Question

The number of radial nodes present in the radial probability distribution curves for the orbital wave function with quantum numbers it \[n = 4,l= 0\] and \[m = 0\] is :
A. 4
B. 3
C. 2
D. 1

Answer 1

Hint: A point or plane of an orbital where the electron density of the electron of an orbital is zero is called a node. Nodes can be two types, one is a radial node and another one is an angular node. To find out the node, the concept of quantum number should be known.
Formula used: \[(n - l - 1)\]

Complete step by step answer:
Quantum numbers are characteristic quantities that are used to describe the various properties of an electron in an atom-like position, energy, or spin of the electrons. There are four quantum numbers namely, principal quantum number, azimuthal quantum number, magnetic quantum number, and spin quantum number.
Now for an orbital, the total number of nodes is \[(n - 1)\] . Where n is the principal quantum number. The principal quantum numbers of the shells are, for K shell is 1, for L is 2, for M shell is 3, for N shell is 4, etc. it cannot be fraction number.
From the given values of the quantum numbers, \[n = 4,l = 0\] and \[m = 0\] , the orbital is 4s orbital
Now the number of angular nodes is equal to the azimuthal quantum number denoted by ‘l’.
Therefore, the total number of radial nodes is, \[(n - l - 1)\] .
Now, for 3p orbital, the value of n is 3, and the azimuthal quantum number is 1. Therefore, the number of radial node is,
 \[\begin{gathered}
  (n - l - 1) \\
   = (4 - 0 - 1) \\
\end{gathered} \]
=3
So, the number of radial nodes in the 3p orbital is 3.

So the correct answer is, B.

Note:
The possible values of four quantum numbers are,
Principle quantum number (n) where the value is, \[n = 1,2,3,4...\] any integer.
Azimuthal quantum number (l) the value is, \[l = 0 to (n - 1)\]
Example: for the value of \[n = 3\] the value of \[l = 0,1,2\] , Where, \[l = 0\] for s orbital, \[l = 1\] for p orbital \[l = 2\] for d orbital, \[l = 3\] for f orbital.
Magnetic quantum number (m): the value is \[m = - l{\text{t}}o + l\]
Example: for the value \[l = 3\]
 \[m = - 3, - 2, - 1,0, + 1, + 2, + 3{\text{ }}\]
Spin quantum number(s): the value is \[ \pm \dfrac{1}{2}\] . for every value of m.
Example: for \[m = - 3, - 2, - 1,0, + 1, + 2, + 3{\text{ }}\]
 \[s = \pm \dfrac{1}{2}, \pm \dfrac{1}{2}, \pm \dfrac{1}{2}, \pm \dfrac{1}{2}, \pm \dfrac{1}{2}, \pm \dfrac{1}{2}, \pm \dfrac{1}{2}\]
Where the + sign means clockwise spin rotation of electron and the – sign means anti-clockwise spin rotation of electrons.