad pm ' ad 1
2-Adamantyl Sulfonates ,
k/k H ad a
k 1
Apm·
-
-
0. 714
1. 159
1. 225
0.0315
1.16
0. 026
1. 153
1. 228
0.0595
1. 22
0.0041
1. 16
1. 22
0. 207
90E
1. 08
3. 35
50E
1.11
97TFE 99TFA
a
-
F r o m references 22 23 and 24 9 0 E is 90 vol % ethanol 10 vol. % water; 50E 9 7 T F E is 97 wt. % 2 9 9 T F A is 99 vol. % trifluoroacetic acid, 1 vol. % water. °The sulfonate esters used were: in 90E the p-bromobenzenesulfonate; in 50E and 9 7 T F E isopropyl-p-bromobenzenesulfonate and 2,2,2trifluoroethanesulfonate, pinacolyl p-bromobenzenesulfonate and 2-adamantyl 2, 2 , 2-trifluoroethanesulfonate; in 99TFA, isopropyl and p-bromobenzenesulfonate, pinacolyl p-toluenesulfonate and g-adamantyl p-toluenesulfonate and p-bromobenzenesulfonate. Rate relative to that of the corresponding pinacolyl ester. compounds in four different solvents, and, for isopropyl and ada mant y 1 esters, the ratio of their solvolysis rates to the c o r r e s ponding faster solvolyzing pinacolyl esters. The results in the four solvents are interesting to compare because on progression through the series, 90 vol. % ethanol, 50 vol. % ethanol, 97 wt. % trifluoroethanol, 99 vol. % trifluoroacetic acid, the tendency towards slower nucleophilic reactions and more facile carbonium ion forming reactions becomes more and more pronounced. Adding water to ethanol affects the reaction p r i m a r i l y through increasing the solvent dielectric constant and its ability to ionize covalent bonds. Trifluorethanol is about 1000 times less basic than ethanol or water (and therefore less nucleophilic) but it is nevertheless a good ionizing solvent because of its abilitv to sta bilize anions by hydrogen bonding. (25) In trifluoroacetic acid these trends are extended considerably further. (26) Despite this wide range of nucleophilicity and ionizing ability we find that pinacolyl esters show almost the same α-d effect on the solvoly sis rate in each solvent! We have argued (23) that the esters of this alcohol all react by the same mechanism in each of these solvents and that this involves rate-determining formation of the tight ion pair; this tight ion pair does not return but rather r e arranges rapidly by methyl migration to form the tertiary c a r bonium ion pair:
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
180
CH ROBs
CH
3
3
—• C — i
CH
3
AND
+
CHEMICAL
CH
+
CH — __
CH
CH
3
3
—
OBs
PRINCIPLES
3
C — CH — ι _ CH OBs
CH
3
3
I
products The tetiary carbonium ion is so much more stable than the secondary ion that it does not return but forms products. Es sentially all of the products found are those of rearranged car bon skeletal structure. Return to the secondary structure seems highly improbable since all reactions which are expected to go through the pinacolyl cation structure never give significant yields of pinacolyl derivative arranged carbon skeleta seems arguable about this mechanism is whether or not rear rangement is concerted with ionization; if so, it is expected to accelerate ionization but relative rate comparisons in several solvents (see below) do not suggest accelerated rates. Also, the Y-d derivative does not show an appreciable isotope effect. (23) If a CD group were migrating in the rate-determining step 9
3
CD CD
3
—
3
C — I CD 3
CH ι OBs
CH
3
the HCC bending force constants should be lower in the transition state and an isotope effect should be observed. In any event, the extent of participation by methyl during ionization should be quite small, if not entirely absent. The α-d rate effect on pina colyl sulfonate solvolyses are seen to be constant at 1.15 - 1.16 for all four solvents listed in Table 3. 2-Adamantyl sulfonates on the other hand, do not rearrange appreciably during solvoly sis, apparently because to do so would introduce significant ring strain into the relatively strainless structure. They are also not readily subject to nucleophilic attack because the compact t r i cyclic structure hinders nucleophilic approach from the rear. Thus we believe that 2-adamantyl sulfonate esters undergo ioni zation to the tight ion pair and return,with rate determining for mation of the solvent separated ion-pair which is nucleophilically attacked to form product. The α-d effects for 2-adamantyl sulfonates in the solvents shown in Table 3 are all in the range 1. 22 - 1. 23 (24) It is interesting to note that the 2-adamantyl sulfonates react from five to 32 times slower than corresponding pinacolyl esters. One would expect the adamantyl esters to ionize somewhat faster because the adamantyl structure has four carbon atoms in the gamma position relative to the reaction cen ter (two on each side) while pinacolyl has only three. The cor responding difference in inductive effect on ionization should
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
8.
SHINER
Isotope Effects and
Reaction Mechanisms
181
contribute a factor of around two to the ionization rate ratio lead ing to estimated ratios of internal return to solvolysis for the 2adamantyl derivatives of — 10 in 9 9 % trifluoroacetic acid, — 34 in 97% trifluoroethanol and —65 in 5 0 % ethanol. If pinacolyl solvolyses were accelerated by rearrangement these factors would be even smaller; the factor of — 10 in T F A certainly could not be much smaller without causing kj to be partly rate deter mining for 2-adamantyl and causing the α-d effect to be lower than 1.22. Thus, the idea that rearrangement acceleration in pinacolyl ionization is small is reinforced. Return from 2-ada mantyl tight ion pairs is probably less in T F A than in T F E and still less in ethanol because Η-bonding of the more acidic solvent reduces the nucleop^hilicity of the leaving group and slows i|s re combination with R relative to the further separation to R /Χ~· The situation with the ing: a very low α-d effect This can only mean that an S^2 reaction is largely, but probably not completely, the reaction pathway. It is significant that in this solvent isopropyl sulfonates react about three times faster than the corresponding pinacolyl esters. Since the isopropyl ester must ionize slower than the pinacolyl ester due to the dif ference in inductive effects of the substituents, this faster obser ved rate can only be caused by an S ^ process. In 50% ethanol the isopropyl esters solvolyze at rates comparable to the cor responding pinacolyl esters and show increased α-d effects. This must be due to the formation and reaction of tight ion pairs in addition to some fraction of S 2 reaction. In 97%> trifluorethanol the pinacolyl esters react about forty times faster than the corresponding isopropyl esters and the α-d isotope effect for the isopropyl esters rises to 1. 16. In this solvent the reaction must be going almost exclusively through tight ion pairs and the only way that the isopropyl compounds can be reacting so much slower is for the ion pairs to be undergoing return about four times more rapidly than they are nucleophilically attacked. In trifluoroacetic acid the very low nucleophilicity of the solvent reduces the rate of nucleophilic attack on the tight ion pairs of the isopropyl sulfonate esters so much that the rate of ester sol volysis is now 244 times slower than the rate of solvolysis of the corresponding pinacolyl esters, suggesting that internal return takes place about twenty times faster than solvolysis. In addition, in this solvent, attack on the tight ion pair is so slow that the rate determining step is now separation of the isopropyl sulfon ate tight ion pair to the solvent separated ion-pair and the α-d effect is 1.22, the same as observed for 2-adamantyl sulfonates in all three solvents! Thus isopropyl sulfonate solvolyses are truly "borderline" in character and can shift over a range of me chanisms with different solvents. There are many other obser vations which buttress these interpretations but no need or space to go into more detail here. It is hoped that these illustrations will show, in part at least, how secondary deuterium isotope
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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rate effects can be used in the intricate process of sorting out reaction mechanistic pathways. Because isotopic substitution involves a minimal perturbation of the reacting system and because of the solid basic theory underlying our understanding of these effects they allow us an unparalleled tool with which to attack mechanistic problems. I hope that this brief review has conveyed some flavor of the intellectual adventure and satisfaction which can be involved in such studies. A c kn owl ed gm en t The preparation of results reported here were supported from the National Science Foundation. thank Dr. W. E. Buddenbaum and Mr. the manuscript before publication and and suggestions. Thank London for permission
this paper and the new in part by grant G P 32854 The author wishes to Richard Seib for reading for valuable discussion
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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SHINER
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Isotope Effects and Reaction Mechanisms
Literature Cited 1. F o r a more detailed discussion of theory and the interpre tation of results from a variety of reactions see: "Isotope Effects in Chemical Reactions", C. J. Collins and N. S. Bow man, eds., Van Nostrand Reinhold, New York, 1970 and references cited therein. 2. Bigeleisen, J. and Mayer, M. G., J. Chem. Phys., (1947), 15, 261. 3. Melander, L. Arkiv. Kemi., (1950), 2, 211. 4. Wolfsberg, M. and Stern M 8, 225. 5.
J.
Pure Appl Chem.
(1964)
Wolfsberg. M. and Stern, M. J., Pure Appl. Chem., (1964), 8, 325.
6. Wolfsberg, M. and Stern. M. J., J. Pharm. S c i . , (1965), 54, 849. 7. Bigeleisen, J., Can. J. Chem., (1952), 30, 443. 8. Willi, Α. V., Can. J. Chem., (1966), 44, 1889. 9· Katz, A. M. and Saunders, W. Η., J r . , J. Amer. Chem. Soc., (1969), 91, 4469. 10. Williams, R. C., and Taylor, J. W., (1974), 96, 3721.
J. Amer. Chem. Soc.,
11. Melander, L a r s , "Isotope Effects on Reaction Rates," pg. 12, Ronald Press, New York, 1960. The approxima tions involved are also discussed on pg 35. See also ref. 6, pg. 850, equation 6: k /k 1
2
=
x (VP) x (EXC)x( Z P E )
For the diatomic model (VP) = (ν /ν ) Therefore, k /k = (EXC) x (ZPE) = sinh l/2 u ÷sinh l/2 u , where u = hν/ kT. 2L
1
2
12. Westheimer, F. Η., Chem. Rev.,
1
1L
2
(1961), 61, 265.
13. Melander, L. "Isotope Effects on Reaction Rates," pps. 24-32, Ronald Press, New York, 1960. 14. Bell, R.P. and Cox, B. G., J. Chem. Soc., Β, (l971), 783.
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
9 Isotope Chemistry and Biology J O S E P H J.
KATZ,
R O B E R T A. U P H A U S , and H E N R Y L . C R E S P I
Chemistry Division, Argonne National Laboratory, Argonne, Ill. 60439 M A R T I N I. B L A K E Department of Pharmacy, University of Illinois, College of Pharmacy, Chicago, Ill. 60612
I.
