Lab #1 Supplement: Scientific notation logarithms, pH, and metric conversions
1. A quick refresher on scientific notation:
Scientific notation is a convenient way of expressing very large or very small numbers. In this method,
numbers are expressed as a root value multiplied by some factor of 10 (10, 100, 0.1, etc). To use this
method effectively, though, you must be comfortable with exponents and be able to switch from scientific
notation to decimal notation and vice versa when needed.
Numbers expressed as scientific notation are presented as a single-digit value (followed by any
fractional values) multiplied by some factor of 10 (for example, 6.25 × 1010). There should be one and
only one digit to the left of the decimal (for example, 62.5 × 109 is incorrect notation, as is 0.925 x 10-2).
You can think of the exponent in scientific notation as indicating the number of digits that
separate the first non-zero number from the decimal point. For example:
1 × 102 = 100.0
Note that there are two zeros separating “1” from the decimal point. Here are some other examples
1 × 100 = 1
1 × 101 = 10
5 × 102 = 500
6.1 × 104 = 61,000
If the exponent is negative, it indicates the number of digits the first non-zero number is
positioned to the right of the decimal point.
1 × 10-1 = 0.1
5 × 10-2 = 0.05
7 × 10-3 = 0.007
3.4 × 10-4 = 0.00034
You can also try a technique of moving the decimal point. For large numbers (10 or greater),
move the decimal point over to the left until there is only a single non-zero digit to the left of the decimal
point. The number of digits you had to move the decimal point to the left is equal to the exponent. For
example, we might convert 256,000,000 into scientific notation in the following manner:
256000000.0 = 2.56 × 108
8 7 6 5 4 3 2 1
For numbers less than 1.0, move the decimal point to the right until one non-zero digit is to the left of the
decimal point. Each digit you moved the decimal point to the right is equal to -1 in the exponent. For
example, 0.000000292 could be converted into scientific notation as such:
0.000000292 = 2.92 × 10-7
1 2 3 4 5 6 7
Give it a try on your own. If you want to use a calculator to check your answers, often scientific
calculators have an ‘EE’ (or ‘EXP’) button that will allow you to enter in numbers in scientific notation
(EE = “times 10 to the power”, so don’t multiply the value by 10 first then hit EE). Enter in the nonzero values, then press EE and enter the exponent. To do the opposite (convert scientific notation into
decimal form), you would simply use the opposite method—move the decimal point to the right for the
number of digits equal to the exponent if the exponent is positive, or to the left if the exponent is negative.
Sample problems: (Answers are provided at the end of this sheet).
Express the following in scientific notation (i.e. × 10x)
1. 0.004
2. 126,900
3. 0.00000001
4. 50
5. 0.167
Express the following in decimal form
6. 6 × 104
7. 1.35 × 10-5
8. 1 × 101
9. 1 × 10-7
10. 6.02 × 1023
2. Logartihms and Calculation of pH
Remember, pH is an index of hydrogen ion concentration in solution. It is calculated in the following
manner:
pH = log(1/[H+]).
Logarithms scare students at times, but they are actually quite easy to understand once you are
comfortable with scientific notation. Basically, a logarithm (specifically, a base-10 logarithm) is the
exponent by which you would multiply 10 by itself in order to get a particular value. For example:
100 = 102 and log 100 = 2
1000 = 103 and log 1000 = 3
Notice that in these examples the logarithm’s first non-zero value is equal to the exponent of the value
expressed in scientific notation. Effectively, a logarithm’s first digit indicate the number of digits the
value is positioned to the left or right of the zero. Here are a couple of parallel number lines to
demonstrate:
Decimal
0.01
0.1
1
10
100
Scientific
1×10-2
1×10-1
1×100
1×101
1×102
Logarithm
-2
-1
0
1
2
The logarithm can therefore indicate the size of a value in factors of 10. For example, a value between 10
and 100 will have a logarithm value between 1 and 2 (e.g. log 50 = 1.69897), and a value between 100
and 1000 will have a logarithm value between 2 and 3 (e.g. log 450 = 2.65321).
