01 Chemical Foundations 6.pdf

1

Chapter 1

Chemistry:

The Science of Change

Alpharetta High School Dr. Sonha Payne

Why Should You Care

About Chemistry?

Chemicals are all around you!

What is Chemistry?

Chemistry is the study of matter (“stuff”) and the changes it can undergo.

Many Familiar Processes are

Chemical Reactions

Combustion

(aka burning) Rusting

Baked goods “rise” due to a chemical

reaction that produces CO2(g) CO2

bubbles

Scientists Seek to

Understand the

World Around Them

Scientists Follow

The Study of Chemistry

Scientists follow a set of guidelines known as the scientific method:

• gather data via observations and experiments

• identify patterns or trends in the collected data

• summarize their findings with a law

• formulate a hypothesis

• with time a hypothesis may evolve into a theory

Process at Center of Scientific

Inquiry is the Scientific Method

1. Make An Observation

An experiment is an opportunity to make an observation. (qualitative or quantitative)

2. Apply the Scientific Method

The scientific method requires making hypotheses about sets of observations and testing the hypotheses.

3. Revise the Hypothesis

When new observations conflict with a current hypothesis, the hypothesis is revised or completely discarded.

Scientific laws, hypotheses, and theories are all part of a way to organize observations about the real world.  Observation

Something that is witnessed and can be recorded.

 Hypothesis

Tentative explanation of observations.

 Theory (MODEL)

 An attempt to explain WHY something happens.

 An encompassing idea that provides a full explanation for known observations.

 Explains observations that have already been made and makes predictions about future observations.

 Law

 summarizes WHAT happens, this is a concise statement that summarizes a fundamental relationship of nature.

States of Matter

 Matter is anything that has mass and occupies space.

 A solid is a substance with a fixed volume and shape.

A liquid is a fluid that has fixed volume but not a fixed shape.

 The shape of a liquid is dictated by the shape of the container.

 A gashas no fixed volume or shape.

 The gas in a balloon adopts the shape of the balloon.

We can convert a substance from one

state to another without changing the

identity of the substance.

Molecular-Level Illustrations of

a Solid, Liquid, and Gas

Solid particles are held closely together in an

ordered fashion.

Liquid particles are close together but are

not held rigidly in position.

Gas particles have significant separation

from each other and move freely. Solids do not conform

to the shape of their container.

Liquids do conform to the shape of their

container.

3

Classification of Matter

 Pure substances are made of one component and are either:  elements - composed of only one type of atom or

 compounds - composed of two or more chemically bonded atoms

 A mixture is a combination of two or more substances in which each substance retains its distinct identity.

 Homogeneous - uniform throughout  Heterogeneous - has distinguishable components

1.2

Homogeneous

Versus

Heterogeneous

Mixtures

In a homogeneous mixture (aka solution),

 uniform composition throughout

 the different parts of the mixture are not visible

In a heterogeneous mixture,

 the composition is not uniform; it varies from one part of the mixture to another

 different parts of the mixture are visible

Mixtures

 A mixture is matter that consists of two or more substances that are physically mixed, not chemically combined

 A mixture can be separated by physical means into its components without changing the identities of the components.

Pure Substances

Elements

and

Compounds

 A Puresubstance is classified as



Matter with a specific composition



An

element

when composed of

one type of atom

simplest kind of matter

Cannot be broken down into simpler substance



A

compound

when composed of

two or more

elements

combined in a definite ratio (H

2

O)

can be broken down by chemical methods

When they are broken down, the pieces have completely different properties than the compound.

Elements are Pure Substances that

Contain Only One Type of Atom



Examples:



Copper (Cu)



aluminum foil (Al)

Elements have abbreviations

called element symbols



Element symbols consist of one or two

letters.



The 1

st

letter of an element symbol is

always capitalized and the 2

nd

letter, if

present, is never capitalized.



H and Al



Na, Au, Sn, Ag, Cu

Compounds Contain Two or More

Elements in a Definite Ratio



Examples:



Salt (NaCl)



Table sugar (C

12

H

22

O

11

)



Water (H

2

O)

Compounds

are Different

from Their

Elements

All

Solutions

are

Homogeneous Mixtures



Mixed molecule by molecule



Every part keeps the properties of the

components.



Can occur between any state of matter.



Solid in liquid- Kool-aid



Liquid in liquid- antifreeze



Gas in gas- air



Solid in solid - brass



Liquid in gas- water vapor

To

tell whether a substance is a

pure substance

or a mixture, it is necessary to determine whether

it can be physically separated into two or more

pure substances.

MATTER

Can it be physically

separated?

Homogeneous Mixture (solution)

Heterogeneous

Mixture Compound Element

MIXTURE

PURE

SUBSTANCE

yes no

Can it be chemically decomposed? no yes

Is the composition uniform? no yes

Colloids Suspensions

Separating mixtures by Physical Processes

(a)

Filtration

is a technique

that uses a porous barrier

to separate a

solid from

a

liquid

.

(b)

Distillation

separates

liquids

based on

differences in their boiling

points.

