HomeenUsing Avogadro's number to calculate the number of atoms and molecules

# Using Avogadro’s number to calculate the number of atoms and molecules

Avogadro’s number, or Avogadro’s constant (N A ), represents the number of carbon atoms in exactly 12 grams of a completely pure sample of the carbon-12 isotope . At the same time, it represents the number of units contained in 1 mol of any substance and has a value of 6,022 .10 23 mol -1 .

In short, understanding Avogadro’s number and knowing how to use it to carry out calculations in chemistry is the most direct way to understand the concept of the mole, which is central to this branch of science. That is why, in this article, we will show, step by step, how to solve two typical chemistry problems that involve the use of Avogadro’s number.

We’ll start with a simple problem to explain the necessary bases, and then move on to a more complex problem that involves several separate calculations.

## problem 1

### statement

Determine the number of water molecules in a drop of this liquid, knowing that it weighs 0.500 g. Data: PA H = 1 amu, PA O = 16 amu.

### Solution

As always when we are going to solve any problem, we must begin by analyzing the statement and extracting the relevant data. In this case, we only have as information the fact that it is water, the mass of the drop and the atomic weights of hydrogen and oxygen.

m water = 0.500g

The molecular formula of water is H 2 O, so its molecular weight is:

The unknown is the number of water molecules, which is represented by the capital letter N. In this way it differs from the number of moles that is represented by the lowercase n . That is to say:

N water = ?

To solve this problem, as well as most problems that involve Avogadro’s constant, the relationship between the number of particles and the number of moles is used, which is the following:

In this particular case, we are interested in finding N, so we need to rearrange this equation. In addition, it is always advisable to identify both the numbers of moles that we are calculating and the numbers of particles with the particular substance, atom or ion in question, to avoid confusion when calculating moles or numbers of particles of several substances in the same problem (which we will do in the next problem).

So, the formula that we will use to find the number of water particles will be:

As you can see, to calculate the unknown we want we need the number of moles of water. Fortunately, these can be calculated from the mass of water using the following equation:

Since we have the molecular weight of water (PM) which is numerically equal to its molar mass (but with different units), then we already have everything we need to solve the problem. We can calculate the moles first and then substitute them into the formula for number of particles, or we can substitute the expression for moles into the above equation and carry out a single calculation.

In this case, we will do the second:

So, in a 0.500 g drop of water there are 1,673.10 22 water molecules. Note that the number of molecules, N, is a pure number. That is, it has no units. We must place the units at the end as appropriate to what we are calculating, in this case, water molecules.

## problem 2

### statement

Determine the number of sulfate ions and the number of total oxygen atoms present in a 10-mg sample of hydrated aluminum sulfate whose formula is Al 2 (SO 4 ) 3 .18H 2 O. The molar mass of the salt is 666.42 g.mol -1 .

### Solution

Again we wish to determine a number of particles, but in this case it is not the whole compound (as in the case of water) but some parts of the substance. We must begin by transforming the mass to grams since we have the molar mass in grams per mole :

With these data we can calculate the number of molecules or formula units of the salt that are present in the sample in the same way as we did in the previous problem. But this is not what we want to determine.

However, from the molecular formula we can establish the simple stoichiometric relationships that will allow us to calculate what we need:

Now, we can see from the formula that there are 3 sulfate ions for every formula unit of salt. So we can convert units of salt to sulfate ions simply by multiplying by this stoichiometric ratio:

For the number of oxygen atoms, we need to add all the oxygens present in the sulfate ions and those present in the water molecules:

With this relationship, we calculate the number of oxygens in the sample from the number of formula units as we did with sulfate ions: