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Question #42310

Difficulty: 8/10 | 🟢 Free

The Question

One mole of an ideal gas is enclosed in a horizontal cylinder equipped with a piston of area $A$ . The gas expands quasi-statically and isothermally at a constant temperature $T_0$ from an initial volume $V_0$ to $2V_0$ . During the expansion, the piston moves against a rough surface, experiencing a variable frictional force $F_f = \frac{A R T_0}{2 V_0^2} (V - V_0)$ , where $V$ is the instantaneous volume of the gas and $R$ is the universal gas constant. If all the heat generated by the friction is completely absorbed by the gas, what is the heat $Q_{\text{res}}$ supplied by the external thermal reservoir to maintain the isothermal process?

Options

A.
$R T_0 \ln 2$
B.
$R T_0 \left( \ln 2 - \frac{1}{4} \right)$
C.
$R T_0 \left( \ln 2 + \frac{1}{4} \right)$
D.
$R T_0 \left( \ln 2 - \frac{1}{2} \right)$

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

Variable Force Integration|Isothermal Process Work|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the First Law of Thermodynamics in a quasi-isothermal process involving an evolving non-conservative constraint.

Students commonly fall into the trap of assuming that the heat supplied by the external reservoir is simply $Q = W_{\text{gas}} = nRT \ln(V_2/V_1)$ , ignoring the fact that the work done against friction is dissipated as heat.

Because this dissipated heat is re-absorbed by the gas, it acts as an internal heat source, thereby reducing the thermal energy required from the external reservoir.

This highlights the critical distinction between the total heat required by a thermodynamic system and the net heat supplied by an external reservoir in the presence of internal mechanical dissipation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Variable Force Integration
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free