Question

The ratio of rate of diffusion of helium with respect to methane under similar conditions of constant temperature and pressure: (a) 2 (b) \(0.5\) (c) 16 (d) 4

Short answer

The answer is (a) 2.

Step-by-step solution

Understanding the Problem

We are given two gases: helium and methane. We need to find out the ratio of their rates of diffusion under similar conditions of constant temperature and pressure.

Apply Graham's Law of Diffusion

Graham's Law of Diffusion states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass:\[\frac{\text{Rate of diffusion of helium}}{\text{Rate of diffusion of methane}} = \sqrt{\frac{\text{Molar mass of methane}}{\text{Molar mass of helium}}}\]

Calculate Molar Masses

The molar mass of helium (He) is approximately 4 g/mol, and the molar mass of methane (CH4) is approximately 16 g/mol.

Substitute Molar Masses into Graham’s Law

Plug the molar masses into the formula from Graham's Law:\[\frac{\text{Rate of diffusion of helium}}{\text{Rate of diffusion of methane}} = \sqrt{\frac{16}{4}} = \sqrt{4} = 2\]

Conclusion

The ratio of the rate of diffusion of helium with respect to methane is 2, meaning helium diffuses twice as fast as methane under the same conditions.

Key concepts

Rate of Diffusion

The rate of diffusion refers to how fast a substance spreads out or moves through another substance. This concept is particularly important when dealing with gases, as it dictates how quickly one gas can move through another or fill a space. The rate of diffusion is influenced by several factors, including temperature, pressure, and the nature of the gases involved. However, under constant temperature and pressure, Graham's Law tells us that the rate is primarily dependent on the molar mass of the gases. Generally, lighter gases diffuse more rapidly than heavier gases. This principle simplifies the comparison between different gases and helps us predict their behavior in various applications. For example, in our exercise, we determined that helium diffuses faster than methane because it is lighter, making its rate of diffusion higher.

Molar Mass

Molar mass is a crucial concept when examining the rate of diffusion, as it is the mass of a given substance (a molecule) divided by the amount of substance (moles). It's expressed in grams per mole (g/mol) and is used to quantitatively compare the densities of gases. According to Graham's Law, the rate of diffusion is inversely proportional to the square root of its molar mass. This means the lighter the gas (lower molar mass), the faster it will diffuse, and vice versa. When we talk about molar masses of helium and methane:

• Helium (He) has a molar mass of approximately 4 g/mol.
• Methane (CH₄) has a molar mass of approximately 16 g/mol.

Understanding molar mass is essential to solve problems like the one we're discussing. It provides a way to quantify and compare the diffusion rates of different gases using their molecular weights.

Helium and Methane Comparison

A comparison between helium and methane using Graham's Law offers a clear illustration of how molar mass affects the rate of diffusion. Helium and methane are both common gases, yet they behave differently in terms of how swiftly they can spread out or move through a medium.

• Helium is much lighter (with a smaller molar mass) than methane, which means it diffuses more rapidly.
• The calculation of their diffusion rates using Graham's Law shows that helium diffuses at a rate that is twice that of methane under the same conditions.

This faster diffusion rate of helium is due to its smaller molar mass, making it an excellent demonstration of the practical use of Graham's Law of Diffusion. In practical scenarios, this knowledge is useful in fields like chemistry and environmental science when predicting how gases will behave in different conditions.