Problem 83 - Chemistry

Question

The average distance traveled by a molecule between successive collisions is called mean free path. For a given amount of a gas, how does the mean free path of a gas depend on (a) density, (b) temperature at constant volume, (c) pressure at constant temperature, (d) volume at constant temperature, and (e) size of the atoms?

Step-by-Step Solution

Answer

The mean free path depends inverse proportionally on density, pressure at constant temperature and size of the atoms, but is directly proportional on volume at constant temperature. There's no significant effect of temperature at constant volume on the mean free path.

Step 1: (a) Dependency on Density

The mean free path of a gas is inversely proportional to its density. When the density of a gas increases, the number of molecules per unit volume also increases, resulting in more frequent collisions and hence, a shorter mean free path.

Step 2: (b) Dependency on Temperature at Constant Volume

The mean free path is not directly affected by the temperature of a gas at constant volume. This is due to the fact that a change in temperature at constant volume does not significantly alter the density of the gas, thus it does not affect the average distance that molecules travel between collisions.

Step 3: (c) Dependency on Pressure at Constant Temperature

The mean free path of a gas is inversely proportional to its pressure at constant temperature. When the pressure of a gas increases at constant temperature, the density of the gas also increases, leading to more frequent collisions and hence, a shorter mean free path.

Step 4: (d) Dependency on Volume at Constant Temperature

The mean free path of a gas is directly proportional to its volume at constant temperature. When the volume of a gas increases at constant temperature, the density of the gas decreases, leading to less frequent collisions and hence, a longer mean free path.

Step 5: (e) Dependency on Size of the Atoms

The mean free path of a gas is inversely proportional to the size of its atoms. The larger the size of the atoms, the greater the chances of collisions, and thus, the shorter the mean free path.

Key Concepts

Density

Density refers to how closely packed the molecules in a substance are, usually measured as mass per unit volume. In gases, density plays a crucial role. When a gas's density increases, the number of molecules within a specific volume also rises.
This leads to more frequent molecular collisions. Because the mean free path, which is the average distance a molecule travels between collisions, depends on how often molecules collide, it becomes shorter when density is high.

• Higher density = more molecules per unit volume.
• More collisions occur, reducing the mean free path.

A simple way to understand this is imagining a crowded room versus a less crowded one. In the crowded room, it is harder to move without bumping into others, just like gas molecules in a high-density environment.

Temperature

Temperature is a measure of the average kinetic energy of molecules in a substance. High temperatures mean molecules are more energetic and move faster. However, at constant volume, temperature doesn't directly impact the mean free path.
This is because at constant volume, a temperature change doesn't significantly alter the gas's density.

• Temperature increases, but volume remains constant.
• Molecule speed increases, possibly leading to more energetic collisions, but not more frequent.

Consequently, the average number of collisions over distance doesn’t change and neither does the mean free path.

Pressure

Pressure in gases is the force exerted by gas molecules when they collide with the walls of their container. At a constant temperature, increasing pressure often means more molecules are squeezed into a given volume, hence increasing density. Consequently, this leads to more frequent collisions and a shorter mean free path.

• Increased pressure = more molecules in a certain volume.
• Higher density, leading to increased collision frequency.

Therefore, we can say that the mean free path is inversely proportional to the pressure at constant temperature.

Volume

Volume is the space a gas occupies. At constant temperature, an increase in volume tends to cause a decrease in gas density. This means molecules have more room to move around, encountering fewer collisions. As a result, the mean free path increases.

• Larger volume = reduced density of molecules.
• Fewer collisions per unit distance, resulting in longer mean free path.

Think of a balloon expanding; the molecules now have more space, reducing collision chances just like in a larger volume of gas.

Atomic Size

Atomic size refers to the physical size of an atom or molecule in a gas. It influences how likely molecules are to collide when they move around. Larger molecules or atoms increase the probability of collisions because they occupy more space.
This means that as atomic size increases, the mean free path decreases.

• Larger atomic size = higher chance of collision.
• Shorter mean free path because molecules "bump" into each other more.

Consider marbles in a box; larger marbles make it harder to move without touching each other, similar to larger gas molecules.