Partition Function: Which Energy Relationship?

[touqra]

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

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

 

[Rainbow Child]

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 $$\beta=\frac{1}{k\,T}$$

 

[touqra]

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 realize 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

 

[Rainbow Child]

It's simply wrong!

 

[mjsd]

Yes, Rainbow Child is right.