Chapter 2 – Analyzing Data

2.1 Units and Measurement [and standard problem solving technique illustrated using density]

2.2 Scientific Notation and Dimensional Analysis

2.3 Uncertainty in Data

2.4 Representing Data

Section 2.1 Units and Measurements

• Know the first five of the seven base units in the SI system (time, length, mass, temperature and amount of substance) including their units, symbols, and the basic physical objects or phenomena on which they are defined.

• Convert between the Kelvin and Celsius temperature scales.

• Distinguish between and give examples of base units and derived units.

Chemists use an internationally recognized system of units to communicate their findings.

Objectives

Section 2.1 Units and Measurements

• Use the equation for density to solve for an unknown quantity using the problem solving process described on page 38 in example problem 2.1.

• Understand both the concepts and experimental procedure involved in the Mini Lab on page 39 (Determine Density) and be able to answer the analysis questions listed at the end of the lab.

• Know all the prefixes from giga to nano in table 2.2 including their symbols and associated powers of ten and be able to use them in a numerical problem.

Objectives (cont)

Section 2.1 Units and Measurements

Key Concepts• SI measurement units allow scientists to report

data to other scientists.

• Adding prefixes to SI units extends the range of possible measurements.

• To convert to Kelvin temperature, add 273.15 to the Celsius temperature. K = °C + 273.15

• Volume and density have derived units. Density, which is a ratio of mass to volume, can be used to identify an unknown sample of matter.

Units of Measurement

SI - Systemè Internationale d’Unités

7 base units

Base unit is based on an object or an event in the physical world

Base unit is independent of other units

Quantity Unit Abbrev.

Time second s

Length meter m

Mass kilogram kg

Temperature kelvin K

Amount of substance mole mol

Current ampere A

Luminous Intensity candela cd

The 7 SI Base Units (see Table 2.1)

Only base unit with prefix

Prefixes

To better describe range of possible measurements, base units (and other units) are modified by using prefixes

Correspond to a particular power of ten

See table 2.2 (following)

Memorize value and the symbol (including upper /lower case) of prefixes in 10-9 to 106 range

SI Prefixes – Page 33

Capital letters Greek letter

SI Prefixes – Expanded

Base Unit - TimeThe Second

In 1967, the 13th General Conference on Weights and Measures first defined the SI unit of time as the duration of 9,192,631,770 cycles of microwave light absorbed or emitted by the hyperfine transition of cesium-133 atoms in their ground state undisturbed by external fields

Length

The Meter

Distance light travels through a vacuum in 1/(299 792 458) of a second

Masskilogram (kg)

Only base unit whose standard is a physical object

Defined by platinum-iridium metal cylinder kept in Sèvres, France

Temperaturekelvin (K)

The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water

Temperature

Kelvin scale• Unit kelvin (small k), abbreviation K (but

no degree sign) • Absolute zero = zero K

Celsius scale• Unit °C (with degree sign)• T(C) = T(K) - 273.15

Kelvin scale important for certain formulas we will use in later chapters

Temperatures – K vs CBoiling point water373.15 K 100.00 C

Freezing point water273.15 K 0.00 C

Absolute zero0.00 K -273.15 C

The Mole and Avogadro’s Number

Abbreviation is mol

SI base unit for amount of substance

Defined as number of representative particles (carbon atoms) in exactly 12 g of pure carbon-12

Mole of anything contains 6.022 X 1023 representative particles

6.022 X 1023 = Avogadro’s number

Picky Details - Units

Abbreviations are avoidedProper

• s or second• cm3 or cubic centimeter• m/s or meter per second

Improper• sec• cc• mps

Picky Details - Units

Unit symbols are unaltered in the plural

Proper• l = 75 cm

Improper• l = 75 cms

Derived Units

Defined by a combination of base units

Velocity v = l/t m/s

Volume V = l l l m3

liter (L) cubic decimeter dm3

Density d = mass / V kg/m3

Derived Unit - Density

Density = mass volume

Most common unit is g/cm3

Will return to using density in a word problem later in presentation

Chapter 2 – Analyzing Data

2.1 Units of Measurement [and standard problem solving technique illustrated using density]

2.2 Scientific Notation and Dimensional Analysis

2.3 Uncertainty in Data

2.4 Representing Data

Section 2.2 Scientific Notation and Dimensional Analysis

• Express numbers correctly in standard scientific notation, convert them to and from numbers not expressed in scientific notation, and perform standard arithmetic operations using them.

