chapter 2 graph of global temperature rise in 20...

Roy Kennedy

Massachusetts Bay Community College

Wellesley Hills, MA

Introductory Chemistry, 3rd Edition

Nivaldo Tro

Chapter 2

Measurement and

Problem Solving

2009, Prentice Hall

Graph of global

Temperature rise in 20th

Century. Cover page

Opposite page 11.

Outline

Tro's "Introductory Chemistry",

Chapter 2

2

2.1 Measuring Global Temperature

2.2 Scientific Notation: Writing Large and Small Numbers

2.3 Significant Figures: Writing Numbers to Reflect Precision

2.4 Significant Figures in Calculations

2.5 The Basic Units of Measure

2.6 Converting One Unit to Another

2.7 Solving Multistep Conversions

2.8 Units Raised to a Power

2.9 Density

2.10 Numerical Problem Solving Strategies and the

Solution Map

2.1 Measuring Global

Temperature


Chapter 2

3


Chapter 2

4

Scientists have Measured the Average

Global Temperature Rise over the Past

Century to be 0.6 °C

• 0.6 – a number, quantitative:between 0.5 and 0.7 °C.

• °C – unit of a standardized scale.

• 0.60 °C - indicates a number between 0.59 and 0.61°C

• Number + unit = a measurement

Chapter 2 -Measurement and

Problem Solving Topics in this

Chapter.• Measurements

• Precision

• Basic Units

• Converting Measurements

• Derived Units


Chapter 2

5

Tro's Introductory Chemistry, Chapter

2

6

Scientific Notation

A way of writing large and small

numbers.

A small number :

Radius of a hydrogen nucleus is about

0.00000000000005 m

(13 zeros between decimal and 5)

A large number:

Distance to nearest star other than the sun is

24,500,000,000,000 miles

24.5 trillion miles

(245 + 11 zeroes)


Chapter 2

7

Scientific Notation

• Expresses a number as a number between 1

and 10 multiplied by a power of 10.

• Example: 3.89 x 105 = 389,000

• More compact way of writing large or small

numbers

• Easier to compare numbers

• Easier to determine precision


Chapter 2

8

Exponents or Powers of Ten

• Positive exponent = larger number

• Negative exponent = smaller number

• Write out powers

10 = 101 1/10 = 10-1

100 =102 1/100 = 10-2

1,000= 103 1/1,000 = 10-3

10,000 = 104 1/10,000 = 10-4

1 = 100

2.2 Scientific Notation: Writing

Large and Small Numbers


Chapter 2

9


Chapter 2

10

Scientific Notation

• Parts of a number in scientific notation

Figure from the center of page 12 showing the parts

Of a number written in scientific notation

exponent

1.35 x 10-8

exponential part (10)

decimal point of number between 1 and 10


Chapter 2

11

Writing Numbers in Scientific Notation

for Numbers Greater than 1. 1. Move the decimal to give a number between 1 and 10. Only

one digit to the left of the decimal.

2. Decimal from right to left, the number is now smaller than

the original number so you must multiply by a number

greater than 1. This means increase in the exponent.

3. Example 34,780 move decimal 4 places to left ()

4. 34,780 = 3.4780 x 10,000 = 3.4780 x 104

5. Decimal to left, increase exponent, the number of places you

moved the decimal point.

Our Examples (1)• A large number:

• Distance to nearest star other than the sun is

• 24,500,000,000,000 miles

• 24.5 trillion miles

• (245 + 11 zeroes)

• To get a number between 1 and 10 the decimal is moved --

(left) by 13 places. 2,45 is smaller than the actual

number so we need a positive exponent.

• 2.45. x 10+13 miles


Chapter 2

12


Chapter 2

13

Writing Numbers in Scientific Notation

for Numbers Less than 1.

1. Move the decimal to give a number between 1 and 10. Only

one digit will be to the left of the decimal.

2. Decimal from left to right, the number is now larger than the

original number so you must multiply by a number less than

1. This means decrease in the exponent.

3. Example 0.0067 move decimal 3 places to right ()

4. 0.0067 = 6.7 x 1/1,000 = 6.7 x 10-3

5. Decimal to right, decrease exponent, the number of places

you moved the decimal point.

