Examining the Equilibrium Constant for an Ideal Gas Mixture

[ehrenfest]

Homework Statement

Consider an ideal-gas mixture of atoma A, atoms B and molecules AB, undergoing the reaction AB <---> A+B. If n_A, n_B, and n_{AB} denote their respective concentrations, then show that, in equilibrium

$$\frac{n_{AB}}{n_A n_B} = V \frac{f_{AB}}{f_A f_B} = K(T)$$

(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.

Homework Equations

The Attempt at a Solution

I got the partition function of the system and I got the Helmholtz free energy. The problem is that I just don't know what to do with the Helmholtz free energy i.e. do I minimize it, mazimize it, set it equal to something? Why? Pathria and my other textbook do little more than define it. They never say what to do with it or give any example of how it is useful!

 

[ehrenfest]

anyone?

 

[daschaich]

Sorry for the slow reply. You should be able to show

$$n_A = e^{\mu_A\beta}\frac{f_A}{V}$$

straight from the grand canonical partition function (taking the appropriate derivative of $$\ln \mathcal Z$$). The desired result then follows from showing the chemical potentials cancel out.