Question

Quantitatively compare the rates of effusion for the following pairs of gases at the same temperature and pressure: a. hydrogen and nitrogen b. fluorine and chlorine

Short answer

The rate of effusion of hydrogen to nitrogen is 7:1. The rate of effusion of fluorine to chlorine is approximately 1.67:1.

Step-by-step solution

Understanding Graham's Law

Graham's Law of effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, \(\frac{r_1}{r_2} = \frac{\text{Molar mass}_2}{\text{Molar mass}_1}\).

Find the Molar Masses of the Gases

Determine the molar masses of the gases involved. Hydrogen (H_2) has a molar mass of 2 g/mol, nitrogen (N_2) has a molar mass of 28 g/mol, fluorine (F_2) has a molar mass of 38 g/mol, and chlorine (Cl_2) has a molar mass of 71 g/mol.

Calculate Rate of Effusion for Hydrogen and Nitrogen

Use the values found to calculate the ratio of effusion rates for hydrogen and nitrogen. \(\frac{r_{\text{H}_2}}{r_{\text{N}_2}} = \frac{\text{Molar mass}_{\text{N}_2}}{\text{Molar mass}_{\text{H}_2}} = \frac{28}{2} = \frac{28}{2} = \frac{1}{\frac{1}{7}} = 2\frac{1}{1.87} .\)

Calculate Rate of Effusion for Fluorine and Chlorine

use the values found to calculate the ratio of effusion rates for fluorine and chlorine. \(\frac{r_{\text{F}_2}}{r_{\text{Cl}_2}} = \frac{\text{Molar mass}_{\text{Cl}_2}}{\text{Molar mass}_{\text{F}_2}} = \frac{71}{38} = 2.08chez0.5.\)

Key concepts

Rate of Effusion

Effusion is the process by which gas particles pass through a tiny opening. The rate of effusion describes how quickly this process happens.
To quantify, the rate of effusion is represented by how many particles move through the opening per unit time.
According to Graham's Law of effusion, the rate at which a gas effuses is related to its molar mass.
Essentially, lighter gases effuse faster than heavier gases. For example, hydrogen molecules, being very light, will effuse much faster than nitrogen molecules, which are heavier.

Molar Mass

Molar mass is a measure of the mass of one mole of a substance, usually expressed in grams per mole (g/mol).
Different gases have different molar masses. For example:

• Hydrogen (H_2) has a molar mass of 2 g/mol.
• Nitrogen (N_2) has a molar mass of 28 g/mol.
• Fluorine (F_2) has a molar mass of 38 g/mol.
• Chlorine (Cl_2) has a molar mass of 71 g/mol.

The molar mass of a gas is essential when calculating the rate of effusion, as it appears in the denominator of the equation used in Graham's Law.

Inverse Proportionality

Graham's Law of effusion establishes that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.
Mathematically, you can express it as \(\frac{r_1}{r_2} = \frac{\text{Molar mass}_2}{\text{Molar mass}_1}\).
This relationship indicates that if a gas has a high molar mass, it will effuse more slowly than a gas with a low molar mass.
In the given exercises, let's calculate the ratio of effusion rates for hydrogen and nitrogen:

• \(\frac{r_{\text{H}_2}}{r_{\text{N}_2}} = \frac{\text{Molar mass}_{\text{N}_2}}{\text{Molar mass}_{\text{H}_2}} = \frac{28}{2} = \frac{28}{2} = \frac{7}{1}\)

For fluorine and chlorine:

• \(\frac{r_{\text{F}_2}}{r_{\text{Cl}_2}} = \frac{\text{Molar mass}_{\text{Cl}_2}}{\text{Molar mass}_{\text{F}_2}} = \frac{71}{38} = 1.87\)

It becomes clear that lighter gases effuse faster due to the inverse square root relationship between rate of effusion and molar mass.