An ideal gas is expanding such that $p T^2=$ constant. The coefficient of volume expansion of the gas is
An ideal gas is expanding such that $p T^2=$ constant. The coefficient of volume expansion of the gas is
$\frac{1}{T}$
$\frac{2}{T}$
$\frac{3}{T}$
$\frac{4}{T}$
Solution
$p T^2=$ constant
$\begin{array}{ll}\therefore \quad & \left(\frac{n R T}{V}\right) T^2=\text { constant } \\ \text { or } \quad T^3 V^{-1}=\text { constant }\end{array}$
Differentiating the equation, we get
$
\frac{3 T^2}{V} \cdot d T-\frac{T^3}{V^2} \cdot d V=0
$
or
$
3 \cdot d T=\frac{T}{V} \cdot d V
$
From the equation, $d V=V \gamma \cdot d T$
$\gamma$ = coefficient of volume expansion of gas $=\frac{d V}{V \cdot d T}$
From Eq. (i) $\gamma=\frac{d V}{V \cdot d T}=\frac{3}{T}$
$\therefore$ correct answer is (c)