(c-d) Separation of the

components of ink using

chromatography

Identify Each of the Following as an

Element

,

Compound

,

Homogeneous

Mixture

, or

Heterogeneous Mixture

 Pasta and tomato sauce

 Aluminum foil

 Helium

 Air

 Graphite

 Pepper

 Sugar

 Sugar water

Paint

 Heterogeneous mixture  Element

 Element

 Homogeneous mixture  Element

 Heterogeneous mixture  Compound

5

Properties



Words that describe matter (adjectives)



Physical

Properties-

properties that can be

observed

and measured

without changing

the identity of the substance.



Includes

color, melting point, boiling point



Chemical

Properties-

characteristics that

describe how a substance will react

chemically.



Includes

flammability, reactivity,

etc.

Quantitative versus Qualitative Properties

Quantitative

property

 A property that is

described in terms of a number(with units)

 Involves a measurement

 Mass  Volume  Temperature  Melting point  Boiling point

Qualitative

Property

 A property that is described

in terms of appearance.

 Color  Odor  Taste

Intensive Versus Extensive Properties

Intensive

properties

 do NOT depend on the amount of

substance.  Density  Temperature  Boiling point  color

Extensive

properties

 DEPENDon the amount of

substance.

 Mass  volume

Classify the Physical Properties

of Copper

Qualitative

or

Quantitative

?

Intensive

or

Extensive

?

Density and Magnetism are

Physical Properties

Physical

or

Classify as a

Chemical

property



Melting point



Density



Magnetic



Tarnishes in air



Gasoline is a liquid a room temp.



Gas burns in air.



Gas has a pungent odor.



Physical



Physical



Physical



Chemical



Physical



Chemical

Classify as a

Physical

or

Chemical

property



Iron and oxygen form rust.



Iron is more dense than aluminum.



Magnesium burns brightly when ignited

.



Oil and water do not mix.



Mercury melts at -39

°

C.

Heat Conduction by Diamond:

Physical or Chemical Property?

Physical and Chemical Changes



Physical

Changes – the

state of matter

changes, but the identity of matter does not

change

.



Change of state, appearance, shape, or size



Liquid water differs from ice in appearance but not

composition. Both consist of water molecules.



Chemical

changes – changes in which a

new

form of matter is formed

.



Products have different properties and original

substances no longer exist



Examples: digestion, combustion, oxidation

Physical Changes can be Dramatic!

Chemical Changes:

When one or more

substances are changed into new substances.



Reactants

- stuff you start with



Products

- the NEW stuff you make



NEW PROPERTIES



Not easily reversed

Iron Fe

Iron (III) oxide Fe2O3

Signs of a Chemical Change

change in color or odor

formation of a gas

formation of a precipitate

(solid)

7

Ammonium Dichromate Volcano: What is

the Evidence of a Chemical Change?

Magnesium Reacts with Dry Ice (CO

2

):

What is the Evidence of a

Chemical Change?

Touch Sensitivity of NI

3

: What is the

Evidence of a Chemical Change?

Sugar and Sulfuric Acid: What is the

Evidence of a Chemical Change?

Classify Each of the Following as a

Physical

or

Chemical

Change

 Burning a candle

 Ice melting on the street

 Toasting a marshmallow

 Cutting a pizza

 Polishing a silver bowl

 Rusting iron

 Dissolving in water

 Grinding spices

 Crushing an Al can

 Chemical  Physical  Chemical  Physical  Chemical  Chemical  Physical  Physical  Physical

Measurements in Chemistry



Need a

number

and a

unit

(most of the time)

157.5 centimeters

Number Unit

R e p r e s e n t s a q u a n t i t y F o r e x a m p l e : 2 m e t e r s 2 i s n u m b e r M e t e r s i s u n i t L e n g t h i s q u a n t i t y U n i t s c o m p a r e w h a t i s b e i n g m e a s u r e d t o a d e f i n e d m e a s u r e m e n t s t a n d a r d



Measurements represent a

quantity

37°C

37 is number

°C is unit

Temperature is quantity

SI Measurement



Le Systeme International d’Unites : SI



System of measurement agreed on by

scientists from all over the world in 1960

(related to the metric system)



A decimal system based on 10



Contains 7 base units

Base Units in the Metric and SI Systems

In the metric and SI systems, one unit is used for each type of measurement:

Quantity Metric Unit SI Unit

Length meter (m) meter (m) Volume liter (L) cubic meter (m3₎

Mass gram (g) kilogram (kg) Time second (s) second (s)

Temperature Celsius (C) Kelvin (K) Amount mole (mol) mole (mol)

Energy calorie (cal) joule (J)

Length Measurement

Length

 uses the unit meter(slightly longer than a yard)

 is measured using a meter stick

 (m)in both the metric and SI systems

 use cm a lot in chemistry (about width of little finger)

 km is used instead of miles for highway distances and car speeds in most countries

Inches and Centimeters

The unit of an inch

 is equal to exactly 2.54 centimeters in the metric (SI) system

1 in. = 2.54 cm

Mass

 Measure of the amount of matter in an object or sample

 uses the unit gram (g)in the metric system (454 g = 1 lb)

 uses the unit kilogram (kg) in the SI system (2.2 lbs = 1 kg) 1 kg = 1000 g = 1 x 103 _g

 mass vs. weight

 weight is the measure of gravitational pull on matter  mass does not depend on gravity

 on a new planet, mass would be same but weight could change

Derived SI Units

 come from combining base units

 combine using multiplication or division

Example:

Area: A = length x width = m x m

= m2

9

Area and Volume are Derived From Length

1 m 1.5 m

2.

