11/11/2015 1 the numerical side of chemistry chapter 2

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The numerical side of chemistry

Chapter 2

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Outline

• Precision and Accuracy• Uncertainty and Significant figures• Zeros and Significant figures• Scientific notation• Units of measure• Conversion factors and Algebraic

manipulations

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Accuracy and Precision

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Precision and AccuracyPrecision and Accuracy• AccuracyAccuracy refers to the agreement of a refers to the agreement of a

particular value with the particular value with the truetrue value.value.

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

Neither accurate nor

precise

Precise but not accurate

Precise AND accurate

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Types of ErrorTypes of Error

• Random ErrorRandom Error (Indeterminate Error) - (Indeterminate Error) - measurement has an equal probability of measurement has an equal probability of being high or low. being high or low.

Systematic ErrorSystematic Error (Determinate Error) - (Determinate Error) - Occurs in the Occurs in the same directionsame direction each time each time (high or low), often resulting from poor (high or low), often resulting from poor technique or incorrect calibration. technique or incorrect calibration. This can This can result in measurements that are precise, result in measurements that are precise, but not accurate.but not accurate.

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Uncertainty in MeasurementUncertainty in Measurement

• A digit that must be A digit that must be estimatedestimated is is called called uncertainuncertain. A . A measurementmeasurement always has some degree of always has some degree of uncertainty.uncertainty. Measurements are performed with instruments No instrument can read to an infinite number of decimal places

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Nature of MeasurementNature of Measurement

•

Part 1 - Part 1 - number number Part 2 - Part 2 - scale (unit) scale (unit)

Examples: Examples: 2020 grams grams 34.5 mL34.5 mL45.0 m45.0 m

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

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Significant figures or significant digits

• Digits that are not beyond accuracy of measuring device

• The certain digits and the estimated digit of a measurement

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Rules

• 245 • 0.04• 0.040• 1000• 10.00• 0.0301• 103

• 3 significant digits• 1 significant digit• 2 significant digits• 1 significant digit• 4 significant digit• 3 significant digit• 3 significant digit

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Rules for Counting Rules for Counting Significant Figures - DetailsSignificant Figures - Details

• Nonzero integersNonzero integers always count always count as significant figures. as significant figures.

34563456 hashas

44 sig figs.sig figs.

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

significant figures. significant figures.

– 0.04860.0486 has has

33 sig figs. sig figs.

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• Zeros Zeros -- Captive zeros Captive zeros always count always count

as as significant figures. significant figures.

– 16.07 16.07 has has


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• Zeros Zeros Trailing zerosTrailing zeros are significant only are significant only if the number contains a decimal if the number contains a decimal point. point.

9.3009.300 has has


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• Exact numbersExact numbers have an infinite have an infinite number of significant figures. number of significant figures.

11 inch = inch = 2.542.54 cm, exactlycm, exactly

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

1.0070 m

5 sig figs

17.10 kg 4 sig figs

100,890 L 5 sig figs

3.29 x 103 s 3 sig figs

0.0054 cm 2 sig figs

3,200,000 2 sig figs

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

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

6.8 + 11.934 = 6.8 + 11.934 =

18.734 18.734 18.7 ( 18.7 (3 sig figs3 sig figs))

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Sig Fig Practice #2Sig Fig Practice #2

3.24 m + 7.0 m

Calculation Calculator says: Answer

10.24 m 10.2 m

100.0 g - 23.73 g 76.27 g 76.3 g

0.02 cm + 2.371 cm 2.391 cm 2.39 cm

713.1 L - 3.872 L 709.228 L 709.2 L

1818.2 lb + 3.37 lb 1821.57 lb 1821.6 lb

2.030 mL - 1.870 mL 0.16 mL 0.160 mL

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Rules for Significant Figures in Rules for Significant Figures in Mathematical OperationsMathematical Operations

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

6.38 x 2.0 = 6.38 x 2.0 =

12.76 12.76 13 (2 sig figs)13 (2 sig figs)

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Sig Fig Practice #3Sig Fig Practice #3

3.24 m x 7.0 m

Calculation Calculator says: Answer

22.68 m2 23 m2

100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 4.22 g/cm3

0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2

710 m ÷ 3.0 s 236.6666667 m/s 240 m/s

1818.2 lb x 3.23 ft 5872.786 lb·ft 5870 lb·ft

1.030 g ÷ 2.87 mL 2.9561 g/mL 2.96 g/mL

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Why do we use scientific notation?

• To express very small and very large numbers

• To indicate the precision of the number

• Use it to avoid with sig digs

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

1 mole = 6020000000000000000000001 mole = 602000000000000000000000

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

Mass of an electron = Mass of an electron = 0.000000000000000000000000000000091 kg0.000000000000000000000000000000091 kg

Scientific NotationScientific Notation

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2 500 000 000

Step #1: Insert an understood decimal pointStep #1: Insert an understood decimal point

.

