Exercises — Modern periodic law — based on atomic number

Atomic number ($Z$). Why: The modern law (Moseley, 1913) says properties are a periodic function of $Z$, the proton count. Mendeleev's older law used mass. Neutrons never define the ordering (they change between isotopes of the same element). See Atomic Number and Atomic Mass for what $Z$ means.

$\sqrt{\nu }=a(Z-b)$

• $\nu$ = frequency of the characteristic X-ray,
• $Z$ = atomic number,
• $a$ = slope constant of the chosen spectral line,
• $b$ = screening constant (≈ 1 for the K-series). A graph of $\sqrt{\nu }$ (up the page) against $Z$ (across the page) is a straight line. Look at Figure s01.

Ar–K, Co–Ni, Te–I. In each, the lighter element chemically belongs after the heavier one — impossible to justify by mass, but perfectly natural by $Z$. See Mendeleev's Periodic Table.

Step 1 — order by $Z$, not mass. Compare $18$ vs $19$. Since $18<19$, Argon comes first. Why this step: The modern law orders strictly by atomic number. Step 2 — check chemistry. Argon is an unreactive noble gas; Potassium is a violently reactive alkali metal. Placing Ar (with the noble gases) before K (with the alkali metals) matches their chemistry. ✅ So the "swap" Mendeleev did by hand happens automatically under the modern law.

Step 1 — write the ratio. Using $b=1$: $\frac{\sqrt{{\nu }_B}}{\sqrt{{\nu }_A}}=\frac{a(Z_B-1)}{a(Z_A-1)}=\frac{Z_B-1}{Z_A-1}.$ Why this step: The constant $a$ is the same line for both atoms, so it cancels. Step 2 — set equal to 2 and solve. $\frac{Z_B-1}{13-1}=2\Rightarrow Z_B-1=24\Rightarrow Z_B=25.$ Why this step: We are told the ratio equals $2$, so we substitute $2$ for the left side. To get $Z_B$ alone we undo each operation wrapped around it — first multiply both sides by $12$ (to clear the division by $13-1=12$), giving $Z_B-1=24$; then add $1$ to both sides (to undo the "$-1$"), giving $Z_B=25$. Isolating the unknown by reversing its surrounding operations is the standard way to solve any linear equation. $Z=25$ is Manganese (Mn).

Step 1 — plug in. $\sqrt{\nu }=5.0\times 10^7\times (26-1)=5.0\times 10^7\times 25=1.25\times 10^9$. Why this step: Direct substitution — $Z-b=25$. Step 2 — square to get frequency. $\nu =(1.25\times 10^9{)}^2=1.5625\times 10^{18}\text{Hz}.$ The square root undoes the squaring in Moseley's derivation, so to recover $\nu$ we square back.

An X-ray comes from an inner electron falling toward the nucleus. From the Bohr Model of the Atom, the transition energy scales as $(\text{effective charge}{)}^2=(Z-b{)}^2$. That energy is carried away by the X-ray, and the energy of a light wave equals Planck's constant $h$ times its frequency, $E=h\nu$ (this is what $h$ does — see the symbol box: it turns a frequency into an energy). So $h\nu \propto (Z-b{)}^2$, and taking a square root of both sides gives $\sqrt{\nu }=a(Z-b)$ — linear in $Z$ (the constant $a$ swallows up $h$ and the other fixed factors). Atomic mass, however, grows irregularly: it depends on neutron count, which jumps unevenly from element to element and even varies between isotopes. The nucleus's pull on electrons is set by proton count ($Z$), not by mass. So the physics locks $\nu$ to $Z$, not to mass — hence only the $Z$-plot is a clean line. Compare Figure s01 (straight) with Figure s02 (scattered).

Look up atomic numbers: $Z_{Co}=27$, $Z_{Ni}=28$. Because $27<28$, Cobalt precedes Nickel — automatically, by $Z$. Under Mendeleev's mass rule this looked like a forced "exception" (Co heavier yet placed first). Under the modern law there is nothing to excuse: ordering by $Z$ already gives the chemically correct sequence. An "exception" only exists relative to a wrong rule; switch to the right rule and it vanishes.