Introduction
The p a s t decade has seen a keen and growing intere s t in t h e application o f s t a b l e i s o t o p e s t o c h e m i c a l and biological problems ( 1 , 2 ) . There have been t h r e e main driving f o r c e s f o r this interest: (a) The applic a t i o n s o f magnetic resonance t e c h n i q u e s t o many complex c h e m i c a l and biological problems a r e greatly facilitated by j u d i c i o u s use o f s t a b l e i s o t o p e s ; ( b ) , t h e utility o f gas chromatographic-mass s p e c t r o m e t r i c t e c h n i q u e s i n m e t a b o l i c and e n v i r o n m e n t a l t r a c e r s t u d i e s is greatly enhanced by adjustment o f t h e isotopic c o m p o s i t i o n ; and, ( c ) , t h e availability of fully d e u t e r a t e d compounds and o r g a n e l l e s from fully d e u t e r a t e d m i c r o o r g a n i s m s (3) has made p o s s i b l e t h e investigation o f many r e f r a c t o r y problems o f biological interest. These, t o g e t h e r w i t h an e v e r i n c r e a s i n g availability o f C, N, O and d e p l e t e d C and N in h i g h isotopic p u r i t y have t r i g g e r e d a wide r e s p o n s e ( 4 ) . D e s p i t e t h e many uses t o which s t a b l e i s o t o p e s c a n be p u t , certain i n h e r e n t limitations still prevail. Deuterium as heavy water is available in large q u a n t i t i e s a t a relatively low p r i c e , but i n c o r p o r a t i o n o f deuterium i n t o living organisms can have s e v e r e t o x i c effects on h i g h e r p l a n t s and animals because o f p o s s i b l e l a r g e kinetic i s o t o p e e f f e c t s (5). The heavy i s o t o p e s o f c a r b o n , n i t r o g e n and oxygen a r e far less t o x i c t o living organisms t h a n is d e u t e r ium, b u t t h e quantities a v a i l a b l e a r e limited and t h e price tends t o be q u i t e h i g h . Hydrogen o f mass 2, d e u t e r i u m , was d i s c o v e r e d by Urey and his co-workers in 1932 (6). I n t h e decade f o l l o w i n g t h i s d i s c o v e r y t h e r e d e v e l o p e d a l a r g e body o f o f t e n c o n f u s i n g d a t a (7,8) c o n c e r n i n g t h e b i o l o g y o f heavy w a t e r . These e a r l y s t u d i e s were i n t e r p r e t e d t o i n d i c a t e t h a t d e u t e r i u m is toxic, and t h a t h i g h 13
12
15
18
14
184
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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concentrations are fundamentally incompatible with c e l l u l a r growth and r e p r o d u c t i o n . However, i n I 9 6 0 , a u t o t r o p h i c f r e s h - w a t e r green and b l u e - g r e e n a l g a e were s u c c e s s f u l l y c u l t u r e d i n heavy w a t e r c o n t a i n i n g 9 9 . 7 atom p e r c e n t H. T h i s work was s h o r t l y f o l l o w e d by t h e c u l t i v a t i o n o f a wide v a r i e t y o f f u l l y d e u t e r a t e d h e t e r o t r o p h i c m i c r o o r g a n i s m s (3,8). These s u c c e s s e s l e d to a considerable e f f o r t to achieve f u l l deuteration of h i g h e r p l a n t s and mammals, b u t t o d a t e t h e s e more com p l e x systems have r e s i s t e d f u l l r e p l a c e m e n t o f ^H by H. T o x i c e f f e c t s i n mammals, f o r example, a r e appar e n t even a t 1 0 - 1 5 atom p e r c e n t H i n t i s s u e f l u i d s . The marked b i o l o g i c a l e f f e c t s o f d e u t e r i u m a r e n o t o b s e r v e d when C i s s u b s t i t u t e d f o r C , o r N f o r The s o u r c e o f t h i differential effect i likel the much l a r g e r k i n e t i d e u t e r i u m , as compared t o the heavy, s t a b l e i s o t o p e s o c a r b o n and n i t r o g e n . The s u b s t i t u t i o n o f a heavy i s o t o p i c s p e c i e s i n a c h e m i c a l bond may change the r a t e o f any r e a c t i o n t h a t i n v o l v e s s c i s s i o n o f t h i s bond. In simple terms, the e f f e c t on the r e a c t i o n r a t e w i l l de pend on the mass r a t i o o f t h e i s o t o p i c atoms i n question. Thus, the mass r a t i o f o r the hydrogen i s o t o p e s , ^H/^H, i s 2, w h i l e the 1 3 c / l C r a t i o i s o n l y 1 . 0 8 , and t h a t of 1 5 N / N i s only 1 . 0 7 . Consequently, k i n e t i c isotope e f f e c t s perhaps an o r d e r o f magnitude l e s s would be e x p e c t e d f o r heavy carbon and n i t r o g e n t h a n f o r heavy hydrogen. The s u b s t i t u t i o n o f H for H also affects equi l i b r i u m c o n s t a n t s , p a r t i c u l a r l y the i o n i z a t i o n constants o f weak a c i d s and bases d i s s o l v e d i n D 0 (10_, 1 1 ) . The r a t e s o f a c i d - b a s e c a t a l y z e d r e a c t i o n s may be g r e a t l y d i f f e r e n t i n ! H 0 as compared t o H 0 (2,12). Deuter ium s u b s t i t u t i o n w i l l t e n d t o i n c r e a s e s l i g h t l y t h e s t r e n g t h o f hydrogen bonds, and d e u t e r i u m has a s i g n i f i c a n t l y s m a l l e r s t e r i c r e q u i r e m e n t t h a n does ^H. Thus, r a t e s o f c o n f o r m a t i o n a l i n t e r c h a n g e i n d e u t e r a t e d b i o p o l y m e r s can be markedly d i f f e r e n t from those o f normal i s o t o p i c c o m p o s i t i o n . I t i s t h e r e f o r e not at a l l s u r p r i s i n g t h a t the o v e r a l l b i o l o g i c a l e f f e c t o f deu t e r i u m can be e x c e e d i n g l y complex. The i s o t o p e s o f c a r b o n and n i t r o g e n may be e x p e c t e d t o have q u a l i t a t i v e l y s i m i l a r e f f e c t s , but the magnitude o f t h e e f f e c t s o f t h e s e i s o t o p e s a r e g e n e r a l l y s m a l l enough t o be w i t h i n the range o f the normal c e l l u l a r c o n t r o l mechanisms. 2
2
2
1 3
1 2
l 5
2
1 4
2
1
2
2
2
2
II.
Deuterium
1930,
Through t h e p a s t 45 y e a r s , s i n c e i t s d i s c o v e r y i n t h e s t a b l e , r a r e hydrogen i s o t o p e o f mass two
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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(deuterium) has been an i n c r e a s i n g l y u s e f u l t o o l i n c h e m i s t r y and b i o l o g y . The h i s t o r y o f d e u t e r i u m has shown a p r o g r e s s i o n from the s i m p l e t o the complex, from t r a c e r a p p l i c a t i o n s t o massive s u b s t i t u t i o n , from k i n e t i c i s o t o p e e f f e c t s i n simple m o l e c u l e s t o i s o t o p e e f f e c t s i n l i v i n g organisms, and from i t s use t o s i m p l i f y the p r o t o n magnetic resonance ( Hmr) spectrum o f e t h y l a l c o h o l t o the s i m p l i f i c a t i o n o f the iHmr s p e c t r a o f proteins. These most r e c e n t a p p l i c a t i o n s r e s u l t from the a b i l i t y t o grow a l g a e and o t h e r m i c r o o r g a n i s m s i n heavy water, H 0 , whose hydrogen c o n t e n t i s 99.8 atom p e r c e n t H and o n l y 0.2 atom p e r c e n t 2
2
2
Algae. A f t e r a succession of f a i l u r e s i n other l a b o r a t o r i e s , Chorney and co-workers (13) succeeded i n c u l t u r i n g two s p e c i e heavy water. T h i s wor s u c c e s s f u l c u l t u r e o f a number o f o t h e r a l g a e i n heavy water. The e x t r a c t i o n o f o r g a n i c s u b s t r a t e s (14) from t h e s e H - a l g a e then made p o s s i b l e the c u l t i v a t i o n o f a number o f h e t e r o t r o p h i c b a c t e r i a and f u n g i i n f u l l y deut e r a t e d form. Many k i n d s o f a l g a e r e q u i r e d a l e n g t h y p e r i o d o f a d a p t a t i o n b e f o r e r o u t i n e c u l t u r e i n heavy water was p o s s i b l e . O f t e n a s m a l l n u t r i t i o n a l s u p p l e ment ( i n the form o f y e a s t e x t r a c t ) h e l p e d the a l g a e t o overcome the problems o f t o t a l k i n e t i c r e o r g a n i z a t i o n i n the new c u l t u r e medium. Near a n a e r o b i c c o n d i t i o n s were a l s o b e n e f i c i a l , as t h e r e were i n d i c a t i o n s t h a t r e s p i r a t i o n i n a d a p t i n g organisms was u n c o n t r o l l e d . A f t e r a d a p t a t i o n , however, t h e a l g a e grew i n a normal manner b u t a t a s l o w e r than normal r a t e . A k i n e t i c i s o t o p e e f f e c t ( k / k ) o f 3.5 was observed i n the l i g h t s a t u r a t e d growth r a t e o f s e v e r a l green and b l u e - g r e e n algae. However, because the l a r g e - s c a l e p r o d u c t i o n o f i s o t o p i c a l l y - a l t e r e d a l g a e u s u a l l y i n v o l v e s growth under l i g h t l i m i t i n g c o n d i t i o n s , t h i s l a r g e i s o t o p e e f f e c t i s n o t a major h a n d i c a p i n the p r o d u c t i o n o f l a r g e amounts o f f u l l y d e u t e r a t e d a l g a e . 2
H
D
Deuterium Organisms i n E s r . The a v a i l a b i l i t y o f f u l l y d e u t e r a t e d a l g a e has p e r m i t t e d some s i g n i f i c a n t b i o l o g i c a l problems t o be a t t a c k e d i n new and e f f e c t i v e ways. Deuterium had been used as a t o o l t o a i d i n the s i m p l i f i c a t i o n and i n t e r p r e t a t i o n o f e l e c t r o n s p i n resonance (esr) and n u c l e a r magnetic resonance (nmr) s p e c t r a o f r e l a t i v e l y s i m p l e o r g a n i c m o l e c u l e s f o r many
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
9. κΑτζ
ET
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AL.
