Construct the phase diagram for benzene near its triple point at 36 Torr and \(5.50^{\circ} \mathrm{C}\) using the following data: \(\Delta_{\text {fus }} H=10.6 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta_{\mathrm{vap}} H=30.8 \mathrm{~kJ}\) \(\mathrm{mol}^{-1}, \rho(\mathrm{s})=0.891 \mathrm{~g} \mathrm{~cm}^{-3}, \rho(\mathrm{l})=0.879 \mathrm{~g} \mathrm{~cm}^{-3}\)

Understanding phase transitions is fundamental in material science, and the Clapeyron equation provides a window into this world. This simple yet powerful formula offers a way to relate the pressure and temperature changes during a phase transition.

The Clapeyron equation is expressed as: \( \frac{\Delta P}{\Delta T} = \frac{\Delta H}{T \Delta V} \), where \(\Delta P\) and \(\Delta T\) represent the changes in pressure and temperature, respectively. \(\Delta H\) is the latent heat involved in the transition, which could be either the enthalpy of fusion or vaporization, and \(\Delta V\) is the change in molar volume.

This fundamental equation enables students to calculate the slope of the phase boundaries on a phase diagram, providing insight into how different phases, such as solid, liquid, and gas, coexist at various temperatures and pressures. By plotting these slopes, one can visually interpret the conditions under which a substance will transition from one phase to another.

During the phase change from solid to liquid, a substance absorbs energy without changing temperature. The enthalpy of fusion, \(\Delta_{\text{fus}} H\), quantifies this energy required to overcome the forces holding the solid structure together.

The value of \(\Delta_{\text{fus}} H\) for benzene in the exercise is 10.6 kJ/mol. This number is crucial in calculating the slope of the solid-liquid coexistence line in a phase diagram using the Clapeyron equation. A higher \(\Delta_{\text{fus}} H\) would mean a substance requires more energy to melt, indicating stronger intermolecular forces within the solid.

For students working on phase diagrams, it's essential to understand that the enthalpy of fusion is a key variable that determines how a substance behaves as it transitions from solid to liquid phase.

The energy absorbed when a liquid turns into a gas is captured by the enthalpy of vaporization, \(\Delta_{\text{vap}} H\). It's a direct measure of the energy needed to break the intermolecular forces that bind the liquid phase.

In our exercise, benzene's \(\Delta_{\text{vap}} H\) is given as 30.8 kJ/mol, indicating it takes more energy to vaporize benzene than it does to melt it. This higher value is also indicative of the larger energy change involved in the gas phase transition. This concept is equally important when using the Clapeyron equation to find the slope of the liquid-vapor coexistence line in the phase diagram, serving as a critical factor in predicting the behavior of benzene under various temperature and pressure conditions.

Molar volume describes the volume occupied by one mole of a substance and changes with its phase––solid, liquid, or gas. It's a key part of applying the Clapeyron equation because phase changes are often accompanied by changes in volume.

The calculation of the molar volumes of solid and liquid benzene were demonstrated in the steps provided. Solid benzene has a defined and typically lower molar volume than liquid benzene due to the closer packing of molecules. The difference in molar volumes, \(\Delta V\), is what drives the process described by the Clapeyron equation.

Appreciating the significance of molar volume can help students grasp why different phases exist under different conditions and the role it plays in the construction of phase diagrams. Learning to calculate this value from density and molar mass is a fundamental skill in the heart of many exercises involving phase changes.