Unit 2:Unit 2:

Significant Figures,Significant Figures,Scientific NotationScientific Notationand Dimensional and Dimensional

AnalysisAnalysis

Cartoon courtesy of NearingZero.net

Nature of MeasurementNature of Measurement

1)1) - - numbernumber

2)2) - - unitunitYou must You must ALWAYSALWAYS include the unit!!! include the unit!!!

Examples:Examples:2020 pounds pounds 20 20 ouncesounces

Mean entirely different weights Mean entirely different weights

Measurement - quantitative (based Measurement - quantitative (based on on numbers) numbers) -made up of 2 parts-made up of 2 parts

The Fundamental SI UnitsThe Fundamental SI Units (le Système International, SI)(le Système International, SI)

Physical Quantity Name Abbreviation

Mass kilogram kg

Length meter m

Time second s

Temperature Kelvin K

Electric Current Ampere A

Amount of Substance mole mol

Luminous Intensity candela cd

SI UnitsSI Units

SI PrefixesSI PrefixesCommon to ChemistryCommon to Chemistry

Prefix Unit Abbr. ExponentKilo k 103

Deci d 10-1

Centi c 10-2

Milli m 10-3

Micro 10-6

Uncertainty in MeasurementUncertainty in Measurement

A A digit that must be digit that must be estimatedestimated is is called called uncertainuncertain. A . A measurementmeasurement always has some degree of always has some degree of uncertainty.uncertainty.

Why Is there Uncertainty?Why Is there Uncertainty? Measurements are performed with instruments No instrument can read to an infinite number of decimal placesWhich of these balances has the greatest

uncertainty in measurement?

Precision and AccuracyPrecision and AccuracyAccuracyAccuracy refers to the agreement of a refers to the agreement of a particular value with the particular value with the truetrue value.value.

PrecisionPrecision refers to the degree of agreement refers to the degree of agreement among several measurements made in the among several measurements made in the same manner.same manner.

Neither accurate nor

precise

Precise but not accurate

Precise AND accurate

Types of ErrorTypes of Error

Random ErrorRandom Error (Indeterminate Error) - (Indeterminate Error) - measurement has an equal probability of measurement has an equal probability of being high or low.being high or low.

Systematic ErrorSystematic Error (Determinate Error) - (Determinate Error) - Occurs in the Occurs in the same directionsame direction each time each time (high or low), often resulting from poor (high or low), often resulting from poor technique or incorrect calibration.technique or incorrect calibration.

Rules for Counting Rules for Counting Significant Figures - DetailsSignificant Figures - Details

Nonzero integersNonzero integers always count always count as significant figures.as significant figures.

34563456 hashas

44 sig figs.sig figs.

Rules for Counting Rules for Counting Significant Figures - DetailsSignificant Figures - Details

ZerosZeros-- Leading zerosLeading zeros do not count do not count as as

significant figuressignificant figures..

0.04860.0486 has has

33 sig figs. sig figs.

Rules for Counting Rules for Counting Significant Figures - DetailsSignificant Figures - Details

ZerosZeros-- Captive zeros Captive zeros always always

count ascount assignificant figures.significant figures.

16.07 16.07 hashas

44 sig figs. sig figs.

Rules for Counting Rules for Counting Significant Figures - DetailsSignificant Figures - Details

ZerosZerosTrailing zerosTrailing zeros are significant are significant only if the number contains a only if the number contains a decimal point.decimal point.

9.3009.300 has has

44 sig figs. sig figs.

Rules for Counting Rules for Counting Significant Figures - DetailsSignificant Figures - Details

Exact numbersExact numbers have an infinite have an infinite number of significant figures.number of significant figures.

11 inch = inch = 2.542.54 cm, exactlycm, exactly

Sig Fig Practice #1Sig Fig Practice #1How many significant figures in each of the following?

1.0070 m

5 sig figs

17.10 kg 4 sig figs

100,890 L 5 sig figs

3.29 x 103 s 3 sig figs

0.0054 cm 2 sig figs

3,200,000 2 sig figs

Rules for Significant Figures in Rules for Significant Figures in Mathematical OperationsMathematical Operations

Multiplication and DivisionMultiplication and Division:: keep keep the same number of sig. figs. In the the same number of sig. figs. In the answer as the least number of sig. answer as the least number of sig. figs in the problem.figs in the problem.

