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!" # $"%'&
(
i)
ni RT , V
EY$?$?1#1hN:F^EY+>5,Q#$.F0Ac&:1Z;V)l5v)#X>N:AWF0H0:$./8AW)e&:H8$.N:AWA & d pg¢F8 !1@)=C,+>AW$.&TAW&i/8&:EV$P(.-092C.1YRS3w1#&:+>; $.A%C7ª361@"%5,/8)#36$?AW1Z&:92N41k1!R
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pi +* V
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p
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pi =
i
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pi = p
n=
X
ni ,
i
ni = pxi , n
xi = ni /n
i
µi
µi (T, pi ) = µ0i (T ) + RT ln(pi /p0 ),
pi
)
!N8O'+>&:EY;BAW& AWN:EY$,5,F0&Y; -0/i),N:+>AWH8(*$.)4AW36&:$.Ac1#AW&:+'; 1hF8#-8N:/25,1@$n; Qs/8),&:+>;i$.QjAW&YN:-0AWAW&:H0$.$.AWAc&:1>$.pA%1Qgd 1#-8/8:(.(\$.)#AW/8&i&:$'(*NG)=F8+.1!R-0&:/2;1hb 36Ac1hF8#N p.+.)e-0/21hF0&:;V5 !" # #
p0
µ0i (T )
pi = pxi
µi (T, p, xi ) = µ0i (T ) + RT ln(p/p0 ) + RT ln xi ,
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r
Y
y1 , . . . , y r yi =
Y
∂Y ∂ni
,
T,p,nj6=i
86
i = 1, . . . , r ,
T
(.R
5=90N2EYO5V-8N:C.EY5=RS1#;¯+>$.A%$'A%C!+.M3&:rC>- (*)#EM1 )#bx/EY5,$?; 1#N:EM-2q1>phQ'_4-0&w/21@"%AW"WN:)#EY$'X,&e5Vd ;
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m
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m=1
Y =
r X
ni yi .
i=1
Y = VP yi = vi V = i ni vi ni vi
vi
i
vi∗
i
Y
T
p
YM
Y
M
=Y −
r X
ni yi∗
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)
r X
ni (yi − yi∗ ),
r X
ni (vi − vi∗ )
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i=1
yi∗
VM =V −
r X i=1
ni vi∗ =
i=1
;<)#X>43F06:/8A)e$?H8N:1@Ag; EMpn1hF0F^1@;
V
T
P
p
∗ i ni vi
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QB(?1@/8&:92901@;<;<)=&:/0+>905,5$?/81@&:;B90;
0
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0
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!#
!1(
)
! (
0
2
n1 , . . . , n r
)
G = U − T S + pV.
*
G
F = U − TS G
86
dG = −SdT + V dp +
*
G = H −T S
*
r X
dU
µi dni .
i=1
G
)
µi =
∂G ∂ni
,
i = 1, . . . , r,
T,p,nj6=i
µ1 , . . . , µ r G
)
G=
r X
ni µi .
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GM =
r X i=1
86
ni (µi − µ∗i ),
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5
86
GM = RT
r X
'3
0
5
(
GM
ni ln xi = RT n
r X
xi ln xi .
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i=1
SM
) S = −∂G/∂T
S
M
VM
VM V = ∂G/∂p
)
r X ∂GM = −nR xi ln xi , =− ∂T p,n1 ,...,nr i=1 M ∂G = = 0. ∂p T,n1 ,...,nr
C!)#"W&:Axp(*)#$.AWp>&:3w361@AWY_N&:JS$nQ./2$.1@C!"%(.NGF^Ac&ª1 ;BAW&Y-8EM1@$.Ac1B+>"c1/p5>1TN8OT&:$'36&:AW90&: e"%C!Ac1K)eH83UN:Ag&:3636$'5=:$'/0)e90-0EYEM$? 1 *
H = G + TS
U = H − pV
H M = GM + T S M = 0,
(./8&:90;BEY5{(\&:N:92EY1@5=/0;®Q.90E436& 5=C!A6)#(\90)=EY5=+s-8/2-01@/2"%1!AWR
H05,;V;µ5N:Jx1@AWH8C,$./4AWp*&:$._4&
SM > 0
HM
= 0
xi < 1
VM =0
)
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5 98)##_X>YF0$':&{/8)e)#H8IX'-8EM1@d 90571@3 /8)#(.;B90AcE41#&Y-0-0/Z/090X.E4&:92$.1@CAxptEY1k;B(\Ac1@)k$n5rEY;B)#AWX>&YF^-0:EM/81@)e$.H8A%N:QPAEMA 1h&F^JL;V&:C=Q#/0F8Q 3N:AW&: n(.1@"%$'EYAW&:&q !)D+.)e36-8AcC!@)#_4$?&t1!-8R
AW5,U; E6d X.921@C=AW&:;`)=+.+>EYAc1!R
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UM = 0
V
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Q#&F0$.&6AWE6&6N:;lEM#1D-037/8&YN:/EY5=Ce; /898&:#_ !Y$n)5>-N81@OB;<-0C'&: !RS1#).+>p $.N:A%EYC!5s43 A%$.-2$'t&:), !N:)EY5=-0C'36RSAW1#H8+>N:AW$.&UA%C#981>)#EYdU98¶#_Y)#$.$.AcAW1@&:"%3$'1@&e_lph/8F0) &Y+>&:$n$?/01@98C )#(.N:AcEM1 #Q.-0/8C'&YRSN:1#EY+>C=Q A 36$?1!EYR
985>)eN8-OgRS1@p!X,/SFY5!depgE4$?&61]36/8EY&G eFD"WY-2+>1@;
1'3'015
*
86
T
p
i
µi = µ∗i (T, p) + RT ln xi ,
)
!p+>1EYAW& F0&: !)Q¢RS1@F^;B&Y-0C,/lAW&:(\;a)#/8&:;<$')#NG"WF8)41!36R&:5=; ; dN2O'|r&:;BAW+>AWEYN:A%EY;i$n55=;36AWN:YEYNe5=p -0_4/8&<&: !3`)98-0)#C.EYRS1#/3U+>$.)#90A%CeE41& +.)e-0C!)#$?1!R
5=; EM1@"W&:_Y5k/5,"%C!))=+s/8&:;B(\&:921@/0Q.905!p N:AWH0$.AW&:$.Ac1A6Q¢RS1@;BCe1 µ∗i (T, p)
i
xi
µi
;<"W)436)#"W5>)4N83wO&: !A%$.) $n5>bxN8/8O]&: !-0)C'-0RS1#C'+>RS1#$.+>A%C!$.43A%C#q1>d.pe $.AWN:&EY5,EM361@"WAW&:H8_YN:5AW&$?$?1@1@/8"W)#&:;B_Y5BAc1#(.-0/U90EY)=5B+B/Q¢5=;yRS1@;B(?C!1@;B43 AW:;5,_
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35
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5
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!
T = const p = const 86
Y
dY =
r X
T = const p = const.
yi dni ,
i=1
dY =
r X
(yi dni + ni dyi ),
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r X
ni dyi = 0,
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xi dyi = 0,
)
dY
T = const p = const.
i=1
*
n
T = const p = const.
i=1
r r−1
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Y
r X i=1
xi dµi = 0,
G
T
p)
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)
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5'.1$# 5
*
)
T
p∗A
B p∗B
A
B p
A
*
p = p A + pB A pB
pA
B
)
pA =
pB =
xcA xcB xcB = 1 − xcA
p∗A xcA , p∗B xcB ,
)
p = p∗B + (p∗A − p∗B )xcA .
*
xgA
pA =
A
xgA p
)
w3NY)1@ e(*A)N:AW(*&Y),N:E:+.bx-0(?/21@1492361
xgA p/p∗A ,
)
p=
*
p∗A p∗B . p∗A + (p∗B − p∗A )xgA
*
X
Y
e
!" # $"%'&
T = const
%&('
p
)!
p∗A
&+*
"!$#
p∗B g xcA (p) xA (p)
5>-0Q.$'&:Ck~>d%
xA = 1 xA = 0 % &# . % "0/ 21(/ /3. 54$6"% 2/ 07 % % 98:1$/ +; <)/ , 9. 0 />= $1 436% ?1+/ + =+;@. A 1 9/ .>=+/ 4 3.B=3. 0; #DC !E1 A +F4B # 1 !" # HG+/ 1(/ H. " = " 0 ; 3 G G # !I1# # F4 %KJ! # 0G ML # 0/ $/ 1+/ H .>=04 3 8 % % HG A c xA c # O 1+/ " =Dp;@. B4 3. x = x (p) N . P; # <1+/ 3. $Q / $" # S R A TU'V L A # ; #< / 9 1+/ .>=4 3 ?1$ / HG W R / $ " # X R Z Y $ U ' U 3 8 +; %K J! # % 8 g g g c , x .>=A4 = x[ 1 A (p) 1+/ %K 7% \ ; # ; #x] A6 3x 8A(= % % M [ ; #] 6 3 8 (=^1# / 6 $1 3. # K = % %9 8:1# / (; % 3. % & 3 8 8 _ K J! # ∗ ∗ (1 p #=5 4$const ` . p A 1 a p/ B .>=+/
T
q 8 ?¡£t¤8¤¥ *¡ ¡¥ 8 ¢ NY)1q(.)#90 !E4nR&:*36EMAW1#+>N8Q#O'F0)M&T3w;<1@$.)=AW+.&ª&:-0"AW6;B;BAW&YAW-0&YEM-81@EM$.1@A%$.$,A%57$l90+.E4)e&Y-0N:C!EY)#5=$?361!AWR-0&G/FY5>dwN8O³UEM981##_-8Y&:$.; A¢-0AW/8& ),98+#_Y/8$.&: !AWNY).& p -21@E4&Y$?)#/01«98)#/(.5=A%"WA&{"%Q.+>XQ._4E^&eF8p1Y36_4&{AW-0C!(.)l98)MN:3wE4Y1#H8+>N:EMAW)4 3w+.&GFt)«;B$'AW)4&Y36-0EM5>1@N8"%O$')eE^H8F0N:14A36N:AWAW-0&YC\N:pEY5!F8d1@CE^F81Y36AW-0C!) 9b
dc
Oe
Qf
hg
0F-0¶1@$')@C{)eF0H8YX?N:N:Aw1@AW&U98;<+>JQ.E4),)$.+.C!&:3 NG"W)MF^RSA=3U1#$?-8&G$'F1#)e+>;BH8;BN:AWA Ac&Y1@-0;B98EM)41@AW3U&Y$.-0&GA%EM$nFY1@p5s$.A%+.$,)e5jpM-0C!3690)#E4(.$?&Y98N:1!)MEYR3w5=&GFY1#36d+>AW¶-0EM/81D)#&G$.FD-8AWAW98&:#_3w3sY1@$._]NYAc&: JS"%QBQ.-0$.AW)#
0
5
1)
!#5
YE
-980EM#_1@Y$.$.AWAcNY1+>+>"c"c11ª;BJQ.AW$.&YC!-0EYNGF^$.A'A%$,;B5AW&Y+.-0EM)e1@-0$.C!)#Ac1>$?dn1!| R&GFZEY-236Ac@EYEY$?C=1@Ql$'Et&T/365,; 5=-8/2+.1@& 98'N:$.EY5A%Q#F^)#&:C,;i98&Y5H0"%JAWQ.IT$.C,/8 b NGF0ª$?1#+>;BAc1@98)43w+>"c136AW&:"%C!)eH8N:A F81@C!) !" # \/ % 3.
86
Y
!C=+>AW&GEYF AW& -21@;<&GF /8F0&:&Y;B-0/(*JS&:Q.92$.1@C!/0Q.NGF090DE4&e;BpgAWN:&YAW-0H0EM$.1@AW$.&:$.Ac1qA%Q+>"cA1q-0;BC'RSAW1#&Y+>-0EMEYAW1@& $.F0A%$n1@5
M Ydosk
g = G/n
M g E = g M − gdosk = g M − RT
1 (*)#$.AW&:3w1@_ 36AWYN
X
gM =
i
gE =
X
X
xi ln xi ,
i
xi (µi − µ∗i ),
xi (µi − µ∗i − RT ln xi ) =
X
xi µE i ,
!N8O'+>&:EY;BAW& AWN:EY$,5,; bx/8&: !)q-0C'RS1#+>$.A%C#1>de"c1F^&Y$?-81#/B+>$?;B1#Ac+>1@;B98)MAc3U1@&G98FY)4p!36;<5=;)#"W)43w(*)#&GF/8&:&:$'$nNG/0F0981!)#R(.&:;A%Axp )#X>F0:/8)eH8N:AgAg&:$n/21@"%(.A%A;l1@;V5!p¢),+>(*)436AW&Y+>$.AW) i
∗ µE i = µi − µi − RT ln xi i
i
)
M sE = s M − s M dosk = s + R
)#921@E
p
v E = v M hE = h M
p.X*)
A
X
xi ln xi ,
i
M vdosk = 0 hM dosk = 0
d
.+N:¶EM&:)#1@;°/8;V&:$'F^5{&:NG !F0-01!)V5=R;V3wN8X*1@O'90)#&:/8"W;B)e&:;H8AWN:N:A*EY$,375d (*-0\&:C')#36RSX?1#$n+>1#5=N:$.; EYA%C#5,1]-0;i/21@5!;B$.p*AWAW_4&Y& &V-8EM)=361@+>$.5=$.A% !AW$n&Y),5!-0+>AWp &:$'$.&Ac1pgF0&Y,)#-03wC=/981@&Y3690H0/8(."c)e1@98H8;i)MI 3w5/81#+>36)#E4EYEY&: e$?$."W1hAW& bb 1@C=/5=36$')eH8N:Axp p>E4+.& '$.AW)M3w1@$'&GF F81@C!)
"0
&
!- "
(
0
01!#
5
"01'& (!
µi
0 *
µi
ai
86
µi = µ0i + RT ln ai .
&$'&G:F:EYp=3698#_5,Y;B$.AcAW1@NY98D)M;B3w1DAWY36+>EYAW&:5 "%C!)eH2IA d=(*| )ª(.)#98 !)e#"-0/0%$'Ql)e)#H2C=N:98Ag&Y36H0"c5,1>X*p##936-0-0Ce/21@1@"%$nA.Q "W)#)= n+>1@$.905,AW&Y/0-0;BAW&:AW$.N:E:Ac1 b EM3w1@1@"W$,&:_Y5B5736)=5,+ !)=C!+')#D$.&:C=C>98-0&:(\/0$'&:&G90FZ5=;<-05=&:/0$nQ?/21#1@NG/8FG)#Aª921>&:dC=-8(\1D&:90(.5=90;<EY5,&:C'$nRS/21#1@+<"%$'-0&G/2Fl1@$'AU&:F0;y&Y-0)=/l+>(*$.),AW&Y+>-05=AW&:C=$./8)@Ac1 b ;
!" # $"%'&
µi
ai µ0i
µ0i = µ∗i
µi = µ∗i + RT ln xi ai = x i
86
γi
(.N:EY905,EY5<$.$.N:A%EYC5,; 1@C,/375,36;B$'AW&Y)e-0H2EMN:1@A$.EMA%1Y$.36AW&AW&:+.92)e1]-0C!N41!)#RS$?1!RA%&G$>F JL)#90;l1#NGF^d *)]1@Qn),3+.1@N2On_Y;V5="W5!&:$.p>A%_4Q&t3t)=+{-8(\+.nR)@cbb -0C!)#$?1!R)eH8N:Axp.;l1@;i5]X\)M36AW&:; ai = γ i xi ,
γi = 1
*
)
µE i = RT ln(ai /xi ) = RT ln γi .
|})# !#"%$')eH2N:A EM1@"W&:_Y5),+-0C'RS1#+>Q ;BAW&Y-0EM1@$.A%$,5!d γi
|$.AW&:$.+>AW36&:; Q'-0C.N:EMRS1##+>-8/0$.C!A%C!)436)M3U5=&G; F;B-8C.AWRS&Y1#-0+>EM$.1@$.A%CeA%$.1 AW&$?+.1#)e+T-0C!98)#)#$?EY1!/3UR&G)#FV98EY&:36; Ac@A¢E4F^&:&:Cj !)Z;B-0AW/8Y:+>_4EY&:5j$.AWN:&:AW; H^b 3P98)#EY/3w)#90E4&i)#C=98&YH0"W)#$n5F0&Y-0/6(.90E4&:Eq(.92143U) 1#)#Q."%/21
1(
$#
01!#5
5 5
5'.1$# 5
i
pi = p∗i xci .
)
|EY36Ac@(.E490&:EYC\5,p(?1#E +>/C=5=Q¯;;B_4AW&T&Y-0Q¢EM1@RS1@$.;
aci
pi = p∗i aci = p∗i γic xci .
f_4&:1@;VC{5]36(.AWY90E4NB&YN4+.1!-0R
/2C!14)43636AWI6A%/8F8&l1@C!N:AW) H0$.AW&:$.AW&(?1@905T$?1#-05>NY)#$'&GFª$?1#+K98)#EY/3U)#98&:; ;<)@b ±L~>d% !² 36$n¶5>)#AWYN8$.NOvAW&:3w),$.+.1@AWN2_ &*OnF^5=3t&Y"W-8&:-0/g(*$."%AcnRA%1#$.N8N:AWO)4EY35,),$.6+$.JSA%Q.C=(.$.A92C!1Y1@NG3wF0C=1 /5=361#$'d )#)e{Q.H8N:"%43/2Ai61>$?pA%;V1 !5q+>)#5 ) !nR+.),"W+'EM&:1@1@_Y/0"W5!&:$.p _MAWN2r),O+>)=(*1@+"%)4X*36)6-0AWC'&YQ#RS+>F01#$.&:+>;AWQg).b pp (*| )436(.5=90EY_45,&GFU(?1#1@+>"%X\C=)VQl(\+>)#Q.$._YA%5>_4N8&GOZF "%)=A%$.+.A%N8A¢O,(.5,98"W)e&:-0ol/8&GEMF 1@"W),&:+>_Y(*$')4)e36H8IAc1#N:+'AWH01h$.F8AW#&:NY$.&GAcF 1q(.),92+B1436-0C'Q RS1#+>1#Q<)#Q.(."%90/2EY1>5 d Q';B-0AW/2&Y1@I"W)#;l$'&G1@FTC>-0/8A%&:;V;BQ.(*; &:921@1@/0"%Q.X*90)«E4&«;B$.A%$.AW&A%;V;iQ.;}Q'-0A+>Xn"c15>Ij(*&:;<36)#$'$'&:)# !/8))#$.-0AWC'N:EYRS1#$?+>1>Qgpªdw"W&Y|vN:E«M3U;
)
c c c c p = pA + pB = p∗A γA xA + p∗B γB xB .
xcA
p
p
86
gE
g E = gAB (T, p)xA xB ,
gAB
/-05>$.EM1@N8$nOg"%5=$'p*;)e1BH8IlE^$.F8AWN:1Y&tAW36&YF^N:AW&YEY-0-8C=5{/AW&:)#3 ; X\&Y(*pN:&:C=$'36/8&V$n#9385=&j; ;B36EMAW5>&Y1@-0-0C=EM/898:1@&Y(.$.-0Q#A%AW$?&BF0&1#-0N2C'O3 RS1#+.+.;B)e43-0AW&YC!-8)#EMA$?1@1!/8$.&:R
A%5=;B$?N21#(\OgN8&:p!O92F01@&Y90/0-0E4Q./ &Y94$.N:dEYAW¶&:5,;B3690EYAWAW5,&-0bbb C'-0EMRS1@1#"%+.$')4)e36H8N:5KAn+>(\Ac)#1@$. eA%92_41@&G; F /8&:Jx;B1@E4(*)4&:36925T1@/0+>Q."c901Z5 +>3U=Np:8EYO3wN:1@AW$'&YN:&GEYFg5! !p\#9C,0$?/8#968/8&V&:;B;l(\1hF0&:l921@"%/0Q.Q.C!92it;BC,90AW5,&bb /C'5>RS1>N:EYp $?/S>FYdUp=(.-8C.90E4RS&Y1#+.+>-0$./2A%1YC=36A Ac;B1 AW&Y-85=EM-M1hd,F0~>kd>-0d=AWK¶)M3 365,+._4)M&G3UF )#"%$n5>N8AWO-0/0$.(.AW98&G)#F0&(\)#F0&Y98+>NGF8$?1#1DN8OgJx1@d EMª1i1@N:/8AW)@&bb ;BAc1#-0/D(*)#$.A%_4&GF -8C.RS1#+>$.A%C=A$.AW&;BAW&Y-0EM1hF0]-8AWi3(*&:36$n5=; EM1@C,98&Y-8AW&
$%.
)
Tgkr
!#
5$ 0
'& (!
Tgkr
Tgkr
-$n0C'5>RSN81#O¯+.43-0`C.RS1#A +')#1#X'N2-8Og&:d9036{Q#)#F0&:_Y;V"%A%3w513tF^-0&Y(*-8nR/
AW/8-0&:/0_k$.AW&:-0$.5=/0AW&
Tdkr
p = const
:
Tgkr
0 β
a
(
α
:
α+β xβA xVkr A
5>-0Q.$'&:C ~>d
xα A
xA = 1 xA = 0 % % 98 % % ;K=6 !0; # 0J!%h1 F4I. 1 9/ 8 ) ! # % @. 3 8 #!/>= = 8 Tgkr T < Tgkr [/ / ; !K7 $ " % 3 8% & α β +
>. =(/
! Y & > + , $ , 5 ; j E w 3 @ 1 0 9 . Q . $ ! C M3 w 3 : & 6 3 ' $ : 0 / 0 9 Y E ' $ G & T F 0 2 / @ 1 . X % A % " ' $ e ) 8 H : N V A . Q ' C S R # 1 > + Q 6 3 W A : & W " @ ) b ;B-0C'AWRS&Y1#-0+>EM$.1@A%$.C!A%)4$n3U5&:+> !36)DQ'F^&Y-0C'-0/wRS1#3w+>1@$.90A%Q.C!$')43w&:C]&GF -0/[email protected]%"%$')eH8;
A+B ∂2g > 0, ∂x2A T,p
g = G/n nA + n B
xA
)
n=
∂g ∂xA
¶-0/8)=)e+.H8ID)#X.;l$.1@AW&/8&:F0901@A%C]A6±S3-0/21@(.X.90EYA%"%5=$'(?)e1#H8+>IiC,;
= µA − µB .
;lQ¢RS1@1@ n;<1>&:p6C{1@;<Xn5)#"W(*)436)#/85«&:$'±SNGN:F8EY1!5=Rq"%AN2C!O')#&:$';BNYAW&:N:$nEY/0$n925 1#NG-0F8C.1RS1#/8&:+> !$.)]A%Ce-01C.RS981#e-+>$.RGp A%Ce !1!+>²5d*98)e"c1lH8$.;BAW&
A+B
xA
µA − µ B
xB = 1 − x A
κT > 0
∂2f ∂ρ2
f = F/V = −p + µρ
∂2f ∂ρ2
=
T
∂µ ∂ρ
=
T
xA
κT
µA
µB
> 0,
T
ρ = 1/v ) dµ + sdT − vdp = 0
∂f ∂ρ
86
f
= µ.
T
(∂µ/∂p)T v 1 = = 2 , (∂p/∂ρ)T (∂p/∂ρ)T ρ κT
1 κT = − v
∂v ∂p
1 = ρ
∂ρ ∂p
369843AW6YN$,Q#3wF0#1@N90Q.±L$'~>&:d%C4!²tE±L~>d% !²F0p¢&Y36-0/]AW+>98EY43A%;V6$'5!)4p\3_4&q1@_Y3$n1@590Q.3$.1@C,90AQ.-0$./2C!1@)4X.36A%"%A]$')e±LH8~>N:d%A !Q.²C'dtRS1#¶+>)@Q b EY;<36)=Ac+>@5,EM1@$?$'1@&w;B-2AWN:EYEU$,+>5=90N2Q.Og ep'A%3;BA=/5=(*;´),N2O'36),5,+>(?$,1#5,+>;BC=QA=),&:+>$'(*&:)49036 eA%AWAg&Y-0+>3w$.)#AWN2X\O)=(*+>)#$'/8&G&:Ft$'ªNGF8A%1!X.RX'43-2«1i/8"%&:Q.9^X b &:"%;BO')#"%/0EM1>d,|)# !#"%$')eH8N:A*-2V)#$'&tC!)#$'-8&:C=3U&:$'NGF8V+>90Q. eAW&GF EM1#-1#+>5B/8&:9^b ;<)=+>5,$?1@;BA%C=Axd'¶)#$.AW&:3w1@_3w1@90Q.$.C=A-0/[email protected]%"%$')eH8N:A-)#;l1Y36Ac1@$'&i3P36AW&:"%Q κT > 0
T
,
T
(\/8&:)= !+>)98YEMN:1@EY n$.1#A%Ce+>1#$.N8AWO&:$.EAc1k/83&:90;<-0(*)=)e+>-85,#X$?r1@;B-05>A%C=-0/8Axp&:;l3 1@//85=&GFN:EYC=$,-05!Acp@_41NY&]F^&Y$.+>AW5=&$.+>AW&{5>-0-0C=5,Q. e/0$?Q#1@F^"%&:A%EY;iQ>5 b F^&:;i5V¶) F0&:/8 !&G)BFDAW+>-0/05,$. eAW98&:&Y$.-SF^AW&qA 36$?198=I(.:90;VEY5,5jC'+.RS1#)+>EYEMAW1@&q n1#36+>5=$.X.AW92&:1@$.$nAc1 5>N8;BOKAWQ.&Y-0C.EMRS1@1#$.+.A%M3$kq+>d 36Q'-8C.RS1#+=b $.±L~>A%C!d%4)4!36²w5>;
*
)
L± ~>d% !² !-0EM+>1@EY$.AW& Ac1>p4 !+>5=_ 365=921@E "%A%$.AW)M365q3Q¢F^&YRS1@-8;B/;
∂ 2 gM ∂x2A
T,p
T,p
> 0,
g M = g −xA µ∗A −xB µ∗B
xA T = T kr xA = xkr A
*
M gdosk
∂ 2 gM ∂x2A
= 0.
= RT (xA ln xA + xB ln xB )
∂ 2 gM ∂x2A
T,p
RT + = xA xB
gE
gE
∂ 2 gE ∂x2A
M +g E g M = gdosk
.
T,p
κT = ∞
e~ ± 4 E # ) g X D d L ± > ~ % d , 0 ² ² 6 d f « ) # ) Y E ? $ # 1 : N M E > 1 p 4 _ { & ¢ (
R = 5 $
3 . ( . Q . $ ! C : N W A & C=;B90A%5,$./A%5=;lN:EY1@$,"%$'5,& ; EY;B-0/21hAc1@F0$,&V5 -0AWN:VAWH0$.$.AW&YAW&:-0$.C!Ac)#1o'N:(*E4)4&:3U$.)=AW&+>Q#H8N:F8AWH0"%EYA%36$?5!1#N:p¢EYA%$.$'$n&5=EY;B;BAAc1@-R$n)M5s365! !p\Y-0$?/81Y)e3UH2N:&:Ax/ d /"Wf&YR1@-8)DNYC=Q.&:AW&q$'"W&:NGEM nF81#1D1N2-8O'C=A%)490"%$'5,3&:/1@5=;V$.N:QlAWEY&i$?98(¢1!)#²R
EY5=d (.$n92Q{1#-83EM1@$.(.A%Q.Q<$.$?C!1N:AW&V'Q.C,C=90/05=Q?/5>1#NGN:F0EY1#$nN25=Ol; !Y-0-0(./892)e14H836N:AAc1>±p'/0EY_43q&qdnH8)#36(?Ac1h1hbb 8#" #
(∂ 2 f /∂ρ2 )T = 0
q l£t¦£t¤8£ edt|ap6(*)«&:36;l$'1#&G-0FgAW&j;BAWN:&YEM-0#EM-01@/8$.&YA%N:$.EYAWC!&w)MN:3UEM&G#F -0/0C!)431)#X>F^:/8 )ehH8;$.A%Ce1 N:; h;
A 74 3 118
MA = 58 B
3
vB = 80 xB = 0, 45
vA = MB = B
0, 85
)
∂hi ∂p
= −T 2
∂vi /T ∂T
hi
,
! -0C.+>RSEY1#AW+>&]EYA%AW$'&e+.d &:C=- )#EY$?1#N:EM1>p _4&]98#_Y$.AWN:EYC=Q#F^&:;i5«(.90EY5sQ'-0/21@"W)#$,5,; >dC!¶43)#Ce1@;BEM1#AWIe&Yp-0EM_41@&ª$.F0A%&Y$,H05K"%A;l(*1h)#F8/8Z&:$'(\NG)eF01!-0/2R
51#I N8O'&:;BAWN:EY$'&Z3t-0EY5>-0/0C,AWN2Ojp\ !-0+>C'EYRS1#AW& +>$.Azb F^ &Y-0Q./
>d NG*F81!1!RR
#_5Y;iN2O'5!&:p ;B_4AW&N:EY3$'&i;B-8AWC.&YRS-01#EM+>1@$.$.A%A%C!$.M3AW&B+>;l361hQ'F0-0BC'RS(\1#)e+>-0$./2A%1#C!I )43U&GF *
A+B
(*)#/8&:$>b
)
µA = µ∗A + RT ln xA + W (xB ), µB = µ∗B + RT ln xB + W (xA ),
!N:EY+>5=EY$.AW& $.A%C,Aw36AW&:F0"W&Y)#-0;B/36Ac1@AW$n&:Q "W)#;BAc1@$'&:;p?F0p!1@(.C90EY985BM36N:$.EY5=AW&:; _ A pde-2]|jJS-8Q.(\$.nRCncbb _4NGF8&:1@X,A ;B5ZA.p.X,/85.N:&:EYR;B)5="%(\-0A (*&:92&M1@R
/0$.Q.AW)#905$'&ªA'98N:43AWH06$.$?AW&:1@$.$.AcAW1>&idnªn1@A%X.C#DX'-2(\1h)eb-0/2Q.1#IO'&:;i;lQ'1-0A.(.;B90EYAW5Z&YI -0/21!R
5>N8O p >d ;<¢$?)#1@"."W;B& 4AWIw&Y-0&:EM$n1@/2$.1@A%"%$,(.5AWw-03U)#X*)=E4+>+.$?& 6'$.;BAW)MAW&Y3w-0EM1@1@$'$.&GFAc1t3s(.90(*EY)#5,(.(?901#E4+'&Y+>1hF8$.#A%;`N4tEM$?1#1+'F^1@&Y$.+.A%&:Qg$ d =dt)#X>"cF01:/8(*)e&:H836N:A\$';<&GF)#;B"W)4AW3U&Y-0&UEMF81@1@$.C!A%)D$,5JSQ.+>$.36C!NGQ'F^-0&C.-0RSC'1#+>RS1#$.+>A%C!QZ)M)#3UC=&G98F &YH0"W)#$'&-2DN:36EM#E4-0)#/092C!1@)4;B3wA & W (x)
W (0) = 0 µ∗A µ∗B
W (x)
W (x)
T
p
*
xA dµA + xB dµB = 0 A+B
86
A+B
)
L± ~>d% !² !¶+>)#EYC#AW1@& EM1#Iep#p _4&w+>)#"c1 921@Q'E -0/21@p "W)#$n-25>ªN8OJSQ.$.C!A NGF01@N:;BEM#A*-8/8/0&:C!;B)43U(*&:&692)#1@X>/0Q.F0:90/85l)eA*H8N:N:A>AWH0;<$.)#AW&:"W$.)43wAc1>& d +'p?1@$'1&U$?EY1#36-0/8Ac@:(.EYC#$.1@AW&q;BEYA $?±L1@~>"Wd%& 4 !IV² -8)#(\X>&MF0R
:$./8)eAc1hH8F8Iª6;<98M3)#6"W)4$?31@<$.AW;B& AW&Y-0EM1@$.Ac1>d >d X."%AWN:EY5>I;<)#"W)43w&:$,/098)#(.AW;BAW&Y-0EM1@$.Ac1{+>"c1 (*)436AW&:/090EM1{){-0C.RS1hb p*(.90EY5hF^;VQ#F8#Nep _4+>&EYAWF0& &Y-0//8)B;BAW&Y-0EM1@$.A%p $?1<+.)e-0C!)#$?1!RS1>d p ~>d698| &:; +'1@$'&GFD/8&:36;B5=(\$'&:)e92-01@A /0Q.90E4&N:AWH0$.AW&:$.AW&l(?1@d 90*5{1@$?C'1#RS1#+k+'N:1hAWF8&:#C.NeR
pn5=_4;®&)#98X\)#EY)M/363UAc)@hbb EY+'Q#1hF0F8&#NY(.&D921Y/83U&:;V) QKN:1#AW)#H0$.Q.AW"%&:/2$.1>A%p\QgEYd $?1@"W& 4IB-0C'RS1#+>5KN:AW&YN:EY5TA(?1@905{),+>(*)436Ac1hb 2 ∗ vA = v A + ax2B − (a − b)x3B , 3 2 ∗ 2 vB = vB + bxA + (a − b)x3A , 3
∗ v∗ vA B
a b
T p
xA dvA +xB dvB =
0
) xN2 = 0, 781 xO2 = 0, 210 xAr = 0, 009
A+B
p = (p∗A + p∗B )/2
*
4 ~> >dEY $?31h1DF^+>N:Q#AWF0&:&qC'R-0&tAWD-0C'(*RS)=1#+ +>$.-0A%/2C,1!A R
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8#" #
A B
p
p = 1
A
B
360 25
*
A
)
B
∆h∗A
TA∗ = 340 = 20
TB∗ = ∆h∗B =
T = 350
A+B *
A
B
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pB B
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p∗B
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25 ◦ C
3, 3 × 107
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*
B
A
A B
A
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)
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B
A
A+B
Π A+B
A
A
B
Π
m
M
ρ
V
h
T
*
A
B
p
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A xcA = xcA (T )
A B
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8#" #
A
B
A
TA∗
p
A
∆h∗A
∆h∗A
B
353, 2
Ke = 2, 53
−1
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64 100
MB = m=
B
A
A
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A
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xcA = xe
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A
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A B
TA∗
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/
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% 3. % & $= #3. %F4 3 0 / " . 1 9 / .>=+/
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,
xe
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#
#
A TA∗
B
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A B
*
µE A
g E = gAB xA xB
µE B
T p γA γB
A
A+B
B 0, 029
xcA = 0, 2
pA = 0, 018
pB = 0, 023
*
A+B
xcA = xaA = 0, 70
T = 298 gAB *
gAB
T
gAB (T ) = 0, 9RT
gAB
p∗A /p∗B = 3
(
p∗A /p∗B = 2, 5
4~
!" # $"%'&
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&Y/6N:-8EYAWC,5ª&:90$.EYAWAc-05=1/03U$.±&GAW/8&GF&:F^&;B;B"%(\AWQ.&Y&:C#-092EM11@1@;B/0"%Q.$'AW92)@&1 bb H928N:1@Ax/0p'Q.A%9t"W&C=F^90&Y5=-8//65>N:(.EYQ.$n$.5>C,N8/8OgM3d C=905,/5=N:EY$,5=N2OTA.F81@C,AW&D-3w1@90/8)eH8N:A/8&:;B(\&b q °§t¡ t¡£t¤8£ ed ¢ !)=+>$.AW&DEq+.& '$.AWNGF8BN:EM#-0/0C!)4365>N8O{36AW&:"%C!)eH8N:Ag;<)#"W)M365=N2OT;l1@;i5 gAB (T )/RT = −2(T − T0 )2 / T0
f
2
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(
*
86
!+'+>1@$'EYAW&e& p(\)#F0E4&Y)e-0-8/D/21h)#F^X>&lF0:36/8)e5,H8EYN:$?Ac1#ZN:98EY)#5>EYI /3U)#90Qgd¶)#$.AW&:31@d _ 1#-2p 198)#EYA /3U)#90-2Qg p p.+'1@$?1F0&Y-0/6EY36Ac@EYC=AW&:; h;<)#" h;<)#" 36AWYN +>"c1 C, ;l1@;V5 ;<)#"%Axd,"c1 +.)e-0/21hF0&:;V5 N:; N:; >d6|¶)#5=$.(.AW&:983w)431@1#_+ &:Y$n;V/21@5"%(.$?Ac1h1 F^(.AW&:903P/8&:$EY36Acp'@-8E4C#&:#C+ +>"c1N:EY5>-0/8&GFw-8Q.X'-0/21@$'NGFGAxd V = nA vA + nB vB = n(xA vA + xB vB ),
V
v A vB
xB
n = n A + nB
)
m
m = nA MA + nB MB = n(xA MA + xB MB ) = n(0, 55 · 58 + 0, 45 · 118) = n · 85 m = 0, 85
n = 10
V = 10(0, 55 · 74 + 0, 45 · 80)
,
)
V
3
= 767
H = U + pV
dH = T dS + V dp + µdn,
3
.
