Question

The Schrodinger wave equation for hydrogen atom is:
\[{{\Psi }_{2s}}=\dfrac{1}{4\sqrt{2}\pi }{{\left( \dfrac{1}{{{a}_{0}}} \right)}^{3/2}}\left[ 2-\dfrac{{{r}_{0}}}{{{a}_{0}}} \right]{{e}^{-r/{{a}_{0}}}}\]
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.$\dfrac{{{a}_{0}}}{2}$
B.$2{{a}_{0}}$
C.$\sqrt{2}{{a}_{0}}$
D.$\dfrac{{{a}_{0}}}{\sqrt{2}}$

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:
 \[{{\Psi }_{2s}}=\dfrac{1}{4\sqrt{2}\pi }{{\left( \dfrac{1}{{{a}_{0}}} \right)}^{3/2}}\left[ 2-\dfrac{{{r}_{0}}}{{{a}_{0}}} \right]{{e}^{-r/{{a}_{0}}}}\]
where, ${{a}_{0}}$ is Bohr radius, ${{r}_{0}}$ is the radial node and $\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
 \[|{{\Psi }_{2s}}{{|}^{2}}=0\]
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 $\left( 2-\dfrac{{{a}_{0}}}{{{r}_{0}}} \right)$ has to be equal to zero.
Since, $\dfrac{1}{4\sqrt{2}\pi }$ is a constant which cannot be equal to zero and the value of ${{\left( \dfrac{1}{{{a}_{0}}} \right)}^{3/2}}$ and ${{e}^{-r/{{a}_{0}}}}$ will always be greater than zero.
So, therefore, we can write
 $2-\dfrac{{{r}_{0}}}{{{a}_{0}}}=0$
On further simplifying, we get,
 $\Rightarrow {{r}_{0}}=2{{a}_{0}}$

Therefore, the correct option is (B) $2{{a}_{0}}$.

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, $\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.