What is Scientific What is Scientific Notation?Notation? Scientific notation is a way of Scientific notation is a way of

expressing really big numbers or expressing really big numbers or really small numbers.really small numbers.

It is most often used in It is most often used in “scientific” calculations where the “scientific” calculations where the analysis must be very precise.analysis must be very precise.

For very large and very small For very large and very small numbers, scientific notation is numbers, scientific notation is more concise.more concise.

Scientific notation Scientific notation consists of two parts:consists of two parts:A number between 1 and 10A number between 1 and 10

A power of 10A power of 10

N x 10N x 10xx

To change standard To change standard form to scientific form to scientific notation…notation… Place the decimal point so that there is Place the decimal point so that there is

one non-zero digit to the left of the one non-zero digit to the left of the decimal point.decimal point.

Count the number of decimal places Count the number of decimal places the decimal point has “moved” from the decimal point has “moved” from the original number. This will be the the original number. This will be the exponent on the 10.exponent on the 10.

If the original number was less than 1, If the original number was less than 1, then the exponent is negative. If the then the exponent is negative. If the original number was greater than 1, original number was greater than 1, then the exponent is positive.then the exponent is positive.

ExamplesExamples

Given: 289,800,000Given: 289,800,000 Use: 2.898 (moved 8 places)Use: 2.898 (moved 8 places) Answer:Answer: 2.898 x 102.898 x 1088

Given: 0.000567Given: 0.000567 Use: 5.67 (moved 4 places)Use: 5.67 (moved 4 places) Answer:Answer: 5.67 x 105.67 x 10-4-4

To change scientific To change scientific notation to standard notation to standard form…form… Simply move the decimal point Simply move the decimal point

to the right for positive to the right for positive exponent 10. exponent 10.

Move the decimal point to the Move the decimal point to the left for negative exponent 10.left for negative exponent 10.

(Use zeros to fill in places.)(Use zeros to fill in places.)

ExampleExample

Answer: Answer: 5,093,0005,093,000 (moved 6 places to (moved 6 places to the right)the right)

Given: 1.976 x 10Given: 1.976 x 10-4-4

Answer: Answer: 0.00019760.0001976 (moved 4 places to (moved 4 places to the left)the left)

Given: 5.093 x 10Given: 5.093 x 1066

Learning CheckLearning Check

Express these numbers in Express these numbers in Scientific Notation:Scientific Notation:

1)1) 405789405789

2)2) 0.0038720.003872

3)3) 30000000003000000000

4)4) 22

5)5) 0.4782600.478260

Chapter 5 - Chapter 5 - Chemistry IChemistry I

Working with NumbersWorking with Numbers

Significant DigitsSignificant Digits

In science numbers are not just numbers they are measurements, and as we have already discovered ALL measurements have some degree of uncertainty inherently in them.

Because of this, when we combine certain measurements we must have the ability to reflect are uncertainty in our final results.

Scientists’ Answer: SIGNIFICANT DIGITS

Significant Digits Significant Digits (Cont.)(Cont.)

Significant Digits are determined in measurements by following four distinct rules.

Rule 1: ALL non-zero digits are significant. (1-9)

Rule 2: Zeros preceding (coming before) the first non-zero number are NEVER significant. (Leading

Zeros)

Rule 3: Zeros in between non-zero numbers are ALWAYS significant. (Trapped Zeros)

Rule 4: Trailing zeros (zeros at the end of a number) are only significant if a decimal is present.

Significant Digits Significant Digits (Cont.)(Cont.)

Rule 1: ALL non-zero digits are significant.

Example:

12.345 has 5 significant digits since all numbers are non-zero numbers.

Significant Digits Significant Digits (Cont.)(Cont.)

Rule 2: Zeros preceding (coming before) the first non-zero number are NEVER significant. (Leading)

Zeros

Example:

0.0123 has only 3 significant digits. The zeros preceding the number 1 are just keeping space in the number.

Significant Digits Significant Digits (Cont.)(Cont.)

Rule 3: Zeros in between non-zero numbers are ALWAYS significant. (Trapped Zeros)

Example:

10,023 has only 5 significant digits. The zeros between the numbers 1 and 2 are a part of the measurement and must be counted.

Significant Digits Significant Digits (Cont.)(Cont.)

Rule 4: Trailing zeros (zeros at the end of a number) are only significant if a decimal is present.

Example:

100 has only one significant digit since there is no decimal present in the number.

100. Has three significant digits, however, since there is a decimal present.

WHY?

Significant Digits Significant Digits (Cont.)(Cont.) How Many Sig. How Many Sig.

Digs. Do the Digs. Do the following following numbers have?numbers have?

