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

Question

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

Option (c) \( \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \) is correct.

Step 1: Understanding Types of Velocities

In kinetic theory of gases, the most probable velocity, average velocity, and root mean square (RMS) velocity are different measures of the speed of gas molecules.- **Most Probable Velocity (\(v_{mp}\))** is the speed at which the largest number of molecules are moving- **Average Velocity (\(v_{avg}\))** is the mean velocity of all molecules- **Root Mean Square Velocity (\(v_{rms}\))** is the square root of the mean of the squares of individual velocities.

Step 2: Mathematical Expressions for Velocities

The mathematical expressions for these velocities for an ideal gas are: - \(v_{mp} = \sqrt{\frac{2kT}{m}}\)- \(v_{avg} = \sqrt{\frac{8kT}{\pi m}}\)- \(v_{rms} = \sqrt{\frac{3kT}{m}}\)where \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of a molecule.

Step 3: Finding the Ratios

To determine the ratios, we compare the equations of each velocity form. 1. **Most Probable Velocity**: \(v_{mp} = \sqrt{2}\cdot \sqrt{\frac{kT}{m}}\), 2. **Average Velocity**: \(v_{avg} = \sqrt{\frac{8}{\pi}}\cdot \sqrt{\frac{kT}{m}}\), 3. **RMS Velocity**: \(v_{rms} = \sqrt{3}\cdot \sqrt{\frac{kT}{m}}\).The ratio becomes: \[v_{mp} : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3}\].

Step 4: Matching with Given Choices

Compare the calculated ratio \(\sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3}\) to the options provided. The correct choice matches with option (c) which is \(\sqrt{2} : \sqrt{(8 / \pi)}: \sqrt{3}\).

Key Concepts

Ideal Gas

An ideal gas is a theoretical concept in physics and chemistry that describes a hypothetical gas composed of many randomly moving particles that are not subject to intermolecular forces. It provides a simplified model that facilitates the understanding of real gases under certain conditions. This model is governed by the ideal gas law, which is expressed as: \[ PV = nRT \]where:

• P is the pressure
• V is the volume
• n is the amount of substance in moles
• R is the ideal gas constant
• T is the temperature in Kelvin

The simplicity of the ideal gas law allows us to predict the behavior of gases under a range of conditions, although it is most accurate for gases at high temperatures and low pressures.
In the context of kinetic theory, the motion of the molecules in an ideal gas is described by the three types of velocities: most probable, average, and root mean square velocities. Understanding these velocities helps us analyze and describe the dynamic behavior of gases, which is important for a variety of practical applications.

Most Probable Velocity

Most probable velocity, often symbolized as \(v_{mp}\), refers to the velocity at which the largest number of gas molecules are moving in an ideal gas. It's a useful measure because it gives insights into the distribution of molecular speeds.
This velocity can be calculated mathematically by:\[ v_{mp} = \sqrt{\frac{2kT}{m}} \]where:

• k is the Boltzmann constant
• T is the absolute temperature
• m is the mass of a gas molecule

Compared to other velocity types, the most probable velocity is often lower because it represents the peak of the speed distribution curve. It tells us how fast the majority of molecules are moving and helps in understanding how temperature influences molecular speeds in a gas.

Average Velocity

Average velocity, denoted as \(v_{avg}\), is another fundamental concept in the kinetic theory of gases. This measure looks at the mean speed of all molecules in the gas.
The formula for calculating the average velocity is:\[ v_{avg} = \sqrt{\frac{8kT}{\pi m}} \]where:

• k is the Boltzmann constant
• T is the absolute temperature
• m is the mass of a molecule

The average velocity provides an insight into the general speed of particles and is an intermediate value when compared to the most probable and root mean square velocities. It helps predict how molecular motion varies with changes in temperature and mass.

Root Mean Square Velocity

Root mean square velocity, abbreviated as RMS velocity and represented as \(v_{rms}\), is a measure that accounts for all individual speeds and provides a kind of average magnitude.
It is calculated using the formula:\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]For:

• k as the Boltzmann constant
• T as the absolute temperature
• m as the mass of a molecule

RMS velocity is the highest among the three velocities because it is derived from the average of the squares of the velocities. This makes it particularly useful in understanding the kinetic energy of gas molecules since kinetic energy is related to the square of the velocity.
In conclusion, the ratio of most probable, average, and root mean square velocities provides crucial insights into the distribution of molecular speeds, helping us better understand gas behavior under different conditions.