Question

Derive an expression for estimating the pressure at which graphite and diamond exist in equilibrium at \(25^{\circ} \mathrm{C}\) in terms of the specific volume, specific Gibbs function, and isothermal compressibility of each phase at \(25^{\circ} \mathrm{C}, 1 \mathrm{~atm}\). Discuss.

Short answer

At equilibrium, \( P = 1 \text{ atm} + \frac{ g_{\text{graphite,0}} - g_{\text{diamond,0}} }{ v_{\text{diamond}} - v_{\text{graphite}} } \).

Step-by-step solution

- Understand the Equilibrium Condition

For graphite and diamond to be in equilibrium at a given temperature, their Gibbs free energies must be equal. Therefore, at equilibrium, the specific Gibbs function of graphite \((g_{\text{graphite}})\) is equal to the specific Gibbs function of diamond \((g_{\text{diamond}})\).

- Write the Gibbs Free Energy Expression

The Gibbs free energy \((G)\) can be expressed as a function of pressure and volume: \[ g = g_0 + v(P - P_0) \] where \(g_0\) is the Gibbs free energy at the reference pressure \(P_0 = 1 \text{ atm}\), and \(v\) is the specific volume.

- Set Up Equilibrium Condition

At equilibrium, \( g_{\text{graphite}} = g_{\text{diamond}} \). Using the Gibbs free energy expressions for both phases, set: \[ g_{\text{graphite,0}} + v_{\text{graphite}}(P - 1 \text{ atm}) = g_{\text{diamond,0}} + v_{\text{diamond}}(P - 1 \text{ atm}) \]

- Solve for Pressure

Rearrange to solve for pressure \((P)\): \[ P = 1 \text{ atm} + \frac{ g_{\text{graphite,0}} - g_{\text{diamond,0}} }{ v_{\text{diamond}} - v_{\text{graphite}} } \]

Key concepts

specific Gibbs function

The specific Gibbs function, or specific Gibbs free energy, is an intensive property that represents the Gibbs free energy per unit mass or per mole. It's a measure of the chemical potential in a system for different phases such as graphite and diamond. At equilibrium, the specific Gibbs functions of the two phases must be equal. This ensures that no net change occurs in either phase over time. To understand this concept clearly, always focus on:

• This equality condition, which signifies equilibrium.
• The role of specific Gibbs functions in determining the phase behavior of materials.

The expression for the Gibbs free energy is given by:


\[ g = g_0 + v(P - P_0) \]
Here:

• \(g\) is the specific Gibbs function at a given pressure \(P\),
• \(g_0\) is the specific Gibbs function at the reference pressure (1 atm),
• \(v\) is the specific volume.

This formula helps in understanding how changes in pressure affect the specific Gibbs function and thereby the phase equilibrium.

isothermal compressibility

Isothermal compressibility, denoted as \(\kappa_T\), describes how the volume of a material changes with pressure at a constant temperature. It is mathematically expressed as:


\[ \kappa_T = -\frac{1}{v} \left( \frac{\partial v}{\partial P} \right)_{T} \]
Where:

• \(v\) is the specific volume,
• \(P\) is the pressure,
• The partial derivative indicates the rate of change of volume with respect to pressure at constant temperature \(T\).

This is important because it tells us how compressible a phase is under pressure when the temperature doesn't change. For the equilibrium of graphite and diamond, knowing their isothermal compressibilities helps us understand how their volumes and specific Gibbs functions adjust with pressure.


Role in Equilibrium: Different phases have different compressibilities, which can affect their specific volumes and therefore their Gibbs functions as pressure changes. This concept aids in solving equations related to phase equilibrium.

specific volume

Specific volume is an intensive property defined as the volume occupied by a unit mass of a substance. It is typically denoted by the letter \(v\). Specific volume is crucial in understanding the behavior of materials under different conditions, such as the pressure at equilibrium between graphite and diamond.


The reference equation involving specific volume is:

\[ v \]
Factors influencing specific volume include:

• Pressure — As pressure increases, specific volume generally decreases.
• Temperature — Specific volume can also change with temperature.

Related to Gibbs Function: Specific volume directly influences the Gibbs function as seen in the equation:

\[ g = g_0 + v(P - P_0) \]
Variations in specific volume impact equilibrium conditions. For example, the pressure at which graphite and diamond coexist is found by balancing their specific Gibbs functions, incorporating their specific volumes:

\[ P = 1 \text{ atm} + \frac{ g_{\text{graphite,0}} - g_{\text{diamond,0}} }{ v_{\text{diamond}} - v_{\text{graphite}} } \]
Understanding specific volume helps in accurately determining equilibrium pressure, ensuring the phases exist harmoniously at a given temperature.