Chapter Objectives

Chapter Objectives

• Understand:

  • How and why measurements are made

  • How and what units are used

  • How to predict whether reported measurements are valid

  • How to use standard notation for large and small numbers

• Be able to:

  • Distinguish between measured numbers and exact numbers

  • Make conversions from one set of units to another

  • Choose problem-solving techniques that simplify calculations

  • Calculate the density of a substance and use density to convert between mass and volume

  • Understand and use the terms for physical quantities described in measurements

Readiness Key Math Skills

• Identifying Place Values (1.4A)

• Using Positive and Negative Numbers in Calculations (1.4B)

• Calculating a Percentage (1.4C)

• Writing Numbers in Scientific Notation (1.4F)

Units of Measurement

• The metric system is the standard system of measurement used in chemistry.

• Learning Goal: Write the names and abbreviations for the metric or SI units used in measurements of length, volume, mass, temperature, and time.

Units of Measurement - Metric and SI

• Scientists use the metric system of measurement and have adopted a modification of the metric system called the International System of Units as a worldwide standard.

• The International System of Units (SI) is an official system of measurement used throughout the world for units of length, volume, mass, temperature, and time.

Length: Meter (m) and Centimeter (cm)

• 1 m = 100 cm

• 1 m = 1.09 yd

• 1 m = 39.4 in.

• 2.54 cm = 1 in.

Volume: Liter (L) and Milliliter (mL)

• 1 L = 1000 mL

• 1 L = 1.06 qt

• 946 mL = 1 qt

• Graduated cylinders are used to measure small volumes.

Mass: Gram (g) and Kilogram (kg)

• 1 kg = 1000 g

• 1 kg = 2.20 lb

• 454 g = 1 lb

• The mass of a nickel is 5.01 g on an electronic scale.

Temperature: Celsius (°C) and Kelvin (K)

• Water freezes: 32 °F 0 °C

• The Kelvin scale for temperature begins at the lowest possible temperature, 0 K.

• A thermometer is used to measure temperature.

Time: Second (s)

• The second (s) is the correct metric and SI unit for time.

• The standard measure for 1 s is an atomic clock.

• A stopwatch is used to measure the time of a race.

Study Check

• Identify the SI units for each of the following:

  • A. volume

  • B. mass

  • C. length

  • D. temperature

Solution

• Identify the SI units for each of the following:

  • A. The SI unit for volume is the cubic meter, m^3.

  • B. The SI unit for mass is the kilogram, kg.

  • C. The SI unit for length is the meter, m.

  • D. The SI unit for temperature is the kelvin, K.

Measured Numbers and Significant Figures

• Learning Goal: Identify a number as measured or exact; determine the number of significant figures in a measured number.

• Length is measured by observing the marked lines at the end of a ruler. The last digit in your measurement is an estimate, obtained by visually dividing the space between the marked lines.

Measured Numbers

• Measured numbers are the numbers obtained when you measure a quantity such as your height, weight, or temperature.

• To write a measured number:

  • observe the numerical values of marked lines

  • estimate value of the numbers between marks

  • use the estimated number as the final number in your measured number

Writing Measured Numbers for Length

• The lengths of the objects are measured as:

  • (a) 4.5 cm

  • (b) 4.55 cm

  • (c) 3.0 cm

Significant Figures

• In a measured number, the significant figures (SFs) are all the digits including the estimated digit.

• Significant figures are:

  • used to represent the amount of error associated with a measurement

  • all nonzero digits and zeros between digits

  • not zeros that act as placeholders before digits

  • zeros at the end of a decimal number

• Core Chemistry Skill: Counting Significant Figures

Scientific Notation and Significant Zeros

• When one or more zeros in a large number are significant, they are shown clearly by writing the number in scientific notation.

• In this book, we place a decimal point after a significant zero at the end of a number.

• For example, if only the first zero in the measurement 300 m is significant, the measurement is written as 3.0 × 10^2 m.

Scientific Notation and Significant Zeros

• Zeros at the end of large standard numbers without a decimal point are not significant.

