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 \(\mathrm{rms}\) velocity of the gas is \(\mathrm{ms}^{-1 ?}\) (a) \(100 \sqrt{8 / 3}\) (b) \(100 \sqrt{30}\) (c) \(100 \vee 0\) (d) \(800 / 3\)

Step-by-Step Solution

Answer

The RMS velocity is \(100\sqrt{10} \ \text{ms}^{-1}\), matching option (b).

Step 1: Understanding RMS Velocity

The root mean square (RMS) velocity is a measure of the speed of particles in a gas, which is calculated by taking the square root of the average of the squares of the individual velocities of the gas particles.

Step 2: Listing Given Velocities

The velocities given for the gas molecules are: \(v_1 = 100 \ ms^{-1}\), \(v_2 = 200 \ ms^{-1}\), and \(v_3 = 500 \ ms^{-1}\).

Step 3: Calculating the Square of Each Velocity

Calculate the squares of each individual velocity: - \(v_1^2 = (100)^2 = 10000\)- \(v_2^2 = (200)^2 = 40000\)- \(v_3^2 = (500)^2 = 250000\)

Step 4: Finding Average of the Squares

Calculate the average of the squared velocities: \[ \text{Average} = \frac{v_1^2 + v_2^2 + v_3^2}{3} = \frac{10000 + 40000 + 250000}{3} = \frac{300000}{3} = 100000 \]

Step 5: Calculating the RMS Velocity

Take the square root of the average obtained in the previous step to find the RMS velocity: \[ \text{RMS velocity} = \sqrt{100000} = 100\sqrt{10} \]

Step 6: Comparing to Given Answers

Compare \(100\sqrt{10}\) with the options provided. It matches option (b), which is \(100\sqrt{10}\).

Key Concepts

Molecular Velocities

Molecular velocities help us understand how fast particles in a gas are moving. This speed is important because it influences how gases interact and react. Each particle in a gas does not move at the same speed due to their different kinetic energies.
Some particles move very fast while others move slowly. The distribution of these different speeds is often described by the Maxwell-Boltzmann distribution. For example, in the given problem, the molecules have velocities of 100, 200, and 500 ms^-1. These different speeds give us insights into the behavior of the gas particles overall.

Root Mean Square Calculation

The root mean square (RMS) calculation is a valuable method for finding an average velocity when considering individual particle speeds. To calculate the RMS velocity, you follow these steps:

• Square each of the individual velocities of the molecules.
• Find the average of these squared values.
• Take the square root of this average.

This method considers both the magnitude and the direction squared of each speed, making it an effective way to measure the effect of all particles. In the example given, the RMS velocity is calculated by first finding the squares of given velocities, summing them up, and then dividing by the number of particles. The final step involves taking the square root of this average to get the RMS velocity, which represents a form of "average" velocity for the entire collection of gas particles.

Gas Particle Speeds

Gas particle speeds can vary widely between individual particles in a sample. Unlike solids or liquids where particles have restricted mobility, gas particles move quickly and independently. They frequently collide with each other and the walls of their container.
This results in a range of velocities as observed with our problem's values of 100, 200, and 500 ms^-1. RMS velocity provides a useful average speed because it takes into account the squared velocities of these particles – allowing more energy-intensive speeds to impact the average more significantly.
Understanding these speeds is crucial for predicting the diffusion and reaction rates of gases, and it can also help in determining the temperature and pressure relationships in a gas sample.