Question

The rate of diffusion of a gas is proportional to: (a) \(\frac{P}{\sqrt{d}}\) (b) \(\frac{P}{d}\) (c) \(\sqrt{\frac{P}{d}}\) (d) \(\frac{\sqrt{P}}{d}\)

Short answer

The rate of diffusion of a gas is proportional to: (a) \frac{P}{\[d}}.

Step-by-step solution

Understand Graham's Law of Effusion

Graham's Law states that the rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass. The molar mass can be represented as 'd' in the given options. Hence, the rate should be proportional to the inverse of the square root of the molar mass.

Relate to the pressure factor

The rate of diffusion is also directly proportional to the pressure 'P' of the gas. Combining the effects of pressure and molar mass, we need to find an expression that shows direct proportionality to pressure and inverse proportionality to the square root of molar mass.

Choose the correct expression

Looking at the given options, we need to find one that combines these relationships correctly: rate should be directly proportional to 'P' and inversely proportional to the square root of 'd'. The correct mathematical representation is the expression where 'P' is divided by the square root of 'd'.

Key concepts

Rate of Diffusion

Understanding the rate of diffusion is fundamental in grasping how different gases behave under various conditions. Diffusion refers to the spontaneous spreading of gas molecules from an area of higher concentration to an area of lower concentration until equilibrium is reached. This movement is intrinsic to the nature of gases and is significantly influenced by their molar mass, a concept that will be discussed further below.

According to Graham's Law, the rate of diffusion is inversely proportional to the square root of the gas's molar mass. This means that if you compare two gases under the same conditions, the lighter gas will diffuse faster than the heavier one. A practical illustration would be the quicker scent spread of perfume (lighter molecules) versus the slower diffusion of a foul odor from decay (heavier molecules).

In mathematical terms, if gas A diffuses at rate 'Ra' and has molar mass 'Ma', and gas B has rate 'Rb' and molar mass 'Mb', then the relationship between their rates of diffusion can be described by the equation \( \frac{Ra}{Rb} = \sqrt{\frac{Mb}{Ma}} \) where gas A is lighter, should diffuse faster, and thus have a higher rate of diffusion compared to gas B. Factoring in the gas pressure, if both gases are at the same pressure, the rate of diffusion remains solely dependent on the molar mass.

Molar Mass

The molar mass is essentially the weight of one mole of a substance, typically measured in grams per mole (g/mol). It has a pivotal role in understanding how gases diffuse and effuse. Since a mole corresponds to Avogadro's number (approximately \(6.022 \times 10^{23}\)) of molecules, the molar mass can be thought of as the collective mass of a vast number of these molecules.

As you may deduce from Graham's Law, gas molecules with lower molar mass will have higher rates of diffusion and effusion because they have less inertia and can move more freely. It's important to note that when considering the rate of diffusion in relation to molar mass, we look at the square root of the molar mass, which means that the relationship isn't linear—doubling the molar mass doesn't halve the rate of diffusion but has a less marked effect.

In a classroom setting, a teacher could demonstrate the influence of molar mass on diffusion rate by releasing balloons filled with helium (lighter molar mass) and another gas like carbon dioxide (heavier molar mass) and observing the rate at which they deflate through tiny pores.

Gas Pressure

Gas pressure relates to the force that gas molecules exert on the walls of their container. This force originates from the constant random motion and collisions of the gas molecules. Higher pressure means more frequent or forceful collisions, hence it is a direct measure of the gas's propensity to move or spread out.

According to kinetic molecular theory, at a given temperature, all gases have the same average kinetic energy, so the pressure exerted by a gas in a closed container is not dependent on the molar mass. However, when it comes to the rate of diffusion, the pressure does play a role, as established by Graham's Law. Higher pressure corresponds to a greater concentration of molecules that are ready to diffuse once a pathway is available.

The relationship between gas pressure and the rate of diffusion can be intuitively understood: If you increase the pressure of a gas in a container, the rate at which it escapes when given an avenue will be higher because there are more molecules pushing to get out. This direct proportionality is crucial for students to remember as they analyze scenarios involving the movement of gases.