Unit I: MeasurementsUnit I: MeasurementsA.A. Significant figuresSignificant figuresB.B. Rounding numbersRounding numbersC.C. Scientific notationScientific notationD.D. Using electronic calculatorsUsing electronic calculatorsE.E. Using sig figs in arithmetic operationsUsing sig figs in arithmetic operationsF.F. The metric systemThe metric systemG.G. Problem solving with unit analysisProblem solving with unit analysisH.H. Derived unitsDerived unitsI.I. Practical conversionsPractical conversionsJ.J. DensityDensityK.K. Applications of using densityApplications of using density

Significant FiguresSignificant Figures

The digits in a decimal number that are The digits in a decimal number that are warranted by the accuracy of the means warranted by the accuracy of the means of measurement. of measurement. http://www.thefreedictionary.comhttp://www.thefreedictionary.com

Every measurement device has a usefull Every measurement device has a usefull range of measurement and some level of range of measurement and some level of accuracy associated with it.accuracy associated with it.

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Accuracy versus PrecisionAccuracy versus Precision‘True’ Value

Measured Values

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Significant FiguresSignificant Figures

““Significant figures in a number are all the Significant figures in a number are all the digits of which we are absolutely certain, digits of which we are absolutely certain, plus one additional digit, which is plus one additional digit, which is estimated and regarded as uncertain.”estimated and regarded as uncertain.”

-Backus, B, CH150 text-Backus, B, CH150 text

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Example of Example of measurements/scalesmeasurements/scales

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Examples of measurementsExamples of measurements

Example of measurementsExample of measurements

A Note on UnitsA Note on Units

Every measured quantity has units. If the Every measured quantity has units. If the units are not given then the measurement units are not given then the measurement is not correct.is not correct.

In this course an answer given without the In this course an answer given without the proper units is WRONG.proper units is WRONG.

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Rules for determining significant figuresRules for determining significant figures

1.1. All non-zero digits are significantAll non-zero digits are significant2.2. Zeros to the left of a nonzero number are NOT significantZeros to the left of a nonzero number are NOT significant3.3. Zeros between significant digits are significantZeros between significant digits are significant4.4. Zeros at the end of a number Zeros at the end of a number andand right of a decimal point right of a decimal point

are significantare significant5.5. Zeros to the right of a number and left of an implied Zeros to the right of a number and left of an implied

decimal are NOT significantdecimal are NOT significant6.6. Counted numbers, and exact conversions are all considered Counted numbers, and exact conversions are all considered

significant but are exempt from the rulessignificant but are exempt from the rules7.7. When zeros follow a number with a terminal decimal point, When zeros follow a number with a terminal decimal point,

they are significantthey are significant

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Significant FiguresSignificant FiguresLecture Problem I.1 (pg 26)Lecture Problem I.1 (pg 26) Determine the number of Determine the number of

significant figures in each of the following measurements.significant figures in each of the following measurements.

a: a: 0.00500 m0.00500 m b: 220,103 kgb: 220,103 kg c: 0.00108 in.c: 0.00108 in.

d: 0.0010320 sd: 0.0010320 s e: 12,000 me: 12,000 m f: 1.60 cmf: 1.60 cm

g: 140.0 Lg: 140.0 L h: 3.0040 x 10h: 3.0040 x 107 7 mgmg i: 25 pencilsi: 25 pencils

Rounding NumbersRounding Numbers

1.1. If the digit dropped is >5, round up the If the digit dropped is >5, round up the final digitfinal digit

2.2. If the digit dropped is <5, leave the final If the digit dropped is <5, leave the final digit unchangeddigit unchanged

3.3. If the digit to be dropped =5 (with no If the digit to be dropped =5 (with no following digits)following digits)

round up if the round up if the precedingpreceding digit is odd digit is odd leave as is if the leave as is if the precedingpreceding digit is even digit is even

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Lecture problem I-2 (page 26):

a. Round 0.005674 L to two significant figs.

Step 1: How may digits are significant now?

0.005674 L Has four sig figs.

0.005674 L The 7 and digits to the right will be dropped.

Step 2: Which is the first digit to be dropped?

Step 3: Apply the appropriate rounding rule.

