# Helmholtz Free Energy

1. Nov 8, 2006

### stunner5000pt

$\tau = k_{B} T$
a) Find the expression for the free energy as a function of the temperature of the system with two states - one iwth eneryg zero and one with energy $\epsilon_{0}$

b) From the free energy find the expressions for the energy and entropy of the system

c) Plot the average energy and the entropy as a function of tau. $\tau = k_{B} T$

Ok for a) wek now that
$$F = U - T \sigma$$

Partition fun ction $Z = \sum_{s} \exp(-\epsilon_{s}/\tau) = 1 + \exp(-\epsilon_{0}/\tau)$

so then
$$U = \frac{\epsilon_{0} \exp(-\epsilon_{s}/\tau)}{1 + \epsilon_{0} \exp(-\epsilon_{s}/\tau)}$$

but im not quite sure how to proceed with the calculation of the entropy, sigma ...

for b)
for entropy use this
$$\sigma = \left(\frac{\partial F}{\partial \tau}\right)_{V}$$

but not sure about how to find the nergy for hte system... is it simply the expression wh9ich doesnt involve tau??

I was thiking a bit more

isnt helmholtz free enryg given by simple
$$F- F(0) = -\tau \log Z$$??

Last edited: Nov 8, 2006