Chapter 2Chapter 2Measurement and Measurement and Problem SolvingProblem Solving

HomeworkHomework

Exercises (optional) Exercises (optional) 1 through 27 (odd)1 through 27 (odd)

ProblemsProblems 29-65 (odd)29-65 (odd) 67-91 (odd)67-91 (odd) 93-99 (odd)93-99 (odd)

Cumulative ProblemsCumulative Problems 101-117 (odd)101-117 (odd)

Highlight Problems (optional)Highlight Problems (optional) 119, 121119, 121

2.2 Scientific Notation: 2.2 Scientific Notation: Writing Large and Small NumbersWriting Large and Small Numbers

In scientific (chemistry) work, it is not unusual In scientific (chemistry) work, it is not unusual to come across very large and very small to come across very large and very small numbersnumbers

Using large and small numbers in Using large and small numbers in measurements and calculations is time measurements and calculations is time consuming and difficultconsuming and difficult

Recording these numbers is also very prone to Recording these numbers is also very prone to errors due to the addition or omission of zeroserrors due to the addition or omission of zeros

A method exists for the expression of A method exists for the expression of awkward, multi-digit numbers in a compact awkward, multi-digit numbers in a compact form: form: scientific notationscientific notation

2.2 Scientific Notation: 2.2 Scientific Notation: Writing Large and Small NumbersWriting Large and Small Numbers

Scientific NotationScientific Notation A system in which an ordinary decimal A system in which an ordinary decimal

number (m) is expressed as a product number (m) is expressed as a product of a number between 1 and 10, of a number between 1 and 10, multiplied by 10 raised to a power (n)multiplied by 10 raised to a power (n)

Used to write very large or very small Used to write very large or very small numbersnumbers

Based on powers of 10Based on powers of 10

n10 m

2.2 Scientific Notation: 2.2 Scientific Notation: Writing Large and Small NumbersWriting Large and Small Numbers

Numbers written in sci. notn. consist of Numbers written in sci. notn. consist of a a numbernumber ( (coefficientcoefficient) followed by a ) followed by a power of 10 (power of 10 (x 10x 10nn))

Negative exponentNegative exponent: number is : number is less than less than 11

Positive exponentPositive exponent: number is : number is greater than greater than 11

210 7.03

coefficient ordecimal part

exponential termor part

exponent

2.2 Scientific Notation:2.2 Scientific Notation:Writing Large and Small NumbersWriting Large and Small Numbers

In an ordinary cup of water there are:In an ordinary cup of water there are:

Each molecule has a massEach molecule has a mass of:of:

0.0000000000000000000000299 gram0.0000000000000000000000299 gram

In scientific notation:In scientific notation:7.91 7.91 хх 10102424 molecules molecules

2.99 2.99 хх 1010-23-23 gram gram

7,910,000,000,000,000,000,000,000 molecules7,910,000,000,000,000,000,000,000 molecules

To Express a Number in Scientific To Express a Number in Scientific Notation:Notation:

For small numbers (<1):For small numbers (<1):

1)1) Locate the decimal pointLocate the decimal point

2)2) Move the decimal point to the Move the decimal point to the rightright to give a to give a number (coefficient) between 1 and 10 number (coefficient) between 1 and 10

3)3) Write the new number multiplied by Write the new number multiplied by 1010 raised raised to the “nto the “nthth power” power”

wherewhere “n”“n” is the number of places you is the number of places you moved the decimal point so there is one moved the decimal point so there is one non-zero digit to the left of the decimal. non-zero digit to the left of the decimal.

If the decimal point is moved to the If the decimal point is moved to the rightright, , from its initial position, then the exponent from its initial position, then the exponent is a negative number (× is a negative number (× 1010-n-n) )

To Express a Number in Scientific To Express a Number in Scientific Notation:Notation:

For large numbers (>1):For large numbers (>1):

1)1) Locate the decimal pointLocate the decimal point

2)2) Move the decimal point to the Move the decimal point to the leftleft to give a to give a number (coefficient) between 1 and 10 number (coefficient) between 1 and 10

3)3) Write the new number multiplied by Write the new number multiplied by 10 raised 10 raised to the “nto the “nthth power” power”

wherewhere “n”“n” is the number of places you is the number of places you moved the decimal point so there is one moved the decimal point so there is one non-zero digit to the left of the decimal. non-zero digit to the left of the decimal.