4~
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A
. T,n
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2
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.
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)
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86
hi =
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hi 6= h = H/n
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n
)
r X i=1
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T,n1 ,...,nr
i=1
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.
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∂hi ∂p
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2
∂vi /T ∂T
,
! 3w+>&GEYF:AWp &B36d 5=C!)#90EY5>-0/21@"%AWH0;V5j+.& '$.AWNGF0
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r X
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r X
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p
r X
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,5R
Xn3r5 N4-01!Q.R
5, !;&:98)MEM3w1@1#C,I98&Y36-0AWE4	-0C.±LRS~>1#+.d M3!²qp!d F^&Y$'H0+."%A&:C>98)#- EY/3U#9;lª1D$.(*AW&)=+>F0&YC,-098/&YH0+."%)eAWIe-8p#C!_4)@& b $.$.±L~>AWAW&Y)#d -0$?C!!²1U)#tF^o')#&YN:X*-0E4/)4)#(*36$')4&:Ac@36 !EY) 5=Q#_4F098-0&V)#EME41/5,N:EMAW"%&:C!1@o')Z"W&:N:_Y3¯E4$'&:)e$. eH8Ac921>Ie1@ped $.$?AWN:1@5 EY)#5=EY/3w3U1ª#9-04AWpDw+>98)#"cp¢17EYN:/EY3UC,5=/8)#"%#9A988&:3°&:; ! e) +.921@)e-0$.-8(*C!AW&MN:)@Rc5 bb $?$.A%1!C#R
1>5,p; 98p )#-8E4/8N:)eAW-0&:Q#o'F^N:&BE4)#-8$nAWi5=; (.92d143U| ) /21@C=1#)#A%; Q."%/2981 )#EY/3U)#90E4&]+.) 98p*)#EY$?(.1@Q'/8)#-0E4;BN:EMAc1#1@-8"z/b +.)D-0Q.X'-0/21@$'NGF^deA.|j98)#-8E4(\N:AWnR&:o'N:N:EYE45=)#$.$'$.&GA%F:Cp (.98p#)#-0(\/8)#)e98-0NGQ#F0F0)#&$?-01@AW"%$')e(.H892N:1YA 3w) &:$'$.)e90-05A?$?&: !1@E:) b 3Uq-0/21!R&GF &:$.905 &: !).p.(.90EY5]N:EY5=; ±E4)#Xgd.EM1#+ dg4!²d RT dxA + xB dµB = 0,
xA → 1
dxA = −dxB
dµB = RT dxB /xB ,
xB → 0
µB = µ ∞ B (T, p) + RT ln xB ,
µ∞ B (T, p)
*
µB µ∞ B
µA
B
A
∞ xB → 0
) pA = p∗A xcA
B
pB = KB xcB
KB 6= p∗B
KB
)
!" # $"%'&
>d y(*)e-0/2A 1#N:A(\)#/8&:$'NGF8;l1!R1@43;Vµ5 N2O'&:;BAWN:EY$n5>N8Oy365=$.A%Ce1>p _4&s+>"c1 *
T =
const p = const
xA dµA + xB dµB = RT dxA + xA W 0 (xB )dxB
!+>EYAW&
)#AEY$?1#N:EM1t(\)=N8O')=+>$?+.>)ed4-0¶/21h)tF0Qn&:;V36EY5 e"WY+>$.AW&:$.A%QEY36Ac@EYC!M3 + RT dxB + xB W 0 (xA )dxA ,
W 0 (x) xA + xB = 1 dxB = −dxA
0-¶E4)#&:$. !AW)<&:3w98431@6_ $?1@$.Ac1BEY$.A%Ce1>p> !+>5
)
p*36AWYNB(.92143w1-8/098)#$?1](\)M365,_:b
xA dµA + xB dµB = [(1 − xA )W 0 (xA ) − xA W 0 (1 − xA )]dxA . P n W (x) = N n=1 Wn x N X
n−1 nWn [(1 − xA )xA − xA (1 − xA )n−1 ] = 0.
1@/3U)]p(*36)#C#AWY1@N]EM1#36IeAWp?&:_4"W)#&V;B36Ac5=1@92$1@_4;V&:$.Q'AW-8&iA63°;BAW$?&YI]1436(\Ac)e1#-0-2/21#1#N8IOEY$.A%Ce15 p !+>EYAW&D3)# !#"%$')eH2N:A EM1@"W&:_Y5),+ A d >d ¢ !)=+>$.AW&DEq+.& '$.AWNGF8B-03U)#X*)=+>$'&GF6&:$,/21@"%(.A%A;BAW&Y-0EM1@$.Ac1B;l1@;V5 n=1
) W (x) = W2 x2
n = 2
W2
*
T
p
86
)
GM /n = xA (µA − µ∗A ) + xB (µB − µ∗B ) = RT (xA ln xA
8/¶-0)#/2)]1h_4F0-2Qn&:1@36;V;<EY5 )e eH2"WN:YA+>$.AW&:$.A%Qj(*)e-0/21#N:A36AW&:"W)#;BAc1@$nQ
+ xB ln xB ) + xA W (xB ) + xB W (xA ).
)
) W (x) = W2 x2
p\p¢)#+.921@)@E b
) xA x2B + xB x2A = xA xB (xA + xB ) = xA xB
GM /n = RT (xA ln xA + xB ln xB ) + W2 xA xB .
=d AWN:EY5=;V5
xA dvA + xB dvB
p*(*),+.-0/21436Ac1hF0#N365=921@_4&:$.Ac1]$?1 A vA
vB ) xA dvA + xB dvB = xA [2axB − 2(a − b)x2B ]dxB + xB [2bxA + 2(a − b)x2A ]dxA = 2xA xB [−a + (a − b)(1 − xA ) + b + (a − b)xA ]dxA = 0.
>
$ ]6 #
X>F^:/8)eH8Iª;<)#"W)M3w1l;BAW&Y-0EM1@$.Ac1B+'1@$?1F0&Y-0/636E4)#98&:;
)
∗ ∗ v M = V M /n = xA (vA − vA ) + xB (vB − vB ),
!+.+>)eEY-0/2AW&1hF^&:;i5 F0&Y-0/)#X>F0:/8)eH8N:Ac;BAW&Y-8EM1@$.Ac1>dM¶)t(*),+.-0/21436AW&:$.A%Q A VM
vA vB
)
v
M
= x A xB
2 2 2 axB + bxA + (a − b)(xA − xB ) . 3
1#-0/8:(.$.AW&ª365=C!)#90EY5>-0/0Q#F^&:;i5]EY36Ac@E4&:C
vM =
xA + x B = 1
p'NY)B+'1hF0&
xA xB [(2a + b)xA + (a + 2b)xB ]. 3
>d \&D36E4)#90Q $?1<&:$,/098)#(.AWq;BpM+.AW&Y)e-0-0EM/21@1h$.F0&:Ac1l;V5V+>"c(\1)6;B(*AW)=&Y+.-0EM-01@/2$.1Y36A%$,AW5T&:$.+.A%Q)e-03wC!)#1@$?90/81!)eRH8&GN:FYA p Q¢RS1@;BC!43;<)#"W)M365=N2O >± [;<)#"
² >± y;<)#"
² ~>d ,C.RS1#+rpN:AWC=&Y/8N:#9EY05!5kp +>"c1 p EY$?1hF0+>Q#F0&:;V5E4&K36E4+')#1h90QF^& E4Ei&:(.Cl92;B143wAWY1 +>EY5]1#)#-0Q.C."%RS/21#1 +.&:;¬N:AW&YN:EY5!p p?p>AA (.-092C'14RS3w1#1]+.&:D;´1@"%(?/8)#1@$?905!1 p p>d36¢5,36$.A%AcCeh1 b p -0C##+
*
S M = −nR
P
i xi ln xi
)
S M /n = −8, 314 · (0, 781 · ln 0, 781 + 0, 21 · ln 0, 21 + 0, 009 · ln 0, 009) = 4, 682
(
xcA
p∗B )xcA
p =
) pA =
(p∗A
+ p∗B )/2 xcA ∗ pA xcA
xgA = xcA
.
) p = p∗B + (p∗A − * c
xA = 0, 5 xgA g ) pA = pxA
p∗A p∗ = ∗ A ∗ . p pA + p B
>d/8!)&YH0N:"%ADAWH0$.Q'AW-0&:/2$.1@AW"W&l)#$'1@&/0;
A+B
xcA
xgA
e
!" # $"%'&
¢N:AW !&Y)=N:EY+>5$.AW+'&1@E$?1(.92F01Y&Y3U-0/&:;EY36Ac@EY1#C,)#AWQ.&:"%; /21±E4)#Xgd ±L~>d !²At±L~>d e²0² "%A%$.Ac1]-0C.RS1#+>Q ±L~>d>M² !N:AW+>&YEYN:AWE:&bx(?N:1@AWH092$.1{AW&:+>$."cAc1K1 N:EY5=-0/5=N2O«A -8C.RS1#+>$.A%)=C!+>M3(\)M36Ac1#A +'1hF0T3 3t/8-0&:(*;BnR(*
AW&:-092/01@$./0AWQ.&:$.90E4A%Q & d>|5=EY$?1#N:EM1hF0#NqEZ±L~>d>M² p.+.)e-0/21hF0&:;V5 ±L~>de!² NEY:nAW$?1@&YC1#N:I EY365ª"%A%AW$.F0+'&YAW1#&-0Ie/3tpJS-0Q.(.(*$.90nREYC!
5{NGAWF0-0DQ'/0$.-8/8/2AW&:&:1@;B$."W)#Ac(\1t$n&:5=N:92;aAW1@&Y/0N:Q.E:N:90bxAW5!(?H0$.d1@AW92&:1 &:$.Xn+>A%Q5ª"c1tF8E4ªN:&:EY36365>$'-05=/:EY5=/0$?90N21#EYOi$nN:-0EY5=C'5=; RSI1#;V+>$.Q'-8A%-0C!C.A%43RS;i1#q+5 d n³U1@"cC°1@(*36&:5,AW98&:)#;V$?5!1rpD36+>"c5=1v$.A%N:C#EY1>5>p-0/8_4&G&KFKF0-0&YQ.H0X'"%A-0/2&:1@$,$'/2NG1@F^"%A(.AcEk1 98(?M31@698$?)41@3$.1@Ac$.1Ac1>³Up "c1@Q'-0A%pQ'$.-21hAW& b EMN:AW1@&Y"WN:&:E:_Ybx5«(?1@),92+«1>p /8&:;Bp'(*+'&:1@921@$'/0&Q.F090&Y5!-0p//836){E4N:)#AW98H8&:$.;aAW&:$.±AWE4&])#Xg$?d'1T98)#"%A%E4$.+>A%AE!d 3t -0(*EMnR1#+
AWdg-0/0 $.!²AW&:$.Ac1 !+.+>)EYN:AW& EY5>-0/8&:F0 !&Y)-0/ -0/8C'&:RS;B1#+>(*$.&:A%92Ce1@1 /0Q.92p (*),+>),$.+.AW-8&Y/2-01YAW36&:$.Ac1hAc1>F8#d NB*1#EM-01 /8)e-0Q#F^N:;VAWH85l$.AW/8&:&:$.$AW& 36E4#9 p 1/0tQ.EM981 <)=+>$.&:AW$n&Y-0/2AW1@&:"%$.(.AcAW1]w(?(.1@90EY98)M5h3wF^;B1@$.A FGAc;i15{N:EY/85=&:-8;B/8&:(* !&:) 921@/0pQ.98B3690dhE4n&:1@$.C!Ac) 1T/8N:&:EY;B5=(\-0/8&:&:92 !1h) b (/8&:;B*(*)=+&:92N:1@AW/0H8Q.$.90AW5]&:$.36AW&:90; E4&:$.Ac1BA*)#;lEY$?1@;V1#N:5 EY;i5F0-05=;VX*)#p."WEM&:1@; /8&:; d K+.& '$.AWNGFGA ¶)e-0/8:(.Q#F8#Ni1@$?1@"W)# eAWN:EY$.AW&qE4&V-0C.RS1#+>$.A%C=AW&:; p>+.)e-0/21hF^&:;i5 !$.+>AW&:EY$.AW&AW&:; )#d.EY¶$?)B1#N:(*EM),1<+./8-8&:/2;B1Y36(\AW&:&:92$.1@A%/0Q Q.98V3690E4&:A $.Ac1
*
)
p=
p∗B (T )
p∗A (T )
+ [p∗A (T ) − p∗B (T )]xcA , p∗B (T )
A
B
xcA
T
xcA =
p − p∗B (T ) . − p∗B (T )
p∗A (T )
p
∆h
pw
pw (T ) = pw (T0 ) exp[(∆h/R)(1/T0 − 1/T )], *
T0
A
∆h
A
A
p∗A
pw
∆h∗A
TA∗
p
p = p∗A (TA∗ ) ∗
*
86
∗
p∗A (T ) = p e(∆hA /R)(1/TA −1/T ) . B
p∗B (T ) = p e
TB∗ p )
∗ −1/T ) (∆h∗B /R)(1/TB
B
p∗A (T ) p∗B (T ) ∗
xcA =
,
∗
1 − e(∆hB /R)(1/TB −1/T ) . ∗ ∗ ∗ ∗ e(∆hA /R)(1/TA −1/T ) − e(∆hB /R)(1/TB −1/T )
/!3U&Y-0)#/90QK/8)K(\)=-0EY+Q.Q'Ce1@-0/2$n1@5"W)#EY$n365=Ac; @E4N:&:AWCH8$.;BAW&:AW$.YAW+>&:EY; 5j/8&:Ag;B-0(*C'&:RS1#921@+./0&:Q.; 92{N:AW36&YN:90EYE45!&:$.d Ac1K98)#E:b &E4&:XnEY5v36AcEY@$?EY1@C,"WQ & 4I-0C.RS1#+ (?1@905r±(.92$?141#3U+ )T98D)#EY1@/"%3w/8)#)#$?981!&:;²pp !+>EYAW&pC!)#90EY)#5=EY-8$?/21#1@N:;VEM51 N:EM1##)#-0Q./0C!"%/2)41 3w&lN:AWH0$.AW&:$.AW&B(?p.1@-090Ce5#+ -0C'RS1#+>$.A%C#1 pg1]$?1#-0/8:(.$.AW&BEB(.92143w1
$ ]6 #
p
xgA
xgA
= pA /p
pA
A
) pA = p∗A xcA
∗
∗
NGFGEYA5>-0-0C.CeRS1@1#$.+>AWQ&),N:AW+>&Y36N:EY985)#/0"%$,Q.5=X{N2Oi(?1@EM901@5!"W&:p¢_Y36$'5=)e;lH8N:Ax1@ph nN:1lEY5,98"%)#A,EY/836&:;BA%$.(\AWY&:N:92Ac1@1/21@$./8AW&YN:N:E4EY&GFY$.pMAW&6/SFY;ld 1!R
5!d!¶)#$.AW&:p43w(*)=1@_+q3w1@90Q.$.C=AW&:; p@_4&w1@90 eQ.;<&:$,/ F0&Y-0/ p 36AWYNª$.(gd'+>"c1 C=lA +.)e-0/21hF^&:;i5 xgA = e(∆hA /R)(1/TA −1/T ) xcA .
ey ≈ 1+y
y
1/TA∗ −1/T TA∗ = 300
∆h∗A = 20
∆h∗A = 0, 027 ∗ 2 R(TA )
−1
≈ (T
−TA∗ )/(TA∗ )2
.
36 5,"c1tC'98RS1##_+>Y$.$.AWAWN:N:5E4&GF.F^&Y-8/g90E436&YN:5=EY$'5,)e36-0EMAWH2#N:NYAW&G& Fg$.C=AWA%&:"%C#361AW&:-0"%/8C=)#Ax(.d $.A>161@90/8 e)6Q.;B;
TA∗
xcA = xcA (T )
TB∗
xgA = xgA (T )
ey ≈ 1 + y
) p∗A (T )/p = 1, 224 p∗B (T )/p = 0, 788
A+B
xgA
) xcA = 0, 486
= 0, 595
A
A
#
!" # $"%'&
K361 AW/0Y"WN&:$ -0/8/8:)_4&:-0$.C'AWRS&{1#+>/0"W$.&:A%$,C Q3 d¶98)#)#EY$./AW3U&:3w)#901@E4_K&j983 )#EY/983wM36#9D$'F^)4&Y3-81#/+>98E4)#&kE4N:N:AWAW&:&Yo'N:N:E:E4bx)# n$,1@5!E!pp p'-0Ce#++.)e-0/21hF0&:;V5 B
xcB 1
!+>EYAW& #) EY$?" h1#;
∗ vA
VA
= 0, 018 nB /VA = 1, 25 × 10−4
2
K+.)e$'-0/2)#1h_MF0#&:N ;V5qN:AW(.H8$.90E4AW&:&:$.EAW&-0/2N:1!EMRS# -0/0 C!&:)4$.3w90&w5/0&:"W !&:$n)lQV+>$?"c1l1#+q/0"W(\&:$n)M36Q{AW3&:90E43wN8O.),$.+>AcEY AW&e3Up )=+>5 p f9 fg9 $.!&YAc1-8/ /0/8"W&:)q$,N:Q AWH03$.AW&:3w$.),AW& +>EY$.AWAW&D&:EY3PX*Y/8+>&:;B$'&(*+.&:92)q1@Q./0Q./09090EYE45=& ;l1@$.Ac1Dd EM1!R)#_4)#$'&: !)-0/8:_4&b >dt"c&:1$.90985 )#EY&:/ !3U))#90±SQ+.)+.)e-8Q.-0C!X')#-0$?/21@1!$'R&:NG !FGA6)<9898)#)#EYE4(.N:Q'AW&:-8o'E4N:N:E4E4)#)#$'$'&:&G !F )]-0²/8d)e-0,Q#/8F0:&D_4&:-0$.AWªAW&(.92 n141@3UEYQ) 3N:AW&YN:EY5]365=$')e-0A 36AWYN xcB = 0, 018 · 1, 25 × 10−4 = 2, 25 × 10−6 .
xcB
KB
)
pB = 3, 3 × 107 · 2, 25 × 10−6
= 74, 25
.
25 ◦ C
B
(
!)#EY+>$?EYAW1#& N:EM1]-0/2/81!)sRS N:AW H0$.&:AW$.&:90$.5AW&&: !N:).EMd #-0¢/0 !C!)=)4+>3w$.& AW&E$?(.921#14+v3U98&:)#;®EY/D3w)#1@98"%/8&:)#; $?p1 1 98$.&:A%;C p#pg1 !F0+>&Y-0EY/AW& Q¢-0Q.RS1@X';BF0-0&Y/2C=-01@/VAW$'&:NG;N4F81!R
;
pB
pxgB
xgB
B
*
p
A xgB = 1
KB pB =
(
B
xgA = 0
pB = p
xcB = p/KB .
(
xcB 1
xcB ≈
nB VB v ∗ = ∗ A, nA vB VA
#
!AWN8+>OEYAW)#& X>F^:/8p )eH8N:Ac1@-2;BA)#X>;
$ ]6 #
∗ v∗ vA B
VA VB
∗ = RT pvB
#
∗ ∗ VB vA VB pvA p = = , ∗ vB VA VA RT KB
-0C##+
RT VB = ∗ . VA KB vA
¶"h)=;<+.)#-0"x/2p'1Y36+>"cAc1 1hF8#ND31@90/8)eBH8N:p?A +.)e-0/21hF0&:;V5j±^ fg9 f9t )#>921@p eE e¶1!² h;<)#" ¶1 " h;<)#" 'd3[³UEY985=M3-06/5T$')4983)#EY1#(.+>Q'E4-0&TE4N:3¬EM1@"%)#$.X*A%&YC{N:$'A)e98H8)#N:EYA6/3wX¢#9R)#q$n/5s3w)#(*90nREM
(.90+>E436&:(.AW&Q'-0JLE41@N:EYEM5!1@p\"%$'C=/8&GFY#9d8&¶-2)@ b $.9843AW&:63w$'1@)e_DH8N:X¢A,R(\)#)#$?/81&:$'(.NG90F8E41!&:R(.43Q's-0E4N8N:O'EM&:1;B/AW5,N:"%EYC!$,)B5=N2N:OEM#+.-8)#/8&Y/5=N:EYN:C,EY5ªA /5,p>"%C!36)AWY/8N&: !3)1@-090C'Q.RS$'1#&:+=C b $.A%C#1 >;l1B)#$(*)e-0/21#I ±L~>d !² !)#EY+>$?EYAW1#& N:EM1t(*)#)#/8EY&:$?$'1#NGN:F01!EM1R?N8O'(\&:)#;B/8&:AW$'N:NGEYF8$,1!5 RBN2O'3&:;B98)#AWN:EYEY/3U$n5)#90E4N:&EY5>E -0/8&: !d *) 1@Qn3wp1@1 _Y;i5!p _4&
) KB = 3, 3 × 107
T = 298
VB /VA =
3, 3
8, 314 · 298 · 133, 322
× 107
∗ = 0, 018 vA
)
· 0, 018
= 0, 031 .
A
)
µ∗A (T, p) = µA (T, p + Π),
µ∗A
A
Π
B
#*
A
µA
) T p µ A µB
T
B
A
)
µA (T, p + Π) = µ∗A (T, p + Π) + RT ln xA .
µA
!" # $"%'&
f1@C36AWYNq3w1@90Q.$'&:Cj±L~>d !²w(.90EY5hFG;iQ#F^&D(*)e-0/21#I
L± ~>d !² $.ªAW1#&Y-0+>/8Q.:_M(.1$.AWA&{98)#C!EY)#3690EYA F05=1@-8;i/21@5 ;V57E3UEM)#1!C!RnR)#_4&:$.Ac1>p_4&K98#_Y$.AWN41sN:AWH0$.AW&:o«F^&Y-8/ )
µ∗A (T, p + Π) − µ∗A (T, p) = −RT ln xA .
µ∗A
Π = 0) ∗ ∂µA ∗ ∗ µA (T, p + Π) − µA (T, p) ≈ Π ∂p T 1 ∂ 2 µ∗A 1 ∗ 2 ∗ + κ Π ≈ vA Π, Π = v Π 1 − T A 2 ∂p2 T 2
±L~>d e²
! 1 +>EYAW& F^&: !)BF^H8&YN:-0AW/BH8"%)#A%3UX>)eF0:H8/8N:)eAcH8N:A%AcE4)#/8;<&:90)#;B"W)MAW3wN:EYj$?>N:EYd 5>-01/8&:)# ! !)nR98)#(\EY)#(.;BQ'-8A F8E41N:EM-01@AW"%ª$.36A%C#5n1>b p 92 1@-0E/21@C,/83&Y1#N:EY+>$.921@AW&e/8p\)436N:AW5H0$.3 AW&:$.AW&ip,NY)e)-0;
κT
Π
κT Π/2 1
)
Π=−
RT ln xA , ∗ vA
(
Π
Π = ρgh,
g
ln xA = ln(1 − xB ) ≈ −xB = −nB /n ≈ −nB /nA =
A
∗ −nB vA /V,
Π=
B
RT m RT nB = . V VM
)
|±L~>5=de-8~!)#²wC!)eA H2±LIw~>d->R
Q.M²(?p'1-0Ce3U#)=+ +>5i3 Ce1@(.A%"c1@90E4&ep p#365=EY$?1#N:EM1@;V5E698436$?1@o
$ ]6 #
h
RT m . V M ρg
*N:E41@&:Qn$.3wAc11@_Y;l;i1#5!-0p5]_4N:&<EM36#-0E4/8#9&YqN:EY/8C!&:)4$j3w;<&GF:)#p _Y$?1]p.;B/8&:Ac_<1@$'36)M5=36C!AW)#N:90AW&EY5>-0/21#IB+.)365=EY$?1hb h=
*
M
M=
RT m . V ρgh
>d (?,Q.1@92X'1-0/2(.1@90$'EYNG5MF0F^1 ;VQ#F0&DF^&Y(*-0/)e-0$./2AW1#&:I "W)#/0$?1>p?36AWYNV3w1@90Q.$'&:C 98M36$')43w1@ eA N:AW&YN:E:b ±L~>d e!² X\N8O')&:;B(.90AWEYN:5hEYF^$n:5<"%AWH0N:;iEY5>5 -0/8&GF-0Q.X'-0p./21@N:$'EY5,NGF^"%AA 3sJx1@EYAW&6 n1@E4)#)4EY3w$?&G1#F:N:d!EM1"c(*1ª)#98/8)#&:EY$'/NG3UF8)@1!bR 90Q +.)e-0C!)#$?1!R&: !)<;l1@;i5(*)#$?1#+>/8) ±L~>d !² p (.90AWEY)#592#-0N]/21!$?R
1#5=-8; /8:(.$.p.AW+.&)e-098/2#_1hYF0$.&:AW;VN:EY5 C!T)#X.Q7-8/098)#$ 98436$')eH8N:A ±L~>d ,² (.90EY5sN:EY5=;-0C!)#90EYp 5> !-0/2+>1@EY"%AWAW& H0;i5 F^&YE4-8&/V&:EY$n36/0Ac98@)#EY(.C=AcQ7T/8;<&:90)#;<"W)4)=3+>>5,d $?«1@;B98M3AW6N:EY$'$')e&:H8 !N:) A ±L~>d ,²w365,$.A%Ce1EY36Ac@E4&:C ±L~>d !² JL!1@+>EYEY5{AW& n1@E4)43w&GFVAUN:AW&:C'R&GFq+>"c)#1 EY$?N:EY1#5>N:EM-0/81K&GFq98#_-0YQ.$.X'AW-0NY/21@$'&:$nNGF^/0A 98)#(.A%3A6;92N81hO b /092Q.1@/090Q.E4&905B(.d 90ªE4&:1@;B"W&:Ac_Y1@5B$n5lF0&YN:+>AW$?&YN:1@E:Cbx(?(?1@1@92;B1AW+>:"c/21q1#IeN:p\EY5>_4&-0/8&: !)V98#_98Y)#$.EYA(.-0Q'AW-0iE4N:),EM+T1@"%/8$.&:A%;BC#1 (\&b (
B
)
µcA (T, p, xcA ) = µgA (T, p, xgA ) = µ∗g A (T, p), xgB = 0 xgA = 1 µ∗g A A )
c µcA = µ∗c A (T, p) + RT ln xA .
µcA = µ∗g A
p
∗g c c (−s∗c A + R ln xA )dT + RT d ln xA = −sA dT,
(∂µ/∂T )p = −s #
∂ ln xcA ∂T
*
s
)
p
=−
∆s∗A + R ln xcA , RT
∗c ∆s∗A = s∗g A − sA
T
#
A
T
A
!" # $"%'&
(*98)#),EY+k(.Q'N:-0AWH0E4$.N:EMAW&:1@$."%$.AW&:A%;Ce1!²d pN:E4EY&Y5=N:"%EYA 5,36AWH8N:±/8AW&&:;BE(*(*&:)n921@RS#/0N:Q.E492&:1{$.Ac36190E4±L~>&:$.d eAc!1{²tN:E EY5=±L-0~>/8d &: !!) ² 365=$.A%C#1>p>_4& p
TA∗
∗c c ∆µ∗A = µ∗g A (T, p) − µA (T, p) = RT ln xA < 0,
>3+ "c1 /8&:;B(*&:921@/0Q.p 90E4N:& EY5,"%p,A 3 C=/8#98&GF EM1#N8O')=+>EYAg3690E4&:d$.AWf&ª)798)#)#EYEY/$?3U1#)#N:90EMQ 1>(*p ),_4+ & E4N:AW)MH03U$.p?&GAW/8&:FY)<$.dgAW!(*&:&Y;)#H0/8"%&:A\$'F0pUNG&YF0+>N:1!$?EYR 5=1@N8-0CKO'/5&:/8;B&:98;BAW)#N:(*EYEY(.&:$n92Q'5 1@-0/0E4Q.N:92EM1 1@"%$.A%Cr$.AWXn&)=5'+>98R
#_(\XnY5j)M$.36AF^AcQ.1#-0_TAW+'1<3 EY/0XnEYJx5=3q1@/0EY$.d?AWAW&T(.)90 nE4),1h&+ bb e/8&:9090EM;<1@$')=&G+>FU5,N:$?AW&Y1@N:;BEY5!AWN:d=EYf$'1@&GFBCe1iEN:(?AW&Y1@N:92E >$.pAW !& +>F^5=&Y_-0/U-036/8)e(.-092Q.1Y$.3tC!+>)MEY3UAW&)3v$.AW98&:43366AW&:$'"%)4C,3AW&1#+>EME41h& b X.N:EYQ.590E437&:$.(?AW&D1@98;<epn)#$._4AW&V&:;B-0(\$.)MAW3U&GF )=;<+.)4)#3_Y1#$?Iq1 eF8D3w)#1!/0R
/890)MEY365=;l$?<1#I(.3790E4&:3w;B1@Ac901@Q.$'$.VCe1#N4N81!O]R&GF6"c1@N:X\AW)@&bb 92C=1@/8/8#9)#890&G5hFFG(*$,5=)#/8N2&:Og$'dgNG¶F81!),R+.N8)#O'X.&:$.;BAW&BAWN:;<EY$,)#5]_Y$?F^1&Y-0Q./DEY365=5=-0Ce_41#-0EYI5(.90),E4+&YN8(*O¢)#R/8)=&:+>$'E4NG)#F8$?1!R
Q(?N81@O'98&eb p ;BAWN:EY$'&: !)ZN:AW&YN:EY5)B/8&GF6-21@;<&GF/8&:;B(\&:921@/0Q.90E4&VAgN:AWH0$.AW&:$.A%Qgd *N:EM1@1@Q,"%3w$.A%1@Ce_Y1i;V;l5j1@+';V1@5 "W&GFYp\_4&+>"c1Z&:$,/21@"%(.A%p,A 36;
∗g µ∗c A (T, p) > µA (T, p)
T
p
T
TA∗
µ∗c A (T, p)
*
h∗A = T s∗A + µ∗A
T
∆h∗A = T ∆s∗A + ∆µ∗A = T ∆s∗A + RT ln xcA .
¶)B3t-0/21436AW&:$.A%QT/8&: !)BEY36Ac@EYC=QT+.)T±L~>d !²w;l1@;i5)e-0/21@/8&YN:EY$.AW& ±L~>d !² 3698#_)#90Y90E4$.;l&:AW$.NY1@Aci"%17$.;BAW98&ªAW)#Y3vEY+>/EY3w/55=)#;[90/8Qg&:;B36p E4(\)#&:d9092E41@!& /0&YQ.+>92$?1@1@;BCPAXn;<365'R90)#)iE4_4&:&:"%$.AW;VN:AcE41l5)#98$'(*)#&)#EY/;B+>3U"cA%1V)#$?90/8#Q&:Ij;BA$.(*N:AWEY&:&:5=92361@-0AW/0/8&:Q.&:"% !Ce905) 98/8)#&:;BEY(.(*Q'&:-092E41@N:/0EMQ.1@90"%E4$.& A%Ce1>pwd At(*),+.-0/21436AWIT31@90/8)eH8I )=+>(\)M36Ac1#+'1hF0#N4
∂ ln xcA ∂T
p
=−
∆h∗A . RT 2
∆h∗A
T
TA∗
∆h∗A
e~ =d $.TAW&:Q' !-0)BA%;iEM5r1#+'(*1@)s$.Ac1(.98(.)e90-0EY/05]QEM-21!N4R1!)#R
_4C!&:)4$.3wA%Qg1#p'I_498& 436$?1@$.AW$.&vAW&q±LEM~>1@d "W&:!²_Y]5]E)=+](*)#/8(.&:;B90E4(*&Y+=&bb 921@/0Q.905!d¢³1!R
C,Q#F8#Nq)=+]/8&:;B(\&:921@/0Q.905 +.) p.+.)e-8/21hF^&:;i5
$ ]6 #
∆h∗A TA∗
T
)
L± ~>d e² ¶+'1@)#/0$.$.AW&:Ac1>3ph1@36_AWY&:Nw$n/2E41@&w"%(.36Ac1ªE4)#(?901@QT98)M±L~>3wd 1@$.e²Ac1ª36985,)#$.EY(.A%CeQ'1>-8p@E4_4N:&wEM1@98"%)#$.EYA%/C#3U1>#9p 3690E4&UpF03s&Y-0/36+.5=)@_:bb -0± E4*&G1@FQn/83w&:1@;B_Y(*;i&:5!92p=1@_4/0Q.&t90/2E41@&C=A¢$.-2A%_1@; N:EY5>36-0E4/#95 9836)#EY5=(.$.Q'A%C#-01ªE4N:E EM1@98"%)#$.EYA%36C*Acpg@N:EMEY1@5,$."%AcA 1iEM1#+'1@$.Ac1 d -0 /2>p1@>;V !5+>5Z/8&:(*92)n1@REV)#_Y+>5,3w;i1<5 EM1!R)#_4&:$.Ac1 p.N:EY±^5=M²"%A F^&Y-8/6;l1!R&ep.36AWd Y²UN |5=C!)#90EY5,b ±L~>d !² A±L!p>² EM/81@&:/8;B&:; (*&:921@/0Q.9213690E4&:$.Ac1>p p=)=+>(\)M36Ac1#+'1hF0#N41 F^&Y-0/wX."%AW-0Ce1 ±L~>d e~!² !$.+>Ac1lEYAW98& )#EY/3U)#90Qj3 -0/8)e-8Q.)#$.C,C,98&YQH0"c+.1i)(./890&:E4;B&Y-0(\Q.&:$.92AW1@Y/0N:Q.AW&ª905K/8&:36;B90(*E4&:&:92$.1@Ac/01]Q.N:90EY55=36-8/890&:E4 !&) b F^98&:)#;iEY(.5 Q'-0E4N:EM1@"%$.A%C#1>d «(\)nRS#N:E4&:$.Ac1±L~>d e²pU±L~>d !²A±L~>d e~!²ª+.)e-8/21hb ±L~>d !² |EM1@(.(.AW-092Q#1@F^C=&D/5>-8NYAWD&tN:EYE436YAc-0@/8E4)ª&:F8CB1@(*C!))#;BAWY+>EY5 A?-0/8:_4&:$.AW&:;[-0Q.X'-0/21@$'NGF^A ±L~>d 'M² !K+>)#EY"cAW1@& "%$')eH8I)#/8EY)K$?"%1#AWN:N:EYEMX?11{;<;<)#"c)#1@"%"%A6$'-0)eQ.H8X'IK-0-0/2Q.1@$'X'NG-0F^/2AU1@$'98)#NGF^EYA6(.Q'98)#-0EYE4(.N:E4Q')#-0$'E4N:&GFE4)#(.$'90&GEYFY5=p (?1hb d +'1hF0#N41$?1F0&Y+>$')e-0/0C!D;l1#-85Z98)#EY(.Q'-0E4N:EM1@"%$.A%Ce1>p?N:EY5="%A ln xcA
∆h∗A = R
1 1 − ∗ T TA
.
