Scientific Notation &Scientific Notation &Significant Figures in Significant Figures in

MeasurementMeasurement

Dr. Sonali SahaDr. Sonali SahaChemistry Chemistry

HonorsHonorsFall 2014Fall 2014

In science, we deal with some In science, we deal with some very very LARGELARGE numbers: numbers:

1 mole = 6022570000000000000000001 mole = 602257000000000000000000

We also deal with some very We also deal with some 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 kg 91 kg x 602257000000000000000000x 602257000000000000000000

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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 between is a number between 11 and and 10 10 nn is an integer is an integer

For really large and really small numbers, a For really large and really small numbers, a convenient way to represent them is by convenient way to represent them is by

using:using:

2 500 000 000

Step #1: Insert a decimal pointStep #1: Insert a 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.

Note the exponent is positive because the original number was greater than 1.

0.00005790.0000579

Step #1: Decide where the decimal Step #1: 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 #2: Count how many places you Step #2: Count how many places you bounce bounce the decimal pointthe decimal pointStep #3: Re-write in the form M x 10Step #3: 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 original number was less than 1.

Characteristics of Characteristics of MeasurementMeasurement

Part 1 - Part 1 - numbernumber Part 2 – Part 2 – unit of measureunit of measure

Examples: Examples: 2020 gramsgrams

20 20 mLmL

Measurement - quantitative Measurement - quantitative observation observation consisting of 2 partsconsisting of 2 parts

Why Is there Uncertainty in Why Is there Uncertainty in Measurements?Measurements?

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 AccuracyPrecision and Accuracy

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

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

Neither accurate nor

precise

Precise but not accurate

Precise AND accurate

More on Precision…• Precision also refers to the smallest

calibrated markings on laboratory instruments and equipment.– The precision of our electronic scale is

1/10th of a gram.– The precision of a graduated cylinder is

1 milliliter (1 mL), of a beaker it is 10 mL, of a burette is 0.1 mL

Significant FiguresSignificant Figures

• When reporting a measurement, how many significant figures/digits should you use?

The significant figures of a number are those digits that carry meaning contributing to its precision.

Rules for using significant Rules for using significant figures to express lab figures to express lab

measurementmeasurementThe measurement represented by the

smallest graduated marking

• For example you used a beaker and measured 20 mL (one sig. fig.) while the mass of the water was 18.2 g (three sig. figs) how would you report your density value?

Multiplication and DivisionMultiplication and Division:: The number of sig figs in The number of sig figs in the result equals the number in the least precise the result equals the number in the least precise measurement used in the calculation. measurement used in the calculation.

18.2 (3 sig.fig) 18.2 (3 sig.fig) ÷ 20 (1 sig.fig.) = calculator says 0.91 20 (1 sig.fig.) = calculator says 0.91

REALLY IS = 0.9 g/mLREALLY IS = 0.9 g/mL

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

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

6.8 + 11.934 = 6.8 + 11.934 =

18.734 18.734 18.7 ( 18.7 (1 place after 1 place after the decimalthe decimal))

Rules for Counting Rules for Counting Significant FiguresSignificant Figures

• Exact numbersExact numbers (definitions and (definitions and counting numbers)counting numbers) have an infinite have an infinite number of significant figures.number of significant figures.

11 inchinch == 2.542.54 cm, exactlycm, exactly

Rules for Counting Rules for Counting Significant Figures - ZerosSignificant Figures - Zeros

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

significant figuressignificant figures..

0.04860.0486 hashas

33 sig figs.sig figs.

Rules for Counting Rules for Counting Significant Figures - ZerosSignificant Figures - Zeros

-- Captive zerosCaptive zeros always always count as count as

significant figures.significant figures.

16.0716.07 hashas

44 sig figs.sig figs.

Rules for Counting Rules for Counting Significant Figures - ZerosSignificant Figures - Zeros

Trailing zerosTrailing zeros are significantare significant if if the number contains a decimal the number contains a decimal point.point. Little bit ambiguous!!!! Little bit ambiguous!!!!

9.3009.300 hashas

44 sig figs.sig figs.

93009300 hashas

22 sig figs.sig figs.

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

following measurements?

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

Sig Fig Practice #2Sig Fig Practice #2

3.24 m + 7.0 m

Calculation Calculator says:CorrectAnswer

10.24 10.2 m

100.0 g - 23.73 g 76.27 76.3 g

0.02 cm + 2.371 cm 2.391 2.39 cm

713.1 L - 3.872 L 709.228 709.2 L

1818.2 lb + 3.37 lb 1821.57 1821.6 lb

2.030 mL - 1.870 mL 0.16 0.160 mL

Sig Fig Practice #3Sig Fig Practice #3

3.24 m x 7.0 m

Calculation Calculator says:Correct Answer

22.68 23 m2

100.0 g ÷ 23.7 cm3 4.219409283 4.22 g/cm3

0.02 cm x 2.371 cm 0.04742 0.05 cm2

710 m ÷ 3.0 s 236.6666667 240 m/s

1818.2 lb x 3.23 ft 5872.786 5870 lb·ft

1.030 g ÷ 2.87 mL 0.358885017 0.359 g/mL