Chapter 2 Measurements Cartoon courtesy of Lab-initio.com.

Chapter 2 MeasurementsChapter 2 Measurements

Cartoon courtesy of Lab-initio.com

2.1 Units of Measurement

Learning Goal Write the names and abbreviations for the metric or SI units used in measurements of

length, volume, mass, temperature, and time

Key Math Skills

•Identifying Place Values (1.4A)

Chapter 2 ReadinessChapter 2 Readiness

• Chemists use the metric system and the International System of Units (SI) for measurement when they

• measure quantities

• do experiments

• solve problems

The International System of Units (SI)

Units of Measurement and Their Abbreviations

Metric System

Prefixes convert the base units into units that are appropriate for the item being measured.

Length is measured in

•units of meters (m) in both the metric and SI systems

•units of centimeters (cm) by chemists

Length

Useful relationships between units of length include:

1 m = 1.094 yd

1 m = 39.37 in.

1 m = 100 cm

2.54 cm = 1 in.

Length

Volume

The most commonly used metric units for volume are the liter (L) and the milliliter (mL).» A liter is a cube

1 decimeter (dm) long on each side.

» A milliliter is a cube 1 centimeter (cm) long on each side.

Volume, the space occupied by a substance,

• is measured using units of m3 in the SI system

• is commonly measured in liters (L) and milliliters (mL) by chemists

Volume

Useful relationships between units of volume include:

1 m3 = 1000 L

1 L = 1000 mL

1 mL = 1 cm3

1 L = 1.057 qt

946.3 mL = 1 qt

Volume

The mass of an object, a measure of the quantity of material it contains,

•is measured on an electronic balance

•has the SI unit of kilogram (kg)

•is often measured by chemists in grams (g)

Useful relationships between units of mass include:

1 kg = 1000 g

1 kg = 2.205 lb

453.6 g = 1 lb

Temperature

By definition temperature is a measure of the average kinetic energy of the particles in a sample.

Temperature

In scientific measurements, the Celsius and Kelvin scales are most often used.

The Celsius scale is based on the properties of water.» 0 C is the freezing

point of water.» 100 C is the boiling

point of water.

Temperature

The kelvin is the SI unit of temperature.

It is based on the properties of gases.

There are no negative Kelvin temperatures.

K = C + 273.15

Temperature

The Fahrenheit scale is not used in scientific measurements.

F = 9/5(C) + 32C = 5/9(F − 32)

Time is based on an atomic clock and is measured in units of seconds (s) in both the metric and SI systems.

For each of the following, indicate whether the unit describes

(1) length, (2) mass, or (3) volume

A. A bag of onions has a mass of 2.6 kg.

B. A person is 2.0 m tall.

C. A medication contains 0.50 g of aspirin.

D. A bottle contains 1.5 L of water.

Learning Check

For each of the following, indicate whether the unit describes

(1) length, (2) mass, or (3) volume

A. A bag of onions has a mass of 2.6 kg. (2)

B. A person is 2.0 m tall. (1)

C. A medication contains 0.50 g of aspirin. (2)

D. A bottle contains 1.5 L of water. (3)

Solution

Identify the measurement that has an SI unit.

A. John’s height is _____.

(1) 1.5 yd (2) 6 ft (3) 2.1 m

B. The mass of a lemon is _____.

(1) 12 oz (2) 0.145 kg (3) 0.6 lb

C. The temperature is _____.

(1) 85 ºC (2) 255 K (3) 45 ºF

Learning Check

Identify the measurement that has an SI unit.

A. John’s height is______. (3) 2.1 m

B. The mass of a lemon is _____. (2) 0.145 kg

C. The temperature is _____. (2) 255 K

Solution

Density

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

volume of a substance.

Derived Units

Density is a physical property of a substance.It has units (g/mL, for example) that are derived

from the units for mass and volume.

Density

Substances that have

•higher densities contain particles that are closely packed together

•lower densities contain particles that are farther apart

Metals such as gold and lead have higher densities because their atoms are packed closely together.

Density, Units

In the metric system, densities of solids, liquids, and gases are expressed with different units.

The density of a solid or liquid is usually given in

•grams per cubic centimeter (g/cm3)

•grams per milliliter (g/mL)

The density of a gas is usually given in grams per liter (g/L).

