Problem 64 - Chemistry The Molecular Science

Question

The reaction of \(\mathrm{SO}_{2}\) with \(\mathrm{Cl}_{2}\) to give dichlorine oxide is $$\mathrm{SO}_{2}(\mathrm{~g})+2 \mathrm{Cl}_{2}(\mathrm{~g}) \longrightarrow \mathrm{SOCl}_{2}(\mathrm{~g})+\mathrm{Cl}_{2} \mathrm{O}(\mathrm{g})$$ Place all molecules in the equation in order of increasing rate of effusion.

Step-by-Step Solution

Answer

Order of increasing rate of effusion: SOCl2, Cl2O, Cl2, SO2.

Step 1: Understand Effusion and Graham's Law

Effusion is the process by which gas molecules escape through a tiny hole into a vacuum. According to Graham's Law, the rate of effusion of a gas is inversely proportional to the square root of its molar mass, \( r \propto \frac{1}{\sqrt{M}} \). Hence, the lighter the molecule, the faster it effuses.

Step 2: Determine Molar Mass of Each Gas

First, calculate the molar masses of \(\mathrm{SO}_{2}\), \(\mathrm{Cl}_{2}\), \(\mathrm{SOCl}_{2}\), and \(\mathrm{Cl}_{2}\mathrm{O} \):- \(\mathrm{SO}_{2}\): \( M = 32.07 + 2 \times 16.00 = 64.07 \) g/mol.- \(\mathrm{Cl}_{2}\): \( M = 2 \times 35.45 = 70.90 \) g/mol.- \(\mathrm{SOCl}_{2}\): \( M = 32.07 + 35.45 + 2 \times 16.00 = 119.97 \) g/mol.- \(\mathrm{Cl}_{2}\mathrm{O} \): \( M = 2 \times 35.45 + 16.00 = 86.90 \) g/mol.

Step 3: List in Order of Increasing Molar Mass

Arrange the gases in order of increasing molar mass, which will correspond to increasing rate of effusion:1. \(\mathrm{SO}_{2}\): 64.07 g/mol2. \(\mathrm{Cl}_{2}\): 70.90 g/mol3. \(\mathrm{Cl}_{2}\mathrm{O} \): 86.90 g/mol4. \(\mathrm{SOCl}_{2}\): 119.97 g/mol.

Step 4: Order of Increasing Rate of Effusion

Based on the inversely proportional relationship between effusion rate and molar mass, the order of increasing rate of effusion is the reverse of the order of increasing molar mass:1. \(\mathrm{SOCl}_{2}\)2. \(\mathrm{Cl}_{2}\mathrm{O}\)3. \(\mathrm{Cl}_{2}\)4. \(\mathrm{SO}_{2}\).

Key Concepts

Understanding Molar Mass

The concept of molar mass is fundamental in chemistry and is pivotal to understanding Graham's Law of Effusion. Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is calculated by summing the atomic masses of all atoms in a molecule. For example, the molar mass of sulfur dioxide (\(\mathrm{SO}_{2}\)) is calculated by adding the atomic mass of one sulfur atom (32.07 g/mol) to twice the atomic mass of oxygen (16.00 g/mol each), giving a total of 64.07 g/mol.

• Each element in a compound contributes to the overall molar mass.
• The periodic table provides the necessary atomic masses to calculate molar mass.
• Knowing the molar mass of gases is crucial for predicting their behavior in effusion processes.

Understanding the molar mass helps determine how fast or slow a gas will effuse through a tiny opening because its rate is inversely related to this property.

Rate of Effusion Explained

Effusion is a fascinating process where gas molecules escape through a small opening into a vacuum. Understanding the rate at which this happens is essential in chemistry. The rate of effusion is governed by Graham's Law, which states that the effusion rate of a gas is inversely proportional to the square root of its molar mass. This means that lighter gases will effuse more quickly than heavier gases.

• Graham's Law formula: \( r \propto \frac{1}{\sqrt{M}} \)
• Gases with lower molar masses have higher effusion rates.
• The order of effusion can be derived by calculating and comparing molar masses.

For instance, in the reaction given, sulfur dioxide with a molar mass of 64.07 g/mol will effuse faster than sulfuryl chloride (\(\mathrm{SOCl}_{2}\)) with a molar mass of 119.97 g/mol due to its lower molar mass.

Behavior of Gas Molecules

Gas molecules are constantly moving and their behavior is influenced by two main factors: temperature and pressure. In the context of effusion, however, a crucial factor is the velocity of gas molecules, which relates directly to their molar mass. As previously discussed, lighter gas molecules move faster, hence they effuse more rapidly. This behavior is due to the kinetic energy of gas molecules, which depends on their mass and velocity.

• Gas molecules are in constant random motion, colliding with each other and the walls of their container.
• The kinetic theory of gases suggests that at a given temperature, all gases have the same average kinetic energy.
• Faster-moving lighter molecules can escape more quickly through small openings than heavier, slower-moving molecules.

This principle helps us predict and explain the sequence by which gases "leak" through microscopic holes, known as effusion. Understanding this characteristic is crucial when studying reactions and practical applications involving gases.