Question

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.

Short answer

In the given states, the liquid state of benzoic acid has a lower density (1.08 g/cm³) than the solid state (1.266 g/cm³) which means the average distance between molecules is greater in the liquid state. The difference in densities can be explained by the difference in kinetic energies of the molecules. In the liquid state at a higher temperature (130°C or 403 K), the molecules have more kinetic energy than in the solid state at a lower temperature (15°C or 288 K). This causes the molecules in the liquid state to be more spread apart, resulting in a lower density compared to the solid state.

Step-by-step solution

Compare densities

First, we need to compare the densities in the given states. At ${15}^{\circ }C$, solid benzoic acid has a density of 1.266 g/cm³, and at ${130}^{\circ }C$, liquid benzoic acid has a density of 1.08 g/cm³.

Determine the state with a greater average distance between molecules

Since the density of liquid benzoic acid at ${130}^{\circ }C$ (1.08 g/cm³) is less than the density of solid benzoic acid at ${15}^{\circ }C$ (1.266 g/cm³), the average distance between molecules in the liquid state is greater. Answer for part (a): The average distance between molecules is greater in the liquid state.

Explain the difference in densities in terms of the relative kinetic energies of the molecules

In terms of kinetic energy, we know that when the temperature increases, the average kinetic energy of the molecules increases as well. This is represented by the equation $KE=\frac{3}{2}kT$, where KE is the average kinetic energy, k is the Boltzmann constant, and T is the temperature in kelvins. At higher temperatures, such as ${130}^{\circ }C$ (403 K) for liquid benzoic acid, the molecules have more kinetic energy than in the solid state at lower temperatures, such as ${15}^{\circ }C$ (288 K). As a result, the molecules in the liquid state move faster and are more spread apart, creating a greater average distance between individual molecules. This is reflected in the lower density of liquid benzoic acid compared to its solid state. Answer for part (b): The difference in densities at the two temperatures can be explained by the differences in the relative kinetic energies of the molecules. The higher kinetic energy in the liquid state due to the higher temperature causes the molecules to be more spread apart, resulting in a lower density compared to the solid state at a lower temperature.

Key concepts

Density Comparison

Density is an essential property of a substance since it tells us how compact a substance's molecules are. When comparing solid and liquid benzoic acid, the exercise tells us that the density of solid benzoic acid at ${15}^{\circ }\text{C}$ is $1.266{\text{ g/cm}}^3$, while the liquid benzoic acid at ${130}^{\circ }\text{C}$ has a lower density of $1.08{\text{ g/cm}}^3$. This means that the molecules in the solid state are packed more tightly together compared to the liquid state.### Greater Average Distance Between Molecules- Since the density of the liquid is lower, it indicates that the average distance between molecules is greater in the liquid state.- In simple terms, imagine having a box filled with tightly packed balls representing the molecules. - If you remove some balls, the remaining ones have more space to spread out—this is analogous to the lower density.

Molecular Kinetic Energy

Kinetic energy is the energy of motion. The molecules in a substance have kinetic energy because they are constantly moving.The exercise illustrates that kinetic energy correlates with temperature. As temperature increases, the kinetic energy of molecules also increases. This can be described mathematically by the equation: $$KE=\frac{3}{2}kT$$Where:

• $KE$ is the average kinetic energy.
• $k$ is the Boltzmann constant, a physical constant that relates temperature to energy.
• $T$ is the temperature (in Kelvins).

### Implications of Increased Kinetic Energy- At ${15}^{\circ }\text{C}$, the molecules have less kinetic energy, which means they vibrate but generally stay close to their neighboring molecules.- At ${130}^{\circ }\text{C}$, or in other words, at higher temperature levels, the increase in kinetic energy leads to increased movement.- This movement causes molecules to separate more, which affects the substance's state and helps explain density differences.

Temperature Effects on Density

Temperature has a profound effect on the density of substances like benzoic acid.### How Temperature Affects Density- When you heat a substance like benzoic acid, you are adding energy to it. This addition of energy increases the molecular movement.- As molecules move more due to increased kinetic energy, they spread out, making the substance less dense—this is why liquid benzoic acid at ${130}^{\circ }\text{C}$ has a lower density than solid benzoic acid at ${15}^{\circ }\text{C}$.### Summary- Solid benzoic acid with less temperature has molecules that are compact, leading to higher density.- On the other hand, heating leads to higher kinetic energy, spread-out molecules, and thus a lower density.Understanding how temperature affects density is crucial in comprehending many physical phenomena and is particularly essential in chemistry and materials science.