0.005674 L 7>5, so we apply the first rule and round up.

Answer = 0.0057 L

Rounding NumbersRounding Numbers

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Lecture problem I-2 (page 26):

b. Round 446,500 m to three significant figs.

Step 1: How may digits are significant?

446,500 m Has four sig figs.

446,500 m The 5 and digits to the right will be dropped.

Step 2: Which is the digit to be dropped?

Answer = 446,000

Step 3: Apply the appropriate rounding rule.

446,500 m 5=5, so we apply the third rule.

Rounding NumbersRounding Numbers

Scientific NotationScientific Notation

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Provides a convenient way to express very large or very small numbers using powers of ten

106,000,000,000,000 can be written as 1.06 x 1014 using scientific notation

Likewise 0.000004812 can be can be expressed as 4.812 x 10-6

10 = 101 100=1 1/10 = 10-1

Scientific NotationScientific Notation

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Example 1:

Write 417,500,000 in scientific notation

Step 1: Place a decimal to the right of the first non-zero digit, and write the significant figures after the decimal

4.175

Step 2: Determine what power of 10 needs to be multiplied by 4.175 to obtain 417,500,000

4.175 x 100,000,000 (or 108) = 417,500,000

Step 3: Make sure that the number expressed in scientific notation has the same number of significant digits as in decimal form.

Scientific NotationScientific Notation

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Lecture Problems I-3 (page 28):

a. Express 52,080,000 in scientific notation

b. Express 0.00000002050 in scientific notation

= 5.208 x 107

= 2.050 x 10-8

Using Electronic CalculatorsUsing Electronic CalculatorsEE or EXP means 10x on most calculators so

entering ’15’, ‘EE’, ‘4’ into your calculator will give you: 15 x 104 or 150,000.

Most calculators have a X2 button which automatically squares a number

The yx or ^ button raises a number to an exponent

Try problem #10 on the online homework after class to check your calculator 1-D

Significant Figures in CalculationsSignificant Figures in Calculations

Rule 1: When multiplying or dividing Rule 1: When multiplying or dividing measured numbers, the product or measured numbers, the product or quotient cannot have more significant quotient cannot have more significant figures than the value in the operation figures than the value in the operation having the least number of significant having the least number of significant figuresfigures

(See lecture problem #4 on page 30)(See lecture problem #4 on page 30)

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Significant Figures in CalculationsSignificant Figures in CalculationsRule 2: For adding or subtracting measured Rule 2: For adding or subtracting measured

numbers, the sum or difference can only be as numbers, the sum or difference can only be as accurate as the least accurate value in the accurate as the least accurate value in the arithmetic operationarithmetic operation

In other words the number of In other words the number of decimal placesdecimal places in in the answer must be equal to the least number of the answer must be equal to the least number of places in any of the numbers being added or places in any of the numbers being added or subtractedsubtracted

(See lecture problem #5 on page 31)(See lecture problem #5 on page 31)

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Significant Figures in CalculationsSignificant Figures in Calculations

Rule 3: For combined operations, find the Rule 3: For combined operations, find the significant figures for each part of the significant figures for each part of the operation, and determine the significant operation, and determine the significant figures of the result based on the least figures of the result based on the least accurate number.accurate number.

Do not round numbers between operations.Do not round numbers between operations.(See lecture problem #6 on page 32)(See lecture problem #6 on page 32)

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The Metric SystemThe Metric SystemThe modern version of the metric system is known The modern version of the metric system is known

as the SI system.as the SI system.

The metric system is a decimal based system, The metric system is a decimal based system, unlike the Imperial, or English system.unlike the Imperial, or English system.

Two types of units are used in the metric system: Two types of units are used in the metric system: basic unitsbasic units, and , and derived unitsderived units..

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Basic unitsBasic units represent independent physical represent independent physical properties or dimensions which can be properties or dimensions which can be measured with an appropriate gauge.measured with an appropriate gauge.

