Derive the work done in an adiabatic process.

III. Long Answer Questions · Q16

Q. Derive the work done in an adiabatic process.

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Step 1. For a quasi-static adiabatic process, $PV^\gamma=\text{constant}$ , so $P=\dfrac{\text{constant}}{V^\gamma}$ . The work done is $W=\int_{V_i}^{V_f}P\,dV=\text{constant}\int_{V_i}^{V_f}V^{-\gamma}\,dV=\text{constant}\left[\dfrac{V^{1-\gamma}}{1-\gamma}\right]_{V_i}^{V_f}.$

Step 2. Evaluating and simplifying (using $\text{constant}=P_iV_i^\gamma=P_fV_f^\gamma$ ) gives $W=\dfrac{P_iV_i-P_fV_f}{\gamma-1}.$

Step 3. Using the ideal gas law, $P_iV_i=\mu RT_i$ and $P_fV_f=\mu RT_f$ ; substituting gives $W_{adia}=\dfrac{\mu R}{\gamma-1}(T_i-T_f).$ …

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Q.Derive the work done in an adiabatic process.

Concept understanding — Adiabatic Process

Q.Derive the work done in an adiabatic process.

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