Measurement andMeasurement andSignificant FiguresSignificant Figures

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Steps in the Scientific MethodSteps in the Scientific Method

1.1. Pose problem; Make Pose problem; Make ObservationsObservations-- quantitative (numerical) Ex: quantitative (numerical) Ex:

length, height, mass. Benefits: non-length, height, mass. Benefits: non-biasedbiased

- - qualitative (descriptive) Ex: qualitative (descriptive) Ex: color, smell, taste. Drawback: biased color, smell, taste. Drawback: biased (opinion)(opinion)

2.2.Formulating hypothesesFormulating hypotheses- - possible explanation for the possible explanation for the

observationobservation3.3.Performing experimentsPerforming experiments

- - gathering new information to gathering new information to decidedecide

whether the hypothesis is validwhether the hypothesis is valid

Outcomes Over the Long-TermOutcomes Over the Long-Term

Theory (Model)Theory (Model)- - A set of tested hypotheses that A set of tested hypotheses that give an overall explanation of some give an overall explanation of some natural phenomenon.natural phenomenon.

Natural LawNatural Law-- The same observation applies to The same observation applies to many different systemsmany different systems

Law vs. TheoryLaw vs. Theory

A A lawlaw summarizes what happens summarizes what happens

A A theorytheory (model) is an attempt to (model) is an attempt to explain explain whywhy it happens. it happens.

Einstein's theory of gravity describes gravitational forces in terms of the curvature of spacetime caused by the presence of mass

Nature of MeasurementNature of Measurement

Part 1 - Part 1 - numbernumberPart 2 - Part 2 - scale (unit)scale (unit)

Examples:Examples:20 20 gramsgrams

6.63 x 106.63 x 10-34-34 Joule·secondsJoule·seconds

A measurement is a quantitative A measurement is a quantitative observation consisting of 2 parts:observation consisting of 2 parts:

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

SI UnitsSI Units

Derived UnitsDerived Units

A derived unit is calculated by doing some mathematical operation. For Example:

• Volume = L x w x h Ex: m3 or cm3

Volume is a 3-dimensional measurement that measures how much space an object occupies

A common volume measurement used in Chemistry is the Liter (L) and/or the milliliter (ml)

• Area = L x w Ex: m2

Celsius & KelvinCelsius & Kelvin

SI Prefixes Common to ChemistrySI Prefixes Common to Chemistry

Uncertainty in MeasurementUncertainty in Measurement

A digit that must be A digit that must be estimatedestimated is is called called uncertainuncertain. A measurement . A measurement alwaysalways has some degree of has some degree of uncertainty.uncertainty. Measurements are performed

with instruments No instrument can read to an infinite number of decimal places

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

Percent Error:Percent Error:

The percent error indicates how far off an observed result (collected data) is from the The percent error indicates how far off an observed result (collected data) is from the

actualactual value. If a mass of 15.4 is obtained from a lab scale and the actual value should have value. If a mass of 15.4 is obtained from a lab scale and the actual value should have been 20.0g, the percent error is calculated as:been 20.0g, the percent error is calculated as:

%error = /%error = /observed – expectedobserved – expected/ x 100 or // x 100 or /15.4 g – 20.0g/15.4 g – 20.0g/ x 100 = 23% x 100 = 23%expected 20.0gexpected 20.0g

Absolute Error or “Error DigitAbsolute Error or “Error Digit” ” Every measurement in the laboratory comes with some Every measurement in the laboratory comes with some uncertainty. The uncertain digit is always the rightmost digit, or the last digit in the uncertainty. The uncertain digit is always the rightmost digit, or the last digit in the measurement. When using a device such as a graduated cylinder or metric ruler, the actual measurement. When using a device such as a graduated cylinder or metric ruler, the actual measurement may fall between two graduation marks on the device. Therefore, the last number measurement may fall between two graduation marks on the device. Therefore, the last number is estimated and is “uncertain.” The measurement lies between the graduations and is written is estimated and is “uncertain.” The measurement lies between the graduations and is written as +/- the distance of the graduated unit. For Example:as +/- the distance of the graduated unit. For Example:

The reading on the The reading on the graduated cylinder belowgraduated cylinder below The reading in this The reading in this

beakerbeakerwould be recorded as: is 48 ml +/- 10 ml would be recorded as: is 48 ml +/- 10 ml

36.2 ml +/- 1 ml36.2 ml +/- 1 ml where the 8 is estimatedwhere the 8 is estimatedThe 2 is estimated The 2 is estimated

Examples of Absolute Error in readings that come from a measuring device such as a scale, graduated

cylinder or ruler.

