Foundations — Atomic radius — covalent, metallic, van der Waals; trends across period and group

Before you can read the parent note on atomic radius, you must own every word and symbol it throws at you. Below, each idea is built from nothing, given a picture, and justified: why does the topic need this?

1. The atom: nucleus + electron cloud

Plain words. An atom is a tiny dense dot of positive charge — the nucleus — surrounded by a spread-out haze of negative charge — the electrons. The haze is thickest near the nucleus and thins out as you move away, but it never suddenly stops.

The picture. Look at the figure. The dark dot in the centre is the nucleus. The shading is the electron cloud: dense (dark) near the middle, fading to nothing at the edges. There is no circle you can draw and say "the atom ends here."

Why the topic needs it. This fuzziness is the entire reason atomic radius is tricky. If atoms had crisp edges, you'd measure one atom with a ruler and be done. Because they don't, the whole topic is a clever workaround. Hold this picture — everything else is built to dodge the "no edge" problem.

2. Distance and "half of it": the symbol $d$ and the radius $r$

Plain words. Since we can't measure one fuzzy atom, we measure between two atoms: the straight-line distance between their two nuclei. We call that distance $d$. Then we share it out — each atom gets half.

What the symbols mean.

• $d$ = the internuclear distance, the gap between two nuclei, centre-to-centre.
• $r$ = the radius we assign one atom, always half of $d$ for two identical atoms: $r=\frac{d}{2}$

The picture. Two fuzzy balls side by side. A double-headed arrow runs nucleus-to-nucleus — that is $d$. Snip it exactly in the middle and each half is $r$.

Why the topic needs it. This one move — "measure between two, take half" — is the definition of every atomic radius. The parent note writes $r_{cov}=d_{A-A}/2$, $r_{met}=d_{M\cdots M}/2$, $r_{vdW}=d_{A\cdots A}/2$. They are all the same recipe; only which two atoms you pick changes.

The dash-vs-dots notation is a silent code: dash = bonded, dots = touching. Watch for it — the next section shows exactly what physically separates a metal-lattice touch from a van der Waals touch.

2a. What actually distinguishes the three contacts?

The three radii differ because the physical reason two atoms sit at a given distance differs. Look at the figure — three panels, three kinds of "why do they stop here?"

Why the topic needs it. This is the physical content behind the parent note's ordering $r_{cov}<r_{met}<r_{vdW}$: stronger, more localised pull → closer nuclei → smaller radius. The dash-vs-dots notation is just shorthand for "bonded and pulled in" vs "unbonded and merely touching."

2b. But what about a bond between two different atoms?

The problem. The "take half" recipe only makes sense when both atoms are the same, because then each fairly owns half the gap. In real chemistry almost every bond is $A-B$ (two different elements), so you can't just halve — the smaller atom would be handed too much.

The picture. Look at the figure. On the left is a symmetric $A-A$ bond: the midpoint splits it fairly, $r_A=d/2$. On the right is a lopsided $A-B$ bond: atom B is bigger, so the fair split is not at the middle.

How chemists fix it. They build the table of radii self-consistently. Measure many bonds; if you already know $r_A$ from an $A-A$ molecule, then for an $A-B$ bond you subtract: $r_B\approx d_{A-B}-r_A$ Read it: "B's radius is roughly what's left of the bond length after you remove A's known share." Do this across hundreds of molecules and average, and a consistent set of covalent radii falls out that reproduces real bond lengths as $d_{A-B}\approx r_A+r_B$.

3. Picometre — the unit (pm)

Plain words. Atoms are absurdly small, so we need a tiny unit. A picometre is one trillionth of a metre.

$1\text{pm}=10^{-12}\text{m}$

The picture. If you blew up a single atom to the size of an orange, that orange would be about as wide as… the whole Earth compared to the real atom. Every radius in this chapter (57 pm, 265 pm, …) lives in the range roughly 30–300 pm.

Why the topic needs it. All the numbers — Cl at 99 pm, Cs at 265 pm — are in picometres. If you don't know the unit, the numbers are meaningless. Bigger pm number = bigger atom. That's all you need.

4. Protons and the symbol $Z$

Plain words. $Z$ is simply the count of protons in the nucleus — the atomic number. It is also each element's ID on the periodic table.

The picture. In our nucleus dot, imagine counting the little $+$ charges. That count is $Z$. Hydrogen has $Z=1$, carbon $Z=6$, chlorine $Z=17$.

