1.2: Measurement- SI Units and Scientific Notation
- Page ID
- 509842
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All reported measurements must include an appropriate unit of measurement because to say that a substance has “a mass of 10,” for example, does not tell whether the mass was measured in grams, pounds, tons, or some other unit. To establish worldwide standards for the consistent measurement of important physical and chemical properties, an international body called the General Conference on Weights and Measures devised the Système internationale d’unités (or SI). The International System of Units is based on metric units and requires that measurements be expressed in decimal form. Table \(\PageIndex{1}\) lists the seven base units of the SI system; all other SI units of measurement are derived from them.
Base Quantity | Unit Name | Abbreviation |
---|---|---|
mass | kilogram | kg |
length | meter | m |
time | second | s |
temperature | kelvin | K |
electric current | ampere | A |
amount of substance | mole | mol |
luminous intensity | candela | cd |
By attaching prefixes to the base unit, the magnitude of the unit is indicated; each prefix indicates that the base unit is multiplied by a specified power of 10. The prefixes, their symbols, and their numerical significance are given in Table \(\PageIndex{2}\). To study chemistry, you need to know the information presented in Tables \(\PageIndex{1}\) and \(\PageIndex{2}\).
Prefix | Symbol | Value | Power of 10 | Meaning |
---|---|---|---|---|
tera | T | 1,000,000,000,000 | 1012 | trillion |
giga | G | 1,000,000,000 | 109 | billion |
mega | M | 1,000,000 | 106 | million |
kilo | k | 1000 | 103 | thousand |
hecto | h | 100 | 102 | hundred |
deca | da | 10 | 101 | ten |
— | — | 1 | 100 | one |
deci | d | 0.1 | 10−1 | tenth |
centi | c | 0.01 | 10−2 | hundredth |
milli | m | 0.001 | 10−3 | thousandth |
micro | μ | 0.000001 | 10−6 | millionth |
nano | n | 0.000000001 | 10−9 | billionth |
pico | p | 0.000000000001 | 10−12 | trillionth |
femto | f | 0.000000000000001 | 10−15 | quadrillionth |
Units of Mass, Volume, and Length
The units of measurement you will encounter most frequently in chemistry are those for mass, volume, and length. The basic SI unit for mass is the kilogram (kg), but in the laboratory, mass is usually expressed in either grams (g) or milligrams (mg): 1000 g = 1 kg, 1000 mg = 1 g, and 1,000,000 mg = 1 kg. Units for volume are derived from the cube of the SI unit for length, which is the meter (m). Thus the basic SI unit for volume is cubic meters (length × width × height = m3). In chemistry, however, volumes are usually reported in cubic centimeters (cm3) and cubic decimeters (dm3) or milliliters (mL) and liters (L), although the liter is not an SI unit of measurement. The relationships between these units are as follows:
\[1 L = 1000 mL = 1 dm^3\]
\[1 mL = 1 cm^3\]
\[1000 cm^3 = 1 L\]
Scientific Notation
Chemists often work with numbers that are exceedingly large or small. For example, entering the mass in grams of a hydrogen atom into a calculator requires a display with at least 24 decimal places. A system called scientific notation avoids much of the tedium and awkwardness of manipulating numbers with large or small magnitudes. In scientific notation, these numbers are expressed in the form
\[ N \times 10^n\]
where N is greater than or equal to 1 and less than 10 (1 ≤ N < 10), and n is a positive or negative integer (100 = 1). The number 10 is called the base because it is this number that is raised to the power n. Although a base number may have values other than 10, the base number in scientific notation is always 10.
A simple way to convert numbers to scientific notation is to move the decimal point as many places to the left or right as needed to give a number from 1 to 10 (N). The magnitude of n is then determined as follows:
- If the decimal point is moved to the left n places, n is positive.
- If the decimal point is moved to the right n places, n is negative.
Another way to remember this is to recognize that as the number N decreases in magnitude, the exponent increases and vice versa. The application of this rule is illustrated in Example \(\PageIndex{1}\).
Example \(\PageIndex{2}\)
Convert each number to scientific notation.