6.38 x 2.0 =6.38 x 2.0 =

12.76 12.76 13 (2 sig figs)13 (2 sig figs)

Sig Fig Practice #2Sig Fig Practice #2

3.24 m x 7.0 m

Calculation Calculator says: Answer

22.68 m2 23 m2

100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 4.22 g/cm3

0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2

710 m ÷ 3.0 s 236.6666667 m/s 240 m/s

1818.2 lb x 3.23 ft 5872.786 lb·ft 5870 lb·ft

1.030 g ÷ 2.87 mL 2.9561 g/mL 2.96 g/mL

Rules for Significant Figures Rules for Significant Figures in Mathematical Operationsin Mathematical Operations

Addition and SubtractionAddition and Subtraction: The : The number of decimal places in the number of decimal places in the result equals the number of decimal result equals the number of decimal places in the least precise places in the least precise measurement.measurement.

6.8 + 11.934 =6.8 + 11.934 =

18.734 18.734 18.7 ( 18.7 (3 sig figs3 sig figs))

Sig Fig Practice #3Sig Fig Practice #3

3.24 m + 7.0 m

Calculation Calculator says: Answer

10.24 m 10.2 m

100.0 g - 23.73 g 76.27 g 76.3 g

0.02 cm + 2.371 cm 2.391 cm 2.39 cm

713.1 L - 3.872 L 709.228 L 709.2 L

1818.2 lb + 3.37 lb 1821.57 lb 1821.6 lb

2.030 mL - 1.870 mL 0.16 mL 0.160 mL

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

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Scientific Scientific Notation:Notation:A method of representing very large A method of representing very large

or very small numbers in the or very small numbers in the form:form:

M x 10nM x 10n MM is a number between is a number between 11 and and 1010 nn is an integer is an integer

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 M x 10Step #4: Re-write in the form M 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 M x 10Step #4: Re-write in the form M 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.

PERFORMING PERFORMING CALCULATIONCALCULATION

S IN S IN SCIENTIFIC SCIENTIFIC NOTATIONNOTATION

ADDITION AND ADDITION AND SUBTRACTIONSUBTRACTION

ReviewReview::Scientific notation Scientific notation expresses a number in the expresses a number in the form:form: M x 10M x 10nn

1 1 M M 1010

n is an n is an integerinteger

4 x 104 x 1066

+ 3 x 10+ 3 x 1066

IFIF the exponents the exponents are the same, we are the same, we simply add or simply add or subtract the subtract the numbers in front numbers in front and bring the and bring the exponent down exponent down unchanged.unchanged.

77 x 10x 1066

4 x 104 x 1066

- 3 x 10- 3 x 1066

The same holds The same holds true for true for subtraction in subtraction in scientific scientific notation.notation.

11 x 10x 1066

Scientific Notation #10Scientific Notation #10

4 x 104 x 1066

+ 3 x 10+ 3 x 1055

If the exponents If the exponents are NOT the are NOT the same, we must same, we must move a decimal to move a decimal to makemake them the them the same.same.

4.00 x 104.00 x 1066

+ + 3.00 x 103.00 x 1055

Student AStudent A40.0 x 1040.0 x 1055

43.0043.00 x 10x 1055 Is this Is this good good

scientific scientific notation?notation?

NO!NO!

== 4.300 x 104.300 x 1066

To avoid To avoid this this problem, problem, move the move the decimal on decimal on the the smallersmaller number!number!

4.00 x 104.00 x 1066

+ + 3.00 x 103.00 x 1055

Student BStudent B

.30 x 10.30 x 1066

4.304.30 x 10x 1066 Is this Is this good good

scientific scientific notation?notation?

YESYES!!

A Problem for A Problem for you…you…

2.37 x 102.37 x 10-6-6

+ 3.48 x 10+ 3.48 x 10-4-4

Scientific Notation #14Scientific Notation #14

2.37 x 102.37 x 10-6-6

+ 3.48 x 10+ 3.48 x 10-4-4

Solution…Solution…002.37 x 10002.37 x 10--

66

0.0237 x 100.0237 x 10--

44

3.5037 x 103.5037 x 10-4-4

Direct ProportionsDirect Proportions The quotient of two variables is a constant As the value of one variable increases, the other must also increase As the value of one variable decreases, the other must also decrease The graph of a direct proportion is a straight line

Inverse ProportionsInverse Proportions The product of two variables is a constant As the value of one variable increases, the other must decrease As the value of one variable decreases, the other must increase The graph of an inverse proportion is a hyperbola