0.00267001 m0.00267001 m 19.0550 kg19.0550 kg 3500 V3500 V 1,809,000 L1,809,000 L

AnswersAnswers

6 significant digits6 significant digits 6 significant digits6 significant digits 2 significant digits2 significant digits 4 significant digits4 significant digits

Significant Digits Significant Digits (Cont.)(Cont.) In scientific calculations we must In scientific calculations we must

account for significant digits account for significant digits because of our uncertainty in because of our uncertainty in measurement.measurement.

We have two separate rules for We have two separate rules for Addition/Subtraction and Addition/Subtraction and Multiplication/DivisionMultiplication/Division

Significant Digits Significant Digits (Cont.)(Cont.)

Rule for Addition/SubtractionRule for Addition/Subtraction The number of significant digits allowed The number of significant digits allowed

in our calculated answer depends on the in our calculated answer depends on the number with the largest uncertainty.number with the largest uncertainty.

Example:Example: 951.0 g 951.0 g + 1407 g + 1407 g + 23.911 g+ 23.911 g + 158.18 g+ 158.18 g 2539.091 g2539.091 g

Significant Digits Significant Digits (Cont.)(Cont.) 951.0 g 951.0 g +1407 g +1407 g

+ 23.911 g+ 23.911 g + 158.18 g+ 158.18 g

2539.091 g2539.091 g

4 sig digs4 sig digs 4 sig digs4 sig digs 5 sig digs5 sig digs 5 sig digs5 sig digs 7 sig digs7 sig digs

The answer is 2540. g with 4 sig digs. We can only express our answer to the most uncertain measurement that we have. In this case, the ones spot.

Significant Digits Significant Digits (Cont.)(Cont.) Rule for Multiplication/DivisionRule for Multiplication/Division The measurement with the The measurement with the

smallest number of significant smallest number of significant digits determines the number of digits determines the number of significant digits in the answer.significant digits in the answer.

Example: V = (3.052 m)(2.10 m)Example: V = (3.052 m)(2.10 m)(0.75 m)(0.75 m)

Significant Digits Significant Digits (Cont.)(Cont.) V = (3.052 m) x (2.10 m) x (0.75 V = (3.052 m) x (2.10 m) x (0.75

m) m) (4 sig figs)(3 sig figs)(2 sig (4 sig figs)(3 sig figs)(2 sig

figs)figs) V = 4.8069 mV = 4.8069 m3 3 (5 sig figs)(5 sig figs)

V = 4.8 mV = 4.8 m33

Significant Digits Significant Digits (Cont.)(Cont.) One Last RuleOne Last Rule

Any numbers that are exact, do not Any numbers that are exact, do not affect the number of significant affect the number of significant digits in the final answer.digits in the final answer.

Exact numbers are constants: Exact numbers are constants: 12 inches/foot; 3.14, 2.54 cm/inch12 inches/foot; 3.14, 2.54 cm/inch

Dimensional Dimensional Analysis in Analysis in ChemistryChemistry

UNITS OF UNITS OF MEASUREMENTMEASUREMENT

Use Use SI unitsSI units — based on the metric — based on the metric systemsystem

Length Length

MassMass

VolumeVolume

TimeTime

TemperatureTemperature

Meter, mMeter, m

Kilogram, kgKilogram, kg

Seconds, sSeconds, s

Celsius degrees, ˚CCelsius degrees, ˚Ckelvins, Kkelvins, K

Liter, LLiter, L

Some Tools for Some Tools for MeasurementMeasurement

Which tool(s) Which tool(s) would you use would you use to measure:to measure:

A.A. temperature temperature

B.B. volume volume

C.C. time time

D.D. weight weight

Learning CheckLearning Check

Match Match L) lengthL) length M) mass M) mass V) V) volumevolume

____ A. A bag of tomatoes is 4.6 kg.____ A. A bag of tomatoes is 4.6 kg.

____ B. A person is 2.0 m tall.____ B. A person is 2.0 m tall.

____ C. A medication contains 0.50 g Aspirin.____ C. A medication contains 0.50 g Aspirin.

____ D. A bottle contains 1.5 L of water.____ D. A bottle contains 1.5 L of water.

M

L

MV

Learning CheckLearning Check

What are some U.S. units that are What are some U.S. units that are used to measure each of the used to measure each of the following?following?