Scientists often express numbers in scientific notation.

Section 2.2 Scientific Notation and Dimensional Analysis

Key Concepts

• A number expressed in scientific notation is written as a coefficient between 1 and 10 multiplied by 10 raised to a power. To add or subtract numbers in scientific notation, the numbers must have the same exponent.

• To multiply or divide numbers in scientific notation, multiply or divide the coefficients and then add or subtract the exponents, respectively.

Scientific Notation

Expresses a number as a multiple of two factors:

• 1 number 10 3.1 -7.9• 10 raised to a power 103 10-7

100 = 1 10n > 1 n positive integer0 < 10-n < 1 n positive integer

Scientific Notation

3.1 104 = 31000

3.1 10-4 = 0.00031• Have moved decimal point 4 places in

both cases

Some exponents match one of the standard SI prefixes

• 4.27 10-6 s = 4.27 s (microseconds)

Scientific NotationAddition/Subtraction

• Must be same power of ten

Multiplication/Division• Multiply first factors• Add exponents if multiplying • Subtract exponents if dividing

Put result back into standard form• z.yyy 10n z = any non-zero digit• One digit in front of decimal point

Practice – Scientific NotationGeneral

Problems 11(a-h), 12(a-d) page 41Problems 76(a-d), 77(a-d) page 62

Addition & SubtractionProblems 13(a-d), 14(a-d) page 42Problems 78(a-j) page 62

Multiplication & DivisionProblems 15(a-d), 16(a-d) page 43Problems 79(a-f) page 62

Chapter 2 – Analyzing Data

2.1 Units of Measurement [and standard problem solving technique illustrated using density]

2.2 Scientific Notation and Dimensional Analysis

2.3 Uncertainty in Data

2.4 Representing Data

Section 2.3 Uncertainty in Data

• Define and compare accuracy and precision and correctly identify which term or terms apply to a given value based on a description of how the value was determined.

• Calculate the percent error associated with a given measurement.

• Determine the number of significant figures associated with a given number.

Measurements contain uncertainties that affect how a result is presented.

Section 2.3 Uncertainty in Data

• Determine the appropriate number of significant figures to record when using an analog measuring device such as a ruler.

• Determine the number of significant figures associated with a result obtained from simple arithmetic operations (addition, subtraction, multiplication, division) on numbers.

• Round a number to a specified number of significant digits.

Objectives (cont)

Section 2.3 Uncertainty in Data

• Show all work for a problem following the Problem-Solving Process described on page 38 in example problem 2.1 while utilizing dimensional analysis, significant figures, rounding and standard algebra skills.

Objectives (cont)

Section 2.3 Uncertainty in Data

Key Concepts

• An accurate measurement is close to the accepted value. A set of precise measurements shows little variation.

• The measurement device determines the degree of precision possible.

• Error is the difference between the measured value and the accepted value. Percent error gives the percent deviation from the accepted value.

error = experimental value – accepted value

Section 2.3 Uncertainty in Data (cont.)

Key Concepts

• The number of significant figures reflects the precision of reported data.

• Calculations should be rounded to the correct number of significant figures.

• The rules for determining the number of significant figures in a number produced in a mathematical operation are different for multiplication/division and addition/subtraction.

Accuracy and Precision

Accuracy refers to agreement of particular value with true value (sometimes true value is difficult to determine; may require appropriate calibration)

Precision refers to degree of agreement among several measurements of same quantity

Calibration[Formal definition – don’t need to know]The set of operations which establish, under specified conditions, the relationship between values indicated by a measuring instrument or measuring system, and the corresponding standard or known values derived from the standard.

