Law of Multiple Proportions

The Law of Multiple Proportions, proposed by English chemist John Dalton in 1803, states that when two elements combine to form more than one compound, the masses of one element that combine with a fixed mass of the other element are in a ratio of small whole numbers. This law is one of the foundational Laws of Chemical Combination and provided crucial experimental evidence for Dalton's Atomic Theory, reinforcing the idea that matter is composed of discrete, indivisible atoms that combine in definite integer ratios. A classic illustration of this law involves the oxides of carbon. Carbon and oxygen together form two distinct compounds: carbon monoxide ( $\ce{CO}$ ) and carbon dioxide ( $\ce{CO2}$ ). If we fix the mass of carbon at 12 grams, then in carbon monoxide, 16 grams of oxygen combine with it, whereas in carbon dioxide, 32 grams of oxygen combine with it. The ratio of oxygen masses (16 : 32) simplifies to the small whole number ratio 1 : 2 — precisely what the Law of Multiple Proportions predicts. This elegant simplicity is a direct consequence of atoms combining in whole-number ratios at the molecular level. This law is distinct from the Law of Definite Proportions (which states that a given compound always contains the same elements in the same mass ratio) because it compares different compounds made from the same two elements. Understanding the Law of Multiple Proportions deepens a student's appreciation for the atomic nature of matter and provides a logical framework for interpreting molecular formulas, stoichiometry, and the existence of chemical isomers and homologous series.

Key takeaways

Identify Two Compounds Formed by the Same Two Elements
Fix the Mass of One Element
Determine the Mass of the Second Element in Each Compound
Compute the Ratios and Verify Whole Numbers
Connect to Dalton's Atomic Theory

Identify Two Compounds Formed by the Same Two Elements

Begin by selecting two compounds that share the same pair of elements. For example, nitrogen and oxygen form five different oxides: nitrous oxide ( $\ce{N2O}$ ), nitric oxide ( $\ce{NO}$ ), dinitrogen trioxide ( $\ce{N2O3}$ ), nitrogen dioxide ( $\ce{NO2}$ ), and dinitrogen pentoxide ( $\ce{N2O5}$ ). The existence of multiple stable compounds from the same two elements is the very phenomenon this law addresses.

Fix the Mass of One Element

Choose one element and fix its mass at a convenient reference value — typically the mass corresponding to one mole of that element or simply a round number like 14 g of nitrogen (one mole of N). This step ensures a fair comparison between the two compounds, because you are holding one variable constant while observing how the other changes.

Determine the Mass of the Second Element in Each Compound

Using the known molar masses and molecular formulas, calculate how many grams of oxygen combine with the fixed mass of nitrogen in each compound. For 14 g of nitrogen: in $\ce{N2O}$ , 8 g of oxygen combines; in $\ce{NO}$ , 16 g; in $\ce{N2O3}$ , 24 g; in $\ce{NO2}$ , 32 g; and in $\ce{N2O5}$ , 40 g. These values come directly from the atomic masses and the stoichiometric ratios embedded in each formula.

Compute the Ratios and Verify Whole Numbers

Divide all oxygen masses by the smallest value to obtain the simplest ratio. Using the nitrogen oxides: 8 : 16 : 24 : 32 : 40 = 1 : 2 : 3 : 4 : 5. This is a perfect sequence of small whole integers, confirming the law. If these ratios had been irrational numbers or large fractions, it would contradict the atomic model. The whole-number result arises because atoms are discrete units — you cannot have half an oxygen atom bonded to a nitrogen atom in a stable molecule.

Connect to Dalton's Atomic Theory

The Law of Multiple Proportions is elegantly explained by Dalton's postulate that atoms of a given element are identical and combine in simple integer ratios. If one N atom bonds to one O atom, you get $\ce{NO}$ ; if one N bonds to two O atoms, you get $\ce{NO2}$ . The ratio of oxygen in $\ce{NO}$ to $\ce{NO2}$ must therefore be 1 : 2. Any non-integer ratio would be physically impossible under the atomic model, making this law both a consequence and a confirmation of atomic theory.

The Key Equation

\frac{m_{\text{O in compound 2}}}{m_{\text{O in compound 1}}} = \frac{p}{q}, \quad p, q \in \mathbb{Z}^+

m_{\text{O in compound 2}} — Mass of oxygen in compound 2 (grams (g)): The mass of oxygen that combines with the fixed reference mass of the other element in the second compound being compared.
m_{\text{O in compound 1}} — Mass of oxygen in compound 1 (grams (g)): The mass of oxygen that combines with the fixed reference mass of the other element in the first compound being compared.
p — Numerator integer (dimensionless): A small positive integer. Values like 1, 2, 3, 4 are typical; very large integers would not conform to the spirit of the law.
q — Denominator integer (dimensionless): A small positive integer forming the denominator of the ratio. Together p/q must simplify to a simple whole-number fraction.

Worked Example

Problem. Sulfur and oxygen form two well-known compounds: sulfur dioxide ( $\ce{SO2}$ ) and sulfur trioxide ( $\ce{SO3}$ ). Show that these compounds obey the Law of Multiple Proportions by fixing the mass of sulfur at 32 g (one mole of sulfur, atomic mass = 32 g/mol) and comparing the masses of oxygen in each compound.

\frac{m_{\text{O in }\ce{SO3}}}{m_{\text{O in }\ce{SO2}}} = \frac{48\text{ g}}{32\text{ g}} = \frac{3}{2}

Answer. The ratio of oxygen masses combining with 32 g of sulfur is 3 : 2 (small whole numbers), confirming the Law of Multiple Proportions.

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Diagram

Origins · 1 minJohn Dalton and how this was discovered

John Dalton

1766–1844 · British (English)

John Dalton formulated the Law of Multiple Proportions in 1803 as part of developing his Atomic Theory, the first modern scientific model of atomic structure. His theory proposed that all matter consists of tiny, indivisible atoms; atoms of the same element are identical in mass; and compounds form when atoms combine in definite whole-number ratios. The Law of Multiple Proportions was both a prediction and a confirmation of this framework. Dalton's meticulous analysis of gas mixtures and compound compositions laid the quantitative groundwork for modern chemistry. Dalton was one of the first scientists to study and describe color blindness — a condition he himself had (inability to distinguish red from green). The condition is sometimes called 'Daltonism' in his honor.

Law of Definite Proportions — The Law of Definite Proportions (Proust's Law) states that a given compound always has the same elemental mass ratios — a prerequisite for understanding why Multiple Proportions arise when comparing different compounds of the same elements.
Law of Conservation of Mass — Dalton's Atomic Theory that underpins Multiple Proportions was also built to explain conservation of mass in chemical reactions; both laws together form the classical Laws of Chemical Combination.
Dalton's Atomic Theory — The Law of Multiple Proportions is a direct experimental consequence and confirmation of Dalton's postulates that atoms combine in simple whole-number ratios to form compounds.
Mole Concept and Molar Mass — Calculating the mass ratios required to verify Multiple Proportions relies on molar masses and the mole concept to convert between atomic ratios and measurable gram quantities.
Empirical and Molecular Formulas — Molecular formulas directly encode the whole-number atomic ratios that produce the whole-number mass ratios described by the Law of Multiple Proportions; understanding both concepts is mutually reinforcing.