For example: For example: • LengthLength metersmeters mm• TemperatureTemperature CelciusCelcius °C°C• MassMass gramgram gg• TimeTime secondsecond ss• Electric currentElectric current AmpereAmpere AA

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The Metric SystemThe Metric System

Area = length x lengthArea = length x length (length)(length)22

Volume = length x length x lengthVolume = length x length x length (length)(length)33

Velocity = length / timeVelocity = length / time length/(time)length/(time)

Acceleration = length / time / timeAcceleration = length / time / time length/(time)length/(time)22

Force = mass x acceleration mass x length/(time)Force = mass x acceleration mass x length/(time)22

Force has SI units of Newtons (N)Force has SI units of Newtons (N)

Pressure = force / area Newton/(length)Pressure = force / area Newton/(length)22

Pressure in N/mPressure in N/m22 is called a Pascal (Pa) is called a Pascal (Pa)

Derived unitsDerived units are made by combining are made by combining basic units.basic units.

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In the metric system units are combined with In the metric system units are combined with a prefix to describe the magnitude of a a prefix to describe the magnitude of a measurement relative to the base unit of measurement relative to the base unit of measure.measure.

For Example: the prefix kilo means 1000.For Example: the prefix kilo means 1000.

Combining this prefix with the meter gives a new Combining this prefix with the meter gives a new unit, the kilometerunit, the kilometer

The kilometer represents a length 1000 scale The kilometer represents a length 1000 scale times larger than the metertimes larger than the meter 1-F

The Metric SystemThe Metric System

PrefixPrefix AbbreviationAbbreviation ExponentExponent NumberNumber MeaningMeaning

gigagiga GG 101099 1,000,000,0001,000,000,000 One billionOne billion

megamega MM 101066 1,000,0001,000,000 One millionOne million

kilokilo kk 101033 1,0001,000 One thousandOne thousand

hectohecto hh 101022 100100 One hundredOne hundred

dekadeka dada 101011 1010 tenten

Base unit (meter, gram, liter, etc.)Base unit (meter, gram, liter, etc.)

decideci dd 1010-1-1 1/101/10 One tenthOne tenth

centicenti cc 1010-2-2 1/1001/100 One hundredthOne hundredth

millimilli mm 1010-3-3 1/10001/1000 One thousandthOne thousandth

micromicro μμ 1010-6-6 1/1,000,0001/1,000,000 One millionthOne millionth

nanonano nn 1010-9-9 1/1,000,000,0001/1,000,000,000 One billionthOne billionth

Prefixes in the metric system

Shamelessly borrowed from:

http://en.wikipedia.org/wiki/File:English_length_units_graph.png

The Metric SystemThe Metric System

English to SI conversions you should knowEnglish to SI conversions you should know1 inch (in.)1 inch (in.) 2.54 cm (exact)2.54 cm (exact)1 pound (lb.)1 pound (lb.) 454 g (3 sig figs)454 g (3 sig figs)1 quart (qt.)1 quart (qt.) 0.946 L (3 sig figs)0.946 L (3 sig figs)1 mile (mi.)1 mile (mi.) 1.61 km (3 sig figs)1.61 km (3 sig figs)

Your text has several other useful conversions on Your text has several other useful conversions on page 34.page 34.

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The Metric SystemThe Metric System

One final example of why we should use the One final example of why we should use the metric system. metric system.

The ease with which we can relate various The ease with which we can relate various physical properties and quantities!physical properties and quantities!

For example, water...For example, water...

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Problem Solving with Unit AnalysisProblem Solving with Unit Analysis

Problem solving using unit analysis is one of Problem solving using unit analysis is one of the most important skill you will learn in the most important skill you will learn in this class.this class.

You will use this technique all throughout You will use this technique all throughout future chemistry classes and in all future chemistry classes and in all branches of science, engineering, or branches of science, engineering, or medicinemedicine

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Unit FactorsUnit Factors

A unit factor, or conversion factor, shows A unit factor, or conversion factor, shows the relationship between units in a the relationship between units in a numerator and denominator.numerator and denominator.

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2.54 cm = 1 in. Can be re-written as 2.54 cm 1 in

or 1 in 2.54 cm

100 legs 1 centipede or

1 centipede 100 legs2 sides coin

or coin 2 sides

Unit FactorsUnit FactorsUnit factors come from:Unit factors come from: Conversion factors known by definitionConversion factors known by definition Conversion factors that accurately describe relationshipsConversion factors that accurately describe relationships Conversion factors defined by this manual. Conversion factors defined by this manual.