Purpose is to determine and/or improve the accuracy of the device being calibrated

Calibration - ExampleTo calibrate an electronic balance, might use following approach:a) Assign a value of zero to the reading of the

output of the balance when nothing is on the balance pan

b) Assign a value equal to the value of a calibration mass (e.g., 1000.00 g) to the reading of the output of the balance when the calibration mass is placed on the balance pan

c) Assume linear behavior of the reading of the balance between these two calibration points

Calibration - ExampleFor calibration to be possible, need a calibration mass (e.g., a 100.00 g mass)

The actual (true) value of the calibration mass has to be determined by the organization/company supplying the mass – this ultimately requires tracing back to the primary standard of mass (making comparisons that are linked to the kilogram standard mass)

It is only through calibration that the accuracy of a measurement device (i.e., a balance) can be determined and improved

Accuracy and PrecisionRefer to figure 2.10

Accuracy and PrecisionRefer to figure 2.10

precise and accurate

precise but not accurate

Precision and Accuracy

Precision and Accuracy

random error

Not precise, individual measurement not generally accurate but average is

systematic error

May be precise (only marginally for this example); average not accurate

Percent Error

% Error = 100 (actual – measured) actual(actual – measured) = errorCan ignore plus or minus signs for now – need only absolute value of errorActual = “true value”In some cases, true value unknown(Often compare a predicted value from some model with measured value)

Practice

Accuracy and PrecisionProblems 46, 48, page 54Problem 87 page 63

Percent ErrorProblems 32-34 page 49Problems 49, 51 page 54Problems 93(a-d), 94(a-d) page 63

Significant Figures2 different types of numbers:

Exact

Measured

Exact numbers have infinite precision

Measured - obtained from measuring device, have error and limited precision

Measured number written to reflect both its numerical value and the precision to which it was measured

Significant Figures and Measurement Precision

For lab data, sig figs determined by precision of measurement device

For digital device, last number to right in display is limit to precision

For analog device, precision limited by the estimated digit obtained from “eyeballing” reading between markings

Examples to follow

Balances and Precision

Electronic analytical 0.0001 g

(digital)

Student 0.1 g(digital)

Triple beam 0.01 g(analog)

Significant Figures

Mass of object measured on student balance (precision ± 0.1g) is 23.6 g

This quantity contains 3 significant figures, i.e., three experimentally meaningful digits

If same measurement made with analytical balance (precision ± 0.0001g) , mass might be 23.5820 g (6 sig. fig.)

2 measurements of mass of same object

Same quantity described at two different levels of precision or certainty

Significant Figures and Precision

Significant Figures Determined by Measuring Device

32.33 C 32.3 C

Significant Figures Determined by Measuring Device

0.1 mL graduationsCan estimate to 0.01 mL

Reading 1.70 mL3 significant figures

Significant Figures Determined by Measuring Device

0.1 cm graduationsCan estimate to 0.01 cm

Reading 5.22 cm3 significant figures

What Is Nail Length?

~6.33 cm

Learning Check

Length of wooden stick?1) 4.5 cm 2) 4.54 cm 3) 4.547 cm

Measurement of Volume

Graduated Cylinder

Volumetric Flask

Buret VolumetricPipet

Syringe

Most accurate but only useable for one volume

20.16 mL ±0.01mL

Measurement of volume using buret

Read at bottom of liquid curve (called the meniscus)

Significant Figures

Rules for recognizing which digits are significant – see the “Problem Solving Strategy” on top of page 51 and the following slide

Recognizing Significant (Sig) Figures

1. Non-zero numbers always significant• 72.3 (3 sf) 4.737x10-8 (4 sf)

2. Zeros between non-zero numbers are significant• 60.5 (3 sf) 7.3002x10-4 (5 sf)

3. All final zeros to right of decimal point are significant• 6.20 ( 3 sf) 5.47000x109 (6 sf)

4. Zeros acting as placeholders not significant• 0.00253 (3 sf) 43200 (3 sf)

Recognizing Significant (Sig) Figures

5. Counting numbers (integers) and defined constants or relationships have an infinite number of significant figures (all are exact numbers or relations)• 60 s = 1 minute • 1 foot = 12 inches• 6 molecules•

Significant Digits Practice

45.8736

.000239

.00023900

4.8000 104

48000

3.982106

1.00040

6

3

5

5

2

4

6

All digits count

Leading 0’s don’t

Trailing 0’s do

Trailing 0’s count

0’s don’t count w/o decimal

All digits count

0’s between digits count as well as trailing in decimal form

Determine number of significant digits in each of following:

How many significant figures in each of following measurements?

5.13

100.01

0.0401

0.0050

220,000

1.90 x 103

153.000

1.0050

3

5

3

2

2

3

6

5

?

