A system can exist in two nondegenerate states: a ground state energy ${}_1$ and a higher energy state $2.$ Develop expressions for the average energy, heat capacity, entropy and Helmholtz energy of the system as a function of temperature.

$2$ Repeat problem one but this time assume that the energy state ${}_1$ and ${}_2$ has a degeneracy of ${}_1$ and ${}_2,$ respectively.

Considering argon to be a monoatomic gas, compute the thermodynamic properties: Helmholtz free energy, internal energy and chemical potential of $1$ mole of argon at $325K$ and $\frac{1\mathrm{.}5}{b}ar.$

For a simple ideal gas at $\frac{0\mathrm{.}5}{b}ar$ and $30C,$ what would be the relative fluctuations of internal energy, $\frac{((\text{:}(E{)}^2\text{:}){)}^{\frac{1}{2}}}{(\text{:}E\text{:})},$ in a volume of $3$ microliters of such gas.