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 \(\mathrm{u}_{\mathrm{rms}}\) is (a) \(\mathrm{u}_{\mathrm{rms}}=\sqrt{(3 \mathrm{E} / 2 \mathrm{M})}\) (b) \(\mathrm{u}_{\mathrm{rms}}=\sqrt{(2 \mathrm{E} / 3 \mathrm{M})}\) (c) \(u_{r m s}=\sqrt{(2 E / M)}\) (d) \(\mathrm{u}_{\mathrm{rms}}=\sqrt{(\mathrm{E} / 3 \mathrm{M})}\)

Step-by-Step Solution

Answer

(c) \(u_{rms} = \sqrt{(2E/M)}\) is the correct relation.

Step 1: Understanding RMS Velocity

The root mean square (RMS) velocity of a gas is given by the formula \(u_{rms} = \sqrt{\frac{3RT}{M}}\), where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass.

Step 2: Relating RMS Velocity and Kinetic Energy

The average kinetic energy \(E\) for one mole of a monoatomic gas is given by \(E = \frac{3}{2} RT\).

Step 3: Substituting E in the RMS Formula

Substitute \(RT = \frac{2}{3}E\) into the RMS velocity equation: \(u_{rms} = \sqrt{\frac{3 \cdot (2E/3)}{M}}\).

Step 4: Simplifying the Expression

By simplifying \(u_{rms} = \sqrt{\frac{2E}{M}}\), we find the relationship between \(u_{rms}\) and \(E\).

Key Concepts

Kinetic Energy of Gases

In the world of gases, kinetic energy plays a fundamental role. Imagine countless tiny particles, each moving quickly and colliding with each other. For a gas, kinetic energy is the energy of motion of these particles.
It's directly linked to the temperature of the gas. The higher the temperature, the higher the kinetic energy of the particles. This is because temperature is a measure of how vigorously the particles are moving.

For a single mole of a monoatomic gas, the average kinetic energy is given by the equation \[E = \frac{3}{2} RT\] where

• \( E \) is the average kinetic energy,
• \( R \) is the ideal gas constant,
• and \( T \) is the temperature in Kelvin.

This equation shows that kinetic energy is only dependent on the temperature of the gas.

RMS Velocity Formula

The root mean square (RMS) velocity is a way to describe the speed of particles in a gas. Even though particles move in various directions at different speeds, the RMS velocity gives a kind of average speed that is useful for calculations.

For one mole of a gas, the RMS velocity is given by the equation:\[u_{rms} = \sqrt{\frac{3RT}{M}}\]In this equation:

• \( u_{rms} \) is the root mean square velocity,
• \( R \) is the ideal gas constant,
• \( T \) represents the temperature in Kelvin,
• and \( M \) is the molar mass of the gas.

This formula tells us that the RMS velocity depends on both the temperature and the molar mass of the gas. As temperature increases, the RMS velocity also increases. Meanwhile, if the molar mass is larger, the particles move slower on average.

Molar Mass and Velocity

Molar mass significantly influences the velocity of gas particles. Within the RMS velocity formula, you'll notice that molar mass \( M \) is in the denominator under a square root. This means it inversely affects the velocity.
To visualize this, imagine two different gases at the same temperature. A gas with smaller molar mass will have particles moving faster than a gas with a larger molar mass, because lighter molecules are easier to move.

Therefore, if the molar mass of a gas is doubled, the root mean square velocity would become:\[u_{rms} = \sqrt{\frac{3RT}{2M}}\]indicating a decrease in speed. Understanding this relationship is vital for predicting how gases will behave under various conditions.