# Number of microstates in multi-particle system

1. Nov 16, 2016

### Kara386

1. The problem statement, all variables and given/known data
Find the number of accessible microstates for a multi-particle system whose energy depends on temperature as $U = aT^n$ where a is a positive constant and $n>1$. Use the fact that
$S = \int \frac{C_v}{T}dT$

2. Relevant equations

3. The attempt at a solution
$U = nC_vdT$

So $\frac{U}{n} = C_v dT = \frac{aT^n}{n}$

$S = \frac{a}{n} \int T^{n-1}dT = \frac{a(n-1)^2}{n}T^{n-2}$

$=kln(\Omega)$
Rearranging and raising both sides to the power of e gives
$\Omega = e^{\frac{a}{k}(n-1)^2T^{n-2}}$

I'm slightly suspicious of that answer and in particular of whether the internal energy U in the equation $U=nC_vdT$ is the same as the U in the question, and the n in the equation $U=nC_vdT$ is the same n as the one in the equation for energy. Because the n in the given energy equation isn't defined. Is what I've done ok?

Last edited: Nov 16, 2016
2. Nov 16, 2016

### Staff: Mentor

Your starting point should be the definition of $C_v$:
$$C_v = \frac{dU}{dT}$$
Also, you should check that integration you did.

3. Nov 16, 2016

### Kara386

Because I differentiated. How ridiculous. Ok, I'll try again!