Question

The number of radial nodes and angular nodes for d orbital can be represented as.
A) Radial and Angular nodes are equal
B) Radial node= \[n - l - 1\] and Angular node= \[l\]
C) Angular node= \[n - l - 1\] and Radial node= \[l\]
D) Radial node= \[n\] and Angular node= \[l\]

Answer 1

Hint: We have to know that the angular node represents orbital angular momentum quantum number i.e. \[l\]. There are 4 different quantum numbers \[n,l,{\text{ }}s,{m_s}\]. Nodes usually represent the plane or a point having zero electron density in an orbital and also they are surrounded by lobes.

Complete step by step answer:
We are considering 3d orbital as an example to learn about radial and angular nodes.
The following quantum number represents as follows:
- Principal Quantum number (\[n\]): It describes the most accurate distance between electron and nucleus.
\[n\] can be any integer \[1,2,3,4 \ldots \]
- Azimuthal Quantum number (\[l\]) : It describes the shape of an orbital.
\[l\] = \[n\] \[ - 1\]
\[l\] can be \[0,1,2,3 \ldots \]
\[l\] \[ = 0\] (s orbital), \[l\] \[ = 1\] ( p orbital ) , \[l\] \[ = {\text{ }}2\] (d orbital) , \[l\] \[ = 3{\text{ }}\]( f orbital)……
So for 3d we can say that
\[n\] \[ = 3{\text{ }}\]and \[l\] \[ = {\text{ }}2\] (i.e. \[n\] \[ - 1\])

Now we will discuss about radial and angular nodes in d orbital
Radial node is the surface of a sphere where there is an impossibility of finding an electron.
It can be calculated as using the formula i.e.
Radial node = difference between total no. of nodes and angular nodes.
Angular node is nothing but the azimuthal quantum number i.e. \[l\] which can also be referred to as orbital angular momentum quantum no.
Angular node = \[l\] (Azimuthal quantum number)
Considering 3d orbital, we can calculate its nodes
Angular node (for 3d orbital ) \[ = {\text{ }}2\]
Radial node= \[n - l - 1\]
For 3d orbital= \[\left( {3 - 2} \right) - 1 = 0\]

Option A) is incorrect since both these nodes cannot be equal.
Option B) is correct since angular node represents l i.e. azimuthal quantum number and radial node represents the difference between total no. of nodes and angular nodes.
Option C) is incorrect since its just reverse of the correct option.
Option D) is incorrect as we have explained above that radial nodes = \[n - l - 1\] which cannot be \[l\].

Hence, the correct option is B.

Note: We also remember the following formulae
Radial node = difference between total no. of nodes and angular nodes.
Angular node = \[l\] (Azimuthal quantum number)