Biology
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2
years.* In t h e s e c a s e s H was i n t r o d u c e d by c h e m i c a l synthesis. A s i m i l a r t e c h n i q u e can now be a p p l i e d t o some e x t r e m e l y complex b i o l o g i c a l m o l e c u l e s . One o f the more i m p o r t a n t r e s u l t s o b t a i n e d by s e l e c t i v e i n c o r p o r ation of and H has been the e l u c i d a t i o n o f t h e c h l o r o p h y l l f r e e r a d i c a l species t h a t i s i n v o l v e d i n the l i g h t c o n v e r s i o n a c t o f p h o t o s y n t h e s i s (15). E s r s t u d i e s on H - c h l o r o p h y l l s and H - o r g a n i s m s have shown t h a t the f r e e r a d i c a l s p e c i e s formed by the e x c i t i n g l i g h t i s comprised o f a p a i r o f c h l o r o p h y l l m o l e c u l e s arranged i n a s p e c i a l c o n f i g u r a t i o n which p e r m i t s them t o a c t as an energy t r a p . 2
2
2
D e u t e r a t e d P r o t e i n s i n Nmr, Deuterated p r o t e i n s can a l s o s e r v e as t h the h i g h l y complex ^Hm Three b a s i c experiments t o t h i s end have been d e s c r i b e d i n the l i t e r a t u r e (16). (1) An a u t o t r o p h i c organism growing i n H 0 can be i n d u c e d t o u t i l i z e an lH-amino a c i d ; t h e o r g a n i s m t h e n b i o s y n t h e s i z e s H - p r o t e i n con t a i n i n g !H-amino a c i d r e s i d u e s embedded i n i t ; ( 2 ) , an 1 H p r o s t h e t i c group can be bound t o a f u l l y d e u t e r a t e d a p o p r o t e i n , making i t p o s s i b l e t o d e t e c t e a s i l y the p r o s t h e t i c group by Hmr; and ( 3 ) , the s l o w l y exchange a b l e amide p r o t o n s can be o b s e r v e d i n o t h e r w i s e f u l l y d e u t e r a t e d p r o t e i n s by o b s e r v i n g t h e time dependence o f t h e iHmr spectrum when t h e p r o t e i n i s d i s s o l v e d i n Ή α We d e s c r i b e here an example o f a t y p e (3) experiment, i e . , t h e o b s e r v a t i o n by J-Hmr o f amide p r o t o n s i n a f u l l y deuterated protein. ^ The p r o t e i n resonance l i n e s o b t a i n e d i n an Hmr experiment a r e g e n e r a l l y q u i t e b r o a d . The l a r g e p r o t e i n m o l e c u l e s tumble s l o w l y i n s o l u t i o n , and many d i p o l e - d i p o l e magnetic i n t e r a c t i o n s and c h e m i c a l s h i f t a n i s o t r o p i e s a r e n o t averaged o u t . In g e n e r a l , amide p r o t o n resonance peaks i n m o l e c u l e s w i t h m o l e c u l a r w e i g h t s o f 10-20,000 d a l t o n s have l i n e w i d t h s o f the o r d e r o f 30-35 Hz. However, i n a d e u t e r a t e d p r o t e i n , the d i p o l e - d i p o l e i n t e r a c t i o n i s much d e c r e a s e d and most amide p r o t o n s have l i n e w i d t h s i n t h e range o f 15-18 Hz. These amide p r o t o n s appear 6 t o 11 ppm downf i e l d from t e t r a m e t h y l s i l a n e , and t h e p r o t o n resonances 2
2
2
1
2
Η has a s p i n o f 1/2 and a magnetic moment o f 2.793 n u c l e a r magnetons. H has a s p i n o f 1 and a magnetic moment o f 0.857 n u c l e a r magnetons. Thus, when a deu t e r i u m atom i s s u b s t i t u t e d f o r a p r o t i u m atom, c o u p l i n g c o n s t a n t s a r e r e d u c e d by a f a c t o r o f 6.5 and t h e f r e quency a t w h i c h energy l e v e l t r a n s i t i o n s t a k e p l a c e i s a l s o reduced by a f a c t o r o f 6.5. 2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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t h a t o r i g i n a t e from t h e a r o m a t i c amino a c i d s i d e chains t h a t a r e n o r m a l l y p r e s e n t i n t h i s r e g i o n o f t h e Hmr spectrum a r e o f course absent i n t h e f u l l y d e u t e r a t e d protein. Thus, a c l e a r , w e l l - r e s o l v e d view o f t h e sLowl y exchangeable amide p r o t o n s can be had. These protons are p a r t o f t h e hydrogen-bonded s t r u c t u r e o f t h e h e l i x and p l e a t e d s h e e t p o r t i o n s o f p r o t e i n s and thus have considerable s t r u c t u r a l significance. F i g u r e 1 shows an example o f an ^Hitir study o f amide p r o t o n s i n a d e u t e r ated a l g a l f e r r e d o x i n . 1
Deuterated N u c l e i c A c i d s . Deuterated algae are a s o u r c e o f d e u t e r a t e d s u b s t r a t e s f o r t h e growth o f bac t e r i a , y e a s t and molds (14). D e u t e r a t e d b a c t e r i a have been found u s e f u l a a c i d s f o r use i n u l t r a c e n t r i f u g a a l y s i s o f t h e i n t e r a c t i o n s o f homologous DNA m o l e c u l e s . F u l l y d e u t e r a t e d DNA ( a l l C- H bonds) has a bouyant d e n s i t y 0.04 g/cm g r e a t e r than normal (normal DNA s have bouyant d e n s i t i e s i n t h e range o f 1.70-1.71 g/cm ) . T h i s t e c h n i q u e has been e x p l o i t e d r e c e n t l y i n t h e study o f t h e i n t e r a c t i o n o f b a c t e r i a l and b a c t e r i o p h a g e n u c l e i c acids (17). 2
3
1
Deuterated M e t a b o l i t e s . A number o f s t u d i e s on the p r o d u c t i o n o f e x t r a c e l l u l a r m e t a b o l i t e s by d e u t e r a t e d organisms have been completed i n r e c e n t y e a r s . These s t u d i e s have i n v o l v e d t h e p r o d u c t i o n and c h a r a c t e r i z a t i o n o f a l k a l o i d s (18), a n t i b i o t i c s (19), and t h e v i t a min r i b o f l a v i n produced by organisms grown i n 99.8 atom p e r c e n t H 0 (20_) . The use o f v a r i o u s *Η ( n a t u r a l abundance) s u b s t r a t e s i n c o n j u n c t i o n w i t h iHmr a n a l y s i s o f t h e p r o d u c t has made i t p o s s i b l e t o o b t a i n a q u a n t i t a t i v e assessment o f t h e e x t e n t o f s o l v e n t p a r t i c i p a t i o n i n t h e b i o s y n t h e s i s o f these n a t u r a l p r o d u c t s . As might be e x p e c t e d , heavy water s t r o n g l y i n h i b i t s t h e production o f the e x t r a c e l l u l a r m a t e r i a l i n a l l o f the e x p e r i m e n t s . More r e c e n t l y , however, i t has been ob s e r v e d t h a t r i b o f l a v i n p r o d u c t i o n by t h e fungus Eremothecium a s h b y i i i s s t i m u l a t e d by heavy water (Table 1 ) . E v i d e n t l y , t h e r e i s an i n v e r s e i s o t o p e e f f e c t on t h i s p a r t i c u l a r m e t a b o l i c pathway. 2
2
H i g h e r P l a n t s . The e f f e c t s o f d e u t e r i u m on h i g h e r p l a n t s have a t t r a c t e d p a r t i c u l a r a t t e n t i o n . The e a r l y l i t e r a t u r e has been reviewed by Morowitz and Brown Π ) and more r e c e n t l y by Flaumenhaft e t a l . (Β). The f i r s t d e t a i l e d study o f t h e e f f e c t s o f e x t e n s i v e s u b s t i t u t i o n o f hydrogen by d e u t e r i u m on t h e growth and development o f h i g h e r p l a n t s was d e s c r i b e d by B l a k e e t a l . (21) who
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
9. ΚΑτζ E T A L .
Isotope Chemistry and Biology
e
ι
.
ι
1
Ferredoxin Oxidized Full Back Exchange
1
189
1
Γ
I
Out Exchanged Three Hours
ι—ι
Π
1
ι
ι
ι
ι
10
9
8
7
6
I
ppm Figure 1. Proton magnetic resonance spectra at 220 MHz (Varian HR 220 spectrometer in the fast fourier transform pulse mode) of fully deuterated algal ferre doxin. About 26 slowly exchangeable protons (Spectrum I) are observable after back exchange in H 0 buffer at pll 9. After 3 hrs at 60°C in Ή Ό buffer, pH 7.2, 11 of these protons have completely out exchanged and 15 have been quite resistant to exchange (Spectrum II). These data indicate that this ferredoxin molecule contains two segments of helix (or pleated-sheet) secondary structure. 1
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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Table I S t i m u l a t i o n o f R i b o f l a v i n P r o d u c t i o n by E. a s h b y i i S u b c u l t u r e d from ^-E O i n t o V a r i o u s L e v e l s o f ^H 0 n
Atom P e r c e n t Deuterium i n N u t r i e n t Water 0 25 50 99
o
Day 0 0 0 0 0
Riboflavin Production (mg/liter) Day Day Day 22 15 8 2.7 3.5 8.5 15.8
4.6 5.4 13.0 23.5
5.6 7.3 15.0 25.1
used peppermint (Menth Above the 20 p e r c e n t i n h i b i t o r y e f f e c t on growth w i t h i n c r e a s e i n the D0 c o n t e n t o f the n u t r i e n t as r e f l e c t e d i n e l o n g a t i o n growth o f the a x i a l s h o o t . V e r y pronounced r e p r e s s i v e e f f e c t s were o b s e r v e d a t the 50 p e r c e n t l e v e l , and growth e s s e n t i a l l y s t o p p e d i n 70 p e r c e n t D 0. In a subsequent r e p o r t the h i s t o l o g i c e f f e c t s o f d e u t e r a t i o n on peppermint were r e p o r t e d by Crane e t a l . (22). The major e f f e c t o f d e u t e r i u m on the growth o f peppermint appeared t o be i n h i b i t i o n o f c e l l d i v i s i o n . In the deut e r a t e d p l a n t s parenchyma c e l l s were e n l a r g e d , w h i l e t h e r e appeared t o be a r e d u c t i o n i n the v a s c u l a r t i s s u e . In g e n e r a l , t h e e f f e c t s o f d e u t e r a t i o n were more appare n t i n a c t i v e l y growing t i s s u e s t h a n i n t i s s u e exposed t o D 0 i n the n u t r i e n t a f t e r d i f f e r e n t i a t i o n had occurred. The e f f e c t s o f c e r t a i n growth r e g u l a t o r s on peppermint grown a t d i f f e r e n t l e v e l s o f D 0 i n the nut r i e n t s o l u t i o n were r e p o r t e d by B l a k e e t a l . (23). The i n h i b i t o r y e f f e c t s o f d e u t e r i u m on the c e l l u l a r l e v e l were not r e v e r s e d by g i b b e r e l l i c a c i d , naphthal e n e - a c e t i c a c i d or i n d o l e a c e t i c a c i d . In some i n s t a n c e s the i n h i b i t o r y e f f e c t s on growth were even g r e a t e r than t h a t a t t r i b u t a b l e t o the D 0 i n the nutrient. I t i s i n t e r e s t i n g to note t h a t maleic hydraz i d e , u s u a l l y c o n s i d e r e d t o be a p l a n t growth i n h i b i t o r , a c t u a l l y s t i m u l a t e d the growth o f peppermint c u l t u r e d i n d e u t e r a t e d media. Perhaps the most i n t e n s i v e l y s t u d i e d h i g h e r p l a n t i s Lemna p e r p u s i l l a (duckweed), which was s u b j e c t e d t o e x t e n s i v e d e u t e r i u m r e p l a c e m e n t by Cope e t a l . (24). A l a r g e number o f growth f a c t o r s , i n d i v i d u a l l y and i n c o m b i n a t i o n , were i n c l u d e d i n the n u t r i e n t s o l u t i o n and t h e i r e f f e c t on the growth o f duckweed a t h i g h l e v e l s o f d e u t e r a t i o n was o b s e r v e d . A t d e u t e r i u m l e v e l s i n 2
2
2
2
2
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
9.
ΚΑτζ E T A L .
