# Relation between heat capacity and internal energy

1. Mar 13, 2009

### trelek2

The task is to derive an expression for the heat capacity of a system at constant volume and show that at high temperatures it is inversely proportional to the square of temperature.

As far as I'm concerned the relation between internal energy and heat capacity is:
Cv=dE/dT
However with with i cannot get a reasonable answer.

The expression for the total internal energy of a system is given by:
$$E= \frac{N \Delta E}{exp( \Delta E/kT) +1 } }$$

Where N is the total number of particles, T is the temperature, k is the Boltzmann constant.

When i differentiate it I get an expression which is negative and inversely proportional to the temperature, not to the square temperature....

2. Mar 13, 2009

### derek e

I do not think you are differentiating correctly. I took the partial in a big T limit and got a positive inverse T relationship. What are your steps?

3. Mar 13, 2009

### trelek2

But it is clearly supposed to be proportional to inverse square T...
I use the quotient rule E'=(u'v-uv')/v^2
and take u=NdeltaE, u'=0
v=exp(deltaE/kT)+1 v'=(deltaE/kT)exp(deltaE/kT).
This clearly gives a negative inverse T proportional.
Show me what I am supposed to do.

4. Mar 13, 2009

### derek e

Sorry, typo. I meant to say positive inverse square T. Your v' is incorrect. Use the chain rule.

5. Mar 14, 2009

### trelek2

oh ****, you're right. How stupid I am.

6. Mar 14, 2009

No worries