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Question #42555

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The Question

A quantum engine uses $N$ non-interacting particle pairs as its working substance. Each pair consists of two particles of masses $m$ and $3m$ , confined in a 1D infinite potential well. The system is restricted to its lowest two energy states. The engine operates in a cycle consisting of:

Isothermal expansion at $T_H$ from well width $L$ to $\sqrt{2}L$ .
Isochoric (constant width) cooling to $T_L$ .
Isothermal compression at $T_L$ back to width $L$ .
Isochoric heating back to $T_H$ . Given that the temperatures are high enough such that $kT \gg \Delta E$ (where $\Delta E$ is the energy gap between the two states), what is the efficiency $\eta$ of this engine?

Options

A.
$\frac{3(T_H - T_L)}{8T_H - 5T_L}$
B.
$1 - \frac{T_L}{T_H}$
C.
$\frac{3(T_H - T_L)}{5T_H - 2T_L}$
D.
$\frac{T_H - T_L}{2T_H - T_L}$

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

Carnot Engine Efficiency|Reduced Mass Effect|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question traps students who blindly assume any thermodynamic cycle operating between two temperatures has the Carnot efficiency $\eta = 1 - T_L/T_H$ .

By defining a Stirling-like cycle for a quantum 2-level system, students must derive the macroscopic thermodynamic state variables ( $U$ , $S$ , $F$ ) from the microscopic partition function $Z$ .

The trap lies in the high-temperature Taylor expansion: failing to expand to the second order in $\beta \Delta E$ results in a constant entropy, leading to zero isothermal heat transfer.

Furthermore, the reduced mass modifies the energy gap, linking quantum mechanics to statistical thermodynamics, ensuring only those with a deep understanding of the fundamental relations $F = -kT \ln Z$ and $S = -\partial F/\partial T$ can extract the correct efficiency.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Carnot Engine Efficiency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free