Problem 41 - A Complete Resource Book in Chemistry for JEE Main

Question

At constant volume and temperature conditions, the rates of diffusion \(\mathrm{D}_{\mathrm{A}}\) and \(\mathrm{D}_{\mathrm{B}}\) of gases \(\mathrm{A}\) and \(\mathrm{B}\) having densities \(\rho_{A}\) and \(\rho_{B}\) are related by the expression (a) \(\mathrm{D}_{A}=\left[\mathrm{D}_{\mathrm{B}} \rho_{\mathrm{A}} / \rho_{\mathrm{B}}\right]^{1 / 2}\) (b) \(\mathrm{D}_{\mathrm{A}}=\left[\mathrm{D}_{\mathrm{B}} \rho_{\mathrm{B}} / \rho_{\mathrm{A}}\right]^{1 / 2}\) (c) \(\mathrm{D}_{A}=\mathrm{D}_{\mathrm{B}}\left[\rho_{\mathrm{A}} / \rho_{\mathrm{B}}\right]^{1 / 2}\) (d) \(\mathrm{D}_{\mathrm{A}}=\mathrm{D}_{\mathrm{B}}\left[\rho_{\mathrm{B}} / \rho_{\mathrm{A}}\right]^{1 / 2}\)

Step-by-Step Solution

Answer

Option (d) is correct based on Graham's law of diffusion.

Step 1: Understand the Context of Diffusion Rates

The problem deals with the concept of diffusion rates of gases under constant temperature and volume conditions. According to Graham's law of diffusion, the diffusion rate of a gas is inversely proportional to the square root of its density.

Step 2: Identify Graham's Law of Diffusion Formula

Graham's law states that the rate of diffusion of a gas (\(D\)) is inversely proportional to the square root of its density (\(\rho\)). Mathematically, this can be expressed as \(\frac{D_A}{D_B} = \sqrt{\frac{\rho_B}{\rho_A}}\).

Step 3: Solve Using Graham's Law Formula

Using the equation from Graham’s law: \(\frac{D_A}{D_B} = \sqrt{\frac{\rho_B}{\rho_A}}\), we solve for \(D_A\):\[D_A = D_B \times \sqrt{\frac{\rho_B}{\rho_A}}\]

Step 4: Compare with Given Options

Compare the expression derived for \(D_A\) with the options provided:- Option (a): \(D_A = \left[ D_B \frac{\rho_A}{\rho_B} \right]^{1/2}\)- Option (b): \(D_A = \left[ D_B \frac{\rho_B}{\rho_A} \right]^{1/2}\)- Option (c): \(D_A = D_B \left[ \frac{\rho_A}{\rho_B} \right]^{1/2}\)- Option (d): \(D_A = D_B \left[ \frac{\rho_B}{\rho_A} \right]^{1/2}\)The expression derived matches with Option (d).

Step 5: Confirm Correct Answer

Confirm that Option (d): \(D_A = D_B \left[ \frac{\rho_B}{\rho_A} \right]^{1/2}\) is correct based on the application of Graham's law.

Key Concepts

Diffusion Rates of Gases

Diffusion is a fascinating process in which gas molecules spread out or mix within a space. When we're talking about gases, this spreading or mixing is called the rate of diffusion. In simple terms, the rate of diffusion is how fast gas molecules move through a space. Graham's Law of Diffusion is a key principle that helps us understand this behavior.

The law states that the diffusion rate of a gas is not random but depends on certain factors, primarily its density. According to Graham's law, the diffusion rate is inversely related to the square root of its density. This means lighter gases, which have lower densities, will diffuse faster than heavier gases with higher densities. The formula representing this relationship is:

• \( \frac{D_A}{D_B} = \sqrt{\frac{\rho_B}{\rho_A}} \)

Here, \( D_A \) and \( D_B \) are the diffusion rates of gases A and B, respectively, while \( \rho_A \) and \( \rho_B \) represent their densities. It gives us a quantitative way to compare how different gases will behave when they diffuse, understanding that their rate is tied to the properties of the gases themselves.

Density and Diffusion Relationship

The relationship between density and diffusion is crucial in understanding the behavior of gas molecules. Density, regarding gases, usually refers to how much mass the gas has in a given volume. A key takeaway from Graham's law is that gases with lower density diffuse faster.

So, how does density play such a significant role?
A gas with lower density has molecules that are not packed closely. They have less mass, leading to quicker movement and more frequent dispersion, which translates to a faster diffusion rate. This can be illustrated as:

• If we have two gases, A and B, with \( \rho_A < \rho_B \), gas A will diffuse more quickly because its molecules are lighter and can move more rapidly through space.

This inverse square relationship binds together how we understand gases in any medium under consistent conditions. It's fascinating to see how the simple measure of gas density can dictate a gas's dynamic behavior!

Constant Temperature and Volume Conditions

To accurately apply Graham's Law of Diffusion, it’s essential to maintain constant temperature and volume conditions. These conditions ensure that any changes observed in diffusion rates are due exclusively to the properties of the gases themselves, not external factors.

Why does this matter?
When temperature and volume remain constant, we discard their potential influence on the rate of diffusion. Temperature, for instance, can affect gas molecules' energy, making them move faster or slower. A change in volume could alter how gas molecules collide and spread out. Thus, keeping these conditions constant means:

• We have a controlled environment where the diffusion behavior is only due to the gases’ densities.
• The comparison of diffusion rates from Graham’s Law remains valid and purely dependent on the inherent properties of the gases involved.

In studying gas diffusion, such controlled parameters ensure that predictions and calculations made with Graham's law are accurate and reliable, providing clear insights into the intrinsic nature of gases.