0

m

3.0 m2

0.75 m

1

m

0.38 m3

1 m

Area = l x w

1.5 m´2.0 m = 3.0 m2

Volume = l x w x h

0.75 m´1.0 m´0.50 m = 0.38 m3

Volume

 amount of space occupied by object  uses the unit liter (L) in the metric system 1 L = 1.06 qt

 uses the unit cubic meter (m3_{)in the SI} system (m3 _{= m x m x m)}

 use cm3 _{in lab a lot (cm x cm x cm)}

1 liter (L) = 1000cm3 = 1000mL 1 cm3 = 1 mL

Temperature

 Is a measure of how hot or cold an object is compared to another object.

 Indicates that heat flows from an object with a higher temperature to an object with a lower temperature

 Is measured using a thermometer

Temperature: How hot or cold an object is

Absolute zero: 0 K, the temperature at which all molecular motion stops.

Temperature:

How hot or cold an object is

C

=

5

9

(

F

-

32

)

F

=

9

5

( )

C

+

32

C

=

K

-

273.15

K=

C

+

273.15

54

 On the Fahrenheit scale, there are 180 °F between the freezing and boiling points, and on the Celsius scale, there are 100 °C.

180 °F = 9 °F = 1.8 °F 100 °C 5 °C 1 °C

 In the formula for the Fahrenheit temperature, adding 32 adjusts the zero point of water from 0 °C to 32 °F. TF = 9 (TC) + 32 °

5

or TF = 1.8(TC) + 32 °

 TC is obtained by rearranging the equation for TF.

TF = 1.8(TC) + 32 °

 Subtract 32 from both sides.

TF – 32 ° = 1.8(TC) + (32 ° – 32 °)

TF – 32 ° = 1.8(TC)

 Divide by 1.8. TF – 32 ° = 1.8 TC

1.8 1.8 TF – 32 ° = TC

1.8

FYI: Where Does the Celsius Formula

Come From?

Normal human body temperature can range over the course of a day from about 36°C in the early morning to about 37°C in the afternoon. Express these two temperatures and the range that they span using the Kelvin scale.

Temperature Conversion from °C to Kelvin

Strategy Use K = °C + 273.15 to convert temperatures from Celsius to Kelvin.

Solution 36°C + 273 = 309 K 37°C + 273 = 310 K

What range do they span?

310 K - 309 K = 1 K

Depending on the precision required, the conversion from

°C to K is often simply done by adding 273, rather than 273.15.

Think About It Remember that converting a temperature from °C to K is different from converting a range or difference in temperature from °C to K.

Temperature Conversion From

°

C to K

What is normal body temperature (37

°

C) in kelvins?

T

K

= T

C

+ 273

= 37

°

C + 273

= 310. K

A body temperature above 39°C constitutes a high fever. Convert this temperature to the Fahrenheit scale.

Temperature Conversion from °C to °F

Solution Temp in °F = ( × 39°C ) + 32°F Temp in °F = 102°F

Think About It Knowing that normal body temperature on the Fahrenheit scale is approximately 98.6°F, 102°F seems like a reasonable

answer.

Temp in °F = ( × temp in °C ) + 32°F

5 9

5 9

Strategy We are given a temperature and asked to convert it to degrees Fahrenheit. We will use the equation below:

Temperature Conversion From

°

C to

°

F

A person with hypothermia has a body temperature of

34.8

°

C.

What is that temperature in

°

F?

TF = 1.8(TC) + 32°

TF = (1.8)(34.8 °C) + 32°

= 62.6 ° + 32 ° = 94.6 °F

Temperature Conversion From

°

F to

°

C

The normal temperature of a chickadee is 105.8

°

F.

What is that temperature in

°

C?

T

C

= T

F

– 32

°

1.8

=

(105.8 – 32

°

)

1.8

=

73.8

°

F = 41.0

°

C

1.8

°

11

Temperature Conversion From

°

F to

°

C

A pepperoni pizza is baked at 455

°

F.

What is that temperature on the Celsius scale?

T

C

= T

F

– 32

°

1.8

=

(455 – 32

°

)

1.8

=

423

°

F = 235

°

C

1.8

°

Why Does Ice Float on Water?

Why Does Aluminum Sink in

Water?

 Ice floats in water because the density of ice is less than the density of water.

 Aluminum sinks in water because its density is greater than the density of water.

 Density is a derived unit from mass and volume

d

=

m

V

The density of

solids

is usually expressed in

g/cm

3

liquids



g/mL

gases



g/L

density of copper

is 8.96 g/cm3 density of _{is 1.0 g/mL} water

Examples:

d m

V

Recall: 1 cm3 _{= 1 mL}

a. In diagram (a), the grey cube has a density of 4.5 g/cm3_{. Is the}

density of the green cube the same, lower than, or higher than the grey cube?

b. In diagram (b), the grey cube has a density of 4.5 g/cm3_{. Is the}

density of the green cube the same, lower than, or higher than the grey cube?

