Question

If \(C_{1}, C_{2}, C_{3} \ldots \ldots \ldots\) represents the speed of \(n_{1}, n_{2}, n_{3}\) .... molecules, then the root mean square of speed is: (a) \(\left(\frac{\mathrm{n}_{1} \mathrm{C}_{1}^{2}+\mathrm{n}_{2} \mathrm{C}_{2}^{2}+\mathrm{n}_{3} \mathrm{C}_{3}^{2}+\ldots}{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}+\ldots}\right)^{12}\) (b) \(\left(\frac{\mathrm{n}_{1} \mathrm{C}_{1}^{2}+\mathrm{n}_{2} \mathrm{C}_{2}^{2}+\mathrm{n}_{3} \mathrm{C}_{3}^{2}+\ldots}{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}+\ldots}\right)^{2}\) (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) \(\left[\frac{\left(\mathrm{n}_{1} \mathrm{C}_{1}+\mathrm{n}_{2} \mathrm{C}_{2}+\mathrm{n}_{3} \mathrm{C}_{3}+\ldots\right)^{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}+\ldots}\right]^{1 / 2}\)

Short answer

Option (d) is correct; it fits the rms speed formula.

Step-by-step solution

Understanding the Root Mean Square Speed Formula

The root mean square (rms) speed is a measure of the average velocity of the molecules in a gas. It is given by the formula: \(C_{rms} = \sqrt{\frac{n_1 C_1^2 + n_2 C_2^2 + n_3 C_3^2 + \ldots}{n_1 + n_2 + n_3 + \ldots}}\). This formula involves taking the sum of the squares of the speeds, weighted by the number of molecules, divided by the total number of molecules, and taking the square root of the whole expression.

Identifying the Correct Answer

Each option given in the problem can be checked against the formula for the root mean square speed. According to the formula, we take the sum of \(n_i C_i^2\) for each molecule group and then divide by the total number of molecules, which is \(n_1 + n_2 + n_3 + \ldots\). Finally, we take the square root of this result to get the root mean square speed.

Matching the Options with the Formula

Let's analyze the provided options: - Option (a) suggests raising the result to the twelfth power, which is incorrect since we need to take a square root. - Option (b) raises the expression to the second power, which is incorrect because the operation required is a square root. - Option (c) attempts individual fractional root calculations, which does not correspond to the rms formula. - Option (d) correctly works with the overall sum of velocities squared divided by number of molecules, and then takes the square root.

Conclusion

Based on the analysis, Option (d) is the correct answer as it matches exactly with the root mean square speed formula by taking the sum, dividing by the total number, and taking the square root.

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases provides a microscopic understanding of gas behavior, allowing us to relate macroscopic properties like pressure and temperature to the motion of molecules within the gas. According to this theory, a gas comprises a large number of tiny particles moving randomly in all directions. These particles are in constant motion, and their collisions with the walls of a container result in the pressure exerted by the gas.

Key assumptions of the kinetic theory include:

• Molecules of a gas are in constant and random motion.
• The volume of the molecules is negligible compared to the volume of the container.
• Collisions between molecules, and between molecules and the walls, are perfectly elastic.
• There are no intermolecular forces between the molecules in a gas.
• The average kinetic energy of the gas molecules is directly proportional to the temperature.

The kinetic theory is vital for understanding concepts such as root mean square speed, which helps in calculating the average speed of gas molecules, as derived from molecular motion and collisions.

Molecular Speed Distribution

The distribution of molecular speeds in a gas describes how speeds vary among molecules at a given temperature. This is illustrated by the Maxwell-Boltzmann distribution, a fundamental concept in the study of gases. The distribution shows that at any given temperature, there is a range of speeds that molecules can have, but a larger number of molecules move at a particular speed called the most probable speed.

The Maxwell-Boltzmann distribution can be visualized as a graph:

• The x-axis represents molecular speed.
• The y-axis represents the fraction of molecules at each speed.
• The peak of the curve indicates the most probable speed, while the tails indicate fewer molecules with very low or very high speeds.

This distribution implies there is diversity in molecular speeds, and this affects how we calculate averages, such as the root mean square speed, essential for deeper analyses in gas behavior.

Average Molecular Speed

Calculation of different average speeds helps to comprehensively describe the molecular motion in gases. Various types of average speeds can be derived from the distribution of molecular speeds:

• Average speed: Simple arithmetic mean of all molecular speeds.
• Most probable speed: The speed that a single randomly chosen molecule is most likely to possess.
• Root mean square speed (\(C_{rms}\)): Provides a measure of the speed of gas particles by considering the square of the speeds, averaging them, and then taking the square root.

The root mean square speed is particularly useful because it closely relates to kinetic energy calculations. Since kinetic energy depends on the square of the speed, \(C_{rms}\) becomes a direct link to expressing kinetic energy in molecular terms.

Understanding these different average speeds helps visualize how kinetic energy and molecular activity relate to macroscopic gas properties.

Gas Laws

Gas laws are essential relationships that define how variables such as pressure, volume, temperature, and number of molecules interact in a gas. These laws are cornerstones in studying gas behavior and help predict gas actions under various conditions.

Important gas laws include:

• Boyle's Law: At constant temperature, the pressure of a gas is inversely proportional to its volume. \( PV = ext{constant} \)
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its absolute temperature. \( V/T = ext{constant} \)
• Avogadro's Law: At constant pressure and temperature, the volume of a gas is directly proportional to the number of moles of gas. \( V/n = ext{constant} \)
• Ideal Gas Law: Integrates the above laws into a single equation. \( PV = nRT \), where \( R \) is the gas constant.

These laws collectively provide a macroscopic view of gas behavior, offering insights into conditions that affect gas characteristics and reinforcing the foundation laid by the kinetic theory of gases.