Page 32 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 2] MATHEMATICAL METHODS IN CHEMISTRY 21

For rounding digits in any whole-number place, use the same rules, except that instead of actually dropping
digits, replace them with (nonsignificant) zeros. For example, to round 6718 to two significant digits
6718 → 6700 (two significant digits)

EXAMPLE 2.26. Round the following numbers to two significant digits each: (a) 0.0654, (b) 65.4, and (c) 654.
Ans. (a) 0.065 (b)65 (c) 650
[Do not merely drop the 4. (b) and (c) obviously cannot be the same.]

Multiplication and Division
In multiplication and division, different rules apply than apply to addition and subtraction. It is the number of
significant digits in each of the values given, rather than their positions, which governs the number of significant
digits in the answer. In multiplication and division, the answer retains as many significant digits as there are in
the value with the fewest significant digits.

EXAMPLE 2.27. Perform each of the following operations to the proper number of significant digits:
3
3
(a) 1.75 cm × 4.041 cm, (b) 2.00 g/3.00 cm , and (c) 6.39 g/2.13 cm .
Ans. (a) 1.75 cm × 4.041 cm = 7.07 cm . There are three significant digits in the first factor and four in the second. The
2
answer can retain only three significant digits, equal to the smaller number of significant digits in the factors.
3
3
(b) 2.00 g/3.00 cm = 0.667 g/cm . There are three significant digits, equal to the number of significant digits in
each number. Note that the number of decimal places is different in the answer, but in multiplication or division,
the number of decimal places is immaterial.
3
3
(c) 6.39 g/2.13 cm = 3.00 g/cm . Since there are three significant digits in each number, there should be three
significant digits in the answer. In this case, we had to add zeros, not round, to get the proper number of
significant digits.
When we multiply or divide a measurement by a defined number, rather than by another measurement, we
may retain in the answer the number of significant digits that occur in the measurements. For example, if we
multiply a number of meters by 1000 mm/m, we may retain the number of significant digits in the number of
millimeters that we had in the number of meters. The 1000 is a defined number, not a measurement, and it can
be regarded as having as many significant digits as needed for any purpose.
EXAMPLE 2.28. Howmanysignificantdigitsshouldberetainedintheanswerwhenwecalculatethenumberofcentimeters
in 6.137 m?

100 cm
Ans. 6.137 m = 613.7cm
1m
The number of significant digits in the answer is 4, equal to the number in the 6.137-m measurement. The (100 cm/m)
isadefinition and does not limit the number of significant digits in the answer.
2.6. DENSITY
Density is a useful property with which to identify substances. Density, symbolized d,isdefined as mass
per unit volume:
mass
Density =
volume
or in symbols,
m
d =
V
Since it is a quantitative property, it is often more useful for identification than a qualitative property such as
color or smell. Moreover, density determines whether an object will float in a given liquid. If the object is less
dense than the liquid, it will float. It is also useful to discuss density here for practice with the factor-label method
of solving problems, and as such, it is often emphasized on early quizzes and examinations.