Calculate the Density of a Substance

Low-density lipoprotein (LDL) contains small amounts of

proteins and large amounts of cholesterol. If a 0.380 g

sample of LDL has a volume of 0.362 cm

3

, what is the

density, in g/cm

3

, of the LDL sample?

m = 0.380 g

V = 0.362 cm

3

D

=

m

V

=

0.380

g

0.362

cm

3

=

1.05

g

cm

3

Use the Density to Calculate the

Mass or Volume of a Substance

The density of silver is 10.5 g/mL. A graduated cylinder contains 18.0 mL of water. What is the new water level, in milliliters, after 35.6 g of silver

metal is submerged in the water?

Strategy:

1. Rearrange d = m/V to solve for V. 2. Plug in given info to solve for V. 3. Use DV = Vf – Vi to find final volume.

 V = m/d = 35.6 g/10.5 g/mL = 3.39 mL

Volume of Regular Solids

V=4

3pr3

V=l´w´h

V=pr2h

V=1

3Ah

V=1

3pr2h

sphere cube cylinder pyramid cone

Calculate the Volume and Density of a

Regularly Shaped Object

Calculate the volume and density of a cube of

lead metal (Pb) that has a mass of 373.6 g and

has edges that measure 3.20 cm.

V = l x w x h = (3.20 cm)(3.20 cm)(3.20 cm) = (32.768 cm3_{) = 32.8 cm}3

d = m/v = 373.6 g/32.8 cm3 _{= 11.39 g/cm}3 _{= 11.4 g/cm}3

Calculating the Volume of Irregular

Solids Using Volume Displacement

Volume

=

D

V

=

V

final

–

V

initial Archimedes’ discovery: a solid submerged in a liquid displaces its own volume.

Experimental:

1. Measure a known volume of water in a graduated cylinder.

2. Submerge the irregular solid in the water. The water level will rise.

3. Measure the new water level. The difference in the two water levels is the volume of the irregular solid. Vinitial

Vfinal

Determine the Density of the

Zinc Object

d = m/v = 68.60 g / (45.0 – 35.5 mL) = 7.22 g/mL

Determine the Density of the

Orange Object

d = m/V

= 8.24 g/(23.1 mL- 18.5 mL) = 8.24g/4.6 mL = 1.8 g/mL

Metric Prefixes



Prefixes are added to the base unit names to

represent quantities smaller or larger

M mega 10

6

1,000,000

larger

k

kilo

10

3

1,000

larger

c

centi 10

-2

1/100

smaller

m milli

10

-3

1/1000

smaller

13

Measuring Length Using

Metric Prefixes

Very Large and Very Small Numbers

(Easier to Write in Scientific Notation!)

Item Value Scientific Notation

Width of human hair 0.000 008 m 8 x 10-6 _m

Hairs on a human scalp 100 000 hairs 1 x 105 _hairs

(0.000 008 m) (100 000 hairs)

Scientific Notation

Used to represent large and small numbers.

Numbers are represented as:

M x 10

n

Where M is a number between 1 and 10 (but not equal to 10).

n is a whole number and may

be positive or negative.

Examples:

5.88 x 10

6

= 5,880,000

3.915 x 10

–4

= 0.0003915

Writing Numbers in Scientific Notation

(Appendix 1)

Numbers are represented in scientific notation as:

M x 10

n

To convert a number to scientific notation, move the decimal point to the right of the first non-zero digit.

The number of places the decimal point moved is the number n.

If the decimal point moved to the left, then n is a positive number.

If the decimal point moved to the right, then n is a negative number.

Examples: 3,253 = 3,253. = 3.253 x 103

0.00026 = 0.00026 = 2.6 x 10–4

Write a Number in Scientific Notation

a. 180 000 000 g b. 0.000 06 m c. 750°C d. 0.15 mL e. 0.024 s f. 1500 cm

a. 1.8 x 108 _g b. 6 x 10-5 _m c. 7.5 x 102°C d. 1.5 x 10-1 _mL e. 2.4 x 10-2 _s f. 1.5 x 103 _cm

Convert Scientific Notation to

a Standard Number

a. 3.6 x 10-5 _L

b. 8.75 x 104 _cm c. 3 x 10-2 _mL d. 2.12 x 105 _kg

Multiplying Numbers in Scientific Notation

Without Using a Calculator

Multiplication

Multiplication step 1:

Multiplication step 2:

a´10y

(

)

(

b´10z

)

Example

Scientific Notation and Calculators

Calculators have a special button for

entering numbers in scientific notation.

Usually it is labeled EE or EXP.

On a scientific calculator, there are two ways to enter numbers in scientific notation:

3.2 3 EE

3.2 x 10^3

Both mean 3.2 x 103

Use one way or the other.

Do not mix and match!

3.2 x 10 3 EE wrong!

