Page 116 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 7] FORMULA CALCULATIONS 105

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
23
6.02 × 10 U atoms
Then we calculate the mass from the number of moles and the molar mass:

238 g
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 238 g
1 U atom = 3.95 × 10 −22 g
23
6.02 × 10 U atoms 1 mol U
Still another solution method:

238 amu 1.00 g
1 U atom = 3.95 × 10 −22 g
23
1 U atom 6.02 × 10 amu
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 8.80 lb
=
100 grains 0.012 lb
x = 73 000

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,

4 mol H
1.25 mol CH 4 = 5.00 mol H
1 mol CH 4