If the decimal point is moved to the If the decimal point is moved to the leftleft, , from its initial position, then the exponent from its initial position, then the exponent is a positive number (× is a positive number (× 1010nn))

ExamplesExamples

Write each of the following in Write each of the following in scientific notationscientific notation12,50012,5000.02020.020237,400,00037,400,0000.00001040.0000104

ExamplesExamples12,50012,500

Decimal place is at the far rightDecimal place is at the far right Move the decimal place to a position Move the decimal place to a position

between the 1 and 2 (one non-zero between the 1 and 2 (one non-zero digit to the left of the decimal) digit to the left of the decimal)

Coefficient (1.25)Coefficient (1.25) The decimal place was moved 4 The decimal place was moved 4

places to the left (large number) so places to the left (large number) so exponent is positiveexponent is positive

1.25x101.25x1044

ExamplesExamples

0.02020.0202 Move the decimal place to a position Move the decimal place to a position

between the 2 and 0 (one non-zero digit between the 2 and 0 (one non-zero digit to the left of the decimal) to the left of the decimal)

Coefficient (2.02)Coefficient (2.02) The decimal place was moved 2 places The decimal place was moved 2 places

to the right (small number) so exponent to the right (small number) so exponent is negativeis negative

2.02x102.02x10-2-2

ExamplesExamples

37,400,00037,400,000Decimal place is at the far rightDecimal place is at the far rightMove the decimal place to a Move the decimal place to a

position between the 3 and 7 position between the 3 and 7 Coefficient (3.74)Coefficient (3.74)The decimal place was moved 7 The decimal place was moved 7

places to the left (large number) places to the left (large number) so exponent is positiveso exponent is positive

3.74x103.74x1077

ExamplesExamples

0.00001040.0000104Move the decimal place to a Move the decimal place to a

position between the 1 and 0 position between the 1 and 0 Coefficient (1.04)Coefficient (1.04)The decimal place was moved 5 The decimal place was moved 5

places to the right (small places to the right (small number) so exponent is negativenumber) so exponent is negative

1.04x101.04x10-5-5

Using Scientific Notation on a Using Scientific Notation on a CalculatorCalculator

1)1) Enter the coefficient (number)Enter the coefficient (number)

2)2) Push the key: Push the key:

Then enter only the power of 10Then enter only the power of 10

3)3) If the exponent is negative, use the If the exponent is negative, use the key:key:

4)4) DO NOTDO NOT use the multiplication use the multiplication key: key:

to express a number in sci. to express a number in sci. notationnotation

(+/-)(+/-)

XX

EXPEXPEEEE or

Converting Back to Standard Converting Back to Standard NotationNotation

1)1) Determine the Determine the signsign of the exponent, of the exponent, nn If If nn is is ++ the decimal point will move to the the decimal point will move to the

right (this gives a number greater than one)right (this gives a number greater than one) If If nn is is –– the decimal point will move to the the decimal point will move to the

left (this gives a number less than one)left (this gives a number less than one)

2)2) Determine the value of the exponent of 10Determine the value of the exponent of 10 The “power of ten” determines the number The “power of ten” determines the number

of places to move the decimal pointof places to move the decimal point Zeros may have to be added to the number Zeros may have to be added to the number

as the decimal point is movedas the decimal point is moved

Using Scientific NotationUsing Scientific Notation To compare numbers written in scientific To compare numbers written in scientific

notation, with the same coefficient, compare notation, with the same coefficient, compare the exponents of each numberthe exponents of each number

The number with the larger power of ten (the The number with the larger power of ten (the exponent) is the larger numberexponent) is the larger number

If the powers of ten (exponents) are the same, If the powers of ten (exponents) are the same, then compare coefficients directlythen compare coefficients directly Which number is larger?Which number is larger?

