Problem 100 - Master Resource Book in JEE Main Physics

Question

\(N\) molecules, each of mass \(m\), of gas \(A\) and \(2 \mathrm{~N}\) molecules, each of mass \(2 m\), of gas \(B\) are contained in the same vessel which is maintained at a temperature \(T\). The mean square velocity of molecules of \(B\) type is denoted by \(V_{2}\) and the mean square velocity of \(A\) type is denoted by \(V_{1}\), then \(\frac{V_{1}}{V_{2}}\) is (a) 2 (b) (c) \(1 / 3\) (d) \(2 / 3\)

Step-by-Step Solution

Answer

The ratio \(\frac{V_1}{V_2}\) is 2, answer (a).

Step 1: Understand the Problem

The exercise asks for the ratio of the mean square velocities of two different gases, A and B, in a mixture. We are given the number of molecules and their masses, and the temperature is constant.

Step 2: Recall the Formula for Mean Square Velocity

The mean square velocity \(V_i\) of gas molecules at temperature \(T\) can be calculated using the formula: \[ V_i = \frac{3kT}{m_i} \] where \(m_i\) represents the mass of a type of gas and \(k\) is the Boltzmann constant. Since both gases are at the same temperature, their mean square velocities only depend on their masses.

Step 3: Calculate Mean Square Velocity for Gas A

For gas A, each molecule has mass \(m\), so the mean square velocity \(V_1\) is given by \[ V_1 = \frac{3kT}{m} \].

Step 4: Calculate Mean Square Velocity for Gas B

For gas B, each molecule has mass \(2m\), so the mean square velocity \(V_2\) is given by \[ V_2 = \frac{3kT}{2m} \].

Step 5: Calculate the Ratio \(\frac{V_1}{V_2}\)

To find the ratio \(\frac{V_1}{V_2}\), substitute the expressions for \(V_1\) and \(V_2\): \[ \frac{V_1}{V_2} = \frac{\frac{3kT}{m}}{\frac{3kT}{2m}} = \frac{2}{1} = 2 \].

Step 6: Select the Correct Answer

From the given options: (a) 2, (b), (c) \(1/3\), (d) \(2/3\), the ratio \(\frac{V_1}{V_2} = 2\), so the correct answer is (a) 2.

Key Concepts

Mean Square Velocity

In kinetic theory, understanding the mean square velocity of gas molecules is essential for analyzing behaviors of gases. Mean square velocity refers to the average of the squares of the speeds of all molecules in a gas sample. It is an indicator of how fast gas molecules are moving on average.

The formula used to calculate the mean square velocity of gas molecules is given by \[ V_i = \frac{3kT}{m_i} \] Where:

• \( V_i \) is the mean square velocity of a molecule type i,
• \( k \) is the Boltzmann constant,
• \( T \) is the absolute temperature, and
• \( m_i \) is the mass of a molecule of the gas.

In the context of the question, both gases A and B are exposed to the same temperature. Thus, their mean square velocity primarily depends on their masses. Gas A has molecules of mass \( m \) and gas B has molecules of mass \( 2m \). This results in different velocities for each type, calculable with the formula above.

Boltzmann Constant

The Boltzmann constant \( k \) is a fundamental factor in statistical mechanics, acting as a bridge between the macroscopic and microscopic worlds. The Boltzmann constant is a proportionality factor that relates the average kinetic energy of particles in a gas with the temperature of the gas.

This is significant in the field of thermodynamics where \( k \) is instrumental in defining the kinetic nature of gases. In the formula for mean square velocity, \( k \) plays a crucial role since it standardizes temperature so it can be related to molecular motion.

• The value of the Boltzmann constant is approximately \( 1.38 \times 10^{-23} \, \text{J/K} \).
• This small number signifies the tiny energies involved at the level of individual atoms and molecules.

Overall, the Boltzmann constant provides a means to understand how temperature influences the speed and energy distribution of molecules.

Gas Mixtures

In the study of gas mixtures, examining the properties of individual gas components is crucial. A gas mixture, such as the one in the exercise, consists of different gases sharing the same volume and temperature. Properties like mean square velocity can vary in mixtures depending on the mass and the number of molecules of each gas.

For instance, if you have a mixture of gases A and B, where gas A is lighter than gas B and we have equal numbers of molecules of both gases, A will have a higher mean square velocity due to its lighter mass. However, in our scenario, there are \(N\) molecules of gas A and \(2N\) molecules of gas B, with differing masses \(m\) and \(2m\) respectively, affecting their velocities differently.

• Such mixtures illustrate how kinetic energy distributes among various gases within a container.
• Consequently, the behavior of each type of molecule contributes to the overall properties of the mixture.

In conclusion, considering different aspects like mean square velocity helps in understanding molecular behavior in gas mixtures.