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

The correct answer is (a).

Step 1: Understanding the Problem

The problem requires identifying the formula for the root mean square (RMS) speed of molecules given their individual speeds and respective quantities. We are given four options with different mathematical expressions.

Step 2: Recalling the RMS Formula

The root mean square speed is calculated by taking the square root of the average of the squares of the individual speeds. For a set of molecules with speeds \( C_1, C_2, C_3, \ldots \) and quantities \( n_1, n_2, n_3, \ldots \), the formula is given as: \[ v_{rms} = \sqrt{\frac{n_1 C_1^2 + n_2 C_2^2 + n_3 C_3^2 + \ldots}{n_1 + n_2 + n_3 + \ldots}} \]

Step 3: Identifying the Correct Formula

Among the options provided, we need to find which one matches the above formula for RMS speed. The correct expression should have the squares of individual speed contributions, summed, and then divided by the total number of molecules, followed by taking the square root of the whole expression.

Step 4: Choosing the Correct Option

Option (a) is \( \left(\frac{n_1 C_1^2 + n_2 C_2^2 + n_3 C_3^2 + \ldots}{n_1 + n_2 + n_3 + \ldots}\right)^{1/2} \). This is exactly the expression used to calculate the RMS speed of a collection of molecules, matching the standard formula.

Key Concepts

Kinetic Theory of Gases

The Kinetic Theory of Gases provides a framework for understanding gas behavior by considering gas molecules as tiny, hard spheres in constant random motion. This theory helps explain fundamental concepts like pressure, temperature, and volume in terms of molecular dynamics.

One core idea is that gas pressure results from collisions of gas molecules with the walls of their container. As temperature increases, these molecules move faster due to increased kinetic energy, leading to more frequent and forceful collisions. This explains why gas pressure depends on temperature.

• Molecules move in random directions.
• Collisions are perfectly elastic, meaning no energy is lost.
• These concepts can help explain phenomena observed in gases, such as diffusion and effusion.

The theory assumes that the volume of the molecules themselves is negligible compared to the volume occupied by the gas. This simplification is crucial for deriving equations like the Ideal Gas Law.

Molecular Speed Distribution

Molecular Speed Distribution describes how the speeds of molecules in a gas are spread over a range of values. This distribution is not uniform, meaning some molecules move faster than others, even at a given temperature.

The Maxwell-Boltzmann Distribution Law gives a statistical way to predict the speed distribution of gas molecules. It indicates that most molecules move at an average speed, but fewer molecules move very fast or very slow. The Root Mean Square (RMS) speed is one way to describe this average speed.

• The RMS speed is a type of average that takes into account all the molecule speeds, squared, and averaged.
• The RMS speed can be calculated using the formula: \[ v_{rms} = \sqrt{\frac{3RT}{M}} \] where \( R \) is the gas constant, \( T \) is the temperature, and \( M \) is the molar mass.

This formula highlights that the RMS speed rises with an increase in temperature and decreases with heavier gas molecules.

Gas Laws

Gas Laws are mathematical relationships that describe how temperature, volume, and pressure interact in a gas. These laws form the basis for understanding how gases will behave under various conditions:

1. **Boyle's Law** states that the volume of a given mass of gas is inversely proportional to its pressure at a constant temperature. Mathematically, \( PV = k \).

2. **Charles's Law** describes how gases expand upon heating. It states that the volume of a gas is directly proportional to its temperature at constant pressure. This can be expressed as \( V/T = k \).

3. **Avogadro's Law** indicates that equal volumes of gases at the same temperature and pressure contain an equal number of molecules, portrayed by \( V/n = k \).

When combined, these laws give the **Ideal Gas Law**: \[ PV = nRT \]This is a comprehensive equation relating pressure, volume, temperature, and quantity of gas, showing the interconnected nature of these variables.

Despite its usefulness, the Ideal Gas Law is an approximation that works best under conditions of low pressure and high temperature, where gas molecules behave ideally. Understanding deviations from this ideal behavior can lead to more nuanced insights into the physical properties of gases.