Law of Multiple Proportions

Short answer

• The law of multiple proportions states that when two elements react to form more than one compound, a fixed mass of one element will react with masses of the other element in a ratio of small, whole numbers. That sentence is the whole law, and the rest of this article explains how to use it. Take carbon and oxygen, the classic example. Carbon monoxide contains 16 grams of oxygen for every 12 grams of carbon. Carbon dioxide contains 32 grams of oxygen for every 12 grams of carbon. Carbon is fixed at 12 grams in both cases, so the oxygen masses can be compared directly: 16 grams against 32 grams. Their ratio is 1 to 2, a ratio of small, whole numbers. That is the law in action. The law is also known as Dalton's law, after the English chemist John Dalton, who formulated it as part of his atomic theory in the early 1800s. It matters because it is evidence for atoms: if elements combined in any smooth, continuous way, there would be no reason for masses to jump in whole number steps, as particles combining one at a time would produce.

The law, stated precisely

OpenStax states the law in its Chemistry 2e textbook in these words: when two elements react to form more than one compound, a fixed mass of one element will react with masses of the other element in a ratio of small, whole numbers. Four conditions must all hold before the law can be applied, and checking them is the fastest way to avoid mistakes.

First, the two compounds must be made from the same two elements. Carbon monoxide and carbon dioxide qualify, because both contain only carbon and oxygen. Carbon monoxide and water do not qualify as a pair for this law, because water contains hydrogen and oxygen, not carbon and oxygen.

Second, there must be more than one compound from that pair. A single compound cannot show multiple proportions, because there is nothing to compare it with. That single compound is covered by a different law, the law of definite proportions, explained later in this article.

Third, one element’s mass must be held fixed across the comparison. In the carbon example, carbon is fixed at 12 grams. You could equally fix oxygen instead and compare the carbon masses, and the law works either way. What you cannot do is compare two samples in which both masses change and then look for a whole number ratio in the raw numbers.

Fourth, the resulting ratio must be a ratio of small, whole numbers, such as 1 to 2, 2 to 3, or 3 to 4. A ratio of 121 to 120 is a ratio of whole numbers, but it is not a small whole number ratio. Real laboratory data give values close to clean ratios, with small differences caused by measurement error.

Reading the law word by word

The phrase “fixed mass” causes more errors than any other part of the law. Fixed does not mean 12 grams, or 1 gram, or any particular number. It means that whatever mass you choose for the first element, you use the same mass in both compounds when you make the comparison. If one sample contains 12 grams of carbon and another contains 24 grams of carbon, you cannot compare their oxygen masses directly. You must first scale both samples to the same carbon mass, by dividing each oxygen mass by its carbon mass. That scaling step appears in every worked example below.

The law is about masses, not about numbers of atoms and not about volumes. You compare grams against grams. The connection to atoms comes afterward, as an explanation: if each atom of the second element has the same mass, then doubling the number of atoms doubles the mass.

The phrase “small, whole numbers” also needs care. Whole numbers here means counting numbers such as 1, 2, 3, and 4, not fractions and not decimals. In real data the ratio is rarely exactly 2.000, because balances have limits and samples are rarely perfectly pure. A measured ratio of 1.99 to 1 still supports the law, and a ratio of 1.5 to 1 is also a small whole number ratio, written as 3 to 2.

Why the fixed mass step cannot be skipped

Consider two samples. Sample A contains 4.27 grams of carbon and 5.69 grams of oxygen. Sample B contains 5.19 grams of carbon and 13.84 grams of oxygen. These are the numbers OpenStax uses in a worked problem. Neither sample has the same carbon mass, so a direct comparison of 5.69 grams against 13.84 grams tells you nothing.

Divide each oxygen mass by its carbon mass. For sample A, 5.69 divided by 4.27 gives 1.33 grams of oxygen per gram of carbon. For sample B, 13.84 divided by 5.19 gives 2.67 grams of oxygen per gram of carbon. Now carbon is effectively fixed at 1 gram in both cases. Divide the larger value by the smaller value: 2.67 divided by 1.33 gives 2. The ratio is 2 to 1, so the data support the law of multiple proportions, and the two samples are different compounds. A pair that fits this relationship is carbon monoxide for sample A and carbon dioxide for sample B, since carbon monoxide has half as much oxygen per amount of carbon as carbon dioxide.