Our Examples (2)

• A small number :

• Radius of a hydrogen nucleus is about

• 0.00000000000005 m

• (13 zeros between decimal and 5)

• To get a number between 1 and 10 the decimal is moved --

(right) by 14 places. 5. is bigger than the actual number

so we need a negative exponent.

• 5. x 10-14 m


Chapter 2

14


Chapter 2

15

Writing a Number in Scientific Notation,

Other Examples Continued

• The U.S. population in 2007 was estimated

to be 301,786,000 people. Express this

number in scientific notation.

• 301,786,000 people = 3.01786 x 108 people


Chapter 2

16

Example with a Number that

Already has a Power of Ten

• 458.98 x 105

• Move decimal 4.5898 This is left () 2 places, so exponent increase by 2 (5 +2 = 7)

• 4.5898 x 107


Chapter 2

17

Inputting Scientific Notation into a Calculator

• Learn how your calculator handles scientific

notation.

• Usually through a key marked EXP


Chapter 218

Reporting Measurements

• Measurements depend upon the equipment

being used.

• All the digits except the last are certain.

• The last is an estimate.

24 25 cm24.7? cm ? Is a guess 24.72cm

3 certain one

digits + guess

= four significant digits

Range is 24.71 to 24.73 cm

2.3 Significant Figures: Writing

Numbers to Reflect Precision


Chapter 2

19


Chapter 2

20

Significant Figures

Writing numbers to reflect precision.

Why do we need significant figures?

All measures have some degree of

uncertainty.

Precision Increased by Dividing?

Consider traveling 237 miles in 3.7 hours.

237 mi

3.7 h


Chapter 2

21

= 64.05405405405 mi/h

Significant Figures in a Single

Measurement

• Nonzero numbers are significant (23.89 cm)

• Zeros

Interior zeros – always significant (1.5067 in)

Trailing zeros after a decimal – always significant

(45.900 L)

Leading zeros – never significant (0.0000456 m)

Trailing zeros before a decimal – ambiguous,

avoid by using scientific notation.

\


Chapter 2

22

Exact Numbers have Unlimited

Significant figures

• Counted numbers (3 trials of an experiment)

• Defined numbers (1 in = 2.54 cm)

• Numbers as part of a formula (2pr = C)


Chapter 2

23


Chapter 2

24

Examples of —Determining the Number of

Significant Figures in a Number

2.4 Significant Figures in

Calculations


Chapter 2

25


Chapter 2

26

Multiplication and Division with

Significant Figures

• The overall answer is limited by the measurement

with the fewest number of significant figures.

5.02 × 89,665 × 0.10 = 45.0118 = 45

3 sig. figs. 5 sig. figs. 2 sig. figs. 2 sig. figs.

5.892 ÷ 6.10 = 0.96590 = 0.966

4 sig. figs. 3 sig. figs. 3 sig. figs.


Chapter 2

27

Rounding

• Find the number that is the last significant figure. Look at the number to the right.

1. For 0 to 4, keep the last significant figure the same .

34.8244 to four significant figures 34.82

1. For 5 to 9, add 1 the last significant figure the same.

45.967346 to four significant figures 45.97


Chapter 2

28

Examples of Multiplication and Division


Chapter 2

29

Addition and Subtraction with

Significant Figures

• The overall answer is limited by the measurement

with the fewest number of decimal places.

5.74 2 decimal places

+0.823 3 decimal places

+2.651 3 decimal places

9.214 2 decimal places rounds to 9.21


Chapter 2

30

Both Multiplication/Division and

Addition/Subtraction with

Significant Figures.

• Follow the standard order of operations.

Please Excuse My Dear Aunt Sally.

Keep track of number of significant figures at each step, but round at the end. Example next page.

- n


Chapter 2

31

Example 1.6—Perform the Following Calculations

to the Correct Number of Significant Figures

4555.30015.45120.010.1 a)

5820.100

1.105

355.0

b)For addition and subtraction, it is useful to draw a line

To help determine the correct decimal place.

Addition or Subtraction and

Multiplication and Division

Together


Chapter 2

32

Do the steps in parentheses and apply the rules to determine the

Correct number of significant figures or decimal p[laces in the

Intermediate answer.

Keep track of significant figures and only round at the last step.

Both Multiplication/Division and

Addition/Subtraction with

Significant Figures (cont.)

