Microscopic View of the Equilibrium Constant

The chemical potential can be expressed in terms of the partition function:

To see this we first expand lnQ_j, starting with the fact that Q_j = q_j^Nj/N_j!,

and then take the derivative with respect to N_j,

This result indicates that

If this expression is substituted into Eqn. 1 we have

or

We can express the concentration dependence in the equilibrium constant as

where r_j is the number density of species j, r_j = N_j/V.

which shows that the equilibrium constant can be expressed in terms of molecular partition functions.

Paritition functions in the equilibrium constant

In considering the molecular partition function we must consider the kinematic contributions and electronic contributions. In other words if we write the molecular partition function as

,

the molecular motions, translation, rotation and vibration are kinematic contributions to the available energy space. The electronic partition function is somewhat different since it represents the binding energy of a molecule with respect to constituent atoms. Until now we have simply stated that q_elec = g_elec the electronic degeneracy. This is equivalent to ignoring the energetic contributions to chemical bonds and treating molecules as translating, rotating and vibrating collections of nuclei that are held together by bonds. However, when we deal with chemical reactions there are, by definition changes in bonding. We can accommodate this writing

where D₀ represents the binding energy as shown in Figure below. First consider a quantum chemical calculation of the potential energy of the CO molecule shown in Figure below.

In this example, the equilibrium energy D_e is obtained from a quantum chemical calculation of the binding energy of CO as shown in Figure below.

The binding energy D₀ is equal to the equilibrium energy D_e shown in the figure minus the zero point energy.

D₀ = D_e – hn/2

The zero point energy is shown as the red stripe at the bottom. For CO this energy is calculated to be 1067 cm^-1 and D_e is –96,545 cm^-1. Thus, the zero point energy is typically a small correction since D_e is –95,581 in this case.

By making this subtraction we also “remove” the zero point energy from the vibrational partition function. The vibrational partition function that we have considered up to now is:

The term in the numerator represents the zero point energy. When this term is incorporated into q_elec the vibration partition function becomes.

In this form we can state that for a singly degenerate vibration q_vib = 1 at T = 0 K and the magnitude of the vibrational partition function increases with temperature. Using this separation the significance of the kinematic partition functions is clear. These represent the temperature dependence of occupation of various levels, translation, rotation and vibration, respectively. The electronic partition function, on the other hand, represents the energy of stabilization of the molecule with respect to its constituent atoms. In fact q_elec approaches infinity as T à 0 K. What sense does this make? Well, if we consider q_elec as a contribution to an equilibrium constant we can think of temperature as a parameter that determines the relative stability of the molecule. The bound state of the molecule will be most favored at T = 0 K. As temperature increases there is some thermal tendency for the molecule to dissociate (even though this is small at most temperatures of interest to chemists). In this sense q_elec is quite different from the kinematic partition functions.