(a) How many subshells are associated with n = 4? (b) How many electrons will be present in the subshells havi

Exercises · 2.67

Q. (a) How many subshells are associated with n = 4?

(b) How many electrons will be present in the subshells having

m_s

value of

-\tfrac{1}{2}

for n = 4?

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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 $n = 4$ , the number of subshells equals $n$ itself (giving 4 subshells: s, p, d, f), and exactly half of all electrons in the shell have $m_s = -\tfrac{1}{2}$ , yielding 16 electrons.

Understanding quantum numbers and electron capacity

Every electron in an atom is described by four quantum numbers. The principal quantum number $n$ determines the shell, and within each shell there are subshells labeled by the azimuthal quantum number $\ell$ . For a given $n$ , the allowed values of $\ell$ run from $0$ to $n-1$ . Each subshell can hold a specific number of electrons, and each electron has a spin quantum number $m_s$ that can be either $+\tfrac{1}{2}$ or $-\tfrac{1}{2}$ .

The key insight: the number of subshells in a shell depends only on $n$ , while the electron count with a particular spin depends on the total capacity of the shell.

Part (a): Counting the subshells

1. Identify the range of $\ell$ values

For $n = 4$ , the azimuthal quantum number $\ell$ can take values:

\ell = 0, 1, 2, 3

Each value of $\ell$ corresponds to one subshell.

2. Name the subshells

The conventional notation assigns letters to each $\ell$ value:

$\ell = 0 \rightarrow$ s subshell (4s)
$\ell = 1 \rightarrow$ p subshell (4p)
$\ell = 2 \rightarrow$ d subshell (4d)
$\ell = 3 \rightarrow$ f subshell (4f)

3. Count them

There are 4 subshells associated with $n = 4$ .

Note

A general rule: the number of subshells in the $n$ -th shell is always equal to $n$ .

Part (b): Electrons with $m_s = -\tfrac{1}{2}$

1. Find the total electron capacity of the $n = 4$ shell

Each subshell with azimuthal quantum number $\ell$ contains $2\ell + 1$ orbitals (corresponding to the magnetic quantum number $m_\ell$ values from $-\ell$ to $+\ell$ ). Each orbital can hold 2 electrons (one with $m_s = +\tfrac{1}{2}$ and one with $m_s = -\tfrac{1}{2}$ ).

The capacity of each subshell:

4s ( $\ell = 0$ ): $2(0) + 1 = 1$ orbital $\times$ 2 electrons = 2 electrons
4p ( $\ell = 1$ ): $2(1) + 1 = 3$ orbitals $\times$ 2 electrons = 6 electrons …

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Q.(a) How many subshells are associated with n = 4?

Understanding quantum numbers and electron capacity

Part (a): Counting the subshells

Part (b): Electrons with ms=−12m_s = -\tfrac{1}{2}ms​=−21​

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Part (b): Electrons with $m_s = -\tfrac{1}{2}$

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