Statistical Mechanics · Problem 3.14

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · law-of-mass-action

Problem

Consider an ideal-gas mixture of atoms

A

, atoms

B

and molecules

AB

, undergoing the reaction

AB \leftrightarrow A + B

. If

n_A

n_B

and

n_{AB}

denote their respective concentrations (that is, the number densities of respective atoms/molecules), then show that, in equilibrium,

\frac{n_{AB}}{n_A n_B} = V\frac{f_{AB}}{f_A f_B} = K(T) \quad \text{(the law of mass action)}.

Here,

V

is the volume of the system while the

f_i

are the respective single-particle partition functions; the quantity

K(T)

is generally referred to as the equilibrium constant of the reaction.