What is the melting point of benzene if \(\Delta H_{\text {fusion }}=9.95 \mathrm{~kJ} / \mathrm{mol}\) and \(\Delta S_{\text {fusion }}=35.7 \mathrm{~J} / \mathrm{K}\) -mol? (a) \(278.7^{\circ} \mathrm{C}\) (b) \(278.7 \mathrm{~K}\) (c) \(300 \mathrm{~K}\) (d) \(298 \mathrm{~K}\)

When we talk about phase transition, it's like discussing the different 'personalities' that a substance can exhibit. In the context of benzene, and substances in general, a phase transition is the transformation from one state of matter to another, such as solid to liquid or liquid to gas. Each transition comes with specific conditions like temperature and pressure, where the substance decides it’s time for a change.

For example, the melting point of a solid is the temperature at which it can no longer hold its rigid structure and begins to flow as a liquid. This transformation requires energy, usually in the form of heat, to overcome the forces keeping the molecules in a solid state. Just like people sometimes need a little encouragement to step out of their comfort zones, molecules need an energy push to move into a new phase of matter.

Think of Gibbs free energy as a scientific indicator for a substance's willingness to undergo a phase change or a chemical reaction. It's like a currency for processes, telling us whether or not a change is 'affordable' for a substance under certain conditions. At equilibrium, when the substance is contemplating between two states – say, should it stay a solid or melt into a liquid? – the Gibbs free energy change, represented by \( \Delta G \), is zero.

Why zero? Well, at this 'no-profit-no-loss' point, the substance is perfectly content with either phase, neither gaining nor losing energy. This balance is the golden ticket for calculating the melting point, as shown in the benzene problem: equate the change in Gibbs free energy to zero, and voilà, you've got your melting point, given that you know the change in enthalpy (\(\Delta H\)) and change in entropy (\(\Delta S\)) for the process.

Imagine a treasure chest filled with energy – that's enthalpy for you. It’s essentially the total heat content of a system and an essential player in understanding how phase transitions happen. For a substance to break free from its phase, such as solid benzene turning to liquid, it must either absorb or release energy – this energy change is what we call the change in enthalpy (\(\Delta H\)).

In our benzene example during melting, energy is absorbed to weaken the forces that keep molecules rigidly in place. The amount of energy absorbed, measured in kilojoules per mole (kJ/mol), is critical for determining how much 'heat' benzene needs to transition into a liquid. Heating up the substance increases the energy in the system, and when it hits the right amount, reflected by \(\Delta H\), phase transition occurs.

Entropy is a bit like the randomness or unpredictability in a teenager's room. It's a measure of the disorder or randomness in a system. High entropy means high chaos – molecules are free to move and there’s less rigidity, which is characteristic of gases and liquids. Low entropy, on the other side, corresponds to more order and less freedom, like in a solid.

During a phase transition, such as melting, the entropy (\(\Delta S\)) of a system will increase because the molecules in a liquid can move around much more freely than in a solid. The change in entropy helps scientists calculate the melting point of substances since it's a measure of how much the 'messiness' or freedom of motion for the molecules increases when transitioning from a solid crystal structure to the freer liquid state.