∆h∗A
T > TA∗
*
p∗B = 0
p = pA = p∗A xcA xcB
)
ln xcA = ln(1 − xcB ) ≈ −xcB ,
xcB
T
TA∗
1/T − 1/TA∗ ≈ −∆T /(TA∗ )2 ,
∆T = T − TA∗
*
)
∆T =
R(TA∗ )2 nB R(TA∗ )2 c x ≈ . ∆h∗A B ∆h∗A nA ∆T
∆T = Ke mB ,
mB
B
B
mB =
nB ; nA MA
>
!" # $"%'&
EYN:EM36#5=-0C,/8&Y"W& N:EY(*C!),)M+'3U1h&GF0F & -8AW F8q(\3v)=+>C=EY AW&:h"%;
MA
Ke
)
Ke =
MA R(TA∗ )2 . ∆h∗A
nf 1@1@E4C<)M3w36AWY3Neph F^&Y>H0± "%A*&:;<$n/2)#1@"
"%²(.pMAW/8&:;B(.90(\E4&:&:92;B1@/0Ac1@Q.$,985Z3 365=B92pM1@1EYA%;l;V1#5Z-23v N: EMh#;<-0/8&Y)#N:"xp>EYC!-0/2)M1!3wRS ;B±36Ac5=1!RS9211@_436)#$?5=;BAc3´1@9 h;<>±)#;<"
²)#"(*C,), +>EYAW&:²"%d A%;i5k(.90E4&:Ek >p/8) X\Y+>EYAW& >d {-0Ce#)#+]"c1@"%(*$'))eH8(\ID)=98+.)#-0EY/2/1Y3U36)#AW90&:Q{$.A%QK"%AWN:+'EY5,1@$n;i5>5N8O E4&ª36E4)#90Q«±L~>d 'M²q±E4)#Xgd.EM1#+ dg e²p ;<)#" hC= ;<)#" hC= K+. & h';<$.)#AWNG"¢F^AF^&Y;<-0/ )#"c;l1@"%1#$'-2)eBH8N:N:A EM#;l-0/81@&Y;VN:EY5 C!)M3wX*&:$.E4&:$nQgp?EM1@/8&:; !+>EYAW& d¢¶)#$.AW&:3w1@_i;l1#-2198)#EY/3w)#90QK365=$')e-0A
Ke
−1
mB = ∆T /Ke = 1, 8/2, 53
*
86
= 0, 711
nB /nA = mB MA ,
MA = nB /nA =
78 0, 055
h;<)#" 36AW1#Y-2N61+>$."c1 AW&:"W)#/0$'&: !)l-8 qC.RS;l1#+>1@$.;VA%5Ce1365,$')e-8Ag36AWY;
.
m = MA nA + MB nB = nA (MA + MB nB /nA ) = nA 81, 52 , m = 100
nA = 1, 227
nB = 0, 067
MB nB = 64 · 0, 067 = 4, 29 g.
;<)#"xd
4~>dt3¶)#1@$.90Q.AW&:$'3w&:1@C_ 9843-06C'RS$'1#)4+>3$.1@A%C eAN:AW&Y$.N:AWE:&DbN:F0Ac&Y1!-0R/)Z)#-0X\/21!&YN:R&D$,5j;l3 1(*N:)eAW&:-0"W/2&1#I -0/21!R
5=; p36AWYN B
)
µcA (T, p, xcA ) = µsA (T, p, xsA ) = µ∗s A (T, p),
> e !$.+>A%C#EY1«AW& 3 Jx1@)#EYEYAW$?&1#-0N:/2EM1!1tR&G(\FY)#d/8|&:$'5,NGF8C!1!)#R>90EYN2O'5>&:-0/0;BQ#AWF8N:#EYNj$n5V+.)eN:-0EYC!5>)#-0$?/8&:1! !R)6)eH898I)#EY98(.)#Q'EY-0/E43UN:)#EM901@Qg"zb p +.)e-0/21hF^&:;i5
$ ]6 #
µ∗s A
89ª431@6EB"W&G$'F*$.)436AW3&:5="W1@)#(. e/0Ag98$?)4N: 3wAW&Y1#-8N:Q.+>E:X'E4bx(?&:-0/2$.1@1@92AW&U$'1NG(.+>F090 "cE41q&:X.98)#AW&:EY± nE4/3U)#16Xg)#AW+.d90Q&:EM$,1#N:/+ AW5>&:d N:C'EY R$.]A&: !AW& t)F8 1@98e²)#C EYd3(.KQ'(.-090Q'E4EY-0N:5=A%EM;V(?1@1#"%5$.+>A%F0C=C#&Q1 b +>),+>5,$.(*AW)4&U3636Ac1#AW&:+'"%1hC!F8)e#H8N:N:5,A';B)=+>A(\Jx1@)MEY36AW&iAc1#-0+'/21h1!F0R#&GNYFY&d¢|j n1@)#E4X\)M36&YNVA./8EM&:1# !-0)Z/2!&:(.$,AWIw/21@36"%(.AW&:AW"%C!;<)eH8)#N:"WAc)41@3w;B A (?$.AW1@Y98N:)4Ac3w1>p1@$.Ac1>p p EM1#-0/8:pg(. !Q#+>F0EY&:AW;V&Z5(.90&:E4$n&:/2E 1@"%(.AcT;<)#)#EY"W$?)431#N:EY5,C="%90AWE4H8;V&:(>5 b /0;lQl1@&:;V$n5/21@/"%(.5="%AWC!)Z;<EY)#;B"W)4Ac31@$'qV/8)#EY$?(.$.1@C=AW&:QT$.Ac3¯1>p=-0C=/8/8)e#9-02Q.1w$.F0C=&YQT-0/ +.+.)<)=E^+'F81@1Y/036$.AWAc-01>Ced,1lf361@CB90E436&:$.AWYAc1N 98)#EY/3U)#90Qs±(\)#94d'365,921@_4&:$.AW&l±L~>d e²0²p'/0EY$gd c ∗s µ∗c A (T, p) + RT ln xA = µA (T, p).
A
B
h∗s A
∗c h∗g A − hA ∗c − hA = −∆h∗A
∆h∗A
ln xcA =
∆h∗A R
1 1 − TA∗ T
,
:N±/8EY)#5,(."%A $.AW&:$.Ac1!²wN:EY5=p -0 !/8&:+> !EY)AW& 98)#EY(.Q')#-0EYE4$?N:1#EM1@N:EM"%$.1jA%Ce/81>&:d¢;Bª(*+>&:5Z921@EM/0Q.1Y398{1@90C,/890)eE4H8&:Iq(.-0$.C'AWRSY1#N:+=Ac1 b $.(.A%90C#E41&:;BAc1@3°$n5T98)#N:EYAW&Y/N:3UE:)#bN:90E4Ac1!&R)]F0&Y-0-0/2/t1!;lR&V1!;
TA∗
B
!+>EYAW& F^&Y-8/6;<)#"c1@"%$')eH8N:Ac<-0C'RS1#+>$.A%Ce1 p'1 ∆T = Kk mB ,
mB
B
Kk =
MA R(TA∗ )2 . ∆h∗A
$')#X.)e$.-0A*A%_4$?&:1@$.EYAW3U&DD/8&:-0;B/21!(\R&G&:Fw921@C,/090Q.AW)e905-0C!C=)#90(*E4)4&:3w(.&G$.F:AWdYN:*Ac1@1Q,983w)#1@EY/_Y3U;i)#5!90p'Q{_4&3E4-0&/8)e36-0 eQ."W$.Y+>C=QQ$?+.)1 /8-0Q.&:;B$.AW(\YN:&:AW92&q1@/0/8Q.&:90;B5(*C=&:9092E41@&:/0(.Q.$.905AWYN:F^&YAc1l-8/N:E4EY+.5>-0& /8'&:$. !AW)l)4398)#1@EY$'(.& Q'F0-01@E4C!N:) EM1@"%$.A%C#1>p\/0QK(.90E4&b p N:EY5,"%A>F0&Y-0/t+.)=+'1@/0$.AW&ed *
86
∆T = TA∗ − T
>4 >d;lAW1#N:EY-21<5=;VN:EM5B#$?-0/81h&YF^N:(.EYAWC!&:)4903731 "%AWN:EYX*t;<)#"%A¢ &:h/2;<1@$')#)#" "%Q{±S-0C'RS1#+>$.A%C ²p!C,/8#98&: !) ;<)#" 1"cl1@"%(\$')#)e$.H8AWIV&:3&:/21@1@_i$')#;l"%Q 1#-361<5,98$')#)eEY(.-8A Q'-0E4N:EM1@"%$.A%Ce1±3U)=+>5.²t/8){*C, 'p¢36AWYNV;<)@b ;<)#" hC, P+.)e-0C!/2)#1h"WF0&:&:A ;V5]31@90/8)eH8I BpU36AWYN{+>"c1j-8/21!R&GF
MB = 46
B
)
nB = 24 /MB = 0, 522
mB = 0, 522
*
,
.
∆T = 0, 97 )
Kk = ∆T /mB = 1, 86
−1
.
B
mB = ∆T /Kk = 0, 5/3, 70
,
B
nB = mB · 1, 5
= 0, 203
MB = 58
.
A
B
B
)
c ∗s µcB = µ∗c B (T, p) + RT ln xB = µB (T, p),
> (./3U90EY)#5r90QgN:pUEY)#5=92; 1@ET36_45=&]C!Jx)#1@90EMEY15>-0-0/2/21@1!"%AWRSH01;V/85v)jEMN:1!EYR5=)#-8_4/&:5 $.AW-0&jC'RS) 1#+>+.$.)eA%C-0C!)#$?d1!R¶)e)#H2$.N:ADAW&:983w)#1@E:_ b (."W&:90_YEY5s5MF0)=:"%+AWH0;V/8&:5;B(*(*)#&:$?921@1#/0+>Q./890).5!p!pU_436&tAW&:Y$,NT/21@;<"%(.)#Ac_41q&:/8;i)#5(.$.$?AW&:1@$.(.AcAW1q-21#$?I 1hJEY/2361@"WAc&:@$,E4QZ&:C¯$.AW±&tE4)#EMXg1hb d EM1#+ d 4~!²
$ ]6 #
B
)
∆h∗B = R
1 1 − ∗ TB T
! +>p EYAW1 & )#EYF^$?&GFB1#N:&:EM$n1/21@/8&:"%(.;BAW{(*&:/892)#1@(./0$.Q.AW98&:$.Ac/81>)#d(.$.¢AW36&:$.Ac@Ac1]E4&:N:C EY5=/8-0&:/8$&GFª[email protected]'5=-03w/21k1@$'-0NGAWF^ A $./"%A%3UA%$.C,)#AcQgl90E4d 98&< )#EY±C,36(.98Q'5=&YH8-892"cE41@1lN:_4EM)#)#1@$'$"%&ª$';l)eQ¢H81@RSN:1@C>A ;B-0-05,C=Q.;lAWX'&:-01@;´/2"%$'1@$';<&NG)#-0F^/8A"W)4:36_4&:5=$.3 ; AW&N:-0AW&:Q.C'X'² R
-05,3/2; 1@$'/8&:NG98;BF^)#A EY(\(.&:Q'92-03°1@E4/0N:Q.98EM90)#1@E4E:"z& bb --00/2/8:1!_4R&:&: !$.).Ac1]d p $?N:1#EY+>5=;B"%A Ac-01@/89i:_4-8&:Q.$.X'AW-0&/21@98$')#EYNGFG/A3U)#90Q36$?5=1#/092-8#5=N4NY1)#$'-0&:AW !).3`d ¶(*)4)e36-0/25=1#_4N:&GA FDN:/8Ac&:1! !RS)1 "c1+'1@$,5=N2OTEDEM1#+'1@$.Ac1l+.)e-0/21hF^&:;i5 ln xcB
B
TB∗ ∆h∗B
,
*
B
B
xcB
T ≤ TB∗
B
)
ln xcB =
19000 (1/352, 3 − 1/298) = −1, 18 . 8, 314
ª36AWRSY1@N ;<&:C;<)#"W)4365d $?1hJS/21@"W&:$nQ«3`98)#EY/3U)#90E4&$?1#-05>NY)#$n5=; 365=$')e-0A >d N:*Ac1@1VQnA*3w"%1@A%$._YAc;i5T98)#$?EY1h(.F^Q'(.-0AWE4&:N:903qEM1@p?"%$'_4&q)eH298N:)#A>EYF^98&Y#_-0Y/6$.+.AW&:)e$.-05>AW&iI;BQ.;
*
xcB = 0, 307
A
)
∆h∗A 1 1 − , R TA∗ T ∆h∗B 1 1 c ln xB = − , R TB∗ T ln xcA =
B
>
!" # $"%'&
!+>+>90Q.EYAW e& AW&GF pg/8)K)#/0dn90!EY5,&:_4;l&:"%1@A';i$?51wF0"%&YA%$.+>AW$'&e&GpgFC=)e/8-8#9A?0),5=+?N2OR)#_Y$?5=1#;VN2On5 5="W&:$.Ac1Tp!;l1ª1h$?F81 (.(.9090E4E4&Y&YN:N:A%AW36YN:$,Ac1l5/EY5>$?N81@OC\+>d 3U¶=NQ.8$.O{C="%/KA%$.&:A%AxQ.p?/8(.&:C=90EY/55>N:N:EYEY$n5=5; )=/8+>&:(\;B)M(*36&:Ac921#1@+'/01Q.92(.1 Q.$.C=/8"W)4&:36_Y5 A (*¶)#)]$.(\A%_4)=&G+.F-0/8/2&:14;B36(*AW&:&:$.92A%1@Qj/0Q.+'9 1@/8$n)#5>(.N8$.OjAW&:E$.AcEM1Z1#+'±C,1@90$.E4Ac&:1(.;l$.AW1@YN:;VAc1!5 ² )#X.Ql-0C'RS1#+>$.A%C!43qdp )#921@E d.ª)e-0/21hF^&:;i5-0/2#+ p.N:EY5,"%A p xcB = 1 − xcA 1/T
ln xcA
Te
) xcA = xe = 0, 4
xcB = 0, 6 Te = 0, 9TA∗ = 0, 84TB∗ ∗ ) 1/TB = 0, 84/Te
1/TA∗ = 0, 9/Te
∆h∗A ∆h∗A (0, 9 − 1) = −0, 1 , RTe RTe ∆h∗B ∆h∗B ln 0, 6 = −0, 511 = (0, 84 − 1) = −0, 16 , RTe RTe
ln 0, 4 = −0, 916 =
EM1@/8&:; -8/8)e-0Q.$'&:C&:$,/21@"%(.A%A/8)#(.$.AW&:$.Ac1<365,$')e-0A
)
∆h∗A /∆h∗B = 9, 16/3, 19 = 2, 87.
'd *1@Q,3w1@_Y;V5T$?1hFG(.AW&:903qp._4& #
*
X ∂ RT nj ln xj = RT ln xi , ∂ni
X*)
j
X
nj
X ∂ (ln nj − ln n) = (δij − xj ) = 0, ∂ni
!$?+>1#EY+>AW;B& Ac1@98)M3U)#EY&G$?F 1#&:N:$'EM&:1K90 e+.A%A&:"%-8/83U )#ªX*98),)#+>$'$'&Y&GN8F C!&:92A%1>X.dX'|j-21>)#p X*&YN
j
δij
GE
∂GE = µi − µ∗i − RT ln xi = µE i . ∂ni
¢ !)=+>$.AW&DEq+.& '$.AWNGF898)#EY/3w)#90QK(.98)e-0/8&: !)K±E4)#Xgd±L~>d%eM²0² *
86
GE = ng E = ngAB xA xB = gAB nA nB /n,
86
±L~>d n!²
>
$ ]6 #
-0C##+(*)(\)=+.-0/21436AW&:$.A%QK+.) ±L~>d n!²
2 2 µE A = gAB (nB /n − nA nB /n ) = gAB xB (1 − xA ) = gAB xB ;
0-1@(\$?&M1@R
"W$.)#Ac e1hAWF8N:EY$.98AW43& 6$?1@$.AW&A%X.X'-21hdb,(.Q.92O'1Y&:3t;l+ 1 Y;i5j/8&:921@E!pN:EY5 A 2 µE B = gAB xA
(
µE A
)
µE B
E xA dµE A + xB dµB = 0.
¶ 90EY5Q'-0/21@"W)#$,5=N2O p A 1(*;l)#$.1@AW;i&:3w5 1@_ )#M3921@E b p 6 3 W A Y N 8 9 $?/8&:1@$'$.NGAWF0&D1!R
ª5]A%X.N8O'X'&:-21h;BbAWN:EYQ.$'O'&e&:d;l1F^&Y-8/w-0(*&MR
$.AW)#$'&ª(.90E4&:ED$?1#+>;BAc1@98)43w&q(*)@b |j-0(\nRN:EY5=$.$.A%C=A1@C,/5,36$')eH8N:A365,EY$?1#N:EM1@;i5E4&DEY36Ac@EYC,Q
dµE B
T = 2gAB xA dxA
p
) dµE A = 2gAB xB dxB
dxB = −dxA
)
µE i = RT ln γi ,
-0C##+ p d >d &:Xn5<365=EY$?1#±N:E4EY)#5=XgIªdt1@±L~>C=d/~!5=²036²p$')e !H8+>N:EYAxAWp=& ;VQ'-0A%)#;iEY$?5l1#-0N:C!EM)#190EY1@5>C,-0//25,1#36I$'E4)e& H8EYI 36Acbx@/8EY&:C= !Q) -0C.RS1#+>$.A%Ce13N:AW&YN:EY5!d?"c1-0C'RS1#+>$.A%Ce1 ;l1@;i5 γA = exp(gAB x2B /RT ) γB = exp(gAB x2A /RT )
pi = p∗i aci
aci
i
A
)
acA = pA /p∗A = 0, 018/0, 031 = 0, 58 ,
1+>"c1B-0C'RS1#+>$.A%C#1
B)
acB = pB /p∗B = 0, 023/0, 029 = 0, 79 .
|j(.90E4-0(\&:EUnRQ¢N:RSEY1@5=;<$.$.&:CqA%Cj;<1@)#C,"W/)M5,36365!$'p@)e36H2AWN:YA N )#/090EY5,;iQ#F^&:;i5!p+>EYAW&:"c#N1@C,/5,36$')eH2I p . d j | 8 \ ( nR : N Y E = 5 . $ . $ % A C @ 1 = C / = 5 6 3 ' $ e ) 8 H : N A -0+>C.E4RS)Z1#+>X.$."%AWA%-8CeC,1 A*F0&Y+>p*$'N:EY)e5=H8N:"%AAxp\98)#36EYAW(.YNBQ'-0+>E4"cN:1EM1@/8"%&:$. !A%)Ce1j-0C'±X*RS1#) +>$.A%C#1(.92143U²) F0&Y-01#/)#X?Q.1@"%/29^1 b F^&Y-0/"W&:(.AW&GFt-0(*&MR
$.AW)#$'&ed c γB
=
acB /xcB B
c = ac /xc = 0, 58/0, 2 = 2, 9 ) γA A A
= 0, 79/0, 8 = 0, 99
xcB xcA
>
!" # $"%'&
>d N:AWH0(.$.92AW1Y&:$.3AW1]&:; D1@(?"%1@/890)#5$?1$?1#;l+ 1@98;V)#5EY/3U)#98&:; p¢EM1@/8p :& ; ±E4)#Xgd±L~>p¢d !~!+>²0² EYAW& F^&Y-0/ *
pA = pxgA pB = pxgB
c c pxgA = p∗A γA xA ,
p
c c pxgB = p∗B γB xB .
8F36¶1@)#AWCTY$.N AW-0&:C'3wRS1#1@+K_ 3 N:AW&Y(.N:Q.EY5!$.p*C!N:N:AWEY&5="%1@A E4&Y)#/098)#(\)M365,;[-0C'RS)#1#92+i1@(?E 1@905 F0&Y-0//21@C,A'-21@; p xgA = xcA = xaA
xgB = xcB = xaB
-0Ce#+ /+. 3U)e)#)#-090/290EY1hE45=F0& -0&:/2(.;V1h98F85 )e#-0N]/5=E4; &36±E4E4)#)#Xg90dnQ EM1#$?+ 1{d,3t ,-0²(*)#nR92N:1@EYE 5,E4$.&t$.EYA%Cs36Ac1@@C,EY/C=5,Q 36$')eH8N:A63´98)#E:b p c c p = p∗A γA = p∗B γB ,
c c γB /γA = p∗A /p∗B .
#
xA + x B = 1
)
,/2#+ (
c c ln(p∗A /p∗B ) = ln(γB /γA ) = gAB (1 − 2xaA )/RT.
gAB ln(p∗A /p∗B ) = −2, 2907 , = RT 1 − 2xaA = −5675 T = 298
:36N EYAW5=Y"%NA 3t-0(*nRN:EY5=$.$.A%C,A h1@;
gAB < 0
)
1 − 2xaA =
xaA
ln 3 ln(p∗A /p∗B ) = = 1, 22. gAB /RT 0, 9
43NY)<6)#EY$'$?)e1#H8N:IªEM/21>1ip'_4$.&DAW&ª$.;
0 < xaA < 1
p
> >d JQ.-0/0EY$.5hAWFG&:$'$.&GAWF&3´"%Q.C,(.AiQ.$.;BC!AWN:&YAW-0&EM1@C="%90$'5,)e/H85=N:N:AEY$,)#5,EY; $?1#p N:NYEM)1r),Q.+>/092(*1@)4/836kAc1#-0+'/21@1X.A%3"%$'1@)e90Q.H2N:$.AC!)4+>365,bA ±E4)#Xgd±L~>d% !²wA ±L~>d% e²0²
$ ]6 #
∂ 2 gM ∂x2A
"c198)#EY/3U)#90Q{(.98)e-0/8&: !)
T,p
RT = + xA xB
∂ 2 gE ∂x2A
= 0.
g E = gAB (T, p)xA xB 2 E ∂ g = −2gAB (T, p). ∂x2A T,p
|1@90Q.$'&:CQ./0921@/5-0/[email protected]%"%$')eH8N:A+>5,JQ.EY5hF^$'&GF
p>36AWYN
T,p
)
1 2gAB (T, p) − = 0, xA xB RT
<;N:EY)#5,_4"%Ae&3jX,5=(.Iª90EY-05=(*(?&M1#R
$.+>AWC=)#QV$n5<+.),/+'5,1@"%C!/0).$.p=AWN2 !Oi+>)=5 +.N8O,5,"W&:o),p>+qNY+.)i)e)#-8EYC!$?)#1#$?N:1!EMR1)eH8N:Axd *1@C,b p F^36RS&Y1#5,-0+'/ (?1h1#;BF0#+'$.N1AW&G+'3 F^1@-0"WE4&G&F)=+.+<),+';B1@A%/0$.$.A%AWA;l)e36H21@I 5="%$'$'&G)eF -03A p''1@EM90p'1@/8N:)eQ,EYH83w5=N:1@"%A\A _YJ !;VQ.+>$.5!5 C!pgNG_4F^&qA !+>5 p!C=/8#921 gE > 0
gAB > 0
#*
gAB
2gAB /RT 1/xA xB
xA = 1/2
/8/8))#;B3Ac1@1#90-0Q./6$'98&:43C6-0$'/2)[email protected] A%"%$')eH2N:A>+>5,JSQ.EY5MF^$'&GF'F0&Y-0/EM143t-0E4&6-0(*&MR
$.AW)#$n5!d ª1hb gAB (T, p)/RT < 2,
.X),+>A%"%-0(*$'C')e)4RS36H81#N:+>AcA 1#Q7+'EM1!1tC=RS90/81@5,&:;i/;B5=Q#(\N:F^EY&]&:$'92-01@&:AW/0 !lQ.) 903 E4& C,905=/5>N:EY$'&GdU/F:5=p "%C!"c1k)T-0+>p:C' !"c1
T kr
T kr
T = = 1/2
xkr A
T =
)
gAB (T )/RT = −2
−2
(T − T0 )2 + 5
T kr
−1
xA 6= xkr A
(T − T0 ) = 2,
>
!" # $"%'&
C=(./890#9EY5]8&ªN:;lEY5,1; +>3w1(.AW&:9036Ac1#-0/0C=A A Bp +>"c1 Ag+>"c1 )#921@E +>"c1 d f31@1@C90Q.36$'AW&:YC«N-0+>/2"c1@1ªX./8A%&:"%$';B)e(\H2N:&:At921@+>/05nQ.JS9 Q.$.EYA%5M_4F^-0$'EY&G5=FN2OlF0&Y-0)=/+ -0(*&MR
"%Q.$.X
gAB /RT < 2
T < T1
gAB /RT > 2
T 2 = T0 + 2
T > T2
T 1 < T < T2 T1
T2
T1 <
T < T2
T1
T2
P° q ] ! § 8£ 36N8O'5>&4)=1@+>+>C!EYEYNGAWA'&:F0)#"c981@; &4$.1@AWC!&N2NGO'F81@&:1>"%;BX*pe36)
<e
*
86
H
H = U + pV,
!N:AW+>H0EY$.AWAW& &:$.AW&eF^p &Y-8/ p &:3y$'&:Q.90C' eAcRSD1#+>3UEY&:AW&V36$'F^&Y:-0/0/V90EYQ'$?-0>/2p.1@"W1 )#$'&eF0p&Y/8-0)T/ )#E<X>(.F0:AW&:/890)e3tH8N:-0AcE4ª&GFqQ.C'EM1#RS1#-2+>1#+>Qg5dn!/8&Y&:H09^"%bA ;<)=+>5,$?1@;BA%C=Axp !+>EYAW& )#EY$?1#N:EM1
V
p
∆U = Q + W,
Q
W
e $?1@$?$?1#+]Q.C'RS1#+.&:; p?+.)e-0/21hF0&:;V5
=33 8#"%'& / F% 4!" % &# %
)
!&YH0"%A¢Q.C.p>RS/81#) +Z;<)#_4&t365=C!)#$n5=3w1#I/5="%C!)i(.921#NY )#X>F0:/8)eH8N:AW)43>p>N:EY5,"%A ∆H = ∆U + p∆V = Q + W + p∆V.
−p∆V
W =
(.98&4901@EYC!5KNGFGN:AEY5=&:$'; +.)#/8&:90;BAWN:EY$'+>&GF:"cd1Z98&41@C!NGF^A&: eE4)#/8&:90;BAWN:EY$'&GFD)#921@E +>"c1 , ' C S R # 1 > + . $ % A , C ª A ' Q : N 4 E Y & 0 0 / . $ W A : N M E # Y N & 3 8 9 4 & @ 1 ! C G N ^ F ª A 2 N ' O : & B ; W A : N Y E ' $ G & F ? $ @ 1 Y E = 5 w 3 @ 1 i ; r 5 8 9 & b EY;<1@X. !)#AW&:#9_Y$n$?/211@3t;BEM-01@AxEYd (.5=AW-8-2/01#C,YID+>AWN2EY3PO«AW&:;V98(*&4)e571@-0 !/2F^&:1#&«$,N:/8A )#43EYq$?d 1#N:EM)#1#90I«;l1@A%$'"%$.+.AW&:&C=-2C#&:1@; _4+'K98p&41@C=C!/8NG#9F00]5N8(.O'90&:E4;B&:X.AWN:AWEY&:$? n1 ∆H = Q,
∆H < 0
∆H > 0
(
R
)
X
νR R −→
X
νR R,
! 1@ !+>&:EY$'AW& N:AW& )#EYd$?¶1#)N:EM"W1ª&:3w3t&G-0F(*-0nR/098N:)#EY$.5=AW$.&]$.EMA%C1@(.-8AW/8-8&YQ#N8F^O.&:AW;i)#;<5k&:-0/0Q.905>X'N:-0EY/2$n1@$'5lNG-0F0/8&)@F8Q.#_YN:5,5/8&(.90+.EY){59898&&bb EF^Y1@&:X.C!;iAWNG)#FG5VAx90pQr(.N:98-0EY)=Q.5,+>X'"%Q.At-8C,/0-0/92Q.5!1@X'pe/8-843N:/0qEY925,1@p "%/A.5!-0p Q.36X'-8AWY/2N1@-0$'Q.3¬NGX'F^&6-8-0/0Q.)#92;+) b -0NGF0Q.1;
R∈{ }
νR R
R ∈ {
R∈{
}
}
(
*
*
*
R
e > N:*EYX?5>1@-094/2d>pU«-8+./21@& X.'A%$."%$?AWNGF^A(*/8&G)eF -0/2361#IK5=$.3 A%C#1>+'p*1@_4$'&i&GFZJSQ./8$.&:C!;BNGF^(\&&:92/81@&:/090Q.;<90E4)=&{+>5,A$?(\1@;B)=+rAWN:EYN:$'AWH8&<$.AW3`&:$.-0AW&:/2;1hb $.98&4AW&]1@C!-0NG/2F^1@A$'N8+'O'1@&:98;B+.AW)4N:36EY$,5=5=; N2OEM1@+'"W1@&:$'_M&q{/8),&:+k90;5=(\$?&:1@92;B1@/0AWQ.N:90EY5!$'d&V!)=&Y+>+>$'$?)e1@-0CkA-03 AWª(.EY9036EY5=5=(?C=1#"W&D+>C=+.Q) /8&:;B¢(* !&:)=921@+>/0$.Q.AW90&j5 E(*)436p.5=N:_4EY-05,EY5="%A; e~ EM >1@p%(.AW-8B&:; d 98&41@C!NGF^AVN8O'&:;BAWN:EY$'&GFYp-8/21@$'+'1@9^b +.)43T±;<)#"W)M3w!²&:$n/21@"%(.AWq98&41@C!NGF^Axp p>(.90E4&Y+.-0/21Y36Ac1@;i5F01@C!) A a/ %'&
*
86
25 ◦ C
*
∆h0r
∆h0r =
X
νR h0R −
X
νR h0R ,
!36+>5=EY; AW& d.| )#(*EY)4$?361#5=N:_4EM-01E4;<&GF6)#98"W43)M63w$'ª)e&:H2$,N:/2A1@3t"%(.-0AWEY5>98-0&4/01@C= !AW&ª&:$n3t/2-01 (\nRN:3«EY5=-0$./21@$.$.A%C=AW&wA-0-0/2/81@&Y$'N8O.+'AW1@)#98;<+.)@&bb /0(*90)45>36N:EYAc1#$'+'&61-2 -0/2"%1@AWN:$'EY)4X?361@A\;B(*A'),+.N:),EM@+'/01@C!/0)M$.3UA%;B&:;VAxdeQg|p>5=3692AW1@YEN EY3U36N2AcO'@),EM+>1@EY$nA*5E4E6& -0EYQ.$?X'1@-0C,/0AW92&:1@;[/21@;B;BA%A'$n)=Q'+=-Mb p 36_41l&AW36YEYNq5,$?923w1@1@C{N8EO';BEY)=36+>A%$nAcEY@Q'AEM-VE41@&q-8$n/85KEY)@$?F0Ei1@#C,N:(.5KAW98&:)=; (.+>90Q.E4(.C,&Y"%+/2Q'1@-4;B-0d.Q.|A;l)=5=+> !(\(\),)M+>)36$.-8AWAcQ.&t1#X'+'F^-01l&Y/0-092/6-81@/2(./21@1#90$'EYN8)4Oj5h36F8#A3 IDC!RS#)#C!N:o')#EM$nNY1 )43w3w-0&:AW&:$';VNGF^+.Qge) pp +.& '$.AWNGF^Ag3t-0(*nRN:EY5,$.$.A%Ce1B-0/8&YN2O.AW)#;<&:/0905=N:EY$'&: !)-0Q.X'-0/0921@/0Qgd.| 43wN:EM1#R∈{ }
R∈{ }
h0R
R
86
∆h0r =
X
νR h0R ,
! N:EY+>5=EY;AW& #) EY$?p 1# !N:+>EM5 1r-0Q.F0;<&Y-0)4/V3-81@Q.$.X'AW&s-0/092(\1@)7/8&:3t; -0pEY)#5>92-01@/0C,E AWN2O98&41@ !&:p$, !/2+>1#5N8OgpD(.F^90&YEY-05 / (.98)=+>Q.C,/8&:; d ª1 (.90EY5=C.RS1#+ +>"c1B98&41@C!NGFGA R
P
R
νR < 0
R
νR > 0
)
-0/21@$'+'1@98+.)M3w1Z&:$n/21@"%(.Ac1B98&41@C!NGF^Ag;l1(*)e-0/21#I
A + 2B → 3C + D )
∆h0r = 3h0C + h0D − h0A − 2h0B ,
NY)B;<)#_Y$?1EM1@(.AW-1#IF01@C!)
∆h0r = νA h0A + νB h0B + νC h0C + νD h0D ,
R
ee
=33 8#"%'& / F% 4!" % &# %
5>-0Q.$'&:CT >d%
)
−∆ h0
tw (
0 ( w ∆h t
)
O
∆h0r
produkty
/ 1(/ /3. 54$6% !% &# . % 1+/ ! R ) +; % 3. 3#/% I6 H. 0;K1#K7^/ %F4 "G ]6" R U$U' GI/ %F437 0 $= #3.$/ . 1(/ +=3H. / 3 8 1 # 9. 1 +R UE/ ∆h $= r 3.$/ . { # 1 a /! "3.}# → G+{R U H. }1(/ +=3H. ?/ %F4 1# 9/! 3.# D= 9. 1 . 8 / ! 3. 3#/% H. 0;K1# . 0/ # (G G. / 3 8 $= #3.$/ . 0 #Q 1# 9 /! 3.# E 4 3. 1(/ %! I/ . ∆h tw % &?. + 0/ # (G K7% $= 0 +%'& # =#MC 0; (/ "G 1 H. / F41+/ ∆h # tw X( 1+/ )8 % G " $ H . 0 ;K1# 4 3.O. G ; P. (; 3. =^1 %56.# + R $= #3.$/ . DU #% + <
R"1(/ +=. DU'
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νC = 3 ν D = 1
) νA = −1 νB = −2
)
)
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%F4 !" # #
$ % &
*
∆h0r
*
∆h0tw (R) )
∆h0r =
X
±^ >d%M²
νR ∆h0tw (R)
R∈{ }
R
−
X
νR ∆h0tw (R) = X
|νR |∆h0tw (R),
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R∈{ }
e
f
f
<e
;
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=33 8#"%'& / F% 4!" % &# % )
.(-0Ce90#EY+5,(?1#+>C=Q 98M36$?1@p $.AW&q98&41@C!NGF^Ag3A -0/21@$.AW&D98M36$'d6)43wf1@1@Cr eA36;lAW1YN(*3)e-0/2)#1# !I #"%$n5=; 0 −3H2 (g) − N2 (g) + 2NH3 (g),
νH2 = −3 νN2 = −1
0
νNH3 = 2
)
X
νR R,
.( 90EY5| N:EY/05=92; 1@C!N:AW&<98&41@C!NG+>F^"cA 1EY-0;BQ.X'AW&:-8$./0Ac921h1@F0/8T43-0AWA A%"W)eH8N:A (*)e+>-0"cE41N:E4(.&:98 !),#"+>%Q.$,C=5=/8N243Okq98d &41@ !&:$>b /8$.43AW&:q;Zd\²!p.&:/8_4)&:"%AEY;BQ.C.Ac1@RS$n1#5]+
νR < 0
νR > 0
dnR
νR
)
!90Q.+>$'EYAW&:&tCª(?981@&41@921@C!;
ξ
dξ > 0
dnR < 0
dnR > 0
nR (ξ) = nR (0) + νR ξ,
nR (0)
G = G(T, p, {nR (ξ)})
{nR }
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%F4 !" # #
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G
ξ
G
nR
ξ G T p ξ G ξ T p ) X X ∂G dnR ∂G =− νR µR . A=− =− ∂ξ T,p ∂nR T,p dξ
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R
A
)
)=+>(\)M36Ac1#+'1B36AWYND98M36$?1@$.A%Q
∂G ∂ξ
= 0, T,p
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5]N2O'&:;BAWN:EY$'&V+'1@$'&q-36E4)#98&:; )
A(T, p, ξ) = −
X
νR µR (T, p, ξ) = 0,
R
ξeq
µR
)
! +>EYAW& )#EY$?)#1#EY$?N:EM1#1ZN:EM(*1r)#/8N:&:AW$'H8$.NGF0AW&:1!$.RUAWN8&«O'&:-8/2;B1@AW$nN:Q EY$,-05{/21@)=$'+>+'(*1@)49836+.Ac)41#3w+'&:1h !F8)#N:5 ± /8&:;VQN:AWX?H0$.1@9AW&²b p $$'.A%)4Qg3p?1@1 eAUN2O'F^&:&Y;B-0/tAWN:N:EYAW$'H8&G$.FqAW&:;l$.AW1@&:;V; 5N:98EM43#6-0/0$'C!)e)MH836I 5,; 98&41@ !&:$n/21 d.p |-0Ce#-0/2+1@$.(\AW)]&D(*9843)=+=bb -0/21436AW&:$.A%QT365=921@_4&:$.Ac1$?1 +.)e-0/21hF0&:;V5 ±^ >d ,² µR = µ0R (T ) + RT ln(pR /p0 ),
p0 µ0R (T ) pR
p0 = 1
µR
RT
X R
P
R ν µ = 0 R R R
νR ln(pR /p0 ) = −
)
X R
νR µ0R (T ).
