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 since the density of liquid benzoic acid (\(1.08 \mathrm{~g} / \mathrm{cm}^{3}\)) is lesser than that of solid benzoic acid (\(1.266 \mathrm{~g} / \mathrm{cm}^{3}\)). The difference in densities is due to the difference in the kinetic energies of the molecules in the solid and liquid states - at higher temperatures, molecules have more kinetic energy, resulting in them being further apart and thus a lower density in the liquid state.

Step-by-step solution

We have been given the density of benzoic acid in both the solid and liquid states. The density is defined as mass per unit volume, or: \(Density = \frac{Mass}{Volume}\) We can rearrange this formula to find the volume, which will give us an idea of the average distance between the molecules: \(Volume = \frac{Mass}{Density}\) Notice that as the volume increases, so does the average distance between molecules. So, we just need to compare the densities to determine in which state the average distance is greater. The density of liquid benzoic acid at \(130^{\circ}\mathrm{C}\) is \(1.08 \mathrm{~g} / \mathrm{cm}^{3}\), and the density of solid benzoic acid at \(15^{\circ}\mathrm{C}\) is \(1.266 \mathrm{~g} / \mathrm{cm}^{3}\). Since the density of liquid benzoic acid is lesser than that of solid benzoic acid, the average distance between molecules is greater in the liquid state. #b) Explain the difference in densities in terms of molecular kinetic energies#

The kinetic energy of molecules is directly proportional to the temperature of the substance. As the temperature of a substance increases, the kinetic energy of its molecules also increases. In the solid state, the molecules of benzoic acid are closely packed, and they vibrate around fixed positions. In the liquid state, molecules have more kinetic energy, which allows them to move more freely, causing them to be further apart from each other. The difference in densities between the solid and liquid states of benzoic acid is due to the difference in the kinetic energies of the molecules in those states: - In the solid state at \(15^{\circ}\mathrm{C}\), the molecules have lower kinetic energy, and they are closely packed, resulting in a higher density (\(1.266 \mathrm{~g} / \mathrm{cm}^{3}\)). - In the liquid state at \(130^{\circ}\mathrm{C}\), the molecules have higher kinetic energy, they are moving more freely and are further apart from each other, resulting in a lower density (\(1.08 \mathrm{~g} / \mathrm{cm}^{3}\)).

Key concepts

Density

Density is a measure of how tightly matter is packed together. It is defined as mass per unit volume and mathematically represented as: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]When we talk about the density of a substance, we consider how much mass fits into a given space. A greater density indicates that the particles are closer to each other, which generally occurs in the solid state, as seen with benzoic acid at 15°C with a density of 1.266 g/cm³. In contrast, a lower density, like that of liquid benzoic acid at 130°C (1.08 g/cm³), means that molecules are more spread out.

For benzoic acid, the solid state has a higher density due to tightly packed molecules, whereas the liquid state has a lower density indicating that molecules have more space and are further apart. Comparing densities allows us to infer the relative distances and arrangements of molecules within different states of matter.

Molecular Kinetic Energy

The kinetic energy of a molecule is the energy it possesses due to its motion. In the context of states of matter, molecular kinetic energy is tightly linked with temperature. Higher temperatures give molecules more energy to move, increasing their kinetic energy.

In a solid like benzoic acid at 15°C, molecules have lower kinetic energy. They vibrate around fixed positions and are tightly packed. This results in less movement and a higher density.

• Low temperature = Lower kinetic energy
• Molecules vibrate in place
• High packing density, like in solids

As temperatures rise, such as in the liquid state of benzoic acid at 130°C, molecules gain kinetic energy. This energy increase leads to more movement, causing molecules to spread further apart.

• High temperature = Higher kinetic energy
• Molecules move freely
• Lower density, typical of liquids

Temperature changes affect the kinetic energy and hence the density and arrangement of molecules in different states of matter.

Phase Transitions

Phase transitions describe the change from one state of matter to another, such as solid to liquid or liquid to gas. These changes occur when substance temperature alters, thereby affecting molecular arrangement and density. Each phase transition involves energy changes associated with molecular kinetic energy adjustments.

For benzoic acid, a transition from solid to liquid happens at 122°C. During melting, molecules absorb energy and increase their kinetic energy, allowing them to move past each other and become more spaced out, transitioning from a structured solid into a fluid liquid with lower density.

• Solid to Liquid: Energy absorbed
• Molecules become more active
• Results in lower density

As phase transitions like melting or boiling occur, the arrangement of molecules changes significantly, impacting properties such as density and volume. Understanding these transitions elucidates how temperature and energy shifts drive changes in matter.