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the n u t r i e n t between 50 and 63 p e r c e n t numerous abnor m a l i t i e s were produced, b u t t h e s e were l a r g e l y e l i m i n a t e d by t h e a d d i t i o n o f k i n e t i n . None o f t h e o t h e r ad d i t i v e s produced a b e n e f i c i a l r e s p o n s e . Even k i n e t i n f a i l e d t o evoke a p r o t e c t i v e e f f e c t when t h e D 0 l e v e l exceeded 63 p e r c e n t . The i n c l u s i o n o f Η-glucose i n the n u t r i e n t s o l u t i o n (50 atom p e r c e n t D 0) r a i s e d t h e f i x e d d e u t e r i u m i n t h e p l a n t DNA t o 75 p e r c e n t . I n b e l l a d o n n a p l a n t s d e u t e r a t i o n had a d r a s t i c e f f e c t on f l o w e r development (25). The number o f c a l y x l o b e s , c o r o l l a l o b e s , and stamens, w h i l e i n v a r i a b l y 5 i n c o n t r o l f l o w e r s , i n c r e a s e d t o as many as 9 o r 10 i n p l a n t s grown i n 70 p e r c e n t D 0 medium. Abnormally shaped b e r r i e s formed i n p l a n t s grown i n heavy water. The e x t e n t o f m a l f o r m a t i o o f t h e medium and ho p l a n t t h a t t h e b e r r y formed. The shapes ranged from pear-shaped t o dumbbell-shaped t o c y l i n d r i c a l . The misshapened b e r r i e s r e s u l t e d from t h e t e n a c i t y w i t h which t h e c o r o l l a remained a t t a c h e d t o t h e r i p e n i n g b e r r y i n d e u t e r a t e d p l a n t s . As t h e b e r r y e n l a r g e d , a c o n s t r i c t i o n d e v e l o p e d where t h e c o r o l l a was a t t a c h e d t o t h e b e r r y . The s i z e and number o f seeds were s e v e r e l y reduced i n d e u t e r a t e d b e r r i e s w i t h o n l y a few r u d i m e n t a r y seeds a p p a r e n t i n t h e 70 p e r c e n t b e r r i e s . A s i m i l a r study w i t h A r a b i d o p s i s t h a l i a n a gave much the same r e s u l t s ( 2 6 ) . A replacement c u l t u r e t e c h n i q u e was d e v e l o p e d by Crane e t a l . (27) t o study t h e e f f e c t o f d e u t e r a t i o n on a l k a l o i d production i n Atropa belladonna. P l a n t s were grown t o m a t u r i t y i n an aqueous (H^O) medium and were then t r a n s p l a n t e d t o media c o n t a i n i n g 50, 60, 75 and 99.7 p e r c e n t D 0. The p l a n t s i n 99.7 p e r c e n t D 0 show ed t h e d r a s t i c e f f e c t s o f d e u t e r a t i o n almost immediate l y and a l l p l a n t s d i e d i n s e v e r a l days. Plants trans p l a n t e d i n t o 75 p e r c e n t D 0 s u r v i v e d about 3 weeks, and t h e 50 and 60 p e r c e n t D 0 p l a n t s w i t h s t o o d t h e s t r e s s e s imposed by deuterium. These p l a n t s were h a r v e s t e d a f t e r a growth p e r i o d o f 7.5 months. Alkaloid p r o d u c t i o n was reduced t o from o n e - t h i r d t o o n e - t e n t h t h a t o f t h e c o n t r o l p l a n t s . The a b s o l u t e amount o f a l k a l o i d formed and t h e t o t a l amount o f p l a n t m a t e r i a l produced were t o o s m a l l t o p e r m i t i s o l a t i o n o f a l k a l o i d . I t appeared from t h i s study t h a t a l k a l o i d p r o d u c t i o n was c o m p l e t e l y i n h i b i t e d upon t r a n s f e r o f t h e p l a n t s from normal growth i n H-O t o t h e d e u t e r a t e d medium. G e r m i n a t i o n o f seeds has been the o b j e c t o f study. Crumley and Meyer (28) o b s e r v e d a d e l a y i n t h e i n i t i a t i o n of germination i n four species of p l a n t s , the e x t e n t o f which i n c r e a s e d w i t h t h e d e u t e r i u m 2
2
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In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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c o n c e n t r a t i o n o f the s o l v e n t . However, t h e number o f seeds which f i n a l l y germinated was o n l y s l i g h t l y lower f o r pure D 0 t h a n f o r w a t e r . More r e c e n t l y , S i e g e l e t a l . (29) s t u d i e d the g e r m i n a t i o n o f 11 s p e c i e s o f seeds i n h i g h c o n c e n t r a t i o n s o f D 0, and o b s e r v e d o n l y r y e seeds were c a p a b l e o f g e r m i n a t i o n a t h i g h D 0 concentrations. B l a k e e t a l . (30) found a s i m p l e r e l a t i o n s h i p between the s i z e o f the seed and the g e r m i n a t i o n c a p a c i t y i n high deuterium c o n c e n t r a t i o n s . Larger seeds a r e a p p a r e n t l y more s u c c e s s f u l i n g e r m i n a t i n g because they c o n t a i n l a r g e r h y d r o g e n - c o n t a i n i n g f o o d r e s e r v e s . M e t a b o l i c d i f f i c u l t i e s a s s o c i a t e d w i t h deut e r i u m i n c o r p o r a t i o n a r e t h u s d e l a y e d u n t i l the hydrogen r e s e r v e s o f the seed a r e exhausted. I f the b a s i s f o r the o b s e r v e d e f f e c t i correct the th s p e c i e s s p e c i f i c deuterium e f f e c t e f f e c t of hydrogen-containing y growing t h e embryo i n t h e absence o f the h y d r o g e n - c o n t a i n i n g endosperm. Removal o f the seed s t r u c t u r e s from the embryos reduces t o a minimum the a v a i l a b i l i t y o f ^H. Crane e t a l . (31) s t u d i e d the e f f e c t o f d e u t e r i u m r e placement on t h e e l o n g a t i o n o f e x c i s e d embryos o f seve r a l s p e c i e s o f seeds and noted t h a t a l l embryos s u f f e r e d growth i n h i b i t i o n i n D 0. Added s u c r o s e m i t i g a t e d t h e r e p r e s s i v e e f f e c t s o f d e u t e r i u m on growth. S i e g e l and G a l s t o n (32) demonstrated t h a t b i o s y n t h e s i s does take p l a c e when w i n t e r r y e seeds a r e germinated i n 99.7 p e r c e n t D 0. F i v e i s o p e r o x i d a s e s i s o l a t e d from seeds germinated i n D 0 were shown t o be d e u t e r a t e d , and i t was c o n c l u d e d t h a t they were b i o s y n t h e s i z e d d u r i n g the g e r m i n a t i o n p r o c e s s . In g e n e r a l , h i g h e r p l a n t s show more complex D 0 e f f e c t s than do m i c r o o r g a n i s m s . T h i s i s t o be e x p e c t e d as t h e i r s t r u c t u r e i s more h i g h l y o r g a n i z e d . The r e sponse t o D 0 i n h i g h e r p l a n t s i s a graded one w i t h 60 t o 70 p e r c e n t the maximum l e v e l t o l e r a t e d i n the nut r i e n t medium. A p r i m a r y response t o d e u t e r a t i o n app e a r s t o be s u p p r e s s i o n o f the p r o d u c t i o n o f i m p o r t a n t metabolites including a l k a l o i d s , a n t i b i o t i c s , proteins, c a r b o h y d r a t e s , e t c . Some b e n e f i c i a l e f f e c t s i n h e l p i n g h i g h e r p l a n t s adopt t o d e u t e r i u m have been noted w i t h the p l a n t growth s t i m u l a n t k i n e t i n and w i t h the growth i n h i b i t o r maleic hydrazide. The complex n a t u r e o f h i g h e r p l a n t s makes d i f f i c u l t a s i m p l e e x p l a n a t i o n o f t h e o b s e r v e d e f f e c t s o f growth r e g u l a t o r s i n H 0, and f u r t h e r s y s t e m a t i c s t u d i e s w i l l have t o be undertaken i n o r d e r t o b e t t e r u n d e r s t a n d the phenomena t a k i n g place. The replacement t e c h n i q u e used s u c c e s s f u l l y f o r c u l t u r i n g c e r t a i n f u n g i and molds was u n f o r t u n a t e l y not found t o be a p p l i c a b l e f o r h i g h e r p l a n t s . 2
2
2
2
2
2
2
2
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In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ΚΑτζ E T A L .
9.
III.
Isotope Chemistry and
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193
B i o l o g i c a l E f f e c t s o f Carbon-13
13 C u l t u r e Systems f o r the Growth o f C-Qrganisms. The c u l t u r e o f m i c r o o r g a n i s m s i n systems r e q u i r i n g o n l y the s u b s t i t u t i o n o f d e u t e r i u m p r e s e n t s no complex p r o b lems i n a p p a r a t u s d e s i g n . U s u a l l y , the i s o t o p e i s p r e s e n t e d i n the form of H 0, and o n l y the p r e v e n t i o n o f exchange w i t h i s o t o p i c a l l y normal water i n the ambient atmosphere i s r e q u i r e d . U t i l i z a t i o n of 13 presents somewhat more complex problems, because the i s o t o p e i n i t s most e c o n o m i c a l form i s a v a i l a b l e as 1 3 c o . Of c o u r s e , c a r b o n d i o x i d e i s the optimum s u b s t r a t e f o r t h e p r o d u c t i o n o f p h o t o s y n t h e t i c organisms, b u t i t does p r e s e n t problems i n m a n i p u l a t i o n and c o n s e r v a t i o n . The current p r i c e of t h i p l e t e l y s e a l e d growt i l y t o p r e v e n t i s o t o p i c a t t e n u a t i o n w i t h the e x t e r n a l c a r b o n d i o x i d e , b u t t o p r e v e n t l o s s o f the r a r e , ex pensive m a t e r i a l . Equipment f o r t h e c u l t u r e o f p h o t o s y n t h e t i c micro organisms i n the l i q u i d phase on 13C0 p r e s e n t s no l a r g e problem i n d e s i g n o r c o n s t r u c t i o n . A l l t h a t i s r e quired, i n a d d i t i o n to a s u i t a b l e , t i g h t container f o r t h e a l g a l c u l t u r e , i s a s u f f i c i e n t l y l a r g e volume t o c o n t a i n t h e gas phase a t a r e a s o n a b l e c a r b o n d i o x i d e l e v e l and a c i r c u l a t i n g pump t o c a r r y the gas m i x t u r e o v e r the c u l t u r e . L i g h t i n g and o t h e r arrangements may be a r r a n g e d as i n the c u l t u r e o f t h e same organisms i n 2H 0. A u s e f u l c o n c e n t r a t i o n of carbon d i o x i d e i s around 20%, w i t h n i t r o g e n as a c a r r i e r gas. C u l t u r e of s a p r o p h y t i c microorganisms w i t h C c o m p o s i t i o n s i s o f t e n f a c i l i t a t e d by d i r e c t use o f sub s t r a t e s d e r i v e d from 13c a l g a e . Alternatively, specific m e t a b o l i t e s o r p r e c u r s o r s p r e p a r e d by o r g a n i c s y n t h e t i c methods may be added t o the s u b s t r a t e s o f whatever yeasts, f u n g i , b a c t e r i a , e t c . are d e s i r e d . The t e c h n i q u e s u s e f u l i n many such growth e x p e r i m e n t s have been d e s c r i b e d (33). Growth o f t e r r e s t r i a l h i g h e r p l a n t s from C0 presents s e v e r a l complications i n apparatus design. The system must be c a p a b l e o f s u s t a i n i n g normal p l a n t s growth f o r l o n g p e r i o d s i n a c o m p l e t e l y s e a l e d con dition. O b v i o u s l y , p r o v i s i o n must be made t o a s s u r e i n o r g a n i c n u t r i e n t s u p p l i e s , adequate c a r b o n d i o x i d e l e v e l s , c o n t r o l o f temperature, h u m i d i t y , s o i l m o i s t u r e and l i g h t i n t e n s i t y . Such systems have been c o n s t r u c t e d and o p e r a t e d t o produce t o b a c c o and o t h e r h i g h e r p l a n t s a t l e v e l s o f o v e r 90% e n r i c h e d 1 C, and a r e d e s c r i b e d i n d e t a i l elsewhere (34J. I t i s t o be emphasized t h a t t h e s e growth chambers d i f f e r i n one i m p o r t a n t a s p e c t 2