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(
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X
X
νR µ0R (T ) =
R∈{ }
R
νR µ0R (T ) −
X
R∈{ }
|νR |µ0R (T )
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∆gr0 ,
∆h0r
∆gr0
exp
)
Y R
pR p0
νR
= Kp (T ),
0
Kp (T ) = e−∆gr /RT ,
Kp
*
*
xR
pR = pxR
R
(
Y
p
pR
xνRR = Kx (T, p),
R
Kx (T, p) = (p/p0 )−∆n Kp (T ),
∆n =
P
R νR
)
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%F4 `/ 9. + 0/ % &
Kx
νR
Y
xνRR =
xνRR
Y
.
)
|ν |
Y
xRR .
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R
*
R∈{ }
Kx
Kx
Kx
(
)
xR
xR (ξ) =
nR (ξ) nR (0) + νR ξ xR (0) + νR x = = , n(ξ) n(0) + ξ∆n 1 + x∆n
!±^+>>dEYAW!²& p+.)e-0/21hF0&:;V598436$?1@p$.AW1 &]$?1 98436$')M3w1@d !¶)43),+.-8/231Y1@3690Ac/81h)eF8H8#IN "%AWN:EYXn5j(*+.)@) b -0/8:(.Q 98&41@C!NGF^Axd q ¯£6¨ ' *§t¡ 7§ *£ n(ξ) =
P
R
nR (ξ)
e
f
x = ξ/n(0)
c
gEf
c
xR (ξ)
e
#/3w) )#E492-81#E4N8&:Oy90EY5,N:;iAW&:C'5VR
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) A
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B
)
µci = µgi = µ0i,dosk (T ) + RT ln(pi /p0 ),
i = A, B
pi
i
) pi = p∗i xi
i = A, B
xi
i
G/8N³UF0&:AWt_DH0$./8EM&:AW1@&:;B"W$.&:(\AW_M&qB&:9298/1@M35,/06"%Q.C!$'90)<5!)43d#)=¶1@+])# e/8$.A &:AWN:;B&:AW3w&Y(*N:1@&:E:_t92bx1@(?+>/01@3Q.921901l5!(.p'N:AWEY36&:5>90AW-03tY/8Nª-8&:E4 !;<& )<)#36-0_4C'5=&:RS92;V1#1@+>5EY5D$.$?A%(\Ce1@1>) (.p AW(.-921#1YI pe3wF0&G&YF-0/-0/0JS98Q.)#$.$.CnAW& b ±^ >d !² F^!&Y+>-8EY/ AW&(*)#/8&:$'NGF01!R&:; N8O'&:;BAWN:EY$,5,;¬N:EY5>-0/8&: !)V-8C.RS1#+>d $.*A%1@CeQ,1 3w1@3r_Y;V985!M36p¢_4$'& )431#+>E4& N:AW&YN:E:bx(?1@921>p'/SFYd µci = µ0i,dosk (T ) + RT ln(p∗i /p0 ) + RT ln xi .
p∗i
)
µci = µ0i (T ) + RT ln xi ,
µ0i (T ) = µ0i,dosk (T ) + RT ln[p∗i (T )/p0 ]
*
µ0i (T )
i
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B
A
) pA = p∗A xA
µcA = µ0A (T ) + RT ln xA ,
µ0A (T ) = µ∗A (T, p∗A (T ))
KB
) pB = K B xB
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AW5=)B; [email protected]#Y+'Nq1@-8;i/8)e5!-0p.Q#_4F0&D#N<-0C'±^RS 1#>d+>$.e²A%6C +>"c1 $?1#+]98)#p>EYEq/3UN:)#AWH098$.&:;yAW&:$.F^AW&Y&:-8; /w n1hb )#C=98&YH0"W)#$n5=;¬(.90E4&:Eq(.92143U) &:$.905 &: !).p'+.)e-0/21hF0&:;V5 ±^ >d% !² !±^+>>dEY%AW& !²w-0/8)e-0Q#F8-0AW 3v(.90EY5=(?1#+>C=Qgp> !+>5 p.1d 436$?1@$.Ac1T±^ >d~!²UAd A )#90;l| 981@#_"%$.(.Y$.AW90& EYAcB5=;l(?-81h1#AWeF0+>Vd C,Qr)#$'98&6)#/2EY1@/Ce3wD)#90-2Q1@;l)jD36(*AW)e:-0C>/2-01#E4Ie&GpnFB1"%F^AWN:&YEY+>X.5=AW$.&KAW&6-0C'36RS5=1#+>921@$._4A%&:C!$.43qAc1qpU$?EY1 $?1hFGb +>-0/8Q#)eF8-8#)MNY3w&: !1#)I+>-0AW"c1B3´(*)#/83w&:1@$'90NGQ.F8$.1!RCe431#N8ON8O'98M3&:;B6$'AWN:)4EY3w$,1@5= eN2AON:(\AW&Y)eN:-0E:/2bx1#(?I1@92±^1>>dp/2 !1@²6C=(*_4)=&Z+ ;<3w)#1@_490&:Q.;i$>5 b C=-0/8AW:&:_4; &:$.p*Ac_41#&N2OB3tA=-0EY+.5=) -8/0C#C,1@AW_4&+.&G-0FQ.X'EU-0$./2AW1@N2$'ONGF0-8&/8)e98-0)#Q#EYF^(.&wQ'-0-0AW E4N:(.E492)#1Y$'3U&B) 36 5>&:-0$./890:5(.Q#&:F8 !).p#3¯1 $.+.AW) -8C,98)#AWN2E:O b (.EM1#Q'N8-0O'E4)=N:EM+>1@EY"%A.$.98A%&4Ce1@1 C!NG-0F8/81ª)e-8N8Q#O'F^&:&<;B-8AWAWN:EY(.$?921>1Ype3U/8)q) RS1@/1#3w)#)qQ."%-0/2(.1>92d14!3t&Y+>H0"%EYA AWIe3yp!-0/2/81@)eC,-0Q#A%;®F8#N 98)#3EY1@/3U90Q.)#$'90&:E4C & 98(*43)e6-0/2$'1#)MI 3w1@ eA6N8O'&:;BAWN:EY$'&GFl±^ >d !²p_4&(.921Y3U)K+>EYAc1!RS1@$.Ac1;l1#-D(.90EY5hF^;VQ#F0&
%F4 `/ 9. + 0/ % &
B
i=B
pB
)
µcB = µ0B (T ) + RT ln xB ,
µ0B (T ) = µ0B,dosk (T ) + RT ln[KB (T )/p0 ] xA ≈ 1 x B = 1 − x A ≈ 0 µ0A
µ0B
)
!+>EYAW&q-0/21!RS1B98&41@C!NGFGA
Y
xνRR = K(T ),
R
0
K(T ) = e−∆gr /RT
!
p'1
∆gr0 =
P
0 R νR µR (T )
d
98¶ &4901@EYC!5=NG(\F0&)#;BN2O'$.&:A F^;B;VAW5TN:EY$?$'1h&eFGp(.98AW&:&: e903qQ¢RSp?1K_4&qJL1@3E Q.C'A%X.RS1#X'+'-21T1#N8;lOgp¢1K3(\C=)e/8-8#9/21#05=I{N2OT±E4)#$.XgAW&DdEM98431#6N2O'$?),1@+>$.EMAW&
± >d !²0² Jx!1@+>E!EYpgAW& 1 )#EY$?"%AW1#N:N:EYEMX*1B"%-0AWN:/8EY)#X\(.$.A -0-0C'3URS1#)#+>X*$.)=A%+>C!5!43qp N:p EY5,"%A $."%AWAW&:N:EMEY1@X*"W]&:_Y3t$n5>-0N8(*OknR
(?AW-01@/092$.1@AW;<&GF0#&:/0N:985>43N8O /8$'&:)4903;<1@)= e+>A 5,$? 1@JL;B1@E!AWdN:EY¶$,)M5=36N2Og5,_4p -8C=EY/85j#90EY5=36N2AcO @E4EY&:;BCjAc361@$?5=1{$.A%$.C#1TAW&]E<$?1@Jx1@90C,Q'/0-0QgEM1{p_43w&<1@3[90Q.$.Q.C=C.QRS1#+>98M3EYAW&eb p
.
5
, 5
54
.1
5
5(
(
)
ls = r + 2 − f,
r ls
f
f
)+! (
0
(
3(\)#/8C=&:/8$'#9NG0F85,1!; RVN83tO'&:-0;B(*nRAWN:
AWEY-0$n/0$.5vAW&GCeF81@j_4+.98#_&:Y !$')k&T-0C'JL1@RSEY1#5!+>pU$.A%/8Ce&:1j;B(*;i&:Q'92-01@EM/0jQ.92;B1>pwAW&YN:IKAWH8$./21@AW&:C=$.AW&KAW&{-2)#1@92;<1@E& 3F^w&Y1@-8/t90/8)#)e(.H8N:AW-2A\1@3U$n& 5]3t(.90-0E4EY5>&:E -0/0C=AWN8O<Jx1@(?EM1@1#92N81@Og;
=33 8#"%'& / F% 4!" % &# %
) T p
r+2
µ1 , . . . , µr
)
f
sφ dT − v φ dp +
r X
xφR dµR = 0,
R=1
φ = 1, . . . , f s
v
x1 , . . . , x r
) ls =
r+2−f
f
r X
)
νR µR = 0.
WA N:EYX?1TEY;BAW&:$.dM$n|5>N8Os)# !$.#"%AW$'&:EM)e1@H8"WN:&:A._Y3«$n5>Q.N8C.OsRS1#EY+>;BEY$.AW&wAW&G;
ls = r+1−f
*
r X
c
)
% νR µR = 0,
R=1
% = 1, . . . , c
f +c
> EY"%AW36N:EYAcX?@EY1BC!-043/8q)#(.p$.C,A/8#9-083U&)#;iX*),Q'+>-05EM]36X,5=5=$'IB)e-0-0A (*&MR
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&:5C!$.NGEAcF01@&BQ.$.C'(.Ac1
%F4 `/ 9. + 0/ % &
r+2
(
ls = r + 2 − f − c .
(
r
c
ls
)
2Ag2 O(s) 4Ag(s) + O2 (g).
)
pO2 = p0 Kp (T ),
p O2
p0 Kp (T ) * Ag2 O
pw
p < pw
p > pw
e
=33 8#"%'& / F% 4!" % &# %
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)@5 b -0NY/2&:1#"%QkIq(*36)#(./898&:)M$'3wNGF81#1!+>R
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AWQ]F^N8&:O' !&) b ;B-0/21YAWN:36EYAc$'1@&:;i !)]5]+>(\"c)#1)#$.EYA%/C!3UM3)#90Qj;B+.AW&Y)e-0-0EMC!1@)#$.$?A%$,1!5R&: !F01@)C!(.) 90E4&Y+=b ±^ >d% !² !+>EYAW& 3 )=+>-0(\/21@)M$.36AWAc&1#+'-01ª/21@-0$'/2+'1@1@$'98)4+.36)MA.36-05,/21@; $'p+'361@98AW+.Y)MNZ3U(\&:)e;V-8/2Qg1#d,IZ¶1@)#C=$./AW5=&:3361@$'_t)eH8EN:+.A& EM'1@$."W&:AWNG_YFG5 A 98),RS1@43+D$.6F^Ac$?1&: !1@;l$.)VAc1#361j-45,p'±^X*1@>d)#$?90%1@Qg"W!)#d ²ª e*AW361@N:EY$.5,$?A%$.; 1ZA%Ce+.(\1]) )=),+'±^+{1@ ;V>d921@!²5BEYQC=p.A%;B"%Q'C#)#Ac1D1@ !$'(.#")4%90$.36EYAW5,)#AWC'N:$?AWRS&1]1#+.(*43)e-0/2EM1#1@IQn(.3921@14_Y3w;V15!p=+>_4EY&tAc1hE b ±^ >d% !² JS !3)#+>90EY(.;lAW90&<1@EY5,"%-0$./2(?AW1!1#& RS+>1C=98Qj&4EM1@1@N:C!"WAW&:&YNG_YN:F^5AEY5KEM1@A98N:M3Ac61!RS$'1])V-0)=/2+1!R/8&:&: !;B)]pg(\EM1 &:1@92"W&:1@_Y/0Q.$'90)e5!H2p4IBF01@),C+A?N:)=AWH0+B$.AWN:&:AW$.H0d$.Ac1AW³U&:;<$.O'Ac),)#1>N:_Yp!Ac$?1@/8)1_ $?(?1ª1#+>)#C, !QnR\ nEM1@1@E4$.M3[AW&Y+>±X?E41#)#IeXgd d.ª98M31@6/8)#$?;B1@$.Ac1#AW&Z-0/±^;V >dQ'!²-8A'0²Xnd 5>I6)#$?1Qn36EY e"WY+>$.Ac1@$?1V3 (.90EY5nb
'$%015
'5
5
.101
.
"01 5
!
(
1 )
!%(
0
2
R
aR
µR = µ0R + RT ln aR ,
aR
86
µ0R = 1
*
Y
)
aνRR = K(T, p),
R
0
K = e−∆gr /RT
K
∆gr0 =
P
0 R νR µR
n 1@C36AW&:;V5!p'(*)#/8&:$'NGF81!R n1@EYQT+.)e-0C!)#$?1!R&: !)l;l1(\)e-8/21#I (.1j90EY$.5«AW&{N:)«EY5=-0;µC.RS1# e+>36$.AcA%1@CrE4+>;BCe1jAW&Y-0)#EMEY1@$?$.1#A%N:$nEM5!1>d p,_4/2&T1@$PN8O'-0)=/2+>1@EY$'A +')1@98+.N:EY)M5>36-05/2F0&Y-0-0Q./lX'-0/0/2QP1@$'-0NG/2F01@ $'&:; p )
µ∗R (T, p) = µ0R (T ) + RT ln(p/p0 ), (
R
N$':EY)e5>H8-0ID/8&G nF[email protected]'+.-0)e/21@-0C!$')#NGFG$?A 1!(*R&:)= !+)DN:F0AW&YH8-0$./6AW&:+'$.1@AW$?&:;136E4)#98&:; X?1@94d>|j)#X*&YND/8&: !)<1@C,/5,3b $.AW&:EMª1@"W+>&:5_Y$.AW&VF0)=&Y+-0/ -0F0Q.&YX'+>-0$n/25=1@;y$'NGF^E4Ag&/-83UC.)#RS1#90EM+>#$.NYA%&GC!FtM3v n1@;BE!d AW&Y-8EM1@$.A%$n5< n1@E4M3v+.)e-0C!)#$?1hb nR
5>1@N8EYOgQgp¢p /8) d " $?1@C="W/&:5=_Y365$'(*)e)BH8ID(.-098C')eRS-01#/0QK+>$.EMA%1#C#-01 /2!(.AW36Ii5,N:$'AWH0)e$.-0AWA &:$.36AWAW&:Y; N N:EM#-0/0C!)4365=; /8&: !) ±^ >d% e² ª > + 5 0 F Y & 0 ª / n @ 1 4 E : & ; 0 9 4 E Y & : N Y E = 5 6 3 W A 0 / , 5 ; p 8 / T ) : N W A 0 H . $ W A : & . $ W A < & : N M E # 0 0 / ! C M ) U 3 e & p p EM1#-0/8:(.Q#F0&:;V5"W)#/0$')eH8N:Ac>p p.N:EY5="%A
%F4 `/ 9. + 0/ % &
p0 = 1
)
a∗R = a∗ = p/p0 ,
R p
pR
)
R
aR = pR /p0 .
R
pR
fR
!+.+>)eEY-0C!AW& )#$?1!R&: !F0&Y)B-0/B"W)#"W/0)#$'/0)e$'H2)eIwH8F^N:&YAcj-0/ n(*1@)EYQ7(.983 )e-0/0-8Q/21@98$.43AW6& $?-01B/21@N:$'AWH8+'$.1@AW98&:+.$.)4A%36QT5=N:; EM#dw-0/0 C!)4"c13w&: n;V1@EYQgQ p +'@_Y5+.) npn1@1ªEY(*QT)#+.$.)eAW&:-03wC!)#1@$?_ 1!3R&: ! e).921@p?$.36AWN:AWY5BN $.AW-0C=AW&: !)qN:AWd H0$.AW&:$.Ac1q n1@Et90E4&YN:EY5,36AW-8/5 0 aR = fR /fR ,
0 fR
fR = p R
fR → p R
ª36AW1&:;V(*5!),p@+._4-8& /21Y(\36)#AW/8&«&:$'98NG43F861!R'$?N21@O'$.&:Ac;B1AWN:EYA%$nX.5X'-N:1hEYb5= -0/8Q.&GO'F&:-8;lQ.X'1 -0/2±1@E4$')#NGXgFGAMdF^98&YM3-86/gJS$?Q.1@$.$.C!AWNG&F0t± >/8&:d ,;²0²b (*&:921@/0Q.905AgN:AWH8$.AW&:$.Ac1>p'NY)B;<)#_Y$?1EM1@(.AW-21#I6F01@C!) ±^ >d% !² *-0/21 1@$'-0/2NG1@F^A$(*-8)=/21@+T$'N:+'AW1@H098$.+.AW&:)4$.36AW5&:; N:EY5=-0/8&GF-8Q.X?X'1@-094/2p¢1@36$'NGAWFGYA=Ni(.Ei90EY+.5h& FG';i$.Q#AWF^NG&wF^A -0AWw-0/21@$i/8&GF-0Q.X>b d *$?1i1@Qn$.3AW&:1@36_Y;VAW&:5i"%C#$?1#)#-0X>/8:F0:(./8$.)eAWH8&eIp4_4;<& )#"W)43BN:AW&YN:p4EY+>5<"c1A N:Ac1!R-0/21!R
5=N2Ogp436p=3vAWYN(*E4)#&9843366EY$? e1@"WY$.+>A%QQ EB3U)n)#RSX>1@F^$?:1D/8)e/2H21@N:C#Acª-2;<1@;l)#"W)43wEY;B nAc1@1@$?E4ª43qN:AWp H8$.EY;BAW&:$.Ac1@Ac$?1U1F^&Y(*-0/)#/8EY&:$?$'1#NGN:F0EY1!$.R
QjAW&6N2;BO'$.&:;BAW&GF0AWN:-0EMEY1$'&:3s !) (.9036EY5,5,bb (? n1#1#+>+>$.C=AWQ&:$.N:AcAW1#&YN:N8EYOj5V+.A=)#N:/Ac5=1!N:R¢EM-0#/2N:1!5>R
5>N8OjN8O-8$./21@A%_w$nQ{3«98(.43906EY5=$'(?)431#+>1@C= eQiA N8 nO'1@&:E4;BM3qAWN:deEYf$'1@&GFªCq;<36AW)#Y_YNw$?3«1lEMEM1h1hbb EY365>N:EM1hFªEM1@$.AW&Y+>X?1#IlEM1@"W&:_Y$')eH8Il1@C=/5=36$')eH8N:A )=+N:AWH0$.AW&:$.Ac1+>"c1N:EY5>-0/8&GF
*
*
µ∗R (T, p) = µ0R (T ) + RT ln a∗R (T, p). p0 = 1 dµ = vdp
86
T = const
a∗R (T, p0 ) = 1
-C,0Q.98&YX'-8-0AW/2&D1@$'N:AWNGH0F^$.A AW3&:o Jx1@EYAW&
=33 8#"%'& / F% 4!" % &# %
a∗R (T, p) ≈ a∗R (T, p0 ) = 1.
9 8 # ) Y E / U 3 # ) 0 9 4 E & . + e ) 0 ! C # ) ? $ ! 1
R = 5 ` ; \ ( @ ) b /8&:$'NGF01!RN8O'&:;BAWN:EY$n5Ce1@_4+.&: !)<-0C'RS1#+>$.A%Ce1B+'1@$n5F0&Y-0/636E4)#98&:; ±^ >d !² ,N:EY/25=1@-8$'/8&:&G;F-0Q.-0/2X'1@-0$'/2+'1@$'1@98NG+.F^A )M365,;)l/8&G-0FC'RS-1#1@+>;<$.&GA%C#F1 /8&:;B(\3´&:921@98)#/0Q.EY/903UE4&)#90AE4N:&lAWH0F0$.&YAW-0&:/B$.A%36QAWYNYN )l-098/2)#1@E:$ b /3w#94p#1 F0&: !)1@C=/5=36$')eH8IF^&Y-8/(*)t(.98)e-0/0Q98436$?1 Q¢RS1@;BC!)M36A.;<)#"W)43U&:;iQ
!®|
)
µR = µ∗R (T, p) + RT ln xR .
(
R
R
aR = x R .
)
63;B" AWAW-0&: /$.#5>$.$N8AW%O&e'd.&(98)#|
EY)*(. +(.Q'90,-8EYE45=-N:(?EM1@1#"%+>$.C=A%CZQ A*98%)#-0EYQ..(.X'%Q'-0/2-01@E4&/N:$' EMNG 1@F00"%$.98 A%C#)# 1BEY !(.36Q'®|5,-0E492N:1@E4_4)#&:98$?$.)#AWEY&l//03w±^921@)#>dC=92/01# Q#N8!²F^O]6&ª90-0-0E4/8AW)e&YN:-0),EYQ#+=5nF0&bb 98-0+>AW)#5 wEY'(.$?CeQ'11#-0NG)#E4F8 !N:1qnREM±^1@\ "%+.$.>d)ªA%Ce 981V!²)#*EY+'(*/@3U)#_Y"W)#5l&:90 nQZ+.16)Z-0$?A%e"%1$.d,EMAW&1#"c-8981q/2)#!E4X?(.N:1@AWAW98&:&:+>o'$.EYN:A%QiAWE4&G)#FUQ¢$'-0RS&:1@/8 !:;B_4).C#)#p!1t$n !5>+>;
6
A
µA = µ∗A (T, p) + RT ln aA ,
aA = 1
(
µ∗A
µA = µ0A (T ) + RT ln aA + RT ln a∗A (T, p).
3!&Y+>(.$?901@EYC5=X.(*"%A%)#_4$.&:$.AW&:A%3wQ1@e_]p1@36C,AW/Y5,NK36$')e-8)e/2H81@Il/0$.N:EYA5=36-0/85=&G92Fq1@EN:AW&Y3aN:EY5±^ >d"%Q.eX!²
%F4 `/ 9. + 0/ % &
'+N:EY1@$5=98;V+. )4 5 36 /05i&Q{+> (."c190 E4-0&:Q. E #X'$-8/2%1@'p'&($'36 NG 5=FGA! X.98 AW)#&:92EY 1Z(.Q'-0 -8AWqE4N:A%E4$?)#1#$'N:&GE4 F:&G p@FC=%$./8#9A%%_i2&(t 3° +>0"c(.1t 90 EY)=5= +> (?98#_1#Y+>$.C=AW,QK&:/2$.1@98Ac)#$71 EY)#(.-8EY/2Q'$?1@-0$>1hE:bbb (.;BN:EMQ'AW1@-0N:"%E4EY$.N:$nA%E4Ce5)#1>$';ld?&G¶1FQ.3wF0$.&Y43C=-0w/]/8&:N:98; EM)#1#EY- )=/3U+>(*$.#9)e{AW-0&Y/2-01#+.AWI<&:)e$.-0±C!AcE41l)#)#$?Xg+>1!d."c1R
985v)#EME41#98+>)#N8E!O'E4d¢N:)MAW3w~>&:do'1@$.N:EME4Ac1]1#)#+$n-05!d AWqd,² -8¶Q.)#X'/8-0&:/2$'1@NG$'F81!NGFGRBAN298O')#E:&bb ±^ >d !² !$.+>A%CeEY1 AW&ZA%$'dt+.n&:1@C=CP- 36AW+')#1#EYIe$?p1#N:-0EM/21@1T$°$.AW-0&Y/2-01@C!$')#+'o'1@N:98E4+.&:)M$.36AW&Z5+>)=Q.+>_4&Z(\)M9836)#E4Ac1#N:+'AW&:1o'N:/0E4Q &:$.AW&]-0C'RS1#+=b d fg±^ 90>dE4&: X?!1q²wF0F0&Y&Y+>-0/ª$?1@$?CZ1(?)#1@ !;BnRAW:-8/2(\1#Ie&MR
p¢$._4AW&q)#$,+>5{"c1/5,90E4"%C!&Y)]N:EY3y5=36 eAW-092/1@5>$.N8AWO{N:5 -0Q.X'-0/21@$'NGF^dAgEY³w36)]Ac@36E4AW&:C b NY&GFYp.p=3(*)e)=-0/2+>1#98I #_Y$.AW&:$.EMA%1@QK"W&:)=_Y+]5Z(\/21@)#C=/8_4&:&D$'NG),F8+1!R
QK98)#EYN2O'(.&:Q';B-0E4AWN:N:EMEY1@$'"%&:$. !A%Ce)Z1>N:p'EY365>-0AWY/8Nª&GF63-0Q.)#X' !-0#"/2%1@$'$')eNGH2F^N:AxAp 90N8E4O'&Y&:N:;BEY5=AWN:36EY$,AW-05,/8;&Gd F -0$.Q.$,X'5,-0;B/21@A$'-NGRF^)4d Ax36pe"W5!&YpUN:E-0/2O.1@A%$7(*)#-0/8/2&:1@/$'5>+'N:EY1@$'98+.&GF )M36-0Q.5X'-0$./2AW1@&$'F0NG&YF^A?-0/<)D/0(*Q7)#/8-8&:/2$'1@NG$'F8&:1@; "W& 98+.)#)BEY903 |E4RS&Y1#+> e-0E4$'90)#Q.)e$'$'H2&:N:N: !AgAW)&i n1@90 nE4EY1@QT&YEYN:QEY+.5)e90E4-0F^C!&Y&YN:)#-8EY$?/t5=1!/836R)<&:AW !-0(*/8).),&:d +. !¢).)#36X.p\$?AcC,@1l/8E4#9&:-08C5=&: !/0Q?)1#3 NGF0RS1>1#p!-0F0$'1@)eC]H8N:3AEY(.X.90"%EYA%_M5=1h(?F81#+>-0C=AWQ (
B
)
µB = µ ∞ B (T, p) + RT ln xB ,
B
∞
xB = 1
xB → 0
µ∗B µ∞ B ∗ µ∞ B 6= µB
µ∞ B (T, p)
*
µ(T, p) = µ0 (T ) + RT ln(p/p0 )
F0)=&Y+>-0(\/q)M-036(*Ac&M1#R
+'$.1]AW)#$nN:AW5{H0$./AW5=&:"%$.C!A%)Q 3 e921@$.AWN:X?5 1@94p*+>"c1ZC=d /8!#9&Y8+>&: !$?)Z1@C3 -0RS1#/21@-8$'$j)e-0H8/2N:1@A$' n+'1@1@EY98Q{+.)49036E4&5 b N$.:EYAc15=36;
p→0
µ0 (T ) p0
# ) ! #" % . $ W A : & . $ W A & 6 3 = 5 2 9 @ 1 4 _ : & . $ c A Z 1 ^ ± >d ! ² ? $ D 1 . ( 0 9 Y E = 5 ? ( # 1 . + : & < C 6 3 W A : > C 0 Y E = 5 2 N Z O 0 8 / : 4 _ : & o 63 5,;l1@ n1BEM1#-0/2!(.AW&:$.Ac1BQ¢RS1@;BC#1;<)#"W)M3U&: !)]1@C,/5,36$')eH2N:Ac>p'N:EY5,"%A ±^ >d ,² ¶$'&:)# !98)M3698$n)#Q#E4F8N:#AW&:NTo'N:±^E4 &:>d$. Ac!1>²pN:E]EY5=±^ "%AU>d+> "c,1 ²p\36AW+>EYA%;i5!pp¢;l_4&i1@;V3¯5 e921@$.AWN:5T$.AW&Yd -0C!)#1ho'F^N:36E45,)@bb !A ),EM+>1@(.$.AWAW-0&GQ#FtF8/8#)ZNV361@C,5=/925,1@36EYAW$'Ie)ep¢H236I(.F0981@)MC!3w) 1#+>EM1hF8#N3t-0(*nRN:EY5,$.$.A%CK1@C=/5=36$')eH8N:Axp p ±^ >d !² 3³1!RS1ª*1@p>A%$>;B(.JL90)#Ac1#EY905-0;l/ N:1#Q¢EYNGRS5=F01@1V; ;B)DC#),1l+.;
=33 8#"%'& / F% 4!" % &# %
#
µB = µ ∞ B (T, p) + RT ln aB .
#
xB → 0
a B → xB
γB
aB = γ B xB .
γB
γB → 1
*
xB → 0
B
[B]
B
mB
B
B
cB
[B]
mB
cB
$(
cB
xB = χ cB /c0 , 86
χ cB xB
xB → 0.
c0
)
0 0 0 µB = µ ∞ B +RT ln χ+RT ln(cB /c ) = µB +RT ln(cB /c ),
µ0B = µ∞ B + RT ln χ cB
cB → 0
c0
#
µB = µ0B + RT ln aB .
9. / 4 +
" C,/5,36$')eH8Iep a
p>EM1@(.AW-8Q#F^&:;i5/8&:921@EV3P(\)e-0/21#N:A
±^ >de~!² (.90EY5N:EY5,; +>"c1 ±^ >d !² f+.& 1@'CV$.36AWNGAWF8Y1 N (.1@C,90EY/5,5q36EY$';B)eAcH21@N:$.AAW&±E4-0)#(*Xg)ed -8)#EMX.1#Qi+ d(\ )#!²;BDAc1@)#9092Q1@E<-0/8+.:& _4'&:$.$.AWAcNG1F01TEY;B-0AW/2&:1@$.$nAcQ1 -8-8AW/2 1@$'/21@+'C=1@_49^& b /+.±^5= )436>d3w$'&: !)e!²).H8N:d*+.A*¶)B98)##_±^98Y M3$.>d6Ac $n!Q#-0²AWF8d #¢Ni3r;B36$.AW5=&:AWN2$.92O]1@Ac1ª_4)&:-0$.AWN:wAcEY1{5=/21@$.±^C,$._4>dA%&wC (*!)e²p=-0AUNY/2)V1#±^I 36>dJ5=Q.e$.$.~!A%C!²C#N:p¢1V5h36FGE$?AW+>(\1 EY)=3tA%+.;V-0-0(*5!/21Yp¢nR36_4N:&iAWEY&:5=1@$.$>C,Ac1 bb $.)#$jA%Ce13 RS1@1#C,-0/$'5,)e36H2$'IT)e±^H8 N:>dAxp=F0!²&Y+>d $?1@$?CT1#N:$.E4AW&G&:FDEM1@;<"W&:43_Y6$.AWAc&l#Ne),p+1@C=+./& 5='36$.$'AWNG)eF^H8A Iª-0/8F0:&Y_4-0/D&:$.EMAc141]3t(*-0E4)e&<-0Ac1#/2+'1@C1 36_4&:5>$.-0AcC#1>1@p."W !)M+>3w51@$?-01>/8:p?_4_4&:&D$.AW-0&q/21h-8F^Q.&ªX'-0-0AW/21@$'(.NG98FG)#A (*98)#)#98EYNG(.F^)#Q'$?-81@E4N:"%$?E41l)#$'+.&G)
)
aB = γB cB /c0 ,
γB → 1,
cB → 0.
86
86
χ
*
86
cB /c0
cB
q ¯§t¡ 7§ \¥ '§t¤i§ 7 |v$.(gAWd\&:"W3°&U-0$?Q.1#X'-8-85=/2NY1@)#$'$,NG5,FGA,; 36985=)#-0EY/8/:3w(.Q#)#F890#E4&BN:5>3UN8O)=+>3$n5=98)#; EY/3w-8)#)#"%92A1#C=N8Q'O
)
gEf
c
f
!90Q +>EY3 AW& 98)#EY/3w)#90E4& A 3U)=+>$n5=; )#d EY¶$?1#),N:+.EM)#1hX.F0$.>AWp@&B)=+>F01@(\Ck)M363´AW&Y(.+>90$.EYAW5,).(?p8F01#)#+>$nC=5qQ-8)=-0Q.+>X'QV-0A,/2N81@O.$'"WNG)@F^b A $.AW&GF0)#$')4365>N8Ogp¢(\)#/8&:$'NGF81!R
5N2O'&:;BAWN:EY$'&tF0)#$'43EM1@(.AW-0Q#F^&:;i5]3P(\)e-0/21#N:A NaCl(s) Na+ (aq) + Cl− (aq),
Na+ (aq) Cl− (aq)
)
(.NY&:90$nEY5]/0921#N:NGEYF05,& ; (\)e-0E4N:E4&: !#"%$,5=N2A OF^)#$'M3r3r98)#p=EY !/+>3UEY)#AW&90E4&ep.1 A A)#EY$?1#N:AWN8EMOZ1hF83t-0C!(*)#nR$>cbb N:EY5=$.$.A%C,A61@C,/5,36$')eH2N:Axd!&YH0"%Aw98)#EY/3w#9F^&Y-0/&:"W&:C,/0905>N:EY$.AW&l)#X*)@F0:/0$n5!p/8) µ+ = µ0+ + RT ln a+ ,
µ− = µ0− + RT ln a− ,
a+ = γ + c+ a− = γ − c−
c+ c− γ+ γ−
N41!R
C!)436A%/21<;<)#"W)43w1Z&:$n/21@"%(.Ac1l-03U)#X*)=+>$?1F^)#$'M3qp p>365=$')e-0A
=33 8#"%'& / F% 4!" % &# %
gj
!+>EYAW&
)
gj = µ+ + µ− = gjdosk + RT ln(γ+ γ− ),
),N41!+>RS(*1 )4A%36$>AcJS1#)#+'90;l1r1#98NG)#F8EY1/3U) )#)=98+.)4N836O,AZ5,"W&:+.$.)eAc-01#C!N2)#Ol$?),1!R+&:;V+.)eQ -0C!98)#)#$?E41!N:RAW)e&:o'H8N:N:A#E4F0)#&Y$'-0&:/;iEM1YQg3dD1@f90/21@1C3«36A%AW"WY)@N b N:&:EYC=5,-8(\$.AW&:&905=3t;<-8&:(\$nnR/21@N:"%EY$.5=AW$.&w$.)#A%C,C!9843&YH8"%AW1@I C=)e/-85=)#36X.$'$,)e5=H8N2N:OAxp 36C'RS1#+.43dg«³w(*)]),36N2O'AWY),NY+>&GEMF:#p*N:$.5>AWN8&OB;<)=)#+_YF^$?)@1 b $'Ei43/5,; +.),36+'(.1@98/0)4$.3AWN21#O]+>A¢EM1]Q#F0-0&:AW;B$nH05>98&YN8+>Og$.p.A13tF0&Y-8+>(\5=nR$.N:AW&EY5=AW$.N2O$.A%RSCK#N:1@EYC=$,/5Z5=36&JL$'&:)eC,H8/4N:dnAx|p EY36p'E4Ac@+.EY& C,.Q b $.AW)431@$n5F81@C!)