Why the topic needs it. More protons = more positive pull on the electron cloud = a tighter, smaller atom (all else equal). $Z$ is the "grip strength" knob. But — crucially — the outer electrons don't feel the full $Z$, which brings us to the next idea.

5. Shells and the symbol $n$

Plain words. Electrons don't sit randomly; they live in shells — nested layers around the nucleus, like the layers of an onion. Each shell is labelled by a whole number $n=1,2,3,\dots$. Bigger $n$ = a layer farther out.

The picture. Look at the figure: concentric rings around the nucleus. The innermost ($n=1$) hugs the nucleus; each higher ring sits farther out. The outermost occupied ring is what sets the atom's size.

Why the topic needs it.

• Moving across a period (left→right in the table): you keep filling the same outer shell — $n$ stays the same.
• Moving down a group (top→bottom): you start a brand-new outer shell — $n$ jumps up by 1.

That single distinction — same shell vs new shell — is why the two trends go in opposite directions. $n$ is the "how far out" knob.

6. Shielding, the symbol $S$

Plain words. The outer electrons don't feel the full pull of all $Z$ protons, because the inner electrons sit between them and the nucleus and block (screen) some of the pull. This blocking is called shielding, and we give its size the symbol $S$.

The picture. Imagine the nucleus is a lamp and the outer electron is looking at it. The inner-shell electrons are people standing in the way, casting shade. The more people (inner electrons) and the better positioned they are, the dimmer the lamp looks. $S$ measures how much light is blocked.

Why the topic needs it. Shielding is what stops "more protons" from always meaning "smaller." Inner shells shield strongly; same-shell electrons shield each other only weakly. This weak same-shell shielding is exactly why atoms shrink across a period. (Deeper mechanism: Shielding and Penetration.)

7. Effective nuclear charge $Z_{eff}$ — the tug-of-war made into one number

Plain words. Combine the grip ($Z$) and the blocking ($S$) into the net pull actually felt by an outer electron:

$Z_{eff}=Z-S$

Read it: "the effective (real, felt) nuclear charge equals the total protons minus the shielded-away part."

The picture. Back to the lamp: $Z$ is the lamp's full brightness, $S$ is the shade cast by the crowd, and $Z_{eff}$ is the brightness that actually reaches the outer electron's eyes. That leftover brightness is what pulls the cloud inward.

Why the topic needs it. This is the engine of the whole chapter (see Effective Nuclear Charge). Every trend the parent note explains is really a statement about $Z_{eff}$:

• Across a period: $Z$ up by 1, $S$ up only a little → $Z_{eff}$ rises → tighter grip → smaller.
• Down a group: $Z$ up, but $S$ also up a lot (new inner shell) → $Z_{eff}$ barely changes → the new shell's distance wins → bigger.

8. Where the formula $r\propto n^2/Z_{eff}$ comes from — and what "$\propto$" means

Plain words. "$\propto$" means "is proportional to" — grows or shrinks in step with the thing on the right, ignoring the constant multiplier. Reading $r\propto n^2/Z_{eff}$: radius gets bigger when $n$ gets bigger, and smaller when $Z_{eff}$ gets bigger.

Why this tool and not just a number? We don't need an exact value of $r$ — we only need to know which way it moves along a trend. A proportionality captures direction cheaply: it lets us say "double the felt grip, halve the radius" or "$n$ from 2 to 3 makes $n^2$ jump from 4 to 9" without computing anything hard.

Deriving it from the simplest atom (the Bohr picture)

We are not asked to accept this from the sky. It comes from the one atom simple enough to solve exactly: a single electron orbiting a nucleus of charge $Z$ (hydrogen, or He${}^{+}$, Li${}^{2+}$…). This is the hydrogen-like or Bohr model. We show every step.

Setup and units. Work in Gaussian units, chosen because they make the Coulomb law carry no separate constant — the electron's charge is a single symbol $e$ and the constant $k$ is absorbed. Let:

• $m$ = electron mass, $e$ = size of the electron's (and one proton's) charge, $v$ = its speed, $r$ = orbit radius,
• the nucleus have charge $+Ze$, the electron charge $-e$ (equal size, opposite sign — that "$-1$" earlier just meant "one unit of charge, negative").

Step 1 — the inward force (WHAT). The electron is pulled toward the nucleus by electric attraction. In Gaussian units Coulomb's law is $F_{\text{pull}}=\frac{(Ze)(e)}{r^2}=\frac{Z{e}^2}{r^2}.$ WHY this law: electric attraction strengthens with each charge and weakens with the square of the separation — that inverse-square fall-off is the experimental fact Coulomb measured. Picture: the nucleus reels the electron in along the radius.

