Which of the following statements concerning the quantum numbers are correct? (Note: more than one of the give

NCERT Exemplar · Q21

Q. Which of the following statements concerning the quantum numbers are correct? (Note: more than one of the given options may be correct.)

(i) Angular quantum number determines the three dimensional shape of the orbital.

(ii) The principal quantum number determines the orientation and energy of the orbital.

(iii) Magnetic quantum number determines the size of the orbital.

(iv) Spin quantum number of an electron determines the orientation of the spin of electron relative to the chosen axis.

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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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Quantum numbers each have a distinct physical meaning: n (principal) → size/energy, l (azimuthal) → shape, mₗ (magnetic) → orientation in space, mₛ (spin) → spin orientation. Only statements (A) and (D) are correct.

The four quantum numbers — principal ( $n$ ), azimuthal or angular ( $l$ ), magnetic ( $m_l$ ), and spin ( $m_s$ ) — form a complete address for an electron in an atom. Each one encodes a specific physical property of the orbital or the electron. The trick is to map each quantum number to exactly one property: size, shape, orientation, or spin direction. Confusing these is the most common error.

Let’s examine each statement one by one.

Statement (A): "Angular quantum number determines the three dimensional shape of the orbital."

The angular quantum number $l$ (also called the azimuthal quantum number) defines the subshell — $s$ ( $l=0$ ), $p$ ( $l=1$ ), $d$ ( $l=2$ ), $f$ ( $l=3$ ). Each value of $l$ gives a characteristic shape: spherical for $s$ , dumbbell for $p$ , cloverleaf for $d$ , and more complex for $f$ . So yes, $l$ determines the shape. This statement is correct.
Statement (B): "The principal quantum number determines the orientation and energy of the orbital."

The principal quantum number $n$ determines the size (average distance from the nucleus) and the energy of the orbital (for hydrogen-like atoms, $E_n \propto -1/n^2$ ). Orientation, however, is governed by the magnetic quantum number $m_l$ . So $n$ does not determine orientation. This statement is incorrect.
Statement (C): "Magnetic quantum number determines the size of the orbital." …

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Q.Which of the following statements concerning the quantum numbers are correct? (Note: more than one of the given options may be correct.)

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