This three step pattern, divide, divide again, then check for a small whole number ratio, is the standard method for every problem of this type.

Worked example 1: the carbon oxides

This is the example to learn first, because it appears in almost every textbook and it uses round numbers. It is labelled here as an example, worked from the formulas of the two compounds and the masses given in standard textbook treatments.

Carbon monoxide has the formula CO: one carbon atom combines with one oxygen atom. Carbon dioxide has the formula CO2: one carbon atom combines with two oxygen atoms. Because each oxygen atom has the same mass, the oxygen mass in carbon dioxide must be twice the oxygen mass in carbon monoxide when the carbon mass is the same.

Fix carbon at 12 grams. In carbon monoxide, 12 grams of carbon combine with 16 grams of oxygen. In carbon dioxide, 12 grams of carbon combine with 32 grams of oxygen. The ratio of the oxygen masses is 32 to 16, which simplifies to 2 to 1. Expressed per gram of carbon, carbon monoxide gives 1.33 grams of oxygen per gram of carbon, and carbon dioxide gives 2.67 grams of oxygen per gram of carbon, and 2.67 divided by 1.33 is 2.

The same result follows if oxygen is fixed instead. Fix oxygen at 16 grams: carbon monoxide then contains 12 grams of carbon, and carbon dioxide contains 6 grams of carbon, a ratio of 2 to 1. The law works whichever element is fixed, which is a useful check on arithmetic.

Worked example 2: the copper chlorides

OpenStax gives a second example that is valuable precisely because its numbers are not round. Copper and chlorine form a green, crystalline solid with a mass ratio of 0.558 grams of chlorine to 1 gram of copper, and a brown, crystalline solid with a mass ratio of 1.116 grams of chlorine to 1 gram of copper. Copper is already fixed at 1 gram in both ratios, so the comparison can be made at once.

Divide the larger chlorine mass by the smaller chlorine mass: 1.116 divided by 0.558 equals 2. The ratio is 2 to 1. The brown compound therefore has twice as much chlorine per amount of copper as the green compound.

The atomic explanation, as OpenStax presents it, is that the copper to chlorine atom ratio in the brown compound is 1 to 2, while in the green compound it is 1 to 1. Doubling the number of chlorine atoms per copper atom doubles the chlorine mass per copper mass. The law does not require pretty starting numbers. The starting ratios, 0.558 and 1.116, look unpromising, but their ratio is clean. The law is a statement about the ratio of ratios, not about the tidiness of any single measurement.

Worked example 3: starting from percentages

Percentage composition data can also be tested against the law. The first compound in this standard textbook problem contains 42.9 percent carbon and 57.1 percent oxygen by mass. The second compound contains 27.3 percent carbon and 72.7 percent oxygen by mass.

Work with a 100 gram sample of each compound, so the percentages become grams directly: 42.9 grams of carbon with 57.1 grams of oxygen, and 27.3 grams of carbon with 72.7 grams of oxygen. Carbon is not fixed yet, so divide each oxygen mass by its carbon mass: 57.1 divided by 42.9 gives 1.33 grams of oxygen per gram of carbon, and 72.7 divided by 27.3 gives 2.66 grams of oxygen per gram of carbon. Divide the larger by the smaller: 2.66 divided by 1.33 equals 2. The data are consistent with the law, and the two compounds are the carbon oxides already met in example 1.

Two practical points follow. Percentages are just masses in disguise, because a percentage is a mass per 100 grams. Rounding in the given percentages also produces small wobbles in the final ratio: 1.98 or 2.02 would still support the law, while a result far from any simple ratio, such as 1.37, would argue against it.

Worked example 4: reading ratios straight from formulas

Once the law is understood, formulas alone can predict the ratio, and this is a fast check on laboratory data. These are labelled examples, worked from the formulas, not new laboratory measurements.

Nitrogen monoxide has the formula NO, with one oxygen atom per nitrogen atom. Nitrogen dioxide has the formula NO2, with two oxygen atoms per nitrogen atom. Fix nitrogen. The oxygen mass per fixed nitrogen mass must therefore double from the first compound to the second, giving a ratio of 2 to 1. No atomic weights are needed for this conclusion, because the nitrogen mass cancels out.