γ+ γ−
86
γ±
)
γ± = (γ+ γ− )1/2 ,
)
|3U&:(¢/R
5,33`(.)=90EY+.5=+>(?EY1#Ac1!+>R
C,5=QK3w1@98)#okE4N:&:AW"W&:&:o'C=N:/0E498)e)#-0$n/25>1@N8/O5=N:98EY)#$,EY5=/N23UO )#98F043&Y-0/ªF0$?)#1$')4/365="W5>&BN8O{-0A%"%36$n5!5>p -0/8_4:&(.$?Q#F01h b EYe$?d 1#{N:EY43$'6&ªA.) ),+./5=N2On;°5="WEY&:36$.AcAc1@E4),&:+Ci(\+.)e)#-0;BC!AW)#Y$?+>1!EYR5V)eH8C!N:)#Ax$'p.NYN:&:EY$n5=/0"%A921#NGF8U98F^#_)#Y$'$.M3A «-0AWA=3tAW-0-0/8(*)#nR/0$.N:AWEY&5,),$>+ b $.3wA%1@C,_MAW&:1@;$,5,; /0p*QEY3w(.901@EY$n5,5(?1# e+>921@C=$.QTAWN:+>EY3U$,=N5,8; OlF^(.)#92$'1Y3wM3&:; ;lª1B&:)#Xn$'5!)B& (*1h)eb -0/21#N8I C!&:"c1>d| 98)#E:b ±^ >d e!² !3w+>1@EY90AW/8& )eH8N:AW)MF^&Y3U-0)e/H8+.N:A),+'F0)#1@$'/0$.43qA%; p1 3t-8(\nRN:EY5=$.$.A%C,AW&:;d¶"%92AW14N:3UEYX*){)436/8)T5=; )#X*EM)4361@"WAc&:@_YEY$nQ#5=F0; &Z+>),"c+1 ;l1@"W&T1!R&G(*F )=C!N8)#O'$')=NY+>&:$?$n1 /0921#NGF^AMF0)#$'43q3 d ±^ >d e!F^²\&Y-836/Z5=$.$.AWA%&YC#-01>C!pY)#_4o'& N:E4)#$?1>p6NYpY)k !+>/R
5 Q.;l1#N:EY5 p -C!0(*)#)#$'98NY& &:$n),/0+.921#N2NGOnF^5=A'"WF0&:)#$.$'Ac1 43qd )=+F0&Y+>$')eH8N:A$?143U&:/+>"c1j-0/8)e-8Q.$.C!)43U)«;l1!R
5>N8O µ+ = µdosk + RT ln γ± , +
µ− = µdosk + RT ln γ± . −
γ±
γ±
)
ln γ± = −wc1/2 ,
w
c = c + = c−
$*
dγ± /dc γ±
c = 0
γ± → 1
c→0
e~
9. / 4 +
!$. &:AWN:_4EY&:$n"%A 5]3¬3Q.-0/2C.1!RS1#R&G+>F6EY/8AW&K&:;BEM(*1@;B&:92C=1@$./0Q.AW:90/E45=&D;A (*(.),90+E4&:-0X./2AW1!&:R
n5=1;¬(\N:&:AW36H0$.AW&:AW&:$$.AW(.&:98; )=NYp.&Y/8-l)<-8EY(\;B)#Ac$,1@/2$?1h1 b &:$n/21@"%(.A%A-03U)#X*)=+>$'&GFQ.C'RS1#+>QT-0(*&MR
$.Ac1$.AW&:98436$')eH2I $.$.¶AWA%9836&Y)=)#AWNYX>&&YF0-&::"W/8/2&:)e1@C,H8C=/0N:98AgAW)=)4;
5
5'(2 !
01!#
$#
(
)
!%(
01
)
∆G ≤ 0.
*
) |W | ≤
W
|∆G|
W = ∆G
*
(
(
Y
Y z+
e−
#
=33 8#"%'& / F% 4!" % &# %
5>-0Q.$'&:C >d
−
2+ 4
+
2+ 4
Zn → Zn2+ + 2e− Cu2+ + 2e− → Cu / 3 # 9; .$/ % &# % # / ; H.$/ #B4 3. " =+/ $ / 9). 0 / % & 4 9. 0/ 1 W6% # >61+/ ' 4 # RW. ;]=% ; H.$/ ; @ . % DU'
Y/N 3w)#9-0w(.92RS1#1Y+>36Q.Ac1>$'p>&:CB_4&+.Q.),EY+'5>1@-0/0C,$.Q#AxF0de&f)#1@$?C1
X
*
X z+
CuSO4
)
X z+ (aq) + ze− X(s),
,-21@/0;<90EM)#1!90R
C#EYQ.1/03 $.AW&D(.92(*14)B3UEM)B1@(*$n)#Q.C#90E41@&:EY$.Q#F0A%Q{&C,&:"WAW&:&:C,90Q./098$')=&:+>C*5p=3 3C,98/8)##9EY0/5=3w; )#90E498&e&41@d C!NGF01(.90E4&:X.AW&: n1 ¶ ) * ( n ) S R # : N 4 E : & . $ % A v Q : & W " : & = C 0 / 8 9 = ) + . ( 0 9 4 E : & U 3 = ) > + . $ % A = C W A : & ; A . Q < ; # ) Y _ % " % A 6 3 W A : & . $ % A r Q . ( 0 9 4 E & b (¢R
5,36QqF^)#$'M37(*)#;BAWY+>EY598)#EY/3U)#921@;BA )#Xn3w,+Z&:"W&:C,/0905>N:EY$n5BE4)e-8/21hF^&6EM1hb (
Y (s) Y z+ (aq) + ze− .
' ;B)# eC=$.$.A%3wAW:1#/5qN2OTA,EME41#)eN:-0/2EY1h5,F0$?&1EM(¢1@R
X.5,Q.$?90#E4IU)#$?(.1>92#d + ph"W3&:C=36/0985=)#$.$nA%5]C,Q(¢R
N:5=E4$?&:# !NY)&D98M3),6+$'Q#)4F03&:;B1@ n$.1AW&ª3$?(*1!nRRS1hcbb +.+>Q#)43F861@EY$';B&GF$.&:AW"W&G&:F0C=-0EM/0981@)=$.+>AW& 5 -0AW+.RS1#) +>+.Q.)=$.+'C!1@43/0«$.AWEY) e98$?)#1!;lRS1#1#+.+>)4E43)#1@$n$'5>&GN8FOB&:$?"W&:1C=&:/0"W98&:)=C,+>/05 98)=+'1#(\N8Og)M3UphNY)@) b 37$'43C! )#²6$'$?-2&:1C,&:3w"W&:&:C,$'/0NG98FG)=A¢+>Q.EY90AWQ'& N8O?1@;BAc1i98&41@C!NGF^98&Y+>Q.C!NGFGA±(.90EY5.RS#N:EM1@$.Ac1i&:"W&:C=/098)@b ±^ >d !² Ag98&41@C!NGF0ªQ./0"W&:$.Ac1@$.Ac1T±S)=+.+'1Y31@$.Ac1l&:"W&:C=/098)#$'43 ²6$?1B&:"W&:C,/098)=+>EYAW& ±^ >d ,² /098"W)=&:+'C=1>/098p ),$?+'1]1>p#C=$?/81#98C=&GF/8#9EM81#&GFN2O'EM),1#+>N2O'EYAU),+>Q.EY/0A."W&:98$.&YAc+>1@Q.$.C!AW&NGF81>pe1@$?$'1@)=EY+'5=3w>dg1ªª-0AW),w+'Ce1h1@F8/8#),NB+'98>43pe61ª$?&:1@"W&:$.C,AW& b ±)#^ e>d$.A%36!AW²ª&q+.3P)36±^5=>d$.A%C=,Q²pg(.+.90E4)e&:-0(¢/21hR
5,F0&:36;VQK5{(.9892&4#1@+>C!QTNGF0&:B "W&:-0C,Q./0;l905>1@N:90EY5=$'N:&:EY !$?) EM1#N2O'),+>EM#N4 3 ±^ >d !² |N8O,36)#A% e"%At$.A%),36+>AW36& 98)==N+>:36AWIe92p1#(.N41@90EY"%$n5,5=C';[RS1#+'C,1hAW&:F8#90Q.N$')=&:+>CB(*98)4&4361@C!AW&YNG+>F^A$.±^AW&] >d$?1@!(.² AWY;
#
Y
X
X)
X z+ (aq) + ze− → X(s),
Y)
#
Y (s) → Y z+ (aq) + ze− .
#
)
X z+ (aq) + Y (s) → X(s) + Y z+ (aq).
B;$?i&:AcXn6+.5A,)V/8)#(.&:/0909090;<E4EY&:5=),(¢;l+>R
5=5=1#$.Iª$?AW1@EYY;B36N:AcA%Ac1BCe@>E4RS1#&:p4C+>98)#Q.-0EY$./23wC,1@1@$'Q _Y)4;i365VAc#(.(.N:90592EY1#5l(\NYU)#98#_;<&:Y"W)e&:$.;<-8AWN:/6&:5<(\$n(\)#/21@;B)#90/8$?AW&:Y$'>+>NGp EYF85]1!R&:43"Wp4&:(*C,/0)#98/0p=)=90N:E4N8EYO'&:5,X>&"%bbA
Wel
dq
E
.($nQ#90EYF05&UF0N:qEY5=Q.;¬C.RS1#EY+ $?d 1@C;BC!)#A%"W$n&:Q'A*-(.90)#EYEY5,$?(¢1#R
N:5=EM$.1>AWp?YN:_4AW&V&q(.RS921#1#+>N4Q.1ª$.F0C,&YQ -0/tQ#F036&:;BAc@$?_41>&p'-8 !AW+>5]E(\36)e5=-8C!/8)@bb (*&:;¬98&41@C!NGFGA 3°)# e$.A%36AW&D36E4)#98&:; Wel = −Edq,
*
dq
)
dq = zeNA dξ = zF dξ,
n !+>EYAW& )#EY$?1#N:EM1lRS1#+>Q.$'&:C&:"W&:C=/098)#$nQgp\1A%"W),N:EY5=$ A "%AWN:EYX,5 " #) n1#+>98).p #
=33 8#"%'& / F% 4!" % &# %
e
e
/8+')Z1Y5nF^1>&Yd +>$? 1K"c17E)#(\ e)=$.+.A%-03w/2171Y3w(.)492361#5>N:Q#N8O7F0#NY-8&:/21! !R
)v5=N2)=O +>36'EY925=1#N4N:1@EY$,"%$.5=AWN2&Ogp(.90EYEY3w5P1@$?-0/211!R
-05>/21!N8O RS 1@A921hb (.Q.92C.1#RS1#N41{+>Qg98p?&4N:1@EYC!5,NGFG"%AA N8O'&:;BAWN:EY$'&GFªF^&Y-8/98M36$?1EY;BAc1@$.AW& &:$,/21@"%(.A%A-03w)#X\)=+>$'&GF ±^ >d !² -0Ce#+ ±^ >d e² 3wX\n1@&:1#CEYN:(./(?3U92#1@)]+.;B)MN8AW36O':/2&:5=1@;BN2;iOAWN:5«EY"%Q.$'±XE4&e)#d Xg-0A
dRS$?±^1@&:>dEY"W5=&:!C=3w²0/0²1l98p )#-0;
T
Wel = −EzF dξ = (dG)T,p , EzF = −
E
∂G ∂ξ
= A.
p
T,p
A=−
P
R νR µR
(
I
I
(
E
*
(
)$.# eA%C=$.Q«A%3wRS1>#pN:EM1<#N:90Q'5,N2;aO')4&:36"W&:A C,//0R98)#)=Ce+>15!d ),+>s(*C!)4)#36"W&:Ac1#A +'),1+>(*90Q')4N236OjAW&Y&:+>"W$.&:C,A%C=/0AW98&:)#; $'M3-0A
R
5K3¯E4(.&:9036E4$'&:3w:/0),90+=E:bb $'C=/8&GF#98C!&:)# !$n)D/098;<)#"%)#Q#_YF8$?#1ªNY&GC!Fg)#90$nQ'/0N898Oi)#"W/)MR3w)#C#1#1IF090&YQ'-0N8/gOZ$?1@&:"W(.&:AWC,Y/0N:98AW)#& $'E4M3&:736$'3:/0(.90EY90E4$'&:&e3wp4(.),90+>EY$.5ªA%C=(*Qg)#pn;
#
*
A=−
P
R νR µR
)
±^ >d !² !+>EYAW&D36AW&:"%C!)eH8I ±^ >d e~!² $?EY361@EYAc@5=E43w&:1KC-0(*AW<)#;B$?AW1@Y(.+>AWEYY5N:AW&:-8; /21!RS-0/21@98$'&41@+'C!1@NG98F^+.Ax)4p 365=pg;mA &:)#$n e/2$.1@A%"%3w(.1>AcKd -03UAW)#)#92X*#)=Nl+>(*$?),]+j98&4Qn1@3wC!1@NG !FGAx p p.+.)e-0/21hF^&:;i5 ±^ >d !² fEY3w1@C1@"c1h36F8AW7YNB36$?5,1EY$?(*1#)=N:+.EY5>-0/2Ij1Y36$?AW1@&B(.AW(*Y)#N:;BAW&«Ac1@-098/2431@[$'+'&:1@"W98&:+.C,)4/03w98)=&kN8O')#&: e;B$.A%AW3N:EY1$,5=$?N21Ogp(\C=)=/8+.#9-08/2&B1436(*)@AW& b EY+>36EM#Ac@NYEY&GFC=Q 3 ±^)#>d e$.A% !36²AWp\&eQ.dEY¶5>)=-0+.C=Q#-0/2F^1Y&:36;iAc5 1hF83w#Nk1@90/8±^)e >dH8I-0!²/21!R+.&G) F ±^ >d+>"c1Z !²98p&4361@C!AWNG+>FGEYAA%;VEM1#5!N8p O'_4)@&b -0/21@$ 98M36$')431@ eAN2O'&:;BAWN:EY$'&GF+>"c198&41@C!NGFGAw±^ >d !²),+>(*)436Ac1#+'1 d RT E =E − ln zF 0
aX aY z+ aY aX z+
,
E 0 = −∆gr0 /zF K
∆gr0 ) ∆gr0 = −RT ln K
E0 =
RT ln K. zF
K
E =0
|/098)=+>(.5!92pe1@C,C,//85=#9NY05 &F036&Y-05=/ !;B),+>AW$.&:90AW&E4)#F0$n&Y5B-0/ª36;<EY e43"W6Y+.AWIl&:; )](\Q.)#;+>5=5$'N:),E4+>&GFª$.AW&:&Y"W-8&:AWC,&bb $.$n5=Ac1>; d| /0QK/(.5=90;aEY5=(?NY1#&:"%+>Q«C,Q{36(.(*98nR)43)# e1#$.+>A%EM3U1)]-0-0AWlC'RS(*1#+')@F01<YN:-0AWAW&<VEi(*nR(*)@)#F^ e&Y$.+>A%5=3w$'1>N:dE4|&GF&:98"W)#&:EYC=3w/0981@),_M+>1h5 b EMNGF81@1!$nRQ.$?90E41@)#"W&:$'_Y&G5]FiEY3yX.Q'98+.)#EY)4/33w1#)#I90E4)#&] e$.-0A%3U3w)#).AWN2p'OT3°F0C,)#$'/8#9430q5=d; +>&:Xn90Q.5j e)#A%; C=98&Y(*H0nR"%AWID)# eF0$.&: !A%3U)&:(*; )#/8F0&:&Y$>-0/b &:"W&:C=/098),+'1ª)=+>$.AW&Y-0AW&:$.Ac1>d@|/8&:$B-0(*)e-8#X;<)#_Y$?1365,EY$?1#N:EY5>Iw(\)#/8&:$'NGF81!R
5
)#fX.1@Q]C«(\36nRAWY)#N e$.A%3q+>p."c1EªC=+'/81@#9$'0&:5= !N2)jO)#EY eX.$.Q'A%+.3)4131@;<$')#&_YF^$?&Y1-8/w(.A%90$,E4/8&Y&:+.98&Y-8/2-0Q#1Y36F8#AWNYIB&DF8$?1@C!1#-6))#98 e#_$.YA%$.3UAWNY). d (\/8),)#+'/8&:1!$'²UNGAF81!"W&:R3w43 (*±L1@)#$';B)=AW+'Y1!+>² EY5K&:"W&:C=/098)=+'(.921Y3w±EY365>N:EM1hF0)43U)BF^&Y-0/$.AcZC#1hb ±^ >d 'M² !+>EYAW& #
=33 8#"%'& / F% 4!" % &# %
E
)
E = E P − EL ,
EP =
EP0
RT − ln zF
aX aX z+
,
EL =
EL0
RT − ln zF
aY aY z+
,
1 A )#EY$?1#d N:EM1hF8D(*)#/8&:$'NGF01!R
5<-0/21@$'+'1@98+.)M3U& (\nR)# e$.A%3qp,(.90EY5BN:EY5,; * @ 1 n Q 3 @ 1 Y _ V ; ! 5 e p 4 _ w & * ( e ) 0 2 / # 1 U I % A W " # ) 2 9 @ 1 Y E B Q @ 1 = C / = 5 6 3 ' $ e ) 8 H : N . A > + c " 1 # ) . X Q * ( nR # ) e . $ % A s 3 6 3 n 5 b .$ A%C#1ED(*)e-0/21#N:Ag98&41@C!NGF^Ag3P(*nR)# e$.A%36AW&qEM1@(.AW-21@$,5=N2O
EP0 EL0 EP0 − EL0 = E 0 *
)
|$')eH898N:&4AU1@±SC!&:NG"WF8&:1#C=N8/0Ok98)#/$'5>43N8¯Oj$.3tAW&D-0(*QnnR36N:EYEY e5="WY$.+>$.$.A%C,Ac1ZA -8-0/8AWM&Y²N8O.p.AW36)#AW;<YNV&:/03t905>-0N:EYEY5>$'-0/0&lC,-2AW&q1@98C=M3/65=$'36&ª$'F^)e&YH8+=N:bA 36375=(*-8nR/8:)#(. eQ#$.F0 A%363 AW&(.3tAW&:-890(\3tnR-0E4N:&GEYF5=(\$.)#$./8A%C,Y+>A\E4-0&e/8&YdnN8!O.&YAW+>)#$?;<1@&:C/090+>5>"cN:1ªEY)#$' !&#";<%$')#&GF nV(*)e-8AW-06/21#98N:#_A.Y98$.&4AW1@IeC!p,NGNYFG) A ;l1¶36)#(¢$.R
AW5,&:33w1@$?_ 1(\)e-8A /21#IªA%;l"W)#1h92F81@iEYQ{/21@1@C#C=i/5=-21@36;l$'V)eH8(\N:AU)e-0±/2E41#)#IeXgp,d.36EMAW1#Y+ Nd (\)#/8&:A $'NG!F8²1!d R(\nRcb )# e$.A%3w1<;<)#_Y$?1B)# !#"%$.AW&D(.90E4&Y+.-0/21436AWIq3(*)e-0/21#N:A ±^ >d n!² !C!+>)43EYAW1@&$?TA%$'+.A &:JSC>)#-0905;<lQ./0"W&:A $.AW)#$?{)#-0EYQ.$?X'1#-0N:/2EM1@1h$'F8NG>F^ptAxdg)=| +>(\)M$?361#-0AW&YEY+>5=$.; AW).(.p690JLEY)#5=90(?;<1#+>C,EYQ98&YJS+>)#Q>9^bb ;B/098),AW&w+>EY5!98p@&Y1w+>JLQ.)#C!90)4;B3wAW1@& $'Q.&G/0F "W&:)=$.+>AW(\)#$')M36&GF Ac1#+'+.1ª)=)#+'X\1@/0)@$.F0:AW& /0$nF0)#5i$n&:5ª"W&:;
Y z+ (aq) + ze− Y (s).
EP
EL
)
RT E = E − ln zF 0
Red
aRed aOx
,
Ox
aRed = 1 *
E = E0 , aOx = 1
| # ) ! #" % ' $ e ) 8 H : N A S J # ) 0 9 l ; Z Y E 8 9 Y & > + . Q ! C 4 ) 3 @ 1 ? $ < ; # ) 4 _ & n X > 5 I ^ F # ) g $ d ª < 1 . ( 0 9 Y E = 5 ' C S R # 1 + p F0&:&:"W&:_4C=&:"%/0A698),98+>)#EY5]/3U(."c#91@l/5=EM$'1Y)4363wAW&:&G92Ft1EM1#F0)#N8O'$n5«)=+>_4EY&:A"c1@98&4EM1@1>C!pNG/8F8){1Q.3¬/0"W)#&:$.X\Ac&Y1@N:$.$'Ac)e1 H8N:98A6&Y+>$.Q.AW&:C!98NG&4F^1@A eQ#F8#NY&GF 9. / 4 +
#
)#90;ld lEY98&Y+>Q.C!)431@$?qF0&Y-0/t36AWYNV/0QBF0)#$
Fe3+ + e− Fe2+ . Fe2+
)
p?1JS)#90;l
Fe3+
¶N:AW)e)4-036E45!N:p!E4&:(. !90#"EY5i%$'N:& EY5=;<; &:/2;B1@"WAW&&GF^;<-8NY)#&w_Y3 $?1{/5,Q.;`(*)#-090E4EM&:#98+>&: eC!QB)43EM1#1@IT"W&:_Y3´5),/0EY+3q3d1@-890E4/8&:)e98H8&:N: A.(*$?)#1@/8(.&:AW$>bb 363UNGF8&:EY1!; eR
Q<"WY3w+.-0/2&:),1@;+.$')#+'98&:1@)M"W98&:36+.C,5,)4/03w;98)=&:+> !± 5=)q¶;<)=|+>&:$./2²1@AWd*&Y"%Qg-0¶AWd!)#&:¶$./8&:Ac)#1j$'/8NG&:F8EY$'1!3NGRwF01@1!-0$'R
/2&G5B1@FZ$'-0-0+'/2/21@1@1@$'98$'+.+'+')M1@1@369898+.5K+.)M)M3U(*36&tnR5,-)#; e$.;B(*A%3wAWnR&:190)#E4 e3w)#$.$')@Az& bb 92A+.1@)#&:/0"W98Q.&:)4C,903wE4/0&:&e98 !)=d )i+>=5<¶F^&Y| (.-0"c/61@-0Q./C'5,;
(
a=1
)
& : , X 5 6 3 , 5 Y E ? $ # 1 : N Y E > 5 q I * ( # ) 8 / : & ' $ G N 0 F ! 1 R 0 2 / @ 1 ' $ ' + @ 1 8 9 . + 4 ) 6 3 T 5 \ ( nR # ) e . $ % A 3 1 ? p ? $ @ 1 W " : & Y _ 5 EY90E4X.&DQ'+.) )4F^3&Y+>1#$'IK)e)#-8 e/0C!$.)4A%3w3U)j&GFEh1@RC=)#/_45=)#36$'&T$')eE H8N:&:A>"WF0&:)#C,$'/09843)=+>5 pp=EM)#1@921@$nEDQ.90E E4)#=$'¶&G|F<3¬d ª981@)#(.EY/AWY3UN:)@AW&b 'EY3U;B$.& AWYAWNG&:;iF^90A5«E4)#(*$'&:)#"W&U/8&:&:C=(\$'/0)#NG98;BF0)=1!+.AWRYZ&:+>; EYN:5=5q-0$.&:/2C!"W1@&:)4$'C=3+'/0{981@),983+.+')41@3698;BM35=A>6; ¶$'/)4(*3wA nR1#+>)#E43 e& $./2A%EV1@3wC,1F0A%)#;¯$?1@)#dU;B e$.nA 1@A%36C!)jAW&gF0(.&Y90-0EY3[/g5=E C.98RS+.)#1#&E:+ bb 90E4/3wE4)4)#36)#$?905=1lE4;®&]3-83w)#98),"%)#A +.EY/)#3w98&:)#; 90E4&i(*d ),C=ª+3w1@1#N:/8-8AW)#QH0$.;BAW&:Ac1#$.-0AW/ &:;µ=¶A<| (*1@)#/0-8E4; /2)e1@d-0$'/2,1h)4Q.F836#;lA6N41@1B&:90"W5>3&:C,N:EY/0C!$?98)#)=1$,+'/2981 1@&4C!1@¶ N:C!AW/VNG&iF8EM1]EV1@$n n+>Q>1h"c1 bb 2H+ (a = 1) + 2e− H2 (1 atm).
X
X X z+ X
6
(
X
ZnSO4
'
Zn2+
(
H2 SO4
(
8/ &: !)<)# e$.A%3w1B/8) #
=33 8#"%'& / F% 4!" % &# %
)
Zn(s) + 2H+ (a = 1) → Zn2+ (a = 1) + H2 (1 atm).
T !+>AW5,&:_90EM#N(*)=)#+>/8$'&:)e$'-0NGAF01!-0AWRU-0+./21@) $'=+'¶1@98|+.)Mpg361l5!pEY !)=+>$.AW&EQ.;<)4336(*5=EY)#$?/8&:1#$'N:NGEMF01@1!;VRw5 -0/21@$>b p +'1@98+.)4365 =¶|u98436$?1l-0AWªE4&:98)B3PCe1@_4+.&GFw/8&:;B(*&:921@/0Q.90E4&ed f1@X\&:"c1 ed
E 0 = EP0 − EL0 ,
(
EL0 (
,
D
9/ #01 %
+ + e− + + e−
+ + e−
+ 2e−
2+ + 2e− 2+ + 2e−
2+ + 2e− 2+ + 2e− + −
2 2+++2ee−
+ − AgCl + e Ag + Cl− 2+ + 2e− 3+ − 2+ + + e− 2 ++2ee−
− 3+ − + 3e 2+
EP0
E 0 /V −3, 045 −2, 925 −2, 714 −2, 37 −0, 763 −0, 44 −0, 136 −0, 126 0, 0 0, 153 0, 222 0, 339 0, 771 0, 799 1, 360 1, 50
|p.;<&:/2/8&:1@$"%A-03w(*&D)e-83t#X-0T(*)#;<;B)#$._Y$?Ac1@1Z$n5 Q.(*-0)#E4&:90EM98&:# l+>C!$?)41@3w(.1#AWYIN:(*AW)4)#36/85&:$'±NGE4F8)#1!XgR
5Kd?f-0/21@1@Xg$'d+'M1@²98d +.¢)M3U$?1h&eb p F^X\)#)M;<36)eAW&:H2;I -0p?E4_4&:&V98&:E4 e&VQ36$?EY1@ e(."WAWYY+>N:Q{AW)43w$?&:1B !EY)ª36;lAc@16E4&:EYC$?1#N:E4&:$.AW&w(.921@C,/5=N:EY$'p'&eEYd $?*1@1@C Q,3w(*1@)#_Y/8;V&:$>5 b 36NG-0F0/2AW1!1@&]R
$?Q1#(.N890-0OjE4/2&:1@-8X.$'/2AW+'1@&:1@$' n981+'+.1@)M-2983U1@+.;<&:)4 !36)#)905>EYN8)#Q.OgC=/0d 98$.&Y*AWH0&e1!"cp 1
E0
*
) E 0 = −∆gr0 /zF
*
X
E0 > 0
X
36/0985=)=/0+>92EY#AWN4& 1@$n5DdEn98&Y)#H0EY"%AU/3w$?)#1@90/8Q)#;B-836Ac1#5=-8N2/O6;
8 #
#
X
X
)
C!)#90EY5=-0/21hF8#NªED$?1#-0/8:(.Q#F8#N:5=N2OT98&41@C!NGF^A
C2 H4 (g) + H2 (g) → C2 H6 (g), )
C2 H4 (g) + 3O2 (g) → 2CO2 (g) + 2H2 O(c), 1 H2 (g) + O2 (g) → H2 O(c), 2 7 C2 H6 (g) + O2 (g) → 2CO2 (g) + 3H2 O(c), 2
.($.90AW).EY5Vp N:EY5,; -0/21@C>$' +'h1@;<98+.)#)M"x3Up &t&:$,/21@"%(.C=AW& h98;<&41@)#C!" NGFG)#A.92361@E 5=$')e-0EM>p#)=+>C>(\ )Mh36;
−285, 8
−1559, 8
N2 O5
)
2NO(g) + O2 (g) → 2NO2 (g), 4NO2 (g) + O2 (g) → 2N2 O5 (g),
,/21@$'+'1@98C>+.)43Uh;<&Z)#&:"$n)#/21@921@"%(.E AW&BC!)#"W&GF^C>$n 5>hN8;
(
−110, 2
N2 (g) + O2 (g) → 2NO(g).
180, 5
) −114, 1
∆H
∆U
#
=33 8#"%'& / F% 4!" % &# %
'd 4& 1@C!NGF01-0(?1@"c1@$.Ac1B(.98)#(?1@$nQT/8) +. )MX.3U"%AW&N:EY&:5>$nID/21@-0"%/2(.1@AW$'&+'/1@3w98)#+.90)4E43&:$.
)
C3 H8 (g) + 5O2 (g) → 3CO2 (g) + 4H2 O(c).
) ∆htw (C3 H8 (g)) = −103, 7
∆htw (CO2 (g)) = −393, 5
∆htw (H2 O(c)) = −285, 8
25 ◦ C
)
3H2 (g) + N2 (g) → 2NH3 (g).
∆htw (NH3 (g)) = −46, 1
)
)
C(s) + 2H2 (g) → CH4 (g)
T = 298 const
)
1 H2 (g) + O2 (g) → H2 O(c), 2
)#921@Eq98&41@C!NGF0q-0(?1@"c1@$.Ac1
C(s) + O2 (g) → CO2 (g),
p = const = 1
V =
)
∆h0tw = −285, 8 kJ/mol, ∆h0tw = −393, 5 kJ/mol,
CH4 (g) + 2O2 (g) → CO2 (g) + 2H2 O(c),
n~ +3U>"c)=1+.C=#9/8t#9A 8&G;
8 #
#
∆h0r = −890, 3
*
∆h0r
T1 p=1
*
∆h0r
(
c0p (T )
T2
)
37NGF^& /8&:;BA (*&:921@-2/0Q.EY90X.E4Q'&+.e)4~ 3 1@°$'& 36Et5,$'N:EM)e#-8-8A /8&YN:E4&:CB"%A%$.AW)436C>5> N8hOg;
∆h0r = −20
A B
(
C
∆h0r
340
A B C
) Kx (T, p) CO(g) + Cl2 (g) COCl(g) + Cl(g), 2SO2 (g) + O2 (g)
2SO3 (g),
3A(g) + B(g) C(g) + 2D(g)
A
B
C
D
ξ = ξeq = 1/4
Kx (T, p)
A(g) + B(g) 2C(g)
=33 8#"%'& / F% 4!" % &# %
(."W& 904E4Ii&:X.-0/8AW&:: n_41&:$.3Ac1<-898/2&41!1@R !&GFU&:$,/8/8&:43;B(*&:3°921@-8/0/2Q.1@90$.E4AW&D&q(\9843)=6+$'N:)MAWH03w$.1@AW e&:$.A AWN2&:O';®&:;B*AWX?N:1@EY$'94d &GF:¢pn$?F^1h&bb H_Y0"%5>A I-8/2+>1!3wRS11]98(.43906EY5=$'(?)41#3+>1@C, eAAw-0+>/8"c:1_4&:/8o&GF(*98)=&4N:1@EMC!@NG/0F^C!A)436365,5>$'N8)eO -8A ±L1!² d )#EY3w1hb p p 90EY5hF8#Iep*_4&-8Q.tX'±-0X¢/21@² $'NGF^& p A p -l n1@EM1@;BAU+.)@b p d ¶ -0C!)#$?1!R
5=;BAxd 4>dt/0¶98)4@F360N85=O._4"W&G)#F 98&:C]JL)e-^JL)#90(.QAWYAN:N8AW),O.N2"W)#O.9 "W)#98&:CJL)e-^JS)#90QrQ."W&: n1j98)#EYC'RS1#+.)436Aª$?1 *
xB (0) = 1/2 xC (0) = 0 xC (0) = 0
Kp = 1 ) xA (0) = 1/2 xA (0) = 1/4 xB (0) = 3/4 A B C
200 ◦ C
)
¶ 90&4E41@&YC!+{NGF0981<)#EY(.(*90E4),&:N:X.E4AWY&:N: nAW&:1l; 3°98/8&4&:1@;BC!NG(\F^&:A92(.1@98/0#XQ..90E4C#&1 ;BAAc1!(*RS),1Z+K;lN:1#AWH0-8$.]AW&:e$.p~
PCl5 320 ◦ C
Veq = 2, 4 PCl5
Kp
Kp (T ) *
Kx (T, p)
Kx
*
)
T
∂A ∂T
= ∆h0r , p,ξ=ξeq
> B;!+>AWEYN:AWEY& $n5=; p?1$?1#-0/8:(.$.AW&D(*)#Ce1@EM1#Iep._4& F0&Y-0/(\)M36A%$')431#N:/3U&:;N2O'&b
8 #
A(T, p, ξ) = −
P
R νR µR
∂H ∂ξ
= ∆h0r
3-0/21@$.AW&D98436$')M3w1@ eAN8O'&:;BAWN:EY$'&GFYp?N:EY5="%A+>"c1 d >d ;B,/2A%1@$.$PA%;V98Q.43;6$'&:)4$n3/21@1@ e"%(.AA%A6N2O'-0&:3U;B)#X*AWN:)=EY+>$'$'&GF:&GpF:pEM1@/8&:;a+>"c1 pt),+>;V(*Q')4-036AUAc1#Xn+'5>1I $.3t-0(\AW-0Ie&ME4p*R
&D$._4AW-0&)#(*$,+>&M5K"cR
1$.3AW n)#1@1@$n90E45!Q.M3d $'y&:C{+.)e-8/2-01@C!X.)#$?A%"%1!$'R
)e5>H8N8N:OjA 31@90Q.$'&:C-0/[email protected]%"%$')eH8d N:A*F0+.&Y)4-03w/ª),EM+=1hbb =d3w¶)M1@3690/8Ac)e@H8EMIK1#Iª"%AWEYN:$?EYXn1@5rC (\)e-0/8:(.Qv98&41@pnC! !NG+>F^EYAxpUAW& E4&KEY$?)#1@EYC=$?AW1#&:N:; EM1iN:98AW43&:6(¢RS$'1k)M3w98&41@1@ !C!)4NG3F^Ax p d | @ 1 ? $ @ 1 W " # ) e W A : N Y E , $ j 5 0 * ( e ) 8 #X \ ( M ) 6 3 c A @ M E # 1 < I 4 E B & 8 # ) ? X 6 3 W A : & z " b F^C!g&&:)eV !H8)N:³UA )#O?X>1@F0/8:&:/8"%AW)e&:H892N:1TAc>±d.E4|)#XgA d.)#EM(?1#1@+ 98dN:A% QK)iA /8 &ª,²EYd 36F^Ac&Y@-0EY/wC=Ag&:$n-^/2JL)#1@90"%(.;VAcQ¢RQ.)MC'3wRS1#1#Ii+>Qg98&:p= e1 Q¢R >d &41@C!NGF8198)#EYC.RS1#+>Q T,p
ξ = ξeq
(
(∂G/∂ξ)T,p = 0 ξ = ξeq
2 2 (∂ G/∂ξ )T,p > 0
(∂ξeq /∂T )p
(∂H/∂ξ)T,p ) (∂ξeq /∂p)T
ξeq
(∂V /∂ξ)T,p H
V
)
F^Q'&Y-0-0/2/w1@"%&:A
$'R*+.-0)#AW6/8&:-090/2;B1@$BAWN:98EYM3$?61>$'d>)4| 31@/8 e&:Ax;Bp,3 (*&:C=92/81@#9/0Q.05=90E4;`& Q¢RS1@;
A
p0
xA (T0 , p0 )
)
T1 > T 0 p1 > p 0
p = p0 T = T0
*
Kp T0
∆h0r
Kp T0
T1 > T 0
T1
e >d 36,/25=1@$'$')e+'-01@A 98+.)M3w1j&:$,/21@C>"%(. Ach1;<98)#&4"x1@dgC! NGF^-0A6EM1#3´NY)4/83&:1#;BI(\3w&:921@1@90/0/8Q.)e90H8E4I& -0/21!R&GF98&41@C!NG F^A 3P/8&:;B(\&:921@/0Q.90E4& `3-8/8)e-0Q.$.C,QK+.) d >ed6|u-0Q.X'-0/21@$'NGF^A EM1#N2O'),+>EYA98&41@C!NGF0198)#EYC'RS1#+>Q
(
∆h0r
=33 8#"%'& / F% 4!" % &# %
T0 = 298
= 32
Kp
T1 = 310
Kp (T0 )
)
A
(.+.90)eEY-05
T1 = 300
A T2 = 315
40%
42%
∆h0r
∆h0r
A
)
A(g) νB B(g) + νC C(g).
