Problem 50 - A Complete Resource Book in Chemistry for JEE Main

Question

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

Step-by-Step Solution

Answer

The ratio of the rate of diffusion of helium to methane is 2, option (a).

Step 1: Identify Variables

First, identify the relevant gases and their molar masses. Here, we have helium (He) and methane (CH4). The molar mass of helium is approximately 4 g/mol, and the molar mass of methane is approximately 16 g/mol.

Step 2: Apply Graham's Law

Graham's Law of effusion states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it is expressed as \( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \) where \( r_1 \) and \( r_2 \) are the rates of diffusion of the gases and \( M_1 \) and \( M_2 \) are their molar masses.

Step 3: Substitute Known Values

Substitute the values for the molar masses into Graham's Law. For helium (1) and methane (2), \( M_1 = 4 \) and \( M_2 = 16 \), respectively. Thus, \( \frac{r_1}{r_2} = \sqrt{\frac{16}{4}} \).

Step 4: Calculate the Ratio

Simplify the square root expression to find the ratio of diffusion rates. \( \frac{r_1}{r_2} = \sqrt{4} = 2 \).

Step 5: Verify

Check that the steps have been accurately followed and calculations are correct. The calculated ratio is 2, matching option (a).

Key Concepts

Understanding Molar Mass

Molar mass is an essential concept in chemistry. It refers to the mass of a given substance (chemical element or chemical compound) divided by the amount of substance in moles. This measurement is commonly expressed in grams per mole (g/mol). The molar mass is crucial because it allows us to convert between the mass of a substance and the amount in moles, which is fundamental for stoichiometric calculations used in chemical reactions. For example, in the context of gases like helium (He) and methane (CH₄), we consider their molar masses to predict how they will behave under different conditions. The molar mass of helium is approximately 4 g/mol, while methane's molar mass is about 16 g/mol. These values are used in various calculations, such as determining rates of diffusion and effusion according to Graham’s Law.

Effusion and Its Significance

Effusion is a process where gas molecules escape through a small opening into a vacuum. It's a vital concept because it helps us understand how gases behave under pressure and in different environments. Graham's Law of Effusion uses the principle that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. This relationship allows us to compare how different gases effuse or escape through a given small hole. In practical applications, effusion is used to separate isotopes, as gases with different molar masses will effuse at different rates. In our example with helium and methane, Graham’s Law can determine which gas will effuse faster, based on their molar masses.

Rates of Diffusion Explained

The rate of diffusion refers to how quickly a gas spreads throughout a space. It's driven by molecular movement and is influenced by temperature, pressure, and the gas's molar mass. According to Graham's Law, the rate of diffusion of a gas is directly related to its molar mass. Specifically, the rate of diffusion is inversely proportional to the square root of the molar mass. This means that lighter gases diffuse faster than heavier ones. For example, using Graham’s Law, we determined that helium diffuses twice as fast as methane under constant conditions, since helium has a lower molar mass than methane. This difference in diffusion rates is particularly important in fields like respiratory therapy and environmental science, where understanding gas behavior in different contexts is crucial.