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Consider the molecule diphosphorus pentaoxide P2O5 write as many unit factors as possible relating the molecule to the atom which it is comprised of

2 5 2 5

2 5

1 molecule P O 1 molecule P O 5 atoms O or or

2 atoms P 5 atoms O 1 molecule P O

2 atoms P 5 atoms O or

5 atoms O 2 atoms P

Unit FactorsUnit FactorsProblem solving with unit factorsProblem solving with unit factors

1.1. Write down the units of the answer on the right sideWrite down the units of the answer on the right side

2.2. Write down the “given” including units on the leftWrite down the “given” including units on the left

3.3. Write down the unit factors that apply to the problemWrite down the unit factors that apply to the problem

4.4. Insert the appropriate unit factors between the “given” Insert the appropriate unit factors between the “given” and the “answer” so that all units will cancel except the and the “answer” so that all units will cancel except the unit of the answerunit of the answer

5.5. Calculate the answer paying attention to significant Calculate the answer paying attention to significant figures where necessaryfigures where necessary

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Unit FactorsUnit Factors

Lecture problems I-7 (pg. 37)Lecture problems I-7 (pg. 37)

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a. How many cm are in 7.98 in?

7.98 in x = cm2.54 cm

in= 20.2692 cm 20.3

b. How many mm are in 9.37 yds?

3 ft 12 in 2.54 cm 10 mm9.37 yds 8567.928 mm

yd ft in cm´ ´ ´ ´ =

After rounding 9.37 yds = 8,570 mm

PrefixPrefix AbbreviationAbbreviation ExponentExponent NumberNumber MeaningMeaning

gigagiga GG 101099 1,000,000,0001,000,000,000 One billionOne billion

megamega MM 101066 1,000,0001,000,000 One millionOne million

kilokilo kk 101033 1,0001,000 One thousandOne thousand

hectohecto hh 101022 100100 One hundredOne hundred

dekadeka dada 101011 1010 tenten

Base unit (meter, gram, liter, etc.)Base unit (meter, gram, liter, etc.)

decideci dd 1010-1-1 1/101/10 One tenthOne tenth

centicenti cc 1010-2-2 1/1001/100 One hundredthOne hundredth

millimilli mm 1010-3-3 1/10001/1000 One thousandthOne thousandth

micromicro μμ 1010-6-6 1/1,000,0001/1,000,000 One millionthOne millionth

nanonano nn 1010-9-9 1/1,000,000,0001/1,000,000,000 One billionthOne billionth

Prefixes in the metric system

Group Practice ProblemsGroup Practice ProblemsHow many significant figures in each of the following numbers?

3.140 m 0.0017 L 1,900 in. 0.1004 s

Round the following to 3 significant figures.

4185 mi. 0.10172 kg 9.0501 yd.

Find the answer and round the correct number of sig figs.

14.22 cm + 2.035 cm + 120.1 cm =

How many inches are in 4.415 m ?

4 2 2 4

4180 mi. 0.102 kg 9.05 yd.

136.4 cm

173.8 in.

Derived UnitsDerived Units

Derived units are created by combining base Derived units are created by combining base units or other derived units.units or other derived units.

Common examples include:Common examples include:AreaArea (length x length)(length x length)VolumeVolume (length x length x length)(length x length x length)VelocityVelocity (length / time)(length / time)PressurePressure (force / area)(force / area)DensityDensity (mass / volume)(mass / volume)EnergyEnergy (force x length)(force x length)PowerPower (energy / time)(energy / time)

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Derived UnitsDerived Units

Frequently derived units deal with area or Frequently derived units deal with area or volume. volume.

For ExampleFor Example 1 liter is defined as 1 cubic decimeter, so how 1 liter is defined as 1 cubic decimeter, so how

many cubic centimeters would be in one liter?many cubic centimeters would be in one liter?

