Ch. 2 “Scientific Measurement & Problem Solving”

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Types of Observations and Measurements

• We make QUALITATIVE observations of reactions — Describes using wordsEx. Odor, color, texture, and physical state.

• We also make QUANTITATIVE observations, using numbers- measurements

• Ex. 25.3 mL, 4.239 g

Standards of Measurement

When we measure, we use a measuring tool to compare some dimension of an object to a standard. For example, at one time the

standard for length was the king’s foot. What are some

problems with this standard?

Accuracy – how close a measurementcomes to the true value of what is measured

Precision – is concerned with the reproducibility of the measurement

Our measurements must be bothAccurate & precise!

Three targets with three arrows each to shoot.

Can you hit the bull's-eye?

Both accurate and precise

Precise but not accurate

Neither accurate nor precise

How do they compare?

Stating a Measurement

In every measurement there is a

¨Number followed by a

¨ Unit from a measuring device

The number should also be as precise as the

measurement!

SI Measurement• Le Systeme International d’Unites : SI

Metrics• System of measurement agreed on all over the

world in 1960• Contains 7 base units• units are defined in terms of standards of

measurement that are objects or natural occurrence that are of constant value or are easily reproducible

• We still use some non-SI units!

• SI units — based on the metric system

• The only countries that have not officially adopted SI are Liberia (in western Africa) and Myanmar (a.k.a. Burma, in SE Asia), but now these are reportedly using metric regularly

• Among countries with non-metric usage, the U.S. is the only country significantly holding out. The U.S. officially adopted SI in 1966.

Information from U.S. Metric Association

Le Système international d'unités

The 7 Base Units of SI

S.I. (the ones you’re responsible for knowing!)

Selected S.I. base (or standard) units

Mass kg

Length m

Time sec

Temperature KAmount of Substance mole mol

Derived Unitsmade by combining Base Units!

• Volume length cubed m3 cm3

• Density mass/volume g/mL kg/L g/ cm3

• Speed length /time mi/hr m/s km/hr

• Area length squared m2 cm2

We use prefixes to expand on the base units!

S.I. prefixes you must memorize!

Prefix Abbreviation Value

kilo k 103

deci d 10-1

centi c 10-2

milli m 10-3

micro m10-6

nano n 10-9

Metric System• These prefixes are based on powers of

10. • From each prefix every “step” is either:

• 10 times larger or

• 10 times smaller• For example

• Centimeters are 10 times larger than millimeters

• 1 centimeter = 10 millimeters kilo hecto deca

Base Unitsmetergramliter

deci centi milli

Metric System

• An easy way to move within the metric system is by moving the decimal point one place for each “step” desiredExample: change meters to centimeters1.00 meter = 10.0 decimeters = 100.

centimeters

kilo hecto decameterlitergram

deci centi milli

mass – measure of the quantity of matter

SI unit of mass is the kilogram (kg)

1 kg = 1000 g = 1 x 103 g

Not to be confused with -

weight – mass + gravity

force that gravity exerts on an object

Mass does not vary from place to place!

A 1 kg bar will weigh

1 kg on earth

0.1 kg on moon

A 1 kg bar has a mass of 1 kgon earth and on the moon

Volume – Amount of space occupied by matter

SI derived unit for volume is cubic meter (m3)

1 L = 1000 mL = 1000 cm3 = 1 dm3

1 mL = 1 cm3

1 dm3 = 1 L

We often use the Liter (L) when working with liquid volumes!

m3 = m x m x m

Temperature Scales• Fahrenheit• Celsius• Kelvin

Anders Celsius1701-1744

Lord Kelvin(William Thomson)1824-1907

TEMPERATURE SCALES

In Chemistry, the terms heat and temperature are often used to describe specific properties of a sample.

HEAT is the most common form of energy in nature and is directly related to the motion of particles of matter.

The faster the motion of particles in a sample the greater its heat content.

A forest fire and a lit match may both be at the same temperature, but there is a large difference in the amount of heat each possess.

TEMPERATURE is associated only with the intensity of heat and is not affected by the size of the sample.

Heat always spontaneously flows from a hotter system (higher temp.) to a colder system (lower temp.).

