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Chapter 6
Reaction Equilibrium in
Ideal Gas Mixtures
Physical Chemistry
Chapter 6
Chemical Potentials in an Ideal Gas Mixture
Chemical Potential of a Pure Ideal Gas
VdPSdTdG (4.36)
dPP
RTdPVd m
isothermal, pure ideal gas
2
1
12
1
P
PdP
PRTd
1
212 ln),(),(
P
PRTPTPT isothermal, pure ideal gas (6.1)
oo
P
PRTT ln)( pure ideal gas, Po1 bar (6.2)
dPVdTSddG mmm
(4.86)*im
PT
Gn
G,
,ii
one-phase pure substance
Physical Chemistry
Chapter 6
Chemical Potentials in an Ideal Gas Mixture
Fig. 6.1 T is fixed
P/Po
- o
01 2 3
RT
-RT
-2RT
mmm TSHG
)(TfHm
)(PfSm
0 Pas
For a pure ideal gas
Physical Chemistry
Chapter 6
Chemical Potentials in an Ideal Gas Mixture
Fig. 6.2
),(),(),,,,( **21 iiiii PTPxTxxPT
An ideal gas mixture
Ideal gas mixture at T and P
PxP ii
*iP
Pure gas mixture at
and T
membrane
(1) RTnPV tot
(2) The mixture is separated by thermally conducting rigid membrane permeable to gas i only, and PxP ii
barPo 1
ideal gas mixture (6.3)
Physical Chemistry
Chapter 6
Chemical Potentials in an Ideal Gas Mixture
Fig. 6.2
),(),(),,,,( **21 iiiii PTPxTxxPT
Ideal gas mixture at T and P
PxP ii
*iP
Pure gas mixture at
and T
membrane
ideal gas mixture(6.3)
oio
ii P
PRTT ln)(
ideal gas mixture (6.4)*barPo 1
The fundamental thermodynamic equation
No intermolecular interaction, no effect on from other gases in the mixture.
i
Physical Chemistry
Chapter 6
Chemical Potentials in an Ideal Gas Mixture
Fig. 6.1 (modified)
Pi/Po
i - i o
01 2 3
RT
-RT
-2RT
)()( , TGT oim
oi
)(Tfoi
barPo 1
For an ideal gas mixture
Physical Chemistry
Chapter 6
Ideal Gas Reaction Equilibrium
For the ideal-gas reaction
dDcCbBaA (4.36)
the equilibrium condition i ii 0
DCBA dcba 0 BADC badc
oio
ii P
PRTT ln)( (6.4)*
0lnln
lnln
oBo
BoAo
A
oDo
DoCo
C
P
PbRTb
P
PaRTa
P
PdRTd
P
PcRTc
Physical Chemistry
Chapter 6
Ideal Gas Reaction Equilibrium
(6.5))lnlnlnln(oB
oA
oD
oC
oB
oA
oD
oC
P
Pb
P
Pa
P
Pd
P
PcRT
badc
0lnln
lnln
oBo
BoAo
A
oDo
DoCo
C
P
PbRTb
P
PaRTa
P
PdRTd
P
PcRTc
oB
oA
oD
oCi
i
oii
oiTmi
oT badcTGG )(,,
b
oB
a
oA
d
oD
c
oCo
P
P
P
P
P
P
P
PRTG lnlnlnln
Physical Chemistry
Chapter 6
the equilibrium condition
Ideal Gas Reaction Equilibrium
(6.6)
(6.7)
)lnlnln(lnb
oB
a
oA
d
oD
c
oCo
P
P
P
P
P
P
P
PRTG
b
oB
a
oA
d
oD
c
oC
o
PP
PP
PP
PP
RTG ln
b
oB
a
oA
d
oD
c
oC
oP
PP
PP
PP
PP
K
barPo 1
Physical Chemistry
Chapter 6
oP
o KRTG ln
Ideal Gas Reaction Equilibrium
oP
o KRTG ln
(6.7)b
oB
a
oA
d
oD
c
oC
oP
PP
PP
PP
PP
K
barPo 1
0]ln[ ,
o
eqioi
ii
iii P
PRT
0ln)( ,
io
eqioi
ii P
PRTT (6.8)
i i iii ii i iii accaandbaba )(
Physical Chemistry
Chapter 6
0i
iithe equilibrium condition
Ideal Gas Reaction Equilibrium
(6.9))(,, TGG oi
ii
oiTm
ii
oT
(6.10)
n
iinn
n
ii aaaaaaaa
12121
1
ln)ln(lnlnlnln
oiTm
oi GT ,,)(
io
eqi
o
eqi
ii
oT
i
P
PRT
P
PRTG
,, lnln
Physical Chemistry
Chapter 6
n
ini aaaa
121 (6.11)*
Ideal Gas Reaction Equilibrium
(6.12)
in
io
eqioT P
PRTG
1
,ln
n
ini aaaa
121 (6.11)*
i
io
eqioP P
PK
,
oP
oT KRTG ln
(6.13)*
(6.14)*
RT
GoP
o
eK
(6.15)
Physical Chemistry
Chapter 6
n
ii
n
ii aa
11
lnln
i
i
o
eqioT P
PRTG
,ln (6.10)
Ideal Gas Reaction Equilibrium
RT
GoP
o
eK
(6.15)
(6.9))(,, TGG oi
ii
oiTm
ii
oT
Since depends only on T, for a given ideal-gas reaction is a function of T only.
