Percent Composition of an Element

A chemical formula is a recipe by count, and this equation translates it into a recipe by mass. Molar mass is additive — the compound weighs exactly what its atoms weigh, summed — so an element's share of the total is its atoms' contribution divided by the whole. In water, hydrogen brings 2 × 1.008 = 2.016 g/mol out of 18.02, which is 11.2% by mass; oxygen carries the other 88.8%. Two atoms out of three, and barely a ninth of the weight.

A case with real stakes. Ammonium sulfate, (NH₄)₂SO₄, has a molar mass of 132.14 g/mol. Nitrogen appears twice, so $\%N = (2 \times 14.007)/132.14 \times 100 = 21.2\%$. That is not a coincidence of the classroom — it is why the fertilizer is sold as 21-0-0, and why a 25 kg bag delivers 5.3 kg of actual nitrogen to a field.

Run the arithmetic backwards and it becomes analysis rather than description. Burn an unknown compound, weigh the CO₂ and H₂O produced, convert those to masses of carbon and hydrogen, take oxygen by difference, then divide each element's mass percent by its atomic mass and normalise to the smallest result. What comes out is the empirical formula. This is combustion analysis, and it was the central technique of organic chemistry from Liebig's refinement of it in the 1830s until spectroscopy arrived. Behind all of it sits Proust's law of definite proportions: a given compound always holds the same elements in the same mass ratio, which is exactly the claim this equation encodes.

The subscript $a$ is the number of atoms of that element per formula unit, and parentheses are where people lose it. In (NH₄)₂SO₄ the hydrogen count is 8, not 4 — the subscript outside the bracket multiplies everything inside. Hydrates catch people the same way: copper(II) sulfate pentahydrate, CuSO₄·5H₂O, has a molar mass of 249.7 g/mol including its water, so the copper is 25.4%, not the 39.8% you would get from the anhydrous salt. If the bottle says pentahydrate, the water is part of what you weighed.

Two further traps. Percent composition can only ever give you an empirical formula, never a molecular one: formaldehyde CH₂O, acetic acid C₂H₄O₂ and glucose C₆H₁₂O₆ all analyse to identical percentages, and separating them needs an independent molar mass. And on fertilizer bags, only the first number is what it appears to be. The N in N-P-K is genuinely percent nitrogen, but P is reported as P₂O₅ equivalent and K as K₂O — a nineteenth-century convention that never died. A bag marked 0-46-0 is 46% P₂O₅, which works out to about 20% actual phosphorus. Reading it as elemental phosphorus overstates the dose by well over a factor of two.