Problem 97 - A Complete Resource Book in Chemistry for JEE Main

Question

The root mean square velocity of one mole of a monoatomic gas having molar mass \(\mathrm{M}\) is \(\mathrm{u}_{\mathrm{rms}}\) ' The relation between the average kinetic energy (E) of the gas and \(u_{\text {rms }}\) is (a) \(\mathrm{u}_{\mathrm{nms}}=\sqrt{(3 \mathrm{E} / 2 \mathrm{M})}\) (b) \(\mathrm{u}_{\mathrm{rms}}=\sqrt{(2 \mathrm{E} / 3 \mathrm{M})}\) (c) \(\mathrm{u}_{\mathrm{rms}}=\sqrt{(2 \mathrm{E} / \mathrm{M})}\) (d) \(\mathrm{u}_{\mathrm{rms}}=\sqrt{(\mathrm{E} / 3 \mathrm{M})}\)

Step-by-Step Solution

Answer

The correct relation is (c) \( \mathrm{u}_{\mathrm{rms}}=\sqrt{\frac{2 \mathrm{E}}{\mathrm{M}}} \).

Step 1: Recall the formulas

The root mean square velocity \( u_{rms} \) of a gas is given by \( u_{rms} = \sqrt{\frac{3kT}{m}} \) where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of one molecule of gas. For one mole, \( M \) is the molar mass such that \( m = \frac{M}{N_A} \), where \( N_A \) is Avogadro's number.

Step 2: Relate kinetic energy to temperature

The average kinetic energy of one mole of a monoatomic gas is given by \( E = \frac{3}{2}RT \), where \( R \) is the ideal gas constant. For an individual molecule, \( E_{molecule} = \frac{3}{2}kT \).

Step 3: Substitute temperature in terms of kinetic energy

Since for one mole \( E = \frac{3}{2}RT \), we can express temperature as \( T = \frac{2E}{3R} \). Substituting this into the formula for \( u_{rms} \), we get \( u_{rms} = \sqrt{\frac{3k(\frac{2E}{3R})}{m}} = \sqrt{\frac{2Ek}{mR}} \).

Step 4: Simplify using mass relations

Replace \( m \) with \( \frac{M}{N_A} \), thus: \( u_{rms} = \sqrt{\frac{2E \cdot k \cdot N_A}{M \cdot R}} \). Since \( R = kN_A \), the formula simplifies to \( u_{rms} = \sqrt{\frac{2E}{M}} \).

Step 5: Verify the correct expression with given options

The expression \( u_{rms} = \sqrt{\frac{2E}{M}} \) matches option (c) of the given options in the exercise.

Key Concepts

Root Mean Square Velocity

Understanding root mean square velocity involves grasping how the particles in a gas move at various speeds due to thermal energy. This concept is crucial in the kinetic theory of gases, which helps explain the motion of gas particles.
The root mean square velocity, often abbreviated as \( u_{rms} \), is a way to quantify the average speed of particles. Mathematically, it is defined by the formula:
\[ u_{rms} = \sqrt{\frac{3kT}{m}} \]
where:

• \( k \) is the Boltzmann constant, which relates the kinetic energy of particles to their temperature in kelvins.
• \( T \) stands for the temperature of the gas.
• \( m \) is the mass of a single molecule of the gas.

For one mole of a gas, where we consider molar mass (\( M \)) instead of molecular mass, the expression changes slightly. Instead of \( m \), we use \( \frac{M}{N_A} \), where \( N_A \) is Avogadro's number. This adjustment helps in calculating \( u_{rms} \) for a mole of gas rather than a single molecule.
Understanding \( u_{rms} \) gives insight into how fast gas particles, on average, are moving and is an essential aspect of studying gas behaviors under various conditions.

Average Kinetic Energy

The average kinetic energy of gas molecules is foundational to understanding the kinetic theory of gases. This energy is due to the constant, random motion of particles in a gas and can be directly related to the temperature of the gas.
The average kinetic energy \( E \) of a monoatomic gas is given by the equation:
\[ E = \frac{3}{2}RT \]
where:

• \( R \) is the ideal gas constant, linking the microscopic and macroscopic states of a gas.
• \( T \) is the absolute temperature measured in kelvins.

This formula signifies that the kinetic energy of one mole of gas is proportional to its temperature.
In simpler terms, as the temperature increases, the kinetic energy of the gas particles also increases, causing them to move faster. This concept is crucial for predicting how gases behave at various temperatures and pressures, illustrating why gases expand when heated and contract when cooled.

Molar Mass

Molar mass is a key concept often encountered in chemistry and physics when dealing with large numbers of atoms or molecules, like in gases. It represents the mass of one mole of a substance, typically expressed in grams per mole.
In terms of gases, the molar mass \( M \) is vital for calculating various properties, such as the root mean square velocity and average kinetic energy.
For instance, in the root mean square velocity formula, we substitute the molecular mass with molar mass by using the relation:
\[ m = \frac{M}{N_A} \]
where:

• \( M \) is the molar mass.
• \( N_A \) is Avogadro's number, which provides the scale to move from microscopic to macroscopic quantities.

By understanding molar mass in these contexts, we can better comprehend how much energy is involved in the movement of gas molecules, assisting in practical applications and calculations involving gas behavior.