Problem 78 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(\mathrm{C}_{1}, \mathrm{C}_{2}, \mathrm{C}_{3} \ldots \ldots \ldots\) represents the speed of \(\mathrm{n}_{1}\), \(\mathrm{n}_{2}, \mathrm{n}_{3}, \ldots\) molecules, then the root mean square of speed is (a) \(\left(\frac{\mathrm{n}_{1} \mathrm{C}_{1}^{2}+\mathrm{n}_{2} \mathrm{C}_{2}^{2}+\mathrm{n}_{3} \mathrm{C}_{3}^{2}+\ldots}{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}+\ldots}\right)^{1 / 2}\) (b) \(\left(\frac{n_{1} C_{1}^{2}+n_{2} C_{2}^{2}+n_{3} C_{3}^{2}+\ldots}{n_{1}+n_{2}+n_{3}+\ldots}\right)^{2}\) (c) \(\frac{\left(\mathrm{n}_{1} \mathrm{C}_{1}^{2}\right)^{1 / 2}}{\mathrm{n}_{1}}+\frac{\left(\mathrm{n}_{2} \mathrm{C}_{2}^{2}\right)^{1 / 2}}{\mathrm{n}_{2}}+\frac{\left(\mathrm{n}_{3} \mathrm{C}_{3}^{2}\right)^{1 / 2}}{\mathrm{n}_{3}}+\ldots\) (d) \(\left[\frac{\left(n_{1} C_{1}+n_{2} C_{2}+n_{3} C_{3}+\ldots\right)^{2}}{n_{1}+n_{2}+n_{3}+\ldots}\right]^{1 / 2}\)

Step-by-Step Solution

Answer

Option (a) correctly represents the RMS speed formula.

Step 1: Define Root Mean Square Speed

The root mean square (RMS) speed is a measure of the average speed of molecules in a gas and is calculated using their individual speeds. The formula for RMS speed is \( v_{rms} = \sqrt{\frac{\sum n_i C_i^2}{\sum n_i}} \) where \( n_i \) is the number of molecules with speed \( C_i \).

Step 2: Analyze Each Option

We need to find the option that represents the standard RMS speed formula:- (a) \( \left(\frac{\mathrm{n}_{1} \mathrm{C}_{1}^{2} + \mathrm{n}_{2} \mathrm{C}_{2}^{2} + \mathrm{n}_{3} \mathrm{C}_{3}^{2} + \ldots}{\mathrm{n}_{1} + \mathrm{n}_{2} + \mathrm{n}_{3} + \ldots}\right)^{1 / 2} \)- (b) does not represent a square root.- (c) involves terms in a format inconsistent with RMS formula.- (d) does not arrange the speeds in terms of squares. It focuses on the sum of quantities, not squared speeds.

Step 3: Identify Correct Expression

Comparing these options, (a) correctly represents the RMS speed formula because it captures the notion of averaging the squares of speeds and then taking the square root. Thus, option (a) matches the intended calculation for root mean square speed.

Key Concepts

Molecular Speed

Molecular speed refers to the velocity at which individual molecules move within a gas. It's a crucial concept in understanding how gases behave, as gas molecules are always in continuous motion. Since molecules within a gas are moving at various speeds, it is important to talk about their average speeds in statistical terms. There are different ways to define average speed:

• **Average Molecular Speed**: This is the average of the speeds of all molecules in the gas.
• **Most Probable Speed**: The speed at which the largest number of molecules is moving, also known as the peak of the speed distribution graph.
• **Root Mean Square (RMS) Speed**: This is a specific calculation that gives more weight to faster molecules, as it involves squaring the speeds before averaging and taking the square root of the result.

Understanding RMS speed helps provide a more comprehensive picture of the kinetic activity within a gas compared to simply using an arithmetic mean. This is because RMS speed considers the energy associated with each molecule's movement.

Kinetic Theory of Gases

The kinetic theory of gases is a foundational concept that explains the macroscopic properties of gases in terms of the microscopic motions of their molecules. This theory provides insights into the constant activity and interaction of gas molecules.
The basic postulates of the kinetic theory include:

• Gas consists of a large number of small particles (molecules) that are in constant, random motion.
• The volume of the actual gas molecules is negligible compared to the volume the gas occupies. Most of the gas volume is empty space, allowing free movement of molecules.
• Collisions between molecules, and between molecules and the walls of their container, are perfectly elastic, meaning they do not lose energy in the process.
• The average kinetic energy of the gas molecules is proportional to the temperature of the gas, indicating that as temperature increases, so does the speed and energy of the molecules.

Through these principles, the kinetic theory of gases allows us to derive meaningful equations, such as the ideal gas law and understand how macroscopic properties like pressure and temperature relate to the molecular level.

Gas Molecule Movement

Gas molecule movement is inherent to the nature of gases, where molecules are in perpetual motion. This movement is fundamental to various gaseous properties and behaviors. In a gas:

• Molecules move rapidly in straight lines until they collide with each other or the walls of their container. Such collisions are constant and change the direction of the molecules' paths.
• Because gas molecules are spaced far apart and experience negligible intermolecular forces, they can move freely and independently, allowing for easy compressibility and expansion of gases.
• The speed of gas molecules is affected by the temperature of the gas. As temperature rises, molecules move faster. Conversely, a decrease in temperature slows them down.
• The distribution of speeds among molecules in a gas can be described by the Maxwell-Boltzmann distribution, which shows that while some molecules move very slowly or very quickly, most have speeds around an average value, typically the RMS speed.

Understanding how gas molecules move is crucial for explaining phenomena such as diffusion, effusion, and gas pressure, all of which stem from the energetic and random movement of molecules throughout any given volume.