The Mole and Molar Mass 


After completing this unit you should be able to:
We have seen that atoms of different elements have different masses, and that atoms combine in whole number ratios to form compounds. Although as a practical necessity we measure the quantity of matter using mass, it is essential when considering properties and reactions to know the relative number of atoms of each element present in a sample. Even the smallest measurable mass of matter contains trillions of atoms, so chemists use a unit of amount called the mole (abbreviated mol). By definition, one mole is the number of atoms in 12 g of carbon12. This number, called Avogadro's number, has been measured to be approximately 6.022 x 10^{23} (to 4 s.f). Avogadro's number is actually known to about nine significant figures, which means the uncertainty is +/ 100 trillion atoms! Although this might seem very imprecise, in practice it is much more precise than our mass measurement. Chemists use the mole in the same way that grocers use the dozen for groups of 12 and stationers use the ream for groups of 500. By grouping numbers together, we get a smaller number to use in practical situations, 2 gross of paper clips for example, instead of 288 paper clips. Since we are most frequently concerned with relative amounts, we can use the mole without being overly concerned about exactly how many objects it represents, and we can use Avogadro's number to convert it to an actual number if needed. Since one atom of carbon12 has a mass of 12 atomic mass units, and one atom of tin120 has a mass of 120 u, it follows that one mole of tin120 atoms will have ten times the mass of one mole of carbon12 atoms, i.e. 120 g. In general, the mass in grams of one mole of atoms of any element will be numerically equivalent to its atomic mass, i.e. the atomic mass unit is equivalent to the unit grams per mole (g/mol). The number of atoms in a sample of an element can be counted by weighing, in the same way that banks count pennies. As an example, let's determine how many atoms are in a sample of silicon that has a mass of 5.23 g. Consulting a periodic table, we find that the atomic mass of Si is 28.09 u or 28.09 g/mol. We use this as a conversion factor to determine how many moles of silicon are in our sample:
For most purposes we will use this number of moles as the amount of silicon in our sample. If we want to know how many atoms this is, we can use Avogadro's number (6.022 x 10^{23} per mol) for the conversion:
The mole is just a number; it can be used for atoms, molecules, ions, electrons, or anything else we wish to refer to. Because we know the formula of water is H_{2}O, for example, then we can say one mole of water molecules contains one mole of oxygen atoms and two moles of hydrogen atoms. One mole of hydrogen atoms has a mass of 1.008 g and 1 mol of oxygen atoms has a mass of 16.00 g, so 1 mol of water has a mass of (2 x 1.008 g) + 16.00 g = 18.02 g. The molar mass of water is 18.02 g/mol. The
formula mass of water is 18.02 u. This is the average mass of one
formula unit (in this case a molecule) of water. In general, the molar mass of any molecular compound is the mass in grams
numerically equivalent to the sum of the atomic masses of the atoms in the
molecular formula: For more complex formulas it is convenient to use a table. The molar mass of glucose, C_{6}H_{12}O_{6}, is:
The molar mass of ionic compounds can be calculated similarly, by adding together the atomic masses of all atoms in the formula to get the formula mass, and expressing the answer in units of g/mol. Calcium chloride, CaCl_{2}, has a molar mass of 40.08 + (2 x 35.45) = 110.98 g/mol. Molar mass is generally calculated with two places after the decimal point, and can be rounded to four significant figures. Molar mass is used as a conversion factor to relate the amount of a substance to its mass. For example, suppose you need 5.00 moles of magnesium nitride for an experiment. What mass in grams should you weigh out on your balance? First we need the formula for magnesium nitride. Magnesium, an alkaline earth metal, forms cations with charge 2+, and nitrogen, a group 5A element, forms anions with charge 3. So the formula for magnesium nitride must be Mg_{3}N_{2}, and we can then determine the molar mass as (3 x 24.31 g/mol) + (2 x 14.01 g/mol) = 100.95 g/mol. Using this as a conversion factor, we can calculate the mass of 5.00 moles of magnesium nitride:
The formula for a compound gives the ratio of the elements in terms of numbers of atoms, which is the same ratio when expressed in moles of atoms. Using the molar masses of the elements and the compound, we can express the composition in terms of mass percentage of the elements. For example, carbon dioxide has a formula weight of 44.01 u, made up of 12.01 u for the average mass of 1 carbon atom and 32.00 u for 2 oxygen atoms. One mole of CO_{2} has a mass of 44.01 g made up of 12.01 g of carbon and 32.00 g of oxygen. The composition of carbon dioxide is calculated as follows:
The formula for a compound, and its composition expressed as percentage by mass, are fixed and unchanging properties of the compound. Any pure sample of carbon dioxide is 72.70 % oxygen by mass. This property allows us to relate the amount of an element in a compound to the mass of the compound. As an example, let's calculate the number of moles of iron in 2.98 g of iron(III) oxide. First we need the formula for iron(III) oxide. Since iron(III) is Fe^{3+} and oxide is O^{2} the formula must be Fe_{2}O_{3}. We then calculate the molar mass as (2 x 55.85) + (3 x 16.00) = 111.7 + 48.00 = 159.7 g/mol. The percentage by mass of iron in iron(III) oxide can then be calculated:
This percentage by mass can be used to calculate the mass of iron in the sample:
Finally, the mass of iron can be converted to an amount in moles using the molar mass:
We can arrive at the same answer rather more easily using a mole ratio generated from the formula. The formula tells us that 1 mol of Fe_{2}O_{3} is composed of 2 mol of iron ions and 3 mol of oxide ions. First we convert our measured mass of iron(III) oxide to moles using the molar mass of the compound, then we convert this to moles of iron using the mole ratio 2 moles of iron per mole of iron(III) oxide:
Mole ratios are used in many important calculations in chemistry. Be sure to write in your calculation the complete unit, moles of iron atoms for example, not just mole (remember: mole is just a number). When your units cancel correctly you can be sure you have made the calculation you intended. 


