Problem 101 - Master Resource Book in JEE Main Physics

Question

Calculate the rms speed of smoke particles each of mass \(5 \times 10^{-17} \mathrm{~kg}\) in their Brownian motion in air at \(\mathrm{NTP}\left(k=1.38 \times 10^{-23} \mathrm{JK}^{-1}\right)\) (a) \(1.5 \mathrm{~mm} \mathrm{~s}^{-1}\) (b) \(1.5 \mathrm{~ms}^{-1}\) (c) \(1.5 \mathrm{cms}^{-1}\) (d) \(1.5 \mathrm{kms}^{-1}\)

Step-by-Step Solution

Answer

The rms speed is \(1.5 \ \mathrm{cms}^{-1}\), option (c).

Step 1: Understanding RMS Speed Formula

The root mean square (rms) speed of particles in a gas can be calculated using the formula: \( v_{rms} = \sqrt{\frac{3kT}{m}} \) where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of one particle. In this problem, \( k = 1.38 \times 10^{-23} \ \mathrm{JK}^{-1} \), and \( m = 5 \times 10^{-17} \ \mathrm{kg} \). At NTP, \( T = 273 \) K.

Step 2: Substitute Values into Formula

Plug the known values into the RMS speed formula: \[ v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 273}{5 \times 10^{-17}}} \]

Step 3: Calculate Numerator

First, compute the product in the numerator: \( 3 \times 1.38 \times 10^{-23} \times 273 = 1.13 \times 10^{-20} \).

Step 4: Calculate Denominator

The denominator is the mass of the particle: \( 5 \times 10^{-17} \).

Step 5: Divide and Simplify

Divide the numerator by the denominator to simplify: \[ \frac{1.13 \times 10^{-20}}{5 \times 10^{-17}} = 2.26 \times 10^{-4} \].

Step 6: Take the Square Root

Finally, take the square root of the result to find the RMS speed: \( v_{rms} = \sqrt{2.26 \times 10^{-4}} = 1.5 \times 10^{-2} \mathrm{~m/s} \).

Step 7: Convert to Appropriate Units

Convert the result to centimeters per second: \( 1.5 \times 10^{-2} \mathrm{~m/s} = 1.5 \mathrm{~cms}^{-1} \).

Key Concepts

Brownian Motion

When we observe tiny particles moving through a fluid, such as smoke particles in air, we notice they move in a random and unpredictable manner. This is known as Brownian motion. It was first observed by botanist Robert Brown in 1827.
Brownian motion happens due to the collisions between the small particles and the much smaller and faster-moving molecules of the fluid, like air molecules. This constant bombardment makes the path of the particles zigzag and erratic.
Understanding Brownian motion helps scientists study various processes, including diffusion, and provides insights into the nature of molecular forces. This motion is a macroscopic reflection of the kinetic theory of gases, exemplifying the activity of molecules at the microscopic level.

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental scientific theory that describes how gases behave. It postulates that gases are composed of a large number of small particles, either atoms or molecules, which are in constant random motion. This theory helps us understand gas properties and behaviors through three main assumptions:

• Gas particles move continuously and randomly.
• They collide with each other and the walls of their container.
• These collisions are perfectly elastic, meaning there is no loss of kinetic energy in the collision.

The kinetic theory provides an equation to relate the temperature of the gas to the average kinetic energy of its molecules. It explains how temperature, pressure, and volume relate through these particle motions. By using this theory, we can deduce interesting properties like pressure exerted by a gas, diffusion rates, and importantly, the root mean square (RMS) speed of the gas particles.

Boltzmann Constant

The Boltzmann constant (\( k \)) is a fundamental constant in physics that connects the microscopic mechanical energy of particles to their macroscopic state, like temperature. It appears in many fundamental equations, most notably in the formula for calculating the RMS speed of gas particles.
This constant, with a value of approximately \( 1.38 \times 10^{-23} \, \text{J/K} \), is essential in statistical mechanics and thermodynamics. It provides a bridge between microscopic physics and macroscopic thermodynamic quantities like pressure and temperature. The Boltzmann constant essentially sets the energy scale of thermal systems, allowing us to relate the energy of individual particles to the temperature of the system.
In our problem, the Boltzmann constant is crucial for calculating the RMS speed of smoke particles in air, capturing the essence of how much energy each particle has due to the temperature of its environment.