  • 400 000 g is written with one SF as 4 × 10^5 g

  • 850 000 m is written with two SF as 8.5 × 10^5 m

• Zeros at the beginning of a decimal number are used as placeholders and are not significant.

  • 0.000 4 s is written with one SF as 4 × 10^{-4} s

  • 0.000 0046 g is written with two SF as 4.6 × 10^{-6} g

Study Check

• Identify the significant and nonsignificant zeros in each of the following numbers and write each number in the correct scientific notation.

  • A. 0.002 650 m

  • B. 43.026 g

  • C. 1 044 000 L

Solution

• Identify the significant and nonsignificant zeros in each of the following numbers and write each number in the correct scientific notation.

  • A. 0.002 650 m is written as 2.650 × 10^{-3} m four SF

    • The zeros preceding the 2 are not significant.

    • The digits 2, 6, and 5 are significant.

    • The zero in the last decimal place is significant.

  • B. 43.026 g is written as 4.3026 × 10^1 g five SF

    • The zeros between nonzero digits or at the end of decimal numbers are significant.

Solution

• Identify the significant and nonsignificant zeros in each of the following numbers and write each number in the correct scientific notation.

• 1 044 000 L is written as 1.044 × 10^6 L four SF

  • The zeros between nonzero digits are significant.

  • The zeros at end of a number with no decimal are not significant.

Exact Numbers

• Exact numbers are:

  • not measured and do not have a limited number of significant figures

  • do not affect the number of significant figures in a calculated answer

  • numbers obtained by counting: 8 cookies

  • in definitions that compare two units in the same measuring system: 6 eggs, 1 qt = 4 cups, 1 kg = 1 000 g

Study Check

• Identify the numbers below as measured or exact, and give the number of significant figures in each measured number.

  • A. 3 coins

  • B. The diameter of a circle is 7.902 cm.

  • C. 60 min = 1 h

Solution

• Identify the numbers below as measured or exact, and give the number of significant figures in each measured number.

  • A. 3 coins is a counting number and therefore is an exact number.

  • B. The diameter of a circle is 7.902 cm. This is a measured number and the zero is significant, so it contains four SF.

  • C. 60 min = 1 h is exact by definition.

Prefixes and Equalities

• Learning Goal Use the numerical values of prefixes to write a metric equality.

Prefixes

• A special feature of the SI as well as the metric system is that a prefix can be placed in front of any unit to increase or decrease its size by some factor of ten.

• For example, the prefixes milli and micro are used to make the smaller units:

  • milligram (mg)

  • microgram (μg or mcg)

• Core Chemistry Skill: Using Prefixes

Prefixes and Equalities

• The relationship of a prefix to a unit can be expressed by replacing the prefix with its numerical value.

• For example, when the prefix kilo in kilometer is replaced with its value of 1000, we find that a kilometer is equal to 1000 meters.

  • kilometer = 1000 meters

  • kiloliter = 1000 liters

  • kilogram = 1000 grams

Study Check

Fill in the blanks with the correct prefix.

A. 1000 m = 1 ___m

B. 1 × 10^{-3} g = 1 ___g

C. 0.01 m = 1 ___m

Solution

Fill in the blanks with the correct prefix.

A. 1000 m = 1 ___m

The prefix for 1000 is kilo; 1000 m = 1 km.

B. 1 × 10^{-3} g = 1 ___g

The prefix for 1 × 10^{-3} is milli; 1 × 10^{-3} g = 1 mg.

C. 0.01 m = 1 ___m

The prefix for 0.01 is centi; 0.01 m = 1 cm.

Measuring Length

Ophthalmologists measure the diameter of the eye’s retina in centimeters (cm), while a surgeon measures the length of a nerve in millimeters (mm).

Each of the following equalities describes the same length in a different unit:

1 m = 100 cm = 1 × 10^2 cm
1 m = 1000 mm = 1 × 10^3 mm
1 cm = 10 mm = 1 × 10^1 mm

Measuring Length

The metric length of 1 m is the same as 10 dm, 100 cm, or 1000 mm.

