Simplified Statistical Model for Equilibrium Constant
Simplified Statistical Model for Equilibrium Constant
Consider a simple chemical equilibrium with equilibrium constant =. (This can alternatively be written = in terms of the concentrations of and .) The difference in electronic energy for the reaction equals , conveniently expressed in kJ/mol. Let the internal structure of each molecule be idealized as a series of equally spaced energy levels (similar to those of a harmonic oscillator), with the energy increments and . The spacings and relative to are exaggerated in the graphic for easier visualization. The sublevels of each molecular species are assumed to occupy a Boltzmann distribution at temperature . Accordingly, (n)=(T), where (T)==, the molecular partition function for , and analogously for . For a mixture of and , a single Boltzmann distribution can be considered to apply for the composite levels of both molecules. This leads to the formula for equilibrium constant in statistical thermodynamics: =.
A⇌B
K
eq
N
B
N
A
K
c
[B]
[A]
A
B
A→B
ΔE
Δ
ϵ
A
Δ
ϵ
B
Δ
ϵ
A
Δ
ϵ
B
ΔE
T
N
A
N
A
-nΔ/kT
ϵ
A
e
q
A
q
A
∞
∑
n=0
-nΔ/kT
ϵ
A
e
-1
(1-)
-Δ/kT
ϵ
A
e
A
B
A
B
K
eq
q
B
q
A
-ΔE/RT
e
Under constant volume conditions, the equilibrium constant is related to the change in Helmholtz free energy: . An exothermic reaction, with , tends to give >1, implying that the forward reaction is favored. This can be reversed, however, with the endothermic reaction favored if the entropy change is sufficiently negative. This could result if species has a greater number of thermally accessible levels at the given temperature.
ΔA=ΔE-TΔS=-RTln
K
eq
ΔE<0
K
eq
AB
BA
ΔS
A