Question

An ideal gas is expanding such that $P{T}^2=$ constant. The coefficient of volume expansion of the gas is

• A. $\frac{1}{T}$
• B. $\frac{2}{T}$
• C. $\frac{3}{T}$
• D. $\frac{4}{T}$

Answer 1

Answer: (C)

$P{T}^2=$ constant (Given) $\dots$ (i)
According to ideal gas equation,
$PV=nRT$ or $P=\frac{nRT}{V}$
Substituting this value of $P$ in (i), we get
$\left(\frac{nRT}{V}\right)T^2=$ constant
or $\frac{T^3}{V}=$ another constant
Differentiating both sides, we get
$\frac{1}{V}3T^{2}dT-\frac{T^3}{V^2}dV=0$
or $3dT=\frac{T}{V}dV$ or
$\frac{1}{V}\left(\frac{dV}{dT}\right)=\frac{3}{T}\dots$(ii)
According to definition of the coefficient of volume expansion of the gas is
$\gamma =\frac{1}{V}\left(\frac{dV}{dT}\right)\dots$(iii)
From (ii) and (iii), we get $\gamma =\frac{3}{T}$