Question

The P-V relation for monoatomic ideal gas undergoing an adiabatic change is

Moderate

Solution

short cut
$P V^{Î³} = C O N S T A N T$
ALTERNATE METHOD: We know that
$E = \frac{3}{2} NkT or dE = \frac{3}{2} NkdT$
Further, $dW = P â dV = \frac{NkT}{V} â dV$
For Adiabatic change dE = - dW. Hence
$\frac{3}{2} NkdT = â \frac{NkT}{V} â dV$
or $\frac{dT}{T} = â \frac{2}{3} \frac{dV}{V}$ or $\frac{dT}{T} + \frac{2}{3} \frac{dV}{V} = 0$
$â´ \log â¡ T + \frac{2}{3} \log â¡ V = constant$
or ${TV}^{2 / 3} = constant$
or ${PV}^{5 / 3} = constant (Using ideal gas law)$

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