Chemistry 103 Chapter 2 $ ₤ ¥ L m kg ml mm μg + - * / y x.

Chemistry 103

Chapter 2

$ ₤ ¥

L m kg ml mm μg

+ - * / yx

Chapter 2 – Slide 2

General Course structure

Learning Tools

Atoms ---> Compounds ---> Chemical Reactions


Outline

• Mathematics of Chemistry (Measurements)– Units– Significant Figures (Sig Figs)– Calculations & Sig Figs– Scientific Notation– Dimensional Analysis– Density


• Importance of Units

• Job Offer: Annual Salary = 1,000,000.


Measurements

Two components –

Numerical component

and

Dimensional component


Everyday MeasurementsYou make a measurement every time you

• Measure your height.

• Read your watch.

• Take your temperature.

• Weigh a cantaloupe.


• Scientists make many kinds of measurements– The determination of the dimensions, capacity, quantity or extent

of something– Length, Mass, Volume, Density

• All measurements are made relative to a standard• All measurements have uncertainty

Units and Measurements


Systems of Measurement

• English System– Common measurements– Pints, quarts, gallons, miles, etc.

• Metric System– Units in the metric system consist of a base unit

plus a prefix.


Measurement in Chemistry

In chemistry we

• Measure quantities.

• Do experiments.

• Calculate results.

• Use numbers to report measurements.

• Compare results to standards.

Copyright © 2008 by Pearson Education, Inc.Publishing as Benjamin Cummings


Length Measurement

Length

• Is measured using a meter stick.

• Has the unit of meter (m) in the metric (SI) system.



Inches and Centimeters

The unit of an inch is

equal to exactly 2.54

centimeters in the

metric system.

1 in. = 2.54 cm



Volume MeasurementVolume

• Is the space occupied by a substance.

• Has the unit liter (L) in metric system.

1 L = 1.057 qt

• Uses the unit m3(cubic meter) in the SI system.

• Is measured using a graduated cylinder. Copyright © 2008 by Pearson Education, Inc.

Publishing as Benjamin Cummings


Mass MeasurementThe mass of an object

• Is the quantity of material it contains.

• Is measured on a balance.

• Has the unit gram(g) in the metric system.

• Has the unit kilogram(kg) in the SI system.



Temperature MeasurementThe temperature of a substance

• Indicates how hot or cold it is.

• Is measured on the Celsius (C) scale in the metric system.

• On this thermometer temperature is 18ºC or 64ºF.

• In the SI system uses the Kelvin (K) scale.



Units in the Metric SystemIn the metric (SI) system, one unit is used for each

type of measurement:

Measurement Metric SI

length meter (m) meter (m)

volume liter (L) cubic meter (m3)

mass gram (g) kilogram (kg)

time second (s) second (s)

temperature Celsius (C) Kelvin (K)


Metric Base Units


For each of the following, indicate whether the unit describes A) length, B) mass, or C) volume.

Learning Check


Learning Check Identify the measurement with an SI unit. 1. John’s height is

2. The race was won in

3. The mass of a lemon is

4. The temperature is


Measured vs Exact numbers


Exact (Defined) and Inexact (Measured) Numbers

• Exact numbers– Have no uncertainty associated with them

– They are known exactly because they are defined or counted

– Example: 12 inches = 1 foot

• Measured numbers– Have some uncertainty associated with them

– Example: all measurements


Accuracy vs. Precision

Accuracy

How closely a measurement comes to

the true, accepted value

Precision

How closely measurements of the same quantities come

to each other


Significant Figures


Significant FiguresDigits in any measurement are known with certainty, plus one digit that is uncertain.

Measured numbers convey

*Magnitude*Uncertainty

*Units


The Calculator Problem7.83.8Calculator Answer: 2.05263……Is this a realistic answer?Is it 2, 2.0, 2.1, 2.05, 2.06, 2.052, 2.053, 2.0526, etc.? Which is it?Answer must reflect uncertainty expressed in original measurements. Using Significant Figures.We will come back to this later.


Rules for Significant Figures

It’s ALL about the ZEROs


Rules for Sig Figs

• All non-zero numbers in a measurement are significant.

4573

4573 has 4 sig figs


Rules for Sig Figs

• All zeros between sig figs are significant.

