#### Learning Objective

- Convert between the number of moles and the number of atoms in a given substance using Avagadro’s number

#### Key Points

- Avogadro’s number is a very important relationship to remember: 1 mole = [latex]6.022\times10^{23}[/latex]atoms, molecules, protons, etc.
- To convert from moles to atoms, multiply the molar amount by Avogadro’s number.
- To convert from atoms to moles, divide the atom amount by Avogadro’s number (or multiply by its reciprocal).

#### Terms

- moleThe amount of substance of a system that contains as many elementary entities as there are atoms in 12 g of carbon-12.
- Avogadro’s numberThe number of atoms present in 12 g of carbon-12, which is [latex]6.022\times10^{23}[/latex] and the number of elementary entities (atoms or molecules) comprising one mole of a given substance.

## Moles and Atoms

As introduced in the previous concept, the mole can be used to relate masses of substances to the quantity of atoms therein. This is an easy way of determining how much of one substance can react with a given amount of another substance.

From moles of a substance, one can also find the number of atoms in a sample and vice versa. The bridge between atoms and moles is Avogadro’s number, 6.022×10^{23}.

Avogadro’s number is typically dimensionless, but when it defines the mole, it can be expressed as 6.022×10^{23} elementary entities/mol. This form shows the role of Avogadro’s number as a conversion factor between the number of entities and the number of moles. Therefore, given the relationship 1 mol = 6.022 x 10^{23} atoms, converting between moles and atoms of a substance becomes a simple dimensional analysis problem.

## Converting Moles to Atoms

Given a known number of moles (x), one can find the number of atoms (y) in this molar quantity by multiplying it by Avogadro’s number:

[latex]x \ moles \cdot \frac {6.022\times10^{23} atoms}{1 \ mole} = y \ atoms[/latex]

For example, if scientists want to know how may atoms are in six moles of sodium (x = 6), they could solve:

[latex]6 \ moles \cdot \frac {6.022\times 10^{23} atoms}{1 \ mole} = 3.61\times 10^{24} atoms[/latex]

Note that the solution is independent of whether the element is sodium or otherwise.

## Converting Atoms to Moles

Reversing the calculation above, it is possible to convert a number of atoms to a molar quantity by dividing it by Avogadro’s number:

[latex]\frac{x \ atoms}{6.022\times 10^{23} \frac{atoms}{1 \ mole}} = y \ moles[/latex]

This can be written without a fraction in the denominator by multiplying the number of atoms by the reciprocal of Avogadro’s number:

[latex]x \ atoms \cdot \frac{1 \ mole}{6.022\times 10^{23} \ atoms} = y \ moles[/latex]

For example, if scientists know there are [latex]3.5 \cdot 10^{24} [/latex]atoms in a sample, they can calculate the number of moles this quantity represents:

[latex]3.5 \times 10^{24} \ atoms \cdot \frac{1 \ mole}{6.022\times 10^{23} \ atoms} = 5.81 \ moles[/latex]