Problem 13 - Chemistry The Central Science In Si Units

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?

Step-by-Step Solution

Answer

(a) Ideal gas behavior; (b) Decrease; (c) \(\mathrm{Cl}_2\): 35.10 \mathrm{~cm}^3/mol, \(\mathrm{NH}_3\): 20.27 \mathrm{~cm}^3/mol; (d) No, they are different; (e) Closer to solid state.

Step 1: Analyze Molar Volumes at STP

Both \(\mathrm{Cl}_2\) and \(\mathrm{NH}_3\) gases have similar molar volumes (22.06 L for \(\mathrm{Cl}_2\) and 22.40 L for \(\mathrm{NH}_3\)) because gases at standard temperature and pressure (STP) obey the ideal gas law. According to the law, one mole of an ideal gas occupies approximately 22.4 L regardless of its chemical identity, assuming low pressure and high temperature relative to liquefaction point.

Step 2: Determine Changes in Molar Volume upon Cooling

When gases are cooled to \(160\, \mathrm{K}\), they condense into crystalline solids. We expect their molar volumes to decrease because the intermolecular distances in solids are much shorter than in gases, which means that atoms or molecules are packed more closely together.

Step 3: Calculate Molar Volumes for Crystalline Solids

The molar volume can be calculated using the formula:\[\text{Molar Volume} = \frac{\text{Molar Mass}}{\text{Density}}\]For \(\mathrm{Cl}_2\) at \(160 \mathrm{~K}\),\[\text{Molar Volume} = \frac{70.906 \mathrm{~g}/\mathrm{mol}}{2.02 \mathrm{~g} / \mathrm{cm}^3} = 35.10 \mathrm{~cm}^3/\mathrm{mol}\]For \(\mathrm{NH}_3\) at \(160 \mathrm{~K}\),\[\text{Molar Volume} = \frac{17.031 \mathrm{~g}/\mathrm{mol}}{0.84 \mathrm{~g} / \mathrm{cm}^3} = 20.27 \mathrm{~cm}^3/\mathrm{mol}\]

Step 4: Compare Molar Volumes in Solid and Gas States

In the gaseous state, the molar volumes are similar (around 22 L) due to the ideality of gases at STP. However, in the solid state, \(\mathrm{Cl}_2\) (35.10 \mathrm{~cm}^3/mol) and \(\mathrm{NH}_3\) (20.27 \mathrm{~cm}^3/mol) have distinctly different molar volumes. This difference arises due to unique packing efficiency, molecular shapes, and interactions in their respective solid structures.

Step 5: Evaluate Liquid State Molar Volumes

Liquid state molar volumes are typically closer to those in the solid state than in the gaseous state. This is due to the intermediate level of molecular interaction and ordering in liquids compared to solids and the very dispersed molecules in gases. Hence, liquid molar volumes should be closer to the solid state molar volumes.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental principle that helps us understand gas behavior under various conditions. It's given by the formula \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature. At standard temperature and pressure (STP), one mole of an ideal gas occupies approximately 22.4 liters. This principle explains why gases like \( \text{Cl}_2 \) and \( \text{NH}_3 \) have similar molar volumes under these conditions, despite their differences in molecular weights and shapes. At STP, the ideal gas assumption allows us to overlook these differences because the gases are far enough apart for their specific interactions to have minimal effect.

Molecular Weights

Molecular weight refers to the sum of the atomic weights of all atoms in a molecule. It gives us an idea of the mass of a molecule and is usually expressed in grams per mole (g/mol). For example, \( \text{Cl}_2 \) has a molecular weight of approximately 70.906 g/mol, while \( \text{NH}_3 \) weighs around 17.031 g/mol. While molecular weights affect properties like density and molar volume, in the gaseous state under ideal conditions, this influence is not as significant because the ideal gas law dominates the behavior of the gas.

Crystalline Solids

When gases cool and condense, they form crystalline solids. In this state, the molecules are tightly packed in a regular, repeating pattern. This tight packing leads to a decrease in molar volume compared to the gaseous state. For example, both \( \text{Cl}_2 \) and \( \text{NH}_3 \) form crystalline structures at 160 K, resulting in different molar volumes due to their various molecular structures and interactions. The packed molecules in solids occupy less space than when they are in the gas phase, highlighting why we see such a change in their volume when transitioning from gas to solid.

Density Calculations

Density is a measure of how much mass is contained in a given volume, usually expressed as grams per cubic centimeter (g/cm³). To find molar volume from density, use the formula \( \text{Molar Volume} = \frac{\text{Molar Mass}}{\text{Density}} \). For \( \text{Cl}_2 \) at 160 K, with a density of 2.02 g/cm³, the molar volume is calculated as approximately 35.10 cm³/mol. Similarly, \( \text{NH}_3 \) has a density of 0.84 g/cm³, resulting in a molar volume of about 20.27 cm³/mol. These calculations show how different densities result from variances in the packing and molecular structure of substances in the solid state.