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The denominator of eqn. 2.84, which is the probability normalizing factor or the sum of the energy distribution over all accessible states, is called the partition function and is denoted Q:

(2.85)

The partition function is a key variable in statistical mechanics and quantum physics. It is related to macroscopic variables with which we are already familiar, namely energy and entropy. Let's examine these relationships.

We can compute the total internal energy of a system, U, as the average energy of the atoms times the number of atoms, n. To do this we need to know how energy is distributed among atoms. Macroscopic systems have a very large number of atoms (∼10²³, give or take a few in the exponent). In this case, the number of atoms having some energy ε_i is proportional to the probability of one atom having this energy. So to find the average, we take the sum over all possible energies of the product of energy times the possibility of an atom having that energy. Thus, the internal energy of the system is just:

Figure 2.9 Occupation of vibrational energy levels calculated from the Boltzmann distribution. The probability of an energy level associated with the vibrational quantum number n is shown as a function of n for a hypothetical diatomic molecule at 273 K and 673 K.

(2.86)

The derivative of Q with respect to temperature (at constant volume) can be obtained from eqn. 2.85:

(2.87)

Comparing this with eqn. 2.86, we see that this is equivalent to:

(2.88)

It is also easy to show that , so the internal energy of the system is:

(2.89)

For 1 mole of substance, n is equal to the Avogadro number, N_A. Since R = N_Ak, eqn. 2.89, when expressed on a molar basis, becomes:

(2.90)

We should not be surprised to find that entropy is also related to Q. This relationship, the derivation of which is left to you (Problem 13), is:

(2.91)

Since the partition function is a sum over all possible states, it might appear that computing it would be a formidable, if not impossible, task. As we shall see, however, the partition function can very often be approximated to a high degree of accuracy by quite simple functions. The partition function and Boltzmann distribution will prove useful to us in subsequent chapters in discussing several geologically important phenomena such as diffusion and the distribution of stable isotopes between phases, as well as in understanding heat capacities, discussed below.