Download Chapter 6

Chapter 6
Physical Chemistry
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
dG  SdT  VdP
dGm  d  Sm dT  Vm dP
(4.36)
 G 
 i     Gm ,i one-phase pure (4.86)*
substance
 ni T , P
 RT 
d  Vm dP  
dP
 P 
isothermal, pure ideal gas
2
P2 1
1 d  RT P1 P dP
P
 (T , P2 )   (T , P1 )  RT ln 2 isothermal, pure ideal gas (6.1)
P1
P
   o (T )  RT ln o
pure ideal gas, Po1 bar (6.2)
P
Chapter 6
Physical Chemistry
Chemical Potentials
in an Ideal Gas Mixture
-o
For a pure ideal gas
RT
0
-RT
  Gm  H m  TSm
1
2
P/Po
3
H m  f (T )
S m  f (P)
  
-2RT
Fig. 6.1 T is fixed
as
P0
Chapter 6
Physical Chemistry
Chemical Potentials
in an Ideal Gas Mixture
An ideal gas mixture
(1) PV  ntot RT
Ideal gas
mixture at
T and P
Pi  xi P
(2) The mixture is separated by
thermally conducting rigid membrane
permeable to gas i only, and Pi  xi P
membrane
Pure gas
mixture at
Pi *and T
Fig. 6.2
Po  1
bar
i (T , P, x1 , x2 ,)  i* (T , xi P)  i* (T , Pi )
ideal gas mixture
(6.3)
Chapter 6
Physical Chemistry
Chemical Potentials
in an Ideal Gas Mixture
Ideal gas
mixture at
T and P
Pi  xi P
i (T , P, x1 , x2 ,)  i* (T , xi P)  i* (T , Pi )
ideal gas mixture
(6.3)
No intermolecular interaction, no effect on  i
from other gases in the mixture.
membrane
Pure gas
mixture at
Pi *and T
Fig. 6.2
Pi
i   (T )  RT ln o
P
ideal gas mixture P o  1 bar
o
i
(6.4)*
The fundamental thermodynamic equation
Chapter 6
Physical Chemistry
Chemical Potentials
in an Ideal Gas Mixture
i - i o
For an ideal gas mixture
RT
io (T )  Gmo ,i (T )
0
io  f (T )
-RT
1
2
Pi/Po
-2RT
Fig. 6.1 (modified)
3
P o  1 bar
Physical Chemistry
Chapter 6
Ideal Gas Reaction Equilibrium
For the ideal-gas reaction
aA  bB  cC  dD
(4.36)
i i i  0 the equilibrium condition
a A  b B  cC  d D
cC  d D  a A  b B  0
Pi
o
i  i (T )  RT ln o
P
PC
PD
o

d


dRT
ln
D
Po
Po
P
P
 a Ao  aRT ln Ao  b Bo  bRT ln Bo  0
P
P
cCo  cRT ln
(6.4)*
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
PC
PD
o

d


dRT
ln
D
Po
Po
P
P
 a Ao  aRT ln Ao  b Bo  bRT ln Bo  0
P
P
cCo  cRT ln
cCo  d Do  a Ao  b Bo 
PC
PD
PA
PB
(6.5)

d
ln

a
ln

b
ln
)
o
o
o
o
P
P
P
P
GTo  i i Gmo ,T ,i   i io (T ) cCo  d Do  a Ao  b Bo
 RT (c ln
i
the equilibrium condition
c
d
a
b

P
P
P
P
 C
 D
 A
 B 
o
G   RT ln  o   ln  o   ln  o   ln  o  
P 
P 
 P  
  P 
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
c
d
a
b
P 
P 
P 
P 
G o   RT (ln  Co   ln  Do   ln  Ao   ln  Bo  )
P 
P 
P 
P 
  PC c  PD  d 
    
o
o
P
P






o
G   RT ln 
(6.6)
a
b 
  PA   PB  
o
 Po

P






c
d
P
P
 C D
 o  o
P
P
K Po    a   b
(6.7)
P o  1 bar
 PA   PB 
 o  o
P  P 
G o   RT ln K Po
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
c
d
 PC   PD 
 o  o
P
P
K Po    a   b
 PA   PB 
 o  o
P  P 
G o   RT ln K Po
P o  1 bar
(6.7)
the equilibrium condition
 i i  0
 Pi ,eq 
i  i i  i  i [  RT ln  Po ]  0


 Pi ,eq 
o
i  i i (T )  RT i ln  Po   0


i
o
i
 (a  b )   a   b
i
i
i
i
i
i i
and
(6.8)
 ca
i
i
 c i ai
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
io (T )  Gmo ,T ,i
GTo   i Gmo ,T ,i   i io (T )
i
(6.9)
i
i
 Pi ,eq 
 Pi ,eq 
G   RT  i ln  o    RT  ln  o 
i
i
 P 
 P 
o
T
(6.10)
n
n
i 1
i 1
 ln ai  ln a1  ln a2    ln an  ln( a1a2 an )  ln  ai
n
a
i
i 1
 a1a2  an
(6.11)*
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
n
a
i
 a1a2  an
i 1
(6.11)*
i
 Pi ,eq 
o
GT   RT  ln  o 
 ln ai  ln  ai
i
i 1
i 1
 P 
i
 n  Pi ,eq  
o
GT   RT ln   o  
 i 1  P  
i
P
 i ,eq 
o
K P    o 
i  P 
n
n
GTo   RT ln K Po
K Po  e
G o

