Indicate the number of unpaired electrons in: (a) P, (b) Si, (c) Cr, (d) Fe and (e) Kr.

Exercises · 2.66

Q. Indicate the number of unpaired electrons in:

(a) P,

(b) Si,

(d) Fe and

(e) Kr.

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Concept understanding — Electron Configuration

Electron Configuration: Where Do Electrons Actually Live?

Imagine a school building. Students don't just wander randomly — they sit in specific classrooms, on specific floors, in specific rows. Electrons in an atom behave similarly. They don't buzz around the nucleus chaotically. They occupy specific energy levels (floors), sublevels (classrooms), and orbitals (seats).

The electron configuration is simply the address system that tells you exactly which "seats" every electron in an atom is sitting in.

The Intuition: Why Can't Electrons Sit Anywhere?

Two big rules force electrons into this orderly arrangement:

Energy matters. Electrons want to be as close to the nucleus as possible (lowest energy). The first "floor" (n=1) is the most comfortable. Higher floors cost more energy.
No crowding. A famous rule called the Pauli Exclusion Principle says: no two electrons in the same atom can have the exact same set of four quantum numbers. In plain language: each orbital (seat) can hold at most two electrons, and they must spin in opposite directions.

So electrons fill up from the bottom floor upward, like students filling a theatre from the front row back.

The Precise Statement

n

For example, the configuration of carbon (6 electrons) is:

1s^2 \; 2s^2 \; 2p^2

This reads: "Two electrons in the 1s orbital, two in the 2s orbital, and two in the 2p orbitals."

The Filling Order: The Aufbau Principle

Electrons don't fill levels in simple numerical order. Here's the actual sequence (memorise this — it's exam gold):

1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow 5s \rightarrow 4d \rightarrow 5p \rightarrow 6s \rightarrow 4f \rightarrow 5d \rightarrow 6p \rightarrow 7s \rightarrow 5f \rightarrow 6d \rightarrow 7p

Notice: 4s fills before 3d. This is because the 4s orbital is actually lower in energy than 3d. This trips up many students.

Tip

Use the diagonal rule (Madelung's rule) to remember the order: draw arrows diagonally across the $(n+\ell)$ chart. Or just remember the mnemonic: "Silly People Don't Fail" for the sublevel order within each shell.

How to Write Any Configuration (Step-by-Step)

Let's do iron (Fe, atomic number 26).

Step 1: Know the total electrons = 26.

Step 2: Follow the filling order, counting electrons as you go:

$1s^2$ (2 used, 24 left)
$2s^2$ (4 used, 22 left)
$2p^6$ (10 used, 16 left)
$3s^2$ (12 used, 14 left)
$3p^6$ (18 used, 8 left)
$4s^2$ (20 used, 6 left)
$3d^6$ (26 used, 0 left)

Step 3: Write it in order of increasing $n$ (standard notation):

1s^2 \; 2s^2 \; 2p^6 \; 3s^2 \; 3p^6 \; 3d^6 \; 4s^2

Watch out

Many textbooks write configurations in order of filling (4s before 3d), but IUPAC standard lists them by principal quantum number $n$ (3d before 4s). Check which convention your exam uses. For CBSE/ICSE, write in order of increasing $n$ : $1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p...$

The Three Golden Rules (Memorise These)

Rule	What it says	Why it matters
Aufbau Principle	Electrons fill lowest energy orbitals first	Determines the order of filling

Why this formula?

Electron Configuration: Why the Rules Work

Let's build this from the ground up — not just what the rules are, but why they exist.

The Core Question

Why do electrons arrange themselves in specific shells, subshells, and orbitals — and not just pile up anywhere?

The answer lies in three fundamental principles, each rooted in physics and quantum mechanics.

1. The Aufbau Principle: Why "Build Up" in Order?

What it says: Electrons fill orbitals from lowest to highest energy.

Why it holds: Nature seeks the lowest possible energy state (ground state). An atom is most stable when its electrons occupy the lowest available energy levels.

Think of it like water flowing downhill — electrons "fall" into the lowest energy orbitals first.

The energy ordering (for multi-electron atoms) is:

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s...

Why this order? It comes from the $(n + \ell)$ rule:

$n$ = principal quantum number (shell)
$\ell$ = azimuthal quantum number (subshell: $s=0$ , $p=1$ , $d=2$ , $f=3$ )

Orbitals fill in order of increasing $(n + \ell)$ . If two have the same $(n + \ell)$ , the one with lower $n$ fills first.