33

12.95 × (12.96 – 11.346)

2 right 3 right

decimal dcecimal

12.95 × (1.614)

2 right decimal (by subt. rules)

12.95 × (1.614) = 20.9013 20.9

four three three

sig. fig sig. fig. sig. fig

(by multip. Rules)


Chapter 2

34

Example 1.6—Perform the Following Calculations

to the Correct Number of Significant Figures,

Continued

5379904.5233.4526755.45299870.3562.4 c)

1.0142.002.855.084.14 d)


Chapter 2

35

2.5 The Basic Units of Measure


Chapter 2

36

Measurement = Number + Unit• Units - agreed upon standardized quantities

• Scientists use the SI system which is a form of the

metric system

• SI = Système International

• Base units are the units for fundamental quantities


Chapter 2

37

Quantity Unit Symbol

Length meter m

Mass kilogram kg

Time second s

Temperature kelvin K

Base Units in the SI System

38

Length• Measure of a 1-D distance

• SI unit = meter = 39.37 in, 3.37 in longer than a yard

• Commonly used centimeters (cm) = 1/100 of a meter. 1 inch = 2.54 cm (exactly, by definition)

A figure showing a meterstick slightly longer than a yardstick


Chapter 2

39

Mass• Measure of the amount of matter present

in an object.

• SI unit = kilogram (kg) = 2.2 lb

• A replica of the standard mass is shown

• Commonly measure in the lab mass in grams (g) or milligrams (mg).

• 1 kg = 1,000 g = 1,000,000 mg

• 1 oz = 23.35 g

• 1 lb = 453.6 g

• A nickel (5 cent piece) weighs about 5 g

Standard

Mass

Figure 2.6

Top loading

Balance

Page 23


Chapter 2

40

Time

• Measure of the duration of an event.

• SI units = second (s)

• 1 s is as a certain number of repeats

of a Cs atom in a so called atomic

clock.


Chapter 2

41

Temperature• Measure of the average energy of

motion of molecules in a sample.

• Is a result of molecular motions

• Faster molecular motions = higher

temperature

• Heat flows from warmer objects to

cooler objects through molecular

collisions until they reach the same

temperature.

• Unit is Kelvin (K). Water boils at 373.15

K, and freezes at 272.15 K

Three figures

Student in 22 °C

Room temp.

Ice at 0 °C

Death Valley at

40 ° C


Chapter 2

42

Nonbase Units in the

SI System

• Units in SI:

Prefix (a multiplier) + base unit

Example centimeter = centi + meter = centimeter0.01 x meter = 0.01 meter


Chapter 2

43

Common Prefixes in the

SI System (see the stuff to memorize on the

web page)

Prefix SymbolDecimal

EquivalentPower of 10

mega- M 1,000,000 Base x 106

kilo- k 1,000 Base x 103

deci- d 0.1 Base x 10-1

centi- c 0.01 Base x 10-2

milli- m 0.001 Base x 10-3

micro- m or mc 0.000 001 Base x 10-6

nano- n 0.000 000 001 Base x 10-9


Chapter 2

44

Prefixes Indicate What Power of 10 the

Base Unit has been Multipied by

• Kilometer = kilo (1000) + meter = 1000 x meter (km)

• Nanosecond = nano (10-9) + second = 10-9 x second (ns)

• Milligram = milli (10-3) + gram = 10-3 x gram (mg)

• If the multiplier is greater than 1 the new unit will be

larger than the base unit.

• If the multiplier is less than 1 the new unit will be

smaller than the base unit.


Chapter 2

45

Volume, an Example of

a Derived Unit

• Derived unit. A combination of the base units.

• This may be a multiplication, division, addition, subtraction or other mathimatical combination

• Examples Area = length2 Volume = length3

• SI system = Area = square meter (m2)

Volume = cubic meter (m3)

Graduated

Cylinder

A cube 10 cm

On a side which

Is the volume =

1 Liter

More Convenient

Volume Units

• Liter (L) is defined as a cubic decimeter (dm3).

• A dm = 10 cm so a Liter is a cube with 10 cm sides.

• 10 cm x 10 cm x 10 cm = 1000 cm3

• So a Liter also equals 1000 cm3

• Milliliter = 1/1000 L = 1 cm3

• REMEMBER: 1 mL = 1 cm3

• Solids usually use cm3 and liquids usually use mL


Chapter 2

46


Chapter 2

47

Table 2.3 (page 24) Common Units

and Their Equivalents

Length

1 kilometer (km) = 0.6214 mile (mi)

1 meter (m) = 39.37 inches (in.)