Step 2 — circular motion needs exactly this much force (WHAT). For the electron to hold a circle of radius $r$ at speed $v$, Newton says the inward force must equal $m{v}^2/r$ (the centripetal requirement). Setting the available pull equal to the required force: \frac{Ze^2}{r^2} = \frac{mv^2}{r} \quad\Longrightarrow\quad mv^2 = \frac{Ze^2}{r}. \tag{1} WHY: a stable orbit is just "the pull you have = the pull a circle needs." Picture: the electron neither spirals in nor flies out.

Step 3 — the orbit rule, and why it isn't arbitrary (WHAT & WHY). Classically the electron could orbit at any radius — but then, being a moving charge, it would radiate energy and spiral into the nucleus in a flash. Atoms plainly don't collapse. Bohr's 1913 fix, forced by that stability puzzle (and confirmed by the exact hydrogen spectral lines it then predicted), was to allow only orbits whose angular momentum $mvr$ comes in whole-number multiples of a fixed quantum $\hslash$ (a tiny fundamental constant of nature): mvr = n\hbar, \qquad n = 1, 2, 3, \dots \tag{2} So $n$ — our shell label — enters here, as the count of these allowed steps. Picture: only certain ring sizes are "permitted," like frets on a guitar.

Step 4 — solve the two equations for $r$ (WHAT). From (2), $v=n\hslash /(mr)$. Put that into (1): $m{\left(\frac{n\hslash }{mr}\right)}^2=\frac{Z{e}^2}{r}⟹\frac{n^2{\hslash }^2}{m{r}^2}=\frac{Z{e}^2}{r}.$ Multiply both sides by $r^2$ and divide by $Z{e}^2$: r = \frac{n^2\hbar^2}{m e^2 Z}. \tag{3} WHY the algebra: we had two facts (force balance, orbit rule) and two unknowns ($v$, $r$); eliminating $v$ leaves $r$ alone.

Step 5 — read the result (WHAT IT MEANS). Everything except $n$ and $Z$ in (3) is a fixed constant, so $\boxed{r\propto \frac{n^2}{Z}}$ The radius grows as $n^2$ (outer shells balloon fast) and shrinks as $1/Z$ (more nuclear charge reels the electron in). Picture: bump $n$ up one and the orbit jumps out a lot; bump $Z$ up and it pulls in.

The final swap: $Z\to Z_{eff}$. In a real many-electron atom the outer electron doesn't feel the bare $Z$ — the inner electrons shield it (Section 6). So we replace $Z$ by the felt charge $Z_{eff}=Z-S$: $\boxed{r\propto \frac{n^2}{Z_{eff}}}$

9. When the simple trends break (real-world caveats)

The clean "shrink across, grow down" story has famous exceptions. You don't need to master them yet, but you should know they exist, so the tidy rules don't feel like laws of nature.

Why mention this now? Because the parent note teaches the smooth rules, and a sharp student will later meet these clashes and wonder whether the rules are wrong. They are not — the root cause is always the same knob you already own: shielding quality ($S$) and therefore $Z_{eff}$. When shielding is unusually poor ($d$ and $f$ electrons), $Z_{eff}$ rises more than the simple picture predicts, and the atom stays smaller than "down a group" alone would suggest. Keep the smooth trends as your default, and reach for these caveats only when a transition-metal or lanthanide comparison misbehaves. (Mechanism: Shielding and Penetration.)

How these foundations feed the topic

Read it top to bottom: the fuzzy cloud forces the "half-distance" trick; $Z$ and $S$ combine into $Z_{eff}$; $Z_{eff}$ and $n$ set the size; contacts, additivity and size together produce the two trends — and unusual shielding produces the anomalies.

Equipment checklist

Cover the right side and check you can answer each before moving on.

Connections

• Effective Nuclear Charge — the $Z_{eff}$ engine, defined fully.
• Shielding and Penetration — how $S$ arises, and why $d$/$f$ electrons shield poorly.
• Ionic Radius — the fourth ruler, for charged atoms.
• Ionisation Energy — smaller radius ⇒ higher IE.
• Electronegativity — tracks $Z_{eff}$ and radius.
• Metallic Bonding — sets the metallic radius geometry.
• Van der Waals Forces — the weak touch behind $r_{vdW}$.