Logarithms of numbers between 1 and 0 will be negative, and the first number will indicate the
number of digits to the right of the decimal point before the non-zero numbers begin. For example,
values between 0.1 and 1 will have a logarithm between 0 and –1 (e.g. log 0.3 = -0.523), and values
between 0.01 and 0.1 will have a logarithm between –2 and –1 (e.g. log 0.07 = -1.1549). With that under
your belt, calculating pH should be a snap. Try these examples:
Sample Problems: (Answers are provided at the end of this sheet).
Calculate the pH of the following solutions given their [H+]
11.
1 × 10-9
12.
3 × 10-14
13.
5 × 10-5
14.
4 × 10-2
15.
1
3. Metric Conversions
Throughout the semester we will be expressing quantities using metric values1, which are the
standard method of measurement throughout the world (and, since they are all based on each
other and are expressed by factors of ten, are much more useful and frankly make a lot more
sense than the English measurement system now used only in the United States and to a lesser
degree in Great Britain). The most common measurements you will see in this course are
measurements of mass (grams), volume (liters), and linear distance (meters).
The metric system has a standardized series of prefixes that are used to express different
multiples of a particular measurement (see Table 1.1), which is useful for describing these
measurements on different scales. For example, using the English system, we probably would
not say that distance between Kokomo and Indianapolis is ~264,000 feet. Rather, we would
express this distance as ~50 miles (50 multiples of 5,280 feet). Likewise, we would not say that
the width of your hand is ~0.0000631 miles, but would rather express that width as ~4 inches.
The metric system’s prefixes are used to describe quantities at different scales based on factors
of 10 (which are much easier to convert from one unit to another than by factors of 12 or 5,280
or the like). However, this does require some familiarity with the various prefixes. Moreover,
you should be able to make some conversions from multiples at one scale to those of another.
For example, rather than expressing a volume as 0.005 liters, it would be more appropriate to
describe it as 5 milliliters (five one-thousandths of a liter). Table 1.2 (next page) provides
common conversions for multiples used in this course.
Table 1.1. Common prefixes used to express multiples of metric units.
Metric prefix
Tera-(T)
Giga-(G)
Mega-(M)
kilo-(k)
-----deci-(d)
centi-(c)
milli-(m)
micro-(µ)
nano-(n)
pico-(p)
Value
Trillion
Billion
Million
Thousand
----Tenth
Hundredth
Thousandth
Millionth
Billionth
Trillionth
Decimal
1,000,000,000,000
1,000,000,000
1,000,000
1,000
1
0.1
0.01
0.001
0.000001
0.000000001
0.000000000001
Scientific Notation
12
× 10
9
× 10
6
× 10
3
× 10
0
× 10
-1
× 10
-2
× 10
-3
× 10
-6
× 10
-9
× 10
-12
× 10
Table 1.2. Conversions of some common multiples of metric values for mass, volume, and distance.
Mass
kg
=
1000g
1,000,000 mg
1,000,000,000 µg
g
=
0.001 kg
1000 mg
1,000,000 µg
Volume
L
=
1000 ml
1,000,000 µl
ml
=
0.001 L
1000 µl
µl
=
Distance
m
=
100 cm
1000 mm
1,000,000 µm
cm
=
0.01 m
10 mm
10,000 µm
0.000001 L
0.001 ml
mg
=
0.000001 kg
0.001g
1000 µg
mm
=
0.001 m
0.1 cm
1000 µm
µg
=
0.000000001 kg
0.000001 g
0.001 mg
µm
=
0.000001 m
0.0001 cm
0.001 mm
Answer Key to Sample Problems:
1. 4 x 10-3
2. 1.269 x 105
3. 1 x 10-8
4. 5 x 101
5. 1.67 x 10-1
6. 60,000
7. 0.0000135
8. 10
9. 0.0000001
10. 602,000,000,000,000,000,000,000
11. 9
12. 13.523
13. 4.301
14. 1.398
15. 0