# Radial and Angular Distribution Curves or Functions

## Radial and Angular Distribution Curves or Functions

In the atomic orbital, there is probability of finding an electron in a particular volume element at a given distance and direction from the nucleus. This gives two types of probability of finding electrons, the radial probability distribution function i.e. probability of finding an electron at a given radial distance from the nucleus without considering the direction from the nucleus (i.e. how far away from the nucleus the orbital extends and the number of nodes the orbital has).The radial distribution functions depend on both n and l. This means that the number of nodes an orbital has and how far that orbital extends from the nucleus depends on the principle quantum number or energy level of the orbital (the 1 in 1s, the 2 in 2s, the 3 in 3s, etc.) and the type of orbital (s vs. p vs. d).

The number of Radial nodes can be clculated by the given formula-

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

Where n = principal quantum number, l = Azimuthal quantum number

The other is angular distribution curve i.e. the probability of finding an electron in any given direction from the nucleus without considering its distance from the nucleus. The Angular distribution function or curve describes the basic shape of the orbital or the number of lobes in an orbital. The angular distribution functions depend only on the quantum numbers l and m. That is the angular distribution functions of all electrons with the same l and m values are the same. Simply, we can say that all s orbitals have the same basic shape. For example The 2s orbital (n = 2, l = 0, m = 0), the 3s (n = 3, l = 0, m = 0) and the 4s (n = , l = 0, m = 0) have the same basic shape spherical.

The planes or planar areas around the nucleus where the probability of finding an electron is zero are called angular nodes. The value of the angular nodes does not depend upon the value of the principal quantum number. It only depends on the value of the azimuthal quantum number.

Example-

In 3d orbital, the value of Azimuthal quantum number (l)= 2

So, the number of Angular nodes = l = 2

Atom is spherical. If we consider it as composed of layers much like onion. The layer extends from r to r + dr and then, the volume of the thin shell as dV.

The volume of the sphere is given by-

V = 4/3 πr

^{3}

dV/dr = 4/3 π(3r

^{2})

or, dV = 4πr

^{2}dr

or, R

^{2}dV = 4πr

^{2}R

^{2}dr

When the radial portion (R) of the wave function is squared an multiplied by 4πr

^{2}, we get the probability 4πr

^{2}R

^{2}(r) verses the distance from the nucleus(r).

If, r = 0,

Then, 4πr

^{2}R

^{2}= 0

Hence, the volume at the nucleus is approach zero. In between them, both r and R have finite values, so there is a maximumin the curve at r = a

_{o}(Bohr radius).

The above plot shows that the maximum probability occurs at distance of 0.52 Å from the nucleus. This is equal to the Bohr radius. It indicates that the maximum probability of finding an electron around the nucleus is at this distance. However, there is a probability to find the electron at other distances also.

The radial distribution function of 2s, 2p, 3s, 3p and 3d orbitals of the hydrogen atom are represented as follows-

## How many radial nodes do 3p, 3d, and 4f orbitals have?

We know that the number of radial nodes is given by the expression-

n − l − 1 and use it to find the values of n and l.

The 3p orbitals have n = 3 and l = 1 and the number of
radial nodes will be n − l − 1 = 1.

The 3d orbitals have n = 3 and l = 2. Therefore, the number of radial
nodes will n − l − 1 = 0.

The 4f orbitals have n = 4 and l = 3 and the number of radial nodes will be
n − l − 1 = 0.

The 3d and 4f orbitals are the first occurrence of the d and f orbitals so this also indicates that they will have no radial nodes.