23007

23007 has 5 sig figs


Rules for Sig Figs

• In a number less than 1, zeros used to fix the position of the decimal are not significant.

0.000210.00021 has 2 sig figs


Rules for Sig Figs

• When a number has a decimal point, zeros to the right of the last nonzero digit are significant

0.00021000.0002100 has 4 sig figs


Rules for Sig Figs

• When a number without a decimal point explicitly shown ends in one or more zeros, we consider these zeros not to be significant. If some of the zeros are significant, bar notation is used.

_

820000 meters - 3 sig figs 820000


Practice Identifying Sig Figs


Significant FiguresHow many assuming all numbers are measured?

a). 75924

b). 30.002

c). 0.004320

d). 0.000002

e). 46,000


Measured Numbers

A measuring tool

• Is used to determine a quantity such as height or the mass of an object.

• Provides numbers for a measurement called measured numbers.



. l2. . . . l . . . . l3 . . . . l . . . . l4. . cm

• The markings on the meter stick at the end of the orange line are read as

The first digit 2 plus the second digit 2.7

• The last digit is obtained by estimating. • The end of the line might be estimated

between 2.7–2.8 as about half-way (0.5) which gives a reported length of 2.75 cm

Reading a Meter Stick


Known + Estimated DigitsIn the length reported as 2.75 cm,

• The digits 2 and 7 are certain (known)

• The final digit 5 was estimated (uncertain)

• All three digits (2.75) are significant including the estimated digit


Learning Check

. l8. . . . l . . . . l9. . . . l . . . . l10. . cm

What is the length of the red line?

1) 9.0 cm

2) 9.03 cm

3) 9.04 cm


Solution

. l8. . . . l . . . . l9. . . . l . . . . l10. . cm

The length of the red line could be reported as

2) 9.03 cm

or 3) 9.04 cm

The estimated digit may be slightly different.

Both readings are acceptable.


. l3. . . . l . . . . l4. . . . l . . . . l5. . cm

• For this measurement, the first and second known digits are 4.5.

• Because the line ends on a mark, the estimated digit in the hundredths place is 0.

• This measurement is reported as 4.50 cm.

Zero as a Measured Number


Significant Figuresin Measured Numbers

Significant figures

• Obtained from a measurement include all of the known digits plus the estimated digit.

• Reported in a measurement depend on the measuring tool.


Rounding off Numbers

• The number of significant figures in measurements affects any calculations done with these measurements– Your calculated answer can only be as certain

as the numbers used in the calculation


Calculator: Friend or Foe?

• Sometimes, the calculator will show more (or fewer) significant digits than it should– If the first digit to be deleted is 4 or

less, simply drop it and all the following digits

– If the first digit to be deleted is 5 or greater, that digit and all that follow are dropped and the last retained digit is increased by one


Sig Fig Rounding Example:

• Round the following measured number to

4 sig figs:

• 82.56702


Adding Significant Zeros• Sometimes a calculated answer requires more significant

digits. Then one or more zeros are added.

Calculated Answer Zeros Added to Give 3 Significant Figures

41.50.2

12


Practice Rounding Numbers


Significant FiguresRound each to 3 sig figs


When multiplying or dividing, use

• The same number of significant figures in your final answer as the measurement with the fewest significant figures.

• Rounding rules to obtain the correct number of significant figures.

Example:

110.5 x 0.048 = 5.304 = 5.3 (rounded)

4 SF 2 SF calculator 2 SF

Multiplication and Division


When adding or subtracting, use

• The same number of decimal places in your final answer as the measurement with the fewest decimal places.

• Use rounding rules to adjust the number of digits in the answer.

25.2 one decimal place

+ 1.34 two decimal places

26.54 calculated answer

26.5 answer with one decimal place

Addition and Subtraction


Math operations with Sig Figs


Report Answer with Correct Number of Sig Figs

A) 124.54 x 2.2 = 273.98800

B) 3420. + 2400. + 1095 = 6915.0000

C) 3420 + 2400 + 1095 = 6915.0000

D) 98.5564 = 2.1575394 45.68


When Math Operations Are Mixed

If you have both addition/subtraction and multiplication/division in a formula,

-carry out the operations in parenthesis first, and round according to the rules for that type of operation.

