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Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Statement I: Change in internal energy of a system containing n mole of ideal gas can be written as Δ U = n Cv (Tf - Ti) = (nR)/(γ - 1)(Tf - Ti), where γ = (Cp)/(Cv), Ti = initial temperature, Tf = final temperature. Statement II: Relation between degree of freedom f and γ (= Cp/Cv) is (γ = 1 + (2)/(f) ) Choose the correct answer from the options given below

Both A and R are true and R is the correct explanation of A

Kinetic Theory

JEE Main level, single correct, medium

67 of 168

Question 67 of 168

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Statement I: Change in internal energy of a system containing $n$ mole of ideal gas can be written as $\Delta U = n C_v (T_f - T_i) = \dfrac{nR}{\gamma - 1}(T_f - T_i)$ , where $\gamma = \dfrac{C_p}{C_v}$ , $T_i =$ initial temperature, $T_f =$ final temperature.
Statement II: Relation between degree of freedom $f$ and $\gamma (= C_p/C_v)$ is $\left(\gamma = 1 + \dfrac{2}{f}\right)$
Choose the correct answer from the options given below

Show answer and solution

Answer: (B) Both A and R are true but R is NOT the correct explanation of A

Statement I is true because for an ideal gas, the change in internal energy is given by

\Delta U = n C_v \Delta T

. Using Mayer's relation

C_p - C_v = R

and the ratio of specific heats

\gamma = \dfrac{C_p}{C_v}

, we can write

C_v = \dfrac{R}{\gamma - 1}

. Substituting this, we get

\Delta U = \dfrac{nR}{\gamma - 1}(T_f - T_i)

.

Statement II is true because according to the law of equipartition of energy, the molar heat capacity at constant volume is

C_v = \dfrac{f}{2}R

and at constant pressure is

C_p = \left(\dfrac{f}{2} + 1\right)R

. Therefore,

\gamma = \dfrac{C_p}{C_v} = 1 + \dfrac{2}{f}

.

Statement II does not explain Statement I, as Statement I is derived purely from macroscopic thermodynamic relations (Mayer's relation) and is independent of the microscopic concept of degrees of freedom.

Hence, both statements are true but Statement II is NOT the correct explanation of Statement I.

Answer: Both A and R are true but R is NOT the correct explanation of A

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JEE Main Physics practice question on Kinetic Theory (Degree of Freedom and Specific Heat), with the answer and a step-by-step solution. Scan a question from your own book or paper to get a whiteboard video.