This is a worked example chemistry problem using the Law of Multiple Proportions.

### Example Law of Multiple Proportions Problem

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 g of the first compound (100 is chosen to make calculations easier) there are 57.1 g O and 42.9 g C. The mass of O per gram C is:

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

In the 100 g of the second compound, there are 72.7 g O and 27.3 g 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

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

2.66 / 1.33 = 2

Which mean 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.

### Tips for Solving Law of Multiple Proportions Problems

- 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 you 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 101 chemistry 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)!