Problem 129 - A Complete Resource Book in Chemistry for JEE Main

Question

In the given sample of a gas all molecules do not possess same speed. Due to frequent molecular collisions, the molecules move with ever changing speeds and also with changing direction. There are three types of velocities (i) root mean square velocity (ii) average velocity and (iii) most probable velocity. Relation between the three types of velocities, i.e., most probable velocity : average velocity: root mean square velocity is (a) \(\sqrt{3}: \sqrt{2}: \sqrt{\frac{8}{\pi}}\) (b) \(\sqrt{3}: \sqrt{2}: \sqrt{8}\) (c) \(\sqrt{2}: \sqrt{(8 / \pi)}: \sqrt{3}\) (d) \(1: 2: 3\)

Step-by-Step Solution

Answer

(c) \(\sqrt{2}: \sqrt{(8/\pi)}: \sqrt{3}\)

Step 1: Root Mean Square Velocity

The root mean square velocity, \( v_{rms} \), 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 molecule.

Step 2: Average Velocity

The average velocity, \( v_{avg} \), is given by the formula: \( v_{avg} = \sqrt{\frac{8kT}{\pi m}} \). This represents the mean of the molecular speed distribution.

Step 3: Most Probable Velocity

The most probable velocity, \( v_{mp} \), is obtained from: \( v_{mp} = \sqrt{\frac{2kT}{m}} \). This is the speed at which the greatest number of gas molecules are moving.

Step 4: Comparing Ratios of Velocities

To find the relation between the velocities, we analyze their ratios: \( v_{mp} = \sqrt{\frac{2kT}{m}} \), \( v_{avg} = \sqrt{\frac{8kT}{\pi m}} \), and \( v_{rms} = \sqrt{\frac{3kT}{m}} \). This gives the ratios: \( \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \).

Key Concepts

Understanding Root Mean Square Velocity

The root mean square velocity (RMS velocity) is an important concept in the kinetic molecular theory of gases. It describes the square root of the average of the squares of the velocities of the gas molecules. This can be thought of as a type of "average" velocity that is particularly useful in physics since it takes into account the direction and magnitude of velocity. The formula for calculating RMS velocity is \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the absolute temperature in Kelvin, and \( m \) is the mass of a single molecule of the gas. This formula tells us that the RMS velocity depends directly on the square root of temperature and inversely on the square root of molecular mass. Thus, as the temperature of a gas increases, so does the RMS velocity, indicating that the molecules are moving faster. Simultaneously, heavier molecules move slower because their mass has an inverse impact on velocity.

Diving into Average Velocity

Average velocity provides a simple mean speed value for gas molecules and is given through the expression \( v_{avg} = \sqrt{\frac{8kT}{\pi m}} \). This velocity is different from the root mean square velocity as it is derived by taking the arithmetic mean of all possible molecular speeds. Influenced by both temperature and mass of the gas molecules, average velocity increases with rising temperatures, indicating hastier molecular motion, just like RMS velocity. However, the interesting part is that this velocity averages over a greater set of velocities compared to RMS, considering even those at the tails of the velocity distribution. It's crucial because the average velocity gives us deeper insights into how temperature manages to impact the molecular movement in the gas sample, illustrated well by its dependence on \( \sqrt{\frac{8}{\pi}} \), showing a clear distinction from the other types of velocities.

Decoding Most Probable Velocity

The most probable velocity points to the speed at which the maximum number of molecules are moving in a gas sample. Calculated using the equation \( v_{mp} = \sqrt{\frac{2kT}{m}} \), it represents the peak of the Maxwell-Boltzmann distribution curve. This is quite intuitive in nature as it showcases the velocity that is most likely to occur among all possible speeds. Factors such as temperature and molecular mass also affect this velocity significantly: as the temperature increases, molecules tend to move faster, increasing the most probable velocity. Whereas heavier molecules, having more mass, shift the most probable velocity lower. Understanding most probable velocity helps one appreciate the distribution spread among molecules in a gaseous sample and provides practical understanding of temperature's role in kinetic activity distribution.