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Question #42509

Difficulty: 8/10 | 🟢 Free

The Question

A rigid, insulated container holds $N$ atoms of a monoatomic gas, each of mass $m$ , at an initial temperature $T_0$ . A catalyst is introduced, causing all atoms to undergo a fusion process where they pair up to form $\frac{N}{2}$ rigid diatomic molecules. Each fusion event (the formation of one molecule) releases a binding energy $\epsilon = 7 k_B T_0$ , which is entirely absorbed by the gas. If $v_1$ is the initial root-mean-square speed of the monoatomic gas and $v_2$ is the final root-mean-square speed of the diatomic gas, what is the ratio $\frac{v_2}{v_1}$ ?

Options

A.
$\sqrt{2}$
B.
$2$
C.
$\sqrt{\frac{10}{3}}$
D.
$\sqrt{\frac{12}{5}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Molecular Degrees of Freedom|Binding Energy per Nucleon Curve|Energy Distribution

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the First Law of Thermodynamics, kinetic theory, and nuclear/chemical binding energy.

The primary trap lies in failing to account for the change in the degrees of freedom ( $f=3$ to $f=5$ ) and the molecular mass ( $m$ to $2m$ ) post-fusion.

Students often incorrectly assume the gas remains ideal with a constant $f$ , leading to an incorrect final temperature, or they forget that the root-mean-square speed of the new diatomic gas depends on the combined mass $2m$ .

The problem requires a strict application of energy conservation where the released binding energy directly increases the internal energy of a system with a newly altered partition function.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Molecular Degrees of Freedom
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free