(
86
α
nA
A
nd
A
B C
α = nd /np A
np = n A + n d
α
Kp
A B
C
)
"W& C.RS$.1#AW+K&:EMEY1@$?"W&:1h_YF0$,+>5=Q#N2F^OB&(?-0AW1@i921@3°;<-0&:/2/01@98$.43 AW&/898&:43906;<$')=)M+>3w5,1@$? e1@A ;B/8&:AWN:90EY;<$,)=5=+>N2OB5=$?$?1@1@;B"W&:AW_YN:5qEY$'(\&G)@F:b d ;BN:+'AW1#&:AWIe&YC'-0p REM& _41h&:F8X,q5-8AW)#6C,3798&YH0+."%)4AWIB3w-8)#/2"%$n1@$j5>N8/8O]&: !(.)98)#Q.(*C')#RS981#NG+>F01#QgN2p>OgF^p\&Y±H8X¢"%A ²/±L3U1!²ª)#90N:EMAWi&YN:+>E436& AW&6JxA1@EY5 A(c) 2B(c) + C(g).
)
A
B
8 #
'd C'RS1#+]/098@F^-8C.RS1#+>$.A%C!)M365!p'3C=/8#905,;´X.AW&: e$.AW&q98&41@C!NGF01
#
)
A(c) B(c) + C(g),
YE(.$?90EY1h5]F^+>N:Q#EYF05=&; -0AWN:kAW&Y3 N:E4& 98M36$'A )431#/3U+>E4)#&90EMN:lAW&Y98N:)#E:EYbx/ n3U1@#9EsD3 +.)e-0/8C!&:;B)#$?(*1!&:R
925!1@d?/0|Q.905,E492& 1@EYAWI p 3UN:)#AW92&:H01@$. !EV)AW&:_4$.-0&iC'AW&wRS;<1# n+>)#1@$._YEY$?A%QBC#1<1 $?(*1#)#+;Bp 98)#A%$?EY#/p=3UIiEM)#1@)#98C.X*&:RS&Y;1#N:+'$'F01h)e1@F8H8C!#IV) Nep=/8JS_4Q.&: !& $.)ZC!F0&YNG-0-0F^C'/wRS/8N:1#)VAW+>H0$.$. nA%AW1@Ce&:ED1B$.Ac+.3¯1ª)e-0N:98C!EM)#)##EY$?-8//01!3UC!R
)@)@5!bb p 90N:E4AWH0&e$.d'AW¶&:$.90AcEY1 5hF0#ItF01@C!)lA EY$?1@$'&q36+>AW&:"c1"%C!3t)eH8-0N:(*A nR
-0AW/2-01!/0RS$.AW&:98$.&4Ac1@1{C!NGN:F^AWAx&Yp N:E:bx(?1@921{p')#N:92EY1@5nE b -0/5>N8OT-8Q.X'-0/21@$'NGFGA A d >d 1@(.AW-21#Iª31@90Q.$'&:C98436$')M3w1@ eAN8O'&:;BAWN:EY$'&GF+>"c198&41@C!NGF^A T
A B
C pC
)
p∗A (T )
K(T )
p∗B (T ) A B
*
)
CuO(s) + H2 (g) Cu(s) + H2 O(g).
¶ 90EY5MF8#Iep>_4&D365>-0/8:(.Q#F8#NY&D/0Q n1@EY5- n1@EM1@;BA+.)e-0C!)#$?1!R
5=;BAxd >d ;V*1!5qR#_1@YC,;V/5,5!36p*$'_4&)eH2+>I "c-01lC'(*RS1#&:+>36$.$'A%C#&:1 !)98)#±S-0EYQ./3wX'-8)#/2901@Q$'NG+>FG36A!98Q')#-0EYC'(.RSQ'1#-8+>E4$.N:A%E4C!)#)4$'3U&G&:F8 !² ) EY$?1hb NG-0F0AW<ª(*/8&:)e;Bp!-0/2 !(*1#+>&:IZEY92AW1@1@& /0C=Q.3t/905=-05!(\36p>nR$'N:AW)eN:H0H8EY$.N:5=AWA&:$.$.A$.AcA%3t1VCZ-8A'1@(\Q¢C=nRRS/1@5=N:;B36EY5=C!$'$.43)e$.PH8A%N:C#Ax;<1{p )#1@"WC=)MpM36/5=F^5=&Y36N2-8Og/$')ed=EYH8$?,N:1@1@AxCB$?pq !EY+>;BJQ.5AW$.&:EMC,$.1 bA ;B;
) aB =
B
γB xB
γB
B
mB
*
xB
[B]
*
86
µ0B
µ∞ B
#
=33 8#"%'& / F% 4!" % &# %
>d 98*&41@1@(.C!AWNG-2F^1#A I31@90Q.$'&:Ck98M36$')43w1@ eA N8O'&:;BAWN:EY$'&GFB+>"c1T$?1#-8/8:(.Q#F0#N:5>N8O *
)
− NH3 (g) + H2 O(c) NH+ 4 (aq) + OH (aq),
BaSO4 (s) Ba2+ (aq) + SO2− 4 (aq).
|$.A%C=Qv(.AW&:N:90E43t&: !-0)«E4&GF(*)4983t&41@-0C!/21hNGFGF0AV&{ n-1@RS1@EXn5798)#EY&:(."W&:Q'C=-0/0E498N:)#EM"%1A%/4p6-0AW1+>3 90Q.3w n1),+>98EY&4AW1@&eC!pNG3F81k36+.5n)@bb /-85=)#N:"%AxEYd¢5 |j$?-01#Ce-81@5=E4NY43)#6$'Ce&:1> !d)¶9890)#EYEY5h/3wF0#)#Ie90p¢Q_4&3U)=98)#+>EY$'/&:3U !#9)D/03w90Q'),+>+>$'$,)Z5 98)#EY(.Q'-8E4N:EM1@F0"%$'&Y-0&G/F 98/3U)#EY)#/903UE4&q)#983w&:;¬),+>$?$,1#5,-0;y5>NYF^)#&Y$n-85=/t; N41!p>R
A\C!)4_436& AW/0N:90AWQ'&q+>E^$'F0))#$.98A%)#E4EY)4(.3Q'1@-8$?E41>N:EMd 1@"%$?1-8#"*3v98)#E:b e~>d $?1#)#-0 !5>#"NY%)#$.$'AWI &: !(*)v)@F^98Y)#N:EYAW& /3wA%"W)#),90N:Q`EY5=$n3UQi)=+>98)#$'EY&:(. !Q')v-0E4/0N:90EMQ'1@+>"%$'$'))eH8N:98A\)#EY±E4(.)#Q'Xg-0dME4EMN:EM1#1@+ "%dM$' &G !F{² -8+>)#"c"%1 A )36E4)#90E4& p.+>"c1C=/8#98&GFw98&41@C!NGF81B+>5>-8)=NGF01#NGF^A*;l1(\)e-8/21#I BaSO4 (s)
)
Ax By
!$,+>5!dEYAW*& 1#-0/8A)e-2)43w)#1#EYI$?1#36N:EM5=1h(.F098>)4pn3),1#+>+>(*E4)4)#36$'&AW&Y+>36$.5,AW92).1@p_4F0&:)#$.$
*
A B
Ag2 CO3
Ks = 6, 2 × 10−12
25 ◦ C
−0.763
+0, 339
) Cu2+ + Zn Cu + Zn2+
NaCl
Na+
Cl−
&:+>"WEY&:AWC,&M/0²w98EM)=1#+>N8EYO'AW&)=+>(\EY)=Ag+?98RS&4#1@N:C!E4NG)#F8$'1 &GF+.)Q#F0&:;B$'&: !)X.AW&: eQ.$?1X?1@/8&:90A%A ±Ce1@/8)@b
8 #
>+1]EY$?AW&M1T²w&:EM"W1#&:N8C=O'/0)=98)=+>+>EYAgEYAW98&<&41@(*C!),NG+?F81 RS#N:E4)#$'&GFi+.)TX.AW&: eQ.$?1T+.)=+'1@/0$.AW&: !)s±L1@$')@b (./0f90921@E4#C=&:+>X?AtQ1(.(¢98N:),R
E45,NY&:$?&YC#-Z#1#NYIe$?&:pg1@ !_4EY)l&:5=Xn(.3w5901jE4Q.&:-8EqEYAW5> -0-0/8C#&:)#1#"W(.&:I C,AW)#/0$?98 )#BT"%A%-8EM#";<>d &:36/2*1@5=1!"%$'AWRN:)e#_EYY-0$'A;V&: 5! !pw)K"_4-8&d.)=n+>$?1@Q1@C/8$?:+?1]_4R
&:Q.Ce$. !1hAW)& b /8$?),1B+>1@EY$'AW&),+>AEYA%"WAW&& 3 /5=; N:EM1#-0AW&365=(.98)=+>Q.C,Q#F0&B-0AW n1@E4)43w&: !)TN8O."W)#90Q e>dt|(\)M3t)#-0 e/2$.1hA%F^36& AW-0&DA
RS),1q+>&:36"W&:92C=1#N4/0981@)#"%$,;<5,)#;´/8)#EM901#5>N2N:O'EY$?),+>1 EYA 98&41@C!NGF81>p=3 36375,/8$.&:A%;BC=Q](*&:C,92/81@#9/08Q>&GFb 90"W&:E4_Y& $')eH8N:A Ag(*)=)=++ /8N:&:AW;BH0$.(*AW&:&:$.921@AW&:/0; Q.9051@/036;<5,)eEY-^$?JS&:1#90N:5>E4N:)#EY$'$n)T5=; 3w1@d 90{/8)e(\H8IZ)#;B(\Ac)=1@N898O'M3)=°+>$'EM1h&GFb )_4#&X."%3vAWN:EY985=&4IZ1@C!N:NGAWFG&:A\(¢Q.R)T/0"W&:98$.&4Ac1@1@C!$.NGAcF^1A CeEM1@1#_4N2O'+>5l),+>v1@EM/8#)#3 NY;[&GFV;<3[&:/2dh)#1@| e"%$.Q]A%)#36(.(?90AW1@&eE498&YpN:N2(.A%O'Q<90),EY)+>5hEMF^/8;V#&wN:Q#+'5F81@#+.$'Ne)& p 98E4)#)eEY-0//23U1#Iª)#9036Q<5,)=C!+.)#+'90EY1h5>F0&w-0/21@$?&:1"W&:C=$?/01B98)#36$n5,5!C!d#)#n$?1@1@Ce$.1 AWN:&DE4(.YH292I1#N:/85 &: !)ªN:AW&:(¢RS1 $.AW&w;<)#_4& >dt|+.)#98-0)4C'3wRS1#!+²63U(\nRN2O')#), e+>$.EYA%A3&:1D"W&:3UC,)=/098+.)=)#+'981)43w(.&:"c !1@/)Z5=±$'EY)4331@1<$'&:EM !1@)q$,Q./890&:E4_t)#&:$?"W1<&:C=3/098),98+')#EY/3U3w)@)@bb 9036E45,&B; F0)#3U$'43)= +.)#3U98&:)=; +.d)#98¶ )4"c361@/5>5=N8$?OP1 A$.(*AW&Z)#E4X.)eAW-0&:/2901hE4F8&l#N4X*1j&:EY3 (*)eC!H098)#&Y$n+>/2$.1@AWC!&:N: !AW)T&Q'EK+> nEY1@Ac1!E4R
)@Q b 3`(\)#;B98&4AW1@YC!+>NGEYFG5Axp n11@E4F0&Y)4+>365,5=$.; AW&<3wQ¢),RS1@+./)#3698&:Ac1K; Q'A>-0F^/2&:1@ !"W&:)$.F0AW)#&l$?-81@AW;B
2Cl− → Cl2 + 2e− . *
E = 1, 015
*
0 ◦C
E (∂E/∂T )p = −4, 02×10−4
0 ◦C
)
1 H+ (aq) + e− H2 (g), 2
#;B) 92AW1@YE+>EY-5lR
Q.(*_Y)#5/8&:F8$'1@C!NGF0)«1!RC!&:)#;´$n/2(*1@nRC=/T)# e&:$."W&:A%3wC,/01905>3wN:),EY+.$n5!)#98d )M¢3U$?&:1@ !"W).& p'4IjN:AWH8EY$.36AW&:Ac@$.E4AW&:&:;´Cr(* n)@1hbb E4/3U)43U)#90&:E4 !&e)d*3w ¯),A%+."W&D)#90EYQ;B)#AW92&:1@$.EDA1@-0AWC,D/5,/836&:$ $')e(*H8)#N:/8Ac&:t$'F0NG)#F0$'1!43RGp!F^&Y3UH8"%)=A +.3)#98/8)4&:36;B5>(\N8O&:921@3r/0Q.9890)#E4E:& b *
=33 8#"%'& / F% 4!" % &# %
± L ! 1 ² : N W A 0 H . $ W A : & . $ W A K & U 3 = ) . + # ) 0 9 r Q 0 ? ( # 1 > + . $ W A { & ~ 2 9 @ 1 Y E ! 5 p . ( 0 9 Y E 5 ' Q 8 2 / @ 1 W " # ) ' $ G & F 981@)MC=36/5=5=36N2OT$')e36H8N:EYA,98)eF^)#H0$.$'AWM3& r3w92),1@EY+.5!)#p'98)M(.3690EY5=5N2OgQ'pg-8/2±X¢1@"W² )#1@$nC=5=/; 5=36N:$'AWH0)e$.H8AWI &:$.F^)#A%Q $'M3r3w),+.)@b 'd C! )MX.3U"%AW&GN:FgEY35>I 9843(*6)#/8$'&:)M$'3wNGF01#1!+>R'E4(*&wnRE*F0)#)# e$?$.1@A%3w;B1AnN:EY5,X.$.Q'C=+.QD)43w3j1@98$')#&:EY !/)ª3UEw)#90&:E4"W&:&wC,)t/0981@)=C,+>/5,5D36N:5,$'$>)@bb H(*8N:)#A /8&:$'NGF01!R
QK-0/21@$'p'+'3P1@98+./8&:)4;B3w&:(\ !&:) 921@/0Q.90E4& p=(.90EY5hd FG;iQ#F0#ND31@90/8)eH8I >d /0¢98X.)=Q'+>+.5ª)M3U3w)=1@+.$')#)98)4)#3U e$.&GF2A%3U²)#)<92Eh1@RE )#E_4)#+>$'90&VQ. eEDAW&:(\ !nR))#(\ enR$.)#A%3 e$.1
#
aZn2+ = 0, 1
0 EZn
25 ◦ C = −0, 763
*
)
1 H2 (g) H+ (aq) + e− , 2
1$?1B&:"W&:C=/098)=+>EYAW& " #b " ,³U"gEM1#N2O'),+>EYAg98&Y+>Q.C!NGF81 )
AgCl(s) + e− Ag(s) + Cl− (aq).
*31@1#(.+>AWEY-2AW1#II36-05,Q.92;l1@_41@&:90$.5>AWN:&BEY$?$?k1VF098&:&4 !1@)C!NGF^k+>"c1dg /8&:+ !F8)s1@C,)#AW eN2$.OA%3w361sAW&:"%)#C!92)e1@H8EN:A36EM5,1@(."W&:98_Y)@5 b H"%0A%98/0&YQk+>$.±E4AW&:)# !³UXg)VEYd'5983t43-0;<6(*$?)#nR1@_YN:$?$.EYAW1T5,&Z$.36±^$. 5=A%>dC!Ce1V)#>90M1@EY²0C=5>² /-05=/21#36I$')e/8){H8N:A!)#F0 e)#$.$'A%3w43q)p +.){p!36+>5,"cEY1V$?&:1#"WN:&:E4C,&:/0$.98)@Ac1 b *
(
(
γ±
q °§¡ ¡£t¤8£ ed C=A?)#-2EY1@36;[Ac@-0EY(\Q#)eF0#-2N#XDEM1#F0+'1@C1@$.EYAc$?11@/8CB&: !98)M36/5=$'(.)eQgH8N:p#A¢/092A\1@+.C=),/0+'Q#1hF^&:F0;i&:;V5i5B-05=98&4;V1@X*C!NG)#F0" &t/21@3sCªF8/21@1hC b 98+'M31@698+.$?)M1@36$.Ac5,1j;B1@A*"%&: !$n&:/2X.1@92"%1@(.AWAcN:1@EY;B$'&eA*d 98&4" 1@$?C!1@NGF^"W)#Axd eAWN:1@EY/$.3UAW&)-0(*(.)e92-01Y/83t:(.+>Q#EYF^AW&:Ie;ip,_45&E4+.&)=+'-01h/21@F0#$>N b
f
→
-F00#/098N)#/0$?901@E4;B&YN:AUAc>98p'&4+.1@C!)eNG-0F0/21hF0(.&:;VAW&:9053tA%-0$nEM/8K&:98Aw&Y-8+>Q#90F0Q.# nN4>p$?A1#-),+k98&4AW1@N8C!O NGF0-0eQ.d>;i|j5)#X*)=&Y+.Nª&GFG/8;i&: !Q>) b +>"c1/8&GF98&41@C!NGFGA365,$')e-0A
]6"
∆h0r
)
∆h0r = (−1411, 3 − 285, 8 + 1559, 8) kJ/mol
B;/21@F0AW"%&:N:(.;BEYA%$?A$?1>-01j/2p1@X*3$n)TQT1@90&:(\/8$,)e)=/2H8N:1@I EM"%@(./0AcC!1K)4-83w/2&:1@)# !$nEY).$?Q«p?1#C!N:N:EYEM)#5=1>o'"%pwNYAg)4_4Q.3U&
= −137, 3 kJ/mol. ∆h0r
N2 O5
5 N2 (g) + O2 (g) → N2 O5 (g). 2
T1$?$'1#)#-0_M/8#:N(.$.(.AWAW&q&:90+.3t)=-0+'EM1hF0#ANq/090+.E4)B&YN:-8AcAW&:X.EBAW&D363t5,;B-0EYAW5>&:-0$./0AWC,)#AW$,&ª5=/0N290OEY5!98p?&4+.1@)eC!-0NG/2F^A1hF^(.&:90;iE4&:5 EZ>p f1@C]-21@;<)B(*)e-0/8:(.Q#F^&:;i5]Eq&:$n/21@"%(.Ac1@;BAxp¢/0EY$gd 3/86&:AW90Y;BN AWN:EY$?1>p !+>5=_l3[-0/21@$.AW&
)
2N2 (g) + 5O2 (g) → 2N2 O5 (g). )
(−2 · 114, 1 − 110, 2 + 2 · 180, 5) kJ/mol = 2∆h 0tw (N2 O5 (g)), ∆h0tw (N2 O5 (g)) = 11, 3
T = const
pV = nRT
∆H = ∆U + ∆(pV ) = ∆U + RT ∆n,
∆n = nk −np
np =
X
R∈{ }
nR ,
=33 8#"%'& / F% 4!" % &# %
13-8/21@$.AW&ªC!)#o'NY)4365=;
X
nk =
-0Ce#+
nR
R∈{ }
∆n =
X
nR −
X
nR .
'dt$,¶5=),N2+.O-8/2EY1Y3636AcAc@1@EY;VC!M35 -0E4/2&w1@$'EY+'$?1@1@98C,+.AW&:)M; 3U&&:+>$,"c/21t1@"%(.(.98AW)=&s+>Q./3UC,)#/8M390sE4&:$.A Ac1v+>(*"c)e1-0-0E4Q.N:X'E4&:-0 !/092#"1hzbb /8fM31@qCqp!36A'AWEtYNU)=+>-8/2(\1@)M$'36+'AW1@&Y98+>+.$.)4A%3w; 1ª3t&:$,-8(\/21@nR"%(.N:AcEY1 5=$.98&4$.1@A%C!C,NGAW&:FGA>; -0(?-01@/8&Y"c1@N8$.O.AcAW1)#;<(.98&:)#/090(?5>1@N:$nEYQ$n5=36; 5nb d $')e-0A R∈{ }
R∈{ }
+
−
)
∆h0r = (−3 · 393, 5 − 4 · 285, 8 + 103, 7) kJ/mol
*1@Q,3w1@_Y;V5!p\_4& >p!F01@C!)B_4&wF0&Y-0/6/8)(.AW&:9036Ac1#-0/8&:C*d ¶(*)j90E4&Y98&4+K1@98C!&4NG1@F^A C!NG-2F8 <X,5.;
= −2220 kJ/mol. ∆h0tw (O2 (g))
*
)
∆V =
3RT RT ∆n =− = −74, 3 l, p p
!!+>&Y+>EYAW$'& )=N:E4&YH0$.AW&KX?1@(.9490pEY15,X,5.R
5 ;<)#"WF0&e&Ypw-0/DN:EYEY5,;B"%A Ac1@#$?] «"%AWN:N:AWEY&:XnC'5R&GF<;<3U)#"%)=A +> n5!1@pUEYNYQg) d +'1@C!1hNGF0F^& A#F^&Y-8N:/Z; (*)#N:$?AW&Y1#N:+ EY5!de,1@«9CV3621@AWEY+'5r1#Ie;Bp#)#$.X>AWF^&G:F0-0/8EM)e1«H2IU)=3w+r),+>EY5;B(*Ac1@)4$,3t5P-0/2)#1!X>R&GF^F:/8Ew)e98H8&N:bA n1@EYp'Qg1Bp?36N:AWAWH0Y$.NqAW&:;<$.)#AW&_Y$?F0&Y1ª-0/tF8-0/2(*1!)#R&e;Bp>A%36$?#AWYIeN d¢¶)#$.AW&:3w1@_B&:$,/21@"%(.Ac1 p=1 3
∆n = −3
H=U+
pV
∆U = ∆H − p∆V = ∆H + 3RT = (−2220 + 7, 4) kJ = −2212, 6 kJ.
e~ >dt|3-0/8Q.&G;BF98AW&4&V1@C!NG;
]6"
∆H = −46, 1
-0C##+
)
NH3
W = −p∆V = −RT ∆n = 2, 5 kJ/mol,
∆U = ∆H + W = (−46, 1 + 2, 5) kJ/mol = −43, 6 kJ/mol.
>d$.¶AW)D&GFU(*98)#&4;B1@C!$'NG)#F^_4Ag&:/$.3wA%Q<)#9098E4&4&:1@$.C!Ac1NGF^A./3w)#p'90E41&:$.$?Ac1#1q-0/83w:),(.+>$.5AW&V(.),90+@E4F0&:YE N:A%QA'98+.&4)=1@+'C!1@NGFG$.AA%Q<-0(?+.1h) b "c1@$.Ac1;<&:/21@$nQgp¢+.)e-0/21hF^&:;i5Z98&41@C!NGF0D/3U)#90E4&:$.Ac1l;<&:/21@$nQgp'EM1@/8&:; CO2
∆h0tw (CH4 (g)) = 2∆h0tw (H2 O(c)) + ∆h0tw (CO2 (g)) − ∆h0r
ª+>5Dd 98&41@1@C!/8NG)#F0;B1 Ac(.1#90-8E4/&:X.+>AW"c&:1K n198&4(*1@),C!+NGF^A6-0/2EM1!1#R
5=N8O'; )=+>N:AWEMH0#$.NYAW&G&:F$.AW3 &:; -0p@/21!/8)R&GFN:AW&:)#(¢X>RF^) :/8)eH8N:A ±E4)#Xgd.EM1#+ p.d X*!)B² Q.C'RS1#+ $.AW&V365,C!)#$,Q#F^&i_M1#+>$'&GFt(.921#N:5!d 1@;V536AWYN = −74, 8 kJ/mol.
Q=
∆H Q = ∆U
! X.+>AW)#EY92AW#& N:5=N2O°F0Q'&Y-0+>/iEYAc(.1!90RlEY5,3 98)e98-0&4/81@&:C!;aNGF^Ax"%AWd N:EY*Xn1@5Qn3w;<1@)#_Y"%;iAw5!-0Q.pDX'_4-8&/21@F0$'&YNGH0FG"%AiAw n3 1@E498)4&4361@C!5>NGN8F^O A F^Q'EY&Y$?N:-0E41#/&YN:$?-0EY/0$.1$.AWAW&])#N: !EM;BnR$./2EYAW1@&G$?C,F^1#-0_4E4N:&ªEY&$.N:)#AcAW1!&DX>RSF0;B1:/8-0$.)e/2AWH81!&GN:F0RA6-0&ªEY;<"%5Q.)#X$."WA%)MN:_q3UAW&Y+>&eN:"cd E41&e| np>1@/8E4))#43;lAWN21YO36E4&ª[email protected]$'RSEY&G1# eFB+"WY98+.+>&4) 1@QTC!$?NGF^1 A +>N:EY3w5,1"%A ;<)#"W& 3U)=+.)#90Qr-2EMp.1#-8-8/2/2#!+ (.AW)#$'&T(.90E4&:EBF^&Y+.&:$r;<)#"t;<&:/21@$nQgp Q = ∆H − ∆(pV ) ≈ ∆H − RT ∆n,
∆n
*
pV
∆n = 1 − 2 = −1
Q = ∆h0tw (CH4 (g)) + RT = −72, 3 kJ/mol.
=33 8#"%'& / F% 4!" % &# %
=dt¶)#$.AW&:3w1@_;<)#"W)431q(*)@F^&:;B$')eH2I6N:AW&:(."%$?1D(*),+-0/21!R
5=;yN:AWH0$.AW&:$.AW&:; cp (T, p) =
∂h ∂T
,
!$.+>AW&:EY$.AW&Ac1B-8/2F^1@&Y$'-0/D+'1@&:98$,+./2)41@3U"%(.&:Ac !) ;<)#"W)43>X?pg1@369AW;lYNB1@;i3`5 -8E4N:E4&: !#"%$')eH8N:A +>"c1]N:AWH^b p
h
)
p=1
c0p (T ) =
dh0 . dT
YE 36#_AcY@$.E4AW&:N:CjEYC,±Q#98F843#6N$?-01@/2$.1@AW$'& +'ª1@98A%+.98N8)MO.3wO') g1&:$n!/2² 1@"%(.AW98&41@C!NGF^A?(*) pn+.)e-8/21hF^&:;i5
T
)
d∆h0r = ∆c0p , dT
!+>EYAW& X
∆c0p =
νR c0p,R −
X
|νR | c0p,R ,
1:N AWH0$.AW&:$.AW)#&:EY; $?1#*N:X?EM11@94;
c0p,R
R∈{ }
(
R T1
)
T2
∆h0r (T2 )
=
∆h0r (T1 )
+
Z
T2
∆c0p (T )dT.
>d6 e|vA%A
Ac²1#p+._4)#&<;<;<)T)#E "W)4'3EY1{5,C=&:A$'&:-090/2 e1@Ac/1 5=-83w/5>&:N:36EY$'$':&GF/090EY±EM$?1#1T-21# n+'1@1KEYQs&:C,+.36)eA%-0(?C!1@)#90$?/1!5=RNG&:F^ !AU)K&:$'36&:5n9^bb $'/0Q#)eF0-0&:A ;V5 /0Q{N:921@EM#EY5q-0/8"%&YAWN:N:EYEYC!X?1 F0-01@/8C!)#)Z(.$.N:A=Ac1!-83UR)Z)#-0X*EY),/+>5,5q36$'N:EM&V#(\-0/8)#&YEYN:X?EY14C=36A> nAW)#1@$'EY&Q]-0±/0/09092Q.1@C,Cnbb /036Q.AW90Y5ªN 3U&:36$':/090EY$'&GF2²d!³UEM#p-0/8&YN:EYCe16"%A%$.AW)431t;l1 6pª-0$?/8)#1@(./8)#$.;BA=-0Ac3U1#-0)#/jX*),N:EM+>h5!b p -3t0/8-0&YEYN:5>EY-0Ce/01 C=AWN8O{;l n11@E4-43 0/8)#+.(.)e$.-0C!A)#-0$?3w1!)#R
X\5>N8)=O+>5!98p*M36EM1@$?/81@&:$.; AW&B-0/21@$nQ;l1l/8i-2d¢1@;l"c1
T1
6
RT /2
-
UA = 5nA RT /2 UB = 5nB RT /2 C UC = 3nC RT
> (\ n1@)eE4-043/21#I +.)e-0/21hF0&:;V5 d¢"c1]&:$,/21@"%(.A%A ;<)#"W)4365>N8Ok(\)e-0E4N:E4&: !#"%$,5=N2O
]6"
) pV = nRT )
-0C##+])#X."%AWN:EM1@;i5 (.90E4&:EVEY98#_Y$.AWN:EYC!)43w1@$.AW&(*) hA = hB = 7RT /2,
hC = 4RT T)
cp
29n1@1@/0CQ.90365!AWd'+'1#Ie"c1 p .$ AW&q.+ )eEM-01@/2"W1h&:F^_Y&:5 ;i/05 Q1@$.A),+KN:AWH0$.AW&:$.Ac1>p\1@$.A)=+ /8&:;B(*&b cp,A = cp,B = 7R/2,
cp ∆c0p
|j)#X\&YNª/8&: !)
cp,C = 4R.
)
∆c0p = 3cp,C − 2cp,A − cp,B = 3R/2.
-0C##+(*)(\)=+.-0/21436AW&:$.A%Q
Bp
+.)e-0/21hF0&:;V5
∆h0r (T2 ) = ∆h0r (T1 ) + (T2 − T1 )3R/2,
T2 = 340
)
∆h0r (340 K) = −19, 48 kJ/mol.
~>d {+.& '$.AWNGFGA-8/21!R&GF *
T1 = 298
86
Kx (T, p)
;l1@;i5
Kx (T, p2 ) = Kx (T, p1 )
)
p2 p1
−∆n
.
^F 1&Y"cH03P1T"%A (.98&4901@EY5,C!NG(?F^1#Aª+>±LC=1!Qk² p=/8±X¢)² 3 (.90EY365,pEY(?1T981#)e+>+>H8C=$."c1KQAW&q98±L+>&41!1@36²UC!Q.-0NG/2C,F^1!Aª98RS)#±1 /0X¢$.² AW&ed $.AW&EY;BAW&:$.pAg36-0AWAWYeN p >d¶)=N:pEM@/0C!)43w&DA%"W)eH8p N:A\98&41@ !&:$n/8433vp'EM;<1@/8)#&:"c; 1#N2O±E4)#Xgd±^ >d!²0² p ∆n = 0
p2 /p1 = 2
∆n = −1 Kx
Kx
) nA (0) = 2 nB (0) =
1/3 nC (0) = 1 nD (0) = 1/2
¶+.)e)#-0$./2AW1h&:F^3&:1@;i_T5Z3t3-01@EY905>Q.-0$'/0C=&:AWC&]$?"%AW1 N:EYXn5s;<)#"%At;VQ'-8EM{Xn5>I +.),+'1@/0$.AW&ep 36AWYN
nA = 2 − 3ξ, nB = 1/3 − ξ, nC = 1 + ξ, nD = 1/2 + 2ξ. ξ)
−1/4 ≤ ξ ≤ 1/3,
e
=33 8#"%'& / F% 4!" % &# %
C=-0/8/8:#9(.0Q598)#&4C=1@98C!&YNGH0FG"cAx1dg;B¶)=A%$.+.A%-0;l/21Y1@36"%$?Ac1h{F8#AwN ;l1@C=-85,;l1@"%$?T;<3w)#"c1@1>90pg/8)e+.H8)eI-0/2"%1hAWN:F0EY&:X,;V5j5(\±)@3 b ;Q¢
) nA = 5/4 nB = 1/12 nC = 5/4 nD = 1
xA = 15/43 xB = 1/43 xC = 15/43 xD = 12/43
Kx (T, p) =
*
)
xC x2D 144 · 43 = 27, 52. = 3 225 xA xB
eed +>*EY1!AcR1!)#RS_Y1@5=$."%AcAW1
A B C
)
x2C = Kp = 1, xA xB
(.1@C!90EYNGF^5Axp N:EY5=;®Q¢RS1@;BC,AU;<)#"W)43w&Z-2]JQ.$.C!NGF81@;BA "%AWN:EYXn5(*)e-0/8:(.Qj98&b ξ)
!EY;B+>EYAWAW&:& $.$?
d>)e-8/21hF^&:;i5Z-0/2#+98436$?1@$.AW&ªC=3w1#+>921@/8)M3U&V$?1
xA (ξ) = xA (0) − x, xB (ξ) = xB (0) − x, xC (ξ) = 2x, x = ξ/n(0) x)
C=/8#98&D;l1B(.AW&:9036Ac1#-0/0C=A
4x2 = [xA (0) − x][xB (0) − x], )
i p 1h −1 ± 1 + 12xA (0)xB (0) . 6
^"%AWN:F EY&:X?;B1@$,;B5A(.$.AW&:AW&:90Q#36F0Ac&:1#;B-0/8$n&:5=CB;B;VAxp\Q'1l-0A%;i(\)#5l$.)=AW&:+>390EY1@Q'_ N:AWIep!X*)DQ¢RS1@;Bp?C,36A¢AW;<YN )#"W)43w& -2 /5,"%C!pg)-036Ce#/8+&Y+>5!p> !+>5 p1d=-0| /8:_4(.&:90$.EYAc5,1(?31#+>-8C=/2Q1@$.±LAW1!&² 98436$')431@ eAU365=$')@b -0EM d¢|(.pg901<EY5=-0(?/8:1#_4+>&:C=$.Q Ac1l±X¢3°² -8/21@$.AW&V98M36$')@b p \ p 8 # C # + 31@ eAg365,$')e-8EM p p d x=
xC (0) = 0 xC ≥ 0 x≥0 xA (0) = xB (0) = 1/2 xeq = 1/6 ) xA = xB = xC = 1/3 xA (0) = 1/4 xB (0) = 3/4 xeq = 0, 134 ) xA = 0, 116 xB = 0, 616 xC = 0, 268
4>d EYA$?1#N:EYd ;V5V1#-0-2Q.1X'N:-8EM/2#1@-8$'/8NG&YF^N:& EYC!)43w1 p p 365=$'ph)e),-0+>A (*)436AW&Y+>$.AW).p2F8 1@hC!;<) )#"xpp N:EY5,"%A;<(\)=)#"xN:d EM@/0C!C!)#)4"W3&:A,1s"%AW"%N:AWEYN:X\EYX? 1k;<;<)#"%An)#3«"%Aª-0/8/2) 1@$.AW& 98M36$')43w1@ eA'N2O'&:;BAWN:EY$'&GF:p p4+.)e-0/21hF0&:;V5DEU98436$?1@$.Ac1 -0/21@$nQi n1@EYQi+.)e-0C!)#$?1!R&: !)
