Question

Why is the p-V curve for adiabatic process steeper than that for isothermal process?

Answer 1

\[ \begin{array}{l} \text{Adiabatic process: } PV^{\gamma} = \text{constant} \\[6pt] \therefore\ V^{\gamma}dP + \gamma PV^{\gamma-1}\,dV = 0 \\[6pt] \therefore\ \dfrac{dP}{dV} = -\dfrac{\gamma P}{V} \\[10pt] \text{Isothermal process: } PV = \text{constant} \\[6pt] \therefore\ PdV + VdP = 0 \qquad \therefore\ \dfrac{dP}{dV} = \dfrac{P}{-V} \\[10pt] \text{Now, } \gamma > 1 \\[6pt] \therefore\ \left|\left(\dfrac{dP}{dV}\right)_{\text{adiabatic}}\right| > \left|\left(\dfrac{dP}{dV}\right)_{\text{isotherm}}\right| \\[10pt] \dfrac{dP}{dV} \text{ is the slope of the } P-V \text{ curve.} \end{array} \]

∴ The P-V curve for an adiabatic process is steeper than that for an isothermal process.