Problem 109 - Master Resource Book in JEE Main Physics

Question

The average energy and the rms speed of molecules in a sample of oxygen gas at \(400 \mathrm{~K}\) are \(7.21 \times 10^{-21} \mathrm{~J}\) and \(524 \mathrm{~ms}^{-1}\) respectively. The corresponding values at \(800 \mathrm{~K}\) are nearly (a) \(14.42 \times 10^{-21} 3,1048 \mathrm{~ms}^{-1}\) (b) \(10.18 \times 10^{-21} J, 741 \mathrm{~ms}^{-1}\) (c) \(7.21 \times 10^{-21} \mathrm{~J}, 1048 \mathrm{~ms}^{-1}\) (d) \(14.42 \times 10^{-21} \mathrm{~J}, 741 \mathrm{~ms}^{-1}\)

Step-by-Step Solution

Answer

Option (d): \(14.42 \times 10^{-21} \mathrm{~J}, 741 \mathrm{~ms}^{-1}\)."

Step 1: Understanding Average Energy Proportionality

The average energy of a gas molecule is directly proportional to its absolute temperature. Given that the initial average energy at 400 K is \(7.21 \times 10^{-21} \mathrm{~J}\), the energy at 800 K (which is double the temperature) should also double. This means the energy will be \(2 \times 7.21 \times 10^{-21} \mathrm{~J} = 14.42 \times 10^{-21} \mathrm{~J}\).

Step 2: Calculating RMS Speed Relation to Temperature

The root-mean-square (rms) speed \(v_{rms}\) is given by the formula \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(T\) is the temperature. When temperature doubles from 400 K to 800 K, we have \(v_{rms,new} = \sqrt{2} \times v_{rms,old}\). Substituting \(v_{rms,old} = 524\, \mathrm{ms}^{-1}\), we get \(v_{rms,new} = 524\, \mathrm{ms}^{-1} \times \sqrt{2} = 741\, \mathrm{ms}^{-1}\).

Step 3: Comparing with Given Options

Now compare the calculated values: energy of \(14.42 \times 10^{-21} \mathrm{~J}\) and rms speed of \(741\, \mathrm{ms}^{-1}\) with the provided options:- Option (a): \(14.42 \times 10^{-21} \mathrm{~J}, 1048 \mathrm{~ms}^{-1}\)- Option (b): \(10.18 \times 10^{-21} \mathrm{~J}, 741 \mathrm{~ms}^{-1}\)- Option (c): \(7.21 \times 10^{-21} \mathrm{~J}, 1048 \mathrm{~ms}^{-1}\)- Option (d): \(14.42 \times 10^{-21} \mathrm{~J}, 741 \mathrm{~ms}^{-1}\)The correct answer is option (d).

Key Concepts

Kinetic Theory of Gases

The Kinetic Theory of Gases provides a fascinating way to understand how gases behave. It describes a gas as a large number of small particles, typically molecules, that are in constant, random motion. The theory explains essential properties of gases, like pressure and temperature, in terms of these molecular behaviors.
To visualize this:

• The molecules are like tiny billiard balls moving around in the container. They follow Newton's laws of motion.
• The motion is completely random, meaning the molecules are moving in all directions at various speeds.
• The pressure exerted by a gas is due to collisions of the molecules with the walls of the container.
• Temperature is related to the average kinetic energy of the molecules. The higher the temperature, the more energetic (faster) the molecules become.

This theory explains why gases diffuse, fill a container, and are compressible. It's a key concept in thermodynamics and crucial for understanding more complex formulas, like the ones used in the RMS speed and energy calculations.

RMS Speed

RMS, or root-mean-square speed, is a vital concept for understanding how fast the molecules in a gas are moving on average. It's like the speedometer of the molecular world, giving a useful measure of molecular speed that correlates with temperature. To compute RMS speed, we use the equation:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
Where:

• \(v_{rms}\) is the root-mean-square speed
• \(k\) is the Boltzmann constant
• \(T\) is the absolute temperature in Kelvin
• \(m\) is the mass of a single molecule of the gas

The beauty of this formula is in its direct relationship to temperature: if the temperature doubles, the RMS speed increases by a factor of \(\sqrt{2}\).
Therefore, higher temperatures translate to faster molecules, echoing the Kinetic Theory's insights about energy and motion. Understanding this helps us connect the dots between microscopic molecular behavior and macroscopic phenomena like gas pressure and distribution.

Temperature Dependence of Energy

The relationship between temperature and the energy of molecules in a gas is straightforward yet profound. In thermodynamics, we often talk about the average kinetic energy of a molecule. This energy is directly proportional to the absolute temperature of the gas, reinforcing the connection between energy and motion.
Let's break that down:

• Each molecule has kinetic energy due to its motion, calculated as \(KE = \frac{1}{2}mv^2\), where \(m\) is mass and \(v\) is speed.
• The average kinetic energy of the molecules in a gas is given by \(\frac{3}{2}kT\), where \(k\) is the Boltzmann constant and \(T\) is the temperature in Kelvin.
• When the temperature increases, the average kinetic energy increases, causing molecules to move faster.

For example, if the temperature of oxygen gas doubles, as in our exercise, the average energy of each molecule also doubles. This concept not only tells us about molecular speed but helps predict the behavior of gases under different temperature conditions.
Knowing this is crucial for anyone exploring thermodynamics, as it lays the foundation for more complex heat-related phenomena, such as phase changes and reactions.