Page 30 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 2] MATHEMATICAL METHODS IN CHEMISTRY 19

EXAMPLE 2.21. Underline the significant zeros in each of the following measurements, all in kilograms: (a) 7.00,
(b) 0.7070, (c) 0.0077, and (d ) 70.0.
Ans. (a) 7.00,(b) 0.7070,(c) 0.0077, and (d)70.0.In(a), the zeros are to the right of the last nonzero digit and to the
right of the decimal point (rule 3), so they are significant. In (b), the leading 0 is not significant (rule 1), the middle
0 is significant because it lies between two significant 7s (rule 2), and the last 0 is significant because it is to the
right of the last nonzero digit and the decimal point (rule 3). In (c), the zeros are to the left of the first nonzero digit
(rule 1), and so are not significant. In (d), the last 0 is significant (rule 3), and the middle 0 is significant because it
lies between the significant digits 7 and 0 (rule 2).

EXAMPLE 2.22. (a) How many significant digits are there in 1.60 cm? (b) How many decimal places are there in that
number?
Ans. (a) 3 and (b) 2. Note the difference in these questions!

EXAMPLE 2.23. How many significant zeros are there in the number 8 000 000?
Ans. The number of significant digits cannot be determined unless more information is given. If there are 8 million people
living in New York City and one person moves out, how many are left? The 8 million people is an estimate, indicating
a number nearer to 8 million than to 7 million or 9 million people. If one person moves, the number of people is still
nearer to 8 million than to 7 or 9 million, and the population is still properly reported as 8 million.
If you win a lottery and the state deposits $8 000 000 to your account, when you withdraw $1, your balance will be
$7 999 999. The precision of the bank is much greater than that of the census takers, especially since the census
takers update their data only once every 10 years.
To be sure that you know how many significant digits there are in such a number, you can report the number
in standard exponential notation because all digits in the coefficient of a number in standard exponential form are
6
significant. The population of New York City would be 8×10 people, and the bank account would be 8.000 000×10 6
dollars.

EXAMPLE 2.24. Change the following numbers of meters to millimeters. Explain the problem of zeros at the end of a
whole number, and how the problem can be solved. (a) 7.3 m, (b) 7.30 m, and (c) 7.300 m.

1000 mm
Ans. (a) 7.3 m = 7300 mm (two significant digits)
1m

1000 mm
(b) 7.30 m = 7300 mm (three significant digits)
1m
1000 mm

(c) 7.300 m = 7300 mm (four significant digits)
1m
The magnitudes of the answers are the same, just as the magnitudes of the original values are the same. The numbers
of millimeters all look the same, but since we know where the values came from, we know how many significant digits
3
3
each contains. We can solve the problem by using standard exponential form: (a)7.3 × 10 mm, (b)7.30 × 10 mm,
3
(c)7.300 × 10 mm.
EXAMPLE 2.25. Change each of the following measurements to meters. (a) 3456 mm, (b) 345.6 mm, (c) 34.56 mm, and
(d) 3.456 mm.
Ans. (a) 3.456 m (b) 0.3456 m (c) 0.03456 m (d) 0.003456 m
All the given values have four significant digits, so each answer also has to have four significant digits (and they do
since the leading zeros are not significant.)

Significant Digits in Calculations
Special note on significant figures: Electronic calculators do not keep track of significant figures at all. The
answers they yield very often have fewer or more digits than the number justified by the measurements. You
must keep track of the significant figures and decide how to report the answer.