Calculator displays: 6.409 _{or 6.4 E09 or 6.4 09}

But this means: 6.4 x 109

3.2 Exp 3

´

2.0 Exp 6

=

3

.

2

EE

3

´

2

.

0

EE

6

=

OR

Multiplying Numbers in Scientific

Calculation Using a Calculator

Example

Buttons to push to do the problem:

Dividing Numbers in Scientific Notation

Without Using a Calculator

Division

Division step 1:

Division step 2:

Example

Dividing Numbers in Scientific

Calculation Using a Calculator

Example

Buttons to push to do the problem:

OR

6.9

EE

5

¸

2

.

3

EE

–

2

=

6.9 Exp 5

¸

2.3 Exp – 2

(

)

=

Calculator displays: 307 _{or 3 E07 or 3 07}

But this means: 3 x 107

6.9´105

2.3´10–2

Calculators and

Parentheses

Calculators do operations as they are entered, which can lead to problems.

Example:

If you enter 4 ÷ 2 x 8, your calculator gives you the wrong answer 16. It assumed you meant:

4

15

Calculators and

Parentheses

When in doubt, USE PARENTHESES to keep numbers together.

Example:

Enter 4 ÷ (2 x 8) and your calculator gives you the correct answer. It then knows you mean:

Uncertainty in Measurement

There are two types of numbers used in chemistry:

1) Exactnumbers:

a) are those that have defined values

1 kg = 1000 g

1 dozen = 12 objects

b) are those determined by counting

28 students in a class

2) Inexact numbers:

a) measured by any method other than counting

length, mass, volume, time, speed, etc. 1.5

Precision and Accuracy:

Uncertainty in Measurement

Uncertainty

 All measurements will have a degree of uncertainty.

 Precision

 The reproducibility of a measurement.

 Accuracy

 How close the measurement is to the true value.

Accuracy and Precision

Three students were asked to find the mass of an aspirin tablet. The true mass of the tablet is 0.370 g.

Student A: Results are precise but not accurate

Student B: Results are neither precise nor accurate

Student C: Results are both precise and accurate

The true mass of the tablet is 0.370 g.

All Measurements Have an

IMPLIED Uncertainty

An inexact number must be reported so as to indicate its

uncertainty.

When a scale reports a weight of 166 pounds,

what is the actual weight?

166.1 lbs?

166.2 lbs?

165.7lbs?



The actual weight could be anywhere from 165 to 167

lbs.

16

6

±

1

lb

(

the range of uncertainty appears in the last digit

)

Which Measurement is More Precise?

Precision is related to the concept of Random Error.

 Random Error is error associated with the estimation of the last digit of any measurement.

 A measurement has high precision when the values of repeated measurements fall into a small range.

 The last digit of a measurement is always estimated. The higher the confidence in the last digit, the more precise the measurement.

166 lbs which means 166 ± 1lb (a range from 165-167 lbs)

How Long is the Piece of Wood?

Which Measurement is More Precise?

(a) 0.1 cm is estimated (4.5 cm)

(b) 0.01 cm is estimated (4.55 cm) More Precise!

When a weight is measured, how

do you determine if it is correct?

 The accuracy of a weighing scale determines whether the measurement is correct. Accuracy depends on the sophistication of the measuring device.

 It is possible to be precise but not accurate if there is systematic error.  For example, if a scale isn’t calibrated correctly, a systematic error can

occur.

 If a measurement is reproducible to the tenths place but it is always 1 lb off, a systematic error has occurred.

Systematic error: an error inherent to the measurement of a value.

 Ex: A clock that measures time to the millisecond but is 5 minutes fast is quite precise but not accurate.

Significant Figures

 All certain digits plus one estimated digit  The number of significant figures that a

measurement has indicates how good the measurement is, that is, tells us how good the measuring device was that we used to make the measurement.

Which Ruler Gives a Better

Measurement and Why?

1 cm 2 cm 1.7 cm

• In a measured number, the sig figs are all of the numbers including the estimated digit.

• The estimated digit is always one digit past the smallest hash mark on the measuring device (+/-1).

• Better measuring devices give better measurements with more significant figures.

1 cm 2 cm 1.72 cm

Determining the Number

of Sig Figs in a Measurement

The numbers 1–9 (non-zero) are always significant !

Zero is a special case.

Whether a zero is significant or not depends on the location of the zero in the number.

LEADING zeros never count

EMBEDDED zeros always

count

TRAILING zeros only count if

there is a decimal point.

Zero is a Special Case

Buried

zeros

are significant.

(embedded

between non-zero numbers

)

706

Leading

zeros

are not significant

.

(at front of all non-zero numbers)

0.000234

Trailing

zeros

(at the end of non-zero numbers):

If there is a

decimal

, all are

significant

.

27.00

17

Determine the Number of

Significant Figures

 101.02

 20.0

 0.005302

 17000

 4320.

 237,401

 2.02 x 105

 0.01030

 102.30



Embedded 5



Trailing w/ 3



Leading 4



Trailing w/o 2



Trailing w/ 4



Embedded 6



Sci notation 3



All types 4



Trailing w/ 5

Significant Figures: Exact Numbers

Constants and Counting Numbers

There is no uncertainty in an exact number. Exact numbers have an infinite or unlimited number of sig figs.

