# Heat Capacity of a classical ideal gas and SHO

## Homework Statement

Ideal gas. In an ideal-gas model. N molecules move almost indepdently with very weak interactions between, in a three-dimensional box of volume V. Find the heat capacity of the system.

SHO. Consider N independent SHOs in a system. each osciallating about a fixed point. The spring constant is assumed to be k and the mass of oscillator m. FInd the heat capacity.

## Homework Equations

I understand heat capacity can be described as change of energy (E) over time (T) so:

Cv (heat capacity) = (dE/dT * dS/dE)*T
= dS/dT*T

## The Attempt at a Solution

I have S, but I am having trouble with taking the derivative of dS/dT. do i bring the T over so i can take dS/dT?

The S that I have is:

S = N*Kb*ln((V/h^3)*(((4*pi*m*E)/(3N)))*^(3/2)+3/2N

having trouble going from here since if it is S/T, my heat capacity would just be -S/T^2??

Thanks in advance for the help

mfb
Mentor
I understand heat capacity can be described as change of energy (E) over time (T) so:
Temperature, not time. And your equation relates entropy changes to temperature, not energy changes.

With your S, you can calculate dS/dE. But that is just the inverse temperature:
$$\frac{1}{T}=\frac{\partial S}{\partial E}$$
With an expression E(T), you can calculate the heat capacity.