21.8 21.8 хх 10 1033 or 2.05 or 2.05 хх 10 1044

2.18 2.18 хх 10 1044 > 2.05 > 2.05 хх 10 1044

3.4 3.4 хх 10 1044 < 3.4 < 3.4 хх 10 1077

2.3 Significant Figures:2.3 Significant Figures:Writing Numbers to Reflect PrecisionWriting Numbers to Reflect Precision

Two kinds of numbers exist:Two kinds of numbers exist: Numbers that are counted (exact)Numbers that are counted (exact) Numbers that are measuredNumbers that are measured

It is possible to know the exact value of It is possible to know the exact value of a counted numbera counted number

The exact value of a measured number The exact value of a measured number is never knownis never known

Counting objects does not entail Counting objects does not entail reading the scale of a measuring reading the scale of a measuring device device

2.3 Exact Numbers2.3 Exact Numbers Exact numbers Exact numbers occur in definitions or in occur in definitions or in

countingcounting These numbers have no uncertainty These numbers have no uncertainty Unlimited number of significant figures Unlimited number of significant figures

(never limit the no. of sig. figures in a (never limit the no. of sig. figures in a calculation)calculation)

They are eitherThey are either Counting numbers Counting numbers

7 pennies, 6 apples, 4 chairs7 pennies, 6 apples, 4 chairs Defined numbers Defined numbers (one exact value)(one exact value)

12 in = 1 ft 12 in = 1 ft 1 gal = 4 quarts1 gal = 4 quarts1 minute = 60 seconds1 minute = 60 seconds

Measured NumbersMeasured Numbers Unlike counted (or defined) numbers, Unlike counted (or defined) numbers,

measured numbers always contain a measured numbers always contain a degree of uncertainty (or error)degree of uncertainty (or error)

A measurement:A measurement: involves reading a measuring deviceinvolves reading a measuring device always has some amount of uncertainty always has some amount of uncertainty uncertainty comes from the tool used for uncertainty comes from the tool used for

comparisoncomparison A measuring device with a smaller unit will A measuring device with a smaller unit will

give a more precise measurement, e.g., give a more precise measurement, e.g., some rulers show smaller divisions than some rulers show smaller divisions than othersothers

Measured NumbersMeasured Numbers Whenever a Whenever a

measurement is made, measurement is made, an estimate is required, an estimate is required, i.e., the value between i.e., the value between the two smallest the two smallest divisions on a divisions on a measuring devicemeasuring device

Every person will Every person will estimate it slightly estimate it slightly differently, so there is differently, so there is some uncertainty some uncertainty present as to the true present as to the true valuevalue

2.8 to 2.9 cm

2.8 cm 2.9 cm

2.3 Significant Figures: Writing 2.3 Significant Figures: Writing Numbers to Reflect PrecisionNumbers to Reflect Precision

Scientific numbers are reported so Scientific numbers are reported so that all digits are that all digits are certaincertain except the except the last digit which is estimatedlast digit which is estimated

To indicate the uncertainty of a To indicate the uncertainty of a single measurement, scientists use a single measurement, scientists use a system called significant figuressystem called significant figures

Significant Figures: All digits known Significant Figures: All digits known with certainty plus one digit that is with certainty plus one digit that is uncertainuncertain

2.32.3 Counting Significant FiguresCounting Significant Figures

The last digit written in a measurement is The last digit written in a measurement is the number that is considered to be the number that is considered to be uncertain (estimated)uncertain (estimated) Unless stated otherwise, the uncertainty Unless stated otherwise, the uncertainty

in the last significant digit is ±1 (plus or in the last significant digit is ±1 (plus or minus one unit)minus one unit)

The precision of a measured quantity is The precision of a measured quantity is determined by number of sig. figuresdetermined by number of sig. figures

A set of guidelines is used to interpret the A set of guidelines is used to interpret the significance of: significance of: a reported measurement a reported measurement values calculated from measurementsvalues calculated from measurements

