Problem 28 - Arihant AIEEE Physics

Question

Choose the correct order of the root mean square velocity \(\left(v_{\mathrm{rms}}\right)\), the average velocity \(\left(v_{\mathrm{av}}\right)\) and the most probable velocity \(\left(v_{\mathrm{mp}}\right):\) (a) \(v_{\mathrm{mp}}>v_{\mathrm{av}}>v_{\mathrm{rms}}\) (b) \(v_{\mathrm{rms}}>v_{\mathrm{av}}>v_{\mathrm{mp}}\) (c) \(v_{\text {av }}>v_{\mathrm{mp}}>v_{\mathrm{rms}}\) (d) \(v_{\mathrm{m} p}>v_{\mathrm{rms}}>v_{\mathrm{av}}\)

Step-by-Step Solution

Answer

Option (b): \( v_{\mathrm{rms}} > v_{\mathrm{av}} > v_{\mathrm{mp}} \).

Step 1: Understand the Concepts

The root mean square (rms) velocity \( v_{\mathrm{rms}} \), the average velocity \( v_{\mathrm{av}} \), and the most probable velocity \( v_{\mathrm{mp}} \) are measures of different aspects of the speed distribution of gas molecules. The rms velocity is associated with the square root of the average of the squares of the velocities, the average velocity is the mean of all velocities, and the most probable velocity is the speed at which the greatest number of molecules are moving.

Step 2: Recall the Formulas

The root mean square velocity is given by \( v_{\mathrm{rms}} = \sqrt{\frac{3kT}{m}} \), the average velocity is \( v_{\mathrm{av}} = \sqrt{\frac{8kT}{\pi m}} \), and the most probable velocity is \( v_{\mathrm{mp}} = \sqrt{\frac{2kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the molecular mass.

Step 3: Rank the Velocities

By comparing the coefficients in the formulas above, we see that: - \( v_{\mathrm{mp}} = \sqrt{\frac{2kT}{m}} \)- \( v_{\mathrm{av}} = \sqrt{\frac{8kT}{\pi m}} \)- \( v_{\mathrm{rms}} = \sqrt{\frac{3kT}{m}} \)Thus, since \( 2 < \frac{8}{\pi} < 3 \), it follows that the most probable velocity is less than the average velocity, which is less than the rms velocity: \( v_{\mathrm{mp}} < v_{\mathrm{av}} < v_{\mathrm{rms}} \).

Step 4: Choose the Correct Option

From the ranking \( v_{\mathrm{mp}} < v_{\mathrm{av}} < v_{\mathrm{rms}} \), option (b) \( v_{\mathrm{rms}} > v_{\mathrm{av}} > v_{\mathrm{mp}} \) correctly reflects the order of these velocities.

Key Concepts

Understanding Root Mean Square Velocity

Root Mean Square Velocity, often abbreviated as \( v_{\mathrm{rms}} \), is a statistical measure of the speed of particles in a gas. It provides insight into the energy of particles, calculated as the square root of the average of the squares of individual velocities. This measure is particularly relevant when discussing kinetic energy, as it relates directly to the kinetic theory of gases.

For a gas molecule, \( v_{\mathrm{rms}} \) is given by the formula:

• \( v_{\mathrm{rms}} = \sqrt{\frac{3kT}{m}} \)

Here:

• \( k \) is the Boltzmann constant, a fundamental constant in statistical mechanics.
• \( T \) stands for the absolute temperature in Kelvin.
• \( m \) is the molecular mass of the gas particles.

The rms velocity is an important concept because it links the macroscopic temperature of a gas to the microscopic motion of its molecules. A higher temperature leads to a higher \( v_{\mathrm{rms}} \), indicating more energetic particles. In summary, \( v_{\mathrm{rms}} \) helps us comprehend how kinetic energy manifests in a gas system.

Deciphering Average Velocity

Average Velocity, denoted as \( v_{\mathrm{av}} \), is another key parameter in understanding gas molecule dynamics. It represents the mean speed calculated from the velocities of all molecules in a gas sample. While it provides a simplified picture of molecular motion, it doesn't capture the full range of variability.

The formula for the average velocity is:

• \( v_{\mathrm{av}} = \sqrt{\frac{8kT}{\pi m}} \)

Where the terms are:

• \( k \) is the Boltzmann constant.
• \( T \) is the absolute temperature.
• \( m \) is the molecular mass.

The presence of \( \pi \) in the formula indicates that it is derived from a more complex integration of the Maxwell-Boltzmann distribution, which describes the distribution of speeds in a gas. While \( v_{\mathrm{av}} \) is useful for predicting the general trend in molecular speed, it does not suffice to describe the spread or peak of the distribution in detail. That's why, it's always complemented with other measures like rms and most probable velocity.

Exploring Most Probable Velocity

Most Probable Velocity, represented as \( v_{\mathrm{mp}} \), is the speed at which the largest number of molecules are moving in a gas system. Unlike average and rms velocity, \( v_{\mathrm{mp}} \) focuses on the peak of the speed distribution rather than the mean or energy-related measures.

The mathematical expression for the most probable velocity is:

• \( v_{\mathrm{mp}} = \sqrt{\frac{2kT}{m}} \)

This involves:

• \( k \) as the Boltzmann constant.
• \( T \) for temperature.
• \( m \), the molecular mass.

By comparing with the other velocities' formulas, it's clear that \( v_{\mathrm{mp}} \) is generally the smallest of these velocity measures. It specifically pinpoints the speed most common among molecules, which can be crucial for understanding reaction rates and transport properties in gases. Therefore, \( v_{\mathrm{mp}} \) offers a valuable, concrete snapshot of molecular behavior in a broader statistical context.