oTG o
PK
)()( TKKTGG oP
oP
oo
At a given temperature, is constant for a given reaction.oPK
the standard equilibrium constant
the standard pressure equilibrium constant
oPK
Physical Chemistry
Chapter 6
Ideal Gas Reaction Equilibrium
i
eqiPiPK
, (6.19)
dDcCbBaA
Since is dimensionless, the standard equilibrium constant is dimensionless.
oi PP
oPK
has dimensions of pressure raised to the change in mole numbers for the reaction as written.
PK
the equilibrium constant
the pressure equilibrium constantPK
i
io
eqioP P
PK
,
(6.13)*
Physical Chemistry
Chapter 6
Ideal Gas Reaction Equilibrium
badf
o
o
boeqB
aoeqA
doeqD
foeqF
boeqB
aoeqA
doeqD
foeqFo
P
P
RTc
cccc
cccc
PRTcPRTc
PRTcPRTcK
)/()/(
)/()/(
)/()/(
)/()/(
,,
,,
,,
,,
(6.23)
dDfFbBaA i
io
eqioP P
PK
,
(6.13)*
the molar concentrationV
nc i
i (6.21)*
ideal gas mixtureRTcV
RTnP i
ii (6.22)
Physical Chemistry
Chapter 6
Ideal Gas Reaction Equilibrium
badf
o
o
boeqB
aoeqA
doeqD
foeqF
boeqB
aoeqA
doeqD
foeqFo
P
P
RTc
cccc
cccc
PRTcPRTc
PRTcPRTcK
)/()/(
)/()/(
)/()/(
)/()/(
,,
,,
,,
,,
(6.23)
dDfFbBaA
n/mol
same dimension as Po3/1/1 dmmollitermolco
i
io
eqioP c
cK
,
(6.24)
i
io
eqioP P
PK
,
(6.13)*
Physical Chemistry
Chapter 6
Ideal Gas Reaction Equilibrium
i
io
eqioP c
cK
,
(6.24)
i
io
eqioP P
PK
,
(6.13)*
the standard equilibrium constant
the concentration equilibrium constant
ocK
the standard equilibrium constant
the standard pressure equilibrium constant
oPK
3/1/1 dmmollitermolco
barPo 1
Physical Chemistry
Chapter 6
Ideal Gas Reaction Equilibrium
badf
o
o
boeqB
aoeqA
doeqD
foeqF
boeqB
aoeqA
doeqD
foeqFo
P
P
RTc
cccc
cccc
PRTcPRTc
PRTcPRTcK
)/()/(
)/()/(
)/()/(
)/()/(
,,
,,
,,
,,
(6.23)
moln
o
ooc
oP P
RTcKK
/
(6.25)
i
ieqix xK
, (6.26)
moln
oxoP P
PKK
/
(6.27)
Physical Chemistry
Chapter 6
van’t Hoff Equation
RT
GK
ooP
lnFrom (6.14)
dT
Gd
RTRT
G
dT
Kd oooP )(1ln
2
(6.31)
(6.9))(,, TGG oi
ii
oiTm
ii
oT
(6.32) i
oim
ii
oimi
o
dT
dGG
dT
dG
dT
d ,,)(
oim
ii
o GG ,
dT
Gd
RTdT
Td
R
G
dT
Kd oooP )(1/1ln
2
11
TTdT
d
Physical Chemistry
Chapter 6
van’t Hoff Equation
dPVdTSdG mmm From
mP
m ST
G
(6.31)
(6.33)oim
oim S
dT
dG,
,
(6.34)o
i
oimi
o SSGdT
d ,)(
(6.32) i
oim
ii
oimi
o
dT
dGG
dT
dG
dT
d ,,)(
Physical Chemistry
Chapter 6
van’t Hoff Equation
22
ln
RT
STG
RT
S
RT
G
dT
Kd oooooP
ooo STHG
(6.36)*2
ln
RT
H
dT
Kd ooP
(6.34)o
i
oimi
o SSGdT
d ,)(
(6.32) i
oim
ii
oimi
o
dT
dGG
dT
dG
dT
d ,,)(
Physical Chemistry
Chapter 6
Gibbs-Helmholtz Equation
TSHG
T
H
T
G
T
HG
T
G
P
T
GHS
T
H
T
G
T
G
P
PP T
G
TT
T
G
T
G
PPPTT
GT
G
TT
G
T
11
T
G
T
G
TT
G
T
G
TT
G
T PPP
112
Physical Chemistry
Chapter 6
Gibbs-Helmholtz Equation
T
G
T
G
TT
G
T
G
TT
G
T PPP
112
T
H
T
G
TT
P
2T
H
T
G
TP
2T
H
T
G
TP
When it is applied to changes
Physical Chemistry
Chapter 6
