Question

The Schrodinger wave equation for hydrogen atom is:
   $${\mathrm{\Psi }}_{2s}={\displaystyle \frac{1}{4\sqrt{2}}}{\left({\displaystyle \frac{1}{a_0}}\right)}^{{\displaystyle \frac{3}{2}}}[2-{\displaystyle \frac{r_0}{a_0}}]e{-}^{{\displaystyle \frac{r}{a_0}}}$$
Where $a_0$ is Bohr radius. If the radial node in 2s be at $r_0$ , then find r in terms of $a_0$
A.${\displaystyle \frac{a_0}{2}}$
B.$2a_0$
C.$\sqrt{2a_0}$
D.${\displaystyle \frac{a_0}{\sqrt{2}}}$

Answer 1

Hint: The radial node occurs where the radial component $R_{nl}(r)$ of the wave function goes to zero, therefore ${{\mathrm{\Psi }}_{2s}}^2=0$ and At node, the radial node is at $$r_0$$ , So $$[2-{\displaystyle \frac{r_0}{a_0}}]$$ = 0, then we can calculate r in terms of $a_0$

Complete step by step answer:
Given in the question,
The Schrodinger wave equation for hydrogen atom is
 $${\mathrm{\Psi }}_{2s}={\displaystyle \frac{1}{4\sqrt{2}}}{\left({\displaystyle \frac{1}{a_0}}\right)}^{{\displaystyle \frac{3}{2}}}[2-{\displaystyle \frac{r_0}{a_0}}]e{-}^{{\displaystyle \frac{r}{a_0}}}$$
The radial node occurs where the radial component $R_{nl}(r)$ of the wave function goes to zero. ${{\mathrm{\Psi }}_{2s}}^2=0$ since there is no angular component $${Y_I}^{ml}(\theta ,\mathrm{\varnothing })$$ to a wave function for a spherical orbital $$(l=0,ml=0)$$

At node, the radial node is at $$r_0$$

0 = $$[2-{\displaystyle \frac{r_0}{a_0}}]e{-}^{{\displaystyle \frac{r}{a_0}}}$$
Since $$e{-}^{{\displaystyle \frac{r}{a_0}}}\ne 0$$ for r in between 0 and $$\mathrm{\infty }$$ (where nodes can occur), that can be divided out as well.
 $\therefore 2-{\displaystyle \frac{r_0}{a_0}}=0$
 ${\displaystyle \frac{r_0}{a_0}}=2$
 $r_0=2a_0$
Therefore, the correct answer is option (B).

Note: The wave function $$(\mathrm{\Psi })$$ , is a mathematical function which is used to describe a quantum object. The wave function that describes an electron in an atom is a product between the radial wave function and the angular wave function. The radial wave function depends only on the distance from the nucleus and is represented by r.
A node occurs when a wave function changes signs, i.e. when its passes through zero. And a radial node occurs when a radial wave function passes through zero. An electron has the zero probability of being located at a node.