C
2
2
2
1
1
3
3
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In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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from most p r e v i o u s l y d e s c r i b e d " p h y t o t r o n s " o r " b i o t r o n s " (35). Whereas most s p e c i a l growth chambers i n use a r e by n e c e s s i t y n o t s e a l e d and a r e c l o s e l y l i n k e d t o the o u t s i d e environment t o p e r m i t easy c o n t r o l o f tempera t u r e and h u m i d i t y , and i n many c a s e s f o r a c o n t i n u o u s s u p p l y o f carbon d i o x i d e , the growth chambers d i s c u s s e d h e r e were d e s i g n e d w i t h the same p h i l o s o p h y used where manipulation of radioactive or other t o x i c materials must be employed. Our growth chambers thus a r e provided w i t h g l o v e p o r t s f o r i n t e r n a l m a n i p u l a t i o n s , hermetically s e a l e d bulkheads, e t c . The p r i n c i p a l s t r u c t u r a l f e a t u r e s a r e as f o l l o w s : c e i l i n g and w a l l s a r e 1/4" thick methacrylate p l a s t i c ; f l o o r and s t r u c t u r a l mate dimensions a r e s u p p l i e d by e x t e r n a l c h i l l e d water p r o v i d e d temperature and h u m i d i t y c o n t r o l . Sensors b u r i e d i n the v e r m i c u L i t e or sand s u p p o r t medium a l l o w e d m o n i t o r i n g o f s o i l mois t u r e c o n t e n t . Carbon d i o x i d e was p r o v i d e d from e x t e r n a l c y l i n d e r s , and n u t r i e n t was s u p p l i e d t o each p l a n t through a s e a l e d b u l k h e a d . Alternatively, solid i n o r g a n i c n u t r i e n t was p r o v i d e d from a p e l l e t b u r i e d i n the s a n d / v e r m i c u l i t e s o i l medium, and the condensed water o f t r a n s p i r a t i o n r e c y c l e d and d i s t r i b u t e d t o each p l a n t by means o f an a u t o m a t i c d i s p e n s e r c o n t r o l l e d by a p r e s e t t i m e r . T h i s system was p r o b a b l y most s a t i s f a c t o r y from the v i e w p o i n t o f e l i m i n a t i n g p e r i o d i c a d d i t i o n s o f e x t e r n a l l i q u i d n u t r i e n t and subsequent w i t h d r a w a l o f t h e condensed water o f t r a n s p i r a t i o n . Such an automated system i s c a p a b l e of smooth f u n c t i o n i n g f o r l o n g p e r i o d s o f time w i t h o u t a t t e n d a n c e . A view o f t h e s e systems i s shown i n F i g u r e 2; t h e c r o p i s t o b a c c o , e n r i c h e d t o a l e v e l o f > 90% C. 3
13 . . . . . B i o l o g i c a l E f f e c t s of C Substitution i n Living Organisms. The consequences o f s u b s t i t u t i o n o f 1 C f o r i ^ C i n b i o l o g i c a l systems a r e o f i n t e r e s t as a problem i n i s o t o p e b i o l o g y , as w e l l as c a r r y i n g e n t i r e l y p r a g matic i m p l i c a t i o n s . Numerous s u g g e s t i o n s have been made (36) i n v o l v i n g p o s s i b l e uses o f 13c compounds i n c l i n i c a l d i a g n o s i s and o t h e r m e d i c a l a p p l i c a t i o n s . It i s t h e r e f o r e o f paramount importance t o determine the e x t e n t o f any p o s s i b l e d e l e t e r i o u s e f f e c t s o f 13c sub s t i t u t i o n i n l i v i n g organisms. G i v e n the s m a l l d i f f e r e n c e i n the masses o f C and C , r e l a t i v e t o t h o s e o f hydrogen and d e u t e r i u m , i t would be e x p e c t e d t h a t the k i n e t i c i s o t o p e e f f e c t s and t h e r e f o r e t h e d i s r u p t i o n o f m e t a b o l i c f u n c t i o n when t h e s e i s o t o p e s are i n t e r c h a n g e d , would be much l e s s J
1
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pronounced. T h i s e x p e c t a t i o n tends t o be b o r n out by e x p e r i m e n t a l f i n d i n g s . In t h e case o f u n i c e l l u l a r , p h o t o s y n t h e s i z i n g organisms, o n l y r e l a t i v e l y s m a l l d i f f e r e n c e s a r e seen i n growth r a t e , s i z e d i s t r i b u t i o n o r g r o s s morphology (37), when 13c i s s u b s t i t u t e d f o r 12c a t l e v e l s above 90% enrichement. D i s t u r b a n c e o f normal b i o l o g i c a l f u n c t i o n by -*-C s u b s t i t u t i o n i s somewhat more e a s i l y demonstrated when t h i s i s o t o p e i s s u b s t i t u t e d i n c o n j u n c t i o n w i t h d e u t e r i u m and o t h e r s t a b l e isotopes. .^ The e f f e c t o f a h i g h l e v e l o f C s u b s t i t u t i o n on complex organisms, p l a n t s o r a n i m a l s , i s ambiguous a t present. Experiments aimed a t growth o f l ^ C t o b a c c o and o t h e r f l o w e r i n g p l a n t s a t l e v e l s o f i s o t o p i c en richment around 90% c o n t r o l s , the i s o t o p i c a l l s l i g h t l y r e t a r d e d f l o w e r i n g , the p r o d u c t i o n o f fewer f l o w e r s , a b s c i s s i n g more f r e q u e n t l y , and s e e m i n g l y ab normal p o l l e n . Any a b n o r m a l i t y i n t h e r e p r o d u c t i v e c y c l e i s p a r t i c u l a r l y s u g g e s t i v e , i n l i g h t o f the ex p e r i e n c e w i t h d e u t e r i u m s u b s t i t u t i o n and i t s marked e f f e c t s on r e p r o d u c t i o n . E x a m i n a t i o n o f 13c p o l l e n by the t e c h n i q u e o f s c a n n i n g e l e c t r o n m i c r o s c r o p y r e v e a l e d a h i g h degree o f m o r p h o l o g i c a l a b n o r m a l i t y . Subsequent experiments w i t h morning g l o r y tended t o s u p p o r t t h e s e observations. I n v e s t i g a t i o n s o f v a r i o u s 13c p o l l e n s are s t i l l i n p r o g r e s s and any c o n c l u s i o n s a t t h i s time must be r e g a r d e d as t e n t a t i v e . The s u b s t i t u t i o n o f C at high levels i n highly e v o l v e d a n i m a l s evokes q u e s t i o n s o f g r e a t i n t e r e s t , o f b o t h p r a c t i c a l and t h e o r e t i c a l c h a r a c t e r . Conclusive answers are not, as y e t , e v i d e n t . The growth o f 13c tobacco provided p e r i p h e r a l evidence. In the c o u r s e o f c u l t i v a t i o n o f 13c t o b a c c o an a d v e n t i t i o u s i n f e s t a t i o n o f w h i t e f l i e s took p l a c e i n one o f the growth chambers, which had been i s o l a t e d from the e x t e r n a l environment f o r s e v e r a l weeks. S u p e r f i c i a l e x a m i n a t i o n by a t i m e o f - f l i g h t mass s p e c t r o m e t e r o f the waxes e l a b o r a t e d by the wings o f t h e s e i n s e c t s i n d i c a t e d a h i g h l e v e l o f 13c s u b s t i t u t i o n . The mass spectrum peaks are shown i n F i g u r e 3. I t was e s t i m a t e d t h a t the i n s e c t has somewhat more than 50% 13c i n i t s i s o t o p i c c a r b o n make up. T h i s u n s c h e d u l e d event p r o b a b l y produced the f i r s t highly enriched l c insect. Whether the d i s p a r i t y w i t h the s u b s t r a t e was due t o complex m e t a b o l i c f r a c t i o n a t i o n p a t t e r n s o r whether performed embryonic t i s s u e made a s i g n i f i c a n t c o n t r i b u t i o n cannot now be d e c i d e d . An o b v i o u s l e s s o n t o be drawn would suggest the ease o f p r o d u c i n g 13c p a r a s i t i c a n i m a l s , g i v e n a s u i t a b l e 13c plant substrate. 3
3
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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Figure 2.
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Automated culture of isotopically enriched pfonts in a closed growth system
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Mass fragment distribution of insect wax from normal abundance wax ( C) and C enriched wax 12
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Long term experiments i n v o l v i n g 13c s u b s t i t u t i o n i n mice have been attempted, w i t h i n c o n c l u s i v e r e s u l t s on t h e p o s s i b l e d e l e t e r i o u s e f f e c t s o f the i s o t o p e (38). These a n i m a l s were f e d on a d i e t o f performed l c mater i a l s d e r i v e d from l a b o r a t o r y s y n t h e s i s , t o g e t h e r w i t h added v i t a m i n s , e t c . o f i s o t o p i c a l l y normal c o n t e n t . A t p r e s e n t a l o n g term s t u d y i s under way t o determine p o s s i b l e b i o l o g i c a l e f f e c t s o f 13c i n f e t a l mice tissue (16). 3
13 A p p l i c a t i o n s of C S u b s t i t u t i o n . The w i d e s p r e a d use o f h i g h l y e n r i c h e d J-^C f o r use i n i n c o r p o r a t i o n i n compounds and organisms of s p e c i a l i n t e r e s t has o n l y begun, but a l r e a d y i t i s e v i d e n t t h a t 13c w i l l have extensive f u t u r e uses the i n v e s t i g a t o r ' s s y n t h e s i z i n g the m a t e r i a l s d e s i r e d , whether by the b i o s y n t h e s i s i n a s p e c i f i c organism o r by l a b o r a t o r y organic synthesis. I t i s c l e a r , from a l r e a d y r e p o r t e d a p p l i c a t i o n s i n e s r (39) and nmr (40), t h a t the use o f 13c may w e l l become a r o u t i n e t e c h n i q u e i n c e r t a i n k i n d s o f magnetic resonance s t u d i e s . The s i g n i f i c a n t p r o p e r t y o f l c h e r e i s the non-zero n u c l e a r s p i n , b u t o t h e r a p p l i c a t i o n s depend upon the i s o t o p i c mass d i f ference i t s e l f . S e p a r a t i o n o f c o n s t i t u e n t s by d e n s i t y g r a d i e n t u l t r a c e n t r i f u g a t i o n , as f i r s t used i n the c a s e o f 15N (£1) i s an o b v i o u s p o s s i b i l i t y . Another a p p l i c a t i o n , a l r e a d y w i d e l y e x p l o i t e d i s the use o f 13c in mass s p e c t r o m e t r y , e i t h e r w i t h h i g h l y e n r i c h e d systems or a t n a t u r a l abundance l e v e l s . The p r o p o s e d a p p l i c a t i o n s i n c l i n i c a l m e d i c i n e , w i t h mass s p e c t r o m e t r y used i n c o n j u n c t i o n w i t h c h r o m a t o g r a p h i c t e c h n i q u e s , has a l r e a d y been mentioned (42^) . 3
IV.
