Problem 29 - A Complete Resource Book in Chemistry for JEE Main

Question

If a gas contains only three molecules that move with velocities of \(100,200,500 \mathrm{~ms}^{-1}\), what is the rms velocity of the gas is \(\mathrm{ms}^{-1}\) ? (a) \(100 \sqrt{8 / 3}\) (b) \(100 \sqrt{30}\) (c) \(100 \sqrt{0}\) (d) \(800 / 3\)

Step-by-Step Solution

Answer

The rms velocity is \( 100 \sqrt{10} \), not exactly matching any given options directly.

Step 1: Understand the rms Velocity Formula

The root mean square (rms) velocity of gas particles is given by the formula \( v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + \ldots}{n}} \), where \( v_1, v_2, v_3, \ldots \) are the velocities of individual molecules and \( n \) is the total number of molecules.

Step 2: Identify Given Velocities

We are given three velocities: 100 m/s, 200 m/s, and 500 m/s.

Step 3: Substitute the Velocities into the Formula

Substitute the given velocities into the rms velocity formula: \( v_{rms} = \sqrt{\frac{100^2 + 200^2 + 500^2}{3}} \).

Step 4: Calculate the Squares of Velocities

Calculate the square of each velocity: \( 100^2 = 10000 \), \( 200^2 = 40000 \), \( 500^2 = 250000 \).

Step 5: Sum the Squared Velocities

Add the squared velocities: \( 10000 + 40000 + 250000 = 300000 \).

Step 6: Divide by the Number of Molecules

Divide the sum of squared velocities by the number of molecules: \( \frac{300000}{3} = 100000 \).

Step 7: Take the Square Root

Take the square root of the result from Step 6: \( \sqrt{100000} = 100 \sqrt{10} \).

Step 8: Match with Given Options

Identify which of the given options matches \( 100 \sqrt{10} \). Since none of the squared terms in the options directly match, reevaluate logic transformation if needed, often directly or indirectly derived from alternative expressions.

Key Concepts

RMS Velocity

RMS velocity, or root mean square velocity, is a valuable concept in understanding the speed at which gas molecules are moving. It represents the square root of the average of the squares of individual velocities of molecules. This is particularly useful when analyzing gases, as it provides a good average speed of the gas particles, even when they move in various directions and speeds.
To calculate the rms velocity, use the formula:- \( v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + \ldots}{n}} \) - \( v_1, v_2, v_3, \ldots \) are the velocities of the gas molecules - \( n \) is the total number of molecules
This formula helps account for the diverse velocities and give you an average speed in a form that's always a positive number, which makes it useful for comparing with theoretical predictions and measurements.

Molecular Velocities

When discussing molecular velocities, it's crucial to consider that molecules in gases move at different speeds. This means particles constantly zip, collide, and bounce off each other, resulting in a variety of velocities across the gas molecules.
**Key points about molecular velocities:**

• Molecular motion in gases is random and chaotic. Each molecule doesn't have a fixed speed or direction.
• Speeds vary due to factors like energy transfers from collisions.
• The temperature of the gas has a direct relationship with average molecular velocity. Higher temperatures give more energy, thus increasing velocity.

Understanding these velocities helps us make sense of gas behavior and predict how changes like pressure and temperature will influence a gas's dynamics.

Velocity Distribution

Velocity distribution in a gas addresses how different speeds are distributed among molecules. The distribution arises from the random nature of molecular motions where a spread of velocities exists, governed by statistical distribution laws.
The key characteristic of velocity distribution in gases is that:

• It usually follows the Maxwell-Boltzmann distribution, which predicts how molecular speeds disseminate at given temperatures.
• This distribution allows predictions about the fraction of molecules moving at certain velocities.
• It also helps explain phenomena like diffusion rates and energy transfers in thermodynamic systems.

By analyzing velocity distribution, scientists can determine essential properties of gases, such as diffusion coefficients and viscosity, contributing to a deeper understanding of thermodynamic processes.