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Q.
An ideal gas is expanding such that $P T^{2}=$ constant. The coefficient of volume expansion of the gas is
Thermal Properties of Matter
Solution:
$P T^{2}=$ constant (Given) $\dots$ (i)
According to ideal gas equation,
$P V=n R T$ or $P=\frac{n R T}{V}$
Substituting this value of $P$ in (i), we get
$\left(\frac{n R T}{V}\right) T^{2}=$ constant
or $\frac{T^{3}}{V}=$ another constant
Differentiating both sides, we get
$\frac{1}{V} 3 T^{2} d T-\frac{T^{3}}{V^{2}} d V=0$
or $3 d T=\frac{T}{V} d V $ or
$\frac{1}{V}\left(\frac{d V}{d T}\right)=\frac{3}{T} \dots$(ii)
According to definition of the coefficient of volume expansion of the gas is
$\gamma=\frac{1}{V}\left(\frac{d V}{d T}\right) \ldots$(iii)
From (ii) and (iii), we get $\gamma=\frac{3}{T}$