Physics Friday 21

Let us make an idealized model of a binary substitution alloy. We have a crystal with N total atomic sites, each of which is occupied by a type A atom or a type B atom (there are no empty sites). Suppose that a site has energy ε_A if it is occupied by an atom of type A, and energy ε_B if it is occupied by an atom of type B.
We will examine the energy and entropy of the crystal in this model, and approximate the average value at equilibrium, for absolute temperature T, of the number of sites N_A occupied by type A atoms.

First, with N_A of the N sites occupied by type A atoms, we then have N–N_A sites occupied by atoms of type B. Thus, the energy is .

To compute the entropy, we use Boltzmann’s equation for entropy:
.
The number of possible microstates with N_A atoms of type A and N–N_A atoms of type B is just . Thus
.

Now, if the alloy is at equilibrium at a temperature T, we can see from classic thermodynamics that:

Now, using our formula for the energy, noting that N, ε_A, and ε_B are constants, and applying the chain rule:

Thus:
.
(Note that we have assumed N and N_A are large enough that N_A can be treated like a continuous variable).

Now, let us reexamine the entropy formula. We have logarithms of factorials, which are difficult to differentiate (see the digamma function). However, assuming that we have large N and N_A (a realistic assumption for a macroscopic crystal), we can use Stirling’s approximation for the factorial. Stirling’s formula says that for large n,
.
Thus
,
where the last approximation is due to the fact that the logarithm grows much more slowly than the other terms.

Thus, we see that we can approximate the entropy:
.
So we see:
.
So, defining , we have:
,
and thus:
.

Tags: Alloy, Boltzmann's Equation, Entropy, Equilibrium, Friday Physics, Model, physics, Statistical Mechanics, Stirling's Approximation, Thermodynamics