Density of Common Substances

Guide to Calculating Density

Learning Check

Osmium is a very dense metal. What is its density, in g/cm3, if 50.0 g of osmium has a volume of 2.22 cm3?

Solution

Osmium is a very dense metal. What is its density, in g/cm3, if 50.0 g of osmium has a volume of 2.22 cm3?

Step 1 Given 50.0 g; 22.2 cm3

Need density, in g/cm3

Step 2 Plan Write the density expression.

Solution

Osmium is a very dense metal. What is its density, in g/cm3, if 50.0 g of osmium has a volume of 2.22 cm3?Step 3 Express mass in grams and volume

in cm3.

mass = 50.0 g volume = 22.2 cm3

Step 4 Set up problem, calculate.

Link to the Environment

The oil pumped out of the ground is called crude oil, or petroleum. It is mostly made of hydrocarbons, or compounds that contain only carbon and hydrogen.

In April 2010, an oil rig in the Gulf of Mexico exploded, causing the largest oil spill in U.S. history.

It leaked a maximum of 10 million liters a day into the Gulf of Mexico.

Link to the Environment

Because the density of this oil, 0.8 g/mL, was less than that of water, which is 1.00 g/mL, it floated on the surface, spreading a thin layer of oil over a very large surface.

Density of Solids

• The density of a solid is calculated from its mass and volume.

• When a solid is submerged, it displaces a volume of water equal to the volume of the solid.

Density of SolidsIn the figure below, the water level rises from 35.5

mL to 45.0 mL after the zinc object is added.

Learning Check

Scuba divers use lead weights to counteract their buoyancy in water. What is the density of a lead weight that has a mass of 226 g and displaces 20.0 cm3 of water when submerged?

Solution

What is the density of a lead weight that has a mass of 226 g and displaces 20.0 cm3 of water when submerged?

Step 1 Given 226 g lead

20.0 cm3 water displaced

Need density (g/cm3) of lead

Step 2 Plan. Write the density expression.

Solution

Step 3 Express mass in grams and volume

in cm3.

mass of lead weight = 226 g

volume of water displaced = 20.0 cm3

Solution

Density as a Conversion Factor

Density can be used as a conversion factor between a substance’s mass and volume.

Learning Check

If the density of milk is 1.04 g/mL, how many grams of milk are in 0.50 qt?

Solution

Step 1 Given 0.50 qt milk

density of milk is 1.04 g/mL

Need grams of milk

Step 2 Write a plan.

SolutionIf the density of milk is 1.04 g/mL, how many grams of milk are in 0.50 qt?

Step 3 Write equalities, conversion factors.

1 L = 1.057 qt 1 L = 1000 mL

1.04 g = 1 mL

Solution

Concept MapConcept Map

2.2 Scientific Notation

Learning Goal Write a number in scientific notation.

Scientific Notation

Scientific notation is used to write very large or very small numbers such as

•the width of a human hair, 0.000 008 m, which is also written as 8 × 10−6 m•the number of hairs on a human scalp,100 000, which is also written as 1 × 105 hairs

• A number written in scientific notation contains a coefficient and a power of ten.

coefficient power unit

of ten

1.5 × 102 m

• The coefficient is at least 1 but less than 10.

Writing Numbers in Scientific Notation

In science, we deal with some In science, we deal with some very very LARGELARGE numbers: numbers:

1 mole = 6020000000000000000000001 mole = 602000000000000000000000

In science, we deal with some In science, we deal with some very very SMALLSMALL numbers: numbers:

Mass of an electron =Mass of an electron =0.000000000000000000000000000000091 kg0.000000000000000000000000000000091 kg

Scientific NotationScientific Notation

Imagine the difficulty of Imagine the difficulty of calculating the mass of 1 mole calculating the mass of 1 mole of electrons!of electrons!

0.00000000000000000000000000000000.000000000000000000000000000000091 kg91 kg x 602000000000000000000000x 602000000000000000000000

???????????????????????????????????

2 500 000 000

Step #1: Insert an understood decimal pointStep #1: Insert an understood decimal point

Step #2: Decide where the decimal Step #2: Decide where the decimal must end must end up so that one number is to its up so that one number is to its leftleftStep #3: Count how many places you Step #3: Count how many places you bounce bounce the decimal pointthe decimal point

123456789

Step #4: Re-write in the form Step #4: Re-write in the form 2.5 x 102.5 x 10nn

2.5 x 102.5 x 1099

The exponent is the number of places we moved the decimal.