Nitrogen-15
Nitrogen-14 n u c l e i have a q u a d r u p o l e moment, which means t h a t nmr s i g n a l s from N a r e b r o a d and d i f f i c u l t to detect. An a d d i t i o n a l d i s a d v a n t a g e i s the f a c t t h a t c o u p l i n g between N and o t h e r n u c l e i i s o f t e n u n o b s e r v a b l e due t o the v e r y s h o r t r e l a x a t i o n time o f the N n u c l e u s . Thus, 1 4 N - H c o u p l i n g i s not ob s e r v e d i n the p e p t i d e bond. I t i s t h e r e f o r e advanta geous t o p e r f o r m magnetic resonance s t u d i e s w i t h b i o l o g i c a l m a t e r i a l s s u b s t i t u t e d w i t h 1%, which has a s p i n o f 1/2 and no q u a d r u p o l e moment. D e u t e r a t e d p r o t e i n c o n t a i n i n g N a l l o w s d i s c r i m i n a t i o n between hydrogen bonds i n v o l v i n g amide groups o r h y d r o x y l groups. 1 4
i 4
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Recent work by Boxer e t a l . (4_3) has l e d t o the assignment o f f o u r 3* N nmr t r a n s i t i o n s i n c h l o r o p h y l l a and i t s magnesium-free d e r i v a t i v e p h e o p h y t i n . An analy s i s o f the c h e m i c a l s h i f t changes i n d u c e d by complexa t i o n w i t h magnesium r e v e a l e d t h a t the magnesium atom s e l e c t i v e l y p e r t u r b s low l y i n g ηττ* s t a t e s . The charge d e n s i t y i n the f o u r p y r r o l e r i n g s becomes more n e a r l y e q u i v a l e n t so t h a t the energy d i f f e r e n c e s among lowl y i n g π π * s t a t e s are decreased. 5
V. B i o l o g i c a l E f f e c t s of M u l t i p l e Isotope S u b s t i tution Taken t o g e t h e r , compounds o f the elements hydro gen, oxygen, n i t r o g e 99% the mass o f l i v i n g p r o t o p l a s m p o s s e s s s t a b l e , r a r e heavy i s o t o p e s . Although k i n e t i c i s o t o p e e f f e c t s i n i s o t o p e s o f elements o t h e r t h a n hydrogen might be e x p e c t e d t o have g r e a t l y l e s s e n e d k i n e t i c i s o t o p e e f f e c t s , and t h e r e f o r e d i m i n i s h e d b i o l o g i c a l e f f e c t s , i t i s o f i n t e r e s t t o c o n s i d e r the r e s u l t s o f s u b s t i t u t i o n o f more than one element by i t s heavy i s o t o p e . The o v e r a l l p r e s e n t c o s t s o f H , " C , 1% and 18Q p r o h i b i t growth o f organisms on any b u t t h e s m a l l e s t s c a l e , i f a l l o f t h e s e heavy i s o t o p e s are t o be i n c o r p o r a t e d s i m u l t a n e o u s l y . There have been o n l y two e x t e n s i v e i n v e s t i g a t i o n s t o d a t e o f organisms i n c o r p o r a t i n g deuterium plus other s t a b l e isotopes at h i g h enrichment. The b a s i s f o r a l l such s t u d i e s must be a t o t a l l y deuterium-adapted organism, t o which the o t h e r i s o t o p e s o f i n t e r e s t may be added i n the c o u r s e o f c u l t u r e . An o b v i o u s c o m b i n a t i o n i s p r e s e n t e d by d e u t e r i u m and 13c; t h i s was c a r r i e d o u t by Flaumenhaft e t a l . a t e n r i c h ment l e v e l s o f 99% + D and 95% C (44). C . vulgaris was grown i n a l l combinations o f 1H,~~2*H, l ^ C and l ^ C . The v a r i o u s i s o t o p i c a l l y a l t e r e d c e l l s were compared on the b a s i s o f s i z e and shape, g r o s s morphology, and s t r u c t u r a l changes i n s u b c e l l u l a r o r g a n e l l e s , as i n d i c a t e d by c y t o l o g i c a l s t a i n i n g t e c h n i q u e s . The i n c o r p o r a t i o n o f had no o b v i o u s e f f e c t on the growth c y c l e o f the a l g a e . The c - H a l g a e grew a t about the same r a t e as * C - H c u l t u r e s , as the 1 3 c - I H grew a t the r a t e seen i n normal 1 C - ! H c u l t u r e s . An e x a m i n a t i o n o f c e l l s i z e d i s t r i b u t i o n , however, r e v e a l e d t h a t i n the change o f i s o t o p i c c o m p o s i t i o n from H C t o H - C , a marked change t a k e s p l a c e , as seen i n F i g u r e 4. E v i d e n t l y , the s u b s t i t u t i o n o f in a d e u t e r i u m adapted c e l l r e s u l t s i n a tendency t o undo some o f the d i s r u p t i v e e f f e c t s o f deuterium, a t l e a s t 2
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as r e g a r d s such a g r o s s and s t a t i s t i c a l parameter as c e l l size distribution. S t u d i e s on s u b c e l l u l a r morphology and c y t o l o g y tended t o b e a r o u t t h e f i n d i n g s on s i z e d i s t r i b u t i o n . The p a r t i a l r e v e r s a l o f e f f e c t s a t t r i b u t a b l e t o d e u t e r ium s u b s t i t u t i o n were b o r e o u t by i n v e s t i g a t i o n o f t h e following features: (1) N u c l e a r s i z e . Nuclei of C - H c e l l s were l a r g e r t h a n t h o s e o f 1 3 C - 1 H c e l l s , but s m a l l e r than those o f the C - H cells. (2) Nudear shape. The n u c l e a r s i z e range seen i n 1 3 C - 1 H c e l l s i s about t h a t seen i n 1 2 C - 1 H c e l l s ? C - H c e l l s have l a r g e r n u c l e i and a r e l e s s s p h e r o i d a l . (3) Nucleic a c i d c o n t e n t . DNA d i s t r i b u t i o n was d e t e r m i n e d by c y t o chemical s t a i n i n g . The s t a i n i n g f o r DNA and RNA o f 13 H c e l l n u c l e i wa found t b i n t e r m e d i a t betwee those of the 1 C - 1 gards s t a i n i n g o f c y t o p l a s m i , again, of C - H c y t o p l a s m was somewhere between t h a t seen i n 1 3 C - 1 H c y t o p l a s m and t h a t o f 1 C - H . The c o n c l u s i o n s i n r e g a r d t o the RNA was somewhat a t v a r i a n c e t o t h e above p a t t e r n . 1 3 C - 2 H c e l l s produced RNA t a k i n g t h e form o f t h r e a d - l i k e a c c r e t i o n s . These s t r u c t u r e s were never seen i n c e l l s made up o f 1 C , whether i n combin a t i o n w i t h hydrogen o r d e u t e r i u m . (4) P r o t e i n and amino acid content. Staining of c e l l u l a r p r o t e i n material d i s c l o s e d about t h e same p a t t e r n s o f i n t e n s i t y and o c c u r r e n c e as was seen f o r the n u c l e i c a c i d s . The s t a i n i n g f o r f r e e amino a c i d s i n d i c a t e d , a g a i n , t h a t c e l l s w i t h t h e 1 C - H c o m p o s i t i o n s t a i n e d most h e a v i l y . Other f e a t u r e s r e v e a l e d by c y t o l o g i c a l s t u d y i n c l u d e d i n d i c a t i o n s t h a t t h e RNA c o n t e n t may be more n e a r l y normal i n C - H c e l l s than i n t h e p r e v i o u s l y much s t u d i e d 1 C - H c e l l s . T h i s o b s e r v a t i o n must be q u a l i f i e d , as t h e d i s t o r t i o n o f c y t o p l a s m i c RNA was so g r e a t i n c c o n t a i n i n g c e l l s t h a t comparisons were difficult. The most o b v i o u s changes b r o u g h t about by 13c s u b s t i t u t i o n , as seen by phase c o n t r a s t m i c r o s c o p y , was i n the t h i c k n e s s o f c e l l w a l l s . C e l l w a l l s o f both t y p e s o f 13c-grown c e l l s were much t h i c k e r than e v e r seen i n C c e l l s , t h e 1 3 c - H showing up as b e i n g even t h i c k e r than those of 13C-1H c e l l s . A l l these observations i n d i c a t e that p r e d i c t i o n o f l i k e l y changes when two heavy i s o t o p e s a r e s u b s t i t u t e d i n an o r g a n i s m i s n o t p o s s i b l e . Simple assump t i o n s o f l i n e a r a d d i t i v e k i n e t i c i s o t o p e e f f e c t s can n o t a c c o u n t f o r t h e v a r i a t i o n s seen i n t h i s s t u d y . The second c o n c l u s i o n , t h a t s u b s t i t u t i o n o f C for 1 C in d e u t e r a t e d organisms t e n d s t o d e c r e a s e t h e magnitude o f the a b n o r m a l i t y produced a f t e r a d a p t a t i o n t o a i 3
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d e u t e r a t e d m e l i e u i s q u i t e unexpected and cannot be explained at present. The o t h e r s t u d y d e a l i n g w i t h m u l t i p l e i s o t o p i c s u b s t i t u t i o n i n c l u d e d a l l f o u r heavy i s o t o p e s o f hydrogen, carbon, oxygen and n i t r o g e n . Uphaus e t a l . (45) grew c u l t u r e s o f deuterium-adapted C. v u l g a r i s i n v o l umes o f 1-2 ml. The r e s u l t s o f t h i s study a r e o f cons i d e r a b l e i n t e r e s t , when t a k e n as a p r e l i m i n a r y e f f o r t , b u t t h e experiments s u f f e r e d from two s h o r t c o m i n g s . One o f t h e s e a r o s e from t h e use o f o n l y p a r t i a l l y enriched (55% m o l e - p e r c e n t 13c), which made any conc l u s i o n s r e g a r d i n g the p o s s i b l e e f f e c t s o f C substit u t i o n ambiguous. The o t h e r weakness o f t h e study was t h a t t h e g r e a t expense o f t h e i s o t o p e s made i t i m p o s s i ble to follow pattern each i s o t o p e were adde t o p i c enrichments o f t h e m a t e r i a l s used were: H , 99%, C , 55%, N , 98%, 0, 97%. The r e s u l t s o f e x t e n s i v e i s o t o p i c s u b s t i t u t i o n o f a l l f o u r major elements i n t h e biomass r e s u l t e d i n d r a s t i c changes, t h e p r o c e s s e s o r c e l l u l a r m e t a b o l i c d i s r u p t i o n and growth a b n o r m a l i t y p r o g r e s s i n g f a r beyond t h a t seen d u r i n g d e u t e r i u m a d a p t a t i o n . The s i z e d i s t r i b u t i o n s of the i s o t o p i c s p e c i e s studied are given i n F i g u r e 5. V i s u a l o b s e r v a t i o n s by l i g h t m i c r o s c o p y gave the i m p r e s s i o n o f even g r e a t e r h e t e r o g e n e i t y t h a n t h a t i m p l i e d by t h e c u r v e s as determined w i t h a c e l l counter. I t i s p o s s i b l e t h a t the "monster" o r g i a n t c e l l s seen a f t e r complete i s o t o p i c s u b s t i t u t i o n may be p a r t o f the a d a p t i v e p r o c e s s and l i f e c y c l e o f t h e s e cells. C e l l d i v i s i o n may n o t always o c c u r . A l a r g e q u a n t i t y o f c e l l u l a r d e b r i s was p r e s e n t i n mature c u l t u r e s , much more i n mass t h a n c o u l d be accounted f o r by t o t a l d i s i n t e g r a t i o n o f the o r i g i n a l innocum, which was about 1% o f t h e f i n a l c e l l mass. C e l l s grown i n 1 H - 1 8 O - 1 3 C - 1 N media appeared t o resemble more c l o s e l y i s o t p p i c a l l y normal c e l l s than d i d those grown i n H - 0 - C - N media, b u t many o f t h e i r c h a r a c t e r i s t i c s were i n t e r m e d i a t e between the extremes. Cytological studies, using appropriate stains f o r v a r i o u s c e l l u l a r components, produced the f o l l o w i n g observations: 2 _16Q.13 .15 c e l l s : A g r e a t e r amount o f n u c l e i c a c i d was produced by t h e s e c e l l s than any o t h e r type s t u d i e d . L a r g e amounts o f DNA appeared i n b o t h the n u c l e u s and the p e r i p h e r y o f c h l o r o p l a s t s . By cont r a s t , the RNA c o n t e n t o f t h e s e c e l l s was t h e l o w e s t found f o r any system. N u c l e i o f t h e s e c e l l s were greatl y e n l a r g e d and had a u n i f o r m d i s t r i b u t i o n o f DNA, suggesting e i t h e r polyploidy or nuclear degeneration. The f a s t - g r e e n s t a i n used t o v i s u a l i z e p r o t e i n s appears t o I
3
2
1 3
1
1 8
5
5
2
6
1 2
1 4
h
c
n
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ΚΑτζ E T A L .
Isotope Chemistry and Biology
1
J 2
Figure
1 4
5.
