Question

At standard temperature and pressure, the molar volumes of \(\mathrm{Cl}_{2}\) and \(\mathrm{NH}_{3}\) gases are 22.06 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) On cooling to \(160 \mathrm{~K}\), both substances form crystalline solids. Do you expect the molar volumes to decrease or increase on cooling the gases to \(160 \mathrm{~K} ?\) (c) The densities of crystalline \(\mathrm{Cl}_{2}\) and \(\mathrm{NH}_{3}\) at \(160 \mathrm{~K}\) are 2.02 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

The molar volumes of both Cl₂ and NH₃ gases are nearly the same at standard temperature and pressure because, according to the ideal gas law, all gases have the same molar volume under the same conditions. Upon cooling to 160 K, the molar volumes decrease as the kinetic energy of gas molecules decreases. In the solid state, Cl₂ and NH₃ have molar volumes of 35.10 cm³/mol and 20.27 cm³/mol, which are not as similar as in the gaseous state, as the differences in their molecular weights, dipole moments, and molecular shapes play a more significant role in the solid state. We expect that the molar volumes in the liquid state would be closer to those in the solid state due to stronger interactions between molecules in the liquid state compared to the gaseous state.

Step-by-step solution

a) The reason for nearly the same molar volumes

Both \(\mathrm{Cl}_{2}\) and \(\mathrm{NH}_{3}\) have nearly the same molar volumes despite their different molecular weights, dipole moments, and molecular shapes. The reason behind this similarity is because they are both gases at standard temperature and pressure, and according to the ideal gas law, all gases have the same molar volume under the same conditions. The ideal gas law is given by: \[PV=nRT\] Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. At standard temperature and pressure (STP), any gas will occupy a volume of 22.4 L per mole.

b) Molar volumes upon cooling to 160 K

Upon cooling the gases to 160 K, we expect that the molar volumes would decrease. The reason is that the kinetic energy of the gas molecules would decrease as we reduce the temperature, which makes the molecules move slower and occupy a relatively smaller volume.

c) Calculation of molar volumes in the solid state

We can calculate the molar volumes for crystalline Cl2 and NH3 at 160 K using the densities given: Density = \(\frac{Mass}{Volume}\) Molar Volume = \(\frac{Molar Mass}{Density}\) For Cl2: Molar mass of Cl2 = 2 x Molar mass of Cl = 2 x 35.45g/mol = 70.90 g/mol Molar volume of Cl2 = \(\frac{70.90 g/mol}{2.02 g/cm^3} = 35.10 cm^3/mol\) For NH3: Molar mass of NH3 = 14.01g/mol (N) + 3 x 1.008g/mol (H) = 17.030 g/mol Molar volume of NH3 = \(\frac{17.030 g/mol}{0.84 g/cm^3} = 20.27 cm^3/mol\)

d) Comparison of molar volumes in solid and gaseous states

The molar volumes of Cl2 and NH3 in the solid state are 35.10 cm^3/mol and 20.27 cm^3/mol, respectively, while in the gaseous state, they are 22.06 L/mol and 22.40 L/mol, respectively. We can see that the molar volumes in the solid state are not as similar as they are in the gaseous state. The differences in their molecular weights, dipole moments, and molecular shapes play a more significant role in the solid state than in the gaseous state, as the molecules are closer together in the solid state and interact more with each other.

e) Molar volumes in the liquid state

We expect that the molar volumes in the liquid state would be closer to those in the solid state than in the gaseous state. This is because the molecules are closer together in the liquid state compared to the gaseous state and have stronger interactions, similar to the solid state. The difference in molecular weights, dipole moments, and molecular shapes would play a more significant role in determining the molar volumes in the liquid state.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental concept used to describe the behavior of gases under various conditions. It states that the product of pressure (P) and volume (V) is proportional to the product of the number of moles (n) and the temperature (T) of a gas. This relationship is expressed by the equation: \[ PV = nRT \]where R is the ideal gas constant.This law is crucial for understanding why gases like \( \mathrm{Cl}_{2} \) and \( \mathrm{NH}_{3} \) have similar molar volumes at standard temperature and pressure (STP), despite having different molecular characteristics. At STP, one mole of any gas occupies approximately 22.4 liters, illustrating that differences in molecular weight or structure do not influence the behavior of gases under these conditions.
This uniformity is why we consider gases 'ideal' under STP, helping us predict and calculate gas behavior in various scientific and industrial applications.

Molecular Weight

Molecular weight, also known as molecular mass, is the sum of the atomic masses of all the atoms in a molecule. It is measured in atomic mass units (amu) or grams per mole (g/mol). For example, the molecular weight of \( \mathrm{Cl}_{2} \) is calculated by adding the weights of two chlorine atoms, each approximately 35.45 g/mol, resulting in 70.90 g/mol. Similarly, \( \mathrm{NH}_{3} \) is composed of one nitrogen (14.01 g/mol) and three hydrogen atoms (1.008 g/mol each), giving a total molecular weight of 17.03 g/mol.Understanding molecular weight is essential for calculating molar volumes and densities, particularly in solid and liquid states. It influences how densely packed molecules are in a given volume, affecting how substances behave when transitioning from gas to liquid or solid form. Moreover, differences in molecular weight and structure can lead to varying interactions between molecules, particularly in condensed phases.

Density Calculation

Density is a measure of how much mass is contained in a given volume. It is crucial in determining how substances will behave in various states. The formula for density is:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]To calculate the molar volume of a substance from its density, you can rearrange this to:\[ \text{Molar Volume} = \frac{\text{Molar Mass}}{\text{Density}} \]This calculation is demonstrated for crystalline \( \mathrm{Cl}_{2} \) and \( \mathrm{NH}_{3} \) at 160 K. We find the molar volume by dividing the molecular weight by the given density.

• For \( \mathrm{Cl}_{2} \): \( Molar\ Volume = \frac{70.90 \ g/mol}{2.02 \ g/cm^3} = 35.10 \ cm^3/mol \)
• For \( \mathrm{NH}_{3} \): \( Molar\ Volume = \frac{17.03 \ g/mol}{0.84 \ g/cm^3} = 20.27 \ cm^3/mol \)

These calculations highlight how higher density leads to a smaller molar volume, which is consistent with more closely packed molecules in solids than in gases.

Crystalline Solids

Crystalline solids are materials whose constituents, such as atoms, ions, or molecules, are arranged in an ordered pattern. This rigid structure contributes to the distinct properties of solids compared to liquids and gases. When gases like \( \mathrm{Cl}_{2} \) and \( \mathrm{NH}_{3} \) cool to form crystalline solids at 160 K, their molecules come closer together, reflecting a significant decrease in molar volume. The well-organized lattice structure in crystalline solids means molecules have less space between them, increasing density compared to the gaseous state. Consequently, despite \( \mathrm{Cl}_{2} \) and \( \mathrm{NH}_{3} \) having similar molar volumes as gases, their solid-state volumes differ more due to these molecular interactions and arrangements.Understanding crystalline structures aids in predicting the physical properties of materials, including hardness, melting point, and solubility. It also helps in technological applications such as semiconductors and materials science.