2.32.3 Counting Significant FiguresCounting Significant Figures

Four rules (the guidelines):Four rules (the guidelines):1.1. Nonzero integers are always significantNonzero integers are always significant

Zeros (may or may not be significant)Zeros (may or may not be significant) significant zeros significant zeros place-holding zeros (not significant)place-holding zeros (not significant) It is determined by its position in a It is determined by its position in a

sequence of digits in a measurementsequence of digits in a measurement2.2. Leading zeros never count as significant Leading zeros never count as significant

figuresfigures3.3. Captive (interior) zeros are always significantCaptive (interior) zeros are always significant4.4. Trailing zeros are significant if the number Trailing zeros are significant if the number

has a decimal pointhas a decimal point

2.4 Significant Figures in 2.4 Significant Figures in CalculationsCalculations

Calculations cannot improve the Calculations cannot improve the precision of experimental measurementsprecision of experimental measurements

The number of significant figures in any The number of significant figures in any mathematical calculation is limited by mathematical calculation is limited by the least precise measurement used in the least precise measurement used in the calculationthe calculation

Two operational rules to ensure no Two operational rules to ensure no increase in measurement precision:increase in measurement precision: addition and subtractionaddition and subtraction multiplication and divisionmultiplication and division

2.4 Significant Figures in 2.4 Significant Figures in Calculations: Multiplication and Calculations: Multiplication and

DivisionDivision Product or quotient has the same number of Product or quotient has the same number of

significant figures as the factor with the significant figures as the factor with the fewestfewest significant figuressignificant figures

Count the number of significant figures in Count the number of significant figures in each number. The least precise factor each number. The least precise factor (number) has the fewest significant figures(number) has the fewest significant figures

RoundingRounding Round the result so it has the same Round the result so it has the same

number of significant figures as the number of significant figures as the number with the number with the fewest fewest significant figuressignificant figures

2.4 Significant Figures in 2.4 Significant Figures in Calculations: RoundingCalculations: Rounding

To round the result to the correct To round the result to the correct number of significant figuresnumber of significant figures

If the last (leftmost) digit to be removed:If the last (leftmost) digit to be removed: is less than 5, the preceding digit stays is less than 5, the preceding digit stays

the same (rounding down)the same (rounding down) is equal to or greater than 5, the is equal to or greater than 5, the

preceding digit is rounded uppreceding digit is rounded up In multiple step calculations, carry the In multiple step calculations, carry the

extra digits to the final result and extra digits to the final result and thenthen round offround off

2.4 Multiplication/Division 2.4 Multiplication/Division Example:Example:

The number with the fewest The number with the fewest significant figures is 1.1 so the significant figures is 1.1 so the answer has 2 significant figuresanswer has 2 significant figures

0.1021 0.082103 273 1.1

2.0804382 SF

5 SF 3 SF

2.1

4 SF

2 SF

2.4 Multiplication/Division 2.4 Multiplication/Division Example:Example:

The number with the fewest The number with the fewest significant figures is 273 (the significant figures is 273 (the limiting term) so the answer has 3 limiting term) so the answer has 3 significant figuressignificant figures

0.1021 × 0.082103 × 273 = 2.2884812.293 SF

3 SF5 SF4 SF

2.4 Significant Figures in 2.4 Significant Figures in Calculations: Addition and Calculations: Addition and

SubtractionSubtraction

Sum or difference is limited Sum or difference is limited by the quantity with the by the quantity with the smallest numbersmallest number of of decimal decimal placesplaces

Find quantity with the fewest Find quantity with the fewest decimal placesdecimal places

Round answer to the same Round answer to the same decimal placedecimal place

2.4 Addition/Subtraction Example:2.4 Addition/Subtraction Example:

The number with the fewest The number with the fewest decimal places is 171.5 so decimal places is 171.5 so the answer should have 1 the answer should have 1 decimal placedecimal place

171.5 72.915 8.23 236.1851 d.p. 3 d.p. 2 d.p.

236.21 d.p.