Measuring Volume

Volumes of 1 L or smaller are common in the health sciences.

When a liter is divided into 10 equal portions, each portion is called a deciliter (dL).

The following are examples of some volume equalities:

1 L = 10 dL = 1 × 10^1 dL
1 L = 1000 mL = 1 × 10^3 mL
1 dL = 100 mL = 1 × 10^2 mL

The Cubic Centimeter

The cubic centimeter (abbreviated as cm^3 or cc) is the volume of a cube whose dimensions are 1 cm on each side.

A cubic centimeter has the same volume as a milliliter, and the units are often used interchangeably.

1 cm^3 = 1 cc = 1 mL

10 cm × 10 cm × 10 cm = 1000 cm^3 = 1000 mL = 1 L

Measuring Mass

When you visit the doctor for a physical examination, he or she records your mass in kilograms (kg) and laboratory results in micrograms (μg or mcg).

The following are examples of equalities between different metric units of mass:

1 kg = 1000 g = 1 × 10^3 g
1 g = 1000 mg = 1 × 10^3 mg
1 g = 100 cg = 1 × 10^2 cg
1 mg = 1000 μg, 1000 mcg = 1 × 10^3 μg

Study Check

Identify the larger unit in each of the following:

A. mm or cm

B. kilogram or centigram

C. mL or μL

D. kL or mcL

Solution

Identify the larger unit in each of the following:

A. mm or cm

A millimeter is 0.001 m, smaller than a centimeter, 0.01 m.

B. kilogram or centigram

A kilogram is 1000 g, larger than a centigram, 0.01 g.

C. mL or μL

A milliliter is 0.001 L, larger than a microliter, 0.000 001 L.

D. kL or mcL

A kiloliter is 1000 L, larger than a microliter, 0.000 001 L.

Writing Conversion Factors

Learning Goal Write a conversion factor for two units that describe the same quantity.

Equalities

Equalities:

use two different units to describe the same measured amount

are written for relationships between units of the metric system, U.S. units, or between metric and U.S. units

For example,

1 m = 1000 mm
1 lb = 16 oz
2.20 lb = 1 kg

Core Chemistry Skill: Writing Conversion Factors from Equalities

Dimensional Analysis

Dimensional Analysis is a method for converting a quantity in one unit to an equivalent quantity in another unit by the use of conversion factors.

Conversion factors are:

equalities written as a ratio

used in calculations to convert from one unit to another

Some examples of equalities are shown below:

1 dollar = 20 nickels

5280 feet = 1 mile

60 minutes = 1 hour

1000 milligrams = 1 gram

Equalities can be written as conversion factors as shown below:

20 nickels/dollar

1 mile /5280 feet

60 minutes /hour

1 gram /1000 milligrams

Note that these conversion factors can be inverted and the relationships remain accurate. The conversion factors shown above can be used in calculations to convert from one unit to another. To make a unit change, start with the given quantity and unit; then, write a conversion factor that will allow you to convert

Equalities: Conversion Factors and SF

The numbers in:

any equality between two metric units or between two U.S. system units are obtained by definition and are therefore exact numbers

a definition are exact and are not used to determine significant figures

an equality between metric and U.S. units contain one number obtained by measurement and count toward the significant figures

Exception: The equality 1 in. = 2.54 cm has been defined as an exact relationship, and therefore 2.54 is an exact number.

Some Common Equalities

We can write:

metric conversion factors:

metric–U.S. system conversion factors:

Study Check

Write conversion factors from the equality for each of the following:

A. liters and milliliters

B. meters to inches

C. meters and kilometers

Solution

Write conversion factors from the equality for each of the following:

Conversion Factors in a Problem

A conversion factor:

may be stated within a problem that applies only to that problem

is written for that problem only

Example: The car was traveling at 85 km/h.

Example: One tablet contains 500 mg of vitamin C.

Conversion Factors: Dosage Problems

Equalities stated within dosage problems for medications can also be written as conversion factors.

Example: Keflex (Cephalexin), an antibiotic used for respiratory and ear infections, is available in 250-mg capsules.

