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Question #42180

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The Question

One mole of an ideal monatomic gas is confined in a cylinder by a movable piston of area $A$ . The piston contains a magnetic core of volume $V_p$ . The system is placed in an external magnetic field $\mathbf{B} = kx \hat{i}$ , where $x$ is the position of the piston. The gas undergoes a cyclic process consisting of four steps:

Isobaric expansion at pressure $P_0$ from $x_1$ to $x_2$ . The core's magnetization is constant at $M_0 \hat{i}$ .
Isochoric depressurization to $P_0/2$ . The magnetization changes to $3M_0 \hat{i}$ .
Isobaric compression at $P_0/2$ back to $x_1$ . The magnetization is constant at $3M_0 \hat{i}$ .
Isochoric pressurization to $P_0$ . The magnetization returns to $M_0 \hat{i}$ . Let $\Delta V = A(x_2 - x_1)$ and define a dimensionless parameter $\beta = \frac{V_p M_0 k}{P_0 A}$ . Which of the following gives the effective molar heat capacity $C_{eff}$ of the system during the isobaric expansion (step 1), and the net heat $Q_{net}$ absorbed by the system over one complete cycle?

Options

A.
$C_{eff} = R\left(\frac{5}{2} + \beta\right), \quad Q_{net} = P_0 \Delta V \left(\frac{1}{2} - 2\beta\right)$
B.
$C_{eff} = \frac{5}{2}R, \quad Q_{net} = P_0 \Delta V \left(\frac{1}{2} + 2\beta\right)$
C.
$C_{eff} = R\left(\frac{5}{2} - \beta\right), \quad Q_{net} = P_0 \Delta V \left(\frac{1}{2} - 2\beta\right)$
D.
$C_{eff} = R\left(\frac{5}{2} + \beta\right), \quad Q_{net} = P_0 \Delta V \left(\frac{1}{2} + 2\beta\right)$

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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

Molar Heat Capacity (Cv vs Cp)|Hysteresis Loop Area|Force on Current-Carrying Loop

Why This Question Is Tricky

Common Mistakes

This question tests the generalized First Law of Thermodynamics $dU = dQ - dW_{net}$ , where the total work includes both PV work and magnetic work.

The magnetic force on the piston $F_m = m \frac{dB}{dx}$ acts in the direction of expansion, modifying the effective pressure and thus altering the molar heat capacity $C_{eff}$ from the standard $C_p = \frac{5}{2}R$ .

Furthermore, the magnetic cycle traces a hysteresis loop.

The magnetic work done by the system over the cycle is proportional to $\oint M \, dV$ , which is negative, representing hysteresis energy loss (work done ON the system).

Students often fall into the trap of either ignoring this non-PV work in $C_{eff}$ or incorrectly adding the hysteresis loss to the net heat absorbed, failing to realize that dissipation reduces the net work output and thus the net heat absorbed in a cycle.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Molar Heat Capacity (Cv vs Cp)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free