- 637.8
- 0.0479
- 7.86
- 12,378
- 0.00032
- 61.06700
- 2002.080
- 0.01020
Solution
Explanation | Solution |
To convert 637.8 to a number from 1 to 10, we move the decimal point two places to the left: 637.8 Because the decimal point was moved two places to the left, n = 2. |
637.8 = 6.378 × 102. |
To convert 0.0479 to a number from 1 to 10, we move the decimal point two places to the right: 0.0479 Because the decimal point was moved two places to the right, n = −2. |
0.0479 = 4.79 × 10−2. |
this is usually expressed simply as 7.86. (Recall that 100 = 1.) | 7.86 × 100 |
because the decimal point was moved four places to the left, n = 4. | 1.2378 × 104 |
because the decimal point was moved four places to the right, n = −4. | 3.2 × 10−4 |
this is usually expressed as 6.1067 × 10. | 6.106700 × 101 |
2.002080 × 103 | |
1.020 × 10−2 |
Addition and Subtraction
Before numbers expressed in scientific notation can be added or subtracted, they must be converted to a form in which all the exponents have the same value. The appropriate operation is then carried out on the values of N. Example \(\PageIndex{2}\) illustrates how to do this.
Example \(\PageIndex{2}\)
Carry out the appropriate operation on each number and then express the answer in scientific notation.
- \( (1.36 \times 10^2) + (4.73 \times 10^3)\)
- \((6.923 \times 10^{−3}) − (8.756 \times 10^{−4}) \)
Solution
Explanation | Solution |
Both exponents must have the same value, so these numbers are converted to either \((1.36 \times 10^2) + (47.3 \times 10^2)\) or \((0.136 \times 10^3) + (4.73 \times 10^3)\). Choosing either alternative gives the same answer, reported to two decimal places: In converting 48.66 × 102 to scientific notation, N has become more positive by 1 because the value of N has decreased. |
\((1.36 \times 10^2) + (47.3 \times 10^2) = (1.36 + 47.3) \times 10^2 = 48.66 × 10^2 = 4.87 \times 10^3\) \( (0.136 \times 10^3) + (4.73 \times 10^3) = (0.136 + 4.73) \times 10^3 = 4.87 \times 10^3\) |
b. Converting the exponents to the same value gives either \((6.923 \times 10^{-3}) − (0.8756 \times 10^{-3})\) or \((69.23 \times 10^{-4}) − (8.756 \times 10^{-4})\) Completing the calculations gives the same answer, expressed to three decimal places: |
\((6.923 \times 10^{−3}) − (0.8756 \times 10^{−3}) = (6.923 − 0.8756) \times 10^{−3} = 6.047 \times 10^{−3}\) \((69.23 \times 10^{−4}) − (8.756 \times 10^{−4}) = (69.23 − 8.756) \times 10^{−4} = 60.474 \times 10^{−4} = 6.047 \times 10^{−3}\) |
Multiplication and Division
When multiplying numbers expressed in scientific notation, we multiply the values of N and add together the values of n. Conversely, when dividing, we divide N in the dividend (the number being divided) by N in the divisor (the number by which we are dividing) and then subtract n in the divisor from n in the dividend. In contrast to addition and subtraction, the exponents do not have to be the same in multiplication and division. Examples of problems involving multiplication and division are shown in Example \(\PageIndex{3}\).
Example \(\PageIndex{3}\)
Perform the appropriate operation on each expression and express your answer in scientific notation.
- \( (6.022 \times 10^{23})(6.42 \times 10^{−2})\)
- \( { 1.67 \times 10^{-24} \over 9.12 \times 10 ^{-28} } \)
- \({(6.63 \times 10^{−34})(6.0 \times 10) \over 8.52 \times 10^{−2}} \)
Solution
a. In multiplication, we add the exponents:
b. In division, we subtract the exponents:
\[{1.67 \times 10^{−24} \over 9.12 \times 10^{−28}} = {1.67 \over 9.12} \times 10^{[−24 − (−28)]} = 0.183 \times 10^4 = 1.83 \times 10^3 \]
c. This problem has both multiplication and division: