The Partition Function

The sum of all of the probabilities must equal 1. This is called normalization. The normalization constant of the above probability is 1/Q where

Q is called the partition function. The population of a particular state J with energy E_J is given by

This known as the Boltzmann distribution.

The derivation of the Boltzmann distribution relies on the concept of an ensemble of system all in thermal equilibrium. Consider a huge number of systems in thermal contact (an ensemble). Then we ask what is the relative number of systems that will be found in any one state E_i.

The Ensemble Partition Function

We distinguish here between the partition function of the ensemble, Q and that of an individual molecule, q. Since Q represents a sum over all states accessible to the system it can written as

where the indices i,j,k, represent energy levels of different particles.

The molecular partition function, q represents the energy levels of one individual molecule. We can rewrite the above sum as

Q = q_iq_jq_k… or Q = q^N for N particles. Note that q_i means a sum over states or energy levels accessible to molecule i and q_j means the same for molecule j.

The molecular partition function counts the energy levels accessible to molecule i only.

Q counts not only the states of all of the molecules, but all of the possible combinations of occupations of those states. However, if the particles are not distinguishable then we will have counted N! states too many (N! = N(N-1)(N-2)….). This factor is exactly how many times we can swap the indices in Q(N,V,T) and get the same value (again provided that the particles are not distinguishable). If we consider 3 particles we have

i,j,k j,i,k, k,i,j k,j,i j,k,i i,k,j or 6 = 3!.

Thus we write the partition function as

The Molecular Partition Function

We have seen above that the internal energy of the system can be calculated using the partition function. In fact, all thermodynamic quantities can be calculated from molecular properties using the partition function. For an ideal gas of non-interacting particles only the translational partition function matters. Indeed, our focus will be on the effects of translation properties and we will prove later that translation alone defines the properties of an ideal gas. It is important to mention at this point that molecular vibrational, rotational, and even electronic states also can contribute to the molecular partition function.

Molecular energy levels are e = e_a^trans + e_b^vib + e_c^rot + e_d^elec where the indices a, b, c, d run over the levels of one particular molecule. We can write the molecular partition function as a product of partition functions for each type of molecular motion.

The Translational Partition Function

The translational partition function is the most important one for statistical thermodynamics. Pressure is caused by translational motion, i.e., momentum exchange with the walls of a container. For this reason it is important to understand the origin of the translational partition function. Translational energy levels are so closely spaced as that they are essentially a continuous distribution. The quantum mechanical description of the energy levels is obtained from the quantum mechanical particle in a box. The energy levels are

The box is a cube of length a. The average quantum numbers will be very large for a typical molecule. This is very different than what we find for vibration and electronic levels where the quantum numbers are small (i.e. only one or a few levels are populated). Many translational levels are populated thermally. The translational partition function is

The three summations are identical and so they can be written as the cube of one summation

The fact that the energy levels are essentially continuous and that the average quantum number is very large allows us to rewrite the sum as an integral.

The sum started at 1 and the integral at 0. This difference is not important if the average value of n is ca. 10⁹! If we have the substitution a = h²/8ma²kT we can rewrite the integral as

This is a Gaussian integral. The solution of Gaussian integrals is discussed the math section of the Website. If we now plug in for a and recognize that the volume of the box is V = a³ we have

The Rotational Partition Function

The rotational partition function can be derived from rotational energy levels. The sum over levels has the form

If the levels are sufficiently closely spaced relative to thermal energy then the sum can be written as an integral

Then we make the substitution, let u = J(J+1) = J² + J. Then du = (2J+1)dJ so that

is the rotational partition function in the high temperature limit. For most gas phase molecules at room temperature, the high temperature limit is valid.

The Vibrational Partition Function

The vibrational partition function can be derived similarly.