Problem 107 - A Complete Resource Book in Chemistry for JEE Main

Question

The ratio of root mean square velocity to average velocity of a gas molecule at a particular temperature is (a) \(1: 1,086\) (b) \(2: 1.086\) (c) \(1.086: 1\) (d) \(1.086: 2\)

Step-by-Step Solution

Answer

(c) 1.086: 1

Step 1: Understanding the Problem

To find the ratio of root mean square velocity to average velocity of a gas molecule at a particular temperature, we need the formulas for both velocities. The root mean square velocity \(v_{rms}\) is given by the formula \(v_{rms} = \sqrt{\frac{3kT}{m}}\), and the average velocity \(v_{avg}\) is given by \(v_{avg} = \sqrt{\frac{8kT}{\pi m}}\), where \(k\) is the Boltzmann constant, \(T\) the temperature, and \(m\) the mass of the gas molecule.

Step 2: Calculate Root Mean Square and Average Velocity

First, find the expression for \(v_{rms}\) and \(v_{avg}\): \[v_{rms} = \sqrt{\frac{3kT}{m}}\] \[v_{avg} = \sqrt{\frac{8kT}{\pi m}}\].

Step 3: Set Up the Ratio

Now calculate the ratio \(\frac{v_{rms}}{v_{avg}}\): \[\frac{v_{rms}}{v_{avg}} = \frac{\sqrt{\frac{3kT}{m}}}{\sqrt{\frac{8kT}{\pi m}}} = \sqrt{\frac{3\pi}{8}}\].

Step 4: Simplify the Ratio

Simplify \(\sqrt{\frac{3\pi}{8}}\) using a calculator: \[\sqrt{\frac{3\pi}{8}} \approx 1.086\].Thus, the ratio of root mean square velocity to average velocity is approximately 1.086.

Step 5: Identify the Correct Option

From the available options, the correct ratio corresponding to \(1.086:1\) matches option (c).

Key Concepts

Root Mean Square Velocity

The root mean square velocity, often abbreviated as \(v_{rms}\), is an important concept when studying gases. It provides a measure of the typical speed of gas particles in a sample. To calculate the root mean square velocity, we use the formula:

• \(v_{rms} = \sqrt{\frac{3kT}{m}} \)

Here:

• \(k\) is the Boltzmann constant, a fundamental physical constant that relates energy at the particle level with temperature, \(T\) is the temperature in Kelvin, and \(m\) is the mass of a single gas particle.

This formula tells us that the speed of particles depends on the temperature and the mass of the molecule. Higher temperatures mean higher speeds, as molecules move faster when they have more energy. Conversely, heavier molecules move slower at the same temperature. The root mean square velocity gives us a statistical measure that helps in understanding dynamic behaviors of gases, such as diffusion, effusion, and even conducting electricity.

Average Velocity of Gas Molecules

The average velocity of gas molecules, noted as \(v_{avg}\), gives us another way to quantify the motion of gases. This is derived from another statistical measure and provides insight into the overall behavior of gas molecules in a sample. It is calculated using the following formula:

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

Comparing this to the root mean square velocity formula, we notice the dependency on the same variables: the Boltzmann constant \(k\), temperature \(T\), and the mass of a gas molecule \(m\). The factor of \(\pi\) in the denominator slightly adjusts this measure from the root mean square velocity, giving a representation closer to the speeds most particles are likely to be found at. At any given temperature, though, the formulas reflect a similar trend: higher temperatures lead to higher average velocities. The distinction between \(v_{rms}\) and \(v_{avg}\) is subtle but significant, and crucial in predicting the cumulative motion characteristics of gas molecules.

Boltzmann Constant

The Boltzmann constant, denoted by \(k\), serves as a bridge between macroscopic and microscopic physics, playing a vital role in the kinetic theory of gases. It provides a way to relate the average kinetic energy of particles in a gas with the thermodynamic temperature of that gas:

• \(k = 1.38 \times 10^{-23} \text{ J/K}\)

In formulas like those for root mean square and average velocities, \(k\) helps us convert energy at the particle level (where velocities are calculated) to a much bigger scale, that of temperature.
With the Boltzmann constant, we grasp how gas laws apply on the microscopic scale. It enables us to understand temperature as a type of energy measurement, where higher temperatures indicate higher average kinetic energies for particles.