Statistical Thermodynamics and Rate Theories/Chemical Equilibrium

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Chemical Equilibrium from Statistical Thermodynamics[edit]

Consider the general gas phase chemical reaction represented by

where A, B, C and D are the reactants and products of the reaction, and is the stoichiometric coefficient of chemical A, is the stoichiometric coefficient of chemical B, and so on. Each of the gases involved in the reaction will eventually reach an equilibrium concentration according to their equilibrium constant():

The chemical potential of species i is given by the equation

where A is the Helmholtz energy, and is the number of molecules of species i. The Helmholtz energy can be determined as a function of the total partition function, Q:

where is the Boltzmann constant and T is the temperature of the system. The total partition function is given by

where is the molecular partition function of chemical species i. Substituting these definitions into the equation for chemical potential yields:

A variable, , is then defined such that , where j = A, B, C or D and is taken to be positive for products and negative for reactants. A change in therefore corresponds to a change in the concentrations of the reactants and products. Thus, at equilibrium,

From Classical thermodynamics, the total differential of A is:

For a reaction at a fixed volume and temperature (such as in the canonical ensemble), and equal 0. Therefore,

Substituting the expanded form of chemical potential:

For the reaction :

This equation simplifies to

By dividing all terms by volume, and noting the relationship , the following equation is obtained:

Partition Functions[edit]

The molecular partition function, is defined as the product of the transnational, rotational, vibration and electronic partition functions:

These components of the molecular partition function may be defined as follows:

where m is the mass of a single particle and h is Planck's constant, and T is temperature.

where is the reduced mass of the molecule, is the bond length between the atoms in the molecule, and is the reduced Planck's constant.

where is the vibrational frequency of the molecule.

where is the degeneracy of the ground state, and is the bond energy of the molecule.

Thus, the equilibrium constant of a chemical reaction can be expressed in terms of the molecular partition functions and the difference in atomization energies of the products and reactants .

Example[edit]

Calculate the equilibrium constant for the reaction of and at 650 K.

Equilibrium Constant Equation (From Molecular Partition Functions)



A simple problem solving strategy for finding equilibrium constants via statistical mechanics is to separate the equation into the molecular partition functions of each of the reactant and product species, solve for each one, and recombine them to arrive at a final answer.


In order to simplify the calculations of molecular partition functions, the characteristic temperature of rotation () and vibration () are used. These values are constants that incorporate the physical constants found in the rotational and vibrational partition functions of the molecules. Tabulated values of and for select molecules can be found here.

Species (K) (K) (kJ mol-1)
6125 0.351 239.0
808 87.6 431.9
4303 15.2 427.7


















Combining the terms from each species, the following expression is obtained:






At 650 K, the reaction between and proceeds spontaneously towards the products. From a statistical mechanics point of view, the product molecule has more states accessible to it than the reactant species. The sponteneity of this reaction is largely due to the electronic partition function: two very strong H—Cl bonds are formed at the expense of a very strong H—H bond and a relatively weak Cl—Cl bond.