Question

One mole of an ideal gas undergoes a process in which $T=T_0+a{V}^3$, where $T_0$ and ‘a’ are positive constants and V is the volume. The volume for which pressure will be minimum is

• A. ${\left(\frac{T_0}{2a}\right)}^{\frac{1}{3}}$
• B. ${\left(\frac{T_0}{3a}\right)}^{\frac{1}{3}}$
• C. ${\left(\frac{a}{2T_0}\right)}^{\frac{2}{3}}$
• D. ${\left(\frac{a}{3T_0}\right)}^{\frac{2}{3}}$

Answer 1

The correct option is A

${\left(\frac{T_0}{2a}\right)}^{\frac{1}{3}}$

Let us write $T=\frac{PV}{R}$ for n = 1
$\therefore \frac{PV}{R}=T_0+a{V}^3$
or $P=\frac{R{T}_0}{V}+aR{V}^2$
For P to be minimum $\frac{dP}{dV}=0$
$\therefore 0=-\frac{R{T}_0}{V^2}+2aRV\therefore V={\left(\frac{T_0}{2a}\right)}^{\frac{1}{3}}$