Foundations — Atomic - ionic size trends; lanthanide contraction

This page assumes you know nothing. Before you read the parent topic, every letter, arrow, and word it uses is unpacked here, in an order where each idea leans only on the ones before it.

0. The atom picture we will keep pointing at

Everything below refers to one mental image, so let's draw it once.

Look at figure s01: the red dot is the nucleus, the blue specks are electrons, and the faint grey rings are shells. The radius we care about is the distance from the centre dot to the outermost blue speck.

1. Symbol $Z$ — the atomic number (how strong the pull)

Why the topic needs it: every trend "across a series" happens because each step to the right adds one more proton, i.e. $Z$ goes up by 1. That is one half of the tug-of-war.

2. Symbol $S$ — shielding (the bodyguards)

Here is the key subtlety. An outer electron does not feel the full pull of all $Z$ protons, because the electrons between it and the nucleus sit in the way and push it back (they are negative too).

Why call them bodyguards? Inner electrons stand between the "magnet" (nucleus) and the outer electron, absorbing part of the attraction. More/closer bodyguards → bigger $S$ → weaker felt pull → the atom can be bigger.

3. Symbol $Z_{\text{eff}}$ — the pull that actually gets through

Now combine the two ideas above into the single most important quantity on the whole parent page.

Why the topic needs it: the parent's whole story is "does $Z_{\text{eff}}$ go up or down at each step?" Up → tighter grip → smaller atom. Down → looser → bigger atom.

4. Symbol $n$ — the shell number (how far out)

Why the topic needs it: going down a group means the outer electrons live in a higher shell (bigger $n$), which is the main reason atoms lower in a group are bigger.

5. The size formula $r\propto n^2/Z_{\text{eff}}$

Now the two knobs ($n$ and $Z_{\text{eff}}$) combine into the one relation the parent uses.

The symbol $\propto$ means "is proportional to" — grows and shrinks in step with, ignoring the fixed constants. We use a proportionality (not an equals) because we only care about which way size moves, not its exact number in metres.

Figure s03 plots this: hold $Z_{\text{eff}}$ fixed and $r$ climbs with $n^2$ (blue); hold $n$ fixed and $r$ falls as $Z_{\text{eff}}$ rises (orange). Every trend on the parent page is just "which of these two curves wins this step."

6. Subshell letters $s,p,d,f$ — the kinds of rooms

Within a shell, electrons live in differently shaped "rooms" labelled $s,p,d,f$. Their shapes decide how well they penetrate toward the nucleus, and penetration decides shielding power.

Figure s04 shows how much each type's cloud pokes toward the centre: $s$ has a spike right at the nucleus; $f$ barely gets close. An electron that never comes near the nucleus can't stand in the doorway — so it makes a lousy bodyguard.

7. Charge, ions, and the $+$ superscript

Why the topic needs it: the parent compares ionic radii (${\text{La}}^{3+}$ to ${\text{Lu}}^{3+}$) and states ${\text{Fe}}^{3+}<{\text{Fe}}^{2+}$. Both follow from: fewer electrons but the same $Z$ → higher $Z_{\text{eff}}$ per remaining electron → tighter grip.

8. "Coordination number (CN 6)" — why radii need a footnote

Why the topic needs it: the parent quotes "$1.032$ Å at CN 6." The CN 6 tag is not decoration — it tells you which measuring setup produced that number, so you compare like with like.

The prerequisite map

Read it top-down: penetration decides subshell shielding → shielding sets $S$ → $S$ and $Z$ give $Z_{\text{eff}}$ → $Z_{\text{eff}}$ and $n$ feed the size formula → out come all the trends and the contraction.

Where each symbol goes next

• $Z_{\text{eff}}$ and Slater's counting → Effective Nuclear Charge & Slater's Rules.
• The $s>p>d>f$ ranking in the electron layout → d-Block Overview & Electronic Configuration and f-Block (Lanthanides & Actinides).
• The size formula's payoff → Periodic Trends — Atomic & Ionic Radii.
• The consequences (density, twins, basicity) → Density, Melting Point Trends in Transition Metals, Basic Character of Oxides & Hydroxides, Separation of Lanthanides (Ion-exchange).

Equipment checklist

Test yourself — cover the right side and answer before revealing.