Question

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

Short answer

The correct option is (d): \( D_{A} = D_{B} \left[ \frac{\rho_{B}}{\rho_{A}} \right]^{1/2} \).

Step-by-step solution

Understanding Graham's Law of Effusion

To find the relation between the rates of diffusion and densities of gases, we need to use Graham's law. Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass (or density). Mathematically, it can be expressed as: \[ \frac{D_{A}}{D_{B}} = \sqrt{\frac{\rho_{B}}{\rho_{A}}} \] where \(D_{A}\) and \(D_{B}\) are the diffusion rates, and \(\rho_{A}\) and \(\rho_{B}\) are the densities of gases \(A\) and \(B\), respectively.

Rewriting the Expression for \(D_{A}\)

The expression from Graham's law can be rearranged to solve for \(D_{A}\): \[ D_{A} = D_{B} \times \sqrt{\frac{\rho_{B}}{\rho_{A}}} \] This equation now directly solves for \(D_{A}\) using \(D_{B}\), \(\rho_{A}\), and \(\rho_{B}\).

Identifying the Correct Option

Compare the derived equation \( D_{A} = D_{B} \times \sqrt{\frac{\rho_{B}}{\rho_{A}}} \) with the given options. The correct option is: d) \( D_{A} = D_{B} \left[ \frac{\rho_{B}}{\rho_{A}} \right]^{1/2} \) Since this matches the equation derived from Graham's law.

Key concepts

Rates of Diffusion

Diffusion is the process by which molecules spread from areas of high concentration to areas of low concentration. When we discuss the rates of diffusion for gases, we are referring to how quickly the gas particles will spread out in a given environment.
Graham's Law of Diffusion helps us understand how different gases diffuse compared to one another. The rate of diffusion can be influenced by several factors including temperature, pressure, and particularly the type of gas involved. To simplify, if two gases are at the same temperature and pressure, the rate at which they diffuse can be compared based on their molecular properties.

Densities of Gases

In Graham's Law, the concept of density plays a critical role in determining the diffusion rate of a gas. Density (c3A ) of a gas is defined as its mass per unit volume. When considering the diffusion rate, gases with lower density will diffuse faster than gases with higher density.
The formula for the density of a gas can be represented as:

• c3A = Mn/V

where:

• M is the molar mass of the gas,
• n is the number of moles, and
• V is the volume.

Because the density is inversely proportional to the diffusion rate in the context of Graham's Law, knowing the density of gases is essential for calculations involving diffusion.

Molar Mass Relation

Molar mass is the mass of one mole of a given substance and it's pivotal in understanding diffusion through Graham's Law. There is a direct relationship between molar mass and density, as articulated by the ideal gas equation D PV = nRT. Here, "M" in density calculation stands for molar mass.
Since the rate of diffusion according to Graham's Law is inversely related to the square root of the molar mass of a gas, a gas with a smaller molar mass will diffuse more quickly than a gas with a larger molar mass. This principle allows chemists to predict and compare the behaviors of different gases when they are mixing or reacting.

Inverse Proportionality

Inverse proportionality is a key principle in Graham's Law. When one value increases, the other decreases at a rate that keeps their product constant. This relationship can be seen when considering how diffusion rates and molar masses relate.
According to Graham's Law, the rate of diffusion is inversely proportional to the square root of the molar mass of the gas. This means that as the molar mass increases, the diffusion rate decreases. This inverse relationship helps visualize that lighter gases will spread out more rapidly compared to heavier gases, under similar environmental conditions.

Square Root Relationship

The square root relationship is an intrinsic part of understanding Graham's Law. According to the law, the rate of diffusion of a gas is proportional to the inverse of the square root of its molar mass and density. This non-linear relationship signifies that small changes in the square root value of a gas's density or molar mass can lead to more significant changes in the diffusion rate.
The formula

• \(\frac{D_{A}}{D_{B}} = \sqrt{\frac{\rho_{B}}{\rho_{A}}}\)\,

illustrates that if the density of one gas changes, we must consider the square root of this ratio to find how it will affect another gas's diffusion rate in comparison. This provides deeper insight into how different gases behave under comparable conditions.