Step #2: Decide where the decimal Step #2: Decide where the decimal must end must end up so that one number is to its up so that one number is to its leftleftStep #3: Count how many places you Step #3: Count how many places you bounce bounce the decimal pointthe decimal point

123456789

Step #4: Re-write in the form M x 10Step #4: Re-write in the form M x 10nn

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2.5 x 102.5 x 1099

The exponent is the number of places we moved the decimal.

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0.00005790.0000579

Step #2: Decide where the decimal Step #2: Decide where the decimal must end must end up so that one number is to its up so that one number is to its leftleftStep #3: Count how many places you Step #3: Count how many places you bounce bounce the decimal pointthe decimal pointStep #4: Re-write in the form M x 10Step #4: Re-write in the form M x 10nn

1 2 3 4 5

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5.79 x 105.79 x 10-5-5

The exponent is negative because the number we started with was less than 1.

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ReviewReview::Scientific notation Scientific notation expresses a number in the expresses a number in the form:form: M x 10M x 10nn

1 1 M M 1010

n is an n is an integerinteger

SI measurementSI measurement• Le Système international d'unitésLe Système international d'unités • The only countries that have not The only countries that have not

officiallyofficially adopted SI are Liberia (in adopted SI are Liberia (in western Africa) and Myanmar western Africa) and Myanmar (a.k.a. Burma, in SE Asia), but now (a.k.a. Burma, in SE Asia), but now these are reportedly using metric these are reportedly using metric regularlyregularly

• Metrication is a process that does Metrication is a process that does not happen all at once, but is not happen all at once, but is rather a process that happens over rather a process that happens over time. time.

• Among countries with non-metric Among countries with non-metric usage, the U.S. is the usage, the U.S. is the only country only country significantly holding outsignificantly holding out.. The U.S. The U.S. officially adopted SI in 1866.officially adopted SI in 1866.

Information from U.S. Metric Association

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The Fundamental SI UnitsThe Fundamental SI Units (le Système International, SI)(le Système International, SI)

Physical Quantity Name Abbreviation

Mass kilogram kg

Length meter m

Time second s

Temperature Kelvin K

Electric Current Ampere A

Amount of Substance mole mol

Luminous Intensity candela cd

Standards of MeasurementStandards of Measurement

When we measure, we use a When we measure, we use a measuring tool to compare some measuring tool to compare some dimension of an object to a dimension of an object to a standard.standard.

For example, at one time the standard For example, at one time the standard for length was the king’s foot. What for length was the king’s foot. What are some problems with this are some problems with this standard?standard?

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Derived SI units

Physical quantity NameAbbreviation

Volume cubic meterm3

Pressure pascal Pa

Energy joule J

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Metric System

• System used in science• Decimal system• Measurements are related by

factors of 10• Has one standard unit for each

type of measurement• Prefixes are attached in front of

standard unit

Metric PrefixesMetric Prefixes• Kilo-Kilo- means 1000 of that unit means 1000 of that unit

– 1 kilometer (km) = 1000 meters (m)1 kilometer (km) = 1000 meters (m)

• Centi-Centi- means 1/100 of that unit means 1/100 of that unit

– 1 meter (m) = 100 centimeters (cm)1 meter (m) = 100 centimeters (cm)

– 1 dollar = 100 cents1 dollar = 100 cents

• Milli-Milli- means 1/1000 of that unit means 1/1000 of that unit

– 1 Liter (L) = 1000 milliliters (mL)1 Liter (L) = 1000 milliliters (mL)

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SI Prefixes Common to ChemistrySI Prefixes Common to Chemistry

Prefix Unit Abbr. Exponent

Mega M 106

Kilo k 103

Deci d 10-1

Centi c 10-2

Milli m 10-3

Micro 10-6

Nano n 10-9

Pico p 10-12

Metric PrefixesMetric Prefixes

Conversion FactorsConversion Factors

Fractions in which the numerator and Fractions in which the numerator and denominator are EQUAL quantities expressed denominator are EQUAL quantities expressed in different unitsin different units

Example: 1 in. = 2.54 cm

Factors: 1 in. and 2.54 cm 2.54 cm 1 in.

Learning Check

Write conversion factors that relate Write conversion factors that relate each of the following pairs of units:each of the following pairs of units:

1. Liters and mL1. Liters and mL

2. Hours and minutes2. Hours and minutes

3. Meters and kilometers3. Meters and kilometers

How many minutes are in 2.5 hours?

Conversion factor

2.5 hr x 2.5 hr x 60 min 60 min = 150 min = 150 min

1 hr1 hr

cancel

By using dimensional analysis / factor-label method, the UNITS By using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right side up, and the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the numbers!are calculated as well as the numbers!

