Question

With the increase in temperature of a gas the fraction of molecules having velocities within a given range around the most probable velocity would (1) increase (2) decrease (3) remain unchanged (4) initially increase and then decrease

Short answer

The fraction of molecules having velocities within a given range around the most probable velocity decreases with an increase in temperature.

Step-by-step solution

- Understanding the Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds for molecules in a gas. The most probable velocity is where the curve peaks.

- Effect of Temperature on Speed Distribution

When temperature increases, the Maxwell-Boltzmann distribution curve flattens and spreads out. This causes the peak to shift to higher velocities.

- Analyzing the Fraction of Molecules

With an increase in temperature, the fraction of molecules within a given range around the most probable velocity decreases because the distribution becomes broader and less peaked.

- Confirming the Answer

Given the broader and less peaked nature of the distribution at higher temperatures, the fraction of molecules in the specified velocity range will decrease. The correct answer is (2) decrease.

Key concepts

Gas Molecules

Gas molecules are tiny particles that move randomly in all directions. These molecules constantly collide with each other and with the walls of their container. The behavior of gas molecules is crucial to understanding many principles in physics and chemistry. These particles follow the principles of kinetic theory, which relates temperature to the average kinetic energy of these molecules.

The speed of gas molecules varies—some move slowly, while others move very fast. This range of speeds can be represented using different distributions. Understanding these distributions helps us explain and predict the behavior of gases under various conditions. It also provides the basis for more advanced concepts like the Maxwell-Boltzmann distribution.

Temperature Effect

Temperature significantly impacts the motion of gas molecules. Higher temperatures mean more kinetic energy, making molecules move faster.

When the temperature of a gas increases:

• The average velocity of the molecules increases.
• The range of speeds broadens, meaning there is a larger spread in how fast or slow individual molecules are moving.
• The peak of the Maxwell-Boltzmann distribution curve, which shows the most probable velocity, shifts to a higher velocity.

Conversely, if the temperature decreases, the molecules move slower, and the distribution curve narrows and becomes taller.

These changes in molecular speed and distribution illustrate how temperature influences the overall behavior of gas molecules.

Most Probable Velocity

The most probable velocity in a gas is the speed at which the largest number of molecules are moving. It is represented by the peak of the Maxwell-Boltzmann distribution curve. This value provides insight into the behavior of gas molecules at a given temperature.

As temperature increases:

• The most probable velocity increases because molecules have more kinetic energy.
• The peak of the distribution curve shifts to the right, indicating a higher most probable velocity.

However, as the peak shifts, it also becomes less pronounced due to the broader distribution of velocities.

Understanding the most probable velocity helps predict and analyze the movement and interaction of gas molecules under different conditions.

Velocity Distribution

The velocity distribution of gas molecules is described by the Maxwell-Boltzmann distribution. This distribution illustrates how molecule speeds are spread out for a given temperature.

Key points about velocity distribution:

• At lower temperatures, the distribution is sharp and narrow, meaning most molecules have similar speeds.
• At higher temperatures, the distribution becomes broader and flatter, encompassing a wider range of speeds.
• The area under any part of the curve represents the fraction of molecules with speeds in that range.

As the temperature increases, the fraction of molecules within any specific velocity range around the most probable velocity decreases. This happens because the broader distribution spreads the molecules across a larger range of velocities.

Understanding velocity distribution is crucial for predicting gas behavior, especially in changing temperature conditions.