# 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