Problem 151 - A Complete Resource Book in Chemistry for JEE Main

Question

Which one of the following statement is not true about the effect of an increase in temperature on the distribution molecular speeds in a gas? (a) the most probable speed increases (b) the fraction of the molecules with the most probable speed increases (c) the distribution becomes broader (d) the area under the distribution curve remains the same as the under the lower temperature

Step-by-Step Solution

Answer

Statement (b) is not true.

Step 1: Understanding Maxwell-Boltzmann Distribution

The distribution of molecular speeds in a gas at a given temperature is described by the Maxwell-Boltzmann distribution. As temperature increases, the average kinetic energy of the molecules also increases, affecting the distribution.

Step 2: Evaluating Statement (a)

The most probable speed, which is the speed at which the maximum number of molecules reside, increases with an increase in temperature. Thus, the statement in (a) is true.

Step 3: Evaluating Statement (b)

An increase in temperature causes the distribution curve to shift such that the most probable speed occurs at a higher velocity, but the fraction of molecules having this most probable speed actually decreases, not increases. Thus, statement (b) is incorrect.

Step 4: Evaluating Statement (c)

With increased temperature, the distribution curve becomes broader, indicating that there is a wider range of speeds among the molecules. Thus, statement (c) is true.

Step 5: Evaluating Statement (d)

The area under the distribution curve represents the total number of molecules, which remains constant regardless of temperature since we are considering a closed system. Thus, statement (d) is true.

Key Concepts

Molecular Speeds Distribution

The concept of molecular speeds distribution is key to understanding how molecules behave in gases. It is represented famously by the Maxwell-Boltzmann distribution. This distribution curve shows the range of molecular speeds in a gas at a specific temperature. The distribution shows how many molecules have a particular speed and helps us visualize the spread of speeds among the molecules.

• The peak of the curve, which indicates the most probable speed, shows where most molecules are likely to be found speed-wise.
• At lower speeds, the curve rises sharply because few molecules will have such low speeds.
• As the speed increases, the number of molecules decreases after reaching the peak, creating the bell-shaped curve of the distribution.

The distribution curve is crucial in predicting the behavior of gases as conditions change, like when the temperature varies.

Effect of Temperature on Gases

Temperature has a significant impact on the behavior of gases, especially when considering molecular speeds. As temperature increases, the average kinetic energy of the molecules also increases, which directly affects their speeds.

• When gases are heated, the distribution curve for molecular speeds shifts to the right. This means that molecules move faster and their most probable speed increases.
• However, as the curve shifts, it also becomes broader. This indicates a wider range of molecular speeds, which leads to more molecules moving at higher speeds.
• The peak of the distribution curve, corresponding to the most probable speed, decreases in height because the molecules are spread over a wider range of speeds.

The overall area under the curve remains the same, though, as this represents the total number of molecules which does not change.

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental aspect of physical chemistry that provides insight into the nature of gases. It helps to explain how gases behave under different conditions based on the motion of their molecules. According to this theory:

• Gas molecules are in constant, random motion, colliding with each other and the walls of their container.
• These collisions are what create gas pressure.
• The kinetic energy of the molecules is directly proportional to the temperature of the gas. Thus, as temperature increases, so does the average kinetic energy of the molecules.

This theory allows us to relate the macroscopic properties of gases, like temperature and pressure, to the microscopic properties of molecules, such as speed and kinetic energy. Understanding this relationship is crucial for predicting how gases will react to changes in temperature and pressure.