Problem 107 - A Complete Resource Book in Chemistry for JEE Main

Question

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

Step-by-Step Solution

Answer

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

Step 1: Root Mean Square Velocity Formula

The root mean square velocity \( v_{rms} \) of gas molecules can be calculated using the formula: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvins, and \( m \) is the mass of the gas molecule.

Step 2: Average Velocity Formula

The average velocity \( v_{avg} \) of gas molecules is given by: \[ v_{avg} = \sqrt{\frac{8kT}{\pi m}} \] This formula also uses the Boltzmann constant \( k \), temperature \( T \), and molar mass \( m \).

Step 3: Ratio Calculation

To find the ratio of \( v_{rms} \) to \( v_{avg} \), divide the \( v_{rms} \) formula by the \( v_{avg} \) formula: \[ \frac{v_{rms}}{v_{avg}} = \frac{\sqrt{\frac{3kT}{m}}}{\sqrt{\frac{8kT}{\pi m}}} \] Simplifying, this becomes \[ \frac{v_{rms}}{v_{avg}} = \sqrt{\frac{3 \pi}{8}} \approx 1.086 \].

Step 4: Choose the Correct Ratio

The computed ratio \( \frac{v_{rms}}{v_{avg}} \) is approximately 1.086. Therefore, the correct given answer choice is \( (c) \ 1.086: 1 \).

Key Concepts

Root Mean Square Velocity

When we talk about the root mean square (RMS) velocity, we're looking at a way to describe how fast gas molecules are moving in a container. RMS velocity gives us an "average" speed of molecules, but in a specific mathematical sense. It's like squaring each molecule's velocity, taking an average, and then the square root of that total. This method helps us understand the "typical" speed at which molecules travel.

The RMS velocity is calculated using the formula:

• \[ v_{rms} = \sqrt{\frac{3kT}{m}} \]

Here:

• \( k \) is the Boltzmann constant, an important number that links the average kinetic energy of particles in a gas with the temperature of the gas.
• \( T \) represents the temperature of the gas in Kelvin.
• \( m \) is the mass of a gas molecule which affects how fast it moves.

Average Velocity

Average velocity is another method to gauge the speed of gas molecules. Unlike RMS velocity, average velocity simply takes the mean speed of all the molecules. This metric offers a baseline understanding but does not take direction or molecule mass into account in the same way RMS does.

The formula for average velocity is:

• \[ v_{avg} = \sqrt{\frac{8kT}{\pi m}} \]

Notice the formula is quite similar to the RMS velocity, but it involves a different constant and factor:

• \( \pi \), the mathematical constant, appears here, impacting the calculated average speed.

This tiny tweak results in a slightly different number, thereby giving us a slightly different perspective on the movement of gas molecules.

Gas Laws

Gas laws are the rules that describe how gases behave under varying conditions of pressure, volume, and temperature. They form the foundation for understanding complex topics in physics and chemistry.

Some of the foundational gas laws include:

• Boyle's Law: This states that the pressure and volume of a gas have an inverse relationship when temperature is constant.
• Charles's Law: It establishes that the volume and temperature of a gas are directly proportional at constant pressure.
• Avogadro's Law: It explains how volume and number of moles are directly proportional at constant temperature and pressure.

Together, these laws can be expressed in the Ideal Gas Law, \( PV = nRT \), which connects pressure (P), volume (V), number of moles (n), and temperature (T) using the gas constant (R). Understanding these laws helps in analyzing gas behaviors such as those calculated in RMS and average velocities.

Boltzmann Constant

The Boltzmann constant, \( k \), is a cornerstone of the kinetic theory of gases. This constant connects microscopic particle interactions with macroscopic systems. It defines the relationship between temperature and energy at the particle level by furnishing the number of joules per kelvin per particle.

In formulas like those for RMS and average velocity, \( k \) becomes pivotal because it scales temperature to energy:

• \( T \), the absolute temperature, when multiplied by \( k \), provides the average kinetic energy per molecule.

Its value roughly equals \( 1.38 \times 10^{-23} \text{ J/K} \). This number helps scientists and students alike translate temperature into energy, helping identify how temperature affects speed and movement of molecules in gases. Understanding this concept demonstrates how integrated and precise the kinetic theory of gases really is.