An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^{3} =$ constant. The coe

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An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^{3} =$ constant. The coefficient of volume expansion of the gas is $\_\_\_\_$ .

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Sol. [Key Concept: From ${PT}^{3} =$const and $PV = nRT$, $V \propto \ T^{4}$; then $\gamma = \frac{1}{V}\frac{dV}{dT} = \frac{4}{T}$.]$\ {PT}^{3} =$ constant $(C)$$${\frac{nRT}{V} \cdot T^{3} = C }{V = \frac{nR}{C} \cdot T^{4} }$$Volume expansion coeff.$${\gamma = \frac{1}{	ext{ }V}\frac{dV}{dT} }{\gamma = \frac{C}{{nRT}^{4}} \cdot \frac{nR}{C} \cdot 4{	ext{ }T}^{3} = \frac{4}{	ext{ }T} }$$Ans. $\gamma = \frac{4}{	ext{ }T}$

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