A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure Pi = 105 Pa and volume Vi = 10–3 m3 changes to a final state at Pf = (l/32) × 105 Pa and Vf = 8 × 10–3 m3 in an adiabatic quasi-static process, such that P3V5 = constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at Pi followed by an isochoric (isovolumetric) process at volume Vf. The amount of heat supplied to the system in the two-step process is approximately

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A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure P_i = 10⁵ Pa and volume V_i = 10^–3 m³ changes to a final state at P_f = (l/32) × 10⁵ Pa and V_f = 8 × 10^–3 m³ in an adiabatic quasi-static process, such that P³V⁵ = constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at P_i followed by an isochoric (isovolumetric) process at volume V_f. The amount of heat supplied to the system in the two-step process is approximately

588 J

813 J

294 J

112 J

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$P_{0} V_{0}^{γ} = \frac{P_{0}}{32} {(8 V_{0})}^{γ}$

2⁵ = 2^3γ

3γ = 5

dQ = dU_if + dW_if

_{$= \frac{nPdT}{γ - 1} + P_{0} \times 7 V_{0} = \frac{P_{f} V_{f} - P_{i} V_{i}}{γ - 1} + P_{0} \times 7 V_{0}$}

_{$= \frac{\frac{P_{0}}{32} \times 8 V_{0} - P_{0} V_{0}}{\frac{5}{3} - 1} + 7 P_{0} V_{0} = \frac{\frac{P_{0} V_{0}}{4} - P_{0} V_{0}}{\frac{2}{3}}$}

_{$= \frac{3}{2} (\frac{- 3 P_{0} V_{0}}{4}) + 7 P_{0} V_{0} = \frac{- 9}{8} P_{0} V_{0} + 7 P_{0} V_{0}$}

_{$= \frac{(56 - 9) P_{0} V_{0}}{8} = \frac{47}{8} P_{0} V_{0} = \frac{47}{8} \times 10^{5} \times 10^{- 3}$}

_{$= \frac{47}{8} \times 100$}

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