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- Homework Statement
- (a) Find an expression for the free energy as a function of ##\tau## (temperature) of a system with two states, one at energy 0 and one at energy ##\epsilon##. (b) From the free energy, find expressions for energy and entropy of the system. The entropy is plotted in Figure 3.11 (see below).

- Relevant Equations
- $$ Z = \sum_i e^{ frac {-E_i} {k_BT} } $$

$$ F = -k_B T \ln{Z} $$

$$ F = U - TS $$

So just by by using the definition of the partition function...

$$ Z = \sum_i e^{ \frac {-E_i} {k_BT} } = e^{ \frac {-0} {k_BT} } + e^{ \frac {-\epsilon} {k_BT} } = 1 + e^{ \frac {-\epsilon} {k_BT} } $$

And then, a result we obtained in class by using the Boltzmann H factor to solve for ##S## and finding an expression for ## U - TS## gave us ##F = -k_B T \ln{Z} ##. So applying that gives part (a)...

$$ F = -k_B T \ln{ 1 + e^{ \frac {-\epsilon} {k_BT} } } $$

Now for (b), I'm confused. I'm not sure if I have an equation to get S and U out of just this free energy expression. All I have is the free energy definition, ## F = U - TS ##, and if I try to solve using that and my above expression for free energy, I'm going to get something like ## U = U ## or ## S = S##. So I'm not sure which equation/principle to use. I have thought about using differential with constant volume but I don't think that would help and the question seems pretty explicit about using the free energy expression from (a) to obtain both U and S.