The Law of Multiple Proportions

Introduction

Carbon and oxygen can combine to form two different gases. Carbon dioxide is the gas we breathe out; carbon monoxide is a poisonous gas formed when fuels burn in too little air. Both contain only carbon and oxygen, yet they are different compounds. John Dalton noticed a striking pattern in such pairs of compounds, and it became powerful evidence that matter is made of atoms. We call it the law of multiple proportions .

Core explanation

The law. When two elements combine to form more than one compound, the different masses of one element that combine with a fixed mass of the other element are in a simple whole-number ratio.

The carbon oxides. Chemical analysis gives these results:

Compound Mass of carbon Mass of oxygen --- --- --- Carbon monoxide 12 g 16 g Carbon dioxide 12 g 32 g

Fix the mass of carbon at 12 g. The masses of oxygen that combine with it are 16 g and 32 g. Their ratio is 16 : 32 = 1 : 2, a simple whole-number ratio.

Dalton's explanation. Atoms cannot be split in chemical reactions, so a carbon atom can combine with one whole oxygen atom or two whole oxygen atoms, but never one and a third. In carbon monoxide each carbon atom has one oxygen atom; in carbon dioxide each carbon atom has two. With the same number of carbon atoms, carbon dioxide contains exactly twice as many oxygen atoms, and so exactly twice the mass of oxygen. Whole atoms give whole-number ratios.

More examples.

- Water and hydrogen peroxide. For 2 g of hydrogen, water contains 16 g of oxygen and hydrogen peroxide contains 32 g. Ratio 1 : 2, matching H₂O and H₂O₂. - Oxides of nitrogen. For 14 g of nitrogen, dinitrogen oxide (N₂O) contains 8 g of oxygen, nitrogen monoxide (NO) 16 g and nitrogen dioxide (NO₂) 32 g. Ratio 1 : 2 : 4. - Oxides of copper. For 16 g of oxygen, copper(I) oxide (Cu₂O) contains about 127 g of copper and copper(II) oxide (CuO) about 63.5 g. Ratio 2 : 1.

Different from constant composition. The law of constant composition is about one compound: its mass ratio is fixed. The law of multiple proportions compares two or more compounds made from the same pair of elements. Dalton predicted the second law from his theory before it had been widely tested, which made the theory especially convincing.

Step-by-step reasoning

To test data against the law of multiple proportions:

1. Choose one element and fix its mass (use the same mass for every compound). 2. Scale each compound's data so that element has that fixed mass. 3. List the masses of the other element. 4. Divide each by the smallest and check whether the results are simple whole numbers.

Visual explanation

Draw one black ball for carbon. Next to it, draw a second black ball joined to one red ball (CO), and another black ball joined to two red balls (CO₂). Line up many copies of each: every CO₂ row carries exactly twice as many red balls as the CO row, so the red mass doubles too.

Real-world analogy

A café sells sandwiches with one slice of cheese or with two slices. It never sells one with one and a half slices, because slices are not cut. For a fixed number of sandwiches, the "double cheese" order always uses exactly twice as much cheese.

Real-world example

Carbon monoxide detectors are fitted in homes with gas boilers or fires. Incomplete burning of fuel, when the air supply is limited, can produce carbon monoxide instead of carbon dioxide. Knowing that these are two distinct compounds with different amounts of oxygen per carbon helps explain why a good air supply matters for safety.

Why?

Why are the ratios always simple, such as 1 : 2 or 2 : 3, rather than 1 : 1.37? Because the extra element is added in whole atoms. Any change in composition must be a jump of one or more complete atoms, so the masses change in whole-number steps.

Common misconception

"The law means the mass ratio in carbon dioxide is 1 : 2." No. Carbon dioxide's carbon : oxygen mass ratio is 3 : 8. The 1 : 2 ratio compares the oxygen masses in two different compounds for the same mass of carbon.

Worked example

Question: Sulfur forms two oxides. In oxide A, 32 g of sulfur combines with 32 g of oxygen. In oxide B, 16 g of sulfur combines with 24 g of oxygen. Show that the data fit the law of multiple proportions.

Reasoning: Fix sulfur at 32 g. Oxide A: 32 g oxygen. Oxide B: double the data, so 32 g sulfur with 48 g oxygen. Oxygen masses are 32 : 48 = 2 : 3.

Answer: The ratio 2 : 3 is a simple whole-number ratio, so the law is obeyed (the oxides are SO₂ and SO₃).

Quick check

1. For 12 g of carbon, carbon monoxide contains 16 g of oxygen. How much oxygen does carbon dioxide contain for 12 g of carbon? Answer: 32 g, twice as much.

Exam focus

Always fix the mass of one element before comparing, and use the same element for every compound. State the ratio in lowest whole numbers. Examiners often ask you to distinguish this law from the law of constant composition, so learn both definitions precisely.

Advanced insight

For large molecules, such as hydrocarbons with dozens of carbon atoms, the whole-number ratios can become large (for example, for a fixed mass of carbon, octane C₈H₁₈ and nonane C₉H₂₀ contain hydrogen in the ratio 81 : 80, which hardly looks "simple"). The law is still obeyed, but its usefulness as a test of atomic theory is greatest for small molecules, which is exactly what Dalton studied.

Summary

The law of multiple proportions states that when two elements form several compounds, the masses of one element combining with a fixed mass of the other are in a simple whole-number ratio. Carbon monoxide and carbon dioxide give 1 : 2. Dalton explained this by whole atoms combining, and the law became strong early evidence for atoms.

Practice questions

1. State the law of multiple proportions. Answer: 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 simple whole-number ratio. 2. For 2 g of hydrogen, water contains 16 g of oxygen and hydrogen peroxide contains 32 g. What is the ratio, and what does it suggest about the formulae? Answer: 16 : 32 = 1 : 2, suggesting hydrogen peroxide has twice as many oxygen atoms per hydrogen atom as water. 3. For 14 g of nitrogen, NO contains 16 g of oxygen and NO₂ contains 32 g. Find the ratio. Answer: 16 : 32 = 1 : 2. 4. How does this law differ from the law of constant composition? Answer: Constant composition describes the fixed mass ratio within one compound; multiple proportions compares the masses in two or more compounds of the same two elements.