Question

Use the following data to determine the normal boiling point, in kelvins, of mercury. What assumptions must you make in order to do the calculation? $$ \begin{aligned} \mathrm{Hg}(l): & \Delta H_{\mathrm{f}}^{\circ} &=0 \text { (by definition) } \\\ & S^{\circ} &=77.4 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol} \\ \mathrm{Hg}(g): & \Delta H_{\mathrm{f}}^{\circ} &=60.78 \mathrm{~kJ} / \mathrm{mol} \\ & S^{\circ} &=174.7 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol} \end{aligned} $$

Short answer

The normal boiling point of mercury is 625 K.

Step-by-step solution

Understand Required Assumptions

In order to carry out this calculation, the primary assumption is that the boiling process is at normal pressure, which is 1 atmosphere. The second assumption is that the boiling process is in equilibrium, meaning that \(\Delta G^\circ\) is equal to zero.

Apply Gibbs Free Energy Equation for Phase Transition

We know that at equilibrium, \(\Delta G^\circ = 0\) (Gibbs free energy change). So, from \(\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ\), We get that \(T = \(\Delta H^\circ/\Delta S^\circ\). Here, \(\Delta H^\circ\) represents the change in enthalpy, which is the heat of vaporization, \(\Delta H_{vap}^\circ\), and \(\Delta S^\circ\) is the change in entropy due to vaporization, \(\Delta S_{vap}^\circ\). Note that these are standard conditions values, indicated by superscript \(\circ\).

Calculate Change in Enthalpy and Entropy

\(\Delta H_{vap}^\circ\) is given by \(\Delta H_{f(g)}^\circ - \Delta H_{f(l)}^\circ\) which is \(60.78 kJ/mol - 0 = 60.78 kJ/mol\). Similarly, \(\Delta S_{vap}^\circ\) is given by \(\Delta S_{f(g)}^\circ - \Delta S_{f(l)}^\circ\) which is \(174.7 J/K \cdot mol - 77.4 J/K \cdot mol = 97.3 J/K \cdot mol\).

Calculate Normal Boiling Point

Finally, we substitute the calculated values into the equation obtained in Step 2 to find the boiling temperature \(T\). Thus, \(T = \Delta H_{vap}^\circ/\Delta S_{vap}^\circ = 60.78 kJ/mol / 97.3 J/K \cdot mol = 625 K\). Remember to change the unit of enthalpy to J/mol for consistency before doing the calculation.