A. length A. length

B. volume B. volume

C. weightC. weight

D. temperatureD. temperature

SolutionSolution

Some possible answers areSome possible answers are

A.A. length length inch, foot, yard, mileinch, foot, yard, mile

B.B. volumevolume cup, teaspoon, gallon, pint, quart cup, teaspoon, gallon, pint, quart

C.C. weight weight ounce, pound (lb), tonounce, pound (lb), ton

D.D. temperature temperature FF

Metric Metric PrefixesPrefixes

Kilo-Kilo- means 1000 of that unit means 1000 of that unit

– 1 kilometer (km) = 1000 meters (m)1 kilometer (km) = 1000 meters (m)

Centi-Centi- means 1/100 of that unit means 1/100 of that unit

– 1 meter (m) = 100 centimeters (cm)1 meter (m) = 100 centimeters (cm)

– 1 dollar = 100 cents1 dollar = 100 cents

Milli-Milli- means 1/1000 of that unit means 1/1000 of that unit

– 1 Liter (L) = 1000 milliliters (mL)1 Liter (L) = 1000 milliliters (mL)

Metric Metric PrefixesPrefixes

Metric Metric PrefixesPrefixes

Units of Units of LengthLength

? kilometer (km) = 500 meters (m)? kilometer (km) = 500 meters (m)

2.5 meter (m) = ? centimeters (cm)2.5 meter (m) = ? centimeters (cm)

1 centimeter (cm) = ? millimeter (mm)1 centimeter (cm) = ? millimeter (mm)

1 nanometer (nm) = 1.0 x 101 nanometer (nm) = 1.0 x 10-9-9 meter meter

O—H distance O—H distance ==9.4 x 109.4 x 10-11 -11 mm9.4 x 109.4 x 10-9 -9 cmcm0.094 nm0.094 nm

O—H distance O—H distance ==9.4 x 109.4 x 10-11 -11 mm9.4 x 109.4 x 10-9 -9 cmcm0.094 nm0.094 nm

Learning CheckLearning Check

Select the unit you would use to measure Select the unit you would use to measure

1. Your height1. Your height

a) millimeters a) millimeters b) metersb) meters c) kilometers c) kilometers

2. Your mass2. Your mass

a) milligramsa) milligrams b) gramsb) grams c) kilograms c) kilograms

3. The distance between two cities3. The distance between two cities

a) millimetersa) millimeters b) metersb) meters c) kilometers c) kilometers

4. The width of an artery4. The width of an artery

a) millimetersa) millimeters b) metersb) meters c) kilometers c) kilometers

SolutionSolution

1. Your height1. Your height

b) metersb) meters

2. Your mass2. Your mass

c) kilogramsc) kilograms

3. The distance between two cities3. The distance between two cities

c) kilometersc) kilometers

4. The width of an artery4. The width of an artery

a) millimetersa) millimeters

EqualitiesEqualities

State the same measurement in State the same measurement in two different unitstwo different units

lengthlength

10.0 in.10.0 in.

25.4 cm25.4 cm

1. 1000 m = 1 1. 1000 m = 1 ______ a) mm b) km c) dma) mm b) km c) dm

2. 0.001 g = 1 2. 0.001 g = 1 ___ ___ a) mg b) kg c) dga) mg b) kg c) dg

3. 0.1 L = 1 3. 0.1 L = 1 ______ a) mL b) cL c) dLa) mL b) cL c) dL

4. 0.01 m = 1 ___ 4. 0.01 m = 1 ___ a) mm b) cm c) dma) mm b) cm c) dm

Learning CheckLearning Check

Conversion FactorsConversion Factors

Fractions in which the numerator and Fractions in which the numerator and denominator are EQUAL quantities denominator are EQUAL quantities expressed in different unitsexpressed in different units

ExampleExample: : 1 in. = 2.54 cm1 in. = 2.54 cm

Factors:Factors: 1 in. 1 in. and and 2.54 cm 2.54 cm

2.54 cm2.54 cm 1 in. 1 in.

Learning CheckLearning Check

Write conversion factors that relate Write conversion factors that relate each of the following pairs of units:each of the following pairs of units:

1. Liters and mL1. Liters and mL

2. Hours and minutes2. Hours and minutes

3. Meters and kilometers3. Meters and kilometers

How many minutes are in How many minutes are in 2.5 hours2.5 hours??

Conversion factorConversion factor

2.5 hr x 2.5 hr x 60 min 60 min = 150 min = 150 min

1 hr1 hr

cancelcancelBy using dimensional analysis / factor-label method, By using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the side up, and the UNITS are calculated as well as the

numbers!numbers!

Sample ProblemSample Problem

You have $7.25 in your You have $7.25 in your pocket in quarters. How pocket in quarters. How many quarters do you have?many quarters do you have?

