For 3s orbital of hydrogen atom, the normalised wave function is 
${\mathrm{\Psi }}_{3\mathrm{s}}=\frac{1}{81\sqrt{3\mathrm{\pi }}}{\left(\frac{1}{{\mathrm{a}}_0}\right)}^{3/2}\left[27-\frac{18\mathrm{r}}{{\mathrm{a}}_0}+\frac{2{\mathrm{r}}^2}{{\mathrm{a}}_0^2}\right]{\mathrm{e}}^{\frac{-\mathrm{r}}{3{\mathrm{a}}_0}}$ If distance between the radial nodes is d. Calculate the value of $\frac{\mathrm{d}}{1.73{\mathrm{a}}_0}$

$$\begin{gathered} {\mathrm{\psi }}^2=0\Rightarrow 27-\frac{18\mathrm{r}}{{\mathrm{a}}_0}+\frac{2{\mathrm{r}}^2}{{\mathrm{a}}_0^2}=0; \\ \mathrm{Let} \mathrm{\sigma }=\frac{\mathrm{r}}{{\mathrm{a}}_{\mathrm{o}}}, \\ \mathrm{then} 27-18\mathrm{\sigma }+2{\mathrm{\sigma }}^2=0 \\ \mathrm{\sigma }=\frac{18\pm \sqrt{324-4\mathrm{x}27\mathrm{x}2}}{2\left(2\right)} \\ \mathrm{\sigma }=\frac{18\pm \sqrt{108}}{4} \\ \mathrm{\sigma }=\frac{\cancel{2}(9\pm 3\sqrt{3})}{\cancel{2}\mathrm{x}2} \\ \frac{\mathrm{r}}{{\mathrm{a}}_{\mathrm{o}}}=\frac{(9\pm 3\sqrt{3})}{2} \\ \mathrm{r}=\frac{(9\pm 3\sqrt{3})}{2}{\mathrm{a}}_{\mathrm{o}} \mathrm{and} \\ {\mathrm{r}}_1=\frac{(9+3\sqrt{3})}{2}{\mathrm{a}}_{\mathrm{o}} \\ {\mathrm{r}}_2=\frac{(9-3\sqrt{3})}{2}{\mathrm{a}}_{\mathrm{o}} \\ {\mathrm{r}}_1-{\mathrm{r}}_2=\frac{(\cancel{9}+3\sqrt{3}}{2}-\frac{\cancel{9}+3\sqrt{3})}{2}{\mathrm{a}}_{\mathrm{o}} \\ \mathrm{d}={\mathrm{r}}_1-{\mathrm{r}}_2=\frac{\cancel{2}\mathrm{x}3\sqrt{3}}{\cancel{2}}{\mathrm{a}}_{\mathrm{o}} \\ \frac{\mathrm{d}}{1.73{\mathrm{a}}_0}=3 \end{gathered}$$