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) \(\left[\frac{\left(\mathrm{n}_{1} \mathrm{C}_{1}+\mathrm{n}_{2} \mathrm{C}_{2}+\mathrm{n}_{3} \mathrm{C}_{3}+\ldots\right)^{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}+\mathrm{n}_{3}+\ldots}\right]^{1 / 2}\)

Short answer

Option (a) is the correct answer: \( \left(\frac{n_1 C_1^2 + n_2 C_2^2 + n_3 C_3^2 + \ldots}{n_1 + n_2 + n_3 + \ldots}\right)^{1/2} \).

Step-by-step solution

Understanding RMS Speed

The root mean square (RMS) speed of molecules is a measure of the average speed of particles in a gas that takes into account both the mass of the particles and their speed. It is a statistical measure that is important in physics for describing the kinetic energy of particles.

Identifying the RMS Speed Formula

The correct formula for the root mean square speed of a gas is \[ \text{RMS Speed} = \sqrt{\frac{n_1 C_1^2 + n_2 C_2^2 + n_3 C_3^2 + \ldots}{n_1 + n_2 + n_3 + \ldots}} \] This formula takes the number of each type of molecule and their respective speeds, squares those speeds, sums them, and divides by the total number of molecules, finally taking the square root of the entire expression.

Matching with Given Options

Comparing the identified formula with the given options, we see that option (a) matches exactly with the formula for RMS speed: \[ \left(\frac{n_1 C_1^2 + n_2 C_2^2 + n_3 C_3^2 + \ldots}{n_1 + n_2 + n_3 + \ldots}\right)^{1/2} \] Thus, option (a) is the correct answer.

Key concepts

Kinetic Theory of Gases

The Kinetic Theory of Gases provides a framework for understanding the motion and behavior of gas particles. This theory is crucial as it links macroscopic properties such as temperature and pressure to the microscopic characteristics of gas molecules. It is based on several assumptions:

• Gas particles move in random, constant motion.
• The size of gas molecules is negligible compared to the distance between them.
• Collisions between gas molecules are perfectly elastic, meaning they retain energy when bouncing off each other.
• There are no intermolecular forces between the particles, except during collisions.

As a result, the kinetic theory is instrumental in explaining the concept of temperature as a measure of the average kinetic energy of particles. In essence, the faster the particles move, the higher the temperature of the gas. This theory is foundational for computed values like Root Mean Square (RMS) speed, which conveys the average speed of molecules in a gas system.

Molecular Speed in Gases

The molecular speed in gases varies significantly due to the random and chaotic nature of molecular motion. One statistical measure of these speeds is the Root Mean Square (RMS) speed, which gives insight into the average kinetic energy of the gas particles. It is calculated using the formula:\[ \text{RMS Speed} = \sqrt{\frac{n_1 C_1^2 + n_2 C_2^2 + n_3 C_3^2 + \ldots}{n_1 + n_2 + n_3 + \ldots}}\]

• The term \(n_i\) represents the number of molecules with a particular speed \(C_i\).
• Speeds are squared to ensure all values are positive and emphasize faster-moving particles.
• The formula averages these squared speeds by dividing the sum by the total number of molecules \(n_1 + n_2 + n_3 + \ldots\).
• Finally, taking the square root provides a linear interpretation of speed.

Molecular speed in gases is highly dependent on temperature, which directly affects particle velocity due to increased thermal energy, leading to greater molecular speeds.

Statistical Mechanics in Chemistry

Statistical mechanics in chemistry bridges microscopic particle dynamics with macroscopic observable phenomena. It employs probability methods to predict the behavior of molecules in different states of matter, particularly gases.

• It uses statistical sums over configurations known as partition functions, vital for calculating thermodynamic properties.
• Allows analysis of property distributions such as energy, speed, and position among a large ensemble of particles.
• Offers a foundation for understanding thermodynamic laws in terms of molecular behavior.

In chemistry, statistical mechanics provides a deeper understanding of reaction kinetics, phase transitions, and even helps in designing new materials by predicting how molecular arrangements will affect material properties. It's indispensable for interpreting experiments and predicting outcomes in complex systems, as it quantifies how fluctuating atomic motions lead to observable chemical phenomena. This area plays a crucial role in calculating averages like RMS speed, providing insights into gas behavior under varying conditions.