# Statistical mechanics: approximation when temperatures are high

Suppose a diatomic ideal gas system of $N$ particles in a volume $V$ and temperature $T$. Each atom can be described by three hamiltonian: traslation (classic), rotation (classic) and vibration (quantum).

I obtained the energy of the system $E$ and the specific heat capacity $C_V$:

$$E=\frac{5N+\beta \hbar \omega N\coth(\hbar \omega \beta/2)}{2\beta}$$

$$C_V=-\frac{5N}{2k_BT^2}-\frac{\hbar^2N\omega^2}{4k_BT^2\sinh^2(\hbar \omega \beta/2)}$$

where $k_B$ is the Boltzmann's constant, $\omega$ is a frequency of vibration and $\hbar$ is related to the Planck's constant.

Now, I want to approximate these expression to high values of temperature, but I've got no idea how to do it. I thought to take the limit when $T$ goes to infinity, but then I don't get an expression.

• Will you still have molecules at this temperature, as it's a distribution I guess you will have some, will they be significant or can 2 d.o.f be disregarded? Just curious more than knowledgeable :) – user140606 Jan 28 '17 at 18:09
• I suppose that I will still have molecules – user326159 Jan 28 '17 at 18:33