Water has the formula H2O, with one oxygen atom per two hydrogen atoms. Hydrogen peroxide has the formula H2O2, with two oxygen atoms per two hydrogen atoms. Fix hydrogen, comparing equal numbers of hydrogen atoms, and the oxygen masses are again in the ratio 1 to 2.

Methane has the formula CH4, with four hydrogen atoms per carbon atom. Ethane has the formula C2H6, with six hydrogen atoms per two carbon atoms, which is three hydrogen atoms per carbon atom. Fix carbon, and the hydrogen masses per fixed carbon mass are in the ratio 4 to 3. The ratio need not be 2 to 1.

A step by step method for any problem

Step 1: Confirm that the two compounds contain the same two elements and nothing else. If a third element is present, the law in its simple form does not apply to that pair.

Step 2: Decide which element to fix, usually the one whose mass is already the same in both samples, or the one given as 1 gram.

Step 3: For each compound, divide the mass of the other element by the mass of the fixed element. If the data are percentages, first treat them as grams in a 100 gram sample.

Step 4: Divide the larger result by the smaller result. This ratio of ratios is the number to judge.

Step 5: Ask whether the result is close to a small whole number ratio. Values such as 2.00, 1.50, 1.33, and 1.25 correspond to 2 to 1, 3 to 2, 4 to 3, and 5 to 4. If it is far from every simple ratio, the data do not support the law for that pair, or an error has been made.

Step 6: State the conclusion in words, naming the two compounds and the ratio found. A bare number alone does not show that the fixed mass condition was respected.

Practice problem with answer

Problem: A sample of compound X contains 14.13 grams of carbon and 2.96 grams of hydrogen. A sample of compound Y contains 19.91 grams of carbon and 3.34 grams of hydrogen. Do these data support the law of multiple proportions? This problem is adapted from an OpenStax check your learning question, with the same numbers.

Solution: Fix hydrogen. For compound X, carbon per hydrogen is 14.13 divided by 2.96, which is 4.77 grams of carbon per gram of hydrogen. For compound Y, carbon per hydrogen is 19.91 divided by 3.34, which is 5.96 grams of carbon per gram of hydrogen. Divide the larger by the smaller: 5.96 divided by 4.77 equals 1.25, which is 5 to 4 as a small whole number ratio. The data therefore support the law of multiple proportions, and X and Y are different compounds of carbon and hydrogen.

The most instructive mistake here is the reverse case: raw oxygen masses of 16 grams and 32 grams, but carbon masses of 12 grams and 24 grams. Carbon is not fixed there, and per gram of carbon both samples give 1.33 grams of oxygen, a ratio of 1 to 1. They are the same compound in different amounts. Always check the fixed mass before trusting a tempting ratio.

Multiple proportions, definite proportions, and conservation of mass

Three laws of chemical combination are often taught together, and keeping their questions separate prevents most confusion.

The law of definite proportions, also called the law of constant composition, asks about one compound. It states that all samples of a pure compound contain the same elements in the same proportion by mass. OpenStax credits this result to the French chemist Joseph Proust. Water is the standard illustration, with hydrogen and oxygen in the same mass ratio, about 1 to 8, in any pure sample. If two samples of what is claimed to be the same compound give different proportions, at least one sample is impure or is a different compound.

The law of multiple proportions asks about two or more compounds of the same pair of elements. It compares across compounds, while definite proportions stays within one compound.

The law of conservation of mass asks about a chemical reaction as a whole. It states that mass is neither created nor destroyed in a chemical reaction in a closed system, so the total mass of the reactants equals the total mass of the products. It does not compare compounds with each other, and it does not predict ratios.

A fourth law, the law of reciprocal proportions, concerns two elements that each combine separately with a fixed mass of a third element. It is not needed for standard multiple proportions problems, and it is mentioned here only so that readers who meet the name elsewhere are not confused by it.

How the law pointed to atoms

Dalton’s atomic theory, as OpenStax summarizes it, rests on postulates that explain both definite and multiple proportions in one picture. Matter is composed of exceedingly small particles called atoms. An element consists of only one type of atom, with a mass characteristic of that element. A compound consists of atoms of two or more elements combined in a small, whole number ratio. Atoms are neither created nor destroyed during a chemical change, but are rearranged.

