Energy and Entropy 2024
You are given the following Gibbs free energy:
\begin{equation*}
G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right),
\end{equation*}
where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).

Compute the entropy.

Work out the heat capacity at constant pressure \(C_p\).

Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy). Simplify the final expression as much as possible.

Find the internal energy \(U\) from the expression for \(G\) that you were given in the main prompt. Simplify the final expression as much as possible.