Question

The Schrodinger wave equation for hydrogen atom is:
$${\mathrm{\Psi }}_{2s}={\displaystyle \frac{1}{4\sqrt{2}\pi }}{\left({\displaystyle \frac{1}{a_0}}\right)}^{3/2}[2-{\displaystyle \frac{r_0}{a_0}}]e^{-r/a_0}$$
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.$\sqrt{2}{a}_0$
D.${\displaystyle \frac{a_0}{\sqrt{2}}}$

Answer 1

Hint:
Bohr radius is the distance between the nucleus and electron of an atom. The probability of an electron located at a particular point is given by the square value of the wave function. In this equation, $r_0$ is the radial node.

Complete step by step answer:
Here, it is given that the Schrodinger wave equation for hydrogen atom is:
 $${\mathrm{\Psi }}_{2s}={\displaystyle \frac{1}{4\sqrt{2}\pi }}{\left({\displaystyle \frac{1}{a_0}}\right)}^{3/2}[2-{\displaystyle \frac{r_0}{a_0}}]e^{-r/a_0}$$
where, $a_0$ is Bohr radius, $r_0$ is the radial node and $\mathrm{\Psi }$ is the wave function.
When wave function passes through zero, a node occurs. The electron has zero probability of being located at a node. The probability of an electron located at a particular point is given by the square value of the wave function. As we discussed that electron has zero probability of being located at a node, we can say that
 $$|{\mathrm{\Psi }}_{2s}{|}^2=0$$
Now, looking at the above equation, we can observe that, if the square of the value of wave function is equal to zero, then the value of $(2-{\displaystyle \frac{a_0}{r_0}})$ has to be equal to zero.
Since, ${\displaystyle \frac{1}{4\sqrt{2}\pi }}$ is a constant which cannot be equal to zero and the value of ${\left({\displaystyle \frac{1}{a_0}}\right)}^{3/2}$ and $e^{-r/a_0}$ will always be greater than zero.
So, therefore, we can write
 $2-{\displaystyle \frac{r_0}{a_0}}=0$
On further simplifying, we get,
 $\Rightarrow r_0=2a_0$

Therefore, the correct option is (B) $2a_0$.

Additional information:
-Schrodinger wave equation is an equation that is used to calculate the wave function of a quantum – mechanical system. The wave function is used to define the state of the system at each spatial position and time.
-Wave function is defined as the quantum state of an isolated quantum system. It is denoted with a symbol, $\mathrm{\Psi }$

Note: A wave function node generally occurs at a point where wave function is zero, that means, the electron has zero probability of being located at a node.
-Bohr radius is the most probable distance between the electron and the nucleus.