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Question #43802

Difficulty: 8/10 | 🟢 Free

The Question

In a dense gas at absolute temperature $T$ , an atom of mass $m$ absorbs a photon of momentum $p_0$ , acquiring an initial recoil velocity. Due to the high density of the gas, the recoiling atom immediately undergoes a head-on collision with a neighboring atom of identical mass $m$ that is momentarily at rest. The surrounding gas acts as a viscous medium during this micro-scale collision, making it inelastic with an effective coefficient of restitution given by $e(T) = \sqrt{\frac{T_0}{T + T_0}}$ , where $T_0$ is a characteristic temperature constant of the gas. What is the total kinetic energy dissipated into the surrounding gas during this single collision event? (Assume all quantities are in standard SI units)

Options

A.
$\frac{p_0^2}{4m} \left( \frac{T}{T + T_0} \right)$
B.
$\frac{p_0^2}{2m} \left( \frac{T}{T + T_0} \right)$
C.
$\frac{p_0^2}{4m} \left( \frac{T_0}{T + T_0} \right)$
D.
$\frac{p_0^2}{2m} \left( \frac{T_0}{T + T_0} \right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Recoil of Atom|Coefficient of Restitution|Equation of State for Mixtures

Why This Question Is Tricky

Common Mistakes

The primary trap in this problem lies in the misapplication of the coefficient of restitution and conservation of momentum.

Because the problem states the surrounding gas acts as a 'viscous medium', students often incorrectly model the target atom as an immovable part of the macroscopic medium (like a fixed wall).

This leads them to assume the recoiling atom simply slows down with a final velocity governed solely by $e$ (i.e., $v_f = e v_0$ ), resulting in an energy dissipation of $K_0(1-e^2)$ and leading directly to the trap option B.

However, the target is a free atom of mass $m$ , meaning momentum must be conserved for the two-atom system.

The correct dissipation must account for the kinetic energy transferred to the target atom, which introduces a factor of $1/2$ originating from the reduced mass of the two-body system.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Recoil of Atom
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free