Читать книгу Geochemistry - William M. White - Страница 56

Consider a mineral sample, A, in a heat bath, B (B having much more mass than A), and assume they are perfectly isolated from their surroundings. The total energy of the system is fixed, but the energy of A and B will oscillate about their most probable values. The question we ask is what is the probability that system A is in a state such that it has energy E_A?

We assume that the number of states accessible to A when it has energy E_A is some function of energy:

(2.76)

Following the basic postulate, we also assume that all states are equally probable and that the probability of a system having a given energy is simply proportional to the number of states the system can assume when it has that energy:

(2.39)

where C is a constant. Thus, the probability of A being in state a with energy E_A is:

(2.77)

Since the total energy of the two systems is fixed, system B will have some fixed energy E_B when A is in state a with energy E_A, and:

where E is the total energy of the system. As we mentioned earlier, Ω is multiplicative, so the number of states available to the total system, A + B, is the product of the number of states available to A times the states available to B:

If we stipulate that A is in state a, then Ω_A is 1 and the total number of states available to the system in that situation is just Ω_B:

Thus, the probability of finding A in state a is equal to the probability of finding B in one of the states associated with energy E_B, so that:

(2.78)

We can expand as a Taylor series about E:

(2.79)

and since B is much larger than A, E > E_A, higher-order terms may be neglected.

Substituting β for ∂lnΩ(E)/dE (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/kT 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_i is:

(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.