The 250 mg is measured: It has two significant figures

The 1 in 1 capsule is an exact number.

Conversion Factors: Dosage Problem

A patient is prescribed heparin at 80 units/kg (read 80 units of heparin per kg of body weight).

How many units of heparin should be given to the patient if she weighs 112 lb?

Percent as a Conversion Factor

A percent factor:

gives the ratio of the parts to the whole. % = parts/whole × 100

uses the same unit in the numerator and denominator.

uses the value 100.

can be written as two factors.

Example: A sample of ground beef contains 30% (by mass) fat.

30 g fat/100 g beef and 100 g beef/30 g fat

Study Check

Write the equality and its corresponding conversion factors, and identify each number as exact or give its significant figures for the following statement:

Salmon contains 1.9% omega-3 fatty acids, by mass.

Solution

Write the equality and its corresponding conversion factors, and identify each number as exact or give its significant figures for the following statement:

Salmon contains 1.9% omega-3 fatty acids.

Study Check

Write the equality and conversion factors for each of the following:

A. meters and centimeters

B. jewelry that contains 18% gold

C. One gallon of gas is $3.40.

Solution

Write the equality and conversion factors for each of the following:

A. meters and centimeters

B. jewelry that contains 18% gold

C. One gallon of gas is $3.40.

Problem Solving Using Unit Conversions

Learning Goal Use conversion factors to change from one unit to another.

Guide to Problem Solving Using Conversion Factors

The process of problem solving in chemistry often requires one or more conversion factors to change a given unit to the needed unit.

Problem solving requires identification of:

the given quantity units

the units needed

conversion factors that connect the given and needed units

Core Chemistry Skill: Using Conversion Factors

Solving Problems Using Conversion Factors

Example: If a person weighs 164 lb, what is the body mass in kilograms?

STEP 1 State the given and needed quantities.

STEP 2 Write a plan to convert the given unit to the needed unit.

pounds kilograms

ANALYZE
GIVEN
NEED
THE PROBLEM

164 lb
kilograms
U.S.−Metric Conversion Factor

Solving Problems Using Conversion Factors

Example: If a person weighs 164 lb, what is the body mass in kilograms?

STEP 3 State the equalities and conversion factors.

STEP 4 Set up the problem to cancel units and calculate the answer.

Study Check

A rattlesnake is 2.44 m long. How many centimeters long is the snake?

Solution

A rattlesnake is 2.44 m long. How many centimeters long is the snake?

STEP 1 State the given and needed quantities.

STEP 2 Write a plan to convert the given unit to the needed unit.

meters centimeters

ANALYZE
GIVEN
NEED
THE PROBLEM

2.44 m
centimeters
Metric Factor

Solution

A rattlesnake is 2.44 m long. How many centimeters long is the snake?

STEP 3 State the equalities and conversion factors.

STEP 4 Set up the problem to cancel units and calculate the answer.

Using Two or More Conversion Factors

In problem solving,

two or more conversion factors are often needed to complete the change of units

to set up these problems, one factor follows the other

each factor is arranged to cancel the preceding unit until the needed unit is obtained

Using Two or More Conversion Factors

Example: A doctor’s order prescribed a dosage of 0.150 mg of Synthroid. If tablets in stock contain 75 mcg of Synthroid, how many tablets are required to provide the prescribed medication?

STEP 1 State the given and needed quantities.

STEP 2 Write a plan to convert the given unit to the needed unit.

milligrams micrograms number of tablets

ANALYZE
GIVEN
NEED
THE PROBLEM

0.150 mg Synthroid
number of tablets
Metric Factor Clinical Factor

Solving Problems Using Conversion Factors

Example: A doctor’s order prescribed a dosage of 0.150 mg of Synthroid. If tablets in stock contain 75 mcg of Synthroid, how many tablets are required to provide the prescribed medication?

STEP 3 State the equalities and conversion factors.

STEP 4 Set up the problem to cancel units and calculate the answer.

Study Check

How many minutes are in 1.4 days?

Solution

How many minutes are in 1.4 days?