2.5 The Basic Units of 2.5 The Basic Units of MeasurementMeasurement

The most used tool of the chemistThe most used tool of the chemist Most of the basic concepts of chemistry Most of the basic concepts of chemistry

were obtained through data compiled were obtained through data compiled by taking measurementsby taking measurements

How much…?How much…? How long…?How long…? How many...?How many...? These questions cannot be answered These questions cannot be answered

without taking measurementswithout taking measurements The concepts of chemistry were The concepts of chemistry were

discovered as data was collected and discovered as data was collected and subjected to the scientific methodsubjected to the scientific method

2.5 The Basic Units of 2.5 The Basic Units of MeasurementMeasurement

The estimation of the magnitude of an The estimation of the magnitude of an object relative to a unit of measurementobject relative to a unit of measurement

Involves a measuring device:Involves a measuring device: meter stick, scale, thermometermeter stick, scale, thermometer

The device is calibrated to compare the The device is calibrated to compare the object to some standard object to some standard (inch/centimeter, pound/kilogram)(inch/centimeter, pound/kilogram)

Quantitative observation with two Quantitative observation with two parts: A parts: A numbernumber and a and a unitunit Number tells the total of the quantity Number tells the total of the quantity

measuredmeasured Unit tells the scale (dimensions)Unit tells the scale (dimensions)

2.5 The Basic Units of 2.5 The Basic Units of MeasurementMeasurement

A unit is a standard (accepted) quantityA unit is a standard (accepted) quantity Describes what is being added upDescribes what is being added up Units are essential to a measurementUnits are essential to a measurement For example, you need “six of sugar”For example, you need “six of sugar”

teaspoons?teaspoons? ounces?ounces? cups?cups? pounds?pounds?

2.5 The Standard Units (of 2.5 The Standard Units (of Measurement)Measurement)

The unit tells the magnitude of the standardThe unit tells the magnitude of the standard Two most commonly used systems of units Two most commonly used systems of units

of measurementof measurement U.S. (English) systemU.S. (English) system: Used in everyday : Used in everyday

commerce (USA and Britain*)commerce (USA and Britain*) Metric systemMetric system: Used in everyday : Used in everyday

commerce and science (The rest of the commerce and science (The rest of the world)world)

SI Units (1960): A modern, revised form of SI Units (1960): A modern, revised form of the metric system set up to create the metric system set up to create uniformity of units used worldwide (world’s uniformity of units used worldwide (world’s most widely used)most widely used)

2.5 The Standard Units (of 2.5 The Standard Units (of Measurement):Measurement):

The Metric/SI SystemThe Metric/SI System The metric system is a decimal The metric system is a decimal

system of measurement based on system of measurement based on the the metermeter and the and the gramgram

It has a single It has a single base unitbase unit per physical per physical quantity quantity

All other units are multiples of 10 of All other units are multiples of 10 of the base unitthe base unit

The power (multiple) of 10 is The power (multiple) of 10 is indicated by a prefixindicated by a prefix

2.5 The Standard Units: 2.5 The Standard Units: The Metric SystemThe Metric System

In the metric system there is one base In the metric system there is one base unit for each type of measurementunit for each type of measurement lengthlength volumevolume massmass

The base units multiplied by the The base units multiplied by the appropriate appropriate power of 10power of 10 form smaller or form smaller or larger unitslarger units

The prefixes are always the same, The prefixes are always the same, regardless of the base unitregardless of the base unit millimilligrams and grams and millimilliliters both mean liters both mean

1/1000 of the base unit1/1000 of the base unit

2.5 The Standard Units: 2.5 The Standard Units: LengthLength

MeterMeter Base unit of length in Base unit of length in metricmetric and SI system and SI system About 3 ½ inches longer than a yardAbout 3 ½ inches longer than a yard

1 m = 1.094 yd1 m = 1.094 yd

2.5 The Standard Units: 2.5 The Standard Units: LengthLength

Other units of Other units of length are derived length are derived from the meterfrom the meter

Commonly use Commonly use centimeters (cm)centimeters (cm) 1 m = 100 cm1 m = 100 cm 1 inch = 2.54 cm 1 inch = 2.54 cm

