2 - 1 measurement uncertainty in measurement significant figures

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MeasurementMeasurement

Uncertainty in MeasurementUncertainty in Measurement

Significant FiguresSignificant Figures

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MeasurementMeasurement

Observation can be both Observation can be both QUALITATIVE and QUANTITIVEQUALITATIVE and QUANTITIVE

A qualitative observationA qualitative observation is a description in words.is a description in words.

A quantitative observation A quantitative observation is a description with numbers and is a description with numbers and

units.units.

A measurement is a comparison to a standard.A measurement is a comparison to a standard.

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Units are importantUnits are important

45 000 has little meaning, just a number

$45,000 has some meaning - money

$45,000/yr more meaning - person’s salary

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Use of Significant Figures

It is important to realize that a measurement always has some degree of uncertainty, which depends on the precision of the measuring device.

Therefore, it is important to indicate the uncertainty in any measurement. This is done by using significant figures.

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• Every measurement has an uncertainty associated with it, unless it is an exact, counted integer, such as the count of trials performed or a definition.

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• Every calculated result also has an uncertainty, related to the uncertainty in the measured data used to calculate it. This uncertainty should be reported either as an explicit ± value or as an implicit uncertainty, by using the appropriate number of significant figures.

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• The numerical value of a "plus or minus" (±) uncertainty value tells you the range of the result.

• When significant figures are used as an implicit way of indicating uncertainty, the last digit is considered uncertain.

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. A significant figure is one that has been measured with certainty or has been 'properly' estimated.

The significant figures in a number includes all certain digits as read from the instrument plus one estimate digit.

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Significant digits or significant figures

- are digits read from the measuring instrument plus one doubtful digit estimated by the observer. This doubtful estimate will be a fractional part of the least count of the instrument.

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All measurements contain some uncertainty.

•Limit of the skill and carefulness of person measuring

•Limit of the measuring tool/equipmentbeing used

Uncertainty is measured with

AccuracyAccuracy How close to the true value

PrecisionPrecision How close to each other

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PrecisionPrecision

Here the numbersare close togetherso we have goodprecision.

• Poor accuracy.

• Large systematic

error.

How well our values agree with each other.

xxx

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AccuracyAccuracy

Here the average valuewould give aaccurate number but the numbersdon’t agree, are not precise.

Large random error

How close our values agree with the true value.

x x

x

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Accuracy and precisionAccuracy and precision

Our goal!

Good precisionand accuracy.

These arevalues wecan trust.

xx x

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Accuracy and precisionAccuracy and precision

Predict the effect on accuracy and Predict the effect on accuracy and precision.precision.

•Instrument not ‘zeroed’ properly

•Reagents made at wrong concentration

•Temperature in room varies ‘wildly’

•Person running test is not properly trained

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Types of errorsTypes of errors

Instrument not ‘zeroed’ properlyReagents made at wrong concentration

Temperature in room varies ‘wildly’Person running test is not properly trained

Random

Systematic

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ErrorsErrors

SystematicSystematic

•Errors in a single direction (high or low).

•Can be corrected by proper calibration or running controls and blanks.

RandomRandom

•Errors in any direction.

•Can’t be corrected. Can only be accounted for by using statistics.

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ErrorsErrors

Systematic: ACCURACYSystematic: ACCURACY

•Errors in a single direction (high or low).

•Can be corrected by proper calibration or running controls and blanks.

Random: PRECISIONRandom: PRECISION

•Errors in any direction.

•Can’t be corrected. Can only be accounted for by using statistics.

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Significant figuresSignificant figures

Method used to express precision.

You can’t report numbers better than the method used to measure them.

67.2 units = three significant figures

ONLY ONE UNCERTAIN DIGIT IS REPORTED

Certain Digits

UncertainDigit

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The number of significant digits is independent of the decimal point.

255 25.5

2.55

0.255

0.0255

These numbersAll have three

significant figures!

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Significant figures: Significant figures: Rules for zerosRules for zeros

Leading zeros are notare not significant.0.421 - three significant figures

Leading zeroLeading zero

Captive zeros areare significant. 4012 - four significant figures

Trailing zeros areare significant.114.20 - five significant figures

Captive zeroCaptive zero

Trailing zeroTrailing zero

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Zeros are what will give you a headache!Zeros are what will give you a headache!

They are used/misused all of the time.

ExampleExampleThe press might report that the federal deficit is three trillion dollars. What did they mean?

$3 x 1012 meaning +/- a trillion dollars

or$3,000,000,000,000.00 meaning +/- a penny

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In science, all of our numbers are either measured or exact.

• ExactExact - Infinite number of significant figures.

• MeasuredMeasured - the tool used will tell you the level of significance. Varies based on the tool.

ExampleExampleRuler with lines at 1/16” intervals.A balance might be able to measure to

the nearest 0.1 grams.

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Significant figures:Significant figures:Rules for zerosRules for zeros

Scientific notationScientific notation - can be used to clearly express significant figures.

