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 ratio is (c) \(1.086: 1\).

Step 1: Identify Relevant Formulas

To solve this problem, we use the formulas for root mean square velocity \(v_{rms}\) and average velocity \(v_{avg}\) of gas molecules. They are given by:\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]\[ v_{avg} = \sqrt{\frac{8RT}{\pi M}} \]where \(R\) is the ideal gas constant, \(T\) is the absolute temperature, and \(M\) is the molar mass of the gas.

Step 2: Express the Ratio

We need to find the ratio \(\frac{v_{rms}}{v_{avg}}\) which results in:\[\frac{v_{rms}}{v_{avg}} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{8RT}{\pi M}}}\]This simplifies to the ratio of the constants under the square root.

Step 3: Simplify the Ratio

Simplifying this equation gives:\[\frac{v_{rms}}{v_{avg}} = \sqrt{\frac{3}{8/\pi}} = \sqrt{\frac{3\pi}{8}}\]Substituting \(\pi \approx 3.1415\) into the equation, we calculate \(\sqrt{\frac{3 \times 3.1415}{8}}\).

Step 4: Calculate Numerically

Calculate \(\sqrt{\frac{3 \times 3.1415}{8}} = \sqrt{1.178}\) using the approximation \(\sqrt{1.178} \approx 1.086\). This implies:\[ \frac{v_{rms}}{v_{avg}} \approx 1.086 \]

Step 5: Select the Correct Answer

The calculated ratio \(v_{rms} : v_{avg} = 1.086 : 1\) matches option (c). Therefore, the correct answer is option (c) \(1.086 : 1\).

Key Concepts

Root Mean Square Velocity

Root mean square velocity, denoted as \( v_{rms} \), is a measure of the speed of gas molecules. It provides an average velocity, considering the squared velocities of the molecules but scaled back by taking the square root.
This metric is particularly useful because it accounts for the entire distribution of molecular speeds in a gas sample. The formula for calculating \( v_{rms} \) is:

• \( v_{rms} = \sqrt{\frac{3RT}{M}} \)

Here:

• \( R \) is the ideal gas constant.
• \( T \) is the absolute temperature.
• \( M \) is the molar mass of the gas.

The root mean square velocity is particularly significant because it connects thermodynamic quantities like temperature and molar mass to molecular motion. It shows how the thermal energy within the gas affects molecular velocities.

Average Velocity

Average velocity, represented as \( v_{avg} \), is another way to describe the speed of gas molecules.
It is based on the principle of calculating the mean speed of all molecules without any squaring involved, thus it’s a direct mean. The formula to calculate \( v_{avg} \) is given by:

• \( v_{avg} = \sqrt{\frac{8RT}{\pi M}} \)

In this formula:

• \( R \) is the ideal gas constant.
• \( T \) is the absolute temperature.
• \( M \) is the molar mass of the gas.
• \( \pi \) is a mathematical constant approximately equal to 3.1415.

The average velocity provides a simpler but less encompassing picture of the molecular speeds as compared to the root mean square velocity.

Ratio of Velocities

The comparison between root mean square velocity \( v_{rms} \) and average velocity \( v_{avg} \) of gas molecules is often expressed as a ratio. Finding this ratio helps to understand the relationship between different measures of molecular speed, and in the given problem, the aim is to calculate this ratio.
To determine this, you use the expressions:

• \( \frac{v_{rms}}{v_{avg}} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{8RT}{\pi M}}} \)

Simplifying further, we remove the common factors and are left with:

• \( \frac{v_{rms}}{v_{avg}} = \sqrt{\frac{3\pi}{8}} \)

Calculating this yields approximately \( 1.086 \), meaning the root mean square velocity is about 1.086 times the average velocity. This ratio shows the distinct influence of different statistical treatments of molecular velocity.

Ideal Gas Law

The ideal gas law plays a crucial role in understanding gas behavior, particularly when linking macroscopic properties like pressure, volume, and temperature with molecular properties such as velocities. The equation represents the relationship as follows:

• \( PV = nRT \)

In this equation:

• \( P \) stands for pressure.
• \( V \) represents volume.
• \( n \) is the number of moles of gas.
• \( R \) is the ideal gas constant.
• \( T \) is the absolute temperature.

Connecting the dots between the Ideal Gas Law and molecular velocities lies in the temperatures' effect. As temperature increases while observing the ideal gas law, it results in higher molecular velocities, given that the other factors remain constant. Understanding how temperature and other variables interplay with gas molecules helps give a more complex picture of gas behavior and kinetic theory.