7.25 dollars 4 quarters7.25 dollars 4 quarters 1 dollar1 dollar

X

= 29 quarters= 29 quarters

Learning CheckLearning Check

A rattlesnake is 2.44 m long. How A rattlesnake is 2.44 m long. How long is the snake in cm?long is the snake in cm?

a) 2440 cma) 2440 cm

b)b) 244 cm244 cm

c)c) 24.4 cm24.4 cm

Learning CheckLearning Check

How many seconds are in 1.4 days?How many seconds are in 1.4 days?

Unit planUnit plan: days hr min seconds: days hr min seconds

1.4 days x 1.4 days x 24 hr24 hr x x ????

1 day1 day

SolutionSolution

Unit planUnit plan: days hr min seconds: days hr min seconds

1.4 day x 1.4 day x 24 hr24 hr x x 60 min60 min x x 60 sec60 sec

1 day 1 hr1 day 1 hr 1 min 1 min

= 1.2 x 10= 1.2 x 1055

secsec

Wait a minute!Wait a minute!

What is What is wrongwrong with the following setup? with the following setup?

1.4 day x 1.4 day x 1 day 1 day x x 60 min 60 min x x 60 60 secsec

24 hr 1 hr 1 24 hr 1 hr 1 minmin

English and Metric English and Metric ConversionsConversions If you know ONE conversion for each If you know ONE conversion for each

type of measurement, you can convert type of measurement, you can convert anything!anything!

You will need to know and use these You will need to know and use these conversions:conversions:– Mass: 454 grams = 1 poundMass: 454 grams = 1 pound– Length: 2.54 cm = 1 inchLength: 2.54 cm = 1 inch– Volume: 0.946 L = 1 quartVolume: 0.946 L = 1 quart

Learning CheckLearning Check

An adult human has 4.65 L of blood. How An adult human has 4.65 L of blood. How many gallons of blood is that?many gallons of blood is that?

Unit planUnit plan:: L qt L qt gallon gallon

Equalities:Equalities: 1 quart = 0.946 L 1 quart = 0.946 L

1 gallon = 4 quarts1 gallon = 4 quarts

Your Setup:Your Setup:

Steps to Problem SolvingSteps to Problem Solving

Read problemRead problem Identify data Identify data Make a unit plan from the initial unit to Make a unit plan from the initial unit to

the desired unitthe desired unit Select conversion factorsSelect conversion factors Change initial unit to desired unitChange initial unit to desired unit Cancel units and checkCancel units and check Do math on calculator Do math on calculator Give an answer using significant figuresGive an answer using significant figures

Dealing with Two Units – Dealing with Two Units – Honors OnlyHonors Only

If your pace on a treadmill is 65 If your pace on a treadmill is 65 meters per minute, how many meters per minute, how many seconds will it take for you to walk a seconds will it take for you to walk a distance of 8450 feet?distance of 8450 feet?

InitialInitial

8450 ft x 8450 ft x 12 in. 12 in. x x 2.54 cm 2.54 cm x x 1 m 1 m

1 ft1 ft 1 in. 1 in. 100 cm 100 cm

x x 1 min 1 min x x 60 sec 60 sec = 2400 sec = 2400 sec

65 m65 m 1 min 1 min

SolutionSolution

Temperature Temperature ScalesScales

FahrenheitFahrenheit CelsiusCelsius KelvinKelvin

Anders Celsius1701-1744

Lord Kelvin(William Thomson)1824-1907

Temperature Temperature ScalesScales

Notice that 1 kelvin = 1 degree Celsius1 kelvin = 1 degree Celsius

Boiling point Boiling point of waterof water

Freezing point Freezing point of waterof water

CelsiusCelsius

100 ˚C100 ˚C

0 ˚C0 ˚C

100˚C100˚C

KelvinKelvin

373373 KK

273273 KK

100100 KK

FahrenheitFahrenheit

32 ˚F32 ˚F

212 ˚F212 ˚F

180˚F180˚F

Calculations Calculations Using Using TemperatureTemperature

Generally require temp’s in kelvinsGenerally require temp’s in kelvins

T (K) = t (˚C) + 273.15T (K) = t (˚C) + 273.15

Body temp = 37 ˚C + 273 = 310 KBody temp = 37 ˚C + 273 = 310 K

Liquid nitrogen = -196 ˚C + 273 = 77 KLiquid nitrogen = -196 ˚C + 273 = 77 K

Generally require temp’s in kelvinsGenerally require temp’s in kelvins

T (K) = t (˚C) + 273.15T (K) = t (˚C) + 273.15

Body temp = 37 ˚C + 273 = 310 KBody temp = 37 ˚C + 273 = 310 K

Liquid nitrogen = -196 ˚C + 273 = 77 KLiquid nitrogen = -196 ˚C + 273 = 77 K