Question

Four particles have speed \(2,3,4\) and \(5 \mathrm{~cm} / \mathrm{s}\) respectively. Their rms speed is: (a) \(3.5 \mathrm{~cm} / \mathrm{s}\) (b) \((27 / 2) \mathrm{cm} / \mathrm{s}\) (c) \(\sqrt{54} \mathrm{~cm} / \mathrm{s}\) (d) \((\sqrt{54} / 2) \mathrm{cm} / \mathrm{s}\)

Short answer

\((\sqrt{54} / 2) \mathrm{cm} / \mathrm{s}\)

Step-by-step solution

Understand the root mean square (rms) speed

The root mean square (rms) speed is a measure of the magnitude of the velocity of a group of particles. It's given by the square root of the average of the squares of the individual speeds. Mathematically, it is expressed as \( v_{rms} = \sqrt{\frac{1}{n}(v_1^2 + v_2^2 + v_3^2 + ... + v_n^2)} \), where \(v_i\) are the individual speeds and \(n\) is the number of particles.

List the individual speeds

The speeds of the four particles are given as \(2, 3, 4,\) and \(5 \mathrm{~cm} / \mathrm{s}\).

Calculate the square of each speed

Square each individual speed: \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), and \(5^2 = 25\).

Find the sum of squares of the speeds

Add the squared speeds together: \(4 + 9 + 16 + 25 = 54\).

Calculate the average of the squares

As there are four particles, divide the sum by the number of particles \(n=4\): \(\frac{54}{4} = 13.5\).

Determine the rms speed

Take the square root of the average calculated in the previous step to determine the rms speed: \(v_{rms} = \sqrt{13.5} = \sqrt{54} / 2 \mathrm{cm} / \mathrm{s}\).

Key concepts

Understanding Physical Chemistry

Physical chemistry is the branch of chemistry concerned with the interpretation of the physical properties and composition of chemical systems using the principles of physics. It involves understanding how molecules behave and interact on an atomic level, which can be described through equations and models. Key areas within physical chemistry include thermodynamics, kinetics, quantum chemistry, and statistical mechanics. These areas help us understand processes such as reaction rates, energy transfer, and the distribution of molecules in terms of their energy.

When it comes to understanding the velocity of particles, physical chemistry offers a comprehensive mathematical approach known as the root mean square (rms) speed. This concept is highly relevant in fields that deal with the movement of particles in gases, liquids, or solids. Physical chemistry provides the tools to quantify and predict these behaviors, creating a bridge between theoretical models and practical, observable phenomena.

Velocity of Particles: RMS Speed

In the context of velocity of particles, the root mean square (rms) speed plays a central role in quantifying the velocity of particles within a system. It's a statistical measure that tends to give more weight to higher values, as it involves squaring the speeds before taking the average. The rms speed is particularly useful because it corresponds to the standard deviation of the velocities in a population of particles and thus provides insight into their kinetic energy.

The formula for rms speed, as explained in the solution provided, is \( v_{\text{rms}} = \sqrt{\frac{1}{n}(v_1^2 + v_2^2 + v_3^2 + \dots + v_n^2)} \) where \( v_i \) represents the individual speeds of each particle and \( n \) is the total number of particles. The rms speed is often used to describe the motion of particles in gases, where it helps to predict how quickly they spread and move, which has implications for understanding pressure and temperature in gas laws.

JEE Physical Chemistry Problems

Students preparing for competitive exams like the Joint Entrance Examination (JEE) in India often face problems in physical chemistry that require a strong conceptual understanding as well as analytical problem-solving skills. The calculation of rms speed is a type of problem that can frequently be encountered in JEE physical chemistry sections.

Problems like these test the candidate's ability to apply formulae and concepts to practical situations. In the example of the rms speed calculation, understanding each step of the process, from squaring the individual speeds to taking their mean and then the square root, is essential for accuracy and speed in the examination environment. Aspiring students must be proficient in applying these principles quickly and correctly, which often involves practicing an array of similar problems to develop a deep understanding and an intuitive grasp of the subject.