Given those postulates, multiple proportions follows necessarily. Suppose one compound contains one B atom per A atom, and another contains two B atoms per A atom. The B mass per fixed A mass then doubles, giving a ratio of 2 to 1, whatever the atomic masses are, because they cancel in the ratio. Change the atom counts to 2 and 3, and the ratio becomes 3 to 2.

This is why the law was such strong evidence for atoms in Dalton’s time. A continuous view of matter, in which elements could mix in any proportion like paints, would predict smooth variation, not steps. The steps were visible in the balance readings, and atoms explained them. The Royal Society of Chemistry’s account of Dalton makes the same point: his rules essentially lead to the law of multiple proportions, also known as Dalton’s law.

Dalton developed these ideas in the early 1800s, and his atomic theory was published in the first part of his book A New System of Chemical Philosophy in 1808. Secondary sources differ on the year of his first presentation, with some giving 1803 or 1804 and others giving 1807, so this article dates the book rather than claiming a single year for the first formulation. The attribution itself is not in doubt: the law is Dalton’s.

Dalton’s own version of atomic theory was not correct in every detail. Because only one compound of hydrogen and oxygen was known to him, his rules led him to formulate water as HO, rather than H2O. The Royal Society of Chemistry notes this error: his table of atomic weights still implied water as HO years later, even after H2O had been established by others. His atomic weights were also revised substantially by later chemists. None of that overturns the law, which is a statement about measured mass ratios, and later corrections left those ratios standing.

Where the law is used today

The law survives in modern chemistry less as a law to be quoted and more as a working habit. Stoichiometry, the calculation of reactants and products in chemical reactions, depends on the same logic: formulas fix atom ratios, atom ratios fix mass ratios, and mass ratios can be checked against the balance. When an analyst determines the formula of an unknown compound, the procedure follows Dalton’s method: measure the masses of each element, convert to moles, divide by the smallest value, and look for small whole numbers.

The law also underlies quality control in chemical analysis. If a product is supposed to be a single compound, its composition must match the definite proportions for that compound in every batch. A batch falling far from any expected ratio signals contamination, an incomplete reaction, or a misidentified product. In teaching, the law remains the clearest bridge between balances and samples and the invisible world of atoms.

Limits and modern nuances

The law in its classic form assumes that a compound has a fixed formula with whole numbers of atoms. Some solid compounds, called nonstoichiometric compounds, do not behave that way. Their composition can vary within a range because of defects in their crystal structure, so the ratio of their elements is not fixed at one simple value. Iron oxides provide the commonly cited case.

Second, Dalton’s postulate that all atoms of an element have the same mass is not exactly true in modern terms, because isotopes exist. Isotopes are atoms of the same element with different numbers of neutrons and therefore different masses. A compound made with a different mix of isotopes will have a slightly different mass ratio, even though its atom ratio is unchanged. For most ordinary work this effect is tiny, because natural samples contain the same mix of isotopes.

Third, the word small deserves emphasis with very large molecules. Two large molecules that differ by one atom can have compositions so close together that their mass ratio is a fraction with large numbers, such as 121 to 120, rather than a small whole number ratio. The law is therefore most informative for small molecules and simple compounds.

Fourth, the law applies to compounds, not to mixtures. A mixture of two compounds of the same elements can have any overall composition between the two. Testing a mixture against the law will give meaningless results, which is why purity matters in every example above.

Common mistakes and how to avoid them

1. Comparing raw masses without fixing one element. Always divide by the fixed element’s mass first.
2. Mixing up multiple proportions with definite proportions. One compound only means a definite proportions question.
3. Using volumes where masses are required. The law as stated is about masses, so convert to masses before applying it.
4. Expecting exact whole numbers from real data. Laboratory data carry error, so 1.99 or 2.02 should be read as 2.
5. Forgetting that the ratio can be inverted. A ratio of 2 to 1 from one direction is 1 to 2 from the other, so state which direction is meant.
6. Treating a mixture as a compound. If the composition varies without any pattern, suspect a mixture or an impure sample before doubting the law.