There are exactly 12 eggs in a dozen eggs. There are exactly 1000 milliliters in 1 liter.

Classify Each of the Following as

Exact or Measured Numbers

a. A measuring tool is required. b. This is a defined

relationship

c. A measuring tool is used to determine length. d. The number of hats is

obtained by counting. e. The volume of soda is

measured.

a. Measured b. Exact c. Measured d. Exact e. measured

Calculations With Measured Numbers

 In calculations:

 Answers must have the same number of sig figs as the measured numbers.

 Calculator answers must often be rounded off.

 Rounding rules are used to obtain the correct number of sig figs.  When rounding, look at the digit after the one you can keep:

Greater than or equal to 5, ROUND UP

2.4884 rounded to 2 sig figs is 2.5 Less than 5, KEEP THE SAME

45.832 rounded to 3 sig figs is 45.8

 Sometimes a calculated answer requires more sig figs. In this case, one or more zeros are added.

Calculated answer = 4

Zeros added to give 3 sig figs = 4.00

Round off each of the following

calculator displays to 3 Sig Figs



761.

5

0



14.3

3

4



10.

4

4



107

8

9



802

4

.50



203.

5

14



762



14.3



10.4



10800



8020

Using Sig Figs in Calculations:

Addition and Subtraction

The answer obtained by addition or subtraction must contain the same number of decimal places as the quantity in the

calculation with the fewestnumber of decimal places.

Using Sig Figs in Calculations:

Multiplication and Division

The answer obtained by multiplication or division must contain the same number of sig figs as the quantity with the fewestsig

figs used in the calculation.

Example:

Three pennies

each have a mass of

2.5 g

.

What is the total mass?

3

×

2.5 g

= 7.5 g

Exact numbers can be considered to

have an infinite number of significant

figures and do not limit the number of

significant figures in a result.

Exact

(counting number) (measurement) Inexact

Do not round after each step!

Compare the following:

In calculations with multiple steps,

round at the end

of the calculation

to reduce any rounding errors.

1) 3.66 × 8.45 = 30.9 2) 30.9 × 2.11 = 65.2

1) 3.66 × 8.45 = 30.93 2) 30.93 × 2.11 = 65.3

Rounding after each step Rounding at end

In general, keep at least one extra digit

until the end of a multistep calculation.

Perform the following arithmetic operations and report the result to the proper number of significant figures: (a) 317.5 mL + 0.675 mL, (b) 47.80 L – 2.075 L, (c) 13.5 g ÷ 45.18 L, (d) 6.25 cm x 1.175 cm, (e) 5.46x102 g +

4.991x103 _g

Worked Example 1.5

Solution (a) 317.5 mL + 0.675 mL 318.175 mL

(b) 47.80 L - 2.075 L 45.725 L

Strategy Apply the rules for significant figures in calculations, and round each answer to the appropriate number of digits.

← round to 318.2 mL

← round to 45.73 L

Perform the following arithmetic operations and report the result to the proper number of significant figures: (a) 317.5 mL + 0.675 mL, (b) 47.80 L – 2.075 L, (c) 13.5 g ÷ 45.18 L, (d) 6.25 cm x 1.175 cm, (e) 5.46x102 _{g +}

4.991x103 _g

Worked Example 1.5 (cont.)

Strategy Apply the rules for significant figures in calculations, and round each answer to the appropriate number of digits.

Solution

(c) 13.5 g 45.18 L

(d) 6.25 cm×1.175 cm

← round to 0.299 g/L

= 0.298804781 g/L

3 S.F.

4 S.F.

← round to 7.34 cm2

= 7.34375 cm2

19

Perform the following arithmetic operations and report the result to the proper number of significant figures: (a) 317.5 mL + 0.675 mL, (b) 47.80 L – 2.075 L, (c) 13.5 g ÷ 45.18 L, (d) 6.25 cm x 1.175 cm, (e) 5.46x102 _{g +}

4.991x103 _g

Worked Example 1.5 (cont.)

Strategy Apply the rules for significant figures in calculations, and round each answer to the appropriate number of digits.

Solution (e) 5.46 x 102 _g

+ 49.91 x 102 _g

55.37 x 102 _{g = 5.537 x 10}3 _g

Think About It Changing the answer to correct scientific notation doesn’t change the number of significant figures, but in this case it changes the number of places past the decimal place.

Concept Check

You have water in each

graduated cylinder shown. You then add both samples to a beaker (assume that all of the liquid is transferred).

How would you write the number describing the total volume?

3.1 mL

What limits the precision of the total volume?

An empty container with a volume of 9.850 x 102 cm3 is weighed and found to

have a mass of 124.6 g. The container is filled with a gas and reweighed. The mass of the container and the gas is 126.5 g. Determine the density of the gas to the appropriate number of significant figures.

Worked Example 1.6

Solution 126.5 g – 124.6 g mass of gas = 1.9 g

density =

Strategy This problem requires two steps: subtraction to determine the mass of the gas, and division to determine its density. Apply the corresponding rule regarding significant figures to each step.

