Benzoic acid, \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH},\) melts at \(122{ }^{\circ} \mathrm{C}\). The density in the liquid state at \(130^{\circ} \mathrm{C}\) is \(1.08 \mathrm{~g} / \mathrm{cm}^{3}\). The density of solid benzoic acid at \(15^{\circ} \mathrm{C}\) is \(1.266 \mathrm{~g} / \mathrm{cm}^{3}\). (a) In which of these two states is the average distance between molecules greater? (b) Explain the difference in densities at the two temperatures in terms of the relative kinetic energies of the molecules.

In general, density can be defined as the mass of a substance divided by its volume. In this particular case, a higher density means that the molecules are more closely packed together, whereas a lower density means that they are farther apart. Therefore, we can determine the average distance between molecules based on the given densities of solid and liquid benzoic acid.

We are given the following density values: - Solid benzoic acid at \(15^{\circ}C\): \(1.266 \frac{g}{cm^3}\) - Liquid benzoic acid at \(130^{\circ}C\): \(1.08 \frac{g}{cm^3}\) Comparing these two densities, we can see that the density of solid benzoic acid is greater than the density of liquid benzoic acid at their respective temperatures.

Since the density of solid benzoic acid is greater than that of liquid benzoic acid, this means that the molecules in the solid state are more closely packed together. Therefore, the average distance between molecules is greater in the liquid state. So, the answer to part (a) is that the average distance between molecules is greater in the liquid state.

As temperature increases, the kinetic energy of a substance's molecules also increases. In general, when a substance is in its solid state, its molecules have lower kinetic energy than in its liquid state. As a result, the molecules in the solid state can lock into a more organized and tightly packed structure, thus having a higher density.

In the case of benzoic acid, when the temperature is increased from \(15^{\circ}C\) (solid state) to \(130^{\circ}C\) (liquid state), the kinetic energy of the molecules also increases. This added energy allows the molecules to move more freely, allowing the substance to transition to its liquid state. In the liquid state, the molecular structure becomes less organized, and the molecules can be farther apart, hence the lower density compared to solid benzoic acid. So, the answer to part (b) is that the difference in densities between the solid and liquid states can be explained by the higher kinetic energy of the molecules in the liquid state, which causes them to be farther apart and exhibit a lower density.