(exactly)(exactly)

2.5 The Standard Units: 2.5 The Standard Units: VolumeVolume

Measure of the Measure of the amount of three-amount of three-dimensional space dimensional space occupied by a objectoccupied by a object

Derived from lengthDerived from length SI unit = SI unit = cubic meter cubic meter

(m(m33)) Metric unitMetric unit == liter (L) liter (L)

or 10 cmor 10 cm3 3

Commonly measure Commonly measure smaller volumes in smaller volumes in cubic centimeters cubic centimeters (cm(cm33))

Volume = side × side × side

volume = side × side × side

2.5 The Standard Units: 2.5 The Standard Units: VolumeVolume

Since it is a three-Since it is a three-dimensional dimensional measure, its units measure, its units have been cubed have been cubed

SI base unit = SI base unit = cubic meter (mcubic meter (m33))

This unit is too This unit is too large for practical large for practical use in chemistryuse in chemistry

Take a volume Take a volume 1000 times smaller 1000 times smaller than the cubic than the cubic meter, 1dmmeter, 1dm33

2.5 The Standard Units: 2.5 The Standard Units: VolumeVolume

MetricMetric base unit base unit == 1dm1dm3 3 == liter (L) liter (L)

1L = 1.057 qt1L = 1.057 qt Commonly measure Commonly measure

smaller volumes in smaller volumes in cubic centimeters cubic centimeters (cm(cm33))

Take a volume 1000 Take a volume 1000 times smaller than times smaller than the cubic decimeter, the cubic decimeter, 1cm1cm33

2.5 The Standard Units: 2.5 The Standard Units: VolumeVolume

MetricMetric base unit base unit == 1dm1dm3 3 == liter (L) liter (L)

1L = 1.057 qt1L = 1.057 qt Commonly measure Commonly measure

smaller volumes in smaller volumes in cubic centimeters cubic centimeters (cm(cm33))

Take a volume 1000 Take a volume 1000 times smaller than times smaller than the cubic decimeter, the cubic decimeter, 1cm1cm33

2.5 The Standard Units: 2.5 The Standard Units: VolumeVolume

The most commonly The most commonly used unit of volume used unit of volume in the laboratory: in the laboratory: milliliter (mL)milliliter (mL)

1 mL = 1 cm1 mL = 1 cm33

1 L= 1 dm1 L= 1 dm3 3 = 1000 = 1000 mLmL

1 m1 m3 3 = 1000 dm= 1000 dm3 3 = = 1,000,000 cm1,000,000 cm3 3

Use a graduated Use a graduated cylinder or a pipette cylinder or a pipette to measure liquids in to measure liquids in the labthe lab

2.5 The Standard Units: Mass2.5 The Standard Units: Mass Measure of the total quantity of Measure of the total quantity of

matter present in an objectmatter present in an object SI unit (base) = SI unit (base) = kilogram (kg) kilogram (kg) Metric unitMetric unit (base) = (base) = gram (g)gram (g) Commonly measure mass in grams Commonly measure mass in grams

(g) or milligrams (mg)(g) or milligrams (mg) 1 kg = 1000 g1 kg = 1000 g 1 g = 1000 mg1 g = 1000 mg 1 kg = 2.205 pounds 1 kg = 2.205 pounds 1 lb = 453.6 g1 lb = 453.6 g

2.5 Prefixes Multipliers2.5 Prefixes Multipliers One base unit for each type of measurementOne base unit for each type of measurement Length (Length (metermeter), volume (), volume (literliter), and mass (), and mass (gramgram*)*) The base units are then multiplied by the The base units are then multiplied by the

appropriate power of 10 to form larger or smaller appropriate power of 10 to form larger or smaller unitsunits