Differentiation of lnKPo with respect to temperature gives
van’t Hoff Equation
RT
GK
ooP
lnFrom (6.14)
(6.36)*
T
G
dT
d
RdT
Kd ooP 1ln
The differentials are complete because K and G depend only on temperature, not on pressure. Using Gibbs-Helmholtz equation
2T
H
T
G
TP
2
ln
RT
H
dT
Kd ooP
Physical Chemistry
Chapter 6
van’t Hoff Equation
2
1
2
12
)(ln
T
T
oT
T
oP dT
RT
THK
(6.36)*2
ln
RT
H
dT
Kd ooP
dTRT
HKd
ooP 2
ln
2
12
1
2 )(
)(
)(ln
T
T
o
oP
oP dT
RT
TH
TK
TK(6.37)
42 ETCTBTAH oT (6.38)
211
2 11
)(
)(ln
TTR
H
TK
TK o
oP
oP (6.39)
2112
11)(ln)(ln
TTR
HTKTK
ooP
oP
Physical Chemistry
Chapter 6
van’t Hoff Equation
(6.36)*2
ln
RT
H
dT
Kd ooP
R
H
Td
Kd ooP
1ln (6.40)
2
11
TTdT
d
dTTT
d2
11
The van’t Hoff equation is an expression for the slope of a graph of the equilibrium constant (specially, lnK) plotted against the temperature. It may be expressed in either of two ways:
Physical Chemistry
Chapter 6
Ideal Gas Reaction Equilibrium
)(2)(3)( 322 gNHgHgN
The equilibriumextent of reaction
the reaction quotient3
2
22
3
HN
NHP PP
PQ (6.41)
PP KQ The reaction goes to right
PP KQ The reaction reaches equilibrium
PP KQ The reaction goes to left
0eq
0eq0eq
i
eqiPiPK
, (6.19) the equilibrium constant
The extent of reaction
Physical Chemistry
Chapter 6
Simultaneous EquilibriaA system with several simultaneous reactions that have species in common.
(6.47)
molnmoln
molnmolnmoln
COH
COOHCH
2,1
,1,1,1
,0,0
,0,0,0
2
224
23
1,
24
2)(
toto
OHCH
HCOoP nP
P
nn
nnK (6.48)
224 3HCOOHCH
2224 42 HCOOHCH
2
2
4
2, )(
)(
24
22
toto
OHCH
HCOoP nP
P
nn
nnK
Physical Chemistry
Chapter 6
Simultaneous EquilibriaAt 600 K, CH3Cl(g) and H2O(g) react to form CH3OH, and then form (CH3)2O with a simultaneous equilibrium shown:
(1)
,6.10,00154.0 2,1, oP
oP KK
Starting with equal amount of CH3Cl and H2O, find the equilibrium amounts of all species.
)()()()( 323 gHClgOHCHgOHgClCH
)()()()(2 2233 gOHgOCHgOHCH (2)
Physical Chemistry
Chapter 6
Simultaneous EquilibriaSuppose that a system initially contains 1 mole of CH3Cl(g) and H2O(g), x moles of HCl and y moles of (CH3)2O are formed at equilibrium.
(1))()()()( 323 gHClgOHCHgOHgClCH
)()()()(2 2233 gOHgOCHgOHCH (2)
1-x 1-x+y x-2y x
x-2y y 1-x+y
6.10)2(
)1(
00154.0)1)(1(
)2(
22,
1,
yx
yxyK
yxx
xyxK
oP
oP
Physical Chemistry
Chapter 6
Simultaneous EquilibriaSuppose that a system initially contains 1 mole of CH3Cl(g) and H2O(g), x moles of HCl and y moles of (CH3)2O are formed at equilibrium.
6.10)2(
)1(
00154.0)1)(1(
)2(
22,
1,
yx
yxyK
yxx
xyxK
oP
oP
009.0
048.0
y
x
Physical Chemistry
Chapter 6
Simultaneous Equilibria
The equilibrium amounts of all species are
(1))()()()( 323 gHClgOHCHgOHgClCH
)()()()(2 2233 gOHgOCHgOHCH (2)
0.952 0.961 0.03 0.048
0.03 0.009 0.961
Physical Chemistry
Chapter 6