Learning Check

Classify each of following as an exact or measured number or relationship

1 yard = 3 feet

Diameter of red blood cell = 6 x 10-4 cm

There are 6 hats on shelf

Gold melts at 1064°C

?

Ans: exact, measured, exact, measured

Practice

Significant FiguresProblems 35(a-d), 36(a-d), 37 p. 51Problems 47, 50 page 54Problems 85, 88 page 63

Rounding RulesSee “Problem Solving Strategy” on bottom of page 52

d = last significant digit r = digit to right of d

1. r < 5 then d d2. r > 5 then d d + 1

3. r = 5 and digit after r ≠ 0 then d d + 1

4. r = 5 and digit after r = 0 or nothing then

d d + 1 if d odd; d d if d even (d always ends up even using this rule)

(You may see variations on rule 4 in other places)

Rounding

4965.03

 

780,582

 

1999.5

4965 0 dropped, <5

780,600 8 dropped, >5

Note: you must include 0’s

2.000x103 5 dropped, = 5

If wrote as 2000 would have 1 SF

Round following to 4 significant figures:

Rounding

1.5587

.0037421

1367

128,522

1.6683x106

1.56

.00374

1370

129,000

1.67x106

Round following to 3 significant figures:

Practice

RoundingProblems 38(a-d), 39(a-d) page 53Problems 91(a-f) page 63

Sig Figures and Multiplication/Division

Answer must have same number of significant figures as number with fewest number of significant figures

24 x 3.26 x 5.774 = 451.75776 450= 4.5 x 102

2 sig figs allowed

6.38 × 2.0 = 12.76 13 (2 sig figs)

Operations with Significant Figures – Adding or Subtracting

When adding or subtracting, number of decimal places in result should equal smallest number of decimal places in any term in sum

135 cm + 3.25 cm = 138 cm

0 digits after dp

135 cm term limits answer to units decimal value

2 digits after dp

0 digits after dp

Sig Figures and Addition/Subtraction

Answer must have same number of digits to right of decimal point as value with fewest number of digits to right of decimal point (value with lowest precision)

• Note: not concerned with # of sig figures in numbers

11.0 + 5.7732 + 2.01 = 18.7832 18.8

Addition/Subtraction

25.5 32.72 320

+34.270 ‑ 0.0049 +12.5

59.770 32.7151 332.5

59.8 32.72 330

Focus on least significant digit (digit with least precision)

Last example – special caseLeast significant digit prior to decimal point (can use sci. notation to justify)

Addition/Subtraction – Special Case

Use scientific notation to handle case where number has zeros in front of decimal and no digits of any kind afterZeros in front of decimal are placeholders to indicate power of ten320 + 12.5 = 3.2x102 + 0.125x102

Now following ordinary rule, only allowed one digit after decimal point3.325x102 3.3x102 = 330

Addition/Subtraction: No Decimal Point

135000 m + 3250 m = ????

Convert to scientific notation

1.35x105 m + 3.25x103 m

Convert to common exponent (the largest)

1.35x105 m + 0.0325x105 m

Apply standard rule regarding digits after dp

1.35x105 m + 0.0325x105 m = 1.38x105 m

13500 m + 3250 m = 138000 m

Addition/Subtraction – Special Case

82000 + 5.32 = [82005.32] = 82000Special case (focus on last precise digit): 82000 = 8.2x104 5.32 = 0.000532x104

8.2x104 + 0.000532x104 = 8.2x104

Addition/Subtraction

6.8 + 11.934 = 18.734 18.7 (1 place after decimal)

Result has 3 SF even though 6.8 has only 2 SF

0.56 + 0.153 = [0.713] = 0.71

10.0 - 9.8742 = [0.1258] = 0.1

10 – 9.8742 = [0.1258] = 0 [1x101 – 0.98742x101 = 0.01258x101= 0]

Addition/SubtractionLimiting term: one having largest value of the least significant digit

ProblemLeast

significant digit

Answer

234 + 34.65 = 268.65 1’s place 2691.642x106 + 23x106 =

24.642x106 1’s place 25x106

100 + 34.56 = 134.56 100’s place 100150 + 28.57 = 178.57

10’s place 180

Rounding RulesIf doing a multistep calculation, round off after last step provided all steps are either multiplication/division or addition/subtraction (can’t be mixed)

