Page 194 - Physical chemistry understanding our chemical world

THERMODYNAMICS AND THE EXTENT OF REACTION 161

Justification Box 4.5

Consider again the simple reaction of Equation (4.46):

aA + bB = cC + dD
We ascertain the Gibbs energy change for this reaction. We start by saying

G = νG (products) − νG (reactants)
where ν is the respective stoichiometric number; so

(4.50)
G = cG C + dG D − aG A − bG B
O
O
From an equation like Equation (4.43), G = G + RT ln(p/p ), so each G term in
Equation (4.50) may be converted to a standard Gibbs function by inserting a term like
Equation (4.43):

O
O
G = cG + cRT ln p C + dG + dRT ln p D − aG O
C D A
p O p O

p A O p B
−aRT ln − bG − bRT ln (4.51)
B
p O p O
We can combine the G O terms as G O by saying
O
O
O
O
G = cG + dG − aG − bG O (4.52)
C D A B
So Equation (4.51) simplifies to become:

p C p D p A
O
G = G + cRT ln + dRT ln − aRT ln
p O p O p O

p B
− bRT ln (4.53)
p O
Then, using the laws of logarithms, we can simplify further:
O c O d
(p C /p ) (p D /p )
O
G = G + RT ln (4.54)
O b
O a
(p A /p ) (p B /p )
The bracketed term is the reaction quotient, expressed
in terms of pressures, allowing us to rewrite the equation We changed the posi-
in a less intimidating form of Equation (4.49): tioned of each stoi-
chiometric number via
O
G r = G + RT ln Q the laws of logarithms,
r
b
saying b × ln a = ln a .
A similar proof may be used to derive an expression
relating to G O and K c .