Radial Nodes And Planar Nodes

Nodes

Nodes are specific regions in an atomic orbital where the probability of finding an electron is zero. These regions arise due to the wave nature of electrons and are represented by areas where the wave function (ψ\psiψ) changes sign and becomes zero. The presence of nodes affects the shape, size, and electron distribution of atomic orbitals.

The total number of nodes present in an orbital is given by:

Total Nodes $=n-1$

where n is the principal quantum number.

Nodes are classified into two types:

• Radial Nodes: Spherical regions where the probability of finding an electron is zero.
• Angular (Planar) Nodes: Planes or cones passing through the nucleus where the probability of finding an electron is zero.

Planar Nodes

Planar nodes (or angular nodes) are flat, two-dimensional regions where the probability density of finding an electron is zero. They are associated with the angular part of the wave function. The number of planar nodes is equal to the azimuthal quantum number $l$. A planar node is also called the angular node, is also called the nodal plane.

For example:

• An s orbital ( $l=0$ ) has 0 planar nodes.
• A p orbital ( $l=1$ ) has 1 planar node.
• A d orbital ( $l=2$ ) has 2 planar nodes

Also Read:

• Frequency, Time Period And Angular Frequency
• Electronic Configuration Of Elements

Radial Node

A radial node is a spherical surface where the probability of finding an electron is zero. The number of radial nodes increases with the principal quantum number (n).

The formula of Radial Node

Number of Radial nodes $=n-l-1=n-(l+1)$

where n is the principal quantum number, and l is the azimuthal quantum number.

(a) Calculating the number of radial nodes of the 1s orbital;

In 1s orbital, the value of principal quantum number (n)= 1 and the value of Azimuthal quantum number (l)= 0

Number of Radial nodes =$n-1-1=1-0-1=0$

(b) Calculating the number of radial nodes of the 2p orbital;

In 2p orbital, the value of principal quantum number (n)= 2 and the value of Azimuthal quantum number (l)= 1

Number of Radial nodes = $n-1-1=2-1-1=0$

(c) Calculating the number of radial nodes of a 3d orbital;

In 2p orbital, the value of principal quantum number (n)= 3 and the value of Azimuthal quantum number (l)= 2

Number of Radial nodes = $n-1-1=3-2-1=0$

Calculations of the Total Number of Nodes

The total number of nodes is defined as the sum of the number of radial nodes and angular nodes.

Total number of nodes = Number of radial nodes + Number of Angular nodes

$\begin{aligned} & =(n-l-1)+1 \\ & =(n-1)\end{aligned}$

Total number of nodes $=(n-1)$

Example:

Calculating the total number of nodes of the 2s orbital:

In the 2s orbital, the value of the principal quantum number (n)= 2

The value of the Azimuthal quantum number (l)= 0
Total number of nodes = n-1= 2-1= 1

Also Check

Some Solved Examples

Question 1: The number of radial nodes of the 3s and 2p orbitals is, respectively

1) 2, 0

2) 0, 2

3) 1, 2

4) 2, 11

Solution

As we learn

For a given orbital, the number of radial nodes =n−l−1

For 3s orbital

n=3, l=0

Number of radial nodes = 3-0-1= 2

For 2p orbital

n = 2, l = 1

Number of radial nodes = 2-1-1 =0

Hence, the answer is option (1).

Question 2: The number of planar nodes in d_x-y is

1) 0

2) 1

3) 2

4) 3

Solution

As we learn

No. of planar nodes = l where l is the azimuthal quantum number.

Number of planar nodes = l

For d_xy, l = 2

Number of planar nodes = 2

Hence, the correct answer is option (3).

Question 3: The number of radial nodes and 2p orbitals is respectively

1) 2,0

2) 0,2

3) 1,2

4) 2,1

Solution

We know,

The number of Radial nodes for any electron in any orbital is given by the value of (n−ℓ−1)
For 3s electrons,$\mathrm{n}=3, \ell=0 \Rightarrow(\mathrm{n}-\ell-1)=3-0-1=2$

And for 2p electron,

$\mathrm{n}=2, \ell=1 \Rightarrow(\mathrm{n}-\ell-1)=2-1-1=0$

Hence, the correct answer is option (1).

Question 4: A certain orbital has no angular nodes and two radial nodes. The orbital is :

1) 3s

2) 3p

3) 2s

4) 2p

Solution

The number of angular nodes is given by ‘l’, i.e., one angular node for p orbitals, two angular nodes for ‘d’ orbitals, and so on.

Radial modes = n-l-1

The total number of nodes is given by (n–1), i.e., the sum of l angular nodes and (n – l – 1) radial nodes.

Given

A certain orbital has no angular nodes and two radial nodes,

So, l=0,

It will be s orbital; it does not have angular nodes.

And

Radial nodes = 2

n – l –1 = 2

n – 0 – 1 = 2

n= 3

So, the orbital will be 3s.

Hence, the answer is option (1).

Question 5: A certain orbital has no angular nodes and two radial nodes. The orbital is :

1) 3s

2) 2p

3) 2s

4) 2p

Solution:

The number of angular nodes is given by ‘l’, i.e., one angular node for p orbitals, two angular nodes for ‘d’ orbitals, and so on.

Radial nodes = n-l-1

The total number of nodes is given by (n–1), i.e., the sum of l angular nodes and (n – l – 1) radial nodes.

Given

A certain orbital has no angular nodes and two radial nodes,

So, l=0,

It will be s orbital; it does not have angular nodes.

And

Radial nodes = 2

n – l –1 = 2

n – 0 – 1 = 2

n= 3

So, the orbital will be 3s.

Hence, the answer is option (1).

Question 6: For $2 P_x$ and $3 d_{x y}$ nodal plane are respectively [JEE Main 2024]

1) YZ plane & YZ, XY Planes

2) YZ plane & XZ, XY Planes

3) XZ plane & XZ, YZ Planes

4) YZ plane & XZ, YZ Planes

Solution:

In 2P_x orbital, the nodal plane is the YZ plane, and in 3d_xy orbital, the nodal planes are the XZ and YZ.

Hence, the answer is option (3).

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