N particles, Partition Function and finding U and Cv

[TFM]

Homework Statement

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?

Homework Equations

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

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 :

[tex] U = frac{\partial}{\partial \beta} = --\epsilon = \epsilon [\tex]

however this doesn't fit into the next question, find Cv, which needs the formula:

$$C_V = \frac{\partial U}{\partial T}$$ sice this would make Cv 0, thus meaning I can't plot a graph.

Any ideas where I have gone wrong?

Many Thanks,

TFM

 

[anathema]

Dear TFM,

Thank you for your post. It seems like you have made a mistake in your calculation for the internal energy, specifically in your expression for ln z. The correct expression should be:

ln z = -ln N! + Nln(1+4e^(-epsilon/kT))

This will give you an internal energy of U=-Nepsilon/(1+4e^(-epsilon/kT)) which, when differentiated with respect to T, will give you a non-zero heat capacity at constant volume, allowing you to plot a graph.

I hope this helps. Good luck with your calculations!
Scientist

 

[nlightner]

Your approach to finding the partition function and internal energy for a single particle is correct. However, for the internal energy of the system of N particles, you should use the relation U = N * <E>, where <E> is the average energy per particle. This can be found by taking the derivative of the partition function with respect to beta and multiplying by N.

For part c), you can use the relation C_V = (1/N) * (dU/dT) to find the heat capacity at constant volume. This will give you a non-zero result and you can then plot a graph to show its temperature dependence.

For part d), you can use the relation F = -k_BT * ln(z_{total}) to find the Helmholtz free energy of the system.

For part e), you can use the relation S = (1/T) * (U - F) to find the entropy of the system. As you have correctly mentioned, the entropy will approach zero as T approaches zero and will approach infinity as T approaches infinity. The number of microstates accessible in the high-temperature limit can be found using the relation S = k_B * ln(W), where W is the number of microstates. You can use the expression for entropy you found earlier and take the limit as T approaches infinity to find W.

Overall, your approach is correct but you may have made some small errors in the calculations. Make sure to double check your algebra and use the correct relations for finding the internal energy and heat capacity. Good luck!