- #1
TFM
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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 [tex] epsilon [/tex]. The upper level is four-fold degenerate (i.e., there are four excited states with the same energy [tex] epsilon [/tex]).
(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
[tex] z_1 = z_{int} =\sum{e^{E_{int}(s)}/k_BT} [/tex]
The Attempt at a Solution
Okay for, a), I have used:
[tex] z_1 = z_{int} =\sum{e^{E_{int}(s)}/k_BT} [/tex]
this has given me:
[tex] 1+4(e^{\epsilon/k_BT}) [/tex]
now b)
I have used:
[tex] z_{total} = \frac{1/N!}(1+4(e^{\epsilon/k_BT}))^N [/tex]
and:
[tex] U = \frac{\partial}{\partial \beta}ln z [/tex]
[tex] \beta = \frac{1}{k_BT} [/tex]
and I have found ln z to be:
[tex] -2(ln N!) +N ln 4 + N - \beta \epsilon [/tex]
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:
[tex] C_V = \frac{\partial U}{\partial T} [/tex] sice this would make Cv 0, thus meaning I can't plot a graph.
Any ideas where I have gone wrong?
Many Thanks,
TFM