Chapter Introduction

Measurements

Opening Essay

Data suggest that a male child will weigh 50% of his adult weight at about 11 years of age. However, he will reach 50% of his adult height at only 2 years of age. It is obvious, then, that people eventually stop growing up but continue to grow out. Data also suggest that the average human height has been increasing over time. In industrialized countries, the average height of people increased 5.5 inches from 1810 to 1984. Most scientists attribute this simple, basic measurement of the human body to better health and nutrition.

Growth chart showing stature percentiles for 2 to 20 year old boys

Source: Chart courtesy of Centers for Disease Control and Prevention, http://www.cdc.gov/nchs/nhanes.htm#Set%201.

In 1983, an Air Canada airplane had to make an emergency landing because it unexpectedly ran out of fuel; ground personnel had filled the fuel tanks with a certain number of pounds of fuel, not kilograms of fuel. In 1999, the Mars Climate Orbiter spacecraft was lost attempting to orbit Mars because the thrusters were programmed in terms of English units, even though the engineers built the spacecraft using metric units. In 1993, a nurse mistakenly administered 23 units of morphine to a patient rather than the “2–3” units prescribed. (The patient ultimately survived.) These incidents occurred because people weren’t paying attention to quantities.

Chemistry, like all sciences, is quantitative. It deals with quantities, things that have amounts and units. Dealing with quantities is very important in chemistry, as is relating quantities to each other. In this chapter, we will discuss how we deal with numbers and units, including how they are combined and manipulated.

Expressing Numbers

Learning Objective
Learn to express numbers properly

Quantities have two parts: the number and the unit. The number tells “how many.” It is important to be able to express numbers properly so that the quantities can be communicated properly.

Standard notation is the straightforward expression of a number. Numbers such as 17, 101.5, and 0.00446 are expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as 306,000,000, or for very small numbers, such as 0.000000419, standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position.

Scientific notation is an expression of a number using powers of 10. Powers of 10 are used to express numbers that have many zeros:

10⁰	= 1
10¹	= 10
10²	= 100 = 10 × 10
10³	= 1,000 = 10 × 10 × 10
10⁴	= 10,000 = 10 × 10 × 10 × 10

and so forth. The raised number to the right of the 10 indicating the number of factors of 10 in the original number is the exponent (Scientific notation is sometimes called exponential notation.) The exponent’s value is equal to the number of zeros in the number expressed in standard notation.

Small numbers can also be expressed in scientific notation but with negative exponents:

10⁻¹	= 0.1 = 1/10
10⁻²	= 0.01 = 1/100
10⁻³	= 0.001 = 1/1,000
10⁻⁴	= 0.0001 = 1/10,000

and so forth. Again, the value of the exponent is equal to the number of zeros in the denominator of the associated fraction. A negative exponent implies a decimal number less than one.

A number is expressed in scientific notation by writing the first nonzero digit, then a decimal point, and then the rest of the digits. The part of a number in scientific notation that is multiplied by a power of 10 is called the coefficient. Then determine the power of 10 needed to make that number into the original number and multiply the written number by the proper power of 10. For example, to write 79,345 in scientific notation,

79,345 = 7.9345 × 10,000 = 7.9345 × 10⁴

Thus, the number in scientific notation is 7.9345 × 10⁴. For small numbers, the same process is used, but the exponent for the power of 10 is negative:

0.000411 = 4.11 × 1/10,000 = 4.11 × 10⁻⁴

Typically, the extra zero digits at the end or the beginning of a number are not included.

Example 1

Express these numbers in scientific notation.

306,000
0.00884
2,760,000
0.000000559

Solution

The number 306,000 is 3.06 times 100,000, or 3.06 times 10⁵. In scientific notation, the number is 3.06 × 10⁵.
The number 0.00884 is 8.84 times 1/1,000, which is 8.84 times 10⁻³. In scientific notation, the number is 8.84 × 10⁻³.
The number 2,760,000 is 2.76 times 1,000,000, which is the same as 2.76 times 10⁶. In scientific notation, the number is written as 2.76 × 10⁶. Note that we omit the zeros at the end of the original number.
The number 0.000000559 is 5.59 times 1/10,000,000, which is 5.59 times 10⁻⁷. In scientific notation, the number is written as 5.59 × 10⁻⁷.

Test Yourself

Express these numbers in scientific notation.

