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Question #42535

Difficulty: 8/10 | 🟢 Free

The Question

One mole of an ideal monatomic gas is confined in a horizontal cylinder by a movable piston of mass $m$ and cross-sectional area $A$ . The piston is subjected to a conservative force field such that, if it were free, it would experience a non-uniform acceleration $a(x) = \alpha x^2$ directed towards the negative x-direction, where $\alpha$ is a positive constant and $x$ is the position of the piston. The gas expands reversibly, pushing the piston from $x = L$ to $x = 2L$ . If the initial pressure and volume of the gas are $P_0$ and $V_0$ respectively, and the space outside the cylinder is evacuated, what is the total heat supplied to the gas during this process?

Options

A.
$\frac{77}{6} P_0 V_0$
B.
$\frac{23}{2} P_0 V_0$
C.
$\frac{21}{2} P_0 V_0$
D.
$6 P_0 V_0$

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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

Internal Energy Dependency|Non-Uniform Acceleration Constraint|Conservative Field Path Independence

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple conservative kinematics with thermodynamic processes.

A common trap is assuming that because the external field is conservative, the thermodynamic work is either zero or path-independent, leading students to erroneously use $W = P_0 \Delta V$ or ignore the $dW$ integral entirely.

By linking the external pressure to a non-uniform acceleration constraint ( $a \propto x^2$ ), the problem forces the derivation of a specific polytropic path ( $P \propto V^2$ ).

Failing to integrate $P(V)$ correctly over this specific path directly alters the work term in the First Law of Thermodynamics, filtering out students who blindly apply state-function shortcuts to path-dependent variables.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Internal Energy Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free