-complete the calculation by rounding according to the rules for the final type of operation.



_____5.681g_____ =(52.15ml - 32.4ml)




_____5.681g_____ = 5.681g(52.15ml - 32.4ml) 19.8ml



Mixed Operations and Significant Figures

• What is the result (to the correct number of significant figures) of the following calculations? Assume all numbers are measured.

(23 - 21) x (24.4 - 23.1)

(298 - 270) x (322)


Back To The Calculator Problem7.83.8Calculator Answer: 2.05263……Is this a realistic answer?Is it 2, 2.0, 2.1, 2.05, 2.06, 2.052, 2.053, 2.0526, etc.? Which is it? Answer must reflect uncertainty expressed in original measurements.


Scientific Notation

Scientific notation • Is used to write very large

or very small numbers• The width of a human hair,

0.000 008 m is written as:

8 x 10-6 m• A large number such as

2 500 000 s is written as:

2.5 x 106 sCopyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings


Scientific Notation• A number in scientific notation contains a coefficient • (1 or greater, less than 10) and a power of 10.

150 0.000735 coefficient power of ten coefficient power of ten 1.5 x 102 7.35 x 10-4

• To write a number in scientific notation, the decimal point is moved after the first non zero digit.

• The spaces moved are shown as a power of ten.

52 000 = 5.2 x 104 0.00378 = 3.78 x 10-3

4 spaces left 3 spaces right


Some Powers of Ten


Comparing Numbers in Standard and Scientific Notation

Standard Format Scientific NotationDiameter of Earth

12 800 000 mMass of a human

68 kg Length of a pox virus

0.000 03 cm


Comparing Numbers in Standard and Scientific Notation

Standard Format Scientific NotationDiameter of Earth

12 800 000 m 1.28 x 107 m (3 sig figs)Mass of a human

68 kg 6.8 x 101 kg (2 sig figs)Length of a pox virus

0.000 03 cm 3 x 10-5 cm (1 sig fig)

NOTE: The Coefficient identifies or indicates the number of significant figures in the measurement.

Dimensional Analysis

Defining Conversion Factors


Conversion Factors• Conversion factors

A ratio that specifies how one unit of measurement is related to another

• Creating conversion factors from equalities12 in.= 1 ft

1 L = 1000 mL

1 = in 12

ft 1or 1

ft 1

in 12

1 = L 1

mL 1000or 1

mL 1000

L 1



How many seconds are in 2 minutes? ? seconds = 2 minutes 60 seconds = 1 minute ? seconds = 2 minutes x 60 seconds =

1 minute

120 seconds (exactly)


Dimensional AnalysisIf we assume there are exactly 365 days in a

year, how many seconds are in one year?

? seconds = 1 year



• A problem solving method in which the units (associated with numbers) are used as a guide in setting up the calculations.

unitsdesiredinAnswerunitgiven

unitdesiredxunitgivenintMeasuremen

Conversion Factor


Exact vs Measured Relationships

• Metric to Metric – exact

• English to English – exact

• Metric to English –

typically measured

(must consider sig figs)


English to Metric Conversion Factors



What is 165 lb in kg?

STEP 1 Given: 165 lb Need: kg

STEP 2 Plan

STEP 3 Equalities/Factors

1 kg = 2.205 lb

2.205 lb and 1 kg

1 kg 2.205 lb

STEP 4 Set Up Problem

? kg = 165 lb


Learning Check• If a ski pole is 3.0 feet in length, how long

is the ski pole in mm?

(1000mm = 1m, 12 inches=1ft, 1m=39.37inches)


Learning Check

• If a ski pole is 3.0 feet in length, how long is the ski pole in mm?

(1000mm = 1m, 12 inches=1ft, 1m=39.37inches)

3.0 feet mm?

Plan


Learning Check

• If a bucket contains 4.65L of water. How many gallons of water is this?

(1 gallon = 4qts, 1L = 1.057qt)



If Jules Vern expressed the title of his famous book, “Twenty Thousand Leagues Under the Sea” in feet, what would the title be?

(1mile = 5280ft, 1 League = 3.450miles)


Converting from squared units to squared units or cubed units to cubed units

• Warning: This type of conversions give many students difficulties!!!!!

