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:

.

### Like this:

Like Loading...

*Related*

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

This entry was posted on May 23, 2008 at 12:17 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

April 10, 2009 at 12:12 am |

[…] 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 […]

July 31, 2009 at 1:05 am |

[…] 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 […]

December 11, 2011 at 6:47 pm |

[…] 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 […]