Percentage by Mass of an Element

Introduction

Water is made of hydrogen and oxygen, but how much of the mass of a glass of water is actually hydrogen? Only about one-ninth. The rest is oxygen. The percentage by mass of an element tells you what fraction of a compound's mass comes from that element. It is used to compare fertilisers, judge metal ores, check the purity of medicines and confirm the identity of new compounds.

Core explanation

The idea. Every pure compound has a fixed formula, so it always contains the same elements in the same proportions by mass (the law of constant composition). This means we can calculate, from the formula alone, what percentage of the mass each element contributes.

The formula.

% by mass of element = (number of atoms of the element × Ar) ÷ Mr × 100

The top of the fraction is the total mass contributed by that element in one formula; the bottom is the mass of the whole formula.

Example: water, H₂O (H = 1, O = 16). Mr = 2 + 16 = 18. - % H = (2 × 1) ÷ 18 × 100 = 11.1% - % O = 16 ÷ 18 × 100 = 88.9%

Although there are twice as many hydrogen atoms as oxygen atoms, oxygen provides almost 89% of the mass, because each oxygen atom is sixteen times heavier than a hydrogen atom. Counting atoms and comparing masses are different things.

Example: calcium carbonate, CaCO₃ (Ca = 40, C = 12, O = 16). Mr = 40 + 12 + 48 = 100. - % Ca = 40%, % C = 12%, % O = 48%.

The percentages must add up to 100%. This is a useful check. If they add up to 99.9% or 100.1%, the difference is only rounding.

Some useful results:

Compound Mr Element % by mass --- --- --- --- CO₂ 44 C 27.3 NH₃ 17 N 82.4 MgO 40 Mg 60.0 Fe₂O₃ 160 Fe 70.0 NaCl 58.5 Na 39.3

Formulae

% by mass of element = (n × Ar) ÷ Mr × 100, where n is the number of atoms of that element in the formula.

Step-by-step reasoning

To find the percentage of iron in iron(III) oxide, Fe₂O₃ (Fe = 56, O = 16):

1. Calculate Mr: (2 × 56) + (3 × 16) = 112 + 48 = 160. 2. Find the mass due to iron: 2 × 56 = 112. 3. Divide by Mr: 112 ÷ 160 = 0.70. 4. Multiply by 100: 70.0% iron. 5. Check: oxygen is 48 ÷ 160 × 100 = 30.0%, and 70.0 + 30.0 = 100.

Visual explanation

Draw a bar representing the full mass of one formula of water, 18 units long. Shade 2 units for the two hydrogen atoms and 16 units for the oxygen atom. The shading shows at a glance that oxygen fills almost nine-tenths of the bar, even though hydrogen atoms outnumber it.

Real-world analogy

A fruit salad contains ten grapes and one melon slice. Grapes win by number, but the melon slice may make up most of the mass. Percentage by mass asks "how much of the weight is melon?", not "how many pieces are melon?".

Real-world example

Farmers compare nitrogen fertilisers by their nitrogen content. Ammonium nitrate, NH₄NO₃ (Mr = 80), is 35% nitrogen, while urea, CO(NH₂)₂ (Mr = 60), is about 46.7% nitrogen. A tonne of urea therefore delivers more nitrogen to a field than a tonne of ammonium nitrate.

Why?

Why can a percentage be calculated from a formula without weighing anything? Because every formula unit of a pure compound is identical. Whatever fraction of the mass iron makes up in one Fe₂O₃ unit, it makes up the same fraction in a billion units or in a tonne of the pure oxide.

Common misconception

"In H₂O, hydrogen is two-thirds of the compound because two of the three atoms are hydrogen." That is true for the number of atoms, not for mass. Percentage by mass must use the relative atomic masses, which gives only 11.1% hydrogen.

Worked example

Question: Calculate the percentage by mass of nitrogen in ammonium nitrate, NH₄NO₃ (N = 14, H = 1, O = 16).

Reasoning: Count atoms: 2 N, 4 H, 3 O. Mr = (2 × 14) + (4 × 1) + (3 × 16) = 28 + 4 + 48 = 80. Mass due to nitrogen = 28. Percentage = 28 ÷ 80 × 100 = 35%. Remember that the nitrogen appears in two places in the formula.

Answer: 35% nitrogen.

Quick check

1. What is the percentage by mass of magnesium in magnesium oxide, MgO (Mg = 24, O = 16)? Answer: 24 ÷ 40 × 100 = 60%.

Exam focus

Write the formula first, then show Mr, then the fraction. The commonest lost marks come from forgetting to multiply Ar by the number of atoms (for example using 56 instead of 112 for iron in Fe₂O₃) and from missing an element that appears twice in the formula.

Advanced insight

The calculation can be run backwards. If analysis shows a compound is 75% carbon and 25% hydrogen by mass, dividing each percentage by the Ar gives the ratio of atoms: 75 ÷ 12 = 6.25 and 25 ÷ 1 = 25, a ratio of 1 : 4, so the simplest formula is CH₄. This is how the formulae of newly made compounds are found.

Summary

The percentage by mass of an element equals the number of its atoms multiplied by Ar, divided by Mr, times 100. The percentages of all elements add up to 100%. Mass percentages differ from atom counts because atoms have different masses. The method is used to compare ores, fertilisers and other sources of useful elements.

Practice questions

1. Calculate the percentage by mass of carbon in carbon dioxide, CO₂ (C = 12, O = 16). Answer: 12 ÷ 44 × 100 = 27.3%. 2. Calculate the percentage by mass of nitrogen in ammonia, NH₃ (N = 14, H = 1). Answer: 14 ÷ 17 × 100 = 82.4%. 3. Calculate the percentage by mass of copper in copper(II) oxide, CuO (Cu = 63.5, O = 16). Answer: 63.5 ÷ 79.5 × 100 = 79.9%. 4. Explain why hydrogen is only 11.1% of the mass of water although two of every three atoms are hydrogen. Answer: Each hydrogen atom has a relative mass of 1 while the oxygen atom has 16, so the two hydrogens contribute only 2 of the 18 mass units. 5. Which ore contains the greater percentage of iron: Fe₂O₃ (70.0%) or Fe₃O₄ (Mr = 232)? Show your working. Answer: Fe₃O₄: 168 ÷ 232 × 100 = 72.4%, which is greater than 70.0%.