A properly written number in scientific notation always has the the proper number of significant figures.

0.00321321 = 3.213.21 x 10-3

Three SignificantFigures

Three SignificantFigures

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Scientific notationScientific notation

• Method to express really big or small numbers.

Format is Mantissa x Base Power

Decimal part ofDecimal part oforiginal numberoriginal number

DecimalsDecimalsyou movedyou moved

We just move the decimal point around.

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If a number is larger than 1If a number is larger than 1

•The original decimal point is moved X places to the left.

•The resulting number is multiplied by 10X.

•The exponent is the number of places you moved the decimal point.

1 2 3 0 0 0 0 0 0. = 1.23 x 108

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If a number is smaller than 1If a number is smaller than 1

•The original decimal point is moved X places to the right.

•The resulting number is multiplied by 10-X.

•The exponent is the number of places you moved the decimal point.

0. 0 0 0 0 0 0 1 2 3 = 1.23 x 10-7

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Most calculators use scientific notation when the numbers get very large or small.

How scientific notation is displayed can vary.

It may use x10n

or may be displayedusing an E.

They usually have an Exp or EEThis is to enter in the exponent.


+

-1

/

x

0

2 3

4 5 6

7 8 9

.

CE

EE

log

ln

1/x

x2

cos tan

1.44939 E-2

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ExamplesExamples

378 000

3.78 x 10 5

8931.5

8.9315 x 10 3

0.000 593

5.93 x 10 - 4

0.000 000 4

4 x 10 - 7

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Significant figuresSignificant figuresand calculationsand calculations

An answer can’t have more significant figures than the quantities used to produce it.

ExampleExample How fast did you run if youwent 1.0 km in 3.0 minutes?

speed = 1.0 km / 3.0 min = 0.33 km / min +

-1

/

x

0

2 3

4 5 6

7 8 9

.

CE

EE

log

ln

1/x

x2

cos tan

0.333333333

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Significant figures and calculationsSignificant figures and calculations

Addition and subtractionAddition and subtractionReport your answer with the same number of digits to the right of the decimal point as the number having the fewest to start with.

123.45987 g+ 234.11 g 357.57 g

805.4 g- 721.67912 g 83.7 g

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Significant figures and calculationsSignificant figures and calculations

Multiplication and division.Multiplication and division.Report your answer with the same number of digits as the quantity have the smallest number of significant figures.

Example. Density of a rectangular solid.Example. Density of a rectangular solid.25.12 kg / [ (18.5 m) ( 0.2351 m) (2.1m)

]= 2.8 kg / m3

(2.1 m - only has two significant figures)

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ExampleExample

257 mg

\__ 3 significant figures

102 miles


0.002 30 kg


23,600.01 $/yr


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Rounding off numbersRounding off numbers

After calculations, you may need to round off.

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If a set of calculations gave you the following numbers and you knew each was supposed to have four significant figures then -

2.57995035 becomes 2.580

34.2004221 becomes 34.20

Rounding offRounding off

1st uncertain digit1st uncertain digit

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Converting unitsConverting units

Factor label methodFactor label method

•Regardless of conversion, keeping track of units makes thing come out right

•Must use conversion factors- The relationship between two

units

•Canceling out units is a way of checking that your calculation is set up right!

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Common conversion factorsCommon conversion factors

SomeEnglish/ Metric conversions FactorFactor1 liter = 1.057 quarts 1.057 qt/L1 kilogram = 2.2 pounds 2.2 lb/kg1 meter = 1.094 yards 1.094

yd/m1 inch = 2.54 cm 2.54

cm/inch

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ExampleExample

A nerve impulse in the body can travel as fast as 400 feet/second.

What is its speed in meters/min ?

Conversions Needed

1 meter = 3.3 feet1 minute = 60 seconds

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m 400 ft 1 m 60 secmin 1 sec 3.3 ft 1 min

ExampleExample

m 400 ft 1 m 60 secmin 1 sec 3.3 ft 1 min?? = x x

?? = x x

mmin ....Fast7273

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Extensive and intensive propertiesExtensive and intensive properties

Extensive propertiesExtensive propertiesDepend on the quantity of sample measured.

ExampleExample - mass and volume of a sample.

Intensive propertiesIntensive propertiesIndependent of the sample size.Properties that are often characteristic of the substance being measured.

ExamplesExamples - density, melting and boiling points.

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DensityDensity

Density is an intensive property of a substance based on two extensive properties.

Common units are g / cm3 or g / mL.

g / cm3

g / cm3

Air 0.0013 Bone 1.7 - 2.0

Water 1.0 Urine 1.01 - 1.03

Gold 19.3 Gasoline 0.66 - 0.69

Density = Mass

Volume

cm3 = mL cm3 = mL

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Example.Example.Density calculationDensity calculation

What is the density of 5.00 mL of a fluid if ithas a mass of 5.23 grams?

d = mass / volume

d = 5.23 g / 5.00 mL

d = 1.05 g / mL

What would be the mass of 1.00 liters of thissample?