Temperature Scales

Notice that 1 Kelvin = 1 degree Celsius

Boiling point of water

Freezing point of water

Celsius

100 ˚C

100˚C

Kelvin

Fahrenheit

32 ˚F

212 ˚F

180˚F

Temperature Scientists do not know of any limit on how high a temperature may be. The temperature at the center of the sun is about 15,000,000 °C. However, nothing can have a temperature lower than –273°C. This temperature is called absolute zero. It forms the basis of the Kelvin scale. Because the Kelvin scale begins at absolute zero, 0 K equals –273°C, and 273 K equals 0 °C.

Calculations Using Temperature

• Many chemistry equations require temp’s to be

in Kelvin

• K = ˚C + 273• Body temp = 37 ˚C + 273 = 310 K

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

˚C = K - 273

DENSITY –

Density mass (g)volume (cm3)

Mercury

13.6 g/cm3 21.5 g/cm3

Aluminum

2.7 g/cm3

Platinummercuryplatinum

an important and useful physical property (Derived Unit)ratio of mass per unit of volume

DENSITY• Density is an

INTENSIVE property of matter.• Since it is a ratio

of mass to volume -does NOT depend on quantity of matter.

Styrofoam Brick

The density of 1 g of gold =The density of 5 kg of gold!

Problem A piece of copper has a mass of 57.54 g. It is 9.36 cm long, 7.23 cm wide, and 0.095 cm thick. Calculate density (g/cm3).

Get out those calculators!

Copper orePure copper metal

SOLUTION1. Make sure dimensions are in common

units. (all are in cm’s)2. Calculate volume in cubic centimeters. L x W x H = volume

3. Calculate the density.57.54 g6.4 cm3 = 9.0 g / cm3

(9.36 cm)(7.23 cm)(0.095 cm) = 6.4 cm3

Learning Check

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

1) 2) 3)

Solution

(K) Karo syrup (1.4 g/mL), (V) vegetable oil (0.91 g/mL,) (W) water (1.0 g/mL)

Denser materials ‘sink’ inless dense materials!

Most solids sink in their liquid form.Can you think of an exception to this?!

WATER!!

Finding Volume of an IrregularSolid byWater Displacement

A solid displaces a matching volume of water when the solid is placed in water.

25 mL33 mL

Volume of solidis 8 mL

Calculator Time! What is the density (g/cm3) of 48 g of a metal if the metal raises the level of water in a graduated cylinder from 25 mL to 33 mL? a) 0.2 g/ cm3 b) 6.0 g/cm3 c) 252 g/cm3

33 mL 25 mL

Percent Error• Percent Error:

• Measures the inaccuracy of experimental data

• Can have + or – value• Accepted value : correct value based on reliable

references• Experimental value: value you measured in the lab

%100accepted

alexperimentaccepted

Scientific NotationThe number of atoms in 12 g of carbon:

602,200,000,000,000,000,000,000

6.022 x 1023

The mass of a single carbon atom in grams:

0.0000000000000000000000199

1.99 x 10-23

N x 10n

N is a number between 1 and 10(1 non-zero digit to left of dec. pt.)

n is a positive or negative integer

To change standard form to scientific notation…

• Place the decimal point so that there is one non-zero digit to the left of the decimal point.

• Count the number of decimal places the decimal point has “moved” from the original number. This will be the exponent on the 10.

• If the original number was less than 1, then the exponent is negative. If the original number was greater than 1, then the exponent is positive.

Examples• Given: 289,800,000• Use: 2.898 (moved 8 places)• Answer: 2.898 x 108

• Given: 0.000567• Use: 5.67 (moved 4 places)• Answer: 5.67 x 10-4

To change scientific notation to standard

form…• Simply move the decimal point to

the right for positive exponent 10. • Move the decimal point to the left

for negative exponent 10.

(Use zeros to fill in places.)

