Page 132 - Chemical equilibria Volume 4

108 Chemical Equilibria
Δφ
change in the form: ν
k Δ H ) k . Thus, we obtain sums of terms such as those
(Δφ
T
illustrated by:
T Δ C ) φ ( Δ H
ν
Δ H T = Δ H 0 + ∑ ∫ r T P dT + ∑ k Δφ k [4.8]
r
r
T
φ T Δφ, k
Δφ
Expressions [4.7] and [4.8] need to be applied to each individual case.
Next, we can combine expressions [4.7], [4.8], [4.1] and [4.2] and then
integrate relation [4.3] to obtain an expression of the rate constant at
temperature T, which is in the form:
Δ H 0 1 T 1
ln K P (T ) = − r 0 T − ∑ ∫ Δ C P ) φ ( dT − ν Δ Δφ H k
r
RT RT RT ∑ k
φ T Δφ, k
(Δφ ) [4.9]
Δ S 0 T T Δ C () φ T Δ H
r
+ 0 T + ∑ ∫ r P dT + ν Δφ k
( )
R R T R ∑ k T Δφ
φ T Δφ, k
(Δφ )
By combining all these relations, we can also obtain an expression which
links the equilibrium constant at a temperature T to a value of that constant
at a different T 0, and the variations of the molar specific heat capacities in
each temperature interval defined. We finally find a relation of the type:
T
ν
RT ln K P (T ) = RT 0 ln K P ( 0 T ) − ∑ ∫ Δ C P ) φ ( dT − ∑ k Δ H k
r
Δφ
φ T ( ) Δφ, k
Δφ
T Δ C () φ Δ H [4.10]
+ ( −TT 0 )Δ S 0 0 T + T ∑ ∫ r P dT + T ∑ k Δφ Δφ k
ν
( )
r
φ T T Δφ, k T
( )
Δφ
To illustrate this approach, we shall determine an expression of the vapor
pressure of liquid copper at temperature T, which is above its fusion point of
1357 K. As a reference state, we choose the standard state of the pure
substance at a temperature of 298 K and pressure of 1 bar. We have the
following data:
– standard enthalpy of vaporization of liquid copper at 298 K:
−
1
0
Δ h 298 = 19498 J.mole ;
V