Problem 56 - A Complete Resource Book in Chemistry for JEE Main

Question

According to the kinetic theory of gases (a) the pressure exerted by a gas is proportional to mean square velocity of the molecules (b) the pressure exerted by the gas is proportional to the root mean square velocity of the molecules (c) the root mean square velocity is inversely proportional to the temperature (d) the mean translational K.E. of the molecule is directly proportional to the absolute temperature.

Step-by-Step Solution

Answer

Correct statements are (a) and (d).

Step 1: Understanding Kinetic Theory Basics

Kinetic theory of gases relates the macroscopic properties of gases to the microscopic actions and interactions of gas molecules. It involves understanding terms related to temperature, pressure, and molecular speed.

Step 2: Analyzing Pressure and Mean Square Velocity

According to kinetic theory, the pressure \( P \) exerted by a gas is directly proportional to the mean square velocity \( \overline{v^2} \) of its molecules. This can be expressed as \( P \propto \overline{v^2} \). Thus, choice (a) aligns with this concept.

Step 3: Examining Root Mean Square Velocity

Root mean square (RMS) velocity \( v_{rms} \) is derived as \( v_{rms} = \sqrt{\overline{v^2}} \), linking the discussed mean square velocity. Pressure is not directly proportional to RMS velocity, so choice (b) is incorrect.

Step 4: Investigating Temperature and RMS Velocity Relationship

RMS velocity \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( T \) is the absolute temperature and \( k \) is Boltzmann's constant and \( m \) is the molecular mass. Hence, \( v_{rms} \) is directly proportional to the square root of temperature, not inversely; thus, choice (c) is incorrect.

Step 5: Evaluating Kinetic Energy and Temperature

The average translational kinetic energy (K.E.) is given as \( \frac{3}{2}kT \), indicating it is directly proportional to the absolute temperature \( T \). Therefore, choice (d) is correct.

Key Concepts

Mean Square Velocity

In the kinetic theory of gases, the **mean square velocity** refers to the average of the squares of the velocities of gas molecules. It is a useful concept when assessing the behavior of a gas at a microscopic level.
Using mean square velocity, we can better understand how the particles of a gas move randomly, colliding with each other and the walls of their container. This motion results in the observable macroscopic property known as pressure. The relationship is mathematically expressed as \( P \propto \overline{v^2} \), meaning that the pressure \( P \) exerted by the gas is directly proportional to the mean square velocity \( \overline{v^2} \).
This concept is crucial because it forms the basis for understanding how gases behave under different conditions of pressure and temperature. It gives insight into the energy dynamics inside a gas-filled container, allowing physicists and chemists to predict how gases will react when subject to varying external conditions.

Root Mean Square Velocity

**Root mean square (RMS) velocity** is a related concept that offers insights into the kinetic energy of gas molecules. RMS velocity is essentially the square root of the mean square velocity and is given mathematically by \( v_{rms} = \sqrt{\overline{v^2}} \).
Why is RMS velocity useful? It provides a measure of the speed of particles in a gas, accounting for their kinetic energy. This is particularly important because it allows scientists to understand and quantify the energy distribution among gas particles more accurately.
It's important to note that, although RMS velocity connects to temperature and molecular conditions, pressure is not directly proportional to RMS velocity. Therefore, this metric is more relevant for understanding molecular speed and energy than pressure relationships.

Translational Kinetic Energy

The **translational kinetic energy** of gas molecules helps explain how the energy related to movement translates directly from heat energy into motion. In the context of kinetic theory, it is the energy due to motion that contributes to the total internal energy of a gas.
The formula for average translational kinetic energy of gas molecules is \( \frac{3}{2}kT \), where \( k \) is Boltzmann's constant, and \( T \) is absolute temperature. It signifies that the mean kinetic energy increases with temperature.
This equation directly shows the relationship between kinetic energy and temperature, highlighting that as the absolute temperature increases, so does the kinetic energy of the molecules. This relationship is key in explaining phenomena such as heating of gases, expansion, and pressure changes in closed systems.

Absolute Temperature

**Absolute temperature** is a fundamental concept in thermodynamics and kinetic theory. It is measured in Kelvin and represents a scale where 0 indicates a hypothetical absence of all particle movement, known as absolute zero.
Absolute temperature is crucial because it provides a baseline for understanding energy interactions in gases. It dictates how heated or cold a substance is at the molecular level, independently from the material or external appearance.
In relation to kinetic theory, absolute temperature is directly linked to the velocity and kinetic energy of gas particles. As the temperature rises, the available thermal energy increases, leading to faster molecular movement. This connection is mathematically captured in formulas related to mean square velocity, RMS velocity, and translational kinetic energy, where temperature determines the scale of molecular speed and motion. Understanding absolute temperature is essential for applying the principles of thermodynamics accurately across various scientific and engineering fields.