Find an Expression for the Helmholtz free energy

[conorod]

Homework Statement

I am attempting the problem below, I might have the correct answer but would appreciate if someone could confirm this (or tell me where I'm going wrong).
Consider a statement having 2 states, one at energy 0 and one at energy ε. Find an expression for the Helmholtz free energy as a function of the temperature, T.

Homework Equations

F = -k_BT log Z

Z = Ʃ e^-βE_r

The Attempt at a Solution

Z = e^-β.0 + e^-β.ε
Z = 1 + e^-β.ε

F = -k_BT ln(1 + e^-β.ε)

 

[Chestermiller]

Isn't β =1/(k_BT)? Shouldn't you substitute that into your equation?

 

[conorod]

I'm not sure - is that necessary? Or is it OK to just use β?

 

[jeppetrost]

Well you want to at least be consistent. In your final equation, you use both beta and kT.

 

[paranormal]

This looks correct! The Helmholtz free energy is defined as F = U - TS, where U is the internal energy, T is the temperature, and S is the entropy. In this case, U is equal to ε since there are two states with energies 0 and ε. And since the system is in equilibrium, S = kB ln(Z), where Z is the partition function. So, substituting these values into the equation, we get F = -ε - kBT ln(1 + e-β.ε). Great job!