## Introduction

Gases have very large numbers of molecules. By convention, chemists define a mole as the number of atoms or molecules in 12 grams of the isotope carbon 12, which has an atomic mass of 12 u. Moles are very convenient as a unit of measurement for a substance.

**Avogadro’s number** N_{A} is 6.02 x 10^{23 }mol^{-1}, which is the same as 6.02 x 10^{23} atoms or molecules per mole. It is based on a theory by Italian physicist Amedeo Avogadro (1776-1856) . He had learned about Gay-Lussac’s Law, and conjectured that all gases with the same temperature and pressure at the same volume also have the same number of atoms or molecules. This conjecture is known as **Avogadro’s Principle**, and the number has been validated experimentally.

Thus 1 mole = 6.02 X 10^{23} elementary units. When we say 1 mole or mol of oxygen or 1 mol of a compound such as air, we know that we mean 6.02 x 10^{23} atoms of oxygen, or 6.02 x 10^{23} atoms of air.

The **molecular mass** is the mass of one molecule of a substance. A mole of water (dihydrogen oxide or H_{2}O) has two molecules of hydrogen and one molecule of oxygen. As can be seen from a periodic table, Hydrogen has an atomic mass of 1 u and oxygen has an atomic mass of 16 u. Water therefore has a molecular mass of 18 grams, since [(2 x 1) + (1 x 16)] = 18 grams. For a mole of carbon dioxide (C0_{2}), which has one molecule of carbon and two of oxygen, the molecular mass is [(1 x 12) +(2×16)] = 44 grams.

The mass of one mol is referred to as the **molar mass**. The molar mass M of one mole of a substance is related to its molecular mass m and Avogadro’s number by the relationship

1) M = mN_{A}

## The Relationship Between Moles, Sample Mass and Molecular Mass

If n is the number of moles, we say it is equal to the ratio of the mass of a substance sample divided by the substance’s molecular mass.

2) n = M_{sam}/m = sample mass/molecular mass

where for SI units, n is in moles, mass is in grams, and the molecular mass is in grams per mol (grams/mol).

## Examples

Example 1: Suppose you wish to know the number of moles in 3 ounces of water, which is approximately the same as 90 grams. Using equation 2 we have n = (90 g)/(18 g/mol) = 5.0 mol. 5 moles are 5 _{*} 6.02 x 10^{23} = 30 x 10^{23} molecules.

Example 2: How would you calculate the number of moles in 88 grams of C0_{2} ? Answer n = (88 g)/(44 g mol^{-1}) = 2.0 mol.

The number of moles n can also be written as

3) n = N/N_{A}

where N is the number of molecules in a sample and N_{A }is Avogadro’s number, as before.

Avogadro’s number is very useful in simplifying the ideal gas law, which is a very important equation of state in thermodynamics.

## References

Physics for Scientists and Engineers by Douglas Giancoli

Fundamentals of Physics by Halliday, Resnick, and Walker

*Image credits*

*Avogadro from Bulldog.-u-net*

## Resources

Bright Hub Activities to Help Learn About Atomic Mass Part One

Bright Hub Activities to Help Learn About Atomic Mass Part Two