Physical Chemistry
Lecture 27
Temperature Dependence of ∆Gθ and
Ka
Energetics of a reaction
1
N2 (g) +
2
3
H 2 ( g ) → NH 3 ( g )
2
Enthalpy diagram
for ammonia
synthesis
Energy determines reactivity
Must relate energetics and measures of
reaction.
Free-energy change is most significant.
Obtaining ΔGθ at reaction
temperature
∆Gθ (T )
ln K a (T ) = −
RT
Some data tables for temperatures other than 298.15
K exist

Solve for ΔGθ(T) as done at 298.15 K
More likely that one has to calculate how ΔGθ changes
with T


May be able to calculate ΔHθ(T) and ΔSθ(T) at the
temperature by thermodynamic cycles involving a
temperature at which these are known
Calculate ΔGθ(T) by the simple equation
∆Gθ (T )
= ∆H θ (T ) − T∆S θ (T )
Temperature and equilibrium
For a chemical reaction, at equilibrium
∆Gθ (T )
ln K a (T ) = −
RT
Must find ΔGθ(T) to evaluate Ka(T), and vice
versa
Relationship of temperature derivatives

 ∂
ln K a (T ) 

P
 ∂T
1  ∂
= − 
R  ∂T
 ∆Gθ (T )  

 

T

P
Temperature dependence of
the equilibrium constant
By definition of the standard Gibbs energy
and the distribution of the derivative
 ∂

K
T
ln
(
)


a
 ∂T
P
1 ∂
= − 
R  ∂T
=
=
∆H θ (T )
RT 2
∆H θ (T )
RT 2
 ∆H θ (T ) − T∆S θ (T )  

 

T

P
−
∆C Pθ (T )
RT
+
∆C Pθ (T )
RT
A very simple result
Must know how ∆Hθ(T) depends on T to
determine Ka(T)
Temperature dependence of ΔHθ,
ΔSθ, and ΔGθ
Determining temperature dependence involves
evaluation of change for both reactants
and products
T
θ
∆
C
∫ P (T )dT
θ
θ
∆H reaction
(T0 ) +
(T ) = ∆H reaction
T0
θ
∆
C
θ
θ
P (T )
∆S reaction
dT
(T ) = ∆S reaction
(T0 ) + ∫
T
T0
The Gibbs energy change at the temperature is
T
∆Gθ (T )
= ∆H θ (T ) − T∆S θ (T )
θ
∆
C
P
= ∆Gθ (T0 ) − (T − T0 )∆S θ (T0 ) + ∫ ∆C Pθ dT ' − T ∫
dT '
T'
T0
T0
T
T
Temperature dependence of
reaction thermodynamics
H2 (g) + ½ O2 (g) → H2O (l)
5
2.8 10
80
∆H( Tr )
75
∆S ( Tr )
∆G( Tr )
70
65
300
5
3 10
5
3.2 10
350
Tr
350
Tr
140
ln K a ( Tr )
300
120
100
300
350
Tr
Determining equilibrium
constants experimentally
Example:
1
3
O2 ( gas ) → NH 3 ( gas )
N 2 ( gas ) +
2
2
Equilibrium expression
Ka
=
aNH 3
a1N/22 aH3 /22
Have to find expression for the activities in
terms of concentration measures to use as a
means to find equilibrium concentrations
Using ideal activities in
expressions for Ka
•Ideal gases: ai = Pi/Pθ
•Pure solids and liquids near standard pressure:
Example: ammonia synthesis reaction
Ka
=
( PNH 3 / Pθ )
( PN 2 / Pθ )1 / 2 ( PH 2 / Pθ )3 / 2
=
PNH 3
( PN 2 )1 / 2 ( PH 2 )3 / 2
= K P Pθ
Example: Boudouard reaction
Ka
=
( PCO / Pθ ) 2
aC ( PCO2 / Pθ )
=
KP
Pθ
=
2
PCO
 1 
 θ
(1)( PCO2 )  P 
Pθ
ai = 1
Nonideal gas-phase reactions
Must correct for nonideality: ai = γi ai, ideal
Include activity coefficients
Example: Ammonia-synthesis reaction
Ka
γ NH ( PNH / Pθ )
=
γ 1N/ 2 ( PN / Pθ )1/ 2 γ H3 / 2 ( PH / Pθ )3 / 2
3
2
2
3
2
2
= Kγ
PNH 3
( PN 2 )1/ 2 ( PH 2 ) 3 / 2
= K γ K P Pθ
Ka is the equilibrium constant
KP is not necessarily a constant because Kγ depends
on pressure and temperature
Must evaluate Kγ
Pθ
Temperature and pressure
dependencies of KP
KP for Ammonia Synthesis Reaction at
673K, 723K and 773K
Measurement of KP
allows extrapolation to
obtain Ka in the limit of
P=0
-3.5
-4.5
-5
-5.5
-6
0
200
400
600
800
1000
Pressure (atm)
Kγ Pθ for the Ammonia Synthesis Reaction at 673K,
723K and 773K
Knowledge of Ka(T)
and KP allows the
calculation of Kγ at
any conditions
1.5
1
Kγ Pθ
ln (KP)
-4
0.5
0
0
200
400
600
Pressure (atm)
800
1000
Summary
ΔG = 0 at equilibrium
ΔGθ(T) and Ka(T) express the same quality of a chemical reaction
Calculation of ΔGθ(T) and Ka(T)




Directly from tabulations of ΔGθformation(T) at the reaction temperature
Indirectly through calculation of ΔHθreaction(T) and ΔSθreaction(T) at the
temperature of reaction from tables at T
Calculation of ΔHθ(T) and ΔSθ (T) at the temperature of reaction from
knowledge of ΔHθ(T0) and ΔSθ(T0) at a reference temperature and
ΔCPθ(T) as a function of temperature
Various approximations sometimes used in thermodynamic
calculations
 Assume ΔHθ and ΔSθ are independent of temperature
 Assume ΔCPθ is independent of temperature
 Can never be totally correct


Depends on the specific system how good the approximation is
Do the full calculation unless you are sure an approximation applies
Measurements of KP allow determination of Ka as a limiting value
Nonidealities can be quantified if Ka and KP are known