Problem 156 - A Complete Resource Book in Chemistry for JEE Main

Question

For gaseous state if most probable speed is denoted \(\mathrm{C}^{*}\), average speed by \(\overline{\mathrm{C}}\) and and mean square speed by C, then for a large number of molecules the ratios of these speeds are: [2013] (a) \(\mathrm{C}^{*}: \overline{\mathrm{C}}: \mathrm{C}=1: 128: 1.225\) (b) \(\mathrm{C}^{*}: \overline{\mathrm{C}}=1: 1.225: 1.128\) (c) \(\mathrm{C}^{*}: \overline{\mathrm{C}}: \mathrm{C}=1.225: 1.128: 1\) (d) \(\mathrm{C}^{*}: \overline{\mathrm{C}}: \mathrm{C}=1.128: 1.225: 1\)

Step-by-Step Solution

Answer

The correct answer is (d): \(\mathrm{C}^{*}: \overline{\mathrm{C}}: \mathrm{C}=1.128: 1.225: 1\).

Step 1: Identify Given Values and Definitions

Recognize that \(\mathrm{C}^*\) is the most probable speed, \(\overline{\mathrm{C}}\) is the average speed, and \(\mathrm{C}\) is the root mean square speed. We are also provided with multiple choice answers presenting different potential ratios of these speeds.

Step 2: Recall the Formulas for Each Speed

Recall the formulas for each speed. For a gas, the most probable speed \(\mathrm{C}^* = \sqrt{\frac{2RT}{M}}\), the average speed \(\overline{\mathrm{C}} = \sqrt{\frac{8RT}{\pi M}}\), and the mean square speed (which is often referred to as the root mean square speed) \(\mathrm{C} = \sqrt{\frac{3RT}{M}}\). Here, \(R\) is the gas constant, \(T\) is the temperature, and \(M\) is the molar mass.

Step 3: Understand the Relationship Between Speeds

These formulas imply that there is a constant factor relating each speed: - \(\overline{\mathrm{C}} = \mathrm{C}^* \cdot \sqrt{\frac{8}{2\pi}}\)- \(\mathrm{C} = \mathrm{C}^* \cdot \sqrt{\frac{3}{2}}\)Thus, these ratios can be simplified based on the constants in the formulas.

Step 4: Calculate the Ratios

Calculate each ratio:- The ratio of \(\mathrm{C}^*\) to \(\overline{\mathrm{C}}\) is \(rac{\mathrm{C}^*}{\overline{\mathrm{C}}} = \frac{1}{\sqrt{\frac{8}{2\pi}}} = 1.128\).- The ratio of \(\mathrm{C}^*\) to \(\mathrm{C}\) is \(rac{\mathrm{C}^*}{\mathrm{C}} = \frac{1}{\sqrt{\frac{3}{2}}} = 1.225\).- Therefore, the ratios are \(\mathrm{C}^* : \overline{\mathrm{C}} : \mathrm{C} = 1 : 1.128 : 1.225\).

Step 5: Select the Correct Answer

Now compare the calculated ratios to the given multiple-choice answers to determine which one is correct.The corresponding ratios match answer (d): \(\mathrm{C}^* : \overline{\mathrm{C}} : \mathrm{C} = 1.128 : 1.225 : 1 \).

Key Concepts

Most Probable Speed

In the gaseous state, molecules move around with different speeds. One important concept to understand is the **most probable speed**. This is the speed that the highest number of gas molecules will be traveling at under ideal conditions.
Think of it as the peak of the speed distribution curve for gas molecules. It is denoted as \( C^* \) and calculated using the formula:

• \( C^* = \sqrt{\frac{2RT}{M}} \)

Here, \( R \) stands for the gas constant, \( T \) is the absolute temperature, and \( M \) represents the molar mass of the gas.
This formula tells us that the most probable speed depends on the temperature and the mass of the gas molecules. As the temperature increases, the most probable speed also increases, assuming the same type of gas. Conversely, heavier molecules will have a lower most probable speed.
This concept is particularly useful when predicting and understanding the behavior of gas particles under varying conditions such as changes in temperature or mass.

Average Speed

The **average speed** of gas molecules is another critical concept. Average speed, denoted by \( \overline{C} \), provides an arithmetic mean of the velocities of all the particles in a gas sample. It is slightly higher than the most probable speed because it accounts for all molecule speeds, including those that move much faster than average.
The formula to find the average speed is:

• \( \overline{C} = \sqrt{\frac{8RT}{\pi M}} \)

In this formula, \( \pi \) is a constant approximately equal to 3.14159.
Like the most probable speed, the average speed also increases with growing temperature and decreases with increasing molar mass.
Understanding average speed is fundamental when analyzing gas reactions and behaviors since it provides a more comprehensive picture of the kinetic energy of particles in a system. Knowing this speed helps in anticipating how collision rates between molecules might change under different conditions.

Root Mean Square Speed

The **root mean square speed** is a measure of the speed of particles in a gas that reflects their mean kinetic energy. It is denoted by \( C \) and is typically higher than both the most probable speed and the average speed. This speed takes into consideration the square of velocity, making it a useful measure for predicting molecular behavior in terms of energy.
The formula for root mean square speed is:

• \( C = \sqrt{\frac{3RT}{M}} \)

This formula shows that the root mean square speed is related directly to the temperature of the gas and inversely to the molar mass, similar to the other speeds.
The root mean square speed gives insight into the overall energy of a gaseous system, which is crucial in the study of gas dynamics and thermodynamics.
Recognizing how these speeds vary with changes in temperature and mass allows chemists to predict how gases will behave in different environments, aiding in everything from industrial design to environmental science.