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

The average distance between molecules is greater in the liquid state at \(130^{\circ}\mathrm{C}\) due to its lower density. The difference in densities at the two temperatures is due to the difference in the kinetic energies of the molecules. Higher kinetic energy at higher temperature causes the molecules to move more vigorously, increasing the distance between the molecules and decreasing the density.

Step-by-step solution

(Step 1: Analyzing the densities)

We are given the density of benzoic acid in the liquid state at \(130^{\circ}\mathrm{C}\) as \(1.08 \mathrm{~g} / \mathrm{cm}^{3}\) and in the solid state at \(15^{\circ}\mathrm{C}\) as \(1.266 \mathrm{~g} / \mathrm{cm}^{3}\).

(Step 2: Comparing the densities)

Since density is inversely proportional to the volume, a substance with a higher density has a smaller volume. This means the molecules will be closer together in a substance with a higher density and further apart in a substance with a lower density. In our case, the liquid state has a lower density, and the solid state has a higher density. Therefore, the average distance between molecules is greater in the liquid state. So the answer to part (a) is the liquid state at \(130^{\circ}\mathrm{C}\).

(Step 3: Analyzing kinetic energies)

The kinetic energies of molecules depend on the temperature. As the temperature of a substance increases, the kinetic energy of its molecules increases. In the problem, we need to compare the densities at two different temperatures: \(130^{\circ}\mathrm{C}\) (liquid state) and \(15^{\circ}\mathrm{C}\) (solid state).

(Step 4: Explaining the difference in densities)

At the higher temperature (\(130^{\circ}\mathrm{C}\)), the kinetic energy of the benzoic acid molecules increases. This increased kinetic energy causes the molecules to move more vigorously, which increases the distance between them. As a result, the density of benzoic acid in the liquid state at \(130^{\circ}\mathrm{C}\) is lower than in the solid state at \(15^{\circ}\mathrm{C}\). The strength of the intermolecular forces in the solid state keeps the molecules in a more compact arrangement, with a higher overall density. The answer to part (b) is that the difference in densities at the two temperatures is due to the difference in the kinetic energies of the molecules. Higher kinetic energy at higher temperature causes the molecules to move more vigorously, increasing the distance between the molecules and decreasing the density.

Key concepts

Molecular Kinetic Energy

Molecular Kinetic Energy is a key concept to understand the behavior of substances like benzoic acid at different temperatures. At a fundamental level, molecular kinetic energy is the energy that molecules possess due to their motion. When the temperature of a substance rises, so does the kinetic energy of its molecules. This increase in kinetic energy results in molecules moving more rapidly and with greater intensity.

For instance, when benzoic acid is in its liquid state at a higher temperature of 130°C, its molecules have higher kinetic energy compared to when it is in a solid state at 15°C. This high kinetic energy means the molecules are moving faster and are spread further apart, causing the substance to have a lower density. Thus, temperature and molecular kinetic energy are directly related, and this relationship influences how molecules are spaced in different phases of matter.

Intermolecular Forces

Intermolecular forces are the forces that hold molecules together in a substance. In solids, these forces are stronger, pulling molecules tightly together in a compact, structured form, leading to higher density. These forces include hydrogen bonds, dipole-dipole forces, and London dispersion forces.

In the case of solid benzoic acid at 15°C, the strong intermolecular forces cause the molecules to be closely packed, making the substance denser. At higher temperatures, like 130°C for liquid benzoic acid, the molecules have increased kinetic energy, which partially overcomes the intermolecular forces and allows the molecules to move more freely. Consequently, the density is lower because the molecules are further apart.

• Strong intermolecular forces in solids: high density
• Weakened intermolecular forces with increased kinetic energy in liquids: lower density

Understanding these forces helps explain why substances change density with temperature changes.

Density and Temperature

The relationship between density and temperature is an essential concept for explaining the physical properties of substances like benzoic acid. Density is defined as mass per unit volume and changes with temperature due to the movement and arrangement of molecules within a substance.

As temperature increases, typically, density decreases. This is because the molecules absorb more energy (increasing their kinetic energy) and start moving more vigorously, causing the space between them to increase, leading to reduced density. For benzoic acid, its density is higher in the solid state at 15°C compared to its liquid state at 130°C due to the reduced kinetic energy and more compact molecular arrangement at lower temperatures.

• Higher temperature: higher kinetic energy, lower density
• Lower temperature: lower kinetic energy, higher density

This concept is crucial in understanding how substances behave when subjected to temperature changes and is widely applicable in various scientific fields.