William M. White - Geochemistry

Substituting β for ∂ln Ω ( E )/d E ( eqn. 2.48), we have:

and

(2.80)

Since the total energy of the system, E , is fixed, Ω( E ) must also be fixed, so:

(2.81)

Substituting 1/k T for β ( eqn. 2.53), we have:

We can deduce the value of the constant C by noting that , that is, the probabilities over all energy levels must sum to one (because the system must always be in one of these states). Therefore:

(2.82)

so that

(2.83)

Generalizing our result, the probability of the system being in state i corresponding to energy ε iis:

(2.84)

This equation is the Boltzmann distribution law * , and one of the most important equations in statistical mechanics. Though we derived it for a specific situation and introduced an approximation (the Taylor series expansion), these were merely conveniences; the result is very general (see Feynman et al., 1989 for an alternative derivation). If we define our system as an atom or molecule, then this equation tells us the probability of an atom having a given energy value, ε i.This is the statistical mechanical interpretation of this equation; it can also be interpreted in terms of quantum physics. The basic tenet of quantum physics is that energy is quantized: only discrete values are possible. The Boltzmann distribution law gives the probability of an atom having the energy associated with quantum level i .

The Boltzmann distribution law says that the population of energy levels decreases exponentially as the energy of that level increases ( energy among atoms is like money among men: the poor are many and the rich few ). A hypothetical example is shown in Figure 2.9.

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 (∼10 23, give or take a few in the exponent). In this case, the number of atoms having some energy ε iis 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.

According to quantum theory, all modes of motion are quantized. Consider, for example, vibrations of atoms in a hydrogen molecule. Even at absolute zero temperature, the atoms will vibrate at a ground state frequency. The energy associated with this vibration will be: