General expression for entropy and free energy The specific heat capacity of a solid at constant pressure and low temperatures is given by $$c_p=aT+bT^3$$ with $a$ and $b$ are constants.

How can I calculate a general expression for the entropy $S(T,p)$ and Gibbs free energy $G(T,p)$ as a function of $T$, assuming that $S$ and $G$ are zero at $T = 0~\mathrm{K}$?

I started with trying to calculate entropy by
$$\mathrm{d}S = \left(\frac{\partial S}{\partial T}\right)_p \mathrm{d}T = \frac {c_p}{T} \mathrm{d}T$$
and
$$S = \int_0^T \frac{c_p}{T}\mathrm{d}T = (aT + bT^3)\ln T$$
Since G and S are taken to be zero at T = 0, H is also 0 at T = 0.  So you can calculate H by integrating the heat capacity from 0 to T:  $$H=\int_0^TC_pdT=\frac{a}{2}T^2+\frac{b}{4}T^4$$
Then you substitute H into the equation $$\frac{\partial (G/T)}{\partial T}=-\frac{H}{T^2}=-\frac{a}{2}-\frac{b}{4}T^2$$
and integrate between 0 and T.  This will give you G.  You can then solve for S from G = H - TS.  This gives you the same result as integrating directly to get S.