Shapes & Energies Of Orbitals

Visualize orbital shapes, calculate nodes, and determine relative energies using the n+l rule.

7 lessons

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Concept

Shapes of Orbitals and Nodes

Describe s, p, and d orbital boundary surfaces.

Shapes of Orbitals

How do we visualize where an electron lives? We use boundary surface diagrams. These 3D shapes enclose the region in space where the probability of finding an electron is very high (usually around 90%). Because electrons exhibit wave-particle duality, they don't move in simple circles—they exist in these "clouds."

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IMAGE

Shapes & Energies of Orbitals Visual

Must show boundary surface diagrams for s, p, and d orbitals, as text cannot convey 3D shapes adequately.

clean scientific diagram, pastel color palette, elegant typography, precise labeling, translucent layers for cross-secti…

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Calculating Nodes

Formula for total, radial, and angular nodes.

What is a Node?

A node or nodal surface is a region where the probability density function reduces to zero. Electrons are never found here.

Total Nodes

n - 1

Angular Nodes

l

Radial Nodes

n - l - 1

Example: 3p orbital

$n = 3, l = 1$
Radial Nodes: $3 - 1 - 1 = 1$
Angular Nodes: $1$
Total Nodes: $2$

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quiz

Practice: Counting Nodes

Calculate nodes for a specific orbital.

An engineer is mapping the probability density of an electron in a $4d$ orbital to determine regions of zero probability. How many radial and angular nodes are present in a $4d$ orbital?

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Content

Energies of Orbitals and Shielding

Explain effective nuclear charge and energy splitting.

Energies of Orbitals and Shielding

In a simple hydrogen atom (which only has one electron), the energy of an orbital is determined solely by its principal quantum number ( $n$ ).

This means that the $2s$ and $2p$ orbitals have the exact same energy! Orbitals with the same energy are called degenerate.

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The (n+l) Rule

Rule for determining relative orbital energies.

The $(n + l)$ Rule

The lower the value of $(n + l)$ , the lower the orbital's energy.
If two orbitals share the same $(n + l)$ value, the orbital with the lower $n$ has lower energy.
Electrons fill lower energy orbitals first!

Example: 4s vs 3d

4s orbital: $n = 4, l = 0 ⟹ n + l = 4$
3d orbital: $n = 3, l = 2 ⟹ n + l = 5$
Because $4 < 5$ , the 4s orbital has lower energy and fills before 3d.

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sequence

Practice: Ordering Energy Levels

Arrange orbitals by increasing energy.

Order the following orbitals in increasing order of energy using the (n+l) rule.

Drag to reorder

← previousQuantum Numbers & Orbitals next →Electronic Configuration Of Atoms

quiz

Practice: Counting Nodes

Calculate nodes for a specific orbital.

An engineer is mapping the probability density of an electron in a $4d$ orbital to determine regions of zero probability. How many radial and angular nodes are present in a $4d$ orbital?

Content

Energies of Orbitals and Shielding

Explain effective nuclear charge and energy splitting.

Energies of Orbitals and Shielding

In a simple hydrogen atom (which only has one electron), the energy of an orbital is determined solely by its principal quantum number ( $n$ ).

This means that the $2s$ and $2p$ orbitals have the exact same energy! Orbitals with the same energy are called degenerate.

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