Thermodynamics · Equilibrium states, work, heat, and the first law
A closed-system gas is taken through a quasi-equilibrium polytropic process of the form…
Problem
A closed-system gas is taken through a quasi-equilibrium polytropic process of the form P V^n = constant. Measured end states are P1 = 150 kPa, V1 = 0.200 m³ and P2 = 600 kPa, V2 = 0.070 m³. No ideal-gas assumption is required. The exponent n is constant but is not given a priori. (a) Determine the polytropic exponent n from n = ln(P2/P1)/ln(V1/V2). (b) Compute the boundary work done by the gas using Wb = (P2 V2 − P1 V1)/(1 − n). Report the value in kJ and state whether the system does work on the surroundings or the surroundings do work on the system. (c) Evaluate P1 V1^n and P2 V2^n and confirm that they agree to three significant figures. (d) If the same end states were instead connected by a two-step path consisting of a constant-volume process followed by a constant-pressure process, compute the boundary work of that alternative path and compare it with part (b). Use this comparison to state whether work is a thermodynamic property.
Hint
The exponent is a log ratio of the measured end states; different paths to those end states give different \(\int P\,\mathrm{d}V\).
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