23,070
0.0009706

Answers

2.307 × 10⁴
9.706 × 10⁻⁴

Another way to determine the power of 10 in scientific notation is to count the number of places you need to move the decimal point to get a numerical value between 1 and 10. The number of places equals the power of 10. This number is positive if you move the decimal point to the right and negative if you move the decimal point to the left:

\begin{matrix} 5 \underset{4}{\underset{↖}{6}}, \underset{3}{\underset{↖}{9}} \underset{2}{\underset{↖}{0}} \underset{1}{\underset{↖}{0}} = 5.69 \times 10^{4} & 0. \underset{1}{\underset{↗}{0}} \underset{2}{\underset{↗}{0}} \underset{3}{\underset{↗}{0}} \underset{4}{\underset{↗}{0}} \underset{5}{\underset{↗}{2}} 8 = 2.8 \times 10^{- 5} \end{matrix}

Many quantities in chemistry are expressed in scientific notation. When performing calculations, you may have to enter a number in scientific notation into a calculator. Be sure you know how to correctly enter a number in scientific notation into your calculator. Different models of calculators require different actions for properly entering scientific notation. If in doubt, consult your instructor immediately.

Key Takeaways

Standard notation expresses a number normally.
Scientific notation expresses a number as a coefficient times a power of 10.
The power of 10 is positive for numbers greater than 1 and negative for numbers between 0 and 1.

Exercises

Express these numbers in scientific notation.
1. 56.9
2. 563,100
3. 0.0804
4. 0.00000667
Express these numbers in scientific notation.
1. −890,000
2. 602,000,000,000
3. 0.0000004099
4. 0.000000000000011
Express these numbers in scientific notation.
1. 0.00656
2. 65,600
3. 4,567,000
4. 0.000005507
Express these numbers in scientific notation.
1. 65
2. −321.09
3. 0.000077099
4. 0.000000000218
Express these numbers in standard notation.
1. 1.381 × 10⁵
2. 5.22 × 10⁻⁷
3. 9.998 × 10⁴
Express these numbers in standard notation.
1. 7.11 × 10⁻²
2. 9.18 × 10²
3. 3.09 × 10⁻¹⁰
Express these numbers in standard notation.
1. 8.09 × 10⁰
2. 3.088 × 10⁻⁵
3. −4.239 × 10²
Express these numbers in standard notation.
1. 2.87 × 10⁻⁸
2. 1.78 × 10¹¹
3. 1.381 × 10⁻²³
These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
1. 72.44 × 10³
2. 9,943 × 10⁻⁵
3. 588,399 × 10²
These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
1. 0.000077 × 10⁻⁷
2. 0.000111 × 10⁸
3. 602,000 × 10¹⁸
These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
1. 345.1 × 10²
2. 0.234 × 10⁻³
3. 1,800 × 10⁻²
These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
1. 8,099 × 10⁻⁸
2. 34.5 × 10⁰
3. 0.000332 × 10⁴
Write these numbers in scientific notation by counting the number of places the decimal point is moved.
1. 123,456.78
2. 98,490
3. 0.000000445
Write these numbers in scientific notation by counting the number of places the decimal point is moved.
1. 0.000552
2. 1,987
3. 0.00000000887
Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
1. 456 × (7.4 × 10⁸) = ?
2. (3.02 × 10⁵) ÷ (9.04 × 10¹⁵) = ?
3. 0.0044 × 0.000833 = ?
Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
1. 98,000 × 23,000 = ?
2. 98,000 ÷ 23,000 = ?
3. (4.6 × 10⁻⁵) × (2.09 × 10³) = ?
Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
1. 45 × 132 ÷ 882 = ?
2. [(6.37 × 10⁴) × (8.44 × 10⁻⁴)] ÷ (3.2209 × 10¹⁵) = ?
Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
1. (9.09 × 10⁸) ÷ [(6.33 × 10⁹) × (4.066 × 10⁻⁷)] = ?
2. 9,345 × 34.866 ÷ 0.00665 = ?

Click here for Solutions

Expressing Units

Learning Objectives

Learn the units that go with various quantities.
Express units using their abbreviations.
Make new units by combining numerical prefixes with units.

A number indicates “how much,” but the unit indicates “of what.” The “of what” is important when communicating a quantity. For example, if you were to ask a friend how close you are to Lake Erie and your friend says “six,” then your friend isn’t giving you complete information. Six what? Six miles? Six inches? Six city blocks? The actual distance to the lake depends on what units you use.

Chemistry, like most sciences, uses the International System of Units, or SI for short. (The letters SI stand for the French “le Système International d’unités.”) SI specifies certain units for various types of quantities, based on seven fundamental units for various quantities. We will use most of the fundamental units in chemistry. Initially, we will deal with three fundamental units. The meter (m) is the SI unit of length. It is a little longer than a yard (see Figure 1.1 "The Meter"). The SI unit of mass is the kilogram (kg), which is about 2.2 pounds (lb). The SI unit of time is the second (s).

Figure 1.1 The Meter

Image showing comparing a yard and a meter. One yard equals 0.914 meters.

The SI standard unit of length, the meter, is a little longer than a yard.

To express a quantity, you need to combine a number with a unit. If you have a length that is 2.4 m, then you express that length as simply 2.4 m. A time of 15,000 s can be expressed as 1.5 × 10⁴ s in scientific notation.

