Determine the Gibbs Free Energy for an ideal gas and show that, this energy, and its derivative can be related to the enthalpy of the system via Gibbs-Helmholtz equation $$H = -T^2 \left( \frac{\partial (G/T)}{\partial T} \right)$$
So I need to determine the fundamental relation for an ideal gas in the Gibbs free energy representation.
My thoughts:
I know the Gibbs free energy, is defined by $$G(p,T) = U + pV - TS = H - TS,$$ where $H$ is the enthalpy, then \begin{align} \mathrm{d}(U + pV - TS) &= V\mathrm{d}p - S\,\mathrm{d}T + \sum{\mu_i\,\mathrm{d}N_i} - \sum{X_i\,\mathrm{d}a_i}\\ \mathrm{d}G &= V\,\mathrm{d}p - S\,\mathrm{d}T + \sum{\mu_i\,\mathrm{d}N_i - \sum{X_i\mathrm{d}a_i}}\\ \therefore \mathrm{d}G &= V\,\mathrm{d}p - S\,\mathrm{d}T \end{align} Now, if the temperature is constant, and only the pressure change then $\mathrm{d}G = V\,\mathrm{d}p$. It follows that $$G(p)-G(p_0)=\int_{p_0}^{p}{V\,\mathrm{d}p}$$ If $V = \frac{nRT}{p}$, we have $$G(p) = G(p_0) + nRT \ln\left(\frac{p}{p_0}\right)$$
But, I don't know if this is correct.