Page 280 - Advanced thermodynamics for engineers

268 CHAPTER 12 CHEMICAL EQUILIBRIUM AND DISSOCIATION

Thus
d d 1 0 0 0 0
ln K p r ¼ y a m þ y b m y c m y d m d
a
c
b
dT dT <T
(12.96)
d 1
¼ ym 0 ym 0
dT <T R P
0
Consider definition of m
0
m ¼ h 0 þ hðTÞ Tfs 0 þ sðTÞg
Z (12.97)
dh
¼ h 0 þ hðTÞ Ts 0 T :
T
Consider the terms

Z Z
dh hðTÞ dh
hðTÞ T ¼ T (12.98)
T T T
Let
v ¼ hðTÞ dv ¼ dh
1 dT
u ¼ du ¼
T T 2
Integrating by parts gives

Z Z
hðTÞ dhðTÞ hðTÞdT
T ¼ T : (12.99)
T T T 2
Thus
d X y hðTÞ X yhðTÞ
¼ ¼ (12.100)
lnK p r 2 2
dT < T <T
Although S has been used as a shorthand form it does include both positive and negative signs;
these must be taken into account when evaluating the significance of the term.
Thus
1
d
lnKp r ¼ 2 ðy a h a þ y b h b y c h c y d y d Þ
dT <T
(12.101)
1
¼ 2 ½yhŠ ½yhŠ P
R
<T
But, by definition

Q p ¼ y c h c þ y d h d y a h a þ y b h b (12.102)
Hence
d Q p
¼ : (12.103)
lnK p r 2
dT <T
Equation (12.103) is known as the Van’t Hoff equation. It is useful for evaluating the heat of
reaction for any particular reaction since

<d lnK p r
Q p ¼ :
dð1=TÞ