Question

At standard temperature and pressure the molar volume of \(\mathrm{Cl}_{2}\) and \(\mathrm{NH}_{3}\) gases are \(22.06 \mathrm{~L}\) and \(22.40 \mathrm{~L},\) respectively (a) Given the different molecular weights, dipole moments, and molecular shapes, why are their molar volumes nearly the same? (b) \(\mathrm{On}\) cooling to \(160 \mathrm{~K}\), both substances form crystalline solids. Do you expect the molar volumes to decrease or increase on cooling to \(160 \mathrm{~K} ?\) (c) The densities of crystalline \(\mathrm{Cl}_{2}\) and \(\mathrm{NH}_{3}\) at \(160 \mathrm{~K}\) are \(2.02 \mathrm{~g} / \mathrm{cm}^{3}\) and \(0.84 \mathrm{~g} / \mathrm{cm}^{3}\), respectively. Calculate their molar volumes. (d) Are the molar volumes in the solid state as similar as they are in the gaseous state? Explain. (e) Would you expect the molar volumes in the liquid state to be closer to those in the solid or gaseous state?

Short answer

(a) At STP, the molar volumes of gases are determined by the ideal gas law, which depends only on temperature and pressure. Therefore, the nearly equal molar volumes of \(\mathrm{Cl}_{2}\) and \(\mathrm{NH}_{3}\) gases at STP can be explained by the fact that both substances are at the same temperature and pressure, regardless of molecular weight, dipole moment, or molecular shape. (b) When the substances cool to 160 K and form crystalline solids, we expect the molar volumes to decrease due to increased intermolecular interactions in the solid state. (c) The molar volumes of solid \(\mathrm{Cl}_{2}\) and \(\mathrm{NH}_{3}\) at 160 K are \(0.035\) L/mol and \(0.0203\) L/mol, respectively. (d) The molar volumes in the solid state are not as similar as in the gaseous state since they are more dependent on the molecular structure and crystalline lattice arrangement of particles. (e) We expect the molar volumes in the liquid state to be closer to those in the solid state because there are more significant intermolecular interactions compared to the gaseous state, but not as strong as in the solid state.

Step-by-step solution

(a) Understanding molar volumes at STP

At STP, where the temperature is 0 °C (273.15 K) and the pressure is 1 atm, the molar volume of a gas is mainly determined by the ideal gas law: \(PV = nRT\). Here, P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. Since all gases are at the same pressure and temperature, their molar volumes are nearly the same, regardless of their molecular weights, dipole moments, or molecular shapes.

(b) Expectation on molar volumes at 160 K (cooling)

On cooling to 160 K, both substances form crystalline solids. The interaction between the particles in the solid state is more intense than in the gaseous state, leading to a more compact structure. We expect the molar volumes to decrease since the particles will be closer to each other due to increased intermolecular interactions.

(c) Calculating molar volumes of crystalline Cl2 and NH3 at 160 K

We will calculate the molar volumes using the density formula: \(\rho = \frac{m}{V}\), where \(\rho\) is the density, m is the mass, and V is the volume. To convert the densities from \(\mathrm{g/cm^3}\) to \(\mathrm{L/mol}\), we will also need the molecular weight of each substance. For \(\mathrm{Cl}_{2}\): Molecular weight (MW): \(2 \times 35.45 = 70.90 \mathrm{~g/mol}\) Density: \(2.02 \mathrm{~g/cm^3}\) Molar volume: \(\frac{70.90 \mathrm{~g/mol}}{2.02 \mathrm{~g/cm^3}}=35.00 \mathrm{~cm^3/mol}\) Now, we need to convert this to liters: \(35.00 \mathrm{~cm^3/mol} \times \frac{1 \mathrm{~L}}{1000 \mathrm{~cm^3}} = 0.035 \mathrm{~L/mol}\) For \(\mathrm{NH}_{3}\): Molecular weight (MW): \(14.01 + 3 \times 1.01 = 17.04 \mathrm{~g/mol}\) Density: \(0.84 \mathrm{~g/cm^3}\) Molar volume: \(\frac{17.04 \mathrm{~g/mol}}{0.84 \mathrm{~g/cm^3}}=20.29 \mathrm{~cm^3/mol}\) Again, we need to convert this to liters: \(20.29 \mathrm{~cm^3/mol} \times \frac{1 \mathrm{~L}}{1000 \mathrm{~cm^3}} = 0.0203 \mathrm{~L/mol}\)