]6"
B
C
#*
10−2 neq = n(ξeq ) neq =
PCl5 PCl3 Cl2 A PCl5 MA = 208 n(0) = nA (0) = 0, 91 × )
0, 314 × 105 · 2, 4 × 10−3 pVeq = = 1, 52 × 10−2 mol. RT 8, 314 · 593
¢;BAc1@$n5]"%AWN:EYXn5];<)#"%Ag98&41@ !&:$,/843¯3P/0921@C!N:AW&q98&41@C!NGF^A 365,$')e-0EM *
-0C##+ ;<)#"xd@¶ 98),NY&:$,/ N:EM#-0/8p&YN:36E4AW&:YC N
)
nA (ξ) = n(0) − ξ, nB (ξ) = ξ, nC (ξ) = ξ,
pMC=/8#98&wQ."W&: ,R
5i98)#EYC'RS1#+.)436A
n(ξ) = n(0) + ξ
ξeq = neq − n(0) = 0, 61 × ) PCl5
10−2
nA (0) − nA (ξeq ) ξeq = = 0, 67, nA (0) n(0)
:Q¢N EYRS5,1@;B"%A C,Ag;
67%
Kp
)
xA =
ξeq nA (ξeq ) = 0, 1974, xB = xC = = 0, 4013. neq neq
¶)#$.AW&:31@_V3-0/21@$.AW&q98436$')431@ eA xB xC = xA
p p0
−1
Kp (T ),
63 AWYN >d98¶&4)#1@$.C!AWNG&:FG3A1@3t_{-0C=(,Q.5,/8/2&:1@C{$.AWEY&l;BF0Ac&Y1@-0$n/Z5)/8&:C,;BAW&:(\90Q.&:$'921@&:/0C Q.90EY5{;B(*Ac1@),$ +-0N:/2AW1!H0$.R&GAWF<&:$.98AW43&:6; $')M*3wX?1@1@ e94A p ;VQ'-0A%;i5$?1hFG(.AW&:903 365,(.98)M3w1#+>EYAWIEY36Ac@E4&:C(\)#;BAWY+>EY5
Kp (320 ◦ C) = 0, 314 · 0, 8158 = 0, 256.
dKp /dT
#
=33 8#"%'& / F% 4!" % &# %
A'$'-0)M/23w1@1@$' e+'A1@9898&4+.1@)MC!3wNGF^VA.&:F0$n&Y/2-0/61@"%E4(.+.Acª& '98$.&41@AW)4C!3NGF^1@Ax$?d!1D¶90F8EY1@5=C!(\) )#;B$.A F^;V5!p!_4&6-0/21!RS19843b 86
)
!)#X.+>EYQ'AW-0& /098)#$.$,5,F^;m&Y-8/qEY98-0#_/2Y1@$.$'AW+'N:1@EY98C!+.)M)43w31@$.A%&:Qr$n/2/81@&:"% !(.)TAcTEY36-03UAc@)#EYX*C=)=Qs+>$?(\]) 98&41@+.C!)eNGFG-8Ax/2d1F^¶&) b ;i5 ln Kp (T ) = −∆gr0 (T )/RT,
∆gr0
T
)
¶)#$.AW&:3w1@_
d ln Kp 1 = dT RT 2 P ∆gr0 = R νR µ0R
∆gr0 − T d∆gr0 /dT = =
∆gr0
p>36AWYN X R
X
d∆gr0 −T dT
.
νR (µ0R − T dµ0R /dT ) νR h0R = ∆h0r ,
!+>EYAW&D365=C!)#90EY5>-0/21@"%AWH0;V5]/8)#_4-21@;<)eH2I R
)
.$)#92AW&q1@EV+.)e+.-8& /2'1h$.F^AW&:NG;iF0V5-8EY/2361@$'Ac@+'E41@&:98C]+.)4/83U&:90&G;
86
)
∆h0r
d ln Kp ∆h0r , = dT RT 2
C=EY$?/8#91@0C 5$')e-0A$?)#C=1@98EY&Y3wH0"c1K9843C,6AW&:$?90Q.1@$.$'Ac&:1 C«EY1@;B$ Ac/1@$)/8&:;B1>d?(*|&:921@5,/0$.Q.A%Ce98)41B36E5>$.N8OvAW&: !-0)./21!p?R_4&G&F 9890Q.M36$.C=$'Q)43w+.1@), e+'A 1@/0$.A%; d p.KN:EYAc1@5=$'"%A¢)4363 AWN:(.AW92&e14pD3U !).+>p#5F0&Y-098/&41@&:C! eNGE4F8)#1v/8&:X.90;BAW&: eAWN:$?EY#$?N41 1v± 3 C,AW&b ¶ª²)n1@p4R/8/8)#)#)t_4;B&:36$.AcEY1#AW98&-0)e/D98-0M33/g6/8$'&:(.;B)490EY3(*5=1@&:(? e921#A=1@+>/0(.C,Q.90Q90E45q&Y98-8-0&4Q.(\1@$.)MC!AW3U&NGF^)=-0A AW+>&:Q#$'36F^+.& AW)#YEYN/8;B&:390$.;BAW-0&G/0AWF^N:98-0EY)#E4$'$'&:&G$.FZAW-8&UQ.± -0X'/2-01!/0R92&G1@F /843q² dd 36-0Q.EY$.98AW)eY-0N:/VAW& eQ#AUF0&B3 EY-036/0AW98:)#C>$'-0BE4&:(.$.98AW)=& +>Q.C,/8p M3N:qEY5,dg"%!A&Y(.-0/q90E4/8&) b ∆h0r
Kp
∆h0r < Kp
0
∆h0r > 0
Kp
(.3690(¢EYR
5,5,C'3wRS1#&:;¯+i+>N:EYEYAc5=1!$.RS1@$.$.A%C#Ac11698E4&:&: e36Q¢$'R
5:/0g&90EYt$'³U&: !O?) 1@/8EM&:1@"%X.AW&:Q.92901>EMp#1hC,F8/8##9NY2&:1 !);<-043/261@$qAxp@98_443&U6(\$')=)@+ b 3w&JS1@&: eC=A¢/ -8EMC.1@RS1#X.+BQ.9098E4M3&:6$.$'Ac1>)4d 31@ !1 )436(.5l90EYEY5,;BC'RSAW1#&:+$.A\+>-0AW"c617/2981@&4C*1@peC!XnNG5FGAEY&:;B eA%E4$.)#A%/8;l&:901@;B"%A%E4AWN:)MEY3w$'1#&GI F (.3690(¢E4R
&Y5,-03wQ.$.&:;AWYN:36AW&qEY98(*)e)n-0R/0)#QB_4&:/8$.&:;BAc1<(\98&:M39261@/0$'Q.)49035 1@F0 e&YA-0/3¯98M3-06/098$')#)#$'EYV$?-81#Q.N:X'EY$'-0/0&t921@E/8Q.4390°Q'N8(\O')=)@+ b ;B/8&GF\AW&:98$.&4AW1@&:C!; NGF^98A!&4Q.1@C'C!RSNG1#FG+DA?)=EM+>1#N:36EY985,)#$?/016$'&G(\FY)=peN8N:O¢EY5,RS1@"%A'$.Ac&:1#$'Iw+.N:)#AW/8&:&:(¢90R;B).p4AWN:NYEY)6$'(.&GFY90deE4&Y| N:A%3t36+>5,EY$.AcA%1!C=RSQ1 36X.Q.EY9098E4)e&:-0$./8)4Ac136AE4&:/836&:;B$':(*/0&:9092EY1@$'/0&:Q. !90).5!d p A3¯/8&:$j-8(\)e-8#X{EY;B$.AW&GF0-0EM1]&JS&:C=/EM1hb 'd/!5=&Y"%-0C!/ )Z/8_4)7&i(\/8&:)=92+.1@)#EBX.(n$'5=&/21@EM$.1@ nAW& 1#+>F^&Y$.-8AW/&:$.)AW&K36F0(¢1@R
Cv5=33 N:AW(*H8$.)#AW(.&:90$.E4Ac&Y1l+>$.$?A%1;-0/2EM1!1#RSZ+'1@98$.M3A%Qgb p $'36)4AWY3wN 1@ eA*98&41@C!NGF^Axd,¶)#$.AW&:3w1@_DE+.& '$.AWNGFGA p.-0Ce#+ p
]6"
86
ln Kx =
−∆n ln(p/p0 )
d ln Kx dp
T
Kx = (p/p0 )−∆n Kp (T ) + ln Kp (T ) ∆V ∆n =− , p RT
=−
!3U+>)=EY+.AW&)43w1@$?98&41@C!NGF86N8O'&:;B)#EYAWN:$?EY1#$?N:>EMd@1 fEY1@;BCªAc1@36$'AWYN)#36X>F0EY:98/8)e)e-0H8/N:N:A>AWQ.H0$.C.AWRS&:1#$.+>AcQB16-0-0(*(*)@)@bb 3U-0/0)=921@+>/8Q#43F^ &B²EYp;B !$.+>AW5 &GF0-0E4&:$.AW&Z-0/21!pR)#&GF921@EZEY36±(.AW:90C>E4-0&YE4-0&:Q.$.$.AWAW& YN:AW&Z3y±(.-090/0E498&Y)#-0$'Q.Z$.AW-0YQ.N:X>AW& b Q.983«&4_Y1@-05,C!/0398NG)#FG1hA$'F0#w)#N6X>(.F09898:&:)=/8 e+>)eQ¢Q.H8R
IC,5 /8Q.&M3C. RS²1#³Up@+>O? !Q1@+>/8598&:)e"%H8AW&:$.92AW1>& d ± ª1ªd (.{90EY)#5=_YC.$?²RS1 p*1#/8+/8))p:F0/236&Y1@H0C=EY"%98A'_4)e&w37-036/q365,N:5,92AWH01@$.$.EYA%C=AWAW&IeQ b p $.N:EYAc1D5,"%-0A (*Q.)4903UQ')=N8+>O'Q#)#F0;B&tAW(.&:$.90E4AW&&Y-0Q.98&4$.1@AWC!YN:NGF^AW& A98)=43+>636$'98)M)#3w/0$'1@&G eFlA*3rE -0/098)#$'-0Q.X'd -0¢/092;B1@$./843AW&GqFb p [email protected].#90NTE4&:-0$.3UAc1B)@F8E4&:36)#X>$'F0::/0/890)eEYH8$'Ie&:p !).Q.peC'F8RS1#1@C=+7A%;+'@F^_Y&Y5 -0/6+.36)jEY98EY)e;B-8$./AWN:&GAWF0H0-0$.E4AW&:&:$.$.AcAc11>d &JS&:C=/0Q ∆V = RT ∆n/p
Kx
∆V > 0
Kx
∆V < 0
∆V > 0
∆V < 0
>d «+.& '$.AWNGFGAU(\)M36A%$')43w1#N:/3w1N2O'&:;BAWN:EY$'&: !).p
*
86
*
A = −(∂G/∂ξ)T,p
p
=33 8#"%'& / F% 4!" % &# %
365=$.A%C#1>p>_4&q3-0/21@$.AW&D98436$')M3w1@ eAN8O'&:;BAWN:EY$'&GF
X ∂A T =− νR T dµ0R /dT + RT ln(pR /p0 ) ∂T p,ξeq R X 0 =T νR sR − RT ln Kp = T ∆s0r + ∆gr0 = ∆h0r ,
! n+>1@EYEYQ«AW&T+.36)e5,-0C!C!)#)#$?90EY1!5>R&:-0 !/21@)"%AW)#H0;i921@5 EZ(\+.&)e'-0/2$.1#AWINGF0Z(\)#-0/8/2&:1!$'R&GNGFqF81!98R
M3Qr6N2$'O')4&:3;B1@ eAWN:A EY$'&: !)«+>"c1 ^ F Y & 0 w / 0 2 / @ 1 ' $ ' + @ 1 8 9 . + 4 ) 3 : & n $ 0 / 8 9 # ) . ( c A 8 9 4 & @ 1 ! C G N ^ F x A d n $ 2 / @ 1 % " . ( W A Q.98&4C'1@RSC!1#NG+>F^QgAxp p p>p@36;<5,C!)#)#_Y$?90EY1 5>36-0/05,Q#92F81@#EYNDAWI/8)#F8_41@-2C!1@);<JSQ.)e$.H8I C!NGF0 p A="%AWN:EYX,5V(*)e-0/8:(.Q R
86
) ln Kp =
−∆gr0 /RT ∆s0r H ξ
T p
ªF0&:;V1#-05 /8:(.$.AW&Z98#_Y$.AWN:EYC,Q#F8#N (\) (.90EY5-0/21!R
5>N8O A p+.)e-0/21hb H = G + T S = G − T (∂G/∂T )p,{nR (ξ)} .
H
)
1(*)#$.AW&:3w1@_
∂H ∂ξ
A=0
ξ
T
∂A ∂T
3-0/21@$.AW&q98436$')431@ eAxp\36AWYN T,p
= −A + T
∂H ∂ξ
p
,
p,ξ
= ∆h0r
>3+ "c1#1 I6F81@C!)B)# !d #"&%$?:3w<B+.&-0'/098$.)#AW$'NGF0ªq(*N:)4AW&:36(¢5=RS_41B-0E498&G&4F1@C!98NG43F^6Ag$'3)eH8-0N:/2A1@$.;
T,p
86
6
dH =
T p ∂H dξ, ∂ξ T,p
63 AWYpnN/8F0)q&YH0Q."%C'ARS-81#C.+lRS1#(*+ )#X.)=AW+.&:N890O,E45,&"%N:AAW-0&:AW(¢DR)) )=d=+¶)=-0N8C'O'RS1#)=+>+>Q$?1 98M36$')43w1@ !)M3U&:)# !X>).bp F0"%N2AW&YOnN:+>5=E4$')#"c1<)e$?-01-0/0AWC!ª3 $."%AWAW&:N:EYEY$?F0Xn&Y1#5«-0N:/EY(\$.36)eAWAW&i-8Y/8ND:),(.+N:Q«AW&:-0/2(¢981@&4R$,&:1@;¬Q C!NG98F^(*43Axp)#6X. !$'AW+>)M&:5k923w1@1@-0$n eC'5=AxRS;¬d 1#+«(.9890&4E41@&:ED !&:Q.$nC./8M3RS1#+ p>),$?+=1 b dξ dH
ξeq
ξeq
(∂H/∂ξ)T,p
>d¶-0/2)#1#CeIª1@(\_4&:)#;V/8&:5$'NG/8F8)B1!R
X*QT&:EYN2(*O')e&:;BH098AW&YN:+>EY$.$'A%&:;¬ !)Z921#+>N8"cO,1Q. n$.1@C,EYAWQ &:; +.p¢)e-836C!5=)#C!$?)#1!90REY&:5= !-0) /0Q#F8#ND(*)@b
]6"
)
{+.& '$.AWNGFGAg(*)436A%$')M3w1#N:/3w1]N8O'&:;BAWN:EY$'&: !)l;l1@;V5
µR (T, p) = µ0R (T ) + RT ln(p/p0 ) + RT ln xR .
*
86
∂2G ∂ξ 2
=−
∂A ∂ξ
=
X
ªRS1@;BC=A ;<)#"W)43w&i-2E4+.& '$.AW)M3w1@$'&F01@C!) T,p
T,p
νR
R
∂µR ∂ξ
, T,p
86
xR (ξ) =
!¶+>)#EY$.AWAW& &:31@_
n(ξ) =
P
R
nR (ξ) nR (0) + νR ξ = , n(ξ) n(ξ)
nR (ξ) = n(0) + ξ∆n
p1
∆n =
P
R νR
X ∂ ln xR ∂µR νR = RT νR ∂ξ T,p ∂ξ R R X νR ∆n − = RT νR nR (0) + νR ξ n(ξ)
X
36AWYN
R
∂2G ∂ξ 2
" # 2 RT X νR = − (∆n)2 . n(ξ) xR (ξ)
90EY1#5=-0-0/8/0:Q#(.F8$.#NªAW&K/8)#(._490-2E41@&:;
R
P
R0
x R0 = 1 )
2 2 x X ν 2 x R0 X ν2 νR x R0 1 X νR 0 R R R = = + xR xR 2 xR x R0 0 0 R
R,R
R,R
X 1 X (νR xR0 − νR0 xR )2 = + ν R ν R0 . 2 x R x R0 0 0 R,R
R,R
d
*1@Q,3w1@_Y;V5!p\_4&
=33 8#"%'& / F% 4!" % &# %
*
X
EM1@/8&:;
ν R ν R0 =
R,R0
X R
νR ·
X
νR0 = (∆n)2 ,
R0
X ν2 1 X (νR xR0 − νR0 xR )2 R − (∆n)2 = ≥ 0. xR 2 x R x R0 0
#/8)K) 90F0;l&Y+>1@$?"%$.1@AWC7&«(.98M390EY65=$'(?)e1#H8+.I«&:CrEM1#$.N8O'AW& )='+>EYEY5>AxN:pEY !$n+>5!5p6X*)k$.AW&-0(\&MR
$.Ac1k31@90dQ.!$.&YC,-0Q / - n2T1@E4Q#M3F0&:;B+.$'p)e&e-0 !pC!+>)#1T5=$?_](.1!R
3t985>)=-0N8+>(*OQ.nRC,3N:/81@M3EY90¬5,Q.$.$'$.+.&:A%),C=Cs+'At1@-8-0/0/2/8$.1@&YX.AWN8&eA%O.d"%$'AW)#f)e;<1@H8N:C&:A/03690F^5>&YAW-8N:Y/EYNZ$'-83 & (\&M-0(.R
Q.$.90X'EYAW)#-85=/0$,(?925s1#1@+>/8+>43C,"cQ1 3F0t&Y-0-0EY/65>36-0/05=C=(.AWQ.N8O«C.RS+.)#JS(.Q.Q'$.-0C!E4NGN:F8EM 1@"%$n5>(.N890O EY5]3Q'1@-090/2/81@)e"W)#H8N:$,A 5=N2O dfA ){)#d EY$?1#N:EM1>p_4& =d6|1@90Q.$'p.&: !C +>EY98AWM3&D6Eq$')4+.3w& '1@$. eADAWNGF^N2A O'&:;BAWN:EY$'&GFDF^&Y-0/Z)#C,98&YH0"W)#$,5s98436$?1@$.AW&:; R
R,R
xR = const · νR
86
xR ≥ 0
xR
ξ
G
T
p
86
A=0
A(T, p, ξ) = −
∂G ∂ξ
.
*(3)e1@)#-090EY/8Q.36:$'(.Ac&:@Q CEM1@98-0$.&4/21@AW1@&:C!X.;NGA%F^"%A$'98)e43H86N:AU$?1@±E4$.)#Ac1Xgd d.EM1#$n+/21@d "%(.!²Ac1ZF0&Y-0-03U/)#X*9843)=6+>$'$?)M1B3w;V1@ !Q')M-83wA-01r(*&M"%AWR
N:$.EYAcX?1#1I
ξeq (T, p)
T,p
A = 0
)
^± >d !² 13 @$?RS1@1#"WH8)#N: eA%3wAWN:&:EY !$,).5d |j+.))#X*3w&Y1@Ni90Q./8$.&: !C=)QrEM+.1@)="W&:+'_Y1@$'/0)e$.H8AWI)eH8N:A)=H8+N:AWH0"%A%3w;<)e)#H8_YN:$?At1]"%Q.)=X+>36N:AW98&:,N(¢:RSAW1I AtN2O'/0&:92;B1@C=AWN:/8)4EY3$'&G1#FtI ;<)#F8_Y1@$?C!1)36JSAWQ.Y$.NDC!(.NGF^90E4&Y+.-0p /21436A AWI6F81@dC!|) 1@90Q.$'&:C 98M36$')43w1@ eA
ξ
∂2G ∂ξ 2
T,p
=−
∂A ∂ξ
T p
> 0,
T,p
A A
ξ
)
ξeq (T, p) = ξ(T, p, A = 0),
e~
]6"
-0C##+
∂ξeq ∂T
∂ξ ∂T
'$$.AW&G)ª&:F EM(.1@/890"W)#E4&:_4&:_Y-2C>$'1@-0&D;<EYAg/2)e1!EMH8RN:1@NYAx"W&:&:p$._YC,Ac$'1ª/8&e#9(*p.2;B)=17N8AcO'(*1@)=$')#+>)MEY$'363w&G1@AWFN:"cAW1r(.&90EYEM±E451@)#;B-0Xg/2AWd1!&:$.R
± 5==dAc1#; >IsM²098² )#Q."c1@_Y;B5=3wAi1@EY;i;B5AW&:EY$.$?$'1h& b =
p
.
p,A=0
A
)
∂ξ ∂T
(∂A/∂T )p,ξ =− . (∂A/∂ξ)p,T ξ=ξeq
¶)#$.AW&:31@_V3-0/21@$.AW&q98436$')431@ eAxp¢N:EY5="%A+>"c1
p,A=0
∂A ∂T
1 = T
∂H ∂ξ
ξ = ξeq
±E4)#Xgd.EM1#+ d !²p.36AWYNq)e-0/21@/8&YN:EY$.AW&i+.)e-0/21hF0&:;V5 p,ξ
p
T,p
)
^± >d ,² (.$'90&GFiEY5<(*N:)EY5,"W&:;[3U&G36F<5,-0C!/0)#9890)#EY$.5>AW&-0/21@F0&Y"%AW-0H0/B;i365
∂ξeq ∂T
p
(∂H/∂ξ)T,p = T |(∂A/∂ξ)T,p |
,
ξ=ξeq
*
(∂H/∂ξ)T,p ξ=ξeq
)
ξeq
)
∂ξeq ∂p
=
T
∂ξ ∂p
T,A=0
(∂A/∂p)T,ξ . =− (∂A/∂ξ)T,p ξ=ξeq
¶)#$.AW&:3w1@_
=33 8#"%'& / F% 4!" % &# %
^± >d !² F036&:AW;VYN 5 RS#N:EM#N±^ >d !²Ei±^ >d !²A.365,C!)#90EY5>-0/0Q#F8#NV±^ >d !²pn+.)e-0/21hb ±^ >d e² s±^ >d !²t)#921@EV+.& '$.AWNGFGA +.)e-0/21hF0&:;V5
*
∂A ∂p
=−
T,ξ
∂ξeq ∂p
∂V ∂ξ
∂2G ∂p∂ξ
T
86
T
=−
∂V ∂ξ
,
T,p
(∂V /∂ξ)T,p =− . |(∂A/∂ξ)T,p | ξ=ξeq A
=
X
νR
∂µR ∂p
=
X
νR vR ,
!+>98EY43AW&6$')eH8-2N:6A6N:±^EM #>d-0/0C!e²)M636365,$.;BAW)eA>-8)#C,X>Q#F0F0:&:/8;V)eH85!N:p\Ac_41@&q;B36A,;<EY98)#)e"W-8)M/ 36N:5,AWH0;B$.A=AW&:98$.&4Ac1@1l !&:3°$n/8-0M3/2q1hb d 90(.R&GQ.90F$.E4/8&YC=&:-0Qg;BQ.p*$.(*3 AW&:Y92N:C,1@AW&/8/0#9Q.F^090&Y5=E4-0;&w/(.3®;l90E41@&Y-0"W-0/0&GQn98F^3)#&B$'1)#{X>9843(.F06:98/8)=$')e+>)MH8Q.3wIC,1@Q./8 !C'M3tqRS1#N2p+>O'Qg&:1k;Bd !ªAW+>N:+>EY5 $?5 3k/5,;¯C,AW&b 3/8)eH8-8IZ/098;l)#$'1@"W&GF0-0&eQ.pX'36-0/0AW92Y1@N]/8M3)=a+>(\±)M(*36)#Ac$.1#AW+'&:1K3w1@/8_T)K3/21@C=98_4&4&Z1@C!(.NG90FGE4A&Y)=-0Q.+>$.36AW98Y)#N:/0A%Q $'&GF<9843)#6X>$'F^)@bb 3361@Q# eF8#A Ne3Pp\36C,AW5,&:92901@Q._4$.&:C=$.Q]Ac1EY;B±^ $.>dAW&GF^,-0² E4&:A6$.±^Ac1>d)#X>e²F0:/8-8)e/2H81@N:$'A )4Q.36C'AcRS1#+>;lQ¢1@²/8d.&:¶;l)=1@+./-05>Q.N:;<EY$')@& b -^JL)#90;VQ¢R)M3w1@$.AW&i98&: eQ¢R
5 g&i³UO?1@/8&:"%AW&:921>d >d ¢±L1! !²V)=+>9843$.6AW&D$'ED)M3w98&:1@ e nQ¢1RS (.90g&E4V&Y-8³UQ.$.O?AW1@&/8&:-0"%AWAWl&:9231K±C=E4AW)#&:Xg90Q.d.$.EM1#C,+Qgdp 3[e²UC,3/8#90(.5=90;EY5=(?Q.1#C.+>RSC,1#Q+ (*;B),AWN:N2O¢EY$?RS1@1>$.p#Ac361AWYN:NwAW&:98(¢43R6).d$')M¶3w)#1@$. nAW&:1q3w(.1@90_KE4&Y98-0&4Q.1@$.C!AWNG&6F01-0AWw98)#3sEYC.C,RSAW1#&:+>90Q.Q$.F0C=&YQ-0/<36&:AW:$'C=+.-0)#E4/8&:&: !9^) b -0/8:_4&:$.Ac1k(.98)=+>Q.C,/8M3qpN:EYdn5,|"%A3 (.90EY$'5,)M(?361#5,+>;C=Q(*±)nX¢R²)#_498&:43$.6A%$'Q )M3w98431@6 n$'1)4(.3901@E4 e&bA -30Q.$.36AW&B5,$.-8A%AWC=iQk3 98)#C=EYAW&:C'90RSQ.1#$.+>C=Q Qg
*
T,p
R
T,ξ
R
∂V /∂ξ < 0 ∂V /∂ξ > 0
*
xA (T1 , p0 ) < xA (T0 , p0 )
xA (T0 , p1 ) > xA (T0 , p0 )
= 4~>d-0!EY&Y5,H0"%; A?/8(.&:;B90EY(*5=&:X.92"%1@A%_4/0&:Q.$.90A%5 Q«;<)#A _Y$?1$.(.AW&90EY985h#_F0Y#$.IeAcpV_4-8& AWtEYXn5=/0$.$.AWAW&B).p,EM/81@)V"W&:3r_Y5j(.),AW&:+ 903b +F^>&:"c;i1 5 d | 43wN:EM1#-VE<98M36$?1@$.Ac1 1@$ / ) 1]+.)e-8/21hb
]6"
T1 T0
∆h0r
T
T0 < T < T 1 )
T1
∆h0r dT 2 T0 RT ∆h0r 1 1 = ln Kp (T0 ) + − R T0 T1
ln Kp (T1 ) = ln Kp (T0 ) +
Z
N:EY5,36AWH8N:AW&q/8&:$36E4#9)#X\)M36Ac@EYQ#F^&V/21@C,_4&ep> !+>5 d >dF0¶#90N:EY5=5M;´F^;B$?A F^1#;V-65!(.p¢90_4E4&&Y+>EYAc1@"W&q$./8AW&:&i;BEM(*1@&:"W&:92_Y1@5T/0Q.)=94+Tp.36/8&:AWY;BN (*&:921@/0Q.905 3¯A%$n8/ &:98&Y-0Q>b T1 < T 0
∆h0r
ln[Kp (T1 )/Kp (T0 )] =
32000 (1/298 − 1/310) = 0, 50, 8, 314
0- /2#+ d >ed ;l*1@1#Qn-t3w+>1@"c_Y1;i/85!&GFp'_498&&41@C!NGFGA ;l1(*)e-0/21#I Kp (T1 ) = 1, 65Kp (T0 )
*
xB = xC = (1 − xA )/2
X\)
p>36AWYNª(.921Y3w)<+>EYAc1!RS1@$.Ac1
)
(1 − xA )2 = Kx (T, p) = Kp (T ), 4x2A
d,| p¢-0C##+ B p
∆n = 0 T1 = 300 1 − xA = 0, 42
1 − xA = 0, 40
p.1q3
³UAW&:(¢R)B98&41@C!NGFGA"%AWN:EY5,;i5E4&ª36E4)#90Q«±E4)#Xgd.EM1#+ dg4~!² Kp (T1 ) = 0, 111,
-0C##+
∆h0r = ∆h0r = 8, 68
Kp (T2 ) = 0, 131.
R ln[Kp (T2 )/Kp (T1 )] , 1/T1 − 1/T2
C= h;<)#"xd
T2 = 315
)
Bp
# e>dt¶92143U)Z+>EYAc1!RS1@$.Ac1B;l1#-+>"c1/8&GF698&41@C!NGF^A ;l1(*)e-0/21#I
=33 8#"%'& / F% 4!" % &# %
)
(pB /p0 )νB (pC /p0 )νC = Kp (T ), pA /p0
!N:EM+>#EY-0AW&/0C!)M3U&ed!¶)ª(*),p +.-0/21436AW&:$.A%Q
pC = pxC
)
xνBB xνCC = xA
p p0
−∆n
Kp (T ) = Kx (T, p),
!;<+>)#EY"WAW)M& 3U&](.90E4&:E-0/8)#(.AW&:o +>5=-2),dNG¶F81#)#NGE4FG)eAx-8p /21hF^d&T³$?1!1@R
C!;µ)43636A%/25,1921@"%EYAWN:AWI]EYX?Q¢1KRS1@;<;B)#C="%AA 3PQ.C'RS1#+>EYAW&q3-0/21@$.AW&q98436$')431@ eA365=$')e-0A ∆n = νB + νC − 1
α
)
(.90EY5N:EY5,; YE !)=+>$.d AW&ª¶Eª)#$.+.AW& &:'3w$.1@AW_NGF8ET-098/8M3)#6(.$?$.1@Ac1$.Ac+>1k5>-898)=&4NG1@F0C!1#NGNGFGF^AA 365,$.A%Ce1>p _4& p * p 6 3 W A Y N p p -0Ce#+ n = n A + nB + nC , 86
nA = n p −
nd = (1 − α)np nB = ν B nd nC = ν C nd
¶)(\)=+.-0/21Y36AW&:$.A%Q
nB = νB αnp nC = νC αnp
p=+>"c1
pn+.)e-8/21hF^&:;i5
n = (1 − α + νB α + νC α)np = (1 + α∆n)np . ) xR = nR /n
1−α , 1 + α∆n
R = A, B, C
νB α , 1 + α∆n
νC α . 1 + α∆n
f+>5>1@-8Cr)=NG36F01#AWYNGNF^A*;lA-01@/2;i1!RS5 98$?&41#1@-0C!/8NG:F^(.A Q#F0#N:5vEY36Ac@E4&:C (\)#;BAWY+>EY5 -0/8)#(.$.AW&:; xA =
xB =
xC =
)
!-0/2&Y1#H8N:"%A A )
(νB α)νB (νC α)νC (1 + α∆n)−∆n = Kx (T, p). 1−α α 1
p'/8)<(*)4365=_4-0E4&V365,921@_4&:$.AW&i;<)#_Y$?1
)
>d X.AWAW&:N: eEY$.X?AW1j&l-0F0C'&YRS+>1#$?+>1$.A%98C!&4431@ C!NGF03¬1kQ.N8C'O'RS&:1#;B+>AWEYN:AWEY& $?361>p65=$'N:EY)e5=-0A"%A dd | ,Q.X'Q.-8C./2RS1@1#$'+>NGEYF0AW1& F37(.0&Y90EY-0/5=(? n1#1@+>E4C,&:;Qp±L1!1K² -0Q.X'A -0/21@/$'3wNGF0)#& 90EMF^A &Y+>$')#-298K)=+>N:$?AW&YqN:EM;B1@;BAW&Y-8AxEMp1@(.$.90A%EY$'5kep,N:36EY5,AWY; N JL"% AW1@N:N:EYEYEY5X?5,136N:AWAW&:JxH81@C'N:E RAW&e&eppC,3 /8#9 n81@&]pwEYAW-21&]T$?3 3 1#+«(.98439098EY6)#5=$'EY(?/)M1#3w3w+>)#1#C=98+>Q¯&:E4;µ& ±X¢E]EY² $? n1h1@F^E4+>A&:Q#; F8{p-0/36AW3UAWY)#N$.90EMAW&«/+>5=36"%C!AW)& d AQ.N:EMC.#RS-01#d /8+>&YQ«N:AWN:EY3yEYC,X?A198-0M3C'$.6RSAW1#$'&:+>EM)4$.1@3wA%"W1#C#&:_Y+>1 $nE45>& N8pOs/8&:1@90(?"W;<&l1@92),/21@1@+>;
]6"
r = 3 ( c = 1
C
A
B
A B f = 2
A
B
f = 3
C
A
B
#
ls = r + 2 − f − c ls = 3 + 2 − 2 − 1 = 2 ls = 3 + 2 − 3 − 1 = 1 A B
) pA = p∗A (T )xA pB = p∗B (T )xB
pA pB xA xB
*
A B
)
µA = µ0A (T ) + RT ln xA ,
µB = µ0B (T ) + RT ln xB ,
C)
µC = µ0C (T ) + RT ln(pC /p0 ).
