Statistical Thermodynamics and Rate Theories/Chemical Equilibrium

Chemical Equilibrium from Statistical ThermodynamicsEdit

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 when the forward and reverse reaction rates become equal. The distribution of reactants to products at the quilibrium point is represented by the equilibrium constant ( ):

 

If the system is not at equilibrium, a shift in the number of reactants and products will occur to lower the overall energy of the system. The difference in the energy of the system at this non-equilibrated point and the energy of the system at equilibrium for any particular species is termed chemical potential. When both temperature and volume are constant for both points aforementioned, the chemical potential   of species i is expressed 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 the molecular partition funCtion into the equation for helholtz energy yields:

 

and then further substituting this equation into the definition of chemical potential yields:

 

rearranging this equation the following derivative can be set-up:

 

From this point Sterling's approximation

 

can be substituted into the derivative to yield:

 

From here the derivative can be rearranged and solved:


 
 
 
 


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,

 
 
Helmholtz Energy with respect to Equilibrium Shifts


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   where   is referred to a number density, the following equation is obtained:

 

Partition FunctionsEdit

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  .

 
 

ExampleEdit

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 spontaneity 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.