← one place past the decimal point (two sig figs)

1.9 g

9.850 x 102 _cm3 = 0.00193 g/cm3 ← round to 0.0019 g/cm3

Think About It In this case, although each of the three numbers we started with has four significant figures, the solution only has two significant

figures.

Calculating Percentages

Example: A fruit salad contains 3 bananas, 5 melons, 10 grapes, and 6 strawberries. What percentage of the salad is represented by the strawberries?

Ans.

Equalities

 1 m = 100 cm  1 m = 1000 mm  1 cm = 10 mm  1 L = 1000 mL  1 quart = 4 cups  1 gallon = 4 quarts  1 cm3 _{= 1 mL}  1 kg = 1000 g

 12 inches = 1 foot  1 yard = 3 feet  1 km = 1000 m  1 inch = 2.54 cm  60 s = 1 minute  60 minutes = 1 hour  1 lb = 16 oz  2.2 lb = 1 kg

An

equality

states the same measurement

in two different units.

Conversion Factors



A

conversion factor



is a fraction obtained from an equality.



is written as a ratio with a numerator and a

denominator.



Can be inverted to give two conversion factors for

every equality



Every conversion factor is equal to 1.

 Equality: 1 inch = 2.54 cm

 Conversion Factors:

1

inch

2.54

cm

2.54

cm

Conversion Factors in a Problem



A

conversion factor

:



May be obtained from information in a word

problem



Is written

for that problem only

 Example 1: The price of one pound (1 lb) of red peppers is $2.99

 Example 2: The cost of one gallon of gas is $3.55

1

lbredpeppers

$2.99

$2.99

1

lbredpeppers

1

gallongas

$3.55

$3.55

1

gallongas

Percent as a Conversion Factor

 Example: The thickness of the skin fold at the waist indicated 11% body fat. What conversion factors can be written for body fat using kg?

 Percent used as a conversion factor  Gives the ratio of the parts to the whole  Uses the value 100

 Can be written as two factors

 Example: A food contains 18% (by mass) fat.

18

gfat

100

gfood

100

gfood

18

gfat

%

=

parts

whole

x

100

%

=

parts

100

11kgfat

100kgbodymass

100kgbodymass

11kgfat

1. Write the given and needed units.

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

3. Write equalities and conversion factors.

4. Use conversion factors to cancel the given unit and provide the needed unit.

Unit 1 x Unit 2 = Unit 2

Unit 1

Given x Conversion = Needed unit factor unit

Using Conversion Factors

to Change from One Unit to Another:

Dimensional Analysis

Using Conversion Factors

to Change from One Unit to Another

How many minutes are 2.5 h?

Given unit = 2.5 h

Needed unit = ? min

Plan = h min

Set up problem to cancel hours (h).

Given Conversion Needed unit factor unit

2.5 h x 60 min = 150 min (2 SF) 1 h

Using Conversion Factors

to Change from One Unit to Another

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

 Given unit: 2.44 m Needed unit: cm

 Plan: m cm

 Equality: 1 m = 100 cm

 Conversion factors:

 Set up problem:

2.44 m x = 244 cm

1m

100cm

100cm

1m

100cm

1m

Problem Solving Using Conversion Factors

In radiological imaging, such as PET or CT scans, dosages of pharmaceuticals are based on body mass. If a person weighs 164 lb, what is that body mass in

kg? 1. Given: 164 lb Needed: kg 2. 2.20 lb = 1kg

3. Conversion factors:

1. Set up the problem to cancel units and calculate the answer.

2.20

lb

1

kg

1

kg

21

Using Density as a Conversion Factor

 Density can be written as an equality

 For a substance with a density of 3.8 g/mL, the equality is: 3.8 g = 1 mL

 From this equality, two conversion factors can be written for density:

 The density of octane, a component of gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane?  Given: d = 0.702 g/mL V = 875 mL Needed: kg  Plan: mL  g  kg

 Equalities: density 0.702 g = 1 mL and 1kg = 1000 g  875 mL x x = 0.614 kg

3.8g

1mL

1

mL

3.8

g

0.702g

1mL

1kg

1000g

Using Density as a Conversion Factor

If olive oil has a density of 0.92 g/mL,

how many liters of olive oil are in 285 g of olive oil?

 Given: d = 0.92 g/mL mass= 285 g

 Needed: volume in L

 Plan: g mL L

 Equalities: density 0.92 g = 1 mL and 1 L = 1000 mL

 285 g x 1mL x = 0.31 L

0.92g

1L

1000mL

Using Two or More Conversion Factors to

Change Units

 Often, two or more conversion factors are required to obtain the unit required for the answer.

 Unit 1  Unit 2  Unit 3

 Additional conversion factors are placed in the set-up to cancel each preceding unit.

 Given unit x factor 1 x factor 2 = Needed unit

 Unit 1 x

Unit

2

x = Unit 3

Unit

1

Unit

3

Unit

2

Using Two or More Conversion Factors to

Change Units

 How many minutes are in 1.6 days?