Example• Given: 5.093 x 106

• Answer: 5,093,000 (moved 6 places to the right)

• Given: 1.976 x 10-4

• Answer: 0.0001976 (moved 4 places to the left)

Learning Check• Express these numbers in

Scientific Notation:

1) 4057892) 0.0038723) 30000000004) 25) 0.478260

Scientific Notation568.762

n > 0568.762 = 5.68762 x 102

move decimal left0.00000772

n < 00.00000772 = 7.72 x 10-6

move decimal right

Addition or Subtraction

1. Write each quantity with the same exponent n

2. Combine N1 and N2 3. The exponent, n, remains

the same

4.31 x 104 + 3.9 x 103 =

4.31 x 104 + 0.39 x 104 =

4.70 x 104

Scientific NotationCalculations

Multiplication1. Multiply N1 and N2

2. Add exponents n1 and n2

3. Put in proper format, if necessary

(4.0 x 10-5) x (7.0 x 103) =(4.0 x 7.0) x (10-5+3) =

28 x 10-2 =2.8 x 10-1

Division1. Divide N1 and N2

2. Subtract exponents n1 and n2

3. Put in proper format, if necessary

8.5 x 104 ÷ 5.0 x 109 =(8.5 ÷ 5.0) x 104-9 =

1.7 x 10-5

Significant FiguresThe numbers reported in a

measurement are limited by the measuring tool

Significant Figures in a measurement include all certain digits plus one estimated digit

•7.50 cm

•19.5 mL

Significant Figures• All certain digits plus one

estimated digit (used when recording measurements)

Known + Estimated DigitsIn 2.85 cm…

• Known digits 2 and 8 are 100% certain(there are lines on the ruler for these!)

• The third digit, 5, is estimated (uncertain)

• In the reported length, all three digits (2.76 cm) are significant including the estimated one

Figure 5.5: Measuring a pin.There are not really lines on the scale here – just estimates!

Reading a Meterstick. l2. . . . I . . . . I3 . . . .I . . . . I4. . cm

First digit (known) = 2 2.?? cmSecond digit (known)= 0.8 2.8? cmThird digit (estimated) between 0.03- 0.05Length reported = 2.83 cm

or 2.84 cm

or 2.85 cm

Learning Check

. l8. . . . I . . . . I9. . . . I . . . . I10. . cm

What is the length of the line?

1) 9.3 cm

2) 9.40 cm

3) 9.30 cm

How does your answer compare with your neighbor’s answer?

Rules for Counting Significant Figures

RULE 1. All non-zero digits in a measured number are significant.

Number of Significant Figures?

38.15 cm5.6 mL65.6 kg122.55 m

Sandwiched ZerosRULE 2. Zeros between nonzero numbers are

significant. Number of Significant Figures?

50.8 mm

2001 min

.702 mg

400005 m

Leading Zeros (in front)

RULE 3. Leading zeros in decimal numbers are NOT significant. Number of Significant Figures?

0.008 mm

0.0156 g

0.0042 cm

0.0002602 mL

Trailing Zeros (at end)RULE 4. Trailing zeros in numbers

without decimals are NOT significant. They are only serving as place holders.

Number of Significant

Figures?

25,000 m

48,600 mg

25,005,000 kg

Trailing Zeros, cont.

RULE 5. Trailing zeros in numbers with decimals ARE significant.

Number of Significant

Figures?

35,000.0 m

700. s

48.600 L

25,005.000 g

How many significant figures are in each of the following measurements?

24 mL 2 significant figures

3001 g 4 significant figures

0.0320 m3 3 significant figures

6.4 x 104 molecules 2 significant figures

560 kg 2 significant figures

Significant Numbers in Calculations

A calculated answer cannot be more precise than the measuring tool.

A calculated answer must match the least precise measurement.

Significant figures are needed for final answers from 1) adding or subtracting

2) multiplying or dividing

Rounding• Need to use rounding to write a calculation

involving measurements correctly.• Calculator gives you lots of insignificant

numbers so you must round to the correct decimal place

• When rounding, look at the digit after the one you can keep• Greater than or equal to 5, round

up• Less than 5, keep the same

ExamplesRound each of the following measurements

so they have 3 sig figs: 761.50 14.334 10.44 10789 8024.50 203.514

76214.3

10.4108008020204

Series of operations: keep all non-significant digits during the intermediate calculations, and round to the correct number of SF only when reporting an answer.

Ex: (4.5 + 3.50001) x 2.00 =

(8.00001) x 2.00 = 16.0002 → 16

Adding and SubtractingThe answer has the same number of decimal places as the measurement with the fewest decimal places.

