# Occupied lattice sites, determining number of microstates and energy

A solid consisting of $N$ molecules on a lattice of $N$ sites is isolated from its environment, and has energy $E$. Each molecule is fixed in position and independent of all others. It can be in any one of four internal states. Two of these states have energy $0$, while the other two states have energy $b$.

Consider $P=E/b$, so that $P$ is the number of molecules with energy $b$.

(a) Calculate the number of microstates of the solid when it has energy $E$.

(b) The solid is now placed in good thermal contact with its environment at temperature $T$. What is the solid's mean energy?

(c) Suppose $b=0.1 \mathrm{eV}$. What is the energy per molecule at $25^{\circ}C$?

I believe the number of microstates of the solid when it has energy $E$ is ${N/2 \choose bP}$.

However, I am not sure. Can someone clarify/guide, and assist with the latter parts?

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does anyone have any advice or help to offer? – Alex Trent Dec 19 '12 at 5:19

1
Your guess for the number of microstates can't be right just from units. $bP$ has units of energy, when you need arguments that are numbers. Units are always the first check to see if your formulas make sense. – Todd R Dec 19 '12 at 5:54

## 1 Answer

Part B)

Let's find the partition function.

$Z = \Sigma_i{e^{-E_i/Tk_b}}$ over all microstates i.

We have 4 microstates of energy b,b,0 and 0. Therefore:

$Z = 2e^{-b/Tk_b} + 2 = 2e^{-b\beta} + 2$

Now we need to find the energy. To do this we use the formula:

$E = -\frac{d}{d\beta}Z(\beta)$ = $-\frac{d}{d\beta}(2e^{-b\beta} + 2)$

Evaluating gives:

$<E> = 2be^{-b\beta} = 2be^{b/k_bT}$

Part C)

Substitute the numbers given into the above expression for E. This expression is already "per particle"

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Don't forget to use the correct SI units when substituting in the numbers. – Chris Dec 19 '12 at 6:03
Hi Chris, for part A, do you agree with what Todd said? – Alex Trent Dec 19 '12 at 6:13
Part A)  N Choose P = N Choose (E/b) – Chris Dec 19 '12 at 6:17
you have N molecules, and you're choosing the ones that contribute to the total energy E, which is given by P. makes sense. thank you! – Alex Trent Dec 19 '12 at 6:17
Yes, but make sure you express P in terms of energy.  That is, replace P with E/b. – Chris Dec 19 '12 at 6:18
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