# Law of Multiple Proportions – Dalton’s Law

In chemistry, the law of multiple proportions states that when two elements form more than one compound, the ratio of the different masses of one element that combine with a fixed mass of the other element are a ratio of small whole numbers. Another name for the law of multiple proportions is Dalton’s law, as John Dalton was the first chemist who described the law. However, Dalton also formulated Dalton’s law of partial pressures, so the “law of multiple proportions” is the preferred name.

### Examples of the Law of Multiple Proportions

For example, Dalton observed that carbon forms two oxides by combing with oxygen in different proportions. For example, a 100-gram sample of carbon reacts with 133 grams of oxygen and forms one compound or with 266 grams of oxygen and forms the other compound. The ratio of oxygen masses that react with 100 grams of carbon is 266:133 = 2:1. From this data, Dalton predicted the chemical formulas for the two compounds are CO and CO2.

As another example, nitrogen reacts with oxygen, forming five different nitrogen oxides. The masses of oxygen that combine with 14 grams of nitrogen are 8, 16, 24, 32, and 40 grams. The ratio of oxygen masses is 1:2:3:4:5.

### Law of Multiple Proportions Problems

There are two main types of law of multiple proportions problems. The first type of problem tests your understanding of the the concept. The other has you find the small number ratio between elements that form multiple compounds with another element.

#### Problem #1

Which of the following illustrates the law of multiple proportions?

• Ordinary water and heavy water
• Sodium chloride and sodium bromide
• Sulfur dioxide and sulfur trioxide
• Caustic soda and caustic potash

The correct answer is that sulfur dioxide and sulfur trioxide illustrate the law. The reason is because this is one element (sulfur) that combines with a second element (oxygen) and forms more than one compound. Sodium chloride and sodium bromide as well as caustic soda and caustic potash are scenarios that involve two compounds, but these compounds don’t contain the same two elements. Ordinary water and heavy water are the same compound as each other, just using different hydrogen isotopes.

#### Problem #2

Carbon and oxygen form two compounds. The first compound is 42.9% carbon by mass and 57.1% oxygen by mass. The second compound is 27.3% carbon by mass and 72.7% oxygen by mass. Show that the ratios between the oxygen masses are consistent with the law of multiple proportions.

To solve this problem, show that the masses oxygen that combine with a fixed amount of carbon are a whole number ratio. Make life easy on yourself and assume you have 100 grams of each sample. Then, there are 57.1 grams of oxygen and 42.9 grams of carbon in the first sample. So, the mass of oxygen (O) per gram of carbon (C) is:

57.1 g O / 42.9 g C = 1.33 g O per g C

For the second compound, assuming a 100 grams sample, there are 72.7 grams of oxygen (O) and 27.3 grams of carbon (C). The mass of oxygen per gram of carbon is:

72.7 g O / 27.3 g C = 2.66 g O per g C

Setting up the problem this way makes the fixed amount of carbon equal to 1 gram. So, all you do is divide the mass of oxygen per grams of carbon for the two compounds:

2.66 / 1.33 = 2

In other words, the masses of oxygen that combine with carbon are in a 2:1 ratio. This small whole number ratio supports the law of multiple proportions.

Note it does not matter if you perform the calculation the other way (1.33 / 2.66 = 1 / 2 or a 1:2 ratio) because you still get a whole number ratio. Also, in actual experiments, you probably won’t get perfect data and may need to do a bit of rounding! For example, if your ratio comes out as 2.1 : 0.9, then you round the number to get a 2:1 ratio.

### Limitations of the Law of Multiple Proportions

The law of multiple proportions best applies to simple compounds.

It does not work well in all circumstances or even apply to all compounds. Specifically, it fails for non-stoichiometric compounds, oligomers, and polymers. It does not work well for larger molecules that contain hydrogen. Hydrogen has such a small mass that rounding errors often give erroneous ratios, plus the ratios between hydrogen masses are not always small whole numbers.

For example, carbon and hydrogen form the hydrocarbons decane (C10H22) and undecane (C11H24). For 100 grams of carbon, decane has 18.46 grams of hydrogen and undecane has 18.31 grams of hydrogen. The ratio of hydrogen masses between the two compounds is 121:120, which is not a small whole number ratio.

### History

The law of multiple proportions is important because it ties in with Dalton’s atomic theory. However, it is unclear whether Dalton observed the law of multiple proportions and then used it to formulate his atomic theory or whether the theory came first.

While Dalton first described the law, he was not the first chemist to observe it in action. In 1792, Bertrand Pelletier noted that a fixed amount of oxygen forms one type of tin oxide and twice that amount of oxygen (ratio of 1:2) forms a different oxide. Joseph Proust confirmed Pelletier’s observations and measured the relative amounts of tin and oxygen in the compounds. While Proust had the necessary information to discover the law, he did not generalize his findings.

### References

• Pelletier, Bertrand (1792). “Observations sur plusieurs propriétés du Muriate d’Étain” [Observations on various properties of muriate of tin]. Annales de Chimie (in French). 12: 225–240.
• Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). General Chemistry: Principles and Modern Applications (8th ed.). Upper Saddle River, N.J: Prentice Hall. ISBN 978-0-13-014329-7.
• Proust, Joseph Louis (1800). “Recherches sur l’étain” [Research on tin]. Journal de Physique, de Chimie, et d’Histoire Naturelle (in French). 51: 173–184.
• Roscoe, Henry E.; Harden, Arthur (1896). A New View of the Origin of Dalton’s Atomic Theory. Macmillan and Co.