Chemical Equilibrium

We are finally at a point where we can talk about chemical equilibrium we have encountered at the kinetics section. Consider the following reaction, 

          .
From before, the Helmholtz freee energy is 
          ,
where dNj = vjd l.

If we keep T and V constant as we do in an experiment, the free energy can be written as 

          .
At equilibrium, the free energy is zero, therefore we have
Equilibrium Condition (1)

The partition function of the system of mixture of ideal gases should be the product of individual partition function of each reactants and products; it means 

          .
We will relate chemical potential from Eqn. (1) to the partition function by 
          .                      (2)
Substitute Eqn. (2) into Eqn. (1), we can get 
          .
Since the partition function is a function of temperature with constant volume, and therefore q/V which can be considered as a concentration. Then we have an equilibrium constant as a function of temperature only.
Concentration Equilibrium Constant(3)
Equilibrium constant can also be written in the partial pressure form, realizing that pi = rikT. We have
Partial Pressure Equilibrium Constant(4)