JEE Challenger
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PhyMedMain 2026 • 5th Apr - Shift 1

Assertion and Reason on Internal Energy and Degrees of Freedom

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 nn mole of ideal gas can be written as ΔU=nCv(Tf−Ti)=nRγ−1(Tf−Ti)\Delta U = n C_v (T_f - T_i) = \frac{nR}{\gamma - 1}(T_f - T_i), where γ=CpCv\gamma = \frac{C_p}{C_v}, Ti=T_i = initial temperature, Tf=T_f = final temperature.

Statement II: Relation between degree of freedom ff and γ(=Cp/Cv)\gamma \left(= C_p/C_v\right) is (γ=1+2f)\left(\gamma = 1 + \frac{2}{f}\right)

In the light of the above statements, choose the correct answer from the options given below

Options

A

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

B

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

Correct
C

A is true but R is false

D

A is false but R is true

Topics & Concepts

ThermodynamicsFirst Law of ThermodynamicsKinetic Theory of Gases

Step-by-Step Solution

To determine the correct choice, let us analyze both statements individually:

1. Analysis of Statement I (Assertion A)

The internal energy of an ideal gas depends solely on its absolute temperature. For nn moles of an ideal gas undergoing a temperature change from TiT_i to TfT_f, the change in internal energy ΔU\Delta U is given by: ΔU=nCv(Tf−Ti)\Delta U = n C_v (T_f - T_i)

From Mayer's relation for ideal gases, we have: Cp−Cv=RC_p - C_v = R

Dividing both sides by CvC_v: CpCv−1=RCv\frac{C_p}{C_v} - 1 = \frac{R}{C_v}

Since γ=CpCv\gamma = \frac{C_p}{C_v}, this gives: γ−1=RCv  ⟹  Cv=Rγ−1\gamma - 1 = \frac{R}{C_v} \implies C_v = \frac{R}{\gamma - 1}

Substituting Cv=Rγ−1C_v = \frac{R}{\gamma - 1} into the expression for ΔU\Delta U: ΔU=nRγ−1(Tf−Ti)\Delta U = \frac{nR}{\gamma - 1}(T_f - T_i)

Thus, Statement I is True.

2. Analysis of Statement II (Reason R)

According to the law of equipartition of energy, the molar heat capacity at constant volume CvC_v for a gas with ff degrees of freedom is: Cv=f2RC_v = \frac{f}{2}R

Using Mayer's relation, the molar heat capacity at constant pressure CpC_p is: Cp=Cv+R=(f2+1)RC_p = C_v + R = \left(\frac{f}{2} + 1\right)R

The adiabatic index γ\gamma is the ratio of CpC_p to CvC_v: γ=CpCv=(f2+1)Rf2R=1+2f\gamma = \frac{C_p}{C_v} = \frac{\left(\frac{f}{2} + 1\right)R}{\frac{f}{2}R} = 1 + \frac{2}{f}

Thus, Statement II is True.

3. Relationship between Statement I and Statement II

  • Statement I is derived directly from thermodynamic definitions (Cp−Cv=RC_p - C_v = R and γ=Cp/Cv\gamma = C_p / C_v) without requiring knowledge of the degrees of freedom ff.
  • Statement II relates γ\gamma to the microscopic degrees of freedom ff via kinetic theory.

Since Statement I is independent of Statement II, Statement II is NOT the correct explanation for Statement I.

Conclusion

Both Statements I and II are true, but Statement II is NOT the correct explanation of Statement I.

Correct Option: B

Assertion and Reason on Internal Energy and Degrees of Freedom | Physics PYQ Solution - JEE Challenger