Sometimes, a given unit is not an appropriate size to easily express a quantity. For example, the width of a human hair is very small, and it doesn’t make much sense to express it in meters. SI also defines a series of numerical prefixes that refer to multiples or fractions of a fundamental unit to make a unit more conveniently sized for a specific quantity. Table 1.1 "Multiplicative Prefixes for SI Units" lists the prefixes, their abbreviations, and their multiplicative factors. Some of the prefixes, such as kilo-, mega-, and giga-, represent more than one of the fundamental unit, while other prefixes, such as centi-, milli-, and micro-, represent fractions of the original unit. Note, too, that once again we are using powers of 10. Each prefix is a multiple of or fraction of a power of 10.

Table 1.1 Multiplicative Prefixes for SI Units

Prefix	Abbreviation	Multiplicative Amount
giga-	G	1,000,000,000 ×
mega-	M	1,000,000 ×
kilo-	k	1,000 ×
deci-	d	1/10 ×
centi-	c	1/100 ×
milli-	m	1/1,000 ×
micro-	μ*	1/1,000,000 ×
nano-	n	1/1,000,000,000 ×
pico-	p	1/1,000,000,000,000 ×
* The letter μ is the Greek letter lowercase equivalent to an m and is called “mu” (pronounced “myoo”).

To use the fractions to generate new units, simply combine the prefix with the unit itself; the abbreviation for the new unit is the combination of the abbreviation for the prefix and the abbreviation of the unit. For example, the kilometer (km) is 1,000 × meter, or 1,000 m. Thus, 5 kilometers (5 km) is equal to 5,000 m. Similarly, a millisecond (ms) is 1/1,000 × second, or one-thousandth of a second. Thus, 25 ms is 25 thousandths of a second. You will need to become proficient in combining prefixes and units. (You may recognize that one of our fundamental units, the kilogram, automatically has a prefix-unit combination, the kilogram. The word kilogram means 1,000 g.)

In addition to the fundamental units, SI also allows for derived units based on a fundamental unit or units. There are many derived units used in science. For example, the derived unit for area comes from the idea that area is defined as width times height. Because both width and height are lengths, they both have the fundamental unit of meter, so the unit of area is meter × meter, or meter² (m²). This is sometimes spoken as “square meters.” A unit with a prefix can also be used to derive a unit for area, so we can also have cm², mm², or km² as acceptable units for area.

Volume is defined as length times width times height, so it has units of meter × meter × meter or meter³ (m³), sometimes spoken as “cubic meters.” The cubic meter is a rather large unit, however, so another unit is defined that is somewhat more manageable: the liter (L). A liter is 1/1,000th of a cubic meter and is a little more than 1 quart in volume (see Figure 1.2 "The Liter"). Prefixes can also be used with the liter unit, so we can speak of milliliters (1/1,000th of a liter; mL) and kiloliters (1,000 L; kL).

Figure 1.2 The Liter

Image comaring a quart and a liter. One quart equals 0.94 liters

The SI unit of volume, the liter, is slightly larger than 1 quart.

Another definition of a liter is one-tenth of a meter cubed. Because one-tenth of a meter is 10 cm, then a liter is equal to 1,000 cm³ (Figure 1.3 "The Size of 1 Liter"). Because 1 L equals 1,000 mL, we conclude that 1 mL equals 1 cm³; thus, these units are interchangeable.

Figure 1.3 The Size of 1 Liter

Image showing one liter equals one thousand cubic centimeters.

One liter equals 1,000 cm³, so 1 cm³ is the same as 1 mL.

Units not only are multiplied together but also can be divided. For example, if you are traveling at one meter for every second of time elapsed, your velocity is 1 meter per second, or 1 m/s. The word per implies division, so velocity is determined by dividing a distance quantity by a time quantity. Other units for velocity include kilometers per hour (km/h) or even micrometers per nanosecond (μm/ns). Later, we will see other derived units that can be expressed as fractions.

Example 2

A human hair has a diameter of about 6.0 × 10⁻⁵ m. Suggest an appropriate unit for this measurement and write the diameter of a human hair in terms of that unit.
What is the velocity of a car if it goes 25 m in 5.0 s?

Solution

The scientific notation 10⁻⁵ is close to 10⁻⁶, which defines the micro- prefix. Let us use micrometers as the unit for hair diameter. The number 6.0 × 10⁻⁵ can be written as 60 × 10⁻⁶, and a micrometer is 10⁻⁶ m, so the diameter of a human hair is about 60 μm.
If velocity is defined as a distance quantity divided by a time quantity, then velocity is 25 meters/5.0 seconds. Dividing the numbers gives us 25/5.0 = 5.0, and dividing the units gives us meters/second, or m/s. The velocity is 5.0 m/s.

Test Yourself

Express the volume of an Olympic-sized swimming pool, 2,500,000 L, in more appropriate units.
A common garden snail moves about 6.1 m in 30 min. What is its velocity in meters per minute (m/min)?

Answers

2.5 ML
0.203 m/min