Problem 37 - A Complete Resource Book in Chemistry for JEE Main

Question

The densities of two gases are in the ratio of \(1: 16\). The ratio of their rates of diffusion is (a) \(16: 1\) (b) \(4: 1\) (c) \(1: 4\) (d) \(1: 16\)

Step-by-Step Solution

Answer

The ratio of their rates of diffusion is 1:4, matching option (c).

Step 1: Understanding Graham's Law of Diffusion

Graham's Law states that the rate of diffusion of a gas is inversely proportional to the square root of its density. Mathematically, this is expressed as \( r_1 / r_2 = \sqrt{d_2 / d_1} \), where \( r_1 \) and \( r_2 \) are the rates of diffusion, and \( d_1 \) and \( d_2 \) are their respective densities.

Step 2: Set the Density Ratio

From the problem, the densities of the two gases are given in the ratio \(1:16\). This means \( d_1 = 1 \) and \( d_2 = 16 \) (taking \( d_1 \) and \( d_2 \) as the relative densities of the two gases).

Step 3: Calculate the Rate of Diffusion Ratio

Using Graham's Law, substitute the densities into the formula: \( r_1 / r_2 = \sqrt{d_2 / d_1} = \sqrt{16 / 1} = \sqrt{16} = 4 \). Thus, \( r_1 : r_2 = 1:4 \).

Step 4: Match with Given Options

Compare the calculated ratio \(1:4\) with the given options. It matches option (c) \(1:4\).

Key Concepts

Density of Gases

Density of gases is a crucial concept that helps us understand how much mass is present in a given volume of a gas. It's often measured as the mass of the gas per unit volume. Typically, when we talk about the density of gases in experiments or problem-solving, we refer to their relative densities. For instance, if we say one gas is denser than the other by a certain factor, it means it contains more mass per volume compared to the other.

The density of a gas can change depending on factors such as pressure and temperature. For example:

• At higher pressures, gas particles are pushed closer together, resulting in higher density.
• At lower temperatures, gas particles move slower and closer together, also increasing density.
• Conversely, higher temperatures will cause gases to expand, reducing density.

Understanding gas density is vital when studying diffusion, as density affects how gases spread out or mix with each other. This is particularly evident in Graham's Law of Diffusion.

Rate of Diffusion

The rate of diffusion refers to how quickly gas particles spread or move from an area of high concentration to an area of low concentration. This can be thought of as how fast two gases mix with each other when they are brought into contact.

Several factors influence the rate of diffusion of gases:

• The density of the gases, where gases with lower density generally diffuse faster.
• The temperature, as increasing temperature gives gas molecules more kinetic energy, causing them to move and spread more rapidly.
• The molar mass of the gases, with lighter gases moving more swiftly.

Graham's Law specifically connects these concepts by stating that the rate of diffusion is inversely related to the square root of the gas's density. In mathematical terms, if you increase the density of a gas, its rate of diffusion decreases, provided all other conditions are equal.

Inverse Relationship

An inverse relationship in mathematics and science describes a situation where one variable increases while the other decreases. In the context of Graham's Law of Diffusion, this refers to the relationship between the rate of diffusion of gases and their densities.

In Graham's Law, this inverse relationship is captured by the equation:\[\frac{r_1}{r_2} = \sqrt{\frac{d_2}{d_1}}\]Where \(r_1\) and \(r_2\) represent the rates of diffusion, and \(d_1\) and \(d_2\) are the constants representing the densities of the two gases. As you can see, if one gas is denser \(d_2 > d_1\), its rate of diffusion \(r_2\) will be slower compared to a less dense gas \(r_1\).

Grasping this relationship is essential to understand how gases behave in various environments and why certain gases spread faster than others. For instance, lighter gases like helium rapidly escape from balloons because they diffuse faster than heavier gases. Understanding this inverse relationship allows us to predict and control the behavior of gases in various scientific and industrial applications.