We have learned that $pV=nRT$ is used to describe an ideal gas. One way to model non$-$

ideal gas is to use van der Waals equation $(p+\frac{a{n}^2}{V^2})(V-nb)=nRT,$ where a and $b$ are

constants. Such a gas is called van der Waals gas. For $1$mol monatomic van der Waals gas

$(n=1),$ the molar energy is given by $E=\frac{3}{2}RT-\frac{a}{V},$ where $V$ is the molar volume at

temperature $T,$ and $a$ is the constant in the van der Waals equation. Initially $1$ mol

monatomic van der Waals gas is at the temperature $T_1$ and occupies a volume $V_1.$ The gas

is allowed to expand adiabatically into a vacuum so that it occupies a total volume $V_2.$

What is the final temperature of the gas?