Question

At a given temperature and pressure the rate of diffusion of a gas is (1) Directly proportional to the density of the gas (2) Directly proportional to the square root of its density (3) Inversely proportional to the density of the gas (4) Inversely proportional to the square root of its density

Short answer

The rate of diffusion of the gas is inversely proportional to the square root of its density, which matches option (4).

Step-by-step solution

Understand the Problem

You are asked to determine the relationship between the rate of diffusion of a gas and its density under given temperature and pressure conditions.

Identify the Proportionality Laws

Under any given conditions of temperature and pressure, the rate of diffusion of a gas follows Graham's Law. According to Graham's Law, the rate of diffusion is inversely proportional to the square root of its density.

Apply Graham's Law

Graham's Law can be written mathematically as: \[ \text{Rate of diffusion} \propto \frac{1}{\text{square root of density}} \] This shows that the correct proportionality relationship is the inverse proportion to the square root of density.

Choose the Correct Option

From the given options, the one that matches Graham's Law is: (4) Inversely proportional to the square root of its density.

Key concepts

Rate of Diffusion

The rate of diffusion refers to how quickly a gas spreads out and mixes with another gas. Imagine opening a bottle of perfume in a room; the rate at which the scent spreads through the air is an example of diffusion. Graham's Law explains how different gases will diffuse at different rates.

In scientific terms, the rate of diffusion can be thought of as the speed at which the gas molecules move from an area of high concentration to an area of low concentration. This movement is influenced by several factors, including the temperature and pressure of the environment, as well as the properties of the gas itself.

Density of Gas

Density is a measure of how much mass a substance has in a given volume. For gases, density plays a significant role in determining how they behave, including how quickly they diffuse.

According to Graham's Law, the rate of diffusion of a gas is inversely proportional to the square root of its density. This means that lighter gases, with lower density, will diffuse faster than heavier gases with higher density. Mathematically, this relationship is shown as: \[ \text{Rate of diffusion} \ \frac{1}{\text{square root of density}} \] This shows why gases like helium, which are very light, diffuse much faster than heavier gases like carbon dioxide.

Temperature and Pressure Effect

Temperature and pressure greatly affect the rate of diffusion. Higher temperatures increase the energy of the gas molecules, causing them to move faster and thus diffuse more quickly. On the other hand, lower temperatures slow down the diffusion rate.

Pressure also has an impact. In general, higher pressures will compress the gas molecules closer together, which can either speed up or slow down diffusion depending on the conditions. However, Graham's Law is usually applied under constant temperature and pressure conditions to focus purely on the effect of density on the rate of diffusion.

Proportionality Relationship

Understanding proportionality relationships helps us predict how changing one variable affects another. In this case, we use Graham's Law to see how the rate of diffusion relates to the density of the gas. According to the law, the rate of diffusion is inversely proportional to the square root of the gas's density.

In simpler terms:

• If the density of a gas increases, the rate of diffusion decreases.
• If the density of a gas decreases, the rate of diffusion increases.

To visualize this: \[ \text{Rate of diffusion} \ \frac{1}{\text{square root of density}} \] As you can see, doubling the density of a gas would reduce the rate of diffusion by a factor of the square root of 2, and so forth.