Page 196 - Physical Chemistry

lev38627_ch06.qxd 3/3/08 10:07 AM Page 177

177
6.2 IDEAL-GAS REACTION EQUILIBRIUM Section 6.2
Ideal-Gas Reaction Equilibrium
The equilibrium condition for the reaction 0 ∆ n A (where n is the stoichiometric
i
i
i
i
number of species A ) is n m 0 [Eq. (4.98)]. We now specialize to the case where
i
i
i
i
all reactants and products are ideal gases.
For the ideal-gas reaction
aA bB ∆ cC dD
the equilibrium condition n m 0 is
i
i
i
am bm cm dm D
B
C
A
cm dm am bm 0
A
B
D
C
Each chemical potential in an ideal gas mixture is given by Eq. (6.4) as m m°
i
i
RT ln (P /P°), and substitution in the equilibrium condition gives
i
cm° cRT ln 1P >P°2 dm° dRT ln 1P >P°2
C
D
C
D
am° aRT ln 1P >P°2 bm° bRT ln 1P >P°2 0
B
A
A
B
cm° dm° am° bm°
C
B
A
D
RT 3c ln 1P >P°2 d ln 1P >P°2 a ln 1P >P°2 b ln 1P >P°24 (6.5)
B
D
A
C
Since m G for a pure substance, the quantity on the left side of (6.5) is the stan-
m
dard Gibbs energy change G° for the reaction [Eq. (5.38)]
T
¢G° a n G° m,T,i a n m°1T2 cm° dm° am° bm° B
i
A
i
i
C
T
D
i i
The equilibrium condition (6.5) becomes
c
d
a
b
¢G° RT 3ln 1P >P°2 ln 1P >P°2 ln 1P >P°2 ln 1P >P°2 4
D
B
A
C
c
1P C,eq >P°2 1P D,eq >P°2 d
¢G° RT ln (6.6)
a
1P A,eq >P°2 1P B,eq >P°2 b
a
where the identities a ln x ln x , ln x ln y ln xy, and ln x ln y ln (x/y) were
used, and where the eq subscripts emphasize that these are partial pressures at equilib-
rium. Defining the standard equilibrium constant K° for the ideal-gas reaction aA
P
bB : cC dD as
c
1P C,eq >P°2 1P D,eq >P°2 d
K° , P° 1 bar (6.7)
P
a
1P A,eq >P°2 1P B,eq >P°2 b
we have for Eq. (6.6)
¢G° RT ln K°
P
We now repeat the derivation for the general ideal-gas reaction 0 : n A .
i
i
i
Substitution of the expression m m° RT ln (P /P°) for a component of an ideal
i
i
i
gas mixture into the equilibrium condition n m 0 gives
i
i
i
a n m a n 3m° RT ln 1P i,eq >P°24 0
i
i
i
i
i i
a n m°1T2 RT a n ln 1P i,eq >P°2 0 (6.8)
i
i
i
i i
where the sum identities (a b ) a b and ca c a [Eq. (1.50)]
i
i
i
i
i
i
i
i
i
i
i
were used. We have m°(T) G° m,T,i , Therefore
i
¢G° a n G° m,T,i a n m°1T2 (6.9)
T
i
i
i
i i