Measurement and Uncertainty

The SI System of MeasurementThe SI System of Measurement

The Nature of MeasurementThe Nature of Measurement

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

Examples:Examples:2020 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 observation consisting of TWOTWO partsparts

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 I ntensity candela cd

SI DefinitionsSI Definitionsmeter:meter:

The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

meter:meter:The meter is the length of the

path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

kilogram:kilogram:The kilogram is the unit of

mass; it is equal to the mass of the international prototype of the kilogram.

kilogram:kilogram:The kilogram is the unit of

mass; it is equal to the mass of the international prototype of the kilogram.

SI DefinitionsSI Definitions

second:second:The second is the duration of

9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

second:second:The second is the duration of

9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

SI DefinitionsSI Definitions

ampere:ampere: The ampere is that constant

current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length.

ampere:ampere: The ampere is that constant

current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length.

SI DefinitionsSI Definitions

candela:candela: The candela is the luminous

intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

candela:candela: The candela is the luminous

intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

SI DefinitionsSI Definitions

kelvin:kelvin: The kelvin, unit of

thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

kelvin:kelvin: The kelvin, unit of

thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

mole:mole: The mole is the amount of

substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.

mole:mole: The mole is the amount of

substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.

SI Derived UnitsSI Derived Units

SI PrefixesSI PrefixesCommon to ScienceCommon to Science

Metric ConversionsMetric ConversionsggmmLL 1010-1-1 1010-2-2 1010-3-3101011101022101033

BaseBaseunitunit

deci centi millidekahectokilo

Conversions in the metric system are merely a matter of moving a decimal point. The “base unit” means the you have a quantity (ggrams, mmeters, LLiters, etc without a prefix.

•In science numbers will be very small or very large.

•Scientific notation is used for these numbers.

Scientific Scientific NotationNotation

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 10M x 10nn

MM is a number betweenis a number between 11 andand 1010 nn is an integeris 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 Step #4: Re-write in the form M x 10M 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 CALCULATIONS CALCULATIONS IN SCIENTIFIC IN 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

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 + + .30 x 10.30 x 1066

4.304.30 x 10x 1066

Move the Move the decimal decimal on the on the smallersmaller number!number!

4.00 x 104.00 x 1066

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

2.37 x 102.37 x 10-6-6

++ 3.48 x 103.48 x 10-4-4

Solution…Solution…002.37 x 10002.37 x 10--

66

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

Solution…Solution…0.0237 x 100.0237 x 10-4-4

3.5037 x 103.5037 x 10-4-4

Uncertainty and Significant FiguresUncertainty and Significant Figures

Cartoon courtesy of Lab-initio.com

Uncertainty in Measurement

A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty.

Why Is there Uncertainty? Measurements are performed with instruments No instrument can read to an infinite number of decimal places

Which of these balances has the greatest uncertainty in measurement?

Precision and Accuracy

AccuracyAccuracy refers to the agreement of a refers to the agreement of a particular value with the true value.particular value with the true value.

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

Neither accurate nor

precise

Precise but not accurate

Precise AND accurate

Types of Error

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

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

Rules for Counting Significant Figures - Details

Nonzero integers always count as significant figures.

3456 has 4 significant figures

Rules for Counting Significant Figures - Details

Zeros- Leading zeros do not count as

significant figures.

0.0486 has3 significant figures

Rules for Counting Significant Figures - Details

Zeros- Captive zeros always

count assignificant figures.

16.07 has4 significant figures

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 hashas

44 significant figuressignificant figures

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

Exact numbersExact numbers have an have an infiniteinfinite 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

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: : # sig # sig figs in the result equals the 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 1313 (2 sig figs) (2 sig figs)

Sig Fig Practice #2Sig Fig Practice #2

3.24 m x 7.0 m3.24 m x 7.0 m

Calculation Calculator says: Answer

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 0.358885017 g/mL0.358885017 g/mL 0.359 g/mL0.359 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 =

18.73418.734 18.718.7 (3 sig figs)(3 sig figs)

Sig Fig Practice #3Sig Fig Practice #3

3.24 m + 7.0 m3.24 m + 7.0 m

Calculation Calculator says: Answer

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