An electron is in one of the 3d orbitals. Give the possible values of n, l and ml for this electron.

Exercises · 2.25

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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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For an electron in a 3d orbital, the principal quantum number is fixed at $n=3$ , the azimuthal quantum number is $l=2$ , and the magnetic quantum number $m_l$ can take any integer value from $-2$ to $+2$ : $-2, -1, 0, +1, +2$ .

The question asks for the possible quantum numbers of an electron in a 3d orbital. This is a direct application of how atomic orbital labels translate into quantum numbers — a fundamental skill in physical chemistry and atomic physics.

Why this works: Energy Level Quantization

In quantum mechanics, an electron in an atom is described by a set of quantum numbers that arise from solving the Schrödinger equation. These numbers are not arbitrary — they come from boundary conditions that restrict the electron's wavefunction to be physically meaningful.

The principal quantum number $n$ tells you the energy level (shell). The azimuthal quantum number $l$ tells you the shape of the orbital (subshell). The magnetic quantum number $m_l$ tells you the orientation of that orbital in space.

The notation "3d" is a shorthand: the number gives $n$ , and the letter gives $l$ (s=0, p=1, d=2, f=3). So when you see "3d", you already know two of the three numbers. The third, $m_l$ , follows from the allowed range for that $l$ .

Watch out

A common mistake is to think $m_l$ can be any integer from $-l$ to $+l$ including zero — that's correct — but students sometimes forget that zero is included, or they list only positive values. All $2l+1$ values are equally possible.

Step-by-step solution

1. Identify the principal quantum number $n$ from the orbital label.

The number in "3d" is the principal quantum number. So $n = 3$ . This tells us the electron is in the third energy level (M shell).

2. Identify the azimuthal quantum number $l$ from the orbital letter.

The letter "d" corresponds to $l = 2$ . (Recall: s → $l=0$ , p → $l=1$ , d → $l=2$ , f → $l=3$ .) This means the electron is in a d subshell, which has a more complex shape than s or p orbitals.

Tip

The value of $l$ also tells you the number of angular nodes in the orbital: for a d orbital ( $l=2$ ), there are 2 angular nodes (planes where the probability of finding the electron is zero).

3. Determine the possible values of the magnetic quantum number $m_l$ . …

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Q.An electron is in one of the 3d orbitals. Give the possible values of n, l and mlm_lml​ for this electron.

Why this works: Energy Level Quantization

Step-by-step solution

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Q.An electron is in one of the 3d orbitals. Give the possible values of n, l and $m_l$ for this electron.

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