# Partition function

Is the energy given by the first or the second? I have seen both relationships in different websites, and I am confused.

$$E = kT^2 \frac{\partial Z}{\partial T}$$

or

$$E = - \frac{\partial ln Z}{\partial \beta}$$

## Answers and Replies

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The first one should be

$$E = kT^2 \frac{\partial \ln Z}{\partial T}$$

You can go from one to the other by the chain rule since $$\beta=\frac{1}{k\,T}$$

The first one should be

$$E = kT^2 \frac{\partial \ln Z}{\partial T}$$

You can go from one to the other by the chain rule since $$\beta=\frac{1}{k\,T}$$
I did realise you can go from one to the other before I post this up. But the MIT website solution happily used the same equation (1) twice. For example, page 6 of http://web.mit.edu/Physics/graduate/gen1sol_S01.pdf

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It's simply wrong!

mjsd
Homework Helper
Yes, Rainbow Child is right.