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Question: The Schrodinger wave equation for hydrogen atom is: \[{\Psi _{2s}} = \dfrac{1}{{4\sqrt 2 }}{\le...

The Schrodinger wave equation for hydrogen atom is:
Ψ2s=142(1a0)32[2−r0a0]e−ra0{\Psi _{2s}} = \dfrac{1}{{4\sqrt 2 }}{\left( {\dfrac{1}{{{a_0}}}} \right)^{\dfrac{3}{2}}}[2 - \dfrac{{{r_0}}}{{{a_0}}}]e{ - ^{\dfrac{r}{{{a_0}}}}}
Where a0{a_0} is Bohr radius. If the radial node in 2s be at r0{r_0} , then find r in terms of a0{a_0}
A.a02\dfrac{{{a_0}}}{2}
B.2a02{a_0}
C.2a0\sqrt {2{a_0}}
D.a02\dfrac{{{a_0}}}{{\sqrt 2 }}

Explanation

Solution

The radial node occurs where the radial component Rnl(r){R_{nl}}(r) of the wave function goes to zero, therefore Ψ2s2=0{\Psi _{2s}}^2 = 0 and At node, the radial node is at r0{r_0} , So [2−r0a0][2 - \dfrac{{{r_0}}}{{{a_0}}}] = 0, then we can calculate r in terms of a0{a_0}

Complete step by step answer:
Given in the question,
The Schrodinger wave equation for hydrogen atom is
Ψ2s=142(1a0)32[2−r0a0]e−ra0{\Psi _{2s}} = \dfrac{1}{{4\sqrt 2 }}{\left( {\dfrac{1}{{{a_0}}}} \right)^{\dfrac{3}{2}}}[2 - \dfrac{{{r_0}}}{{{a_0}}}]e{ - ^{\dfrac{r}{{{a_0}}}}}
The radial node occurs where the radial component Rnl(r){R_{nl}}(r) of the wave function goes to zero. Ψ2s2=0{\Psi _{2s}}^2 = 0 since there is no angular component YIml(θ,∅){Y_I}^{ml}(\theta ,\emptyset ) to a wave function for a spherical orbital (l=0,ml=0)(l = 0,ml = 0)

At node, the radial node is at r0{r_0}

0 = [2−r0a0]e−ra0[2 - \dfrac{{{r_0}}}{{{a_0}}}]e{ - ^{\dfrac{r}{{{a_0}}}}}
Since e−ra0≠0e{ - ^{\dfrac{r}{{{a_0}}}}} \ne 0 for r in between 0 and ∞\infty (where nodes can occur), that can be divided out as well.
∴2−r0a0=0\therefore 2 - \dfrac{{{r_0}}}{{{a_0}}} = 0
r0a0=2\dfrac{{{r_0}}}{{{a_0}}} = 2
r0=2a0{r_0} = 2{a_0}
Therefore, the correct answer is option (B).

Note: The wave function (Ψ)(\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.