Round after series of additions or subtractions before doing a multiplication or division

Sig Figures and Mixed OperationsSome calculations involve both multiplication/division and addition/subtraction

Must round intermediate result prior to switching to new category of operation

8.52 + 4.1586 18.73 + 153.2 =

= 8.52 + 77.89 + 153.2 = [239.61] = 239.6

(8.52 + 4.1586) (18.73 + 153.2) =

= 12.68 171.9 = [2179.692] = 2.180x103

(2180 has 3 SF, not 4 as required)

Sig Figures and Mixed Operations

Calculate 5.000Tc – 25C, where Tc = Celsius temperature, for Tk = 298.1 K

?

Tc = Tk – 273.15 = 298.1- 273.15 = [24.95]

Tc = 25.0 C 5.000Tc = 125C

125C – 25C = 1.00x102 C

Not 100! (need 3 SF)

Mixed Operations - Percent Error

% Error = 100 (actual – measured) actualCalculating % error always involves a mixed operation – subtraction followed by division

Must round after subtraction prior to dividing

Actual =16.24 g Meas. = 15.8 g % error ?

Error = 16.24 g – 15.8 g = [0.44 g] = 0.4 g

% Error = 100 0.4 / 16.24 = [2.463] = 2%

Not: 100 0.44 / 16.24 = [2.709] = 2.7%

Mixed Operations - Percent Error

Note: Results shown in example problem 2.5 on page 49 are incorrectAll answers in that example should have one significant figure (i.e., 3%, not 3.14%)

Sig Figures and Mixed OperationsAccepted value for density of copper = 8.92 g/cm3. 3 experiments to measure its density resulted in values of 8.74, 9.01, and 8.83 g/cm3. Calculate % error of average value of experiments.Avg = (8.74+9.01+8.83)/3 = 8.86 g/cm3

% Error=100(actual–measured)/actual% Error = 100 0.06 g/cm3/8.92 g/cm3

% Error = 0.7% (1 SF)

Sig Figures and Mixed OperationsResult of performing the following? 5.00 (22 1.85)?22 1.85 = 20.15 rounds to 2.0x101

Units place is significant2.0x101 5.00 = 1.0x102

Answer has 2 significant digits because subtraction operation generated a 2 significant digit intermediate result

Practice

Rounding – Addition/SubtractionProblems 40(a-b), 41(a-b) page 53Problems 92(a,b,d) page 63

Rounding – Multiplication/DivisionProblems 42(a-d), 43(a-d), 44 p. 54Problems 92(c,e) page 63

Chapter 2 – Analyzing Data

2.1 Units of Measurement [and standard problem solving technique illustrated using density]

2.2 Scientific Notation and Dimensional Analysis (Conversions)

2.3 Uncertainty in Data

2.4 Representing Data

Section 2.2 Scientific Notation and Dimensional Analysis

• Use dimensional analysis (aka “the factor-label method” or conversion factor) in a numerical problem to convert a given quantity to different units.

Scientists often solve problems using dimensional analysis.

Section 2.2 Scientific Notation and Dimensional Analysis

Key Concepts

• Dimensional analysis uses conversion factors to solve problems.

Dimensional Analysis(Conversions / Factor-Label)

In dimensional analysis always ask 3 questions:

1. What data are we given?

2. What quantity do we need?

3. What conversion factors are available to take us from what we are given to what we need?

Dimensional AnalysisAlso referred to as factor-label method

1. Start with given quantity with its units2. Multiply by conversion factors until

desired final units are obtained3. Make sure units cancel in converting

Example: convert 48 km to meters48 km 1000 m

1 km

= 48000 m = 4.8 104 m

Dimensional Analysis

Convert quantity 2.3 x 10-8 cm to nanometers (nm)

Determine conversion factorsCentimeter (cm) Meter (m)

1 cm = 0.01 m = 1 x 10-2 m

Meter (m) Nanometer (nm)

1 x 10-9 m = 1 nm

Setup equation so cm and m units cancel out leaving only nm

m

nm

cm

mcm8103.2

Fill-in values for conversion factors and solve equation

nmm

nm

cm

mcm 23.0

101

1

1

01.0103.2

98

Dimensional Analysis

Convert quantity 14 m/s to miles per hour (mi/hr)

Determine conversion factorsMeter (m) Kilometer (km)

