Question

The temperature and volume of one mole an ideal gas during its expansion is related as $T=T_0+\alpha V^2$ where $T_0$ and $\alpha$ positive constants. R is the universal gas constant.
The volume at which the pressure is minimum is;

• A. $V=\sqrt{\frac{T_0}{\alpha }}$
• B. $V=\sqrt{\frac{2T_0}{\alpha }}$
• C. $V=2\sqrt{\frac{T_0}{\alpha }}$
• D. $V=\sqrt{\alpha T_0}$

Answer 1

The correct option is A $V=\sqrt{\frac{T_0}{\alpha }}$

$T=T_0+\alpha v^2\cdots \left(1\right)$
From ideal gas law PV=RT(n=1)
$\Rightarrow T=\frac{PV}{R}\cdots \left(2\right)$
(2) in (1)
$\frac{PV}{R}=T_0+\alpha v^2$
$P=\frac{R{T}_0}{V}+\alpha RV$
From minima $\frac{dP}{dV}=0$
$\Rightarrow V=\sqrt{\frac{T_0}{\alpha }}$