Page 27 - Theory and Problems of BEGINNING CHEMISTRY

16 MATHEMATICAL METHODS IN CHEMISTRY [CHAP. 2

comparison with all units and subunits. Usually, the basic unit is chosen as the standard, but the kilogram (not
the gram) is chosen as the standard of mass because it is easier to measure precisely. Since there can be only
one standard for a given type of measurement, and the cubic meter has been chosen for volume, the liter is not a
standard of volume.]
It is very important that you get used to writing the proper abbreviations for units and the proper symbols at
the beginning of your study of chemistry, so you do not get mixed up later.

EXAMPLE 2.13. What is the difference between mg and Mg, two units of mass?
Ans. Lowercase m stands for milli (Table 2-2), and 1 mg is 0.001 g. Capital M stands for mega, and 1 Mg is 1 000 000 g.
It is obviously important not to confuse the capital M and lowercase m in such cases.

2.4. EXPONENTIAL NUMBERS
The numbers that scientists use range from enormous to extremely tiny. The distances between the stars are
literally astronomical—the star nearest to the sun is 23 500 000 000 000 mi from it. As another example, the
number of atoms of calcium in 40.0 g of calcium is 602 000 000 000 000 000 000 000, or 602 thousand billion
billion. The diameter of one calcium atom is about 0.000 000 02 cm. To report and work with such large and
small numbers, scientists use exponential notation. A typical number written in exponential notation looks as
follows:

3
4.13 × 10 , interpreted as 4.13 × 10 3
Coefficient Base Exponent

Exponential part

The coefficient is merely a decimal number written in the ordinary way. That coefficient is multiplied by the
exponential part, made up of the base (10) and the exponent. (Ten is the only base that will be used in numbers
in exponential form in the general chemistry course.) The exponent tells how many times the coefficient is
multiplied by the base.
3
4.13 × 10 = 4.13 × 10 × 10 × 10 = 4130
Since the exponent is 3, the coefficient is multiplied by three 10s.

5
EXAMPLE 2.14. What is the value of 10 ?
Ans. When an exponential is written without an explicit coefficient, a coefficient of 1 is implied:
1 × 10 × 10 × 10 × 10 × 10 = 100 000
There are five 10s multiplying the implied 1.

0
1
EXAMPLE 2.15. What is the value of 10 ?of10 ?
1
Ans. 10 = 10. There is one 10 multiplying the implied coefficient of 1.
0
10 = 1. There are no 10s multiplying the implied coefficient of 1.
4
EXAMPLE 2.16. Write 2.0 × 10 in decimal form.
Ans. 2.0 × 10 × 10 × 10 × 10 = 20 000
When scientists write numbers in exponential form, they prefer to write them so that the coefficient has one
and only one digit to the left of the decimal point, and that digit is not zero. That notation is called standard
exponential form,or scientific notation.