# Calculating Thermodynamic State Functions For Systems With Degenerate States

## Introduction

In the chapter of "Thermal Physics, Blundell" that introduces calculating state functions from the partitions function of the system, takes into account a system that doesn't have degenerate energy levels and reaches to the usual results. Later, these results are applied on the system of ideal gas. However the ideal gas does have degenerate states. So, I tried to derive the furmulas that connect the partition function of a degenerate system and thermodynamic state functions and see if they are the same.

## Calculation Of Entropy For A Degenerate System

Let's take for example the entropy. Since the system is degenerate, the probability of the system being in the state of energy $E_i$ is:

$$P_i = \frac{g_j e^{-βΕ_i}}{Z}$$ $$ln {P_i}=-βΕ_i + ln{g_i}-ln{Z}$$

Now, the gibbs entropy is expressed as:

$$S=-k_B \sum_{i} p_i ln{p_i}=$$ $$=-k_B \sum_{i} p_i (-βE_i + ln{g_i} - ln{Z})=$$ $$= k_B \sum_{i} p_iE_i -k_B \sum_{i} p_i ln{g_i} + k_Bln{Z}$$

Therefore we get to the conclusion that:

$$S= \frac{U}{T} + k_B \left<ln{g}\right> + k_BlnZ = \frac{U}{T} + k_B\left<ln{ \frac{Z}{g}}\right>$$

## Application To Ideal Gas

Now, the density of states (degeneracy) for the ideal gas is: $$g(k)=\frac{Vk^2}{2π^2}$$

The energy levels $$E(k)=\frac{\hbar^2 k^2}{2m}$$

And The partition Function: $$Z_1 = \frac{1}{\hbar^3} \left(\frac{mk_BT}{2π} \right)^{3/2}$$

Now, the difference between the equations for the degenerate and the non-degenerate system is the term: $$\left<ln{g_i} \right>= \int p_iln{g_i}=\frac{V}{2π^2Z} \int_{0}^{\infty} e^{-β\hbar^2k^2/2m}k^2ln{ \left(\frac{Vk^2}{2π^2}\right)}dk \neq 0$$