# Sackur-Tetrode problem

1. Feb 1, 2009

### kasse

1. The problem statement, all variables and given/known data

Use the Sackur-Tetrode formula to verify that the average kinetic energy of an ideall gas is $$\frac{3}{2}k_B T$$.

2. Relevant equations

Sackur-Tetrode:

$$S_{tot}(E_A) = k_B[N_A(\frac{3}{2}ln \ E_A + ln \ V_A) + N_B(\frac{3}{2}ln(E_{tot} - E_A) + ln \ V_B)] + const.$$

3. The attempt at a solution

The average value is the most probable value, because of gaussian distribution. Derivation gives:

$$0 \ = \ \frac{dS_{tot}}{dE_A} \ = \ \frac{3}{2}k_B(\frac{N_A}{E_A} - \frac{N_B}{E_B})$$

Am I on the right track? What can I do next? Simply set
$$E_A = E_B = \frac{3}{2}k_BT$$
and
$$N_A = N_B$$?

Last edited: Feb 1, 2009
2. Jul 8, 2009

### tinytoon

this is probably too late a response for you, but I'll post it for future readers:

The Sackur-Tetrode equation is:
$$S = Nk (ln \left(\frac{4V^{2/3} \pi mU}{3N^{5/3}h^2}\right) +5/2)$$
Just take the derivative with respect to U:
$$\frac{\partial S}{\partial U} = \frac{1}{T} = \frac{3Nk}{2U}$$
Rearrange and it gives:
$$U = \frac{3}{2} NkT$$