Empirical formulas

An empirical formula is a chemical formula that gives the smallest whole number ratio of atoms in a compound.

As an example, let’s take the molecule ethene, which has two carbons and four hydrogens, so the molecular formula is C2H4.

But notice that both subscripts are divisible by two.

So to figure out the empirical formula for ethene, we have to find the smallest whole-number ratio that describes the relative number of different atoms.

So we divide both the two and the four by two, to get the empirical formula CH2. If we want to get the molecular formula from the empirica

l formula, we need to do the opposite of this: we would multiply all of the subscripts by two.

This is because, for ethene, the empirical formula is different than the molecular formula.

For an unknown compound, we would need to determine the common factor of the subscripts (two in the case of ethene) using an experimental technique such as mass spectrometry.

Sometimes the empirical formula and the molecular formula are the same.

A common example is water, which is H2O.

That means that there are two hydrogens for every one oxygen.

Now, let’s suppose we have a mystery compound that is 84.1% carbon and 15.9% hydrogen and we want to figure out the empirical formula for our mystery compound.

A useful trick is to assume that we start with exactly 100 grams of the mystery compound.

That means we have 84.1 grams of carbon and 15.9 grams of hydrogen.

Our first step is to convert the masses into moles.

We can look up the molar mass for carbon and hydrogen on the Periodic Table.

The molar mass of carbon is 12.0 grams per mole and the molar mass of hydrogen is 1.01 grams per mole.

We take 84.1 grams of carbon and divide it by 12.0 grams per mole.

The grams cancel out of the division, and after we divide the numbers we are left with around 7 moles of carbon.

We next take 15.9 grams of hydrogen, and divide this number by by 1.01 grams per mole.

This tells us that we have 15.7 moles of hydrogen.

We now have the relative number of moles of carbon and hydrogen in our unknown sample.

We next try to divide all of of our numbers by the smaller number, in order to figure out the ratio of these two elements in our sample.

So we divide 7 moles of carbon by 7 moles to get 1.0, and we divide 15.7 moles of hydrogen by 7 moles to get 2.24.

At this point, we might be tempted to write that the empirical formula of our unknown compound is C1H2.24.

But the empirical formula has to have whole numbers as the subscripts.

So we need to figure out the smallest whole number to multiply all of our subscripts by, so that all of them are whole numbers.

2.24 times two is 4.48, which is not a whole number.

2.24 times three is 6.72, which is still not very close to a whole number.

But 2.24 times four is 8.96, which is extremely close to a whole number - 9.

So we multiply both of our subscripts by four, in order to produce the empirical formula C4H9, which is the smallest whole-number ratio of different elements in our sample.

Now let’s say we use a mass spectrometer which tells us the molar mass of the mystery compound is 114 grams per mole.

Next, we need to use the empirical formula to find the molecular formula.

We start by figuring out the total molar mass of our empirical formula.

There are four carbons, each with a mass of 12 grams per mole, and 9 hydrogens, each with a mass of 1 gram per mole.

So the molar mass of our empirical formula is 4 times 12 grams per mole, plus 9 times 1 grams, which is 57 grams per mole.

If we take the molar mass reported by the mass spectrometer and divide it by the total molar mass of our empirical formula, that’s 114 grams per mole divided by 57 grams per mole, which equals two.

This means that to get the molecular formula, we should multiply every subscript by two, which gives us the molecular formula C8H18.