Entropy and Energy Density from a Partition Function

[howin]

Homework Statement

Partition Function: Z = 1/N! [8 pi V (kT/hc)^3]^N
There are several parts to this but here are the parts I'm struggling with:
a) Compute the entropy of the system S.
b) Computer the energy density u and compare the result with the corresponding results for a nonrelativistic gas and radiation (a "photon gas").

Homework Equations

S = Bk (U-F) = - (partial F w/ respect to T)
S = k ln Z

u = U/V

The Attempt at a Solution

I got that U = 3 NkT. So is u just 3Nkt/V? How do I find this for a nonrelativistic gas and radiation?

I also computed cv = 3Nk and cp = 4Nk but don't know what to do from there.

Much appreciate the help!

 

[Jhero]

it is important to approach problems like this with a clear and logical thought process. First, let's break down the problem into smaller parts and address them one by one.

Part a) Compute the entropy of the system S:

To compute the entropy, we can use the equation S = k ln Z, where Z is the partition function given in the forum post. We can rewrite Z as Z = 1/N! [8 pi V (kT/hc)^3]^N = (8 pi V (kT/hc)^3)^N/N!. Taking the natural logarithm of both sides, we get ln Z = N ln (8 pi V (kT/hc)^3) - ln N!.

Using Stirling's approximation, we can rewrite ln N! as N ln N - N. Plugging this into our equation for ln Z, we get ln Z = N ln (8 pi V (kT/hc)^3) - N ln N + N. Substituting this into the equation for entropy, we get:

S = k ln Z = Nk ln (8 pi V (kT/hc)^3) - Nk ln N + Nk

Part b) Compute the energy density u and compare with nonrelativistic gas and radiation:

To compute the energy density, we use the equation u = U/V where U is the internal energy of the system and V is the volume. From the partition function, we know that U = - (partial F w/ respect to T). To find this derivative, we can use the equation F = -kT ln Z, where Z is the partition function.

Taking the derivative of F with respect to T, we get:

(part F w/ respect to T) = -k ln Z - kT (d/dT) ln Z = -k ln Z - kT (1/Z) (dZ/dT)

We can simplify this by noting that dZ/dT = N (partial Z/partial T). Substituting this into our equation, we get:

(part F w/ respect to T) = -k ln Z - kT (N/Z) (partial Z/partial T)

Substituting this into our equation for U, we get:

U = - (partial F w/ respect to T) = k ln Z + kT (N/Z) (partial Z/partial T)

Now, let