Exercises — Mendeleev's periodic table — based on atomic mass

The properties of elements are a periodic function of their atomic masses. "Periodic function" = arrange by increasing atomic mass, and similar properties come back at regular intervals.

• Period = a horizontal row; atomic mass increases as you move left → right.
• Group = a vertical column stacking elements with similar chemical properties. Trace the row and the column in the figure above.

Prefix = eka, meaning "one". Eka-silicon = the element one step below silicon in the table, later discovered as germanium (1886).

Write each element's highest oxide and read off the valency (oxygen = 2 bonds each):

• ${\text{Li}}_2\text{O}$ → Li valency $1$
• $\text{BeO}$ → Be valency $2$
• ${\text{B}}_2{\text{O}}_3$ → B valency $3$
• ${\text{CO}}_2$ → C valency $4$
• ${\text{N}}_2{\text{O}}_5$ → N valency $5$
• (O and F cap the row)
• ${\text{Na}}_2\text{O}$ → Na valency $1$

The valency climbed $1,2,3,4,5\dots$ and then, at sodium, it drops back to 1 with ${\text{Na}}_2\text{O}$ — the same formula shape as ${\text{Li}}_2\text{O}$. Na is also a violent (highly reactive) metal like Li: both fizz in water because both give away one outer electron. So the chemistry returns at Na. Mendeleev "wraps the line" here: start a new row so Li sits directly above Na. See the wrap figure below.

Balance so the total "bonds from $R$" equals the total "bonds from O".

• ${\text{R}}_2\text{O}$: two $R$ share $1\times 2=2$ bonds → each $R=1$.
• $\text{RO}$: one $R$ meets $1\times 2=2$ bonds → $R=2$.
• ${\text{R}}_2{\text{O}}_3$: two $R$ meet $3\times 2=6$ → each $R=3$.
• ${\text{RO}}_2$: $2\times 2=4$ → $R=4$.
• ${\text{R}}_2{\text{O}}_5$: $5\times 2=10$, over two $R$ → $R=5$.
• ${\text{RO}}_3$: $3\times 2=6$ → $R=6$.
• ${\text{R}}_2{\text{O}}_7$: $7\times 2=14$, over two $R$ → $R=7$. So oxide valency rises 1, 2, 3, 4, 5, 6, 7 across the period (the rising bars in the figure below).

Hydrogen contributes valency 1 each, so $R$'s valency = number of H atoms.

• ${\text{RH}}_4\to 4$
• ${\text{RH}}_3\to 3$
• ${\text{RH}}_2\to 2$
• $\text{RH}\to 1$ The hydride valency falls 4, 3, 2, 1 across groups IV → VII — the mirror of the rising oxide valency (coral line in the figure above).

(a) His law orders by increasing atomic mass. Putting the heavier Ar ($39.9$) before the lighter K ($39.1$) is a mass inversion — a direct violation of the ordering rule. (b) Chemistry forced it: argon is an inert noble gas (it barely reacts at all), potassium is a violent (highly reactive) alkali metal (it bursts into flame on water). If he obeyed mass strictly, K would land among noble gases and Ar among reactive metals — nonsense. He trusted properties over mass, keeping each element with its true family. This is an anomalous pair.

(a) The tabulated atomic mass is a weighted average over abundances: $\bar{m}=0.75\times 35+0.25\times 37=26.25+9.25=\mathrm{35.5.}$ So "chlorine $=35.5$" is not any single atom — it is a blend. Why this matters: Mendeleev's ruler (mass) is really this average, so the very number he ordered by is a statistical mix, not a fundamental property of one atom. (b) A table ordered by mass demands different masses = different positions. But ${}^{35}\text{Cl}$ and ${}^{37}\text{Cl}$ are chemically identical chlorine — they must share one slot. Mass-ordering can't grant that. The fix: order by atomic number $Z$. All chlorine isotopes have $Z=17$, so they naturally occupy a single position. This is a core reason mass was the wrong ruler.

Hydrogen wears two masks:

• Like the alkali metals (Group I): it can lose an electron to form ${\text{H}}^{+}$, valency 1 — just like Na, K.
• Like the halogens (Group VII): it can gain an electron to form ${\text{H}}^{-}$, again valency 1 — just like F, Cl. Both fits look equally good, and mass ($\approx 1$) doesn't break the tie. So H floats ambiguously — a genuine defect of the scheme.

Percentage error $=\frac{|\text{predicted}-\text{actual}|}{\text{actual}}\times 100\%$. Mass: $\frac{|72-72.6|}{72.6}\times 100\%=\frac{0.6}{72.6}\times 100\%\approx 0.83\%$. Density: $\frac{|5.5-5.35|}{5.35}\times 100\%=\frac{0.15}{5.35}\times 100\%\approx 2.80\%$. Both under $3\%$ — astonishing for a prediction made 15 years before discovery. This is why the table counts as a law of nature, not a filing trick.

($R$ is again the placeholder "some element".) (a) Oxide ${\text{R}}_2{\text{O}}_5$: oxygen bonds $=5\times 2=10$, over $2$ atoms → oxide valency $=5$. Hydride ${\text{RH}}_3$: three H → hydride valency $=3$. (b) $5+3=8$ ✓ — the rule holds. (c) Oxide valency $5$ = Group V (the nitrogen/phosphorus family, oxide ${\text{R}}_2{\text{O}}_5$).

(a) Tellurium ($127.6$) sits before iodine ($126.9$) even though Te is heavier — a mass inversion. (b) Chemistry demands it: Te belongs with the oxygen-family (Group VI) and I with the halogens (Group VII). Keeping families intact beats obeying mass. (c) The fix is atomic number $Z$: Te has $Z=52$, I has $Z=53$, Xe has $Z=54$. Ordered by $Z$ the sequence rises $52,53,54$ with no inversion — the "defect" vanishes. See the figure below.

Mendeleev's reply: The table is not a list I copy from measured masses — it is a law. Beryllium's chemistry (it forms $\text{BeO}$, valency 2, an alkaline-earth twin of Mg) places it firmly in Group II. A mass of $14$ would shove it into the wrong column, contradicting its chemistry. So the chemistry-fixed position predicts mass $\approx 9$. When later re-weighing gives $\approx 9$, the law made a successful prediction about the data itself — that is the strongest possible evidence a law can offer. The vindication: Moseley's work in 1913 showed the true ordering variable is atomic number, which places Be ($Z=4$) exactly where its chemistry said, confirming Mendeleev's property-first instinct.

• Döbereiner's Triads — grouped elements in threes using atomic mass (middle mass ≈ average of the outer two).
• Newlands' Law of Octaves — every 8th element repeats properties, again by atomic mass.
• Mendeleev — full table ordered by atomic mass, grouped by valency; predicted gaps.
• Modern Periodic Law — based on atomic number — Moseley (1913) reorders by atomic number $Z$, curing all mass inversions and the isotope problem. The through-line: the variable evolved mass → mass → mass → atomic number, each step keeping the "periodic" insight but sharpening the ruler.

Ordering variable — old vs new? ::: Old = atomic mass (Mendeleev); new = atomic number $Z$ (Moseley, 1913).