Q.45 Physics ⭐ 360° Option Elimination Active

NEET 2026 Physics Question 45 Solution

Topic: Kinetic Theory of Gases & Thermodynamics · Official NTA Booklet Code 50

Question

An ideal gas is made of polyatomic molecules. Each of the molecules has three translational, three rotational and f number of vibrational modes. If the ratio of heat capacities Cₚ/Cᵥ of the gas is 8/7, then the value of f is:

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Verified Against NTA Final Answer Key

Option (1) : f = 4

📚 Kinetic Theory of Gases & Thermodynamics Medium

📋 Given Parameters & Problem Setup

Gas Type Polyatomic ideal gas

Translational Modes 3 modes (each contributes (1/2)R to C_v)

Rotational Modes 3 modes (each contributes (1/2)R to C_v)

Vibrational Modes f modes (each contributes 1R to C_v: kinetic + potential)

Heat Capacity Ratio gamma = C_P / C_V = 8/7

Target Variable Number of vibrational modes f

⚡

The 30-Second Exam Hall Shortcut: ⚡ Vibrational Degree-of-Freedom Equipartition Rule

⏱️ Avoids fractional confusion.

Each vibrational mode contributes 1R (NOT R/2) because it stores both kinetic and potential energy!

1 Translational C_v = 3 * (R/2) = 1.5 R.

2 Rotational C_v = 3 * (R/2) = 1.5 R.

3 Vibrational C_v = f * R.

4 Total C_v = 1.5 R + 1.5 R + f R = (3 + f) R.

5 By Mayer's relation: C_p = C_v + R = (4 + f) R.

6 Ratio C_p / C_v = (4 + f) / (3 + f) = 8 / 7.

7 Cross-multiply: 7*(4 + f) = 8*(3 + f) ==> 28 + 7f = 24 + 8f ==> 8f - 7f = 28 - 24 ==> f = 4.

8 Solved in 20 seconds!

🛡️

Why You Can 100% Trust This Shortcut: The Law of Equipartition of Energy states that each quadratic term in the Hamiltonian contributes (1/2)k_B*T. A vibrational mode has two quadratic terms (kinetic (1/2)m*v^2 and potential (1/2)k*x^2), contributing 2 * (1/2)R = 1R to molar heat capacity.

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Formulate Molar Heat Capacity C_v from Equipartition Theorem

According to the law of equipartition of energy, each degree of freedom contributes to the internal energy of one mole of gas:

C_v = 3 \left(\frac{1}{2} R\right) + 3 \left(\frac{1}{2} R\right) + f (R) = \frac{3}{2} R + \frac{3}{2} R + f R = (3 + f) R

• 3 translational degrees of freedom contribute 3 * (1/2)R = (3/2)R.
• 3 rotational degrees of freedom contribute 3 * (1/2)R = (3/2)R.
• f vibrational modes contribute f * R (since each vibrational mode possesses both kinetic and potential energy states).

Step 2

Apply Mayer's Relation to Find C_p

For any ideal gas, C_p and C_v are related by Mayer's equation C_p = C_v + R:

C_p = C_v + R = (3 + f) R + R = (4 + f) R

• Universal relation C_p - C_v = R holds for all ideal gases.

Step 3

Equate Ratio C_p / C_v to 8/7 and Solve for f

Set up the ratio of molar heat capacities:

\gamma = \frac{C_p}{C_v} = \frac{(4 + f) R}{(3 + f) R} = \frac{4 + f}{3 + f} = \frac{8}{7}

• Cross-multiply: 7(4 + f) = 8(3 + f).
• 28 + 7f = 24 + 8f.
• 8f - 7f = 28 - 24 \implies f = 4.
• Matches Option (1) exactly.

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 Option (1)

✓ 100% Correct

Verdict & Scientific Reason:

100% mathematically and physically exact; confirmed by Official NTA Final Answer Key.

💡

Why this is airtight: Correctly applies 1R contribution per vibrational mode and solves (4+f)/(3+f) = 8/7.

Note: Exact kinetic theory physics.

2 Option (2)

✗ Incorrect

Verdict & Scientific Reason:

Yields gamma = 7/6 != 8/7.

⚠️

Student Fallacy (Why candidates pick this trap): Tests f = 3: (4+3)/(3+3) = 7/6 != 8/7.

Note: 7/6 is 1.167, whereas 8/7 is 1.143.

3 Option (3)

✗ Incorrect

Verdict & Scientific Reason:

Yields gamma = 6/5 = 1.20 != 8/7.

⚠️

Student Fallacy (Why candidates pick this trap): Arithmetic miscalculation.

Note: (4+2)/(3+2) = 6/5.

4 Option (4)

✗ Incorrect

Verdict & Scientific Reason:

Yields gamma = 5/4 = 1.25 != 8/7.

⚠️

Student Fallacy (Why candidates pick this trap): Tests f = 1: (4+1)/(3+1) = 5/4.

Note: Ratio must equal 8/7.

⚠️

NTA Trap Alert: The "Vibrational Mode = R/2" Equipartition Fallacy

Students often assume vibrational modes contribute R/2 just like translational and rotational modes, leading to an incorrect f = 8.

🛡️ How to avoid the -1 mark: Remember: Translational and Rotational have ONLY kinetic energy (1/2 R each). Vibrational modes have BOTH kinetic and potential energy ((1/2 + 1/2) R = 1 R each)!

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Core NEET Syllabus Standard

NEET UG Physics Core Standards · Kinetic Theory of Gases Standard Benchmark

Core Scientific Principle Tested:

"Molar heat capacity of polyatomic gas with f vibrational modes: C_v = (3 + f)R and C_p = (4 + f)R. The adiabatic index is gamma = (4 + f) / (3 + f)."

Syllabus Focus: Law of Equipartition of Energy & Specific Heats of Polyatomic Gases

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2026 Physics Question 45 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Vibrational Degree-of-Freedom Equipartition Rule

Step-by-Step Rigorous Workflow & Mathematical Proof

Formulate Molar Heat Capacity C_v from Equipartition Theorem

Apply Mayer's Relation to Find C_p

Equate Ratio C_p / C_v to 8/7 and Solve for f

360° Option Elimination Engine Key Feature

NTA Trap Alert: The "Vibrational Mode = R/2" Equipartition Fallacy

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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