base unit = meter, liter, or gram

2.5 Prefixes Multipliers 2.5 Prefixes Multipliers ((memorizememorize))

Mega (M) 1,000,000 Mega (M) 1,000,000 10 1066

Kilo Kilo (k)(k) 1,000 1,000 10 1033

BaseBase 1 1 10 1000

DeciDeci (d)(d) 0.1 0.1 10 10-1-1

CentiCenti (c)(c) 0.01 0.01 10 10-2-2

MilliMilli (m)(m) 0.001 0.001 10 10--

33

MicroMicro (µ) 0.000001(µ) 0.000001 10 10--

66

Nano Nano (n)(n) 0.000000001 10 0.000000001 10-9-9

× base unit

meter liter gram

2.5 Prefix Multipliers2.5 Prefix Multipliers For a particular measurement:For a particular measurement:

Choose the prefix which is similar in Choose the prefix which is similar in size to the quantity being measuredsize to the quantity being measured

Keep in mind which unit is largerKeep in mind which unit is larger A kilogram is larger than a gram, so A kilogram is larger than a gram, so

there must be a certain number of there must be a certain number of grams in one kilogramgrams in one kilogram

Choose the prefix most convenient Choose the prefix most convenient for a particular measurementfor a particular measurement

n < µ < m < c < base < k < Mn < µ < m < c < base < k < M

2.6 Converting from One Unit to 2.6 Converting from One Unit to Another: Dimensional AnalysisAnother: Dimensional Analysis

Many problems in chemistry involve converting Many problems in chemistry involve converting the units of a quantity or measurement to the units of a quantity or measurement to different unitsdifferent units

The new units may be in the same measurement The new units may be in the same measurement system or a different system, i.e., U.S. System to system or a different system, i.e., U.S. System to Metric and the converseMetric and the converse

Dimensional AnalysisDimensional Analysis is the method of problem is the method of problem solving used to achieve this unit conversionsolving used to achieve this unit conversion

Unit conversion is accomplished by Unit conversion is accomplished by multiplication of a given quantity (or multiplication of a given quantity (or measurement) by one or more measurement) by one or more conversion conversion factors factors to obtain the desired quantity or to obtain the desired quantity or measurementmeasurement

2.6 Converting from One Unit to 2.6 Converting from One Unit to Another: EqualitiesAnother: Equalities

An An equalityequality is a fixed relationship is a fixed relationship between two quantitiesbetween two quantities

It shows the relationship between two It shows the relationship between two units that measure the same quantityunits that measure the same quantity

The relationships are The relationships are exact, exact, not not measuredmeasured 1 min = 60 s1 min = 60 s 12 inches = 1 ft12 inches = 1 ft 1 dozen = 12 items (units)1 dozen = 12 items (units) 1L = 1000 mL1L = 1000 mL 16 oz = 1 lb16 oz = 1 lb 4 quarts = 1 gallon4 quarts = 1 gallon

2.6 Converting from One Unit to 2.6 Converting from One Unit to Another: Dimensional AnalysisAnother: Dimensional Analysis

Conversion factor: Conversion factor: An equality An equality expressed as a fraction expressed as a fraction It is used as a multiplier to convert It is used as a multiplier to convert

a quantity in one unit to its a quantity in one unit to its equivalent in another unitequivalent in another unit

May be exact or measuredMay be exact or measured Both parts of the conversion factor Both parts of the conversion factor

should have the same number of should have the same number of significant figuressignificant figures

2.7 Solving Multistep Conversion 2.7 Solving Multistep Conversion Problems:Problems:

Dimensional Analysis ExampleDimensional Analysis Example(Conversion Factors Stated within a Problem)(Conversion Factors Stated within a Problem)

The average person in the U.S. The average person in the U.S. consumes one-half pound of consumes one-half pound of sugar per day. How many sugar per day. How many pounds of sugar would be pounds of sugar would be consumed in one year?consumed in one year?

