Question

The wave function $(\Psi)$ of $2 s$ -orbital is given by
$\Psi_{2 s}=\frac{1}{2 \sqrt{32 \pi}} \cdot\left[\frac{1}{a_{0}}\right]^{3 / 2}\left[2-\frac{r}{a_{0}}\right] e^{-r / 2 a_{0}}$
At $r=r_{0},$ radial node is formed. The ratio of $r_{0}: a_{0}$ is _______.

Answer 1

$\Psi_{2 s }=\frac{1}{2 \sqrt{32 \pi}} \cdot\left[\frac{1}{a_{0}}\right]^{-3 / 2} \cdot\left[2-\frac{r}{a_{0}}\right] \cdot e^{-r / 2 a_{0}}$

For radial node at $r=r_{0}, \psi_{2 s}^{2}=0 .$

This is possible only when

$\left[2-\frac{r_{0}}{a_{0}}\right]=0$ or $2=\frac{r_{0}}{a_{0}}$