# Number of microstates in multi-particle system

## 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##

## 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?

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

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!