Shapes of Atomic Orbitals:Boundary Surface Diagrams & Nodes

Shapes of Orbitals

Probability Density

ψ gives us the amplitude of wave. The value of ψ has no physical significance.
|Ψ|2 gives us the region in which the probability of finding an electron is maximum. It is called probability density.

Nodal surfaces

The region where this probability density function reduces to zero is called nodal surfaces or simply nodes.
There are two types of Nodes

Radial nodes occur when the probability density of wave function for the electron is zero on a spherical surface of a radius. Number of radial nodes = n – l – 1

Angular nodes or Nodal Planes

Angular nodes occur when the probability density wave function for the electron is zero along the directions specified by a particular angle. Number of angular nodes = l
Therefore ,
Number of angular nodes =l
Total number of nodes =Number of radial nodes + Number of angular nodes= n – 1
Example
2s orbital has n=2 and l=0. The number of angular nodes = l = 0. The number of radial nodes = [(n-1) - l] = [1 - 0] = 1. Total Number of Nodes=1
2p orbital has n=2 and l=1. The number of angular nodes = l = 1. The number of radial nodes = [(n-1) - l] = [1 - 1] = 0. Total Number of Nodes=1

Boundary Surface Diagrams

It is surface in the space where probability density is constant for a given orbital. This gives  a good representation of the shape of the orbital. This shapes encloses the volume or region where probability of finding electron is high

Shape of S-orbital
• All the s -orbital are Spherical shape
• The probability of finding the electron at a given distance is equal in all the directions.
• The size of the s orbital increases with increase in n, that is, 4s > 3s > 2s > 1s and the electron is located further away from the nucleus as the principal quantum number increases.

Shape of P orbitals
• It has 3 possible orientation
• Each p orbital consists of two sections called lobes that are on either side of the plane that passes through the nucleus
• The probability density function is zero on the plane where the two lobes touch each other.
• The size, shape and energy of the three orbitals are same just the orientation is different
• They are given the designations $2p_x$, $2p_y$, and $2p_z$

Shape of D Orbitals
• It has five orientations
• The shapes of the first four D orbitals are similar to each other, where as that of the fifth one is different from others, but all five 3d orbitals are equivalent in energy
• The five d-orbitals are designated as $d_{xy}$, $d_{yz}$,$d_{xz}$,$d_{x^2 - y^2}$ and $d_{z^2}$.

Question 1
Which orbital is dumb-bell shaped
(a) s-orbital
(b) p-orbital
(c) d-orbital
(d) f-orbital
Solution
(b)

Question 2
For the dumb-bell shaped orbital, the value of l is
(a) 3
(b) 0
(c) 1
(d) 2
Solution
(c)

Question 3
Which of the sub-shell is circular
(a) 4s
(b) 4f
(c) 4p
(d) 4d
Solution
(a)

Question 4
The shape of $d_{xy}$ orbital will be
(a) Circular
(b) Dumb-bell
(c) Double dumb-bell
(d) Trigonal
Solution
(c)

Question 5
Number of nodal centres for 2s orbital
(a) 1
(b) 0
(c) 4
(d) 3
Solution
(a)