Example: $4s$ has $(4+0)=4$ , $3d$ has $(3+2)=5$ . So $4s$ fills before $3d$ — even though $4s$ is a higher shell number.

2. Pauli Exclusion Principle: Why Only Two Per Orbital?

What it says: No two electrons in an atom can have the same set of all four quantum numbers.

Why it holds: This is a fundamental law of quantum mechanics — electrons are fermions (spin-1/2 particles). Fermions obey the Pauli exclusion principle, which arises from the antisymmetry of the wavefunction.

The four quantum numbers:

$n$ (shell)
$\ell$ (subshell shape)
$m_\ell$ (orbital orientation)
$m_s$ (spin: $+\frac{1}{2}$ or $-\frac{1}{2}$ )

Since only $m_s$ can differ for electrons in the same orbital, maximum 2 electrons per orbital — one spin-up ( $\uparrow$ ) and one spin-down ( $\downarrow$ ).

Key result: The $s$ subshell ( $\ell=0$ , one orbital) holds 2 electrons. The $p$ subshell ( $\ell=1$ , three orbitals) holds 6 electrons. The $d$ subshell ( $\ell=2$ , five orbitals) holds 10 electrons.

3. Hund's Rule: Why Spread Out First?

What it says: Within a subshell, electrons occupy empty orbitals singly before pairing up — and all unpaired electrons have parallel spins.

Why it holds: Electrons repel each other (Coulomb repulsion). By occupying different orbitals, they stay farther apart, reducing repulsion energy.

The spin alignment (all parallel) comes from exchange energy — a quantum mechanical effect where parallel spins have a slightly lower energy state due to wavefunction symmetry.

Example for carbon ( $1s^2 2s^2 2p^2$ ):

Correct: $\uparrow$ | $\uparrow$ | (two unpaired, parallel)
Wrong: $\uparrow\downarrow$ | | (paired in one orbital — higher repulsion)

The Big Picture: Why These Three Rules Together? …

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The number of unpaired electrons is found by writing the ground-state electron configuration and applying Hund’s rule. The results are: P → 3, Si → 2, Cr → 6, Fe → 4, Kr → 0.

Why electron configuration tells you the number of unpaired electrons

Every atom in its ground state fills orbitals in a specific order — 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so on. Within a subshell, electrons occupy orbitals singly before pairing up (Hund’s rule). The unpaired electrons are simply the ones that sit alone in an orbital. So the task reduces to writing the configuration and counting the singles.

A common trap: the 4s orbital fills before 3d, but when writing configurations for atoms beyond calcium, we list 3d before 4s because 3d is lower in energy once occupied. For counting unpaired electrons, the order of writing doesn’t matter — what matters is which orbitals are singly occupied.

Let’s go element by element.

1. Phosphorus (P, atomic number 15)

Configuration: $1s^2 2s^2 2p^6 3s^2 3p^3$

The 3p subshell has three orbitals ( $p_x, p_y, p_z$ ). With three electrons, Hund’s rule says each goes into a separate orbital, all with parallel spins. So all three are unpaired.

Watch out

Do not pair any electrons in the 3p subshell — three electrons in three orbitals means three singles, not one pair plus one single.

Unpaired electrons in P: 3

2. Silicon (Si, atomic number 14)

Configuration: $1s^2 2s^2 2p^6 3s^2 3p^2$

The 3p subshell has two electrons. They occupy two different p orbitals (say $p_x$ and $p_y$ ), with parallel spins. No pairing yet.

Unpaired electrons in Si: 2

3. Chromium (Cr, atomic number 24)

Here is where the textbook exception lives. The expected configuration would be $[Ar] 4s^2 3d^4$ , but that is not the ground state. Half-filled subshells are extra stable, so one electron from the 4s moves into the 3d, giving:

$[Ar] 4s^1 3d^5$

Now the 3d subshell has five orbitals, each with one electron — all unpaired. The 4s has one electron, also unpaired. …

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Q.Indicate the number of unpaired electrons in:

Why electron configuration tells you the number of unpaired electrons

1. Phosphorus (P, atomic number 15)

2. Silicon (Si, atomic number 14)

3. Chromium (Cr, atomic number 24)

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