 Given unit: days Needed unit: minutes

 Plan: days  h  min

 Set up problem:

1.6 days x x = 2300 = 2.3 x 103 _min

• Be sure to check your unit cancellation in the set up. • The units in the conversion factor must cancel to give the

correct unit for the answer.

24h

1day

60min

1

h

Using Two or More Conversion Factors to

Change Units

 A bucket contains 4.65 L of water. How many gallons of water is that?

 Given unit: 4.65 L Needed unit: gallons

 Plan: L qt  gal

Equalities: 1.06 qt = 1L 1 gal = 4 qt

 Set up problem:

4.65 L x 1.06qt x = 1.23 gal

1L

1

gal

4

qt

Using Two or More Conversion Factors to

Change Units

 If a ski pole is 3.0 feet in length, how long is the ski pole in mm?

 Given unit: 3.0 ft Needed unit: mm

 Plan: ft  in  mm

 Equalities: 1 ft = 12 in 2.54 cm = 1 in 1 cm = 10 mm

 Set up problem:

3.0 ft x x x = 910 mm

• Check initial unit (ft), check needed unit (mm), • Check that units cancel properly.

12

in

1

ft

2.54

cm

1

in

10

mm

Problem Solving Using Conversion Factors



Always consider what unit you are starting and ending with



If you aren’t sure what steps to take, write down all the info you know about the start and end unit to find a connection



Always begin with the number and unit you are given with a 1 below it



Always cancel units as you go



If you are converting between two units that both have prefixes, convert to the plain unit as an intermediate

Converting Between Two Metric Units

that Both have Prefixes

Convert 500,000

μg to kg



Equality:

1,000,000 μg = 1

g

1 kg = 1000

g



Must use

g

as an intermediate

kg

g

kg

g

g

g

0

.

0005

1000

1

000

,

000

,

1

1

000

,

500











Advanced Conversions

 One difficult type of conversion deals with squared or

cubed units

 Be sure to square or cube the conversion factor you are using to cancel all the units

 If you tend to forget to square or cube the number in the conversion factor, try rewriting the conversion factor instead of just using the exponent

 Remember: m2 _{= m x m and m}3 _{= m x m x m}

Advanced Conversions



Convert:

2000 cm

3

to m

3



No intermediate needed

OR





₀

.

002

3

100

1

100

1

100

1

2000

m

cm

m

cm

m

cm

m

cm

cm

cm

















































3 3

3

₀

.

002

100

1

2000

m

cm

m

cm

















Advanced Conversions

 Another difficult type of conversion deals units that are fractions themselves

 Be sure convert one unit at a time; don’t try to do both at once

 Work on the unit on top first; then work on the unit on the bottom

 Be sure to cross out cancelled units as you go

Advanced Conversions

Convert: 350 g/mL to kg/L

No intermediate needed

OR

L

kg

L

mL

g

kg

mL

g

350

1

1000

1000

1

350







L

kg

g

kg

L

mL

mL

g

350

1000

1

1

1000

23

The Food and Drug Administration (FDA) recommends that dietary sodium intake be no more than 2400 mg per day.

Worked Example 1.7

Solution 2400 mg ×

Strategy The necessary conversion factors are derived from the equalities 1 g = 1000 mg and 1 lb = 453.6 g.

1 lb

453.6 g or 453.6 g 1 lb 1 g

1000 mg or 1000 mg 1 g

1 g

1000 mg × 453.6 g 1 lb = 0.005291 lb

Think About It Make sure that the magnitude of the result is reasonable and that the units have canceled properly. If we had mistakenly multiplied by 1000 and 453.6 instead of dividing by them, the result

(2400 mg×1000 mg/g×453.6 g/lb = 1.089×109 mg2/lb) would be

unreasonably large and the units would not have canceled properly.

The Food and Drug Administration (FDA) recommends that dietary sodium intake be no more than 2400 mg per day.

Worked Example 1.7

Solution 2400 mg ×

Strategy The necessary conversion factors are derived from the equalities 1 g = 1000 mg and 1 lb = 453.6 g.

1 lb

453.6 g or 453.6 g 1 lb 1 g

1000 mg or 1000 mg 1 g

1 g

1000 mg × 453.6 g 1 lb = 0.005291 lb

Think About It Make sure that the magnitude of the result is reasonable and that the units have canceled properly. If we had mistakenly multiplied by 1000 and 453.6 instead of dividing by them, the result

(2400 mg×1000 mg/g×453.6 g/lb = 1.089×109 _mg2_{/lb) would be}

unreasonably large and the units would not have canceled properly.

An average adult has 5.2 L of blood. What is the volume of blood in cubic meters?

Worked Example 1.8

Solution 5.2 L ×

Strategy 1 L = 1000 cm3 _{and 1 cm = 1x10}-2 _{m. When a unit is raised to a}

power, the corresponding conversion factor must also be raised to that power in order for the units to cancel appropriately.

1000 cm3

1 L 1 x 10

-2 _m

1 cm

× = 5.2 x 10-3 _m3

3

Think About It Based on the preceding conversion factors, 1 L = 1×10-3

m3_{. Therefore, 5 L of blood would be equal to 5×10}-3 _m3_{, which is close to}