What is the lowest value of n that allows g orbitals to exist?

Exercises · 2.24

Q. What is the lowest value of n that allows g orbitals to exist?

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Concept understanding — Energy Level Quantization

Energy Level Quantization: From Intuition to Precision

Imagine you're climbing a smooth ramp. You can stop at any height — 1 metre, 1.5 metres, 2.1 metres — anywhere you like. That's how we intuitively think about energy in everyday life: continuous, like a slide.

Now imagine a staircase. You can stand on step 1, step 2, or step 3 — but you cannot stand halfway between step 2 and step 3. There's no such place. The steps are discrete, not continuous.

Energy level quantization is the idea that in the microscopic world of atoms and molecules, energy behaves like a staircase, not a ramp. Electrons in an atom cannot have just any energy — they can only occupy specific, allowed energy levels. Everything else is forbidden.

Why does this happen? The core intuition

In the classical world, an electron orbiting a nucleus would continuously radiate energy, spiral inward, and crash — atoms would be unstable. But atoms are stable. Nature solved this problem by imposing a rule: the electron's angular momentum (and therefore its energy) can only take certain discrete values.

Think of a guitar string. It can only vibrate at specific frequencies — its fundamental and harmonics. You can't pluck it to produce a frequency halfway between two harmonics. The string's vibration is quantized by its boundaries. Similarly, an electron bound to a nucleus is confined in space, and that confinement forces its energy to be quantized.

Note

Quantization is not a mysterious extra rule — it emerges naturally whenever a wave (like an electron's matter wave) is confined. Confinement creates standing waves, and standing waves only exist at specific frequencies.

The precise statement

For a bound system (like an electron in an atom), the total energy $E$ of the system can only take certain discrete values:

E = E_1, E_2, E_3, \dots

where each $E_n$ is a specific, fixed number. The integer $n$ (1, 2, 3, …) is called the principal quantum number. The lowest energy level ( $n=1$ ) is the ground state; higher levels ( $n>1$ ) are excited states.

For the hydrogen atom, the allowed energies are given by:

E_n = -\frac{13.6 \text{ eV}}{n^2}

So:

$n=1$ : $E_1 = -13.6$ eV (ground state)
$n=2$ : $E_2 = -3.4$ eV
$n=3$ : $E_3 = -1.51$ eV
and so on, approaching 0 eV as $n \to \infty$ (the ionization limit)

Important

The negative sign means the electron is bound to the nucleus. Zero energy corresponds to the electron being free (ionized). The more negative the energy, the more tightly bound the electron.

How do we know this is real?

The most direct evidence comes from atomic spectra. When an electron jumps from a higher energy level to a lower one, it emits a photon of light with energy exactly equal to the difference:

\Delta E = E_{\text{higher}} - E_{\text{lower}} = hf

where $h$ is Planck's constant and $f$ is the frequency of the emitted light. …

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The g orbital appears when the azimuthal quantum number $\ell = 4$ , which first becomes possible at principal quantum number $n = 5$ .

The structure of an atom's electron cloud is governed by quantum mechanics, where electrons occupy orbitals defined by a set of quantum numbers. Understanding when a particular type of orbital becomes available requires knowing the relationship between these quantum numbers.

The Quantum Number Hierarchy

Every electron in an atom is described by four quantum numbers, but two are crucial for determining orbital types:

The principal quantum number $n$ ( $n = 1, 2, 3, \ldots$ ) defines the energy level or shell
The azimuthal (angular momentum) quantum number $\ell$ defines the orbital shape and subshell

The key constraint is that $\ell$ can only take integer values from $0$ to $(n-1)$ . This means the principal quantum number sets an upper limit on which orbital types can exist at that level.

Each value of $\ell$ corresponds to a familiar orbital designation:

$\ell$	Orbital type
0	s
1	p
2	d
3	f
4	g
5	h

The pattern follows alphabetical order starting from s (skipping j historically).

Finding the Minimum $n$ for g Orbitals

Identify the azimuthal quantum number for g orbitals

From the table above, g orbitals correspond to $\ell = 4$ .
Apply the quantum number constraint

Since $\ell$ must satisfy $\ell \leq n - 1$ , we need:

\ell = 4 \leq n - 1

Solve for the minimum $n$ Rearranging: $n \geq \ell + 1 = 4 + 1 = 5$ …

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Q.What is the lowest value of n that allows g orbitals to exist?

The Quantum Number Hierarchy

Finding the Minimum nnn for g Orbitals

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Finding the Minimum $n$ for g Orbitals

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