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Derived UnitsDerived UnitsLecture problem I-8 (pg. 41):Lecture problem I-8 (pg. 41):

a.a. How many cubic millimeters are in 0.250 cubic How many cubic millimeters are in 0.250 cubic miles?miles?

b.b. How many cubic meters are in 18.4 ftHow many cubic meters are in 18.4 ft22

c.c. How many ml are in 1.75 x 10How many ml are in 1.75 x 10-3-3 km km33

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1.04 x 1018 mm3

0.521 m3

1.75 x 1012 mL

Practical ConversionsPractical Conversions

We sometimes encounter the need to We sometimes encounter the need to convert units in day to day life: distances, convert units in day to day life: distances, currency exchange rates, etc.currency exchange rates, etc.

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Lecture problem I-9 (pg. 44):

You fill up your gas tank in a rented car somewhere in Europe and pay 32.50 Euros for 50.0 L. What were you paying in dollars per gallon? The exchange rate at the time was 0.7939 Euros to the dollar.

= $ 3.10 / gal

DensityDensityDensity is a physical property of a material Density is a physical property of a material

and is defined as the ratio of its mass to and is defined as the ratio of its mass to volume.volume.

d = m / vd = m / v

The units for density are g/mL or g/cmThe units for density are g/mL or g/cm33 (cc) (cc) for liquids and solids. for liquids and solids.

For gasses density is usually expressed in For gasses density is usually expressed in g/Lg/L

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DensityDensity

The density of a material is not a constant! The density of a material is not a constant!

Pressure and temperature effect the density Pressure and temperature effect the density of solid and liquids only slightlyof solid and liquids only slightly

Pressure and temperature have a very large Pressure and temperature have a very large effect on the density of gasseseffect on the density of gasses

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DensityDensity

Lecture problem I-10 (pg.46):Lecture problem I-10 (pg.46):

What is the density, in g/cmWhat is the density, in g/cm33 of a rectangular of a rectangular block of metal measuring 2.25 cm x 4.3 cm x block of metal measuring 2.25 cm x 4.3 cm x 12.0 cm, and having a mass of 0.475 kg ?12.0 cm, and having a mass of 0.475 kg ?

1-J

= 4.1 g/cm3

DensityDensity

Lecture problem I-11: (pg. 47)Lecture problem I-11: (pg. 47)

What is the mass, in grams, of 75.0 cmWhat is the mass, in grams, of 75.0 cm33 of iron? of iron?

1-J

The density of iron is 7.87 g/cm3 (from the table on pg 45)

75.0 cm3 x = g3

7.87 g

cm590.25

= 590. g

After checking sig figs and rounding..

DensityDensity

Lecture problem I-12 (pg 48):Lecture problem I-12 (pg 48):a.a. Calculate the volume in quarts of 439 mg of Calculate the volume in quarts of 439 mg of

ethyl alcoholethyl alcohol

b.b. What is the mass, in lbs., of 2.50 x 10What is the mass, in lbs., of 2.50 x 1044 mm mm33 of of aluminum?aluminum?

1-J

= 5.88 x 10-4 qts.

= 0.149 lbs.

Applications of DensityApplications of DensityDensity is used frequently in many of the calculations Density is used frequently in many of the calculations

commonly done in chemistry.commonly done in chemistry.

The density of a substance can sometimes be used to The density of a substance can sometimes be used to identify it, or to determine if a substance is pure or identify it, or to determine if a substance is pure or not.not.

Because we can relate the mass and volume of a Because we can relate the mass and volume of a substance through density we can often avoid substance through density we can often avoid making difficult measurements.making difficult measurements.

1-K

Group practiceGroup practiceIf molasses has a density of 1.35 g/mL, what is If molasses has a density of 1.35 g/mL, what is

the mass of 1.20 cups of molasses?the mass of 1.20 cups of molasses? (Remember 1 cup is 8 fluid ounces, and 1 fluid ounce = 29.57 mL) (Remember 1 cup is 8 fluid ounces, and 1 fluid ounce = 29.57 mL)

If you have a block of copper that has a mass of If you have a block of copper that has a mass of 245.3 grams, what would be its mass if it was 245.3 grams, what would be its mass if it was gold instead of copper?gold instead of copper? (see density chart on pg 45) (see density chart on pg 45)

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383 g

531 g