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## 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 NA is 6.02 x 1023 mol-1, which is the same as 6.02 x 1023 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 1023 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 1023 atoms of oxygen, or 6.02 x 1023 atoms of air.

The molecular mass is the mass of one molecule of a substance. A mole of water (dihydrogen oxide or H2O) 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 (C02), which has one molecule of carbon and two of oxygen, the molecular mass is [(1 x 12) +(2x16)] = 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 = mNA

## 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.

1. n = Msam/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 1023 = 30 x 1023 molecules.

Example 2: How would you calculate the number of moles in 88 grams of C02 ? Answer n = (88 g)/(44 g mol-1) = 2.0 mol.

The number of moles n can also be written as

1. n = N/NA

where N is the number of molecules in a sample and NA 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

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