25.2 one decimal place (to right of decimal pt.)

+ 1.34 two decimal places (to right of decimal pt.)

26.54Answer: 26.5 (one decimal place)

Using Sig Figs in Calculations• Adding/Subtracting:

• end with the least number of decimal places

Using Sig Figs in Calculations• Adding/Subtracting:

• end with the least number of decimal places

Significant Figures

Addition or Subtraction (con’t,)

89.3321.1+

90.432 round off to 90.4one significant figure after decimal point

3.70-2.91330.7867

two significant figures after decimal point

round off to 0.79

Learning Check

In each calculation, round the answer to the correct number of significant figures.A. 235.05 + 19.6 + 2.1 =

1) 256.75 2) 256.8 3) 257

B. 58.925 - 18.2 =1) 40.725 2) 40.73 3) 40.7

Multiplying and Dividing

Round (or add zeros) to the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures.(Sometimes you’ll need to put the answer into Sci. Notation to get correct # of sig figs!)

Using Sig Figs in Calculations

• Multiplying/Dividing:• end with the least number of sig figs

(Counting sig figs from left)

Using Sig Figs in Calculations• Multiplying/Dividing:

• end with the least number of sig figs

Significant Figures

Multiplication or DivisionThe number of significant figures in the result is set by the original number that has the smallest number of significant figures

4.51 x 3.6666 = 16.536366 = 16.5

3 sig figs round to3 sig figs

6.8 ÷ 112.04 = 0.0606926

2 sig figs round to2 sig figs

= 0.061

Learning Check A. 2.19 X 4.2 =

1) 9 2) 9.2 3) 9.198

B. 4.311 ÷ 0.07 = 1) 61.58 2) 62 3) 60

C. 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.3 2) 11 3) 0.041

For exact numbers (e.g. 4 beakers) and those used in conversion factors (e.g. 1 inch = 2.54 cm), there is no uncertainty in their measurement. Therefore, IGNORE exact numbers when finalizing your answer with the correct number of significant figures.

(Numbers from definitions or numbers of objects are consideredto have an infinite number of significant figures)

The average of three measured lengths, 6.64, 6.68 and 6.70 is:

6.64 + 6.68 + 6.70

3= 6.67333 = 6.67

Because 3 is an exact number

Chemistry In ActionOn 9/23/99, $125,000,000 Mars Climate Orbiter entered Mar’s atmosphere 100 km lower than planned and was destroyed by heat.

1 lb = 1 N

1 lb = 4.45 N

“This is going to be the cautionary tale that will be embedded into introduction to the metric system in elementary school, high school, and college science courses till the end of time.”

Conversion Factors• Ratio that comes from a statement of

equality between 2 different units• every conversion factor is equal to 1

dollarquarters 14

Example:

statement of equality

conversion factor 141

quartersdollar 4 quarters

1 dollar=

Conversion Factors (con’t.)

Fractions in which the numerator and denominator are EQUAL quantities expressed in 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 each of the following pairs of units:1. Liters and mL

2. Hours and minutes

3. Meters and kilometers

1 L. and 1000 mL 1000 mL 1 L

1 hr. and 60 mins. 60 mins. 1 hr

1000 m and 1 km__ 1 km 1000 m .

Conversion Factors

• can be multiplied by other numbers without changing the value of the number (since you are just multiplying by 1)

quartersdollar

quartersdollars 121

Dimensional Analysis Method of Solving Problems

1. Start with the given

2. Determine what unit label is needed on the answer

3. Add conversion factor(s) & cancel units until you are left with the desired unit label!

1 L = 1000 mL

How many mL are in 1.63 L?

1L1000 mL

1.63 L x = 1630 mL

1L1000 mL

1.63 L x = 0.001630 L2

Sample Problem• You have $7.25 in your pocket in

quarters. How many quarters do you have?

7.25 dollars 4 quarters 1 dollar

X = 29 quarters

Learning Check

How many seconds are in 1.4 days?

Unit plan: days hr min seconds

1.4 days x

Solution

Unit plan: days hr min seconds

1.4 day x 24 hr x 60 min x 60 sec 1 day 1 hr 1 min

= 1.2 x 105 sec