Problem 176 - Master Resource Book in JEE Main Physics

Question

On the basis of kinetic theory of gases, the mean kinetic energy of 1 mol per degree of freedom is [BVP Enge, 2006] (a) \(\frac{1}{2} k T\) (b) \(\frac{1}{2} R T\) (c) \(\frac{3}{2} k T\) (d) \(\frac{3}{2} R T\)

Step-by-Step Solution

Answer

The mean kinetic energy of 1 mole per degree of freedom is \( \frac{1}{2} R T \), so the answer is (b).

Step 1: Understand the Kinetic Theory of Gases

The kinetic theory of gases posits that the average kinetic energy of gas particles per degree of freedom is solely dependent on temperature. For a single particle, this is expressed as \( \frac{1}{2} k T \), where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin.

Step 2: Relate to the Per Mole Basis

Since we are considering 1 mole of gas, we need to relate the Boltzmann constant \( k \) with the gas constant \( R \). The relation between them is \( R = N_A k \), where \( N_A \) is Avogadro's number. This will help us express the energy in terms of the gas constant \( R \).

Step 3: Calculate Mean Kinetic Energy Per Degree of Freedom

Using the relation \( R = N_A k \), the mean kinetic energy per degree of freedom for one mole of gas translates to: \[ \text{Energy per mole} = N_A \times \frac{1}{2}k T = \frac{1}{2} R T \].

Step 4: Conclusion Based on Calculation

Based on the calculation, the mean kinetic energy of 1 mole per degree of freedom is \( \frac{1}{2} R T \). This corresponds to option (b).

Key Concepts

Mean Kinetic Energy

The mean kinetic energy of gas particles is an essential concept in understanding their behavior. According to the kinetic theory of gases, this energy depends solely on the temperature of the system. For a single particle, the mean kinetic energy per degree of freedom is given by the formula:

• \( \frac{1}{2} k T \).

Here, \( k \) represents the Boltzmann constant and \( T \) denotes the temperature in Kelvin. This equation signifies that as the temperature increases, the kinetic energy of the gas particles also increases.

In simpler terms, when heated, gas particles move faster, leading to increased energy. When calculating for 1 mole of gas, it is crucial to relate this per single particle energy to Avogadro's number to express it in terms of the gas constant \( R \), leading to \( \frac{1}{2} R T \) as the energy per mole per degree of freedom.

Boltzmann Constant

The Boltzmann constant (\( k \)) is a fundamental parameter representing the relationship between temperature and energy. It is a key component in the kinetic theory of gases and is involved in determining the energy level of particles within a gas.

• The value of the Boltzmann constant is approximately \( 1.38 \times 10^{-23} \, \text{J/K} \).

This constant provides a bridge between macroscopic and microscopic physical phenomena. It links the average kinetic energy of particles in a gas to the temperature by the equation \( E = \frac{1}{2} k T \).

In essence, the Boltzmann constant allows us to translate temperature into the kinetic energy of individual particles, thereby explaining how temperature can affect the dynamics of a gas.

Avogadro's Number

Avogadro's number (\( N_A \)) is another critical concept, which is defined as the number of constituent particles, usually atoms or molecules, present in one mole of a substance.

• The value is approximately \( 6.022 \times 10^{23} \) particles/mole.

This constant provides the basis for the mole concept, essential for connecting macroscopic amounts of material to the number of atoms or molecules they contain.

In the context of the kinetic theory of gases, Avogadro's number is used to relate the Boltzmann constant (\( k \)) to the gas constant (\( R \)) as follows: \( R = N_A k \).

This relationship helps in calculating the total energy of one mole of gas in terms of \( R \), making the gas constant a crucial parameter in broader thermodynamic calculations.