Pressure at which graphite and diamond are in equilibrium


> 
> 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?




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 $p\_{eq}$:


$$p\_{eq}=-\frac{\Delta\_f G\_m^{\circ}}{\Delta V\_m}+p^\circ$$


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