0.00005790.0000579

Step #2: Decide where the decimal Step #2: Decide where the decimal must end must end up so that one number is to its up so that one number is to its leftleftStep #3: Count how many places you Step #3: Count how many places you bounce bounce the decimal pointthe decimal pointStep #4: Re-write in the form 5.79 x 10Step #4: Re-write in the form 5.79 x 10nn

1 2 3 4 5

5.79 x 105.79 x 10-5-5

The exponent is negative because the number we started with was less than 1.

• The number of spaces moved to obtain a coefficient between 1 and 10 is shown as a power of ten.

52 000. = 5.2 × 104

move decimal 4 spaces left

0.003 78 = 3.78 × 10−3

move decimal 3 spaces right

Writing Numbers in Scientific Notation

Some Powers of Ten

Some Measurements in Scientific Notation

Standard Format Scientific Notation Diameter of the Earth

12 800 000 m 1.28 × 107 m

Mass of a human

68 kg 6.8 × 101 kg

Diameter of a virus

Comparing Numbers in Standard and Scientific Notation

You can enter a number written in scientific notation on many calculators using the EE or EXP key.

Scientific Notation and Calculators

When a calculator display appears in scientific notation, it is shown as a number between 1 and 10, followed by a space and the power (exponent).

On many scientific calculators, a number is converted to scientific notation, using the appropriate keys.

0.000 52 2nd or 3rd function key SCI

Key Key

= 5.2 −04 or 5.2−04 = 5.2 × 10−4

Calculator display

Guide to Writing a Number in Scientific Notation

Write the following number in the correct scientific notation, 0.000 058 g.

Learning Check

Step 1 Move the decimal point to obtain a coefficient that is at least 1 but less than 10.0.000 058

Move the decimal 5 places to the right, to give a coefficient of 5.8.

Solution

Step 2 Express the number of places moved as a power of 10.

Moving the decimal 5 places to the right gives a power of −5.

Solution

Step 3 Write the product of the coefficient multiplied by the power of 10 with the unit.5.8 × 10−5 g

Solution

Select the correct scientific notation for each.

A. 0.000 008

(1) 8 × 106 (2) 8 × 10−6 (3) 0.8 × 10−5

B. 72 000

(1) 7.2 × 104 (2) 72 × 103 (3) 7.2 × 10−4

Learning Check

Select the correct scientific notation for each.

A. 0.000 008

(Move the decimal 6 places to right.)

(2) 8 × 10−6

B. 72 000

(Move the decimal 4 places to the left.)

(1) 7.2 × 104

Solution

Write each as a standard number.

A. 2.0 × 10−2

(1) 200 (2) 0.0020 (3) 0.020

B. 1.8 × 105

(1) 180 000 (2) 0.000 018 (3) 18 000

Learning Check

Write each as a standard number.

A. 2.0 × 10−2

(3) 0.020

B. 1.8 × 105

(1) 180 000

Solution

Dimensional Analysis• We use dimensional analysis to convert one

quantity to another.• Most commonly, dimensional analysis utilizes

conversion factors (e.g., 1 in. = 2.54 cm)

2.54 cm

1 in.or

Dimensional Analysis

Use the form of the conversion factor that puts the sought-for unit in the numerator:

Given unit desired unitdesired unit

given unit

Conversion factor

Dimensional Analysis

For example, to convert 8.00 m to inches,» convert m to cm» convert cm to in.

8.00 m 100 cm

2.54 cm 315 in.

Uncertainty in Data

Uncertainty in Measurements

Different measuring devices have different uses and different degrees of accuracy.

Accuracy versus Precision

• Accuracy refers to the proximity of a measurement to the true value of a quantity.

• Precision refers to the proximity of several measurements to each other.

