# Thermal Physics, energy and temperature

1. May 20, 2013

### Cogswell

1. The problem statement, all variables and given/known data
In section 2.5 I quoted a theorum on the multiplicity of any system with only quadratic degrees of freedom: In the high-temperature limit where the number of units of energy is much larger than the degrees of freedom, the multiplicity of any such system is proportional to $U^{Nf/2}$, where Nf is the total number of degrees of freedom. Find an expression for the energy of such a system in terms of its temperature, and comment on the result. How can you tell that this formula for $\Omega$ cannot be valid when the total energy is very small?

2. Relevant equations

$S = k \ln (\Omega)$

$\dfrac{1}{T} = \left( \dfrac{\partial S}{\partial U} \right)$

3. The attempt at a solution

So they said that the multiplicity is proportional to $U^{Nf/2}$

And so I'm assuming $\Omega = A \cdot U^{Nf/2}$ for some constant A

Putting that into the Entropy formula I get

$S = k \ln (A \cdot U^{Nf/2})$

$S = k \ln (A) + \dfrac{Nfk}{2} \ln(U)$

Partial differentiating with respect to U I get:

$\dfrac{1}{T} = \left( \dfrac{\partial S}{\partial U} \right) = \dfrac{Nfk}{2U}$

And so: $T = \dfrac{2U}{Nfk}$
which tells us that T is proportional to U and inversely proportional to f.

Is that right? And what about the last part of the question? If U is very small, then T is very small as well, so why can it not be valid? Is it because the temperature is measured in Kelvin starting from -273.15 and so no amount of U will give you that?