Problem 47 - Chemistry and Chemical Reactivity

Question

In each pair of gases below, tell which will effuse faster: (a) \(\mathrm{CO}_{2}\) or \(\mathrm{F}_{2}\) (b) \(\mathrm{O}_{2}\) or \(\mathrm{N}_{2}\) (c) \(\mathrm{C}_{2} \mathrm{H}_{4}\) or \(\mathrm{C}_{2} \mathrm{H}_{6}\) (d) two chlorofluorocarbons: \(\operatorname{CFCl}_{3}\) or \(\mathrm{C}_{2} \mathrm{Cl}_{2} \mathrm{F}_{4}\)

Step-by-Step Solution

Answer

(a) \(\mathrm{F}_2\), (b) \(\mathrm{N}_2\), (c) \(\mathrm{C}_2 \mathrm{H}_4\), (d) \(\operatorname{CFCl}_3\).

Step 1: Introduction to Graham's Law

To determine which gas will effuse faster, use Graham's Law of Effusion. This law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. It can be expressed with the formula: \[ \text{Rate of effusion} \propto \frac{1}{\sqrt{M}} \] where \( M \) is the molar mass of the gas.

Step 2: Determine Molar Mass for Pair (a)

For \( \mathrm{CO}_2 \), the molar mass is: \[ 12.01 + (2 \times 16.00) = 44.01 \text{ g/mol} \]For \( \mathrm{F}_2 \), the molar mass is: \[ 2 \times 19.00 = 38.00 \text{ g/mol} \]Since \( \mathrm{F}_2 \) has a smaller molar mass, it will effuse faster according to Graham's Law.

Step 3: Determine Molar Mass for Pair (b)

For \( \mathrm{O}_2 \), the molar mass is: \[ 2 \times 16.00 = 32.00 \text{ g/mol} \]For \( \mathrm{N}_2 \), the molar mass is: \[ 2 \times 14.01 = 28.02 \text{ g/mol} \]Since \( \mathrm{N}_2 \) has a smaller molar mass, it will effuse faster.

Step 4: Determine Molar Mass for Pair (c)

For \( \mathrm{C}_2 \mathrm{H}_4 \), the molar mass is: \[ (2 \times 12.01) + (4 \times 1.01) = 28.05 \text{ g/mol} \]For \( \mathrm{C}_2 \mathrm{H}_6 \), the molar mass is: \[ (2 \times 12.01) + (6 \times 1.01) = 30.07 \text{ g/mol} \]Since \( \mathrm{C}_2 \mathrm{H}_4 \) has a smaller molar mass, it will effuse faster.

Step 5: Determine Molar Mass for Pair (d)

For \( \operatorname{CFCl}_3 \), the molar mass is: \[ 12.01 + 19.00 + (3 \times 35.45) = 137.37 \text{ g/mol} \]For \( \mathrm{C}_2 \, \mathrm{Cl}_2 \, \mathrm{F}_4 \), the molar mass is: \[ (2 \times 12.01) + (2 \times 35.45) + (4 \times 19.00) = 170.92 \text{ g/mol} \]Since \( \operatorname{CFCl}_3 \) has a smaller molar mass, it will effuse faster.

Key Concepts

Molar Mass Calculation

Understanding molar mass is crucial for comparing how different gases behave. Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). To calculate it for a compound, you simply add up the atomic masses of all the atoms in its chemical formula.

For example, the molar mass of carbon dioxide (CO\(_2\)) is calculated by adding the atomic mass of carbon (12.01 g/mol) to twice the atomic mass of oxygen (16.00 g/mol), which equals 44.01 g/mol.

When comparing the molar mass of two gases, the one with the lower molar mass will effuse faster. This is because, according to Graham's Law, the rate of effusion is inversely proportional to the square root of the molar mass. By understanding how to calculate molar masses, you can easily predict the effusion rates of different gases.

Gas Effusion

Gas effusion is the process by which gas particles pass through a tiny opening or hole into a vacuum or lower-pressure area. An interesting aspect of effusion is that lighter gases effuse more quickly than heavier ones.

According to Graham's Law of Effusion, the rate of effusion of a gas is inversely proportional to the square root of its molar mass. This can be expressed as:

• \( \text{Rate of Effusion} \propto \frac{1}{\sqrt{M}} \)

For example, if you compare two gases like oxygen (O\(_2\)) and nitrogen (N\(_2\)), nitrogen will effuse faster because it has a smaller molar mass. Understanding this concept helps in predicting how quickly different gases will move through openings, which is important in fields like chemical engineering and environmental science.

Kinetic Molecular Theory

The kinetic molecular theory provides insights into the behavior of gases and is fundamental to understanding effusion. This theory explains gas behavior based on the idea that gas particles are in constant, random motion.

According to this theory, the pressure exerted by a gas is due to collisions between gas particles and the walls of their container. These collisions also cause gases to effuse through small openings.

There are several key assumptions in the kinetic molecular theory:

• Gas particles are in continuous, random motion.
• The volume of gas particles is negligible compared to the volume of their container.
• There are no attractive or repulsive forces between gas particles.
• Collisions between gas particles are perfectly elastic, conserving energy.

This theory helps explain why lighter gas molecules, which move faster due to their lower mass, will effuse more rapidly than heavier ones. By linking kinetic energy to effusion, this framework offers a deeper understanding of gas behavior in different scenarios.