Accuracy and Accuracy and PrecisionPrecision

Low accuracy, high precision

High accuracy, low precision

High accuracy, high precision

Accepted mass of the object is Accepted mass of the object is 10.91 g10.91 g

Trails Mass (g)

1 11.50

2 11.20

3 10.50

4 10.60

5 10.30

mass of object has actual mass of 25.11 gmass of object has actual mass of 25.11 g

Data Set 1 Data Set 2 Data Set 3 Data Set 4

24.06 g 25.12 g 23.76 g 26.51 g

28.09 25.09 g 23.80 g 25.08 g

29.56 g 25.14 g 23.78 g 23.63 g

Significant Figures

• The term significant figures refers to digits that were measured.

• When rounding calculated numbers, we pay attention to significant figures so we do not overstate the accuracy of our answers.

Significant Figures1. A number is a significant figure if it is Not a zero One or more zeros between nonzero digits One or more zeros at the end of a decimal number In the coefficient of a number written in scientific

notation2. A zero is not significant if it is At the beginning of a decimal number written in

scientific notation Used as a placeholder in a large number without a

decimal point

Rules for Counting Rules for Counting Significant FiguresSignificant Figures

55.3255.32 hashas

44 significant figuressignificant figures

1.00041.0004 hashas

0.00050.0005 hashas

6051.006051.00 hashas

Sig Fig In Class PracticeSig Fig In Class PracticeHow many significant figures in each of the following?

0.02 1 sig figs

0.020 2 sig figs

501 3 sig figs

4 sig figs

5000 1 sig figs

5000. 4 sig figs

6051.00

6 sig figs

0.0005 1 sig figs

0.1020 4 sig figs

10001 5 sig figs

142 3 sig figs

0.073 2 sig figs

1.071 4 sig figs

10810 4 sig figs

5.00 3 sig figs

55.320 5 sig figs

4 sig figs

154 3 sig figs

8710 3 sig figs

1.0004 5 sig figs

Significant Figures

• When addition or subtraction is performed, answers are rounded to the least significant decimal place.

• When multiplication or division is performed, answers are rounded to the number of digits that corresponds to the least number of significant figures in any of the numbers used in the calculation.

In calculations

•calculated answers are usually rounded off

•rounding rules are used to obtain the correct number of significant figures

Rounding Off

1. If the first digit to be dropped is 4 or less, then it and all following digits are simply dropped from the number.

2. If the first digit to be dropped is 5 or greater, then the last retained digit is increased by 1.

Rules for Rounding Off

Rounding Off and Significant Figures

Adjust the following calculated answers to give answers with three significant figures:

A. 824.75 cm

B. 0.112486 g

C.8.2 L

Learning Check

Adjust the following calculated answers to give answers with three significant figures:

A. 824.75 cm 825 cm

B. 0.112486 g 0.112 g

C.8.2 L 8.20 L

Solution

In multiplication and division, the final answer is written to have the same number of significant figures (SFs) as the measurement with the fewest SFs.

For example,

24.65 × 0.67 = 16.5155 17 4 SF 2 SF Calculator Final answer

Multiplication and Division with Measured Numbers

When the calculator answer is a small whole number and more significant figures are needed, we can add one or more zeros.

For example,

= 4 4.00

3 SF Calculator Final answer

Adding Significant Zeros

In addition or subtraction, the final answer is written so that it has the same number of decimal places as the measurement having the fewest decimal places.

For example, 2.367 Thousandths place

+ 34.1 Tenths place 36.467 Calculator display 36.5 Answer, rounded off to tenths place

Addition and Subtraction with Measured Numbers

Give an answer for each with the correct number of significant figures.

A.2.19 × 4.2 = (1) 9 (2) 9.2 (3) 9.198

B. 2.54 × 0.0028 = 0.0105 × 0.060

(1) 11.3 (2) 11 (3) 0.041

Learning Check

Give an answer for each with the correct number of significant figures.

A. 2.19 × 4.2 = (2) 9.2

B. 2.54 × 0.0028 = (2) 11 0.0105 × 0.060

Solution

For each calculation, round the answer to give the correct number of digits.

A. 235.05 + 19.6 + 2 = (1) 257 (2) 256.7 (3) 256.65

B. 58.925 – 18.2 =(1) 40.725 (2) 40.73 (3) 40.7

Learning Check

A. 235.05 Hundredths place

+19.6 Tenths place

+ 2 Ones place

256.65 rounds to 257 answer (1)

B. 58.925 Thousandths place

–18.2 Tenths place

40.725 rounds to 40.7 answer (3)

Solution

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