Draw a heating curve (such as the one in Figure 12.36 ) for \(1 \mathrm{~mol}\) of benzene beginning at \(0^{\circ} \mathrm{C}\) and ending at \(100^{\circ} \mathrm{C}\). Assume that the values given here are constant over the relevant tempera- ture ranges. $$ \begin{array}{ll} \hline \text { Melting point } & 5.4^{\circ} \mathrm{C} \\ \text { Boiling point } & 80.1^{\circ} \mathrm{C} \\ \Delta \mathrm{H}_{\text {fus }} & 9.9 \mathrm{~kJ} / \mathrm{mol} \\ \Delta H_{\text {vap }} & 30.7 \mathrm{~kJ} / \mathrm{mol} \\ C_{\text {s, solid }} & 118 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} \\ \mathrm{C}_{\text {s, liquid }} & 135 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} \\ \mathrm{C}_{\mathrm{s}, \text { gas }} & 104 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} \\ \hline \end{array} $$

Enthalpy of fusion, often symbolized as \( \Delta H_{fus} \), is a thermodynamic property that represents the heat required to change one mole of a substance from the solid phase to the liquid phase at constant pressure and temperature. This process is commonly known as melting. For instance, in our chemistry exercise, the enthalpy of fusion for benzene is given as \( 9.9 \text{ kJ/mol} \). During this phase change, the substance does not change temperature despite the addition of heat, which is represented as a horizontal section on a heating curve. This reflects the energy being used to break the intermolecular forces that hold the solid structure together rather than increasing the kinetic energy, which corresponds to temperature increase.

Understanding the concept of enthalpy of fusion is crucial as it helps in calculating the energy needed for the melting process. Knowing this allows for more efficient energy use in industrial processes involving phase changes, such as in the melting of metals for casting or the manufacturing of materials like plastics.

Equally important in chemistry is the concept of enthalpy of vaporization, \( \Delta H_{vap} \), which denotes the amount of energy required to convert one mole of a liquid into gas at constant pressure. In the exercise, benzene's enthalpy of vaporization is \( 30.7 \text{ kJ/mol} \). During the transition from liquid to vapor, known as boiling, there is no temperature increase as heat is added, which is represented as another horizontal line on the heating curve. This portion of the curve indicates that the added heat is overcoming the intermolecular attractions within the liquid, resulting in a phase change to gas.

The enthalpy of vaporization is a critical factor in various industries, including the distillation of spirits, the production of fuels, and the chemical separation processes. This value helps to ensure that processes are carried out efficiently, minimizing energy consumption and costs.

A phase change refers to the physical transformation of a substance from one state of matter (solid, liquid, or gas) to another. In a heating curve, these changes happen at specific temperatures where the substance's temperature remains constant, despite heat being continuously added. For benzene, there exist two important phase changes: melting at \( 5.4^\circ \text{C} \) and boiling at \( 80.1^\circ \text{C} \). The flat sections of the heating curve during these temperatures signify the phase changes.

Understanding phase changes is essential for many scientific and industrial applications, including material synthesis, food preservation, and climate control systems. Recognizing these key transition points aids in precisely controlling the conditions to achieve desired outcomes in both laboratory and industrial settings.

Finally, specific heat capacity, often represented by \( C_s \), is the amount of heat needed to raise the temperature of one mole of a substance by one degree Kelvin or Celsius. This property is pervasive in heating curves, dictating the slope of the lines illustrating the temperature increases within a given phase. In our example, the specific heat capacities for benzene as a solid, liquid, and gas are given as \( 118 \text{ J/mol} \cdot \text{K} \), \( 135 \text{ J/mol} \cdot \text{K} \), and \( 104 \text{ J/mol} \cdot \text{K} \) respectively. Higher specific heat capacities mean that more heat is required to increase the temperature, resulting in a gentler slope.

Knowing the specific heat capacities is not only crucial for our heating curve exercise but also for many practical situations, such as designing thermal systems, managing heat in cooking, and even in climate science to understand the heat retention of oceans versus landmasses.