Visual walkthrough — Modern periodic law — based on atomic number

Prerequisites we lean on (all built here as we go): Bohr Model of the Atom, Atomic Number and Atomic Mass, Effective Nuclear Charge and Shielding, and the experiment itself, Moseley's X-ray Experiment. This page is the visual companion to the parent topic.

Step 1 — What is inside an atom, drawn to scale of idea

WHAT we did: Named the pieces and the one number we care about, $Z$.

WHY: Everything that follows is about how hard the nucleus pulls an electron. That pull is set by the charge $+Z$. So $Z$ is the hero of the whole story.

WHAT IT LOOKS LIKE: A red central charge $+Z$, with electrons parked on rings labelled by their shell number $n=1,2,3,\dots$ (the innermost ring is $n=1$).

Step 2 — Why an X-ray is born: an electron falls

WHAT we did: Turned "X-ray emission" into a concrete motion — an electron dropping from ring $n=2$ to ring $n=1$.

WHY: Energy is never destroyed. The energy the electron loses by falling has to go somewhere; it comes out as the X-ray. So the X-ray's energy = the energy gap between the two rings. If we can compute that gap, we know the X-ray.

WHAT IT LOOKS LIKE: A red arrow diving from the outer ring to the inner hole, with a wavy red X-ray shooting off.

Step 3 — The screen: electrons hide part of the nucleus

WHAT we did: Replaced the full charge $Z$ with the smaller, honest charge $Z-\sigma$.

WHY THIS TOOL and not just $Z$? Because a real atom has many electrons, and they get in each other's way. Using $Z$ alone would over-count the pull. The single correction $-\sigma$ is the simplest honest fix — see Effective Nuclear Charge and Shielding.

WHAT IT LOOKS LIKE: The bright red $+Z$ at the centre, partly greyed out by a shield, leaving a smaller effective red core labelled $Z-\sigma$.

Step 4 — The energy of a fall depends on charge squared

WHAT we did: Wrote the X-ray energy in terms of the felt charge, and reminded ourselves that energy $=h\nu$.

WHY the square? Two effects both grow with charge: a bigger charge pulls the rings closer and pulls harder. Multiply two charge-effects together and you get charge${}^2$. That is why $(Z-\sigma )$ appears to the power 2, not 1.

WHAT IT LOOKS LIKE: A curve of $E$ versus $Z$ that bends upward — a parabola — because of the squaring. Not a line yet.

Step 5 — Take the square root to straighten the curve

$\nu \propto (Z-\sigma {)}^2\stackrel{\sqrt{}}{\to }\sqrt{\nu }\propto (Z-\sigma )$

WHAT we did: Undid the square with a square root, turning the parabola into $\sqrt{\nu }=a(Z-b)$.

WHY THIS TOOL (the square root)? A square root is exactly the operation that undoes a square. We chose it precisely because it converts the awkward $(Z-\sigma {)}^2$ into a clean first-power $(Z-\sigma )$, which draws a line.

WHAT IT LOOKS LIKE: The bent parabola of Step 4 pulled straight — a red line rising steadily, crossing the $Z$-axis a little to the right of zero (at $Z=b$).

Step 6 — Why the line crosses at $Z=b$, not at zero

WHAT we did: Located the exact crossing point of the line.

WHY: This is the degenerate case — what happens when the frequency drops to zero. It pins down the meaning of $b$: it is the atomic number at which the felt charge would vanish.

WHAT IT LOOKS LIKE: A zoom on the bottom of the line, red dot sitting at $(Z=b,\sqrt{\nu }=0)$, clearly to the right of the origin.

Step 7 — The whole point: $Z$ gives a line, mass does not

WHAT we did: Ran the experiment's logic side by side — $Z$ versus mass.

WHY: Ordering elements needs a quantity that changes smoothly and by exactly one from element to element. The straight line shows $Z$ does; the crooked plot shows mass does not.

WHAT IT LOOKS LIKE: Two panels — left, a clean red line ($\sqrt{\nu }$ vs $Z$); right, the same points scattered off a line ($\sqrt{\nu }$ vs mass).

The one-picture summary

Everything above, in one flow: an electron falls ($E=h\nu$), it feels charge $(Z-\sigma {)}^2$, we take a root, and out comes the straight line $\sqrt{\nu }=a(Z-b)$ that fingerprints atomic number.

Active Recall

Connections

• Bohr Model of the Atom — source of the $(Z-\sigma {)}^2$ energy dependence (Step 4)
• Effective Nuclear Charge and Shielding — origin of $\sigma$ and the constant $b$ (Step 3)
• Atomic Number and Atomic Mass — meaning of $Z$ (Step 1)
• Moseley's X-ray Experiment — the experiment we reconstructed
• Mendeleev's Periodic Table — the mass-based law this replaced (Step 7)
• Periodic Trends — the repeating properties that riding on $Z$ makes regular