Steps to Problem SolvingSteps to Problem Solving1. Write down the given amount. Don’t forget the

units!2. Multiply by a fraction.3. Use the fraction as a conversion factor.

Determine if the top or the bottom should be the same unit as the given so that it will cancel.

4. Put a unit on the opposite side that will be the new unit. If you don’t know a conversion between those units directly, use one that you do know that is a step toward the one you want at the end.

5. Insert the numbers on the conversion so that the top and the bottom amounts are EQUAL, but in different units.

6. Multiply and divide the units (Cancel).7. If the units are not the ones you want for your

answer, make more conversions until you reach that point.

8. Multiply and divide the numbers. Don’t forget “Please Excuse My Dear Aunt Sally”! (order of operations)

Learning Check

A rattlesnake is 2.44 m long. How A rattlesnake is 2.44 m long. How long is the snake in cm?long is the snake in cm?

a) a) 2440 cm2440 cm

b)b) 244 cm244 cm

c)c) 24.4 cm24.4 cm

Solution

A rattlesnake is 2.44 m long. How A rattlesnake is 2.44 m long. How long is the snake in cm?long is the snake in cm?

b)b) 244 cm244 cm

2.44 m x 2.44 m x 100 cm 100 cm = 244 cm= 244 cm1 m

Learning Check

How many seconds are in 1.4 days?

Unit plan: days hr min seconds

1.4 days x 24 hr x ?? 1 day

Wait a minute!

What is What is wrongwrong with the following setup? with the following setup?

1.4 day x 1.4 day x 1 day 1 day x x 60 min 60 min x x 60 60 secsec

24 hr 1 hr 1 24 hr 1 hr 1 minmin

Dealing with Two Units Dealing with Two Units

If your pace on a treadmill is 65 If your pace on a treadmill is 65 meters per minute, how many meters per minute, how many seconds will it take for you to walk a seconds will it take for you to walk a distance of 8450 feet?distance of 8450 feet?

What about Square and Cubic units? What about Square and Cubic units?

• Use the conversion factors you Use the conversion factors you already know, but when you square already know, but when you square or cube the unit, don’t forget to cube or cube the unit, don’t forget to cube the number also!the number also!

• Best way: Square or cube the Best way: Square or cube the ENTIRE conversion factorENTIRE conversion factor

• Example: Convert 4.3 cmExample: Convert 4.3 cm33 to mm to mm33

4.3 cm4.3 cm33 10 mm 10 mm 33

1 cm 1 cm ( ) =

4.3 cm4.3 cm33 10 1033 mm mm33

1133 cm cm33

= 4300 mm3

Learning CheckLearning Check

• A Nalgene water A Nalgene water bottle holds 1000 bottle holds 1000 cmcm33 of of dihydrogen dihydrogen monoxide monoxide (DHMO). How (DHMO). How many cubic many cubic decimeters is decimeters is that?that?

SolutionSolution

1000 cm1000 cm33 1 dm 1 dm 33

10 cm10 cm( ) = 1 dm= 1 dm33

So, a dmSo, a dm33 is the same as a Liter ! is the same as a Liter !

A cmA cm33 is the same as a milliliter. is the same as a milliliter.

Temperature ScalesTemperature Scales• FahrenheitFahrenheit

• CelsiusCelsius

• KelvinKelvin

Anders Celsius1701-1744

Lord Kelvin(William Thomson)1824-1907

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Temperature ScalesTemperature Scales

Notice that 1 Kelvin = 1 degree Celsius1 Kelvin = 1 degree Celsius

Boiling point of Boiling point of waterwater

Freezing point Freezing point of waterof water

CelsiusCelsius

100 ˚C100 ˚C

0 ˚C0 ˚C

100˚C100˚C

KelvinKelvin

373 K373 K

273 K273 K

100 K100 K

FahrenheitFahrenheit

32 ˚F32 ˚F

212 ˚F212 ˚F

180˚F180˚F

Calculations Using Calculations Using TemperatureTemperature

• Generally require temp’s in Generally require temp’s in kelvinskelvins

• T (K) = t (˚C) + 273.15T (K) = t (˚C) + 273.15• Body temp = 37 ˚C + 273 = 310 KBody temp = 37 ˚C + 273 = 310 K

• Liquid nitrogen = -196 ˚C + 273 = 77 KLiquid nitrogen = -196 ˚C + 273 = 77 K

• Generally require temp’s in Generally require temp’s in kelvinskelvins

• T (K) = t (˚C) + 273.15T (K) = t (˚C) + 273.15• Body temp = 37 ˚C + 273 = 310 KBody temp = 37 ˚C + 273 = 310 K