Kilometer(km) Mile(mi)

1 mile = 1.6093 km 1000m = 1 km

Seconds (s) Minutes (min)

Minutes (min) Hours (hr)

60 sec = 1 min 60 min = 1 hr

Dimensional Analysis

hr

min

min

s

km

mi

m

km/14 sm

Setup equation so m, km, s, and min cancel out leaving only miles and hours

Dimensional Analysis

Fill-in values for conversion factors and solve equation

hrmi

sm

/31

hr1

min60

min1

s60

km6093.1

mi1

m1000

km1/14

Multiple Conversion Factors

Convert 100 km/h to m/s

100 km 1000 m 1 h 1 min h km 60 min 60 s

= 27.8 m/s

= 30 m/s (to have 1 SF)

Multiple Conversion Factors

Baseball thrown at 89.6 miles per hour

Speed in meters per second?

m/s

1 mile = 1.609 km; 60 s = 1 min; 60 min = 1 h

speed = 89.6mile

h

mile/h m/h

1.609kmmile 103 m

km

= 1.44x105mh 1 h

60 min 1 min

60 s= 40.0

ms

3 SF

Units Raised to a Power

Conversion factor must also be raised to that power

Area of circle = 28 in2 Area in cm2?

1 in = 2.54 cm (1 in)2 = (2.54 cm)2

Area= 28 in2

Area = 1.8 x 102 cm2

in2 cm2

(2.54 cm)2

(1 in)2 = 28 in2 6.45 cm2

1 in2

2 SF

Unit Conversion w Powers: mL to cm3

liter (L) defined = cubic decimeter (dm3)milliliter (mL) = 10-3 L

= 10-3 L 1 dm3/L= 10-3 dm3

= 10-3 dm3 (10-1 m/dm)3

= 10-3 dm3 10-3 m3/dm3

= 10-6 m3

= 10-6 m3 (1 cm/10-2 m)3

= 10-6 m3 1 cm3/10-6 m3

= 1 cm3

Convert 31,820 mi2 to square meters (m2)

Determine conversion factorsMile (mi) kilometer (km)

1 mile = 1.6093 km

kilometer (km) meter (m)

1000 m = 1 km

Dimensional Analysis

Setup equation so mi2 and km2 cancel out leaving only m2 (must have squares)

22

2820,31km

m

mi

kmmi

22

2

1

1000

1

6093.1820,31

km

m

mi

kmmi

Dimensional Analysis

Fill-in values for conversion factors and solve equation

2102

26

2

22 102407.8

1

101

1

5898.2820,31 m

km

m

mi

kmmi

8.2411010 m2

Unit Conversion with Powers

Convert 531 lb/ft3 to units of g/cm3

1 in = 2.54 cm, 12 in = 1 ft, 1 kg = 2.2046 lb

531 lb/ft3 x (1 kg/2.2046 lb) x (103 g/kg)

= 2.409x105 g/ft3

2.409x105 g/ft3 x (ft/12 in)3 x (in/2.54 cm)3

2.409x105 g/ft3 x (ft3/1728 in3) x (in3/16.39 cm3)= 8.507 g/cm3

= 8.51 g/cm3 3 SF

PracticeDimensional Analysis (Conversions)

Practice Problems 17(a-c), 18(a-b), 19(a-h), 20 page 45Practice Problems 21-23 page 46Problems 27, 28, 30 page 46Problems 80(a-f), 81- 84 page 63

Chapter 2 – Analyzing Data

2.1 Units of Measurement [and standard problem solving technique illustrated using density]

2.2 Scientific Notation and Dimensional Analysis

2.3 Uncertainty in Data

2.4 Representing Data

Section 2.1 Units of Measurement

• Show all work for a problem following the Problem-Solving Process described on page 38 in example problem 2.1 while utilizing dimensional analysis, significant figures, rounding and standard algebra skills.

Objectives

Word Problems in Chemistry

Will need to solve variety of such problems

Exists standard approach, which will also be used in physics

Illustrated in text using problem involving density

Word Problem Involving Density

Problem 2.1, page 38

Sample of aluminum placed in 25 mL graduated cylinder containing 10.5 mL water. Water level rises to 13.5 mL.