$b$ is the screening (shielding) constant — see Effective Nuclear Charge and Shielding. Recall from the definition box that a K-series X-ray ends in the innermost (K) shell. That shell holds one other electron beside the one arriving. That single companion electron partly cancels one unit of nuclear charge, so the jumping electron feels roughly $(Z-1)$ protons' worth of pull. Hence $b\approx 1$. If there were no inner electrons, $b$ would be $0$ and the line would pass through the origin.

Step 1 — spacing per unit $Z$. From $\sqrt{\nu }=a(Z-b)$, increasing $Z$ by $1$ raises $\sqrt{\nu }$ by exactly $a$. So on the straight line each element sits one "rung" of height $a$ above the previous. See the equal steps in Figure s01. Step 2 — read a gap. If a measured element's $\sqrt{\nu }$ sat two rungs ($2a$) above its neighbour instead of one, that would mean a whole integer of $Z$ — an undiscovered element — is missing between them. Step 3 — the synthesis. Because the line is straight and evenly stepped, Moseley could count elements by rungs and predict missing atomic numbers (e.g. $Z=43,61,72,75$) before they were isolated. The regularity of the plot turned "how many elements exist up to here?" into simple counting.

Fact (ii): chemical family (group) repeats at regular $Z$-intervals (e.g. alkali metals at $Z=3,11,19,37,\dots$). See Periodic Trends. Fact (i): $\sqrt{\nu }$ rises linearly with $Z$, so a fixed jump in $Z$ would give a fixed jump in $\sqrt{\nu }$ only if the group members were equally spaced in $Z$ — but they are not ($3\to 11$ is a jump of 8, $11\to 19$ is 8, but $19\to 37$ is 18). Because the $Z$-gaps between successive family members grow (periods get longer), the $\sqrt{\nu }$-gaps grow too. Synthesis: the chemistry repeats (periodicity in $Z$), while the X-ray frequency marches steadily upward (linear in $Z$). One property cycles; the other climbs — both driven by the single variable $Z$.

Step 1 — expose the wrong premise. The student is ordering by mass. The modern law orders by $Z$: $Z_{Te}=52$, $Z_I=53$. Since $52<53$, Te correctly precedes I — no error. Step 2 — why mass misleads here. Tellurium has, on average, more neutrons than iodine, so it is heavier despite having fewer protons. Mass counts protons + neutrons; only protons ($Z$) fix the position. Step 3 — Moseley prediction. From $\sqrt{\nu }=a(Z-b)$, smaller $Z$ gives smaller $\sqrt{\nu }$. So Tellurium ($Z=52$) emits X-rays with the lower $\sqrt{\nu }$, iodine ($Z=53$) higher. Measuring exactly this confirms $Z_{Te}<Z_I$ directly, independent of any chemistry — the experimental nail in the coffin. Conclusion: the "error" is in the student's ordering rule, not the table. Moseley's X-ray count settles it objectively. See Moseley's X-ray Experiment.

(a) Slope. Consecutive elements differ by $\Delta Z=1$, so the slope is the step in $\sqrt{\nu }$: $a={\sqrt{\nu }}_2-{\sqrt{\nu }}_1=52-48=4.$ Check the next step: $56-52=4$ ✅ — same rung height. (b) Consecutive? Yes: both gaps equal $a=4$, i.e. exactly one unit of $Z$ each. Equal, single-sized rungs ⇒ no missing element between them. (c) Find $b$. Use element 1: ${\sqrt{\nu }}_1=a(Z_1-b)$. $48=4(12-b)\Rightarrow 12-b=12\Rightarrow b=0.$ Why this step: $b$ is the only unknown left, so we peel away everything wrapped around it. First divide both sides by $a=4$ (undoing the multiplication), which turns $48$ into $12$ and leaves $12-b$ on the right. That gives $12-b=12$, so $b$ must be $0$ (subtracting $0$ from $12$ returns $12$). We always isolate the unknown by reversing the operations attached to it, one at a time.

Order elements by $Z$ (protons), and every anomaly disappears, Moseley's straight line $\sqrt{\nu }=a(Z-b)$ counts them exactly, and chemistry repeats periodically — all because $Z$, not mass, is the atom's true identity.