A(c) B(c) + C(g)
)
xB (pC /p0 ) = K(T ). xA
:N³AWH01!$.R
C!AW&:)4o 36A%N:/8EM&t#N:-0/0AWH0C!$.)4AW36&:$.5>AWN8&O n1@EYQB$?1#+98)#EY/3U)#98&:; p pF0&Y-0/-0Q.;l /090E4&YN8O X?¶1#)#Ie$.p>AW&:363AWY1@N_q)#X\&YN:$')eH8Iª-0C'RS1#+>$.A%Ce1 3v98)#EY/3U)#90E4&q;<)#_Y$?1VEM1@$.AW&Y+=b p
)
p = pA + pB + pC = p∗A xA + p∗B xB + pC . C
xA + xB + xC ≈ xA + xB = 1,
=33 8#"%'& / F% 4!" % &# %
NY)B(*)(*),+.-0/21436AW&:$.A%QT+.)B(.92143w1<+>EYAc1!RS1@$.Ac1<;l1#-t+'1hF0& -0Ce#+ -0/21@/8&YN:EY$.AW&
)
(1 − xA )pC = p0 K(T )xA , pC , + pC
xA =
p0 K
xB =
p0 K . + pC
p0 K
p∗A pC + p∗B p0 K + pC . p0 K + p C
p=
>dt¶92143U)Z+>EYAc1!RS1@$.Ac1B;l1#-+>"c1/8&GF698&41@C!NGF^A ;l1(*)e-0/21#I
)
aCu aH2 O = K. aCuO aH2
¶R&G)#FY$.p.AW;<&:3w)#_Y1@$?_1l³U(.Q¢90 [EY5MF8A #³UIep.Q_4&q-2AWN2N:O{EY5=1@-8C=//5=5=;B36A $'-0)eQ.H8N:X'A-0/2-1@$'98NG43F861@;B$'&ZA3ed?JLf1@1@EYCAW&B36-0/2AWY1hN b 3«NGF^A (. n921@1YE436)4AW36&5>+>N8EYO Ac1!RS1@$.Ac1;l1#-(\)#E4)e-8/21hF0/5="%C!)1@C=/5=36$')eH8N:A.-0Q.X'-0/21@$>b !-0/2+>1!EYRSAW1B&l98364365,C!$')#)4903EY5>1@ e-0/2A1@98"%&4AWH01@;iC!NG5F^A EY36Ac@E4&:C7±^d >d% e²dg| /5=; (.90EY5=(?1#+>C,Q >d -0 /8EY&Y$?N:EY1#C!N:)MEY3w;i5P(.9890)#E4EY&:(.E Q'-0E4N:dEMK1@"%$.)#A%"cCP1@"%$'-0)e5,H8;iIgX\F^&Y)#-8"W&:/g; E4+.& 'p$.AW1K)43F01@&: !$?)s1 F8;l1@C!1#)6-8j"%AWN:N:EYEMX?h1 b ;<36AW)#Y"%NAªF0-0&YQ.+>X'$'-0)e/2-01@/0$'CeNGF^A;<98)#)#"cEY1@(."%$'Q')e-0E4H8N:N:E4A')#F0$'&Y-0&G/FYp p 3 *;
pH O a H2 O = 2 = Kp , a H2 p H2
K = Kp
A
86
MA
B m0 = 1
mB /m0 =
kg nB 1000 nB · = , nA MA mol nA (MA /M 0 )
!+>EYEYAWAW&:& "c#N"%AWN:EY$.A%CA> ;BhAc;<1@$')#)4"ª36F0$.&Y-0A%C/(.F^&Y90+>E4&:$'E)e-0N4/01!CeR
7C!)436;lA%1#/2-0q5°"%AWN:N:EYEMX\#-0/8;<&YN:)#EY"%C!Ax)Mp 3U&GFYd p.+.)e-0/21hF0&:;V5 M0 = 1
n=
nA + n B
)
mB /m0 =
xB 1000 . xA (MA /M 0 )
]6"
*¶1#)#-0$./8AW)e&:-03Q#1@F0_&:;V5$?1hF^(.p.AW&:36903´AWYND(*+.)4)e36-05=/21h_4F0-0&:EY;V5k5 36E4#9B3 e921@$.AWN:5 d !+>5 ;l!+>1@EY;VAW& 5]$?1#-0/8:(.Q#F8#N:5ZEY36Ac@E4&:d C](*1@)#/8;B)#;BAWYAc+>1#EY-85/+>"c1ZA+.)43w)#"%$n5>N8O-0/8:_4&:o *
xB → 0
)
xA → 1
xB = χmB /m0 ,
χ = 10−3 (MA /M 0 )
xB → 0,
xB
mB )
1 -0/2#+ ±^ >d !² "C=/5=36$')eH8IU365,921@_4)#$?1 F81@C!)tJSQ.$'N8ChF01 Q¢RS1@;BCe1 ;<)#"W)43U&: !)ª;l1 (*)@b -0/21#I xB /(1 − xB ) = χmB /m0 , xB =
)
χmB /m0 . 1 + χmB /m0
(.$'90)eEYH85]N:A N:36EY5=5,; 921@_4+.)#)=$'+'&GF1@"%F8AWH01@;iC!)5 JA%Q.$'$.+.C!&:NGC=F8-1;
x
|N:EY5,$. e92$.1@A%C=$.AAWN:1@5sC,/$.5,AW36&Y-0$'C!)e)#H2o'N:N:AE4+'&:$.@_MAW& B+>+.Q.) _4e&: !p>)36AW98Y)#N E4N:AW&:o'N:E4&:$.Ac1)#X?1{3t-0(*nRcb m 0 am B = γB mB /m .
|N:EY5,5,$.$.$.A%CeA%1VC -8/2d>#+|jpn)#_4X*&ª&Y1@NDC,/8/&:5, !36)<$'+>)e"cH21N:A +.)43w)#A "%$'&: !)Z98#_-8Y/8$.:Ac_4&:$.-0AWAc 1/;l5="%1@C!;V)5 )-0/21!R
5 axB = xB = χmB /m0 = χam B. axB am B
)
χ
−1 x x x am B = χ aB = γ B xB = γ B
mB /m0 m = γB mB /m0 , 1 + χmB /m0
!+>EYAW&D3t-0(*nRN:EY5,$.$.A%C1@C,/5,36$')eH2N:A ;l1(\)e-8/21#I m γB
m γB (T, p, mB ) =
)
γ x (T, p, xB ) , 1 + χmB /m0
.(±^ 90>dEY5{ !N:²EYd5=;,1@C«RS1@/$?3w1@)"W&:_Y-05(.92361Y3t5=92+>1@EYEYAWAWIeIp F01@C!)JQ.$.C!pNG !F0+>5 EB98436$?p1@/2$.1@AcC1 F01@CZ(\)M36A%$.$')<Xn5>Ied
xB
m → 1 γB
mB mB → 0
=33 8#"%'& / F% 4!" % &# %
=d {+.& '$.AWNGF^A-0/8:_4&:$.Ac1;<)#"W)M3U&: !) (.C!)M90EY3U5V&BN:)#EYX>5=F0;°:/8)#)eX>H8N:F0A:/8;<)eH8)#IU"W)4983U)#EY&/3w98)#)#EY90(.QgQ'p -8E4N:p4EM;<1@"%)#$._YA%$?C#1 136±S-05=C'92RS1@1#EY+>AW$.I A%(.C 90E4&:²tEN:AEM-0#Q.-0X>/^bb -0/21@$'NGF^A 98)#EY(.Q'-0E4N:E4)#$'&GFi±S-0C'RS1#+>$.A%C ²p A EYAW&:"c#N"%AWp.N:+.EY$.)eA%-0C/21hA>F0;B&:;VAc1@5 $')436$.A%C(.90E4&:EN41!R
C!)436A%/2q"%AWN:EYX\;<)#"%Axp *
86
)
[B] = nB /V, V
A
v A vB )
B
V = n A vA + n B vB .
n=
)
nA + n B
xB xA vA + x B vB
ª +>5 p./8) >p 36AWYN !98)#+>EYEY(.AW& Q'-0E4N:EM1@"%$.A%;
xB → 0 c0 = 1
xA → 1
∗ [B]/c0 = xB /(c0 vA ), ∗ vA
+(
[B] , c0
(.;<90)#EY"c51@"%N:$'EY)e5,H8; N:A6±(*)#(.90E4&Y+>d$.*AW&i1@QnEM31#+'1@1@_Y$.;VAW&M5!²p,p¢_43t&-0A%$?(*1#nRN:E4N:&GEYF5,$.$.$.A%_6A%C 3 (.90EMEY1@5="W&:(?_Y1#5 +>C,/0QQ "W)=N:)MEY+]3U5v/8&GF&:Aq;B)=N:+<(\Ac1!&:N:RS921sAW1@H0$./0-0Q.AW/2&:901!$.5]RAc&:1DAg !).N:;
∗ χ = c 0 vA
*
χ
acB
acB = [B]/c0
axB = xB = χ[B]/c0 = χacB . χ
axB = χacB
x ∞ c µB = µ ∞ B + RT ln aB = µB + RT ln χ + RT ln aB ,
]6"
63 AWYN !+>EYAW&
±^ >d n~!²
µB = µ0B + RT ln acB ,
∞ 0 ∗ µ0B = µ∞ B + RT ln χ = µB + RT ln(c vA )
F^Qn&Y3w-0/1@_Y$';i)45!36pn5=_4;&t !-0+>/25,1@X,$'5B+'1@EY9836+.Ac)M@36E4&:5,C ;(\)#/8&:$'NGF81!R&:; peC=/8N8#9O'0&:5;B+>AW"cN:1DEY$,90E45,&Y; N:EYd 5=*361hAzbb -0-0/8(\&:&M !R
)$.AW)#98$,)#EY5/3U3 )#90N4Q1!R
)#5,X*;})436EMAc1@@C,EYQ#98&YF0&{-8AW&k/5=-0"%/8C!:)k_4&:3 ogp e/892)71@$.-8AW/2N:1@5 $¯-0/21@$'+'1@98+.pU)MXn365'5 R EY$.),Q#+>AW&BF0(*#F0)4N{&Y36-0-0Ac/l/21#1@+')=$P1!+>R
3UX,-0/2)n5v1@R
$'5=-03w+'/8:1@1#_498I&:+.$.)4-0A%36AWQ T5 +.-0)kQ.X'-0/2-01@/2$n1@$'QrNGF^O.AªA%d(\98)#)#f/8EY1@&:(.C7/Q'5=-036N:E4EYAWN:$'YE4N{&G)#F$'N8&G-0O?FYQ.1@pX'9236-01@/2C=5,1@/8 !$'&:)=NG90F^5n+=Axbb p C,3v/8#9982)#1EY/3U+>"c)#190E4+.&D)M$.3UAW)#&Y"%-8$'C!&:)# !o') N:E4-0&:/8$.:AW_4&D&:$.98)#Ac1E4N:EMAW&:1#o'N8N:O'E4)M)#36$nQ#5=F^; &d -0ªAWT1@"W/2&:1@_YC75l-2(.1@90;<EY5Z) /F05=1@; C (\/5>)=NY+>& C,98&YH8"%36AWIe5=p?EY_4$?&1#F^N:&YEM-81/ -0/8AW)ZEM31@X.)#AW(?&: 1@98N:N:EYA%Q 5=-8)/8)<3 JLRS)#1#90-0;l$')e1@H8"%N:$nA 5!98p*)# !EY+>/5=3U_i)#903¯QT(.-8A%92"%1@$.C,AW& b 98)#E4N:AW&:o'N:E4)#$'&: !).dph !K+>AcEY1@AW&$')436AWN:)#AWEY&<$?EM1#1@N:(.EM1AW-8EY3t;V-0(*5jnR1@N:C,EY/5,5,$.36$.$'A%)eCBH21@IBC,3`/5,36(*)e$'-0)e/2H21#N:N:A A 3U43wN:EM1#-E4&q36EY e"WY+>Q $?1 ±^ >d n~!² ±^ >d !² /0E,98/8)#(*:"%_4)#Q#&:;BF^$.&:AcAW;i1@&V985;<43 ±)#EM"W/81@)M&:3UC.90RS&i;<1#+'A),1@+>/8;i&:5=;B$?5 1@(*;B&:92AW1@N:/0EYQ.$n925>1ZN8OT-F^²&Y(?d -81@/8*92&Y1!1@H0;V;
acB = [B]/c0
[B] → 0
[B]/c0 = 1
µ0B
c [B]/c0 acB = γB
c γB
c µB − RT ln([B]/c0 ) = µ0B + RT ln γB .
(
p = const
) µ0B
µB
c γB
[B] = 0
[B] → 0 86
µ0B
*
[B] → 0
µ0B
=33 8#"%'& / F% 4!" % &# %
>d6EM|a1@C'(.RS901#EY+'5,1@(?;V1#5!+>pC=_4Qq&<(.3wAW&:),903t+'1T-0E436&GF 5>98-0&4/81@:C!(.NGQ#F^F^A!&l;l(.1@92;V1@C,5 /t5=N:-8EYC.$.RS1#AW&+>$.3 A%C=(\AxpY)e(.-090/2EY1#5DN:AwN:EYN:5,EY5n; b -$'0/8M3&GyFª-0$?Q.1X'-0F0/2&G1@Fq$'1@NGC,F^Ax/p*5,N:36EY$'5,)e"%AH8Ie(*p)#/S;BF:d%A p F01@;i5K36(¢R
5=3d «$.AWC!&:)#36"W&:AW&:A"%C,1@;
*
aNH3
a H2 O ≈ 1 ≈ pNH3 /p0 pNH3
)
aNH+ aOH− 4
= K.
!NY)#&Y$,H8"%5,A¢; N8O')=/090+>Q'EY+>A*$')V)]+>9890)#Q.EY n(.VQ'98-0&4E41@N:C!EM1@NGF^"%$'epn&GFD/8)q-8)#;<"%Ax)4d 311@F^;V&Y-05T/U)q3698AWY)#NEY/3t3w-0)#(*90E4nR&
AW-0$?/01#$.-0AW5n&bb $.905=AW&ª;(.92981@)#C=EY(./5>Q'N:-0EYE4$.N:AWE4&q)#$'N:EY&5>-0-2/8k&: !N:)BEM#C=-090/85=&YN:-0EYEY/2C,1!AqR
Q -2)#-2"%Ax)#dU"%A ¶ED)#98$.)#AW&:EY3w/3U1@)#_s98&:1@; C,/p?5,3P36$'C,)e/8H8@bI N3:EY1@5>90-0Q./8$'&: !&:C)DN:98Ac431!6RS1ª$')4-03/21!1@R e&:A !)w+.)eF0&Y-8-0/2/1hF^3s&:;i(.590EY5,X."%A%_4&:$.A%Q<98M36$?1Bep@36AWYNF01@C!) pNH3
)
aBa2+ aSO2− = Ks ,
!_4&B+>EY-8AW#"&*(.F0&Y90-0E4/ª&:E /090Q'+>$')#)EY$?98)#1#EYN:(.EYQ'5,"%-0AWE4H0N:;iEM1@5"%$?-01>/21!pRS)#{EY$?98431#6N:EM$'1)M3wF0&G1@F e;lAª1!98&4R&B1@C!-0NG/8F^:Ax_4d &:f$.).AW& p 3¯98)#EY98/)#3UEY)#/3w905)#903UE4&e)=d +>|$'&B(.-8)#90EY"%Ax5=p*(?-01#Q.+>X'C,-0Q/21@$'&:"WNG&:F8C=1Z/09898)#)#"%EYA%(./843Q'q-0E4pgN:+.E4)#)Z$?C,1]/8#93605>5>-0N8/8O:(.$?Q#1@F0"W#&:_MN41 3°|j)#;lX*1!&YR
N5,/8; &: !-0)B/8:-0_4/2&:1!$.RS1 A%Q
Ks
Ks
[Ba2+ ][SO2− 4 ] = Ks ,
j1 (\)#$.AW&:31@_ p 36AWYNK-8/8:_4&:$.AW&TCe1@_4+.&: !)jElF^)@b $')M398)#(.36Q'-05=E4$'N:)eEM-01@A "%$')eH2N:Ad-8)#f"%A1@C3363wAW),YN+>EY-0AW/2&e1!d RS1 +.)e-8/21@98N:EM1 A%$>JL)#90;l1#NGFGA [Ba2+ ] = [SO2− 4 ] 1/2 Ks
Ks
#~ e~>d 1@C,)#/EY5,/363w$'#9)eH8$?IT1#-0365>5,NY$')#)e$n-85tAVF0&Yk-0/±SN:3kEY5>98-0M3/65 $'C=)49035>1#-0+>EYE4/21!&tRLE6²d -8)#|j"c )#X*&YN /8&: !)p@(.C=92/81Y#93U8&G) F +(\>EY)eAc-01!/2RS1#1@Iª$.A%Ac"W1<)=N:;lEY5=1#-ª$nQ +>"c1B98&41@C!NGF^A ;l1 NF0:± #EMNi1@"%/8$nA &:5>$N8O 36-2-8E4t)##91@"%qA*C=$?+./5=1@)36EY985=$')#3w)eEY1H8/N:3U-0Ac)#1@AW 90;BQkA%A4"W),F03U)#N:)=EY$'+>5=43$'$' &:&:² !;´pM)C,98/8-8)##9)#EY0"%(.5ªA Q'+>-0"cE41N:/0EM901@Q'"%$'+>)e$'p\H8)tC,N:98Ax/8d)##9EY,2(.1]/8)eQ'+>-8-8Q>5,E:bbb -8),NGF^Q#F0& 3w&Y+?R
Q. l98M36$?1@$.Ac1
]6"
Ax By (s)
Ax By (s) xA(aq) + yB(aq)
)
axA ayB = Ks
aA aB
(
Ag2 CO3
)
;l1@;V5 (.90EY5N:EY5=; )
Ag2 CO3 (s) 2Ag + (aq) + CO2− 3 (aq),
d'ª)e-0/21hF0&:;V5-0/2#+
Ks = a2Ag+ aCO2− ≈ [Ag+ ]2 [CO2− 3 ], 3
+
[Ag ] =
2[CO2− 3 ]
;<)#" h" )#921@E <;N:EY)#5," "%hA "xd >d 36,C!5,)#;´90EY5=Ag-0-0/2/21@1!;iRSB5 98&4E41@&qC!EYNG36FGAUAc@±EYEY36C=Q{Ac@(\E4&:)#Cj;B±^AWY +>>dEY5 !²0(*² )#/8&:$'NGF81!R&:; -0/21@$'+'1@98+.)@b 3 −12 4[CO2− , 3 ] = Ks = 6, 2 × 10
−4 [CO2− 3 ] = 1, 16 × 10
[Ag+ ] = 2, 32 × 10−4
(
!+.+>)436EYAW5!& p
E0 =
)
RT ln K, zF
³ h;<)#"xp¢1
d'¶)#p¢/81&:$'3tNGF0-01!(*R nR-0N:/2EY1@5,$'$.+'$.1@A%9^C b
F = 9, 6485 × 104 z=2 E 0 = 0, 339 − (−0, 763) V = 1, 102 V ) RT /zF
-0C##+
365=$')e-0A
e³
RT 8, 314 · 298 = × 10−4 zF 2 · 9, 6485 K = eE
0 zF/RT
= 1, 284 × 10−2 V,
= e85,8 = 1, 8 × 1037 .
=33 8#"%'& / F% 4!" % &# %
3f 1@1@Ck90/8)e36H8AWIeYd N-0/21!RS1{98M36$')431@ eA+>"c1{/8&GF98&41@C!NGFGA6;l1{X?1@98+>E4)j+>Q._M f)VEM1#+'1@$.AW& ;<)#p'_Y !$?+>1qEYAW/8& &:_98)#EY36Ac@EM1#I37pA%$.1 $,5l-0(*)e-8#Xgd KAc1@$')4p\36(.AW90N:EYAW5 & N:EY5=; C=C!)M/8#93w8!& ± ²l-0A C'RSE4²1#&K+')#1jEY-0/2$?1@-01#AW$'N: +'EM11@E 98+.(\-0/2)4nR1!3wRS)#&: e !$.98) A%M336(*1j$'nR)4EK)#3w e&:1@$."W e&:A%Aª3wC,/01k+>98"c)=13U+')=98+.&41@)#;BC!98AW)4NG&YF^3w+>At&:EY3 !Ac1@).$?d )#v e $.±S/0A%N:36905,EYAW$>5n&ebb p ;iQ#F^&:;i5]-8/2#+EY36Ac@E4&:C
E 0 = EP0 − EL0
EP0 = 0, 339
EP0 =
RT ln KP , zF
EL0 =
EL0 = −0, 763
RT ln KL . zF
KP KL
)
-0Ce#+
RT RT RT ln K = ln KP − ln KL , zF zF zF K = KP /KL
d.¶),+.-8/21Y36Ac1hF8#NV+'1@$'&ep'+.)e-0/21hF0&:;V5 )
0
KP = eEP zF/RT = e26,4 = 2, 9 × 1011 ,
YN )<+'1hF0& d >ed -8)= +>Ql$?1VCe)=1@+>/8(\),)M+>36EYAcAW&e1#+'pn(.1T90E4 &:E;<-8)##""W)#\; ;VpgQ'-836A¢AW(.YNB90E4_4&:&:(¢X,R
5=5$?Q.#EYIª5=RS-01#Ce+>1#Q.IB$'/2&:1@CCe]+>3UA%"W=N)e8H8O I ;
KL = eEL zF/RT = e−59,4 = 1, 6 × 10−26 ,
K = 1, 8 × 1037
Na
2Na+ + 2e− → 2Na
F · 2 mol = 1, 9297 × 105 C. )
|N2O."W)#/905,Qg; p X\N:EM)]1#$?-0AW1i&F0$?&Y1+>$?1@]$'),N:+>EMEY#AW-0&/8&Y36N:EY5=C!(. 98),+>Q.C=(.Q#90F^EY&5=-0(?AW1#+'V1hF8;<)#]" n&:"W1@&:E4C,)M/03U98)#&:$, !5) )f=+.1@+'C1@36$'&lAWYNq(.90$?E41B&:E 1@$'B),F^+>)#EY$,AW&q5 +.)e-8/21@p$.1 AW&:(.;V90E45 &:(¢>pR
5,~$? l!RDN2O.RS1#"W)#+>90Q.Qg$'d &:C d 1, 9297 × 105 C/10 A = 19297 s ≈ 5, 36 h.
Cl−
Cl2
F · 2 mol
> e>d E4&:&:CBXn5s(\)#98;B)#EYAW36Y+>Ac@EY5lEM1#N:I AW&:/8(¢)R&:EM;´1#+'981@&4$.1@AWC!&eNGp F^A\$?A\1@-0"W&:A
RS_Yi5«&:$?"W&:1hC=F^/0(.98AW)#&:;<903¬)#/8EY)#$?905=1@"WN:& EY4$?IT>dEY36 AcAWh&bb /0-090/2Q'1@$'+>)4$'36)Ac-8#AWNYU&G+.F)#EY;i365=AcH0@"%E4AW&:IeCvp#_4&w(\)#$?;B1@"WAW&:Y_Y+>5VEY5-8C!)#(.9092EY1#5=N4-0s/21#(.Iw92E#+>98Q436&:$'"W)e&:C=H2N:/0A905=±^N: EY>d$'&:!² !) p A(.AWEY-0;BE4&:Ac;V1@$?5BF8&:V$n/0/2Q.1@/2"%(.1hA%FAF8-01@3UC!)#) X*),+>$'&GF6Q.C'RS1#+>Q (.90EY5-0/21!R
5>N8O A d *1hb ±^ >d >M² N36Y)qAW& F0&Y-0/$.Q.AW&ªEM1#EY-;B1#AW+>&:$.$.AWAc)#1<$'-0&eAWp=eF^d'&Y³UH0"%AWA&:(¢3°R)B/092981@&4C!1@N:C!AWNG&FGA (*(*)e)=-0/8+:(.-8/2Q{1!R
985,&4;¬1@C!NGN:FGAWAH0$.3AW&:)#$. eAW&:$.; Azb F^¶&Y)#-0$./AW98&:43361@$'_ &qEY;BAc1@$.AW&&:$n/21@"%(.A%AQ.C'RS1#+>Qgp p.3365=$.A%C,QK98&41@C!NGFGAxd 36AWYND+>"c1 ;l1@;i5
]6"
T
p
−EzF ∆ξ = ∆G,
E
*
∆H
G = H − TS = H + T
∂G , ∂T
)
T = const
∂∆G , ∂T
.X?( 901#EYN85]O N:;<EY5=)#;¬"%AD9898#_&41@Y$. !&:AWN:$nEY/8C!M3)Mq3wpt1@N:$.EYAW& 5,"%F^A &Y-8/t(.90EY5-0/21!R
dw5,¶; ),+.-0A/214-036/21!AcR
1h5>F0N8#OTN "%AWN:E:b ED98436$?1@$.Ac1T±^ >d >M²p¢+.)e-0/21hF0&:;V5 ∆H = ∆G − T
p
ξ = const
∆G
)
" # ∂E ∆H = zF T − E ∆ξ. ∂T p
fN:EY1@5,C="%AA -21@;®EY36Ac@E4&:C;iQ'-0A X,5=IZ-0(\&MR
$.AW)#$,5/21@C,_4&l+>"c198#_Y$.AWN:E4&:C*p # " ∂E dH = zF T − E dξ. ∂T p
U(.³ 90AWEY&:5,(¢(?R)r1#+'981h&4F81@#C!N4NG1{F^A$?(\1l)=F^+¯&Y+.-0&:/2$s1!R
;<5,;)#"N:98&4AWH01@$.C!AWNG&:FG$.AxAWp&:N:;EY5="%/8A )rEY;BAc1@$?1 :& $nd/21@f"%(.1@A%C A 36$.A%AW3wYNB1-0;lEYQ.1C#(*1@)e$,-05T/21#EYI 36Ac@E4&:C(*)#;BAWY+>EY5{N:AW&:(¢R&:;®98&41@C!NGF^AA #) #b (∂H/∂ξ)T,p (
)
∂H ∂ξ
T,p
" # ∂E = zF T −E . ∂T p
e
=33 8#"%'& / F% 4!" % &# %
{)#_4&:;V5/8&:92B1@Elp¢NY(*)l)=+.+'-01h/2F^1Y& 36AWI<+'1@$'&Z+>"c1]/8&:;B(*&:921@/0Q.905
0 ◦C
p N:EY5,"%A
T = 273, 15 ) ∂H = −2 · 96485 (273, 15 · 4, 02 × 10−4 ∂ξ T,p
³ 1@C=-85,;l1@"%$?1 (.921#N41 $.AW&Y)#X>F0:/8)eH8N:AW)431>p¢F01@Ce;<)#_Y$?1TQ.EY5=-8C#1#IBE 36;B)# e5=AW$.&:$'$.A%)e3AW-0)#1]A $?(.1<92$?1#1N:d.Q#f(.F0#1<921#NYN:&:NYE4 !eY)Tp#H8F^IV&Y),-8+>N:/6AW36&:98(¢92431#RS61N4$?1@98"%1B$.&41@AW98&C!#_NGY3 FG$.AxAWp?N:C,5 /8#921B$.AW&ª;IDEM1hb p + 1, 015)
/mol = −217, 05 kJ/mol. T = const
p = const
∆G
∂∆G = zF T ∆H − ∆G = T ∆S = −T ∂T
∂E ∂T
∆ξ.
0F¶&:)D;V(.5!90p.E43P&GF0H8(.N:90A%E4Q<&:"%),AWN:+E4-0&:C!$.)#A%Q{o'N:$?E41])#$,*5=;
)
T
∂S ∂ξ
= zF T
∂E ∂T
= −21, 19 kJ/mol.
,X,5=/21@I$'36)M365,C!A6)#/890){EY5>)#-0C!/21@)n$'R) &+. )V365,N:C!AW&:)#(¢$?RS1@1{$.Ac981V&41@(.C!92NG1#F^N:Axd5¶$.)#(gE4d!)e(.-090/2E41!&:REª& -0~ A%"%$.A%CB;<&:"W)#&:_4Cn& b /F80#905>NY&ªN:EY3P$n5-21@EM;V1#-05=A%"c; 1@$,5-0A%"%E6$.A%/8C=&:Q¢ !)V²d.)#! e&Y$.-0A%/3w/81l)±3w(*1@)#90;B/8)eA F8H21@ID;V;l51@/0C=Ql-85,-0/0;l921@1@/"%$?51>(*p>)4C=3t/8#9-0/281h&GFb $.Ce1hAW&{F0#;
T,p
p
*
>dtF^|)#$,5q98&43U1@C!)=NG+.FG)#A90X.QBAW&:)ª90E41@&iC=/Q'5=+>36EY$'Ac1!)eR H8N: nA 1@E4)M365Tdh|3w5,),92+.1@#9_4D&:$.)AW&N:AW$?H01$.AW(*&:)#$./8A%Q&:$'NGF01!R?)#(*92nR1@cE b );<# e&:$./2A%1@3w"W)413U;<&GFi)#_Y3`$?1 98433665,$'(.)M983w)M1#3w+>1#E4+>&EYAWE4I]&l1@-0$?3w1@)#"WA%)#;B eAWAgN:F0EY)#$.$?AW&q1@;BF81@A C±E4+>)#"cXg1 dU&:±^"W&: C=>d/0n98),!+>²0²5 d /0T905=Q'N:-0EYA%;i$'&e5d K/5=Ac"%1@C!$')«)436Qn36AWN:EYAW&i e"W+>Y+>"c1$.AW98I&41@98C!#_NGYFG$'A &{3t-0(*nRN:EY5,$.$.A%C=AD-0/8&YN8O.AW)#;<&b
]6"
p H2
a H+
1 H2 (g) 2
3t-0(\nRN:EY5=$.$.A%C(.90EY5 n1@E4)M365,; 3U)=+.)#90E4&q365=$')e-0A p?36AWYN H+ (aq) + e−
1/2
EH2 = EH0 2 −
! 1/2
a H2 a H+
RT ln zF
)
,
.((\90)=EY+ 5K$.N:AWEY&:5=EY; Xn5=/]365=-2)#d C,{A%; )#_Y$?N:AW1H0$.(.AW90&:EY$.5hAWF0&:#; Iep\_4&B+>"c1 n1@E4)43w+>&:"c1 !) N:AW3wH8$.),AW+.&:)#$.90AcQ 1 365,921@_4)#$'²&:d !)i«3rC!)#1@"W/0&:;
a H2 = p H2
a H2 = 1
*
)
E H2
1/2
p H2 a H+
RT =− ln F
!
,
(.$'90)eEYH85KIF0N:)#EY$'5=43; 3w),+.)#98)M36p*5=EYN2 !OZ)=+>F0$.&Y-0AW/6&Q'E<-0+./21@& "W')#$.$?AW1>NGF8p? /8)B=¶EY;B|¯Ac1@d $?n1 &YH0"%A 1@C,EY/365,Ac3hbb EM361@5,$?$'1)e-0E4A &ZEY;BAc1@$?TN:AWH0$.AW&:$.Ac1T3U)=+.)#90Q«E $?1 EH0 2 = 0
(
86
p H2 = p 1
E H2 p H2 = p 2
)
RT ln(p1 /p2 ). 2F
,)+ C!)#"W&:Axp= !+>5
∆EH2 = EH2 (p2 ) − EH2 (p1 ) =
a H+ = a 1
a H+ = a 2
∆EH2 = EH2 (a2 ) − EH2 (a1 ) =
RT ln(a2 /a1 ). F
#
"c1 ¬;l1@;V5 +.)e-0/21hF0&:;V5]36AWYN p1T®3 ±X¢²
T = 298, 15 p1 /p2 = 9 = 32
∆EH2 =
=33 8#"%'& / F% 4!" % &# %
d * | L ± ! 1 6 ² l ; @ 1 i ; 5 d| )#X.Q«(.90EY5=(?1#+>Ce1#N8O
RT /F = 0, 0257 a2 /a1 = 3
RT ln 3 = 0, 0282 V. F
'd $. AW)#&:903EY1@5=_ª-0/2JS1@)#;i90;B5AW& E4&BEY9836&Y+>E4)#Q.90C!Qr)43±^1@ >d$'&GnF6!)=²D+>$?(*1)436(\Ac)#1#/8+'&:1$'NGN:F8EY1!5>R-0(*/5lnRN:)#5= e$.$.C
#
aRed = aZn = 1
RT 1 ln aZn2+ = (−0, 763 + 0, 0257 · ln 0, 1) V 2F 2 = −(0, 763 + 0, 0296) V = −0, 7926 V.
0 EZn = EZn +
>dt90ª5>)=N:+'EY$?1hF0 #N]98&41@C!NGF0&]3´(\nR)# e$.A%31#N8OgpU+.)e-0/21hF^&:;i5s98&41@C!NGF^-0Q.;l1hb
)
1 H2 (g) + AgCl(s) Ag(s) + H+ (aq) + Cl− (aq). 2
¶)#/8&:$'NGF01!R
5(*nR)# e$.A%3+'1@$'&q-$?1#-0/8:(.Q#F0#N:5=;BA365=921@_4&:$.Ac1@;BA ±E4)#Xgd.EM1#+ d !²)#921@E
E H2
RT =− ln F
1/2
p H2 a H+
)
!
0 EAg−AgCl = EAg−AgCl −
RT ln F
aAg aCl− aAgCl
.
36¶ )#AWY$.N AW&:3w1@a_B1@)#C, e/$.5,A%363$'1)e36H2N:5=A$'N:)eAc1!-0ARS1<-0/21!R&: !)<(\)#;BA F81@;V5F81@C!)98436$'&lep (
)
RT 0 E = EAg−AgCl − ln F
aH+ aCl− 1/2
p H2
!
.
90nEY1@5=CqI F^[email protected]=_D/5=(.36AW-2$'1@"%)eAWH8H0N:;iA>5Z1@$.3 AW)#98$')#43E4s+>EYA!AcCe1@1@"W&d$'43'qp>ph$."WAW&Y&ªN:Eg;
]6"
a H+ =
γ+ m+ aCl− = γ− m−
! H8N:+>AgEY)#AW& X.QBF0)#$'43-2/21@C=AW&q-21@p ;<1 &Z±S+>"c1(.98)e-0/8)#/5]/0921@C=p/0X\Q#)F^&:;i;<5])#"c/01@Q "%$')@b F0921@1@C!_4))#$'36&GFUAW&:3"%C!)eC,H8 IhX*;<&:)#EY"
36²d>5,;B"cAc11@98N:)MAW3wH0$.qAW&:98$.436Ac1 $?ª"%AWN:EYX*)43w1@/0)D;´;<;<)#"c)#1@_4"%$'&:;i)eH85ZN:A'3636AWY5,N b $?1@(.AW-21#I ±^ >d e!² g&N:EM:1@3w$'1&q-0&:/0C=98-8)#(\$?&:1{905=(*;<)4&:36$n5=/2_41@-0"%$.E4&GAWF& 98436«$')ep H8N:A6p'EMAg14;<36)#AW&:"c921@1"%$')e36H2AWIe&:p"%C!)eH8d'N:fA61@36C5=36EY$?AWY1hN b F^Ek(\)#)#$'±^/8M3&:>dq$'NGep F8!1!²BR
p.Q«+>;<"c-81)#/2_Y1@&:$?$'"W&:1+'C,1@/0)#9898X.+.)#"%)4"%AWA%N:3U/0EYQ &:5= !IK(*).)=d H0+Z98 &Y3+>C!$.1@)#A90"WQ.&:3t$.Aw-8C,(\(\AW)#&:nR/8; &:N:$'p'EYNG5=_4F8&ª$.1!$.R EYA%$?C7-01@/2;V1@1@$'C,5]+'/5,1@3w36981@+.$'90)4/8)e36)eH2H85!N:I Ap . p < ; # ) Y _ ? $ B 1 # ) = C 8 9 Y & 0 H % " W A D I ? $ 1 * ( , ) . + 8 2 / Y 1 6 3 W A D & * ( # ) B ; c A @ 1 8 9 43 +>"c1 pp 9836)#AWEYY/ND3U"W)#&:983wM31l-0/09898)#)#E4$?N:AW1&:o'9843N:E46)#$'$,)e5=H2N2N:OgAd +'¶@_Y)#5$.AW+.&:3w)<1@-0_ /21!R&GFYp.98M36p$' !&G+>F 5 !+>5&:C>-0/0921@(*)#"%Q#F^&:;i5iF8+.)3w1@90/8)eH2N:A d aH+ aCl− = aHCl = γ 2 m2 ,
2 = γ γ γ 2 = γ± + −
m = m + = m−
m
p H2 = 1
)
E+
2RT 2RT 0 ln m = EAg−AgCl − ln γ. F F )
γ
0 EAg−AgCl
(
E
m
*
(
γ → 1
m=0
m → 0 0 EAg−AgCl
k i¶±/dZR
Q.|; d.d Ei"1@/0$.C, 'A%$'d ²-Md p
&/ ep¶| ip|1@98-0EM143w1¬ > i¶d |d "/0C=A%$'-4pw³d "d fg921@(.(gp «d ¶d³1#+>5!p6³dªA%Q.$n/21>p
> l ± / R
Q.&(; d. E1@$. ' d ² d ' 0
+
pY¶| p4|1@98-0EM143w1 d kd 1@9098)43q p
&( !pi¶| pV|1@98-0EM143w1´4~ ±/R
Q.; d.Ei1@$. 'd ²d d d ³1@"%"W&:$g p
& p @$'+&Y+ d%p !)#O.$|vA%"W&:5!p &:3 )#90C4~ >d Bd<Q.;BA%o'-0C=Axp / ep365>+ dB>pB¶| ip|1@98-0EM143w1 4~ #>d d )nR
5>-0/4p "idU¶)#$.AW&:36AW&:98-0C=Axp "dw³UAc1#N8Og! p
)
& " #%$ " '& + ( pn|5>+'1Y36$.AWN:/3U) $.A%3U&:98-85,/8&:/0Q q1@98+>5=$?1!RS1 ,/8&Jx1@$?1|5=-0EY5=o'-0C=AW&: !).p.|1@98-0EM143w1l >d )*
+ &( epe(.921#N41qEYX.AW)#98)431V(*)=+B98&Y+ d d q1@;BAW&:o'-8C,AW&: !).p 365>+ d 'p'¶| ip>|1@98-0EM143w1T4~ >d «d q1@;BA%o'-0Ce1>, p
) $ $ - (# ) /. .10 ep98)#E4+>EYAc1!Rª)/8&:;B(*&:921@/0Q.90E4&T&:;B(.A%905>N:EY$'&GF:p On/0/0( hX*)#X*).d JQ,3qd &Y+>Qgd (." 9SFGX @f&:90;<),+>5=$?1@;BA%C#1
(
)
∼
D; / $
~ " d d q 1@90 /0 Q' -0 EY5, o' -8 C# 1>
p¢ ³U Og d "i& d & :"W&/N: EY Q.C\:p p"i¶d?| id p,/0|98)#1@;V98-0X*EM&:1Y903 '1 p 4~ ±/R
Q.; d?ED98)e-4d ²d id )#$'+.&:(.Q'+>Axp d.¶90A% !)# eA%$'&ep
!& p'n)#O.$ |vA%"W&:5!p &:3 )#90C4~e~e~>d e d ªQ."W&Y-2-21>p . /
Sp@On/0/0( hQ'-8&:98-Md Q#F:d &Y+>Qgd (." C=Q."W&Y-8-21 4 dM¶1@Q."%A%$. 'p .
)
/ ep:ª) &:9¶Q.X."%AWN41@/0AW)#$'-4p $'Ned%p &:3 )#90C4~ >d ifDdU¶&:$.Ce1@"c1>p - 0 -& . ) pU365=+ dU>p ¶| ip |1@9^b -0EM143w1T4~ =ed ³d d>¶90Q./0/8)#$gp \d d 1@98)#$gp $ !
) # & # ) - " &% )
#p 1#N KA%"%"c1@${³w)#;B(?1@$n5!p &:3 )#90C4~ >d td d ,;BA%/0Ogp -
0 /
& &( pD¶ | p |1@98-0EM143w1K4~e~ ±/R
Q.; d.EV1@$. 'd ²d V|dif)#;l1#-8-0Axp -
0
& &( :p¶ | p |1@98-0EM143w1K4~ >d ) Bd *1@"W&:3t-0C=Axp / ) . &( &( p.¶ | p.|1@98-0EM143w1>p4~ >d
+(
)
∼
(
#
(
+
+
*
+
