Pressure at which graphite and diamond are in equilibrium

Question

The standard state Gibbs free energies of formation of graphite and diamond at $T = \pu{298 K}$ are $\pu{0 kJ mol-1}$ and $\pu{2.9 kJ mol-1}$, respectively.

The conversion of graphite to diamond reduces its volume by $\pu{2e-6 m3 mol-1}$.

If graphite is converted to diamond isothermally at $T = \pu{298 K}$, the pressure at which graphite is in equilibrium with diamond, is

(A) $\pu{14501 bar}$
(B) $\pu{58001 bar}$
(C) $\pu{1450 bar}$
(D) $\pu{29001 bar}$

I applied

$$\Delta G_{(p,T)} =\Delta_\mathrm{f}G^\circ + \int_{p_1}^{p_2}V\,\mathrm dp,$$

and since the system is at equilibrium,

$$\Delta_\mathrm{f}G^\circ = -\int_{p_1}^{p_2}V\,\mathrm dp.$$

Now I am stuck. I have not been given any relation between pressure and volume. Is there any assumption I have to make to solve this integral?

Answer 1

For each phase $i$ (graphite or diamond) you can show that

$$\mathrm{d}\mu _i = V_i \mathrm{d}p - S_i \mathrm{d}T$$

or after integration

$$ \mu_i = \mu_i^\circ+\int_{p^\circ}^{p} V_{i} dp $$ at constant T (note that each phase consists of a pure substance and $V_{i}$ refers to the molar volume of phase $i$; below the subscript $m$ is used instead to refer to the molar volume of a pure phase).

We are asked to find the pressure $p=p_{eq}$ at which carbon coexists in the two phases (the Gibbs free energy is equal in both) so that

$$ \mu(diamond) = \mu(graphite)$$

This leads to

$$\Delta \mu^{\circ}=-\int_{p^\circ}^{p_{eq}}\Delta V_mdP$$

and, since $\Delta \mu^{\circ}= \Delta_f G_m^{\circ}$, ultimately to the expression you provided:

$$\Delta_f G_m^{\circ}=-\int_{p^\circ}^{p_{eq}}\Delta V_mdP$$

The approximation you are allowed to make at this point is that the solids are incompressible, such that their $V_m$ are constant with change of pressure. It follows that

$$\Delta_f G_m^{\circ}=-\Delta V_m\int_{p^\circ}^{p_{eq}}dp=-\Delta V_m(p_{eq}-p^\circ)$$

which can be solved for $p_{eq}$:

$$p_{eq}=-\frac{\Delta_f G_m^{\circ}}{\Delta V_m}+p^\circ$$