A scale that gave the reading: 1.456 g. » The error is +/- 0.001 g (the smallest reading on

the scale

» A reading from a graduated cylinder that reads 45.67ml has a error of +/- 0.01 ml, or the smallest reading from the device.

» A length measurement from a device that reads 1.2578 m has an error of +/- 0.0001 m, or the smallest reading from the device.

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

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

34563456 has has 44 sig figs sig figs..

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

ZerosZeros-- Leading zeros Leading zeros do not do not count as significant figures.count as significant figures.

0.04860.0486 hashas33 sig figs.sig figs.

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

ZerosZeros-- Captive zerosCaptive zeros

always count as significant always count as significant figures.figures.

16.0716.07 hashas44 sig figs.sig figs.

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

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

9.3009.300 hashas44 sig figs.sig figs.

Rules for Counting Significant Rules for Counting Significant Figures - DetailsFigures - Details

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

11 inchinch == 2.542.54 cm, exactlycm, exactly

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

1.0070 m 1.0070 m 5 sig figs5 sig figs

17.10 kg 17.10 kg 4 sig figs4 sig figs

100,890 L 100,890 L 5 sig figs5 sig figs

3.29 x 103.29 x 1033 s s 3 sig figs3 sig figs

0.0054 cm 0.0054 cm 2 sig figs2 sig figs

3,200,000 3,200,000 2 sig figs2 sig figs

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

Multiplication and DivisionMultiplication and Division:: # sig figs in the result equals the # sig figs in the result equals the

number in the least precise number in the least precise measurement used in the measurement used in the calculation.calculation.

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 m3.24 m x 7.0 m

CalculationCalculation Calculator says:Calculator says: AnswerAnswer

22.68 m22.68 m22 23 m23 m22

100.0 g ÷ 23.7 cm100.0 g ÷ 23.7 cm33 4.219409283 g/cm4.219409283 g/cm33 4.22 g/cm4.22 g/cm33

0.02 cm x 2.371 cm0.02 cm x 2.371 cm 0.04742 cm0.04742 cm22 0.05 cm0.05 cm22

710 m ÷ 3.0 s710 m ÷ 3.0 s 236.6666667 m/s236.6666667 m/s 240 m/s240 m/s

1818.2 lb x 3.23 ft1818.2 lb x 3.23 ft 5872.786 lb·ft5872.786 lb·ft 5870 lb·ft5870 lb·ft

1.030 g ÷ 2.87 mL1.030 g ÷ 2.87 mL 2.9561 g/mL2.9561 g/mL 2.96 g/mL2.96 g/mL

Rules for Significant Figures in Rules for Significant Figures in Mathematical OperationsMathematical 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 result equals the number of decimal places in the least precise decimal 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 m3.24 m + 7.0 m

CalculationCalculation Calculator says:Calculator says: AnswerAnswer

10.24 m10.24 m 10.2 m10.2 m

100.0 g - 23.73 g100.0 g - 23.73 g 76.27 g76.27 g 76.3 g76.3 g

0.02 cm + 2.371 cm0.02 cm + 2.371 cm 2.391 cm2.391 cm 2.39 cm2.39 cm

713.1 L - 3.872 L713.1 L - 3.872 L 709.228 L709.228 L 709.2 L709.2 L

1818.2 lb + 3.37 lb1818.2 lb + 3.37 lb 1821.57 lb1821.57 lb 1821.6 1821.6 lblb

2.030 mL - 1.870 mL2.030 mL - 1.870 mL 0.16 mL0.16 mL 0.160 mL0.160 mL