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Question #42193

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The Question

A heat engine operates on a cyclic process consisting of three steps:

Isochoric pressure increase from $P_2$ to $P_1$ at volume $V_1$ .
Polytropic expansion following $P V^{3/2} = \text{constant}$ from $P_1$ to $P_2$ .
Isobaric compression at $P_2$ back to volume $V_1$ .

The measured values of the independent parameters are: $P_1 = 8.0 \times 10^5 \text{ Pa} \pm 1\%$ $V_1 = 1.0 \text{ m}^3 \pm 1\%$ $P_2 = 1.0 \times 10^5 \text{ Pa} \pm 2\%$

Assuming the polytropic exponent $3/2$ is exact, what is the maximum percentage error in the calculated net work done per cycle?

Options

A.
$3.8\%$
B.
$19.0\%$
C.
$5.0\%$
D.
$4.0\%$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Cyclic Process Geometry|Ideal Gas Assumptions|Relative Error Propagation

Why This Question Is Tricky

Common Mistakes

The question tests the propagation of errors in thermodynamic cycles where intermediate variables (like $V_2$ ) are correlated with the independent measured variables ( $P_1, V_1, P_2$ ).

A common trap is calculating the intermediate volume $V_2$ and its relative error first, and then treating it as an independent variable in the work equation $W = 2 P_1 V_1 - 3 P_2 V_2 + P_2 V_1$ .

This incorrect assumption of independence ignores the covariance between $V_2$ and the pressures, leading to a massive overestimation of the error ( $19.0\%$ instead of the correct $3.8\%$ ).

Students must express the final quantity entirely in terms of independent variables before applying partial differentiation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Cyclic Process Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free