RT
(6.10)
(6.12)
(6.13)*
(6.14)*
(6.15)
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
GTo   i Gmo ,T ,i   i io (T )
i
i
K Po  e
G o

RT
(6.9)
(6.15)
Since GTo depends only on T, K Po for a given ideal-gas
reaction is a function of T only.
G o  G o (T )
K Po  K Po (T )
At a given temperature, K Po is constant for a given reaction.
K
o
P
the standard equilibrium constant
the standard pressure equilibrium constant
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
aA  bB  cC  dD
i
 Pi ,eq 
o
K P    o 
i  P 
(6.13)*
Since Pi P o is dimensionless, the standard equilibrium
constant K Po is dimensionless.
K P   Pi ,eq  i

i
(6.19)
K P has dimensions of pressure raised to the change in mole
numbers for the reaction as written.
the equilibrium constant
KP
the pressure equilibrium constant
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
aA  bB  fF  dD
i
P
 i ,eq 
o
K P    o 
i  P 
(6.13)*
ni
ci 
the molar concentration
V
ni RT
Pi 
 ci RT
ideal gas mixture
V
K Po 
(6.21)*
(6.22)
(cF ,eq RT / P o ) f (cD ,eq RT / P o ) d
(c A,eq RT / P o ) a (cB ,eq RT / P o )b
(cF ,eq / c ) (cD ,eq / c )  c RT
 o

o a
o b 
(c A,eq / c ) (cB ,eq / c )  P
o
f
o d
o



f  d  a b
(6.23)
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
aA  bB  fF  dD
K Po 
(cF ,eq RT / P o ) f (cD ,eq RT / P o ) d
(c A,eq RT / P o ) a (cB ,eq RT / P o )b
(cF ,eq / c ) (cD ,eq / c )  c RT
 o

o a
o b 
(c A,eq / c ) (cB ,eq / c )  P
o
f
o d
o
c o  1 mol / liter  1 mol / dm3
f  d  a b
(6.23)


n/mol

same dimension as Po

i
P


i ,eq
K Po    o 
i  P 
i
 ci ,eq 
o
K P    o 
i  c

(6.13)*
(6.24)
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
i
 Pi ,eq 
K    o 
i  P 
o
P
K Po
P o  1 bar
the standard equilibrium constant
the standard pressure equilibrium constant
i
c
 i ,eq 
o
K P    o 
(6.24)
i  c

c o  1 mol / liter  1 mol / dm3
K
o
c
(6.13)*
the standard equilibrium constant
the concentration equilibrium constant
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
K Po 
(cF ,eq RT / P o ) f (cD ,eq RT / P o ) d
(c A,eq RT / P o ) a (cB ,eq RT / P o )b
(cF ,eq / c ) (cD ,eq / c )  c RT
 o

o a
o b 
(c A,eq / c ) (cB ,eq / c )  P
o
f
o d
 RTc
K  K  o
 P
o
P
o
o
c



o



f  d  a b
(6.23)
n / mol
(6.25)
i
K x   xi ,eq 
(6.26)
i
 P 
K  Kx  o 
P 
o
P
n / mol
(6.27)
Chapter 6
Physical Chemistry
van’t Hoff Equation
o
d 1
1

G
o
From (6.14)
  2
ln K P  
dT  T 
T
RT
d ln K Po
G o  d 1 / T   1 d (G o )



dT
R  dT  RT dT
d ln K Po G o
1 d (G o )


2
dT
RT
RT dT
(6.31)
GTo   i Gmo ,T ,i   i io (T )
i
(6.9)
i
G o   i Gmo ,i
i
d
d
o
(G ) 
dT
dT
o

G
 i m,i   i
i
i
dGmo ,i
dT
(6.32)
Chapter 6
Physical Chemistry
van’t Hoff Equation
dGm  Sm dT  Vm dP
From
 Gm 

  Sm
 T  P
dGmo ,i
dT
(6.31)
  S mo ,i
d
d
o
(G ) 
dT
dT
(6.33)
o

G
 i m,i   i
i
i
dGmo ,i
dT
d
(G o )   i S mo ,i  S o
dT
i
(6.32)
(6.34)
Chapter 6
Physical Chemistry
van’t Hoff Equation
d ln K Po G o S o G o  TS o



2
dT
RT
RT
RT 2
G o  H o  TS o
d ln K Po H o

dT
RT 2
d
d
o
(G ) 
dT
dT
(6.36)*
o

G
 i m,i   i
i
i
dGmo ,i
dT
d
(G o )   i S mo ,i  S o
dT
i
(6.32)
(6.34)
Chapter 6
Physical Chemistry
Gibbs-Helmholtz Equation
G  H  TS
GH G H
 G 
 

