Before You Begin

Question #42199

Difficulty: 8/10 | 🟢 Free

The Question

One mole of an ideal monoatomic gas is used to model the energy dissipation of an elastic material undergoing a hysteresis cycle. The gas undergoes a three-step cyclic process $A \to B \to C \to A$ :

Expansion (stretching analog) from state $A(P_0, V_0)$ to $B(2P_0, 2V_0)$ following a polytropic process $P \propto V$ .
Compression (relaxation analog) from state $B$ to $C(P_C, V_0)$ following a polytropic process $P \propto V^2$ .
Isochoric return from state $C$ to state $A$ . Let $Q_{\text{rej}}$ be the magnitude of the heat rejected by the gas during the relaxation phase ( $B \to C$ ), and $W_{\text{net}}$ be the net work done by the gas in the entire cycle (representing the hysteresis loss). What is the ratio $\frac{Q_{\text{rej}}}{W_{\text{net}}}$ ?

Options

A.
$\frac{77}{4}$
B.
$\frac{11}{2}$
C.
$\frac{63}{4}$
D.
$\frac{77}{12}$

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

Elastic Hysteresis Loop|Polytropic Process|Approximation (First-order expansion)

Why This Question Is Tricky

Common Mistakes

This question tests the thermodynamic modeling of elastic hysteresis, specifically the coupling of internal energy and macroscopic work.

The primary trap lies in evaluating the heat exchange during the relaxation phase ( $B \to C$ ) governed by a dynamic polytropic index ( $n=-2$ ).

Students often misidentify this as an adiabatic or isothermal process, or incorrectly calculate the molar heat capacity for a negative index.

A secondary trap is dividing the rejected heat by the work done during the specific relaxation phase ( $W_{BC}$ ) rather than the net cyclic work ( $W_{\text{net}}$ ), leading to the incorrect option 11/2.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Elastic Hysteresis Loop
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free