Summary

The law of multiple proportions says that when two elements form more than one compound, the masses of one element that combine with a fixed mass of the other are in a ratio of small, whole numbers. Carbon monoxide and carbon dioxide give 1 to 2, and the copper chlorides in OpenStax give 2 to 1. Percentage data and formula data give the same kinds of ratios when the fixed mass step is respected. Dalton formulated the law as part of his atomic theory, and it remains the clearest early evidence that matter combines in whole atoms rather than in smooth, continuous amounts. Its limits, nonstoichiometric solids, isotopic variation, and very large molecules, mark the edges of its classic form without removing its value for ordinary compounds and stoichiometry.

Frequently Asked Questions

What is the law of multiple proportions in simple words?

When the same two elements make two different compounds, and you keep the amount of one element the same in both, the amounts of the other element compare as small whole numbers, like 1 to 2. Carbon monoxide and carbon dioxide are the standard example.

Who discovered the law of multiple proportions?

John Dalton, the English chemist, formulated the law as part of his atomic theory in the early 1800s. It is therefore also called Dalton’s law, published in the first part of A New System of Chemical Philosophy in 1808.

What is the formula for the law of multiple proportions?

There is no single formula to memorize. For each compound, divide the mass of the second element by the mass of the first element, with the first element’s mass fixed. Then divide the larger result by the smaller result. If that final number is close to a small whole number ratio, the data follow the law.

How is the law of multiple proportions different from the law of definite proportions?

Definite proportions concerns one compound: every pure sample has the same elemental proportions. Multiple proportions concerns two or more compounds of the same pair of elements, where the proportions step between compounds in small whole number ratios.

What is an example of the law of multiple proportions?

Carbon monoxide (CO) and carbon dioxide (CO2): with 12 grams of carbon fixed, they contain 16 grams and 32 grams of oxygen, a ratio of 1 to 2. The copper chlorides in OpenStax give another example: 0.558 grams and 1.116 grams of chlorine per gram of copper, also 1 to 2.

Does the law of multiple proportions work for all compounds?

It works for ordinary compounds with fixed formulas. It does not describe nonstoichiometric solids whose composition varies, and its “small” wording loses force for very large molecules. Isotopic variation can also shift mass ratios slightly, although atom ratios remain whole.

How do you solve law of multiple proportions problems?

Confirm the two compounds share the same two elements, fix one element’s mass, divide each second element mass by its fixed element mass, divide the larger result by the smaller result, and check whether the answer is close to a small whole number ratio.

Why is the law of multiple proportions important?

It provided the first quantitative evidence for atoms. If matter combined continuously, mass ratios would vary smoothly. Their whole number jumps are exactly what atoms predict, and that connection still underlies stoichiometry and formula determination today.

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Sources & references

• OpenStax, Chemistry 2e, Section 2.1 Early Ideas in Atomic Theory: statement of the law of multiple proportions, the copper and chlorine example (0.558 g and 1.116 g chlorine per 1 g copper, ratio 2 to 1), the carbon and oxygen worked problem (4.27 g carbon with 5.69 g oxygen; 5.19 g carbon with 13.84 g oxygen), the carbon and hydrogen check problem (14.13 g carbon with 2.96 g hydrogen; 19.91 g carbon with 3.34 g hydrogen, ratio 5 to 4), Dalton's postulates, and Proust's law of definite proportions. Page opened and read 27 Oct 2026.
• LibreTexts, Introductory Chemistry (CK-12), Section 4.03 Law of Multiple Proportions: carbon monoxide and carbon dioxide masses (16.0 g and 32.0 g oxygen per 12.0 g carbon) and the percentage composition problem (42.9 percent carbon with 57.1 percent oxygen; 27.3 percent carbon with 72.7 percent oxygen). Search result excerpt read 27 Oct 2026; page not opened directly in this pass, so QA should confirm these figures on the page before approval.
• Royal Society of Chemistry, Dalton Transactions, "John Dalton: the man and the myth": the law of multiple proportions as Dalton's law, and Dalton's formulation of water as HO with the consequences for his atomic weights. Search result excerpt read 27 Oct 2026; QA should confirm on the article page before approval.
• Search results for the publication history of A New System of Chemical Philosophy (first part 1808) were used only for the book date; secondary sources differ on the year of Dalton's first presentation, so the article dates the book and does not claim a single year for first formulation. QA should confirm the 1808 book date against an institutional source before approval.