Ionization energy — first, second, …; trends and anomalies (e.g. B - Be): Foundations

This page assumes nothing. Every letter and squiggle used by the parent note the topic note is unpacked here, from the ground up, in an order where each idea leans on the one before it.

1. The atom as a picture

Before any symbol, we need the object every symbol talks about.

Figure — Ionization energy — first, second, …; trends and anomalies (e.g. B - Be)

Why do we care? Because ionization energy is the story of pulling one violet dot off the picture. Everything below is a tool for measuring how tightly that dot is held.

2. Charge, and the symbols $Z$ , $S$ , $Z_{eff}$

Now the letters the parent note throws at you:

But an outer electron doesn't feel the full nuclear pull, because inner electrons stand in the way and partly block (screen) it. That blocking gets its own letter.

The subscript "eff" is just short for effective — meaning effective, i.e. what counts in practice. You'll meet the recipe for computing $S$ in Slater's Rules, and the bigger story in Effective Nuclear Charge & Shielding.

Recall Quick check on

$Z_{eff}$ If $Z = 3$ and $S = 1.7$ , what is $Z_{eff}$ ? ::: $3 - 1.7 = 1.3$

3. Coulomb's law — putting numbers on the pull

We keep saying "stronger pull / weaker pull". To make that precise we need the rule that governs electrical pulling: Coulomb's law (full story in Coulomb's Law).

The distance $r$ here is the electron's distance from the nucleus — which is exactly the atom's radius (see Atomic Radius Trends).

4. The absolute-value bars $∣ \cdot ∣$ — a tiny but essential symbol

Before we talk about energy we need one more piece of notation, because it appears the moment we compute ionization energy.

5. Energy, and why $E$ is negative

Intuition WHY bigger

$n$ means bigger distance $r$ — deriving the Bohr result In the simplest working model of the atom (the Bohr model), the electron rides a circular orbit of radius $r$ around the nucleus. Two facts pin it down:

Fact 1 — force balance. For circular motion the inward Coulomb pull must equal the "wants-to-fly-off" pull. Coulomb gives $F \propto Z_{eff} / r^{2}$ ; the circular-motion requirement is $F \propto v^{2} / r$ (where $v$ is the electron's speed). Setting them equal: $\frac{Z _{eff}}{r ^{2}} \propto \frac{v ^{2}}{r}$ , i.e. $v^{2} \propto \frac{Z _{eff}}{r}$ .
Fact 2 — quantization condition. Bohr's key extra rule: the electron's "angular momentum" $\propto v r$ is only allowed to take whole-number-of- $n$ values, i.e. $v r \propto n$ . So $v \propto n / r$ .

Now substitute Fact 2 into Fact 1: $(\frac{n}{r})^{2} \propto \frac{Z _{eff}}{r}$ , giving $\frac{n ^{2}}{r ^{2}} \propto \frac{Z _{eff}}{r}$ , and cancelling one $r$ : $r \propto \frac{n ^{2}}{Z _{eff}} .$ So doubling the shell number $n$ does not double the radius — it roughly quadruples it ( $n^{2}$ ). This is why an $n = 3$ electron sits far outside an $n = 1$ electron, feels a weaker grip, and is easier to remove. The smallest such radius (hydrogen's $n = 1$ ) even has a name, the Bohr radius ( $\approx 0.53 \times 1 0^{- 10}$ m).

Now we can assemble the energy. Coulomb's law fixed the force; the energy of a bound orbit follows from it.

Formula The hydrogen-like energy formula — and WHERE each power comes from

$E_{n} = - \frac{Z _{eff}^{2}}{n ^{2}} \times 13.6 eV$ Reading the two powers physically:

Why $Z_{eff}^{2}$ (charge squared, not just charge)? Start from energy $\propto Z_{eff} / r$ (the $1/ r$ law we just derived, with charge $\propto Z_{eff}$ ). Substitute the Bohr radius $r \propto n^{2} / Z_{eff}$ : the $Z_{eff}$ upstairs times the $Z_{eff}$ that comes up from dividing by $r$ gives two factors ⇒ $Z_{eff}^{2}$ . Physically: a stronger nucleus pulls harder and drags the electron into a smaller orbit, two reinforcing effects that multiply.
Why $1/ n^{2}$ (falls off with shell number squared)? Same substitution: energy $\propto 1/ r$ and $r \propto n^{2}$ , so energy $\propto 1/ n^{2}$ . So the outer shells crowd toward $0$ energy very quickly — an $n = 2$ electron is only a quarter as tightly bound as the $n = 1$ electron of the same $Z_{eff}$ .
Where does $13.6 eV$ come from? It is not arbitrary. Plug the real constants (Coulomb's $k$ , the electron's charge and mass, and $n = 1$ , $Z_{eff} = 1$ ) into the Bohr model and out pops exactly $13.6 eV$ — the measured energy needed to ionize a hydrogen atom. It is the natural energy scale of "one electron, one proton", confirmed by experiment. Everything else is that scale multiplied by $Z_{eff}^{2} / n^{2}$ .
The minus sign — because bound electrons sit below the zero-energy "escape" line.

The figure shows the "energy well": to ionize means to lift the electron from its negative floor $E_{n}$ all the way up to $0$ . The height you must climb is $∣ E_{n} ∣$ — and that is the ionization energy.

Z

The number of protons = the nucleus's full positive charge.

Meaning of

S

Shielding — how much of the nuclear pull is blocked by other electrons.

The formula linking them

Z_{eff} = Z - S

What

Z_{eff}

physically means

The net nuclear pull an outer electron actually feels.

What

\propto

means

"Is proportional to" — grows/shrinks together, ignoring the constant.

What the bars

∣ x ∣

mean

The absolute value — the size of

x

with any minus sign dropped (

∣ - 5∣ = 5

Coulomb's law in full

F = k \frac{q _{1} q _{2}}{r ^{2}}

, with

k

the fixed Coulomb constant.

Why

r^{2}

(not

r

) matters

Distance is squared, so moving farther weakens the pull fast.

Why energy goes like

1/ r

while force goes like

1/ r^{2}

Energy = force added up (integrated) over distance, which raises the power by one.

What

n

counts

The shell number, from

n = 1

innermost outward — bigger

n

= farther out.

Why bigger

n

means bigger

r

Force balance plus Bohr's quantization (

v r \propto n

) give

r \propto n^{2} / Z_{eff}

Why the energy has

Z_{eff}^{2}

Energy

\propto Z_{eff} / r

and

r \propto n^{2} / Z_{eff}

combine two factors of

Z_{eff}

Why the energy has

1/ n^{2}

Energy

\propto 1/ r

and

r \propto n^{2}

, so energy

\propto 1/ n^{2}

Where

13.6 eV

comes from

The Bohr-model / measured ionization energy of hydrogen (

n = 1

Z_{eff} = 1

Why the hydrogenic formula is only approximate

It ignores electron–electron repulsion; real atoms need quantum mechanics, faked here by

Z_{eff}

Why bound-electron energy

E

is negative

Free electron =

0

; a trapped one sits below that, needing energy to escape.

What

I E

physically is

The energy you must supply to lift an electron out of its well (

∣ E_{n} ∣

Why

I E

is always positive

You add energy to climb up and out — energy goes in.

Meaning of

(g)

X (g) \to X^{+} (g) + e^{-}

Gas phase, so no neighbouring atoms interfere.

Meaning of superscript

+

X^{+}

One unit of net positive charge (one electron removed).

The master trend line

I E \approx 13.6 eV \times Z_{eff}^{2} / n^{2}

Convert

1 eV

to kJ/mol

96.49 kJ mol^{- 1}

Foundations — Ionization energy — first, second, …; trends and anomalies (e.g. B - Be)

1. The atom as a picture

2. Charge, and the symbols $Z$ , $S$ , $Z_{eff}$

3. Coulomb's law — putting numbers on the pull

4. The absolute-value bars $∣ \cdot ∣$ — a tiny but essential symbol

5. Energy, and why $E$ is negative

6. Ionization energy itself: $I E_{1}, I E_{2}, \dots$

7. The proportional-sign toolkit for reading trends

Prerequisite map

Equipment checklist