# I Dimensional analysis involving partial derivatives

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1. Sep 27, 2018

### Wledig

It is mentioned in Reif's book, statistical physics, that trough dimensional analysis it can be shown that: $$\frac{1}{\beta} = kT$$ where $\beta$ equals $\frac{\partial \ln \Omega}{\partial E}$ and k is the Boltzmann constant. I don't quite see how to reach this result, can anyone give me a hand here?

2. Sep 27, 2018

### RPinPA

The dimensions of dA/dB are A/B, for any two quantities A and B. You can see that from the definition of derivative.

The log is dimensionless (arguments of logs, exponentials, trig functions should always be dimensionless). So the units of $\partial{ln (\Omega)}/\partial{E}$ are 1/energy.

Therefore the units of $1/\beta$ are energy, and those are the dimensions of $kT$

3. Sep 27, 2018

### Wledig

Thanks, I wasn't sure about what to do with the ln. Got intimidated, I guess.