• Liquid nitrogen = -196 ˚C + 273 = 77 KLiquid nitrogen = -196 ˚C + 273 = 77 K

Fahrenheit Formula – Fahrenheit Formula –

180°F180°F = = 9°F 9°F == 1.8°F 1.8°F 100°C 5°C 100°C 5°C 1°C1°C

Zero point: 0°C = 32°FZero point: 0°C = 32°F

°F = 9/5 °C + 32°F = 9/5 °C + 32

Celsius Formula – Celsius Formula –

Rearrange to find T°CRearrange to find T°C

°F °F = = 9/5 °C + 329/5 °C + 32

°F - 32 = °F - 32 = 9/5 °C ( +32 - 32)9/5 °C ( +32 - 32)

°F - 32°F - 32 = = 9/5 °C9/5 °C

9/5 9/5 9/5 9/5

(°F - 32) * 5/9 = °C(°F - 32) * 5/9 = °C

Temperature Conversions – Temperature Conversions –

A person with hypothermia has a body A person with hypothermia has a body temperature of 29.1°C. What is the body temperature of 29.1°C. What is the body temperature in °F? temperature in °F?

°F °F = = 9/5 (29.1°C) + 329/5 (29.1°C) + 32

= = 52.4 + 3252.4 + 32

= = 84.4°F84.4°F

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Temperature measurementsKelvin temperature scale is also called absolute temperature scaleThere is not negative Kelvin temperature K=0C + 273.15

0F = 32 + 9/5 0C

0C = 5/9 (0F –32)

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What is temperature?

• Measure of how hot or cold an object is

• Determines the direction of heat transfer

• Heat moves from object with higher temperature to object with lower temperature

Learning Check –

Pizza is baked at 455°F. What is that in °C?

1) 437 °C1) 437 °C

2) 235°C2) 235°C

3) 221°C3) 221°C

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Density: m/v

• Density tells you how much matter there is in a given volume.

• Usually expressed in g/ml or g/cm3

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Densities of some common materialsMaterial Density

g/cm3

Material Densityg/L

Gold 19.3 Chlorine 2.95

Mercury 13.6 CO2 1.83

Lead 11.4 Ar 1.66

aluminum

2.70 Oxygen 1.33

Sugar 1.59 Air 1.20

Water 1.000 Nitrogen 1.17

Gasoline 0.66-0.69

Helium 0.166

Ethanol 0.789 Hydrogen

.0084

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• Intensive properties do not depend on amount of matter (density, boiling point, melting point)

• Extensive properties do depend on amount of matter(mass , volume, energy content).

Intensive and Extensive properties

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Energy

• Capacity to do work • Work causes an object to move (F

x d)• Potential Energy: Energy due to

position• Kinetic Energy: Energy due to the

motion of the object

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The JouleThe Joule

The unit of heat used in modern thermochemistry is the Joule

2

2111

s

mkgmeternewtonjoule

Non SI unit calorie1Cal=1000cal4.184J =1cal or 4.184kJ=Cal

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Law of conservation of energy

• Energy is neither created nor destroyed; it only changes form

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CalorimetryCalorimetry

The amount of heat absorbed or released during a physical or chemical change can be measured…

…usually by the change in temperature of a known quantity of water1 calorie is the heat required to raise the temperature of 1 gram of water by 1 C1 BTU is the heat required to raise the temperature of 1 pound of water by 1 F

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CalorimeteCalorimeterr

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A A Cheaper Cheaper

CalorimetCalorimeterer

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Specific heat

• Amount of heat energy needed to warm 1 g of that substance by 1oC

• Units are J/goC or cal/goC

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Specific Heat Notes

• Specific heat – how well a substance resist changing its temperature when it absorbs or releases heat

• Water has high s – results in coastal areas having milder climate than inland areas (coastal water temp. is quite stable which is favorable for marine life).

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More Specific Heat

• Organisms are primarily water – thus are able to resist more changes in their own temperature than if they were made of a liquid with a lower s

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Specific heats of some common substances

SubstanceWaterIronAluminumEthanol

(cal/g° C) (J/g ° C)• 1.000 4.184• 0.107 0.449• 0.215 0.901• 0.581 2.43

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Calculations Involving Specific Calculations Involving Specific HeatHeat

Tmsq

s = Specific Heat Capacity

q = Heat lost or gainedT = Temperature

change

OR

Tm

qs

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Principle of Heat Exchange

• The amount of heat lost by a substance is equal to the amount of heat gained by the substance to which it is transferred.

• m x ∆t x s = m x ∆t x s heat lost heat gained

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How to calculate amount of heat ?

H= specific heat x mass x change in T

ExampleCalculate the energy required to raise the

temperatureof a 387.0g bar of iron metal from 25oC to 40oC.

Thespecific heat of iron is 0.449 J/goC

11/11/2015 1 the numerical side of chemistry chapter 2

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