Problem 26 - Arihant AIEEE Physics

Question

Four molecules of a gas have speeds \(1,2,3\) and \(4 \mathrm{~km} / \mathrm{s}\). The value of the root-mean square speed of the gas molecules is : (a) \(\frac{1}{2} \sqrt{15} \mathrm{~km} / \mathrm{s}\) (b) \(\frac{1}{2} \sqrt{10} \mathrm{~km} / \mathrm{s}\) (c) \(2.5 \mathrm{~km} / \mathrm{s}\) (d) \(\sqrt{\frac{15}{2}} \mathrm{~km} / \mathrm{s}\)

Step-by-Step Solution

Answer

The root-mean-square speed is \( \sqrt{\frac{15}{2}} \mathrm{~km}/\mathrm{s}\), option (d).

Step 1: Understand the root-mean-square speed formula

The root-mean-square (RMS) speed of gas molecules is given by the formula: \( v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + v_4^2}{n}} \), where \(n\) is the number of molecules and \(v_1, v_2, v_3, \text{and} v_4\) are the individual speeds of the molecules.

Step 2: Square the individual speeds

Calculate the square of each given speed: \(v_1 = 1\), \(v_2 = 2\), \(v_3 = 3\), \(v_4 = 4\). So, \(v_1^2 = 1^2 = 1\), \(v_2^2 = 2^2 = 4\), \(v_3^2 = 3^2 = 9\), and \(v_4^2 = 4^2 = 16\).

Step 3: Sum of squared speeds

Now, add the squared values: \(1 + 4 + 9 + 16 = 30\).

Step 4: Calculate RMS speed

Substitute back into the RMS formula: \( v_{rms} = \sqrt{\frac{30}{4}} \). Simplify the expression to get: \( v_{rms} = \sqrt{7.5} \).

Step 5: Express RMS speed in given options

Notice that \( \sqrt{7.5} \) can be expressed as \( \sqrt{\frac{15}{2}} \), making it match option (d) exactly.

Key Concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental concept that helps us understand the behavior of gases in terms of the movements of their molecules. This theory postulates that gas molecules are in constant, random motion and that their interactions resemble perfectly elastic collisions. The key ideas include:

• Gas molecules are in perpetual and random motion.
• They possess kinetic energy, which is related to temperature.
• The volume of individual molecules is negligible compared to the volume the gas occupies.
• There are no attractive or repulsive forces between the molecules, except during collisions.
• Collisions between molecules, or with the walls of a container, are perfectly elastic.

The kinetic theory provides a framework to derive important equations such as the ideal gas law and expressions for molecular speeds, including root-mean-square speed.

Molecular Speed

Molecular speed refers to the speed at which a molecule in a gas moves. There are different ways to think about or calculate molecular speeds, which can give you different averages:

• Average Speed: The arithmetic mean of the speeds of all molecules.
• Most Probable Speed: The speed at which the largest number of molecules are moving.
• Root-Mean-Square Speed (RMS): A type of average that is useful for understanding kinetic energy.

Among these, the root-mean-square speed is particularly significant as it directly relates to the kinetic energy of gas molecules and is used in physical calculations involving the velocity and energy of molecules in a gas.

Gas Molecules

Gas molecules can be visualized as tiny particles that are in ceaseless motion, zipping through the air in different directions. They are constantly bumping into each other and the walls of their container. This behavior explains observable properties of gases such as:

• Pressure, which is related to the force exerted by molecules as they collide with surfaces.
• Temperature, which correlates with the average kinetic energy of the molecules.
• Volume, as gases expand to fill any container, due to the free space between rapidly moving molecules.

Understanding gas molecules at this microscopic level gives insight into macroscopic phenomena and is central to the study of thermodynamics and fluid mechanics.

Calculation of RMS Speed

The calculation of root-mean-square speed, a critical element of kinetic theory, reflects how gas molecules distribute their speeds in a system. To find the RMS speed (\( v_{rms} \)) we use the formula\(v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + v_4^2}{n}}\), where \(v_1, v_2, v_3, v_4\) represent individual molecule speeds, and \(n\) is the total number of molecules. Here’s a step-by-step view:

• First, square each of the speeds: e.g., if \(v_1 = 1\), then \(v_1^2 = 1\)
• Add these squared speeds together.
• Divide the sum by the number of molecules to find the mean of the squares.
• Take the square root of this mean to find the RMS speed.

This process gives a clearer understanding than simple averages because it relates directly to the molecular kinetic energy, offering insights into the thermodynamic behavior of gaseous systems.