Modern periodic law

Moseley's discovery. In 1913, Henry Moseley bombarded a series of elements with fast electrons and studied the characteristic X-rays each element emitted. He found that the frequency of these X-rays was tied to a whole number that increased by exactly one from element to element, with no gaps and no exceptions. This number is the atomic number ($Z$), equal to the number of protons in the nucleus. Atomic mass, by contrast, does not increase in such a clean, unbroken sequence — it can even step backwards for neighbouring elements.

Statement of the law. The Modern Periodic Law says: the physical and chemical properties of elements are a periodic function of their atomic numbers. This replaced Mendeleev's law, which used atomic mass, and it is the single idea that underlies the entire layout of the table used today.

Why the switch matters — the tellurium-iodine anomaly. Tellurium has atomic mass about 127.6 and iodine about 126.9, so mass-ordering places iodine before tellurium. But tellurium's atomic number is 52 and iodine's is 53, so atomic-number-ordering places tellurium first — and this matches their chemistry. Tellurium behaves like oxygen and sulphur (Group 16), while iodine behaves like fluorine, chlorine and bromine (Group 17). Atomic number, not atomic mass, is the true ordering variable.

Layout of the modern periodic table. The table has 18 vertical groups and 7 horizontal periods. Elements in the same group share the same number of valence (outermost-shell) electrons and therefore similar chemical behaviour; elements in the same period share the same number of occupied electron shells. Group 3 to Group 12, spanning Periods 4 to 7, hold the transition elements; two extra rows placed below the main table hold the inner-transition elements (lanthanides and actinides). At this level you only need to know that these blocks exist and roughly where they sit — their detailed chemistry is a later topic. The electron capacity of each shell is fixed by a simple rule, and this rule is the tool that turns a bare atomic number into a period and a group.

where: $n$ is the shell number counted outward from the nucleus (1, 2, 3, ...), and $n_{\max}$ is the maximum number of electrons that shell can hold before the next shell must start filling. This gives capacities of 2, 8, 18, 32 for shells 1, 2, 3, 4.

Since the Modern Periodic Law is a statement about periodicity, not a computed quantity, the natural "derivation" here is the reasoning that connects electron-shell filling to the table's shape — this is the tool used to place any element once its atomic number is known. Placing an element in the correct period and group from nothing but its atomic number is a two-step act of reasoning that only needs the shell-capacity rule above and a subtraction, and getting it right is the single most testable skill this topic produces.

Step 1. Convert the atomic number into a shell-by-shell electron distribution. Fill shell 1 first, up to its capacity of $2\times1^2 = 2$; then shell 2, up to $2\times2^2 = 8$; then shell 3, up to $2\times3^2 = 18$ — but note that for the elements on this syllabus (up to around $Z = 20$), shell 3 in practice stops filling at 8 once shell 4 begins, so the working capacities to use are 2, 8, 8, 2 in sequence rather than the full $2n^2$ ceiling. Subtract each shell's electrons from the running total until none remain.

Step 2. Count how many shells received at least one electron — this count is the period number, because a period is defined as a row sharing the same number of occupied shells.

Step 3. Read off the number of electrons in the outermost (last-filled) shell — this is the valence electron count, and for the main-group elements on this syllabus it fixes the group number directly (1 valence electron places an element in Group 1, 2 in Group 2, and so on up to Group 18 for a full outer shell of 8).

Step 4. Cross-check the result against known chemical behaviour where possible — elements in the same group should show family resemblance (for example, all Group 1 elements form compounds of the pattern $XCl$, all Group 2 elements form $XCl_2$), which confirms the shell-counting method is consistent with the Modern Periodic Law's promise that properties repeat periodically with atomic number.

Problem. An element X has atomic number 12. State which period and which group it belongs to, using the shell-filling rule, and name a chemical property that confirms the placement.

Step 1. Distribute the 12 electrons into shells. Shell 1 takes 2 (capacity $2\times1^2=2$), leaving $12-2=10$. Shell 2 takes 8 (capacity $2\times2^2=8$), leaving $10-8=2$. Shell 3 takes the remaining 2. Distribution: 2, 8, 2.

Step 2. Count occupied shells: shells 1, 2 and 3 all hold electrons, so X is in Period 3.

Step 3. Read the outermost shell: shell 3 holds 2 valence electrons, so X is in Group 2.

Step 4. Confirm with chemistry: Group 2 elements form chlorides of formula $XCl_2$ (for example $MgCl_2$), consistent with X being magnesium — atomic number 12 is indeed magnesium, and it is a known alkaline-earth metal in Period 3, Group 2.

Answer: element X (Z = 12, magnesium) is in Period 3, Group 2.

Competitive stretch

Try the same method on atomic number 17. Shell 1 takes 2 (remaining 15), shell 2 takes 8 (remaining 7), shell 3 takes the remaining 7. Three occupied shells give Period 3; 7 valence electrons give Group 17. This is chlorine, a halogen — and Group 17 elements are known to form compounds such as $HCl$, matching the family pattern for that group.

• Saying the Modern Periodic Law is based on atomic mass — it is based on atomic number, Moseley's discovery, not Mendeleev's original ordering variable.
• Confusing period with group — a period is a row (same number of occupied shells); a group is a column (same number of valence electrons).
• Using the full $2n^2$ ceiling for shell 3 when placing syllabus-level elements — in practice shell 3 fills to 8 before shell 4 begins for the elements studied here, not to its theoretical maximum of 18.
• Assuming transition-metal or inner-transition-metal chemistry is examinable — the syllabus asks only that you know these blocks exist and roughly where they sit in the table.

Ask the tutor: want to practise finding the period and group for a few more atomic numbers, or see how Moseley's X-ray data actually looked?