 
T
T T
 T  P
 
 G  G

   T 
 T  P T
 T
 

 T
H G
T
H
 G  G

  
T
 T  P T
S
 G 
  
 T  P
 
1  G 
 G 



G

 


 T   P T  T  P
 T
 1 
  
 T  P
   G 
1  G 
G 1  G  G 

    
  2  
  
T  T  P T 
 T  T   P T  T  P T
Chapter 6
Physical Chemistry
Gibbs-Helmholtz Equation
   G 
1  G 
G 1  G  G 

    
  2  
  
T  T  P T 
 T  T   P T  T  P T
 
T 
 T
H
 G 
    
T
 T  P
   G 
H

     2
T
 T  T   P
When it is applied to changes
   G  
H


    2
T
 T  T   P
Physical Chemistry
Chapter 6
van’t Hoff Equation
o

G
From (6.14)
ln K Po  
RT
Differentiation of lnKPo with respect to temperature gives
d ln K Po
1 d  G o 



dT
R dT  T 
The differentials are complete because K and G depend only
on temperature, not on pressure. Using Gibbs-Helmholtz
equation
   G  
H






2

T
T
T

 P

d ln K Po H o

(6.36)*
2
dT
RT
Chapter 6
Physical Chemistry
van’t Hoff Equation
d ln K Po H o
(6.36)*

2
dT
RT
o
o
T2
T2 H (T )

H
o
d ln K Po 
dT
ln
K
dT
P  
2
2

T1
T1
RT
RT
o
T2 H (T )
K Po (T2 )
ln o

dT
(6.37)
2
T
1
K P (T1 )
RT
HTo  A  BT  CT 2  ET 4
(6.38)
K Po (T2 ) H o  1 1 
  
ln o

K P (T1 )
R  T1 T2 
(6.39)
H o  1 1 
  
ln K (T2 )  ln K (T1 ) 
R  T1 T2 
o
P
o
P
Chapter 6
Physical Chemistry
van’t Hoff Equation
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:
d ln K Po H o

dT
RT 2
d ln K Po
H o

R
1
d 
T  d 1
1
 


 
dT  T 
T2
(6.36)*
(6.40)
1
1
d     2 dT
T
T 
Chapter 6
Physical Chemistry
Ideal Gas Reaction Equilibrium
N 2 ( g )  3H 2 ( g )  2 NH 3 ( g )
2
PNH
3
QP 
the reaction quotient
3
PN 2 PH 2
K P   Pi ,eq  i
(6.41)

the equilibrium constant (6.19)
i
The
equilibrium The extent of
extent of
reaction
reaction
eq  0 The reaction goes to right
QP  K P
eq  0 The reaction reaches equilibrium
QP  K P
QP  K P
eq  0
The reaction goes to left
Chapter 6
Physical Chemistry
Simultaneous Equilibria
A system with several simultaneous reactions that have
species in common.
CH4  H 2O  CO  3H 2
(6.47)
CH4  2H 2O  CO2  4H 2
n0,CH 4  1 mol, n0, H 2O  1 mol, n0,CO2  1 mol,
n0, H 2  1 mol, n0,CO  2 mol
K
o
P ,1
2
2
nCO (nH 2 )  P 
nCO2 (nH 2 )  P 
o
 o  K P , 2 
 o  (6.48)

2 
nCH 4 nH 2O  P ntot 
nCH 4 (nH 2O )  P ntot 
3
4
Chapter 6
Physical Chemistry
Simultaneous Equilibria
At 600 K, CH3Cl(g) and H2O(g) react to form CH3OH, and
then form (CH3)2O with a simultaneous equilibrium shown:
CH3Cl ( g )  H 2O( g )  CH3OH ( g )  HCl ( g )
(1)
2CH3OH ( g )  (CH3 ) 2 O( g )  H 2O( g )
(2)
K Po ,1  0.00154 , K Po , 2  10.6,
Starting with equal amount of CH3Cl and H2O, find the
equilibrium amounts of all species.
Physical Chemistry
Chapter 6
Simultaneous Equilibria
Suppose 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.
CH3Cl ( g )  H 2O( g )  CH3OH ( g )  HCl ( g )
1-x
1-x+y
x-2y
x
(1)
2CH3OH ( g )  (CH3 ) 2 O( g )  H 2O( g )
x-2y
y
1-x+y
(2)
( x  2 y) x
K 
 0.00154
(1  x)(1  x  y )
y (1  x  y )
o
K P,2 
 10.6
2
( x  2 y)
o
P ,1
Chapter 6
Physical Chemistry
Simultaneous Equilibria
Suppose 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.
( x  2 y) x
K 
 0.00154
(1  x)(1  x  y )
y (1  x  y )
K Po , 2 
 10.6
2
( x  2 y)
o
P ,1
x  0.048
y  0.009
Physical Chemistry
Chapter 6
Simultaneous Equilibria
The equilibrium amounts of all species are
CH3Cl ( g )  H 2O( g )  CH3OH ( g )  HCl ( g )
0.952
0.961
0.03
0.048
(1)
2CH3OH ( g )  (CH3 ) 2 O( g )  H 2O( g )
0.03
0.009
0.961
(2)