Question

At constant volume and temperature conditions, the rates of diffusion \(\mathrm{D}_{A}\) and \(\mathrm{D}_{\mathrm{B}}\) of gases \(\mathrm{A}\) and \(\mathrm{B}\) having densities \(\rho_{A}\) and \(\rho_{n}\) are related by the expression (a) \(\mathrm{D}_{\mathrm{A}}=\left[\mathrm{D}_{\mathrm{a}} \rho_{\mathrm{A}} / \rho_{\mathrm{h}}\right]^{1 / 2}\) (b) \(\mathrm{D}_{\mathrm{A}}=\left[\mathrm{D}_{\mathrm{B}} \rho_{\mathrm{B}} / \rho_{A}\right]^{1 / 2}\) (c) \(D_{A}=D_{B}\left[\rho_{A} / \rho_{B}\right]^{1 / 2}\) (d) \(\mathrm{D}_{\mathrm{A}}=\mathrm{D}_{\mathrm{B}}\left[\rho_{\mathrm{B}} / \rho_{\mathrm{A}}\right]^{12}\)

Short answer

The correct answer is (b).

Step-by-step solution

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 molar mass or density. Mathematically, it can be expressed as \( \frac{D_A}{D_B} = \sqrt{\frac{\rho_B}{\rho_A}} \). This implies that \( D_A = D_B \sqrt{\frac{\rho_B}{\rho_A}} \).

Comparing with Given Options

Let's compare the expression \( D_A = D_B \sqrt{\frac{\rho_B}{\rho_A}} \) with each of the provided options: (a) \( \mathrm{D}_{\mathrm{A}}=\left[\mathrm{D}_{\mathrm{a}} \rho_{\mathrm{A}} / \rho_{\mathrm{h}}\right]^{1 / 2} \) is incorrect as it involves \( D_A \) itself.(b) \( \mathrm{D}_{\mathrm{A}}=\left[\mathrm{D}_{\mathrm{B}} \rho_{\mathrm{B}} / \rho_{A}\right]^{1 / 2} \) rearranges as \( D_A = D_B \sqrt{\frac{\rho_B}{\rho_A}} \), which matches our derived expression. (c) \( D_{A}=D_{B}\left[\rho_{A} / \rho_{B}\right]^{1 / 2} \) is the inverse of what we want.(d) \( \mathrm{D}_{\mathrm{A}}=\mathrm{D}_{\mathrm{B}}\left[\rho_{\mathrm{B}} / \rho_{\mathrm{A}}\right]^{12} \) uses the wrong exponent.

Key concepts

Rate of Diffusion

The rate of diffusion refers to how quickly one substance, such as a gas, spreads out or mixes with another. In the context of gases, this is explored through Graham's Law of Diffusion. Simply put, diffusion implies the movement of molecules from an area of higher concentration to one of lower concentration until equilibrium is reached. This process is crucial in many natural phenomena and industrial applications.
The rate at which a gas diffuses depends on multiple factors:

• Molecular Weight: According to Graham's Law, lighter gases will diffuse faster than heavier ones. This is because lighter molecules move with higher velocity.
• Temperature: At higher temperatures, gas molecules move faster, leading to an increased rate of diffusion.
• Pressure: Lower pressure generally facilitates faster diffusion since there's less resistance from surrounding molecules.

Understanding the rate of diffusion helps chemists and scientists predict how gases will behave when they interact or when conditions change.

Gas Density

Gas density is an important factor in understanding gas behavior and is defined as the mass per unit volume of a gas. Given the same conditions of temperature and pressure, gases with higher molecular weights will have higher densities. This impacts how gases diffuse according to Graham's Law.
Since the rate of diffusion of a gas is inversely related to the square root of its density, it means:

• High density implies slower diffusion because the gas particles are more "packed" and experience more interaction with one another.
• Low density results in faster diffusion as particles have more room to move freely.

This concept of density influencing the movement of gas particles is crucial in applications like respiratory systems in living organisms and industrial gas mixing processes.

Molar Mass

Molar mass is the mass of one mole of a substance, and in the case of gases, it plays a crucial role in their diffusion rates, as depicted by Graham's Law. It's expressed in grams per mole (6{g/mol}6). The connection between molar mass and diffusion stems from the fact that gases composed of lighter molecules (lower molar mass) diffuse faster than those with heavier molecules (higher molar mass).
To quantify this relationship, Graham's Law uses the inverse square root principle:

• The rate of diffusion of gas is inversely proportional to the square root of its molar mass: \[\text{{rate of diffusion}} \propto \frac{1}{\sqrt{\text{{molar mass}}}}\]
• Hence, if comparing two gases, the lighter gas will diffuse more rapidly.

Understanding molar mass implications is essential not just in theoretical chemistry, but also in practical applications such as the separation of gases and designing effective respiratory systems.