1 meter (m) = 1.094 yards (yd)

1 foot (ft) = 30.48 centimeters (cm)

1 inch (in.) = 2.54 centimeters (cm) exactly


Chapter 2

48

Table 2.3 (cont.) Common Units and

Their Equivalents, Continued

Volume

1 liter (L) = 1000 milliliters (mL)

1 liter (L) = 1000 cubic centimeters (cm3)

1 liter (L) = 1.057 quarts (qt)

1 U.S. gallon (gal) = 3.785 liters (L)

Mass

1 kilogram (km) = 2.205 pounds (lb)

1 pound (lb) = 453.59 grams (g)

1 ounce (oz) = 28.35 (g)

2.6 Converting One Unit to

Another


Chapter 2

49

•Metric metric (easy, a power of 10)

•Metric English (a little harder, a conversion factor)

•Arithmetic operations apply to numbers and units

•Keep track of units by dimensional analysis

Dimensional Analysis

• Using units to help solve problems is called

dimensional analysis

• Units may be operated on mathematically

like numbers


Chapter 2

50

SI to SI Conversions

• This involves a change in the unit that

differs by a power of 10. The number will

change by moving the decimal.

• A way to guide you through the problem is

a plan that your text calls solution map.

• A solution map lists the steps one mus go

through to solve the problem


Chapter 2

51

Example SI to SI

• How many mm are there in 34.9 m?

• Solution map:

m mm

• How do we make the change? Using a

conversion factor


Chapter 2

52

Conversion Factor

• A conversion factor is derived from an

equation. It is a fraction = 1

• Multiplying by a conversion factor does not

change a quantities size, but does change

the unit used to measure it.

• Example from the table 1 mm = 0.001 m

• 1mm 0.001 m

0.001 m 1mmTro's "Introductory Chemistry",

Chapter 2

53

Conversion Factors (cont)

• Notice the equation gives two conversion

factors.

• The one to choose is determined by which

unit is supposed top cancel out

• Back to our problem 34.9 m is how many

mm?


Chapter 2

54

Conversions Convert an Old Unit

to a New Unit

new unit

• old unit × = new unit

old unit

• In the solution map

• Solution map: m mm

• m = old unit and mm = new unit

55

Carrying out the Solution

1 mm

• 34.9 m × = 34,900 mm

0.001 m 3.49 x 104 mm

• Writing in scientific notation to give 3 sig. fig.


Chapter 2

56

Problem to Work in Class

• How many kg are there in 4,392 mm?

• Solution map:

mm m km

• Could be done in one step, but it is easy to

convert to the base unit.


Chapter 2

57

SI to Other Systems of Measure

• A piece of notebook paper is 8.50 in wide,

what is that length in cm?

• Solution map:

in cm

• Need a conversion factor from the

definition 1 in = 2.54 cm

• 1in or 2.54 cm

2.54 cm 1in


Chapter 2

58

Solution

• 8.50 in × 2.54 cm = 21.59 cm

1 in 21.6 cm

( 3 sig. fig. )


Chapter 2

59

2.7 Solving Multistep

Conversions


Chapter 2

60

Some Conversions Require Two

or More Conversion Factors

• Given that 2 Tablespoons (tbsp) = 1 oz, 8 oz

= 1 cup, 4 cup = 1 qt, and 1 L = 1.097 qt

• How many mL are there in 1.00 tbsp?

• Solution map:

tbsp oz cupqtLmL

• Each step represents a conversion. 5 steps.

(Solution next slide)


Chapter 2

61

• 1.00 tbsp × 1 oz = 0.500 oz

2 tbsp

• 0.500 oz × 1 cup = 0.0625 cup

8 oz

• 0.0625 cup × 1 qt = 0.015625 qt

4 in

• 0.015625 qt × 1 L = 0.0142339 L

1.097 in

• 0.0142339 L × 1000 mL = 14.2339 mL

1 L 14.2 mL


Chapter 2

62

2.8 Units Raised to a Power


Chapter 2

63

Units Raised to a Power

• An injection of a drug has a volume of 4.52

cm3. What is the volume in cubic inches

(in3)?

