What are nodes? Explain their types, numbers and Significance.
Node is a region where the probability of finding an electron is zero. In other words, it's a point, line, or surface within an orbital where an electron cannot exist.
Types of Nodes
There are two types of node. One is radial and other is angular. These two nodes are discussed below in details.
Radial Nodes
Radial nodes are the spherical surface region where the probability of finding an electron is zero. This sphere has a fixed radius. Therefore, radial nodes are determined radially. Radial nodes increases as the principal quantum number increases.
When finding radial nodes, the radial probability density function can be used. The radial probability density function gives the probability density for an electron to be at a point located the distance 'r' from the proton. The following equation is used for this purpose.
Ψ(r,θ,Φ) = R(r) Y(θ,Φ)
Where Ψ is the wave function, R(r) is the radial component (depends upon only the distance from the nucleus) and Y(θ,φ) is the angular component. A radial node occurs only when R(r) component becomes zero.
Number of Radial nodes = n – l – 1 = n – (l + 1)
Where n = principal quantum number, l = Azimuthal quantum number
Examples
Number of radial nodes of 1s orbital-
In 1s orbital-
Principal quantum number (n) = 1 and
Azimuthal quantum number (l) = 0
Number of Radial nodes = n – l – 1 = 1 – 0 – 1 = 0
Number of radial nodes of 2s orbital-
In 2s orbital-
Principal quantum number (n) = 2 and
Azimuthal quantum number (l) = 0
Number of Radial nodes = n – 0 – 1 = 2 – 0 – 1= 1
Number of radial nodes of 3s orbital-
In 3s orbital-
Principal quantum number (n) = 3 and
Azimuthal quantum number (l) = 0
Number of Radial nodes = n – 0 – 1 = 3 – 0 – 1= 2
Indian Institutes of Science Education and Research (IISER): 2023
How many radial nodes does Ca+ have in its 4s orbital?
A. 3
B. 0
C. 1
D. 2
Solution:
Hints: Ca+ and Ca both have same number of principal quantum number and azimutha quantum number.
Principal quantum number (n) = 4 and
Azimuthal quantum number (l) = 0
Therefore, the Number of Radial nodes = n – 0 – 1 = 4 – 0 – 1= 3
EAMCET: 2011-E
The number of radial nodes present in the radial probability distribution curves for the orbital wave function with quantum number n = 4, l = 0 and m = 0 is:
A. 4
B. 3
C. 2
D. 1
Solution:
Hints: Given: n = 4, l = 0 and m = 0
So, the Number of Radial nodes = n – 0 – 1 = 4 – 0 – 1= 3
Angular Nodes or Nodal Planes
The planes or planar areas around the nucleus where the probability of finding an electron is zero are called angular nodes. This means we cannot ever find an electron in an angular (or any other) node. While radial nodes are located at fixed radii, angular nodes are located at fixed angles. The number of angular node present in an atom is determined by the angular momentum (Azimuthal) quantum number only. Angular nodes increases as the angular momentum quantum number increases.
Number of Angular nodes = l
Where l = Azimuthal quantum number
Examples
Angular nodes or nodal planes of 1s orbital-
In 1s orbital-
Azimuthal quantum number (l)= 0
Number of Angular nodes = l = 0
Angular nodes or nodal planes of 2s orbital-
In 2s orbital-
Azimuthal quantum number (l)= 0
Number of Angular nodes = l = 0
Angular nodes or nodal planes of 2p orbital-
In 2p orbital-
Azimuthal quantum number (l)= 1
Number of Angular nodes = l = 1
Angular nodes or nodal planes of 3d orbital-
In 3d orbital-
Azimuthal quantum number (l)= 2
Number of Angular nodes = l = 2
Total Number of Nodes
The total number of nodes is the sum of the number of radial nodes and angular nodes.
Total number of nodes = Number of radial nodes + Number of Angular nodes
Total number of nodes = (n – 1) + l = (n – 1)
Total number of nodes = (n – 1)
Note: If the node at r = ∞ is also considered then the number of nodes will be 'n' (not n – 1)
Total number of nodes of 2s orbital-
In 2s orbital-
Principal quantum number (n) = 2 and
Azimuthal quantum number (l) = 0
Total number of nodes = n – 1 = 2 – 1 = 1
Significance of Nodes
The numbers and types of nodes in orbitals helps to explain the structure of atoms and molecules, as well as the distribution of electrons in space.
The presence of nodes affects the spatial distribution of electrons, influencing chemical bonding and molecular properties.
Nodes are directly related to the shape and energy of atomic orbitals. Orbitals with more nodes generally have higher energy.
Number of Nodes Calculator
Enter the principal quantum number (n) and the azimuthal quantum number (l) to calculate the number of radial, angular, and total nodes.
Number of Radial Nodes:0
Number of Angular Nodes:0
Total Number of Nodes:0