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Example.Example.Density calculationDensity calculation

What would be the mass of 1.00 liters of the fluid sample?

The density was 1.05 g/mL.

density = mass / volume

so mass = volume x density

mass = 1.00 L x 1000 x 1.05

= 1.05 x 103 g

mlL

gmL

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Specific gravitySpecific gravity

The density of a substance compared to a reference substance.

Specific Gravity =

•Specific Gravity is unitless.

•Reference is commonly water at 4oC.

•At 4oC, density = specific gravity.

•Commonly used to test urine.

density of substancedensity of reference

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Specific gravity measurementSpecific gravity measurement

Hydrometer

Float height willbe based onSpecific Gravity.

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Measuring timeMeasuring time

The SI unit is the second (s).The SI unit is the second (s).

For longer time periods, we can use SI prefixes or units such as minutes (min), hours (h), days (day) and years.

Months are never used - they vary in size.

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The moleThe mole

Number of atoms in 12.000 grams of 12C

1 mol = 6.022 x 1023 atoms mol = grams / formula

weight

Atoms, ions and molecules are too small to directly measure - measured in uu.

Using moles gives us a practical unit.

We can then relate atoms, ions and molecules, using an easy to measure unit - thethe gramgram.

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The moleThe mole

If we had one mole of water and one mole of hydrogen, we would have the name number of molecules of each.

1 mol H2O = 6.022 x 1023 molecules

1 mol H2 = 6.022 x 1023 molecules

We can’t weigh out moles -- we use grams.

We would need to weigh out a different number of grams to have the same number of molecules

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Moles and massesMoles and masses

Atoms come in different sizes and masses.

A mole of atoms of one type would have a different mass than a mole of atoms of another type.

H - 1.008 u or grams/molO - 16.00 u or grams/molMo - 95.94 u or grams/molPb - 207.2 u or grams/mol

We rely on a straight forward system to relate mass and moles.

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Masses of atomsMasses of atomsand moleculesand molecules

Atomic massAtomic mass

•The average, relative mass of an atom in an element.

Atomic mass unit (u)Atomic mass unit (u)

•Arbitrary mass unit used for atoms.

•Relative to one type of carbon.

Molecular or formula massMolecular or formula mass

•The total mass for all atoms in a compound.

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Molar massesMolar masses

Once you know the mass of an atom, ion, or molecule, just remember:

Mass of one unit - use u

Mass of one mole of units - use grams/mole

The numbers DON’TDON’T change -- just the units.

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Masses of atomsMasses of atomsand moleculesand molecules

HH22OO - water

2 hydrogen 2 x 1.008 u1 oxygen 1 x 16.00 u

mass of molecule 18.02 u18.02 g/mol

Rounded off basedon significant figuresRounded off based

on significant figures

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Another exampleAnother example

CHCH33CHCH22OHOH - ethyl alcohol

2 carbon 2 x 12.01 u6 hydrogen 6 x 1.008 u1 oxygen 1 x 16.00 u

mass of molecule 46.07 u46.07 g/mol

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Molecular mass vs. formula massMolecular mass vs. formula mass

Formula massFormula massAdd the masses of all the atoms in formula

- for molecular and ionic compounds.

Molecular massMolecular massCalculated the same as formula mass

- only valid for molecules.

Both have units of either u or grams/mole.

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Formula mass, FMFormula mass, FM

The sum of the atomic masses of all elements in a compound based on the chemical formula.

You must use the atomic masses of the elements listed in the periodic table.

CO2 1 atom of C and 2 atoms of O

1 atom C x 12.011 u = 12.011 u2 atoms O x 15.9994 u = 31.9988 u Formula mass Formula mass == 44.010 u44.010 u

or or g/mol g/mol

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Example - (NHExample - (NH44))22SOSO44

OK, this example is a little more complicated.

The formula is in a format to show you how the various atoms are hooked up.

( N H ( N H 4 4 ) ) 2 2 S O S O 44

We have two (NH4+) units and one SO4

2- unit.

Now we can determine the number of atoms.

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Ammonium sulfate contains2 nitrogen, 8 hydrogen, 1 sulfur & 4

oxygen.

2 Nx 14.01 = 28.028 H x 1.008 = 8.0641 S x 32.06 = 32.064 O x 16.00 = 64.00

Formula massFormula mass = 132.14= 132.14The units are either u or grams / mol.

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How many atoms are in 20.0 grams of ammonium sulfate?

Formula weight = 132.14 grams/molAtoms in formula = 15 atoms / unit

moles = 20.0 g x = 0.151 mol1 mol

132.14 g

atoms = 0.151 mol x 15 x 6.02 x1023 atomsunit

unitsmol

atoms = 1.36 x1024

2 - 1 measurement uncertainty in measurement significant figures

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