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## Definition of Law of Multiple Proportion

The law of multiple proportion, states that when two elements combine to form more than one compound, the mass of one element, which combines with a fixed mass of the other element, will always be ratios of whole numbers.

### Other Definitions

- Law of multiple proportions, statement that when two elements combine with each other to form more than one compound, the weights of one element that combine with a fixed weight of the other are in a ratio of small whole numbers. For example, there are five distinct oxides of nitrogen, and the weights of oxygen in combination with 14 grams of nitrogen are, in increasing order, 8, 16, 24, 32, and 40 grams, or in a ratio of 1, 2, 3, 4, 5.
- The law was announced (1803) by the English chemist John Dalton, and its confirmation for a wide range of compounds served asthe most powerful argument in support of Dalton’s theory that matter consists of indivisible atoms.

## History of Law of Multiple Proportion

The law of multiple proportions was a key proof of the atomic theory, but it is uncertain whether Dalton discovered the law of multiple proportions by accident and then used atomic theory to explain it, or whether his law was a hypothesis he proposed in order to investigate the validity of the atomic theory.

In 1792, Bertrand Pelletier discovered that a certain amount of tin would combine with a certain amount of oxygen to form one tin oxide or twice the amount of oxygen to form a different oxide.

Joseph Proust confirmed Pelletier’s discovery and provided measurements of the composition: one tin oxide is 87 parts tin and 13 parts oxygen, and the other is 78.4 parts tin and 21.6 parts oxygen. These were likely tin(II) oxide (SnO) and tin dioxide (SnO2), and their actual compositions are 88.1%, tin —11.9% oxygen, and 78.7% tin—21.3% oxygen.

Scholars who have reviewed the writings of Proust found that he had enough data to have discovered the law of multiple proportions himself, but somehow he did not.

With regards to the aforementioned tin oxides, had Proust adjusted his figures for a tin content of 100 parts for both oxides, he would have noticed that 100 parts of tin will combine with either 14.9 or 27.6 parts of oxygen. 14.9 and 27.6 form a ratio of 1:1.85, which is 1:2 if one forgives experimental error.

It seems this did not occur to Proust, but it occurred to Dalton.

## Limitations of Law of Multiple Proportion

- The law of multiple proportions is best demonstrated using simple compounds. For example, if one tried to demonstrate it using the hydrocarbons decane (chemical formula C10 H22) and undecane (C11 H24), one would find that 100 grams of carbon could react with 18.46 grams of hydrogen to produce decane or with 18.31 grams of hydrogen to produce undecane. For a ratio of hydrogen masses of 121:120, which is hardly a ratio of “small” whole numbers.
- This law fails with non-stoichiometric compounds and also doesn’t work well with polymers and oligomers.
- The law does not hold true if the element’s different isotopes are involved in making chemical compounds. The law is not applicable when elements combine in the same ratio, but different compounds are formed.

## Examples of Law of Multiple Proportion

Here are some examples of Law of Multiple Proportion

### Example-1

Let us assume 2 molecules CO (carbon monoxide) and CO_{2} (carbon dioxide).

CO= 12gm of carbon + 16 grams of Oxygen.

CO_{2}= 12gm carbon+ 32 grams of Oxygen.

The ratio of the mass of oxygen in the given two compounds is 16:32=1:2.

Thus law of multiple proportions is proved.

## Example-2

The masses of oxygen which combines with a fixed mass of carbon in CO_{2} and CO are 32 and 16 respectively.

These masses of oxygen bear a simple ratio of 32 : 16 or 2 : 1 to each other. **For example**, sulphur combines with oxygen to form two compounds, namely, sulphur trioxide and sulphur dioxide.

## Example-3

The masses of oxygen which combine with a fixed mass of sulphur in SO_{3} and SO_{2} are 48 and 32 respectively.

These masses of oxygen bear a simple ratio of 48 : 32 or 3 : 2 to each other. This law shows that there are constituents which combine in a definite proportion. These constituents may be atoms. Thus, the law of multiple proportions shows the existence of atoms which combine to form molecules.

## Example-4

Two different compounds are formed by the elements carbon and oxygen. The first compound contains 42.9% by mass carbon and 57.1% by mass oxygen. The second compound contains 27.3% by mass carbon and 72.7% by mass oxygen. Show that the data are consistent with the law of multiple proportions.

**Solution:** The law of multiple proportions is the third postulate of Dalton’s atomic theory. It states that the masses of one element which combine with a fixed mass of the second element are in a ratio of whole numbers.

Therefore, the masses of oxygen in the two compounds that combine with a fixed mass of carbon should be in a whole number ratio. In 100 grams of the first compound (100 is chosen to make calculations easier), there are 57.1 grams oxygen and 42.9 grams carbon. The mass of oxygen (O) per gram of carbon (C) is:

57.1 gO / 42.9 gC = 1.33 gO per gC

In the 100 grams of the second compound, 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 gO / 27.3 gC = 2.66 gO per gC

Dividing the mass O per gC of the second (larger value) compound:

2.66 / 1.33 = 2

This means that the masses of oxygen that combine with carbon are in a 2:1 ratio. The whole-number ratio is consistent with the law of multiple proportions.

## Example-5

While the ratio in this example problem worked out to be exactly 2:1. It’s more likely chemistry problems and real data will give you ratios that are close, but not whole numbers.

- If your ratio came out like 2.1:0.9, then you’d know to round to the nearest whole number and work from there. If you got a ratio more like 2.5:0.5, then you could be pretty certain you had the ratio wrong (or your experimental data was spectacularly bad, which happens too). While 2:1 or 3:2 ratios are most common, you could get 7:5, for example, or other unusual combinations. The law works the same way when you work with compounds containing more than two elements. To make the calculation simple, choose a 100-gram sample (so you’re dealing with percentages). And then divide the largest mass by the smallest mass. This isn’t critically important—you can work with any of the numbers. But it helps to establish a pattern for solving this type of problem.
- The ratio won’t always be obvious. It takes practice to recognize ratios.

In the real world, the law of multiple proportions doesn’t always hold. The bonds formed between atoms are more complex than what you learn about in a chemistry 101 class.

Sometimes whole number ratios don’t apply. In a classroom setting, you need to get whole numbers. But remember there may come a time when you’ll get a pesky 0.5 in there (and it will be correct).

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