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Question #83206

Difficulty: 5/10 | 🟢 Free

The Question

A quantity of $n = 3 \text{ moles}$ of an ideal non-linear polyatomic gas with molar heat capacity at constant volume $C_v = 3R$ undergoes a quasi-static cyclic process. The path of the cycle in the $P-V$ plane is given by the parametric equations $P(t) = P_0 + \alpha \sin(\omega t)$ and $V(t) = V_0 + \beta \cos(\omega t)$ for time $t \in [0, \frac{2\pi}{\omega}]$ . The constants are given as $P_0 = 2 \times 10^5 \text{ Pa}$ , $V_0 = 0.05 \text{ m}^3$ , $\alpha = 0.5 \times 10^5 \text{ Pa}$ , and $\beta = 0.01 \text{ m}^3$ . What is the net change in the total translational kinetic energy of the gas molecules over one complete cycle?

Options

A.
$0 \text{ J}$
B.
$500\pi \text{ J}$
C.
$750\pi \text{ J}$
D.
$1500\pi \text{ J}$

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

Pressure-Kinetic Energy Relation|Cyclic Process Geometry|Integration

Why This Question Is Tricky

Examiner Levers

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Common Mistakes

This question tests the fundamental distinction between state functions and path functions in thermodynamics.

Students often conflate the cyclic integral of work ( $\oint P dV$ ) or heat with the change in internal energy components.

By providing a complex parametric path and extraneous variables like $C_v$ and the cycle's amplitudes, the problem baits students into calculating the area of the $P-V$ loop (yielding $500\pi \text{ J}$ ) or misapplying the pressure-kinetic energy relation as a differential flux ( $\frac{3}{2} \oint P dV = 750\pi \text{ J}$ ).

The trap penalizes those who fail to recognize that translational kinetic energy, being strictly proportional to the state variables $P$ and $V$ , is a state function whose net change over any closed loop is identically zero.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Pressure-Kinetic Energy Relation
Difficulty	5
Cognitive Load	6
Target Solve Time	150 sec
Access	Free