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? $$ \begin{aligned} \mathrm{Hg}(l): & \Delta H_{\mathrm{f}}^{\circ} &=0 \text { (by definition) } \\\ & S^{\circ} &=77.4 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol} \\ \mathrm{Hg}(g): & \Delta H_{\mathrm{f}}^{\circ} &=60.78 \mathrm{~kJ} / \mathrm{mol} \\ & S^{\circ} &=174.7 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol} \end{aligned} $$

Gibbs Free Energy, represented as \text{\(G\)}, is a thermodynamic property that indicates the amount of reversible work obtainable from a chemical reaction at constant temperature and pressure. When a substance is at its boiling point, the Gibbs Free Energy for the phase change from liquid to gas is zero. This is because at equilibrium, which occurs at the boiling point, there is no net change occurring in the system. The relationship between Gibbs Free Energy \text{\(\Delta G\)}, enthalpy \text{\(\Delta H\)}, and entropy \text{\(S\)} is defined by the equation \text{\(\Delta G = \Delta H - T\Delta S\)}, where \text{\(T\)} is the absolute temperature in kelvins.

This relationship is pivotal in determining the normal boiling point of a substance, as it allows us to set \text{\(\Delta G\)} to zero and solve for \text{\(T\)}, when both \text{\(\Delta H\)} and \text{\(\Delta S\)} are known. In the exercise, by setting \text{\(\Delta G\)} to zero, we can isolate and calculate the temperature, which corresponds to the normal boiling point of mercury.

Enthalpy, symbolized as \text{\(\Delta H\)}, is a measure of the total heat content in a thermodynamic system. It is associated with chemical and physical changes--such as phase transitions, like boiling. The enthalpy change of a reaction or a phase change is a crucial component of chemical thermodynamics and signifies the amount of heat absorbed or released by the system at constant pressure.

When a substance transforms from a liquid to a gaseous state, as in boiling, the enthalpy increases because energy is absorbed to overcome the intermolecular forces in the liquid. For the calculation of the normal boiling point, we consider the enthalpy change that occurs when mercury transitions from its liquid form to its gaseous form. This is done by subtracting the enthalpy of liquid mercury from that of gaseous mercury, representing the heat required for the phase transition.

Entropy, denoted as \text{\(S\)}, is another central concept in chemical thermodynamics that measures the disorder or randomness of the particles in a thermodynamic system. During a phase change, such as boiling, the entropy typically increases because the gaseous state is more disordered than the liquid state.

In the provided exercise, the entropy values for both liquid and gaseous mercury are given. The change in entropy \text{\(\Delta S\)} is calculated by subtracting the entropy of liquid mercury from that of gaseous mercury. This change in entropy captures the increase in disorder as mercury transitions from liquid to gas at its boiling point. This change is central to calculating the normal boiling point using the Gibbs Free Energy equation, as it is part of the formula that sets the stage for determining the temperature at which mercury boils under normal conditions.

Chemical thermodynamics is the branch of science that deals with the study of energy changes accompanying chemical reactions and physical changes, such as phase transitions. It involves concepts like Gibbs Free Energy, enthalpy, and entropy to predict the spontaneous direction of a process and determine equilibrium conditions, such as a substance's normal boiling point.

The principles of chemical thermodynamics are exemplified when calculating the normal boiling point of mercury. This process requires an understanding of the energy balance during the phase transition. By applying the data on enthalpy and entropy changes, and recognizing that Gibbs Free Energy will be zero at the boiling point, we obtain crucial insights into the temperature that defines the boiling point of the substance. Students must assume that the given values remain constant up to the boiling point and that the transition happens under equilibrium conditions to successfully determine the boiling point based on these thermodynamic concepts.