(d) Comparing molar volumes in the solid state

We can observe that the molar volumes in the solid state are not as similar as they are in the gaseous state. The molar volumes of solid \(\mathrm{Cl}_{2}\) and \(\mathrm{NH}_{3}\) at 160 K are \(0.035 \mathrm{~L/mol}\) and \(0.0203 \mathrm{~L/mol}\), respectively. This difference is primarily due to the fact that molar volumes in the solid state are more dependant on the molecular structure and the arrangement of the particles within the crystalline lattice.

(e) Expectation on molar volumes in the liquid state

In the liquid state, the particles are still in close proximity, and there are intermolecular interactions, but not as strong as in the solid state. Therefore, we would expect the molar volumes in the liquid state to be closer to those in the solid state than in the gaseous state, as the intermolecular interactions are more significant than in the gaseous state, but not as strong as in the solid state.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle that helps us understand how gases behave under various conditions. It is expressed as \( PV = nRT \), where:

• \( P \) stands for pressure
• \( V \) is the volume
• \( n \) represents the number of moles
• \( R \) is the gas constant
• \( T \) is the temperature in Kelvin

At standard temperature and pressure (STP), where the temperature is 273.15 K (0°C) and the pressure is 1 atm, the molar volume of a gas is approximately 22.4 L. This is because all gases behave similarly as per the ideal gas law, providing they are under the same conditions of temperature and pressure.
This explains why gases with different molecular weights, such as \( \mathrm{Cl}_2 \) and \( \mathrm{NH}_3 \), have nearly the same molar volumes at STP. Here, intermolecular forces and molecular sizes have minimal effect.

Intermolecular Interactions

Intermolecular interactions play a crucial role in determining the properties of substances in different states of matter. These forces include:

• Van der Waals forces
• Dipole-dipole interactions
• Hydrogen bonding

When substances cool and form crystalline solids, like \( \mathrm{Cl}_2 \) and \( \mathrm{NH}_3 \) at 160 K, these interactions become significant.
The particles are attracted to each other more strongly, resulting in a compact structure. This is why molar volumes decrease as gases transition into solids. Different substances will show varying molar volumes in the solid state because the strength and type of intermolecular forces depend heavily on the molecular structure.

Density Formula

The density of a substance is calculated using the formula \( \rho = \frac{m}{V} \), where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume. For crystalline substances, molar volumes can be derived from the density and molecular weight:

• For \( \mathrm{Cl}_2 \): \( \text{Molecular Weight} = 70.90 \text{ g/mol} \), \( \text{Density} = 2.02 \text{ g/cm}^3 \)
• For \( \mathrm{NH}_3 \): \( \text{Molecular Weight} = 17.04 \text{ g/mol} \), \( \text{Density} = 0.84 \text{ g/cm}^3 \)

The molar volume can be determined by dividing the molar mass by the density and converting to liters.
This approach highlights how molecular weight and density influence the volume occupied by one mole of a substance.

Crystalline Solids

Crystalline solids are substances whose molecules are arranged in a highly ordered and repeating pattern. In the solid state, the molecules are closely packed in a lattice, which can be quite different from their gaseous arrangement.
The crystalline structure depends on:

• Molecular arrangement
• Intermolecular forces
• Temperature and pressure conditions

While \( \mathrm{Cl}_2 \) and \( \mathrm{NH}_3 \) have similar molar volumes as gases, their solid-state molar volumes differ significantly due to differences in their molecular structures and interactions in the lattice.
These differences emphasize the impact of molecular structure and intermolecular forces in the solid state.