• The one thing you have to remember:– What does it mean to say that a unit is squared or

cubed?

– m2 = m x m; s3 = s x s x s

• When there are squared or cubed units, you have multiple units to cancel out!


Examples• Convert 127.4 cm3 to m3.

(100cm = 1m)

• Convert .572 miles2 to km2.

(1km = .621miles)


Displacement volume for a stock engine in a 1984 Corvette is specified at 350 in3. What is the displacement in L?


Percent Factor in a ProblemIf the thickness of the skin fold at the waist indicates an 11% body fat, how much fat is in a person with a mass of 86 kg?

percent factor

86 kg mass x 11 kg fat

100 kg mass

= 9.5 kg fat



Even MORE Practice with Conversion Factors

• A lean hamburger is 22% fat by weight. How many grams of fat are in 0.25 lb of the hamburger? (1lb = 453.6g)


Density

• A ratio of the mass of an object divided by its volume

Density = Mass/Volume

• Typical units = g/mL (NOTE: 1mL=1cm3)

• We have an unknown metal with a mass of 59.24 g and a volume of 6.64 mL. What is its density?


Density

• A ratio of the mass of an object divided by its volume

Density = Mass/Volume

• Typical units = g/mL (NOTE: 1mL=1cm3)

• We have an unknown metal with a mass of 59.24 g and a volume of 6.64 mL. What is its density?

Density = 59.24g = 8.92g/mL 6.64mL


Densities of Common Substances

Is Density a Physical or a Chemical Property?


Measuring Density in Lab


What is the density (g/cm3) of 48.0 g of a metal if the level of water in a graduated cylinder rises from 25.0 mL to 33.0 mL after the metal is added?

A) 0.17 g/cm3 B) 6.0 g/cm3 C) 380 g/cm3

25.0 mL 33.0 mL

object

Learning Check


Sink or Float• Ice floats in water

because the density of ice is less than the density of water.

• Aluminum sinks because its density is greater than the density of water.



Which diagram correctly represents the liquid layers in the cylinder? Karo (K) syrup (1.4 g/mL), vegetable (V) oil (0.91 g/mL,) water (W) (1.0 g/mL)

A B C

K

K

W

W

W

V

V

V

K

Learning Check


Osmium is a very dense metal. What is its density in g/cm3 if 50.0 g of osmium has a volume of 2.22 cm3?

a) 2.25 g/cm3 b) 22.5 g/cm3 c) 111 g/cm3

Learning Check


Density can be written as an equality. • For a substance with a density of 3.8 g/mL, the equality is:

3.8 g = 1 mL

• From this equality, two conversion factors can be written for density.

Conversion 3.8 g and 1 mL factors 1 mL 3.8 g

Density as a Conversion Factor


Density Example• You have been given 150.g of ethyl alcohol which

has a density of 0.785g/mL. Will this quantity fit into a 150mL beaker?


DENSITY PRACTICE


The density of octane, a component of gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane?

A) 0.614 kg B) 614 kg C) 1.25 kg

Learning Check


Temperature

Temperature

• Is a measure of how hot or cold an object is compared to another object

• Indicates that heat flows from the object with a higher temperature to the object with a lower temperature

• Is measured using a thermometer



Temperature Scales


Solving a Temperature Problem

A person with hypothermia has abody temperature of 34.8°C. What isthat temperature in °F?

TF = 1.8 TC + 32

TF = 1.8 (34.8°C) + 32° exact tenths exact

= 62.6 + 32° = 94.6°F tenths Copyright © 2005 by Pearson Education, Inc.

Publishing as Benjamin Cummings


Converting between Temperature Scales

• ***Conversions between Celsius and Kelvin (Temperature in K) = (temperature in oC) + 273

(temperature in oC) = (temperature in K) – 273

• Conversions between Celsius and Fahrenheit oF = 9/5 (oC) + 32 or 1.8 (oC) + 32 oC = 5/9(oF – 32) or 1/1.8 (oF – 32)

9/5 = 1.8/1 or 5/9 = 1/1.8

Chemistry 103 Chapter 2 $ ₤ ¥ L m kg ml mm μg + - * / y x.

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