JEE Exam » JEE Study Material » Chemistry » Radial and Angular Nodes Formula

A node refers to a point where the probability of an electron is calculated to be zero. In a given orbital there are two types of nodes. Radial node and angular node. Radial nodes also represent the nodal region. Radial nodes are usually a spherical surface in which the probability of finding an electron is always zero. The number of radial nodes increases along with the principal quantum number (n). Angular node is also referred to as the nodal plane. Angular node refers to a plane that passes through the nucleus. Angular nodes are usually  equal to the azimuthal quantum number (l). The number of angular nodes = l The number of radial nodes = (n – l – 1) Total number of nodes = n – 1.

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Radial nodes are generally spherical regions in which the probability of finding an electron is usually zero. This sphere possesses a fixed radius. Thus, radial nodes can be determined radially. Radial nodes arise when the principal quantum number increases. The principal quantum number refers to electron shells.

While calculating radial nodes, the radial probability density function must be used. The radial probability density function provides the probable density for an electron to be at a point situated at a distance r from the proton. The below equation can be used for this purpose:

Ψ(r,θ,Φ)   =  R Y(θ,Φ)

In this case Ψ represents the wave function, R refers to the radial component (based upon only the distance from the nucleus) and Y(θ,φ) represents the angular component. A radial node is formed when the R component becomes zero.

Radial nodes can be calculated via using the below-written formula:

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

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

What are Angular Nodes?

Angular nodes are usually flat planes (or cones) where the probability of finding an electron is mainly zero. This defines that we cannot ever find an electron in an angular (or any other) node. Whereas radial nodes are situated at fixed radii, angular nodes are placed at fixed angles. The number of angular nodes present in an atom can be calculated via the angular momentum quantum number. Angular nodes are formed as and when the angular momentum quantum number increases.

Formula of Angular Nodes

The value of the angular nodes is not based on the value of the principal quantum number. It is only based on the value of azimuthal quantum number.

Number of Angular nodes = l

Where l = Azimuthal quantum number

Total number of nodes = Number of radial nodes + Number of Angular nodes = (n-l-1) = (n-1)

Therefore, Total number of nodes =(n-1)

Radial Nodes and Angular Nodes: Comparison

Number of Electrons

Radial Nodes: These nodes usually possess spherical areas in which the probability of finding an electron is found to be zero.

Angular Nodes: Angular nodes are generally flat planes (or cones) where the probability of finding an electron is also zero.

Based on Shape

Angular Nodes: Angular nodes possess a planar or cone-like shape.

Based on Characteristic Properties

Angular Nodes: Angular nodes possess fixed angles.

Based on Number of Nodes

Radial Nodes: the number of radial nodes present in an atom can be calculated via the principal quantum number.

Angular Nodes: The number of angular nodes present in an atom is usually calculated via the angular momentum quantum number.

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Conclusion

Nodes are the areas in atoms in which none of the electrons can ever be found. There are generally two types of nodes like those of the radial nodes and angular nodes. The major difference between radial nodes and angular nodes is that radial nodes are spherical while angular nodes are generally flat planes. Here we come to an end of this topic, we hope that you were able to grasp a clear concept of the topic.

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