• Solution map: cm3 in3

• 4.52 cm3 × 1 in = 1.78 in3 (NO!!!!)

2.54 cm

• What does cm3 really mean


Chapter 2

64


• 4.52 cm3 = 4.53 cm × cm × cm

• For squared or cubed units, one must also

square or cube the entire conversion factor

• 4.52 cm3 × 1 in × 1 in × 1 in =

2.54 cm 2.54 cm 2.54 cm

or 3

• 4.52 cm3 × 1 in = 4.52 cm3× 13 in3 =

2.54cm (2.54) 3cm3


Chapter 2

65


• 4.52 cm3 × 1 in3 = 0.276 in3

16.387 cm3


Chapter 2

66


Chapter 2

67

2.9 Density


Chapter 2

68

Mass and Volume• Two main characteristics of matter.

• Mass or volume alone cannot be used to

identify what type of matter something is.

If you are given a large glass containing 100 g

of a clear, colorless liquid and a small glass

containing 25 g of a clear, colorless liquid, are

both liquids the same stuff?

• Even though mass and volume are

individual properties, for a given type of

matter they are related to each other!


Chapter 2

69

Mass vs. Volume of BrassMass

grams

Volume

cm3

20 2.4

32 3.8

40 4.8

50 6.0

100 11.9

150 17.9

This slide was a graph of the

mass as y and the volume as x

resulting in a straight line

this indicates that the ratio

of mass to volume is a constant =

the density of a substance

For these brass samples d = 8.3

g/mL


Chapter 2

70


Chapter 2

71

Density• Ratio of mass:volume.

• Its value depends on the kind of material, not the amount.

• Solids = g/cm3

1 cm3 = 1 mL

• Liquids = g/mL

• Gases = g/L

• Volume of a solid can be determined by water displacement—Archimedes Principle. Explain

• Density : solids > liquids > gases

Except ice is less dense than liquid water!

Volume

MassDensity

For a Table of Sample Densities

see Table 2.4 on page 33


Chapter 2

72

Volume by Water Displacement

• Graduated cylinder – water is added = Vinitial

• A solid object is added to the cylinder.

• Volume of water rises = Vfinal

• Vobject = Vfinal - Vinitial


Chapter 2

73


Chapter 2

74

Using Density in Calculations

Three Possible Formulas

Volume

MassDensity

Density

Mass Volume

Volume Density Mass

Solution Maps:

m, V D

m, D V

V, D m

Density Problems


Chapter 2

75

•Think of density as a conversion factor between

mass and volume of a given substance.

•Problems from text for class 2.96, 2.100,

Problem 2.96


Chapter 2

76

An aluminum engine block has a volume of 4.77 L and a

mass of 12.88 kg. What is the density of the aluminum

in grams per cubic centimeter?

Solution map

Volume, L Volume, mL=cm3

mass

Density =

Mass, kg mass, g volume

Problem 2.96 (cont.)


Chapter 2

77

Volume to cm3:

4.77 L × 1000 mL = 4,770 mL = 4,770 cm3

1 L

Mass to g:

12.88 kg × 1000 g = 12,880 g

1 kg

Density using the formula:

12,880 g = 2.7002096 g/cm3 2.70 g/cm3

4,770 cm3 3 sig. fig.

Problem 2.100


Chapter 2

78

Acetone (fingernail polish remover) has a density of 0.7857 g/cm3.

(a) What is the mass in grams of 28.56 mL of acetone?

(b) What is the volume in milliliters of 6.54 g of acetone?

There are two basic ways to work this problem. 1)Plug the

variables into the formula for density and solve for the unknown.

Or

2) Treat the density as a conversion factor between mass and

volume of a substance.

We will use # 2

Solution maps (a) ml g (b) g mL

Problem 2.100 (cont.)


Chapter 2

79

A density of 0.7857 g/cm3 means that 1 cm3 or mL of the substance

Has a mass of 0.7857 g. (i.e. 1 mL = 0.7857 g of acetone)

From this equation we can derive conversion factors.

(a) The conversion is from mL to g so we want mL to cancel

28.56 mL × 0.7857g = 22.44 g (4 sig. fig.)

1 mL

(a) The conversion is from g to mL so we want g to cancel

6.54 g × 1 mL = 8.32 g (3 sig. fig.)

0.7857g

chapter 2 graph of global temperature rise in 20...

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