Answer
See explanation
Work Step by Step
The definition for the isothermal compressibility is
$$
\alpha=-\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T
$$ The derivative is $$
\left(\frac{\partial P}{\partial v}\right)_T=-\frac{R T}{(\boldsymbol{v}-b)^2}+\frac{a}{T^{1 / 2}} \frac{2 \boldsymbol{v}+b}{\boldsymbol{v}^2(\boldsymbol{v}+b)^2}
$$ Substituting, $$
\alpha=-\frac{1}{v}\left(\frac{\partial \boldsymbol{v}}{\partial P}\right)_T=-\frac{1}{\boldsymbol{v}}\left(\frac{1}{-\frac{R T}{(\boldsymbol{v}-b)^2}+\frac{a}{T^{1 / 2}} \frac{2 \boldsymbol{v}+b}{\boldsymbol{v}^2(\boldsymbol{v}+b)^2}}\right)=-\frac{1}{-\frac{R T \boldsymbol{v}}{(\boldsymbol{v}-b)^2}+\frac{a}{T^{1 / 2}} \frac{2 \boldsymbol{v}+b}{(\boldsymbol{v}+b)^2}}
$$
