Problem 18 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The ratio of root mean square velocity to average velocity of a gas molecule at a particular temperature is (a) \(1.086: 1\) (b) \(1: 1.086\) (c) \(2: 1.086\) (d) \(1.086: 2\)

Step-by-Step Solution

Answer

The correct answer is (a) \(1.086: 1\).

Step 1: Understand the Concept

The root mean square (rms) velocity of gas molecules is a measure of the speed of gas particles and is given by the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of a gas particle. The average velocity (mean velocity) of gas molecules is \( v_{avg} = \sqrt{\frac{8kT}{\pi m}} \). Our task is to find the ratio of these two velocities.

Step 2: Find the Formula for the Ratio

To find the ratio of the root mean square velocity to the average velocity, we need to divide the formula for \( v_{rms} \) by the formula for \( v_{avg} \). This gives: \[\frac{v_{rms}}{v_{avg}} = \frac{\sqrt{\frac{3kT}{m}}}{\sqrt{\frac{8kT}{\pi m}}}.\] Simplify this expression to find the actual ratio.

Step 3: Simplify the Ratio

In the expression \( \frac{\sqrt{\frac{3kT}{m}}}{\sqrt{\frac{8kT}{\pi m}}} \), the terms \( kT/m \) will cancel out each other under the square root, giving: \[\frac{v_{rms}}{v_{avg}} = \sqrt{\frac{3}{\frac{8}{\pi}}} = \sqrt{\frac{3\pi}{8}}.\] Calculating this gives approximately \(1.086\). This means the ratio is approximately \(1.086: 1\).

Step 4: Choose the Correct Answer

The ratio of root mean square velocity to average velocity is found to be approximately \(1.086: 1\). Hence, the correct answer among the given options is (a) \(1.086: 1\).

Key Concepts

Root Mean Square Velocity

Root mean square (RMS) velocity is a statistical measure used in the kinetic theory of gases to determine the average speed of particles in a gas. It is represented by the formula: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] Here, \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of a single gas molecule. The RMS velocity gives an idea of how fast the gas molecules are moving, helping us understand the kinetic energy involved. This concept considers the velocities of all particles, squares them to remove the direction, averages these values, and then takes the square root to bring the units back to speed.

• Measures average speed effectively incorporating all molecules.
• Useful in calculations involving kinetic energy.
• Proportional to the square root of temperature.

Average Velocity of Gas Molecules

The average velocity of gas molecules, often referred to as mean velocity, is another important concept in understanding gas behavior. It provides a simpler but less precise measure compared to RMS velocity and is calculated using the formula: \[ v_{avg} = \sqrt{\frac{8kT}{\pi m}} \] The symbols here remain the same: \(k\) is the Boltzmann constant, \(T\) is the temperature, \(m\) is the mass of the gas particle, and \(\pi\) is a mathematical constant. The average velocity accounts for the vector nature of velocity by considering all speeds, discarding direction altogether, and averaging them to provide a single value.

• Provides simpler calculation than RMS velocity.
• Also relates to temperature and molecular mass.
• Can be less useful for precise kinetic calculations but offers a quick overview.

Boltzmann Constant

The Boltzmann constant \(k\) plays a crucial role in both the RMS and average velocities by linking the microscopic and macroscopic worlds. It's a fundamental constant used in statistical mechanics with a value of approximately \(1.38 \times 10^{-23} \) J/K. This constant appears in several formulas and helps in deriving properties of gases, based on molecular scale physics. Understanding the Boltzmann constant is crucial for grasping how temperature affects molecular speed and energy. As one can see in the formulas for both RMS and average velocity, temperature \(T\) directly relates to the velocities through \(k\). The constant allows us to branch from the particle level to tangible concepts like temperature and energy in thermodynamics.

• Relates temperature to energy at the molecular level.
• Facilitates calculations in thermodynamics and statistical mechanics.
• Embedded in formulas describing particle speed and kinetic energy.