Heat Capacity temperature dependence and Gibbs energy

Question

I got this problem in school as a tool to get ready for International Chemistry Olympiad, but I have some problems with this challenge. So, the problem is to calculate as precise as possible the Gibbs free energy for $\ce{CaCO3}$ decomposition at 1200 K.

I know that $\mathrm{d}G=\mathrm{d}H-T\cdot\mathrm{d}S$.

But $\mathrm{d}S$ and $\mathrm{d}H$ are temperature dependant, so

$$\mathrm{d}H=\mathrm{d}H_0+\int_{T_0}^TC_p\mathrm{d}T=\mathrm{d}H_0+\mathrm{d}C_p\cdot\left(T-T_0\right)$$

and

$$\mathrm{d}S=\mathrm{d}S_0+\ln\left(T/T_0\right)\cdot\mathrm{d}C_p$$

But molar heat capacity at constant pressure is also temperature dependant, and the equation is

$C_p=a+b\cdot T+c/T^2$, where a, b, c coefficients for each compound at specific phase.

But the problem is with both solids, because this equation can not be used for them, because in 1200 K they are not gases and Einstein’s Quantum Mechanics and Debye models do not work, because Einstein’s approximates are maxed at $C_p\approx3R$, which is about 24.9, but $C_p$ at 298 K, 1 atm for both solids in tables are > 24.9.

Where is the problem? I would appreciate any possible lead to mistake/ correct answer.

Answer 1

At constant pressure, $dG=-SdT$. Also, the definition of G is $G=H-TS$, so $S=\frac{H-G}{T}$. Substituting, $$dG=\frac{G-H}{T}dT$$ This equation can be manipulated into the form: $$\frac{d(G/T)}{dT}=-\frac{H}{T^2}$$ So, $$\frac{d(\Delta G/T)}{dT}=-\frac{\Delta H}{T^2}$$ I leave the rest up to you.

A math error was corrected on 10/16/15

Answer 2

The answer given by Chester Miller is completely correct but I think it just causes old problems on new places and does not solve the problem. In the last (van't Hoff) equation an enthalpy is assumed that is independent from temperature. In general it is: $$ \frac{\partial \Delta G/T} {\partial T} = \frac{\Delta H (T)}{T} $$ This leads to: $$ \left. \frac{\partial \ln K}{\partial T} \right|_p = \frac{ \Delta H^0(T)}{RT^2} $$ Also in this case you end up searching for the temperature dependence of the enthalpy hence the heat capacity.

I would just recommend to get experimental data for the polynomials, perhaps CRC handbook. I do not know any theoretical method better than Debye modell that you could still apply just with calculator and pen.