1)1) State the initial quantity State the initial quantity givengiven (and (and the unit): the unit): One yearOne year State the final quantity to State the final quantity to findfind (and (and the unit): the unit): PoundsPounds

2)2) Write a sequence of units (Write a sequence of units (mapmap) which ) which begins with the initial unit and ends begins with the initial unit and ends with the desired unit:with the desired unit:

year day pounds

2.7 Solving Multistep Conversion 2.7 Solving Multistep Conversion Problems:Problems:

Dimensional Analysis ExampleDimensional Analysis Example

3)3) For each unit change,For each unit change,

State the equalities:State the equalities: Every equality will have two Every equality will have two

conversion factorsconversion factors

1 cal 4.184 J1 cal 4.184 J1 cal 4.184 J

year day pounds

0.5 lb sugar 0.5 lb sugar =1day=1day

365 days = 1 365 days = 1 yearyear

2.7 Solving Multistep Conversion 2.7 Solving Multistep Conversion Problems:Problems:

Dimensional Analysis ExampleDimensional Analysis Example State the conversion factors:State the conversion factors:

4)4) Set Up the problem:Set Up the problem:

year1year1day(s) 365

sugar lbs. 183

sugar lb. 0.5day1

day1 sugar lb.0.5 and

day1sugar lb 0.5

Guide to Problem Solving when Guide to Problem Solving when Working Dimensional Analysis Working Dimensional Analysis

ProblemsProblems Identify the known or given quantity and the Identify the known or given quantity and the

units of the new quantity to be determinedunits of the new quantity to be determined Write out a sequence of units which starts Write out a sequence of units which starts

with your initial units and ends with the with your initial units and ends with the desired units (“solution map”)desired units (“solution map”)

Write out the necessary equalities and Write out the necessary equalities and conversion factors conversion factors

Perform the mathematical operations that Perform the mathematical operations that connect the unitsconnect the units

Check that the units cancel properly to Check that the units cancel properly to obtain the desired unitobtain the desired unit

Does the answer make sense?Does the answer make sense?

2.9 Density2.9 Density The ratio of the mass of an object to the volume The ratio of the mass of an object to the volume

occupied by that objectoccupied by that object Density tells how tightly the matter within an object is Density tells how tightly the matter within an object is

packed togetherpacked together Units for solids and liquids =Units for solids and liquids =

1 cm1 cm33 = 1 mL so can also use = 1 mL so can also use Unit for gases = g/LUnit for gases = g/L Density of three states of matter: solids > liquids >>> Density of three states of matter: solids > liquids >>>

gasesgases

Density mass

volume

g/mL

g/cm3

vmd

2.9 Density2.9 Density Can use density as a conversion factor Can use density as a conversion factor

between mass and volumebetween mass and volume Density of some common substances Density of some common substances

given in Table 2.4, page 33given in Table 2.4, page 33 You will be given any densities on tests You will be given any densities on tests

EXCEPTEXCEPT water water Density of water isDensity of water is 1.0 g/cm1.0 g/cm33 at at

room temperatureroom temperature 1.0 mL of water weighs how much?1.0 mL of water weighs how much? How many mL of water weigh 15 g?How many mL of water weigh 15 g?

2.9 Density2.9 Density To determine the density of an objectTo determine the density of an object Use a scale to determine the massUse a scale to determine the mass Determine the volume of the objectDetermine the volume of the object

Calculate it if possible (cube shaped)Calculate it if possible (cube shaped) Can also calculate volume by Can also calculate volume by

determining what volume of water is determining what volume of water is displaced by an objectdisplaced by an object

Volume of Water Displaced = Volume of ObjectVolume of Water Displaced = Volume of Object

Density ProblemDensity Problem Iron has a density of 7.87 g/cmIron has a density of 7.87 g/cm33. If . If

52.4 g of iron is added to 75.0 mL of 52.4 g of iron is added to 75.0 mL of water in a graduated cylinder, to water in a graduated cylinder, to what volume reading will the water what volume reading will the water level in the cylinder rise?level in the cylinder rise?

m 52.4 g

d 7.87 g cm3

Vi 75.0 mL

Vf ?

Density ProblemDensity Problem

volumemassdensity

1 mL iron7.87 g iron

6.658 mL iron52.4 g iron

6.658 mL iron + 75.0 mL water = 81.7 mL total

1 cm3 = 1 mLdensitymassvolume

Solve for volume of iron

End