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 εA if it is occupied by an atom of type A, and energy εB 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 NA occupied by type A atoms.

First, with NA of the N sites occupied by type A atoms, we then have NNA 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 NA atoms of type A and NNA 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, εA, and εB are constants, and applying the chain rule:

Thus:
.
(Note that we have assumed N and NA are large enough that NA 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 NA (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:
.

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3 Responses to “Physics Friday 21”

  1. Physics Friday 67 « Twisted One 151’s Weblog Says:

    […] and the above confirms that systems in thermodynamic equilibrium have the same temperature (see here for a previous post using this relation). In fact, this can be used as a definition of temperature […]

  2. Physics Friday 83 « Twisted One 151’s Weblog Says:

    […] approximation), one could find the energy U for equilibrium at temperature T. Go read this post to see exactly how this is done (let εa=0 and εb=ε). Instead, this time […]

  3. Monday Math 41: The Digamma Function « Twisted One 151's Weblog Says:

    […] to do so. In particular, mathematicians define the digamma function (which I’ve mentioned here), the logarithmic derivative of the gamma function: This is continuous except for the poles at […]

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