Page 268 - Advanced Thermodynamics for Engineers, Second Edition

12.5 CALCULATION OF CHEMICAL EQUILIBRIUM 257

The value of pressure to be used in Eqn (12.49) is the ratio of the partial pressure of the individual
constituent to the datum pressure.
Substituting for m in the equilibrium equation, Eqn (12.48):

0 0 0 0
a b c d
y a m þ<T ln p r a þ y b m þ<T ln p r b ¼ y c m þ<T ln p r c þ y d m þ<T ln p r d (12.51)
which can be rearranged to give

0
0
0
y a m þ y b m y c m y d m 0 d þ<T y a ln p r a þ y b ln p r b y c ln p r c y d ln p r d ¼ 0 (12.52)
a
b
c
Hence
8 9
"( ) #
p p
< y a y b= y a y b
0 0 0 0 r a r b p p b
a
y a m þ y b m y c m y d m ¼<T ln ¼<Tln p ðy c þ y d y a y b Þ
a b c d y c y d y c y d 0
c
: p p ; p p
r c r d d
8 9
"( ) #
p p p p
< y c y d= y c y d
¼ <T ln r c r d ¼ <T ln c d p ðy a þ y b y c y d Þ
y a y b y a y b 0
: p p ; p p
r a r b a b
(12.53)
The left-hand side of Eqn (12.53) is the difference in the standard chemical potentials at the
reference pressure, p 0 , of 1 bar or 1 atm. This is defined as
0
0
0
0
DG ¼ y a m þ y b m y c m m m 0 (12.54)
T
c
b
d d
a
Hence
"( ) #

DG 0 p p p p y d
y c y d
y c
T r c r d c d ðy a þy b y c y d Þ
¼ ln y a y b ¼ ln y a y b p 0 ¼ ln K p r (12.55)
<T p p a p
b
p r a r b
giving
( )
p p p p
y d
y c y d
y c
r c r d c d ðy a þy b y c y d Þ
K p r ¼ ¼ p 0 (12.56)
p
y a y b y a y b
b
p r a r b p a p
is called the equilibrium constant, and is a dimensionless value. Sometimes the equilibrium
K p r
constant is defined as
( )
p p y d
y c
c
d
p a p
K p ¼ y a y b (12.57)
b
i.e.
.
¼ K p p ðy a þy b y c y d Þ p ðy a þy b y c y d Þ (12.58)
0 0
K p r or K p ¼ K p r
K p has the dimensions of pressure to the power of the sum of the stoichiometric coefficients, i.e.
Sy
p . The nondimensional equilibrium constant is defined as
DG 0 T
ln K p r ¼ : (12.59)
<T