Dimensional analysis involving partial derivatives

  • #1
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Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
RPinPA
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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
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Thanks, I wasn't sure about what to do with the ln. Got intimidated, I guess.
 

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