Number of microstates in multi-particle system

[Kara386]

Homework Statement

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
$S = \int \frac{C_v}{T}dT$

Homework Equations

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?

 

[DrClaude]

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

 

[Kara386]

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

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