Problem $3.$ Let $G$ represent carbon in the form of graphite and $D$ represent the diamond crystal form. At $298\mathrm{.}15$ K$,$ the thermodynamic equilibrium constant for $GD,$ based on a standard pressure $p_0=1$ bar, has the value $K=0\mathrm{.}31.$ The molar volumes of the two crystal forms at this temperature are $V_m(G)=5\mathrm{.}3{10}^{-6}{m}^{3}mo{l}^{-1}$ and $V_m(D)=3\mathrm{.}4{10}^{-6}{m}^{3}mo{l}^{-1}.$

$($a$)$ Write an expression for the equilibrium constant $K$ as a function of pressure. Hint: Recall the equations for fugacity of a pure component in a condensed phase $($liquid or solid$)$ in Chapter $10.$

$($b$)$ Use the value of $K$ to estimate the pressure at which the $D$ and $G$ crystal forms are in equilibrium with one another at $298\mathrm{.}15K.($This is the lowest pressure at which graphite could in principle be converted to diamond at this temperature.$)$