Question

If \(C_{1}, C_{2}, C_{3} \ldots\) represent the speeds of \(n_{1}, n_{2}, n_{3} \ldots\) molecules respectively, then the root mean square speed will be: (a) \(\sqrt{\frac{n_{1} C_{1}^{2}+n_{2} C_{2}^{2}+n_{3} C_{3}^{2}+\ldots}{n_{1}+n_{2}+n_{3}+\ldots}}\) (b) \(\sqrt{\frac{\left(n_{1}+n_{2}+n_{3}+\ldots\right)^{2}}{n_{1} C_{1}^{2}+n_{2} C_{2}^{2}+n_{3} C_{3}^{2}+\ldots}}\) (c) \(\sqrt{\frac{\left(n_{1} C_{1}\right)}{n_{1}}+\frac{\left(n_{2} C_{2}\right)}{n_{2}}+\frac{\left(n_{3} C_{3}\right)}{n_{3}}}\) (d) \(\sqrt{\frac{\left(n_{1} C_{1}+n_{2} C_{2}+n_{2} C_{3}+\ldots\right)^{2}}{n_{1}+n_{2}+n_{3}+\ldots}}\)

Short answer

\(a) \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-by-step solution

Identify the Definition of Root Mean Square Speed

The root mean square speed (RMS speed) is a measure of the speed of particles in a gas that is most useful in the context of kinetic theory of gases. The RMS speed is the square root of the average of the squares of the speeds of the particles. It is given by the formula: \(\text{RMS speed} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} v_{i}^{2}}\) where \(\text{N}\) is the number of particles and \(\text{v}_{i}\) is the speed of the ith particle.

Compare Formula Options to RMS Speed Definition

The given options should be compared to the general formula of RMS speed. We know that in the RMS speed formula, each speed is squared, summed up and then divided by the total number of particles, and finally, we take the square root of the entire expression.

Analyze Option (a)

Option (a) \(\root{\sqrt{\frac{n_{1} C_{1}{2}+n_{2} C_{2}{2}+n_{3} C_{3}{2}+\ldots}{n_{1}+n_{2}+n_{3}+\ldots}}\) looks the most like the RMS speed formula, where \(\text{n}_{i}\) is multiplied by the square of respective speed \(\text{C}_{i}{2}\) and then averaged by the sum of \(\text{n}_{i}\) values.

Eliminate Incorrect Options

Options (b), (c), and (d) do not match the general form of the RMS speed equation. Option (b) incorrectly places the sum of particles squared in the numerator instead of the sum of the squared speeds. Option (c) simplifies to just the sum of speeds, which doesn't match the RMS formula. Option (d) has squared terms in the numerator that are not each particle's speed squared, instead, it is the square of the sum of products of number and speed for individual particles.

Select the Correct Option

Therefore, option (a) is the correct representation of the RMS speed for the given scenario with multiple particles with respective speeds and quantities.

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental principle used to explain the behaviors of gases by considering their molecular composition and motion. This theory makes several assumptions: gases are composed of numerous particles in constant, random motion; these particles are far apart relative to their size; and they interact with each other only during elastic collisions, where no energy is lost.

The kinetic theory provides a microscopic explanation for macroscopic phenomena such as pressure and temperature of a gas. Pressure is understood as the result of collisions of gas particles with the walls of a container. The temperature, related to the average kinetic energy of the particles, can be directly linked to the speed of the gas molecules. This theory becomes particularly useful when discussing concepts like root mean square speed, forming a key part of the foundation for problem-solving in physical chemistry, especially in the context of competitive exams like JEE.

Speed of Particles in a Gas

In the realm of physical sciences, the speed of particles in a gas describes the rapidity with which the gas molecules move. These speeds vary greatly, with each molecule traveling at a different velocity vector, depending on its interactions with other particles and the container walls.

Understanding the various speeds is essential for many applications, such as predicting the rate of diffusion, effusion, and the transport properties of gases. When we talk about speed in this context, we frequently refer to measures like the average speed, most probable speed, and the root mean square (RMS) speed. The RMS speed, in particular, gives us valuable insight into the energy of the gas particles because it factors in the square of the velocities, which is directly related to the kinetic energy of the particles.

Physical Chemistry for JEE

Physical chemistry encompasses the study of macroscopic, atomic, subatomic, and particulate phenomena in chemical systems. For students preparing for competitive exams like the Joint Entrance Examination (JEE) in India, an understanding of physical chemistry principles is crucial. It constitutes a significant portion of the chemistry syllabus and includes topics like thermodynamics, equilibrium, kinetics, and the kinetic theory of gases.

In the context of the kinetic theory, mastery of calculations involving the RMS speed is critical. JEE aspirants need to understand this conceptually, beyond rote memorization, as it underpins further topics in thermodynamics and statistical mechanics. The ability to solve exercises related to the RMS speed of gas particles directly impacts a student's performance in the physical chemistry section of the exam.