THE MOLE

7.4. THE MOLE

Atoms and molecules are incredibly small. For hundreds of years after Dalton postulated their existence, no one was able to work with just one atom or molecule. (In recent times, with special apparatus, it has been possible to see the effects of individual atoms and molecules, but this subject will be developed later.) Just as the dozen is used as a convenient number of items in everyday life, the mole may be best thought of as as number

of items. The mole is 6.02 × 10 23 items, a number called Avogadro’s number. This is a very large number: six hundred two thousand billion billion! The entire earth has a mass of 6 × 10 24 kg. Thus, the earth has only

10 times as many kilograms as 1 mol of carbon has atoms. One can have a mole of any item, but it makes little sense to speak of moles of anything but the tiniest of particles, such as atoms and molecules. It might seem unusual to give a name to a number, but remember we do the same thing in everyday life; dozen is the name for 12 items. Just as a grocer finds selling eggs by the dozen more convenient than selling them individually, the chemist finds calculations more convenient with moles. The number of formula units (i.e., the number of uncombined atoms, of molecules of molecular elements or compounds, or of formula units of ionic compounds) can be converted to moles of the same substance, and vice versa, using Avogadro’s number (Fig. 7-1).

Number of

Avogadro’s number

Number of

formula units

moles

Fig. 7-1. Avogadro’s number conversions

Mole is abbreviated mol. Do not use m or M for mole; these symbols are used for other quantities related to moles, and so you will be confused if you use either of them. Note: A mole is referred to by some authors as

a “gram molecular mass” because 1 mol of molecules has a mass in grams equal to its molecular mass. In this terminology, a “gram atomic mass” is 1 mol of atoms, and a “gram formula mass” is 1 mol of formula units. The formula mass of a substance is equal to its number of grams per mole. Avogadro’s number is the number of atomic mass units in 1 g. It is defined in that manner so that the atomic mass of an element (in amu) is

FORMULA CALCULATIONS

[ CHAP. 7

numerically equal to the number of grams of the element per mole. Consider sodium, with atomic mass 23.0:

1g 23.0 g = Na atom

23.0 amu

23.0 amu

23 Na atoms

6.02 × 10 23 amu = mol Na Avogadro’s number appears in both the numerator and the denominator of this expression; the values reduce to

Na atom

1 mol Na

a factor of 1 (they cancel), and the numeric value in grams per mole is equal to the numeric value of the atomic mass in amu per atom.

23.0 amu

23.0 g

Na atom = mol Na atoms

A similar argument leads to the conclusion that the formula mass of any element or compound is equal to the number of grams per mole of the element or compound.

EXAMPLE 7.3. How many feet tall is a stack of a dozen shoe boxes, each 4 in. tall? Ans.

The same number, but in different units, is obtained because the number of inches in 1 foot is the same as the number in 1 dozen. Compare this process to the one above for grams per mole.

Changing grams to moles and moles to grams is perhaps the most important calculation you will have to make all year (Fig. 7-2). We use the term molar mass for the mass of 1 mol of any substance. The units are typically grams per mole.

Molar mass

Number of

Mass

moles

Fig. 7-2. Molar mass conversions

The mass of a substance can be converted to moles, and vice versa, with the molar mass.

EXAMPLE 7.4. Calculate the mass of 1.000 mol of SCl 2 .

Ans.

The formula mass of SCl 2 is given by

S = 32.06 amu 2 Cl = 2 × 35.45 amu = 70.90 amu SCl 2 = total = 102.96 amu Thus, the mass of 1.000 mol of SCl 2 is 102.96 g.

EXAMPLE 7.5. Calculate the mass of 2.50 mol NaClO 4 .

Ans.

The formula mass of NaClO 4 is given by

64.0 amu NaClO 4 122.5 amu

NaClO 4 has a formula mass of 122.5 amu.

CHAP. 7]

FORMULA CALCULATIONS

EXAMPLE 7.6. Calculate the mass in grams of 1 uranium atom. Ans.

First we can calculate the number of moles of uranium, using Avogadro’s number:

1 mol U

1 U atom

1.66 × 10 − 24 mol U

6.02 × 10 U atoms

Then we calculate the mass from the number of moles and the molar mass:

1.66 × 10 − 24 mol U

3.95 × 10 − 22 g

1 mol U

Alternately, we can combine these expressions into one:

1 mol U

1 U atom

3.95 × 10 − 22 g

6.02 × 10 23 U atoms

1 mol U

Still another solution method:

1.00 g

1 U atom

1 U atom

How can we count such a large number of items as Avogadro’s number? One way, which we also can use in everyday life, is to weigh a small number and the entire quantity. We can count the small number, and the ratio of the number of the small portion to the number of the entire quantity is equal to the ratio of their masses.

EXAMPLE 7.7.

A TV show requires contestants to guess the number of grains of rice in a gallon container. The closest contestant after 4 weeks will win a big prize. How could you prepare for such a contest, without actually counting the grains in 1 gal of rice?

Ans. One way to get a good estimate is to count 100 grains of rice and weigh that sample. Weigh 1 gal of rice. Then the ratio of number of grains of rice in the small sample (100) to number of grains of rice in the large sample (which is the unknown) is equal to the ratio of masses. Suppose 100 grains of rice weighed 0.012 lb and 1 gal of rice weighed

8.80 lb. The unknown number of grains in 1 gal x can be calculated using a proportion:

x grains in 1 gallon =

Why not weigh just one grain? The calculation would be simpler, but weighing just one grain might be impossible with the balances available, and the grain that we choose might not have the average mass.

The situation is similar in counting atoms, but much more difficult. Individual atoms cannot be seen to be counted, nor can they be weighed in the ordinary manner. Still, if the mass of 1 atom can be determined (in amu, for example) the number of atoms in a mole can be calculated. Historically, what chemists have done in effect is to weigh very large numbers of atoms of different elements where the ratio of atoms of the elements is known; they have gotten the ratio of the masses of individual atoms from the ratio of the mass of the different elements and the relative numbers of atoms of the elements.

EXAMPLE 7.8. If equal numbers of carbon and oxygen atoms in a certain sample have a ratio of masses 12.0 g to 16.0 g, what is the ratio of atomic masses?

Ans. The individual atoms have the same ratio, 12.0/16.0 = 0.750. (The atomic masses of carbon and oxygen are 12.0 amu and 16.0 amu, respectively.)

The number of moles of each element in a mole of compound is stated with subscripts in the chemical formula. Hence, the formula can be used to convert the number of moles of the compound to the number of moles of its component elements, and vice versa (Fig. 7-3).

EXAMPLE 7.9. How many moles of hydrogen atoms are present in 1.25 mol of CH 4 ?

Ans.

The formula states that there is 4 mol H for every 1 mol CH 4 . Therefore,

FORMULA CALCULATIONS

[ CHAP. 7

Number of

Chemical formula

Number of moles

moles of

of element in

compound

the compound

Fig. 7-3. Chemical formulas mole ratio The number of moles of each element in a compound and the number of moles of the compound as a whole are related by the subscript of that element in the chemical formula.