1 6
1 8
,
j
I 10
I I 12
r
I I I 14 !6 18 2 0 SIZE, MICRONS
I 22
i 24
I 26
I 28
L_ 30
Size distributions of various Chlorella cells after multiple isotopic substitution
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
202
AND CHEMICAL
PRINCIPLES
be d e e p l y and u n i f o r m l y d i s t r i b u t e d throughout the cell. S t a i n i n g f o r carbohydrate d i s t r i b u t i o n revealed t h a t t h e s e c e l l s c o n t a i n e d much more such m a t e r i a l than e i t h e r o f the I H o r H c o n t r o l s o r the H - 1 8 o c e l l s . 2
2
°0- c°N C e l l s . These c e l l t y p e s showed t h e g r e a t e s t v a r i a b i l i t y i n s t a i n i n g o f components and the l e a s t l o c a l i z a t i o n o f a p a r t i c u l a r component b e i n g stained. Young c e l l s w i t h t h i s i s o t o p i c c o m p o s i t i o n tended t o resemble o r d i n a r y H - 1 0 - 1 C - 1 N c e l l s , w i t h round and p e r i p h e r a l l y l o c a t e d n u c l e i which tended w i t h a g i n g t o become m u l t i l o b a t e ; m u l t i p l e n u c l e i were commonly found. C h l o r o p l a s t s o f t h e s e c e l l s appeared t o produce more DNA than t h o s e o f o t h e r t y p e s . Protein staining indicated a larg t f thi material p r o b a b l y l a r g e r than a l s o true of carbohydrat most h e a v i l y s t a i n e d and t h e t h i c k e s t w a l l s o f any c e l l type. J
1
6
2
4
H - 0 - C - N Cells. The t r e n d s i n s t a i n i n g i n t h i s c e l l t y p e i n d i c a t e d t h a t i t s h o u l d be p l a c e d i n t e r m e d i a t e between c e l l s o f normal i s o t o p i c c o n t e n t and those of H - 1 0 - C - 1 % , t e n d i n g t o resemble the l a t t e r more than t h e former. N u c l e i appeared t o c o n t a i n more RNA than d i d the I H - I ^ O c o n t r o l . The most s i g n i f i c a n t d i f f e r e n c e between 1^0 and l ^ o - c o n t a i n i n g c e l l s appeared i n t h e n a t u r e o f t h e c a r b o h y d r a t e d i s t r i b u t i o n and content. Those c o n t a i n i n g the heavy oxygen i s o t o p e showed l e s s c l e a r s t a i n i n g o f the c e l l w a l l s . Perhaps the most i m p o r t a n t g e n e r a l i z a t i o n t o app e a r from t h i s study was t h a t i t i s q u i t e o b v i o u s t h a t d e v i a t i o n s from normalcy appear more and more as heavy i s o t o p e s a r e s u b s t i t u t e d i n t o t h e organism. The t e n dency t o form g i a n t c e l l s , some w i t h a volume s e v e r a l o r d e r s o f magnitude g r e a t e r than normal c e l l s , and a suggestion of increasing s u b c e l l u l a r disorganization, i s q u i t e common. The e x a c t c o n t r i b u t i o n o f each heavy i s o t o p e cannot be a s s e s s e d a t p r e s e n t and must await f u r t h e r s t u d i e s on o t h e r organisms and l a r g e s i z e d cultures. The a p p l i c a t i o n s o f m u l t i p l y s u b s t i t u t e d organisms i s a t p r e s e n t l i m i t e d by the h i g h c o s t o f the heavy i s o t o p e s , e s p e c i a l l y 1 0. The even r a r e r 0 may have g r e a t f u t u r e p o t e n t i a l , because o f i t s non-zero n u c l e a r spin. A s i d e from a r e a s o f o b v i o u s a p p l i c a t i o n , such as mass s p e c t r o m e t r y , t h e r e appears t o be g r e a t p o t e n t i a l f o r t h e study o f b i o l o g i c a l problems by means o f m u l t i p l e resonance t e c h n i q u e s as endor ( e l e c t r o n - n u c l e a r doab l e resonance), with the use of multiply substituted organisms. 1 8
1
2
1 3
6
1 5
1 2
8
i 7
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
INDEX A A values, relative 116 Absorption band, infrared 35 Abstraction, hydrogen-atom 54 Abstraction reactions 167 Activated complex theory 49 Activity, optical 40 Activity, solvent 121 2-Adamantyl trifluoroethanesulfonate .... 178 Additivity, isotopic 20 Additivity, substituent 20 Adiabatic approximation 67 Adiabatic corrections 71,73,75 Aircraft fuel , 91 Algae, H 18 Algal ferredoxin 18 Alkyl sulfonates 17 Amide protons 187 Amino acid content 199 Ammonia-hydrogen exchange reaction . . . 94 Anharmonic corrections 15,105,107 Animals, substitution of C in 195 Aquamolal concentration 120 Aqueous solutions 101 solvent effects 119 solvent structure 126 Argon 82, 110 Arrhenius plots 54 Arrhenius pre-exponential factors ...50,57,59 Asymmetric solvent case 157 Atom abstraction, model hydrogen54 Atom exchange reactions, heavy 74 Atomic spectra 29 Automated culture 196
Bond order 171 Bonding studies, hydrogen 37 Bonding, transition state 169 Born-Oppenheimer approximation 2, 33, 39, 64,105, 142 Born-Oppenheimer corrections 65, 69 Born-von Karman (BVK) lattice calculations 151 lattice spectrum of lithium metal 152 solid 150 vibrational frequency distribution 152 BSVHW model 106, 112
C
2
1 3
Β Barrier double-minimum potential functions, low 38 Eckart 45 one-dimensional 47 permeability of 46 Bases in proton transfer, reacting 170 Belladonna plants 191 Bending forces ...26, 173 Bending frequency 157 Beta calutron separator 80 Bigeleison and Mayer, isotope-exchange reaction theory of 146 Bigeleisen and Mayer, reduced partition function ratio of 9 Biological effects of C substitution . 1 9 3 , 194 Biological effects of multiple isotope substitution 199 Biology, isotope chemistry and 184 Bond extension 173 1 3
biological effects of 193 C reduced partition function ratio .... 22 culture of saprophytic microorganisms with 193 organisms 193 substitution 194,195,197 C1/ C1 leaving group effects 171 C0 193 C - X bond extension 173 Calcium chloride solutions 127 Calcium-isotope-exchange reactions 144 Calutron separator, beta 80 Carbon monoxide, liquid 113 saturated 177 solvolytic nucleophilic displacements on 174 Carbonium ions 176, 179 Cascade ideal separation 89 no-remixing separation 86 simple 85 thermal diffusion 82 Cation solvates 39 Cell(s) degassing 121 double 133 electrochemical 131 iH~ 0- C- N 202 H- 0- C- N 200 H- 0- C- N 202 isotopically substituted chlorella 201 measurements on the 135 quadruple 137 sample 121 Chambers, growth 193, 194 Chemical exchange separation processes 9 exchange system, simple 92 isotope fractionation 7,11 1 2
35
37
1 3
2
18
13
15
2
1 6
1 3
1 5
2
1 8
1 3
1 5
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
ISOTOPES
210 Chemical (continued) isotope separation factors reactions, tunneling in Chemistry and biology, isotope Chlorella cells Chlorophyll free radical species Closed growth system Column, thermal diffusion Complete equation Complex (es) metal organisms theory, activated transition metal Compressors in a gaseous diffusion plant Concerted displacement reaction, α-effect Condensed phase isotope effects Condensed phase modes Configuration interaction Contour map Controlled thermonuclear reactor (CTR) Coriolis interaction Corresponding states, the law of Coulombic force constants Coupling constants effects N-H Covalent substrate Cross over equilibrium Crowding CTR ( controlled thermonuclear reactors ) Culture of isotopically enriched plants of photosynthetic microorganisms of saprophytic microorganisms systems Current reversibility check, voltage 1 4
10 45 184 201 187 196 82 106 37 195 49 37 92 174 101 103 72 47 8 34 108 153 34 118 197 174 23 173 88 196 193 193 193 134
AND CHEMICAL
PRINCIPLES
Deuterium (continued) requirement projections 88 on seed germination, effect of 191 substitution 185 vapor pressure isotope effects 114 α-Deuterium rate effects ..... 177 Diagonal nuclear motion correction 67 Diatomic initial and final states 166 molecular dissociation model 168 molecule spectra 30 molecules 15, 64, 109, 166 Difference method 131 Diffusers in a gaseous diffusion plant .... 92 Diffusion cascade, thermal 82 column, thermal 82 controlled reaction 167 gaseous 83,85,89,92 plant, gaseous 85, 92 Displacement reactions 165, 174 Displacement, solvolytic nucleophilic .... 174 Disproportionation reaction, isotopic .... 23 Dissociation of diatomic molecule 166 effect 173 into free particles 165 model, diatomic molecular 168 photo97,98 Distillation deuterium enrichment by 88 hydrogen 90 process requirements 90 water 90 Distribution functions 4, 8 Double cell 133 Dual-temperature exchange 93, 95 Duckweed 190
D
Ε
«-D rate effects 173, 179 DeBoer's modification of the law of corresponding states 108 Debye-Huckel theory 123 Debye solid 150 Degassing cell 171 Degree of feedom, translational 167 Deuterated algal ferredoxin 189 belladonna plants 191 ethylenes 118 media 190 metabolites 188 nucleic acids 188 organic acids 117 proteins in NMR 187 Deuterium 185 compounds 5 concentration 91,95 effects, hydrogen 112 enrichment 88 on higher plants, effects of 188 kinetic isotope effects 55, 58 organisms in ESR and NMR 186 replacement of duckweed 190
Eckart barrier 45 Eckart function 47 α-Effects 172, 173,174 Effects, isotope (see Isotope effects) .... 131 Electrochemical cells 131 Electrochemical determination of equilibrium constants 131 Electrolyte solutions, VPIEs of some 125 Electron acceptance, inductive 173 release, hyperconjugative 172 spin resonance, deuterium organisms in 186 Electronic isotope effect , 65, 70, 74 Electromagnetic separators 78 Electronic wave-functions 66 Elementary separation factor 84 Energy diagrams, potential 110 fusion 88 level distributions .....101, 105 levels, zero point 166 potential 2, 48 reduced 46 standard molar internal 5 zero point 2, 15, 102, 106, 166
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
INDEX
211
Enriched plants, isotopically 196 Enrichment factor, single-stage 93 Enthalpies, partial molal 121 Enthalpies of transfer, excess 124 Entropies, partial molal 121 Equilibrium constant (s) electronic isotope effect on the 70 exchange 75, 93 isotope effects on 64 isotope exchange 5, 15 for an isotope exchange reaction ...6, 131 prediction 146 cross over 23 exchange of oxygen 24 in ideal gases 1 internuclear distances 75 isotope exchange reaction 17 orthogonal polynomial expansion calculation of 2 solvent isotope effects 11 Esters, pinacolyl 180 Ethylenes, deuterated 118 Exchange deuterium enrichment by 88 effects 158 equilibrium constant 75, 93 isotope (see Isotope exchange) reaction, ammonia-hydrogen 94 reactions, gas phase heavy atom 74 separation processes for uranium isotopes 9 system, dual-temperature 95 system, simple chemical 92 unit 95 Exergonic reaction 167 Expansion, orthogonal polynomial 24 Expansion, Taylor series 10 External condensed phase modes 103 External interactions, internal107
F F-sum rule 36 Ferredoxin, deuterated algal 189 Field, force (see Force field) Final states, diatomic 166 First order rules 20 Flow, interstage 86 Force(s) constants 34,49 Coulombic 153 vibrational 32 fields 34,116 Lennard-Jones 108 state vibrational 163 mean square Ill isotope chemistry and molecular 16 stretching and bending 26 Fractionation, chemical isotope 7, 11, 21 Fragment distribution of insect wax, mass 196 Free carbonium ion 176 Free radical species, chlorophyll 187 Freezing points 122, 127 Frequency distribution, Born-von Karman vibrational 152
Frequency (continued) ratio vibration Fuel, aircraft Functional forms Fusion energy
38 34 91 104 88
G Gas (es) ideal 1 monatomic 109 phase approximation, isolated molecule .... 105 heavy atom exchange reactions 74 isotopic exchange reactions 65 reactions 142 Gaseous diffusion plant 85,92 stages 89 GF-matrix method, Wilson Gibbs-Duhem-Bjerrum integration Graham's law Growth chambers Growth system, closed
