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Professor Justin Lorieau

4. Reversible Compression

Reversible Compression: The Work Extremum Principle
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Reversible Compression: The Work Extremum Principle

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1. Reversible vs. Irreversible Work

Section titled “1. Reversible vs. Irreversible Work”

Compression work on a gas is defined by the external opposing pressure PextP_{\text{ext}}:

w=−∫V1V2Pext dVw = -\int_{V_1}^{V_2} P_{\text{ext}} \, dV
  • Single-Stage Irreversible Compression: A large external weight is placed instantly on the piston (Pext=P2P_{\text{ext}} = P_2). The work done on the gas is: wirrev=−P2(V2−V1)=P2(V1−V2)w_{\text{irrev}} = -P_2 (V_2 - V_1) = P_2 (V_1 - V_2)
  • Reversible Compression: External pressure is adjusted infinitesimally (Pext=Pgas±dPP_{\text{ext}} = P_{\text{gas}} \pm dP) at every step: wrev=−∫V1V2nRTV dV=nRTln⁡(V1V2)w_{\text{rev}} = -\int_{V_1}^{V_2} \frac{n R T}{V} \, dV = n R T \ln\left(\frac{V_1}{V_2}\right)

Because P2(V1−V2)>nRTln⁡(V1/V2)P_2 (V_1 - V_2) > nRT \ln(V_1 / V_2) for any non-zero compression, reversible compression always demands the minimum possible work to reach the final state.