STEP 1 State the given and needed quantities.

STEP 2 Write a plan to convert the given unit to the needed unit.

days hours minutes

ANALYZE
GIVEN
NEED
THE PROBLEM

1.4 days
minutes
Time Factor Time Factor

Solution

How many minutes are in 1.4 days?

STEP 3 State the equalities and conversion factors.

STEP 4 Set up the problem to cancel units and calculate the answer.

Study Check

If your pace on a treadmill is 65 meters per minute, how many minutes will it take for you to walk a distance of 7.5 kilometers?

Solution

If your pace on a treadmill is 65 meters per minute, how many minutes will it take for you to walk a distance of 7.5 kilometers?

STEP 1 State the given and needed quantities.

STEP 2 Write a plan to convert the given unit to the needed unit.

kilometers meters minutes

ANALYZE
GIVEN
NEED
THE PROBLEM

65 meters/min minutes

7. 5 kilometers

Metric Factor Speed Factor

Solution

If your pace on a treadmill is 65 meters per minute, how many minutes will it take for you to walk a distance of 7.5 kilometers?

STEP 3 State the equalities and conversion factors.

STEP 4 Set up the problem to cancel units and calculate the answer.

Density

Learning Goal Calculate the density of a substance; use the density to calculate the mass or volume of a substance.

Objects that sink in water are more dense than water; objects that float are less dense.

Densities of Common Substances

Densities of Common Substances

Density

Density compares the mass of an object to its volume.

Volume by Displacement

A solid:

completely submerged in water displaces its own volume of water

has a volume calculated from the volume difference

45. 0 mL − 35.5 mL = 9.5 mL = 9.5 cm^3

Density Using Volume Displacement

The density of the zinc object is calculated from its mass and volume.

Study Check

What is the density (g/cm^3) of a 48.0-g sample of a metal if the level of water in a graduated cylinder rises from 25.0 mL to 33.0 mL after the metal is added?

Solution

What is the density (g/cm^3) of a 48.0-g sample of a metal if the level of water in a graduated cylinder rises from 25.0 mL to 33.0 mL after the metal is added?

Problem Solving Using Density

If the volume and the density of a sample are known, the mass in grams of the sample can be calculated by using density as a conversion factor.

Core Chemistry Skill: Using Density as a Conversion Factor

Problem Solving Using Density

Example: John took 2.0 teaspoons (tsp) of cough syrup for a cough. If the syrup had a density of 1.20 g/mL and there is 5.0 mL in 1 tsp, what was the mass, in grams, of the cough syrup?

STEP 1 State the given and needed quantities.

STEP 2 Write a plan to calculate the needed quantity.

teaspoons milliliters grams

ANALYZE
GIVEN
NEED
THE PROBLEM

2. 0 tsp syrup
   grams of syrup

density syrup 1.20 g/mL

U.S.−Metric Factor Density Factor

Problem Solving Using Density

Example: John took 2.0 teaspoons (tsp) of cough syrup for a cough. If the syrup had a density of 1.20 g/mL and there is 5.0 mL in 1 tsp, what was the mass, in grams, of the cough syrup?

STEP 3 Write the equalities and conversion factors.

STEP 4 Set up the problem to calculate the needed quantity.

Study Check

An unknown liquid has a density of 1.32 g/mL. What is the volume of a 14.7-g sample of the liquid?

Solution

An unknown liquid has a density of 1.32 g/mL. What is the volume (mL) of a 14.7-g sample of the liquid?

STEP 1 State the given and needed quantities.

STEP 2 Write a plan to calculate the needed quantity.

gramsmilliliters

ANALYZE
GIVEN
NEED
THE PROBLEM

14. 7 g liquid
    volume of liquid

density liquid 1.32 g/mL

Density Factor

Solution

An unknown liquid has a density of 1.32 g/mL. What is the volume (mL) of a 14.7-g sample of the liquid?

STEP 3 Write the equalities and conversion factors.

STEP 4 Set up the problem to calculate the needed quantity.

Concept Map

CHEMISTRY AND MEASUREMENTS