# N particles, Partition Function and finding U and Cv

1. Mar 18, 2009

### TFM

1. The problem statement, all variables and given/known data

Consider a system of N identical particles. Each particle has two energy levels: a ground
state with energy 0, and an upper level with energy $$epsilon$$. The upper level is four-fold degenerate (i.e., there are four excited states with the same energy $$epsilon$$).

(a) Write down the partition function for a single particle.

(b) Find an expression for the internal energy of the system of N particles.

(c) Calculate the heat capacity at constant volume of this system, and sketch a graph to
show its temperature dependence.

(d) Find an expression for the Helmholtz free energy of the system.

(e) Find an expression for the entropy of the system, as a function of temperature. Verify
that the entropy goes to zero in the limit T --> 0. What is the entropy in the limit
T --> infinity? How many microstates are accessible in the high-temperature limit?

2. Relevant equations

$$z_1 = z_{int} =\sum{e^{E_{int}(s)}/k_BT}$$

3. The attempt at a solution

Okay for, a), I have used:

$$z_1 = z_{int} =\sum{e^{E_{int}(s)}/k_BT}$$

this has given me:

$$1+4(e^{\epsilon/k_BT})$$

now b)

I have used:

$$z_{total} = \frac{1/N!}(1+4(e^{\epsilon/k_BT}))^N$$

and:

$$U = \frac{\partial}{\partial \beta}ln z$$

$$\beta = \frac{1}{k_BT}$$

and I have found ln z to be:
$$-2(ln N!) +N ln 4 + N - \beta \epsilon$$

thus U equal the beta derivative, Thus I have found :

$$U = frac{\partial}{\partial \beta} = --\epsilon = \epsilon [\tex] however this doesn't fit into the next question, find Cv, which needs the formula: [tex] C_V = \frac{\partial U}{\partial T}$$ sice this would make Cv 0, thus meaning I can't plot a graph.

Any ideas where I have gone wrong?

Many Thanks,

TFM