32 121 84 193, 194 196
H Ή - 0 - Ο - Ν cells H-algae H - 0 - C - N cells H - 0 - C - N cells Halide constants, methyl Harmonic calculation Harmonic solid lattice oscillations Hartree-Fock limit wavefunctions Heats of solution Henry's law Hybridization tautomerism Hydrogen atom abstraction model bonding studies chloride, isotopic variants of -deuterium VPIEs distillation exchange reaction, ammonia isotope effects, primary -isotope-exchange reactions kinetic isotope effects, primary liquid sulfide-water dual temperature exchange transfer reactions Hyperconjugative electron release 1 8
1 3
1 5
2
2
1 6
1 3
1 5
2
1 8
1 3
1 5
202 186 200 202 34 116 2 70 121,125 120 37 54 37 31 112,114 90 94 49 131, 138 50 91 ·
93 167 172
I Inductive electron acceptance Infrared absorption band Initial state reaetants Initial states, diatomic Insect wax Integrated intensities Intermolecular potential Internal energy, standard molar -external interactions
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
173 35 163 166 196 38 108 5 107
ISOTOPES
212 Internal (continued) partition function 5 Internuclear distances, equilibrium 75 Interstage flow 86 Intramolecular energy level distributions 105 Ion(s) free carbonium 176 isotopic 135 pair 176,179 solvation 39,146 Ionization, photo97, 98 Isolated molecule gas phase approximation 105 Isomers 20 Isopropyl sulfonate solvolyses 181 Isopropyl p-toluenesulfonate 178 Isotope (s) chemistry and biology 184 first order rules for 20 and molecular forces 1 quantum mechanical foundation effect(s) 20 condensed phase 101 for diatomic molecular dissociation model 168 electronic 65,70,74 on equilibrium constants 64 equilibrium solvent 119 kinetic 49,50,185 primary 164, 170 hydrogen 49, 50 and quantum-mechanical tunneling 44 and reaction mechanisms 163 relative tritium-deuterium kinetic ...56, 59 secondary 172 on solute excess thermodynamic properties 127 solvent ...125 127 and spectroscopy 29 virial coefficient 107 vapor pressure (see VPIE) exchange equilibria 5, 15 reaction (s) Bigeleisen-Mayer theory of 146 calcium 144 equilibrium 17 equilibrium constant for 6, 131 gas phase 65 hydrogen 131, 141 in ion solvation studies 146 lithium 132,138 thermodynamics of 131 fractionation 7, 11, 21 light 8 of liquid carbon monoxide 113 neon 4 separation argon 82 factors 10 laser 96 methods of 77 at Oak Ridge through 1972 80 plutonium 79 process 7, 77, 94 scheme 96
A N DCHEMICAL
PRINCIPLES
Isotope separation (continued) by thermal diffusion 81 uranium 79,89 stable 184 substitution 198 uranium 9 Isotopic additivity 20 disproportionation reaction 23 exchange reactions, gas phase 65 ions 135 metals 135 methanes 115,116,117 paleotemperature scale 15 partition functions 6 rate constant ratio 58 solids 148 solute species 151 substitution 20, 29,101,172 on molecular properties 64 Isotopically enriched plants substituted chhrelh cells substituted plants
196 201 195
Κ Kinetic isotope effects
49, 50, 58,185
L Laplacian, mean 109 Laser isotope separation 96 Lattice calculations, BVK 151 constants 151 oscillations, harmonic solid 2 spectrum of lithium metal, BVK 152 Leaving group effects 171,177 Lennard-Jones force field 108 Limit wavefunctions, Hartree-Fock 70 Linear molecules 36,109 Liquid hydrogen 91 Liquid phase 92,193 Lithium-isotope-exchange reactions ....132,138 Lithium metal, BVK lattice spectrum of 152 Living organisms, C substitution in .... 194 1 3
M Magnetic resonance spectra, proton ...... 189 Mass fragment distribution of insect wax 196 Matrix elements for isotopic methanes, F- 115 Matrix method,- Wilson G F 32 Mean Laplacian 109 square force Ill square torque I l l , 113 Mechanisms, reaction 163 Media, deuterated 190 Metabolites, deuterated 188 Metal complexes 37 Metals, isotopic 135 Methane 112 isotopic 115,116,117 Methyl halides constant 34 Microcanonical partition functions ........ 106
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
INDEX
213
Microorganisms, photosynthetic Microorganisms, saprophytic Minimum potential functions, low-barrier double Modulating coefficients Molal enthalpies, partial Molecular forces photo-predissociation process properties, effect of isotopic substitution vibrations Molecule spectra, diatomic Molecule, standard molar internal energy of the Monatomic gases Motion correction, diagonal nuclear Multiple isotope substitution
193 193 38 20 121 16 97 64 19 30 5 109 67 198
Ν N - H coupling 19 N e / N e , reduced partition function of 17 Neon isotopes 4 Nernst equation 132, 141 Nitrogen-15 197 No-remixing separation cascade 86 Nuclear magnetic resonance, deuterium organisms in 186, 187 motion, diagonal 67 shape 199 size 190 Nucleic acid 188, 199 Nucleophilic displacements, solvolytic .... 174
Partition function(s) (continued) reduced 11 translational and rotational 165 Peppermint 190 Permeability of a barrier to a particle ... 46 Phase reactions, gas 142 Photochemical isotope separation processes 94, 96 Photo-dissociation 97, 98 Photo-ionization 97, 98 Photo-predissociation process, molecular 98 Photosynthetic microorganisms 193 Pinacolyl p-bromobenzenesulfonate 168 Pinacolyl esters 180 pK difference in proton transfer 170 Plant, gaseous diffusion 85, 92 Plants belladonna 191 higher 188
1 4
22
20
Ο Oak Ridge, isotope separation at One-dimensional barrier Optical activity Organic acids, deuterated Organisms, C substitution on complex Orthogonal polynomial expansion calculation Oscillations, harmonic solid lattice Osmotic coefficients Oxygen, equilibrium exchange of
80 47 40 117
1 3
194,195 24 2 122 24
Ρ Paleotemperature scale, isotopic 15 Parametrization 119 Parasetic operations 91 Partial molal enthalpies and entropies .... 121 Particle, permeability of a barrier to a .... 46 Particles, free 165 Partition function(s) internal 5 isotopic 6 microcanonical 106 N e / N e , reduced 17 ratio(s) 9,147 of Bigeleison and Mayer, reduced .... 9 C / C reduced 22 for isotopic solids 148 for isotopic solute species 151 reduced 104 22
22
1 3
1 2
terrestrial higher 193 Plutonium isotope separations 79 Polyatomic molecules 32,112 Polynomial expansion calculation, orthogonal 24 Potential energy , 2 diagrams 110 surface, hypothetical 48 function 37, 38 intermolecular 108 Predissociation, molecular 98 Preexponential factors 50, 57, 59 Pressure effects for deuterated organic acids 117 Primary isotope effect 164, 170 Probability vs. reduced energy transition 46 Product rule, Teller-Redlich 33 Protein content 199 Proteins, deuterated 187 Protium compounds 5 Proton (s) amide 187 magnetic resonance spectra 189 transfer 170
Q Quadruple cell Quantum-mechanical foundations of isotope chemistry Quantum-mechanical tunneling
137 1 44
R Radical species Rate constant ratio ., determining step rate effects, α-deuterium Reaetants, initial state Reaction mechanisms Reactors (CTR), controlled thermonuclear Reduced partition function ratio (RPFR) Restoring force
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
187 50,57 176 177 163 163 88 .11, 17 9,22,104 165
ISOTOPES
214 Reversibility check, voltage-current 134 Riboflavin production 190 Rotation spectra, vibration 31 Rotation-translation interaction 109 Rotational partition functions 165 -vibrational interaction 68 wave functions 30 RPFRs (reduced partition function ratios) 9,22,104 S S 2 reactions 169 Salt phases, solid 138 Sample cell 121 Saprophytic microorganisms 193 Schroedinger equation 65 Secondary alkyl sulfonates 178 Secondary isotope effects 172 Secular equation 1 Seed germination 19 Separation cascade 85, 86, 89 factor, elementary 10, 84 isotope (see Isotope separation) Separative work units (S.W.U.) 87 Separator, Beta calutron 80 Separators, electromagnetic 78 Single-stage enrichment factor 93 Sodium chloride solutions 127 Solid Born-von Karman (BVK) 150 Debye 150 isotopic 148 lattice oscillations, harmonic 2 phase model 148 salt phases 138 Solute excess thermodynamic properties 127 Solute species, isotopic 151 Solution (s) aqueous 101 calcium chloride 127 electrolyte 125 heats of 121, 125 thermodynamics 120 Solvates, cation 39 Solvation, ion 39, 146 Solvent activity 121 case 154, 157 effects, aqueous 119 isotope effects 119, 125, 127 -separated ion pair 176 structure, aqueous 126 Solvolysis isopropyl sulfonate 181 mechanisms , 178 relative rates of 179 Solvolytic nucleophilic displacements .... 174 Spectra atomic 29 diatomic molecule 30 lattice 152 proton magnetic resonance 189 of small polyatomic molecules 32 vibrational 31, 37, 153 N
AND CHEMICAL
PRINCIPLES
Spectroscopy, isotope effects and 29 Stable isotopes 184 Standard molar internal energy of the molecule 5 Stern-Lindemann formulation of VPIE 2 Stretching forces 26 frequencies 157 modes 165 motion 167 Substituent additivity 20 Substituted chlorella cells, isotopically .... 201 Substituted plants, isotopically 195 Substitution «•C 195,197 deuterium 185 isotopic (see Isotopic substitution) multiple isotope 198 Subsrtate, covalent 174 Sulfonates, secondary alkyl 178 Symmetric solvent case Symmetry number ratio
151 11
Τ Tautomerism, hybridization 37 Taylor series expansion 10 Teller-Redlich product rule , 33 Temperature dependence of tunneling ... 50 Temperature exchange, dual 93, 95 Terrestrial higher plants 193 Tertiary carbonium ion pair 179 Thermal diffusion 79, 81, 82 Thermodynamic properties, solute excess 127 Thermodynamics of isotope-exchange reactions 131 Thermodynamics, solution 120 Thermonuclear reactors (CTR), controlled 88 Tight ion pair 176 Torque, mean square I l l , 113 Transfer effects 158 excess enthalpies of 124 proton 170 reactions, hydrogen 167 Transition metal complexes 37 probability 46 state 173 bonding 169 theory 167 triatomic 166 vibrational force field 163 Translation interaction, rotation109 Translational degree of freedom 167 Translational partition functions 165 Transmission probability 44 Triatomic molecule, linear 36 Triatomic transition state 166 Tritium-deuterium kinetic isotope effects, relative 55, 58 Tunneling in chemical reactions 45 and kinetic isotope effects 48 quantum-mechanical 44
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
215
INDEX
Tunneling (continued) temperature dependence of Wigner
50 56
U Uranium enrichment 83 Uranium isotope separation 9, 79, 89 Urey-Rittenberg formulation of isotope exchange equilibria 5
V Vapor, pressure ( s ) isotope effects (VPIE) and changes in mean-squared torques for deuterated organic acids hydrogen-deuterium of some electrolyte solutions Stern-Lindemann formulation studies theoretical basis for of isotopic methanes of the neon isotopes Vibration (s) frequencies molecular -rotation spectra Vibrational constants force field, state frequency distribution, BVK
101 113 117 114 125 2 10 10 117 4 34 19 31 32, 75 163 152
Vibrational (continued) interaction, rotationalmodes of cation solvates spectra Virial coefficient isotope effect Voltage-current reversibility check VPIE (see Vapor pressure isotope effects )
68 39 ....37,153 107 134
W Water distillation Water dual temperature exchange, hydrogen sulfideWave functions adiabatic corrections with electronic Hartree-Frock limit rotational Wave packet
90
Wigner tunneling Wilson GF-matrix method
56 32
93 71 66 70 30 44
Ζ Zero point energy (ZPE) approximation difference effect levels Zeta constants
In Isotopes and Chemical Principles; Rock, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
2, 102 106 15 118 166 34
