Problem 8 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

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

Step-by-Step Solution

Answer

The correct ratio is given by option (c): \( C^* : \overline{C} : C = 1 : 1.128 : 1.225 \).

Step 1: Understanding Speed Definitions

In a gaseous state, molecules can have different speeds. The most probable speed \( C^* \) is the speed at which the largest fraction of molecules move. The average speed \( \overline{C} \) is the mean speed of all molecules, and the root mean square speed \( C \) is related to the mean of the square of speeds of molecules.

Step 2: Expressing Relations between Speeds

The theoretical ratios between the speeds are known to be specific values. For a large number of molecules, the most probable speed \( C^* \), the average speed \( \overline{C} \), and the root mean square speed \( C \) are related as follows: \( C^* : \overline{C} : C = 1 : 1.128 : 1.225 \).

Step 3: Identifying Correct Ratio

Compare the theoretical ratio \( C^* : \overline{C} : C = 1 : 1.128 : 1.225 \) with the given choices.- Option (a) presents \( C^* : \overline{C} : C = 1.225 : 1.128 : 1 \), which does not match the theoretical ratio.- Option (b) presents \( C^* : \overline{C} : C = 1.128 : 1.225 : 1 \), which does not match the theoretical ratio.- Option (c) presents \( C^* : \overline{C} : C = 1 : 1.128 : 1.225 \), which matches the theoretical ratio.- Option (d) presents \( C^* : \overline{C} : C = 1 : 1.225 : 1.128 \), which does not match the theoretical ratio.

Key Concepts

Most Probable Speed

In the gaseous state, the motion of gas molecules is random and varies in speed. Among the different speeds that gas molecules exhibit, the **most probable speed** is a key concept.
This speed, denoted as \( C^* \), is where the highest number of molecules are found moving at any given instant. It's like the peak of a bell curve in a speed distribution plot, representing the mode in statistics.
Understanding this concept is crucial in the study of gas behavior because it helps predict how the majority of molecules in a gas sample are moving.

• **Definition**: The speed at which the greatest number of molecules are moving.

In practical terms, if you were to pick a molecule at random, the chances are highest that it would move at this most probable speed.
This concept helps in making initial predictions about the behavior of gases under different conditions.

Average Speed

The **average speed** is another important concept when dealing with gases. The average speed, denoted as \( \overline{C} \), is calculated by adding up all the individual molecular speeds and dividing by the total number of molecules.
This gives an overall picture of how fast molecules are moving on average. It differs from the most probable speed because it considers the speed of every molecule in the gas, thus reflecting the mean value.

• **Definition**: The mean of all the individual molecular speeds in a gas.

Imagine you're timing a race where each competitor runs at a different pace. The average speed would be like calculating the average time it took all the competitors to finish the race.
Knowing the average speed of gas molecules helps in applications where the overall kinetic energy of the system is a factor, such as predicting pressure or temperature changes in a gas container.

Root Mean Square Speed

The **root mean square speed** is a slightly more complex concept that offers valuable insights into the kinetic theory of gases.
Denoted as \( C \), the root mean square speed considers the square of each molecule's speed, finds the mean of these squares, and then takes the square root of that mean. This speed is particularly important because it is tied to the kinetic energy of the gas, providing a measure that's proportional to the temperature of the gas.

• **Definition**: The square root of the mean of the squares of all molecular speeds.

This measure is often larger than both the most probable and average speeds, giving more weight to higher speeds.
In practical applications, the root mean square speed helps predict how molecules within the gas will behave under different conditions and is crucial in calculating properties like pressure and temperature using the kinetic molecular theory.