91Ó°ÊÓ

To understand why graphite and diamond can exist in equilibrium, we need to understand Gibbs free energy. Gibbs free energy (\textbf{G}) is a measure of the maximum reversible work that can be done by a thermodynamic system at constant temperature and pressure. It plays a key role in predicting the direction of chemical processes and phase changes. When two phases, like graphite and diamond, are in equilibrium, their Gibbs free energies are equal. This is represented mathematically as: \[ \text{dG} = \text{G}_{\text{diamond}} - \text{G}_{\text{graphite}} = 0 \]
This implies the chemical potential (\textbf{µ}) of graphite is equal to that of diamond, meaning neither phase is energetically favored over the other under the given conditions. Understanding this equilibrium is crucial in deriving the pressure expression for the coexistence of graphite and diamond.

Isothermal compressibility (\textbf{β}) is a measure of how much a material's volume changes under pressure at constant temperature. Mathematically, it is given by: \[ \beta = - \frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T \]
At constant temperature, \textbf{β} helps us understand how a substance reacts to changes in pressure. For instance, for small changes in pressure, the change in volume (\textbf{\textDelta V}) can be approximated as: \[ \Delta V = -V \times \beta \times (P - P_0) \]
This relationship is essential in deriving pressure conditions for equilibrium. By substituting the compressibility into the volume expression, we can understand how the volumes of graphite and diamond change with pressure and thus find the equilibrium condition.

Specific volume (\textbf{V}) is the volume occupied by a unit mass of a substance. It's essentially the reciprocal of density. In the context of our problem, it's crucial because the difference in specific volumes of graphite and diamond affects their Gibbs free energy under different pressures. When deriving the pressure at which they coexist in equilibrium, specific volumes appear in the following relationship: \[ G_{\text{diamond}} - G_{\text{graphite}} = V_{\text{diamond}} \times (P - P_0) - V_{\text{graphite}} \times (P - P_0) \]
Also, using isothermal compressibility, the equation further evolves. Hence, specific volume helps us understand how different phases will behave under pressure differences, crucial for predicting equilibrium states. Ultimately, this leads us to derive the pressure condition for equilibrium: \[ P = P_0 + \frac{G_{\text{diamond}} - G_{\text{graphite}}}{V_{\text{graphite}} - V_{\text{diamond}}} \] Understanding specific volume and its implications is key to solving such thermodynamic problems.