Mass of aluminum sample?density of Al = 2.7 g/cm3 (also see Table R-7, page 971 for data)

Required Problem-Solving StepsRead problem; make sure you understand what is being askedAnalyze problem to find unknown (mass)State defining equation (d = m/V)Re-write equation to solve for unknown

• mass = volume density• m = V d (use of symbols preferred)

Show all intermediate work with unitsEvaluate final answer – check that units and significant figures are correct

Word Problem 2.1Solution shown on following slide differs from that shown on page 38Symbols (m) are used in place of words (mass)Subscripts used to convey additional information about the symbol (VAl instead of “volume of sample”; VAl vs VW to distinguish volume of aluminum from measured volume of water)Prefer use of symbols in your workSubscripts optional

Review of Problem 2.1 Al=Aluminum W=Water f=final i=initial

dAl = 2.7 g/cm3

VAl = DVw = Vwf – Vwi (water displacement)

VAl = 13.5 mL – 10.5 mL = 3.0 mL

dAl = mAl / VAl (main formula; must include)

mAl = dAl VAl (formula solved for unknown)

mAl = 2.7 g 3.0 mL mL

= 8.1 g of Al

You must show symbols, values, and units in all your work using the above method

Practice

Working word problemsPractice Problems 1 – 3 page 38Problem 9, page 39Problems 66 – 69 page 62Problems 104, 105, 107 page 64

Chapter 2 – Analyzing Data

2.1 Units of Measurement [and standard problem solving technique illustrated using density]

2.2 Scientific Notation and Dimensional Analysis

2.3 Uncertainty in Data

2.4 Representing Data

Section 2.4 Representing Data

• Create graphs to reveal patterns in data.

• Interpret data presented in graphs.

• Identify dependent and independent variables

• Create a properly labeled line graph from supplied data (one with a reasonable number of labeled tic marks, axes with labels, a graph title, and data points and trend lines that are easily viewed and interpreted).

Graphs visually depict data, making it easier to see patterns and trends.

Objectives

Section 2.4 Representing Data

• Determine the numerical value and units of the slope of a straight line that is presented in a line graph.

• Define and distinguish between the processes of interpolation and extrapolation and use them to obtain predicted values from a trend line.

Objectives (cont)

Section 2.4 Representing Data

Key Concepts

• Circle graphs show parts of a whole. Bar graphs show how a factor varies with time, location, or temperature.

• Independent (x-axis) variables and dependent (y-axis) variables can be related in a linear or a nonlinear manner. The slope of a straight line is defined as rise/run, or ∆y/∆x.

• Because line graph data are considered continuous, you can interpolate between data points or extrapolate beyond them.

Pie ChartGood for showing breakdown of quantities that add to 100%

Bar GraphTrends of quantities versus a discrete variable (month, year, sample number, etc.)

Line GraphTrends of quantities versus a continuous variable (mass, time, volume, temperature)

Line GraphIndependent variable (variable deliberately changed by experimenter) on x axisDependent variable on y axisTerm “plot of A versus B” means that B = independent variable (x) A = dependent (y)

Line GraphsGraph should have title, axes labels with units, tick marks, data markers, and legend (if more than one set of data is plotted on same graph)If least squares/best-fit line used, get slope from

Slope = y2-y1 = Dy x2-x1 Dx

or get directly from softwarePay attention to the units of the slope

Line Graphs

Graph displaying both data points and best fit line

Negative slope(Dy < 0)

Line Graph - ExcelIn MS Excel, making a “line” chart (graph) will create a graph with a discrete x axis – not appropriate for most scientific uses

In Excel, make a scatter chart – it will have the necessary continuous x axis

When plotting data, use point markers (circles, etc.) only – do not connect points

When plotting lines use lines (solid, dashed, etc.) but no point markers

If variable have units, make sure they appear in axis labels – “Length (m)”

Interpolation / Extrapolation

If have trendline (does not have to be a straight line), points on trendline curve considered to be continuousInterpolation – reading value from a point on curve that falls between recorded data pointsExtrapolation - reading value from a point on curve that extends beyond recorded data points (potentially dangerous, especially if overdone)

Interpolation / Extrapolation

Extrapolated point at

elevation = 700 m

Interpolated point at

elevation = 350 m

Practice

Line graphsProblems 53, 54, 58 page 58Problem 111 page 64

End of Chapter