Question on the degeneracies of a thermodynamic system

[1v1Dota2RightMeow]

Homework Statement

A system possesses three energy levels $$E_1=\varepsilon$$ $$E_2=2\varepsilon$$ $$E_3=3\varepsilon$$ with degeneracies $$g(E_1)=g(E_3)=1$$ $$g(E_2)=2$$. Find the heat capacity of the system.

Homework Equations

$$\beta=\frac{1}{kT}$$
$$Z=\sum_i g_ie^{-\beta \varepsilon_i} \ $$

The Attempt at a Solution

From a beginner's perspective I know to apply the partition function for a system with degeneracies as the first step in order to be able to obtain more information about the system. Thus,

$$Z=g_ie^{-\beta \varepsilon} + 2g_ie^{-\beta 2\varepsilon} + g_ie^{-\beta 3\varepsilon}$$

But I still don't quite understand what's going on here. Also, the hint at the back of the book (Statistical Physics by F. Mandl) simply says to take the zero of the energy scale at $E_1=0$ and then proceeds to give the final answer as: $$C=2k\frac{x^2e^x}{(e^x+1)^2}$$

But this isn't really helpful in understanding the concept. What do the degeneracies of a system do to the solution of the partition function?

 

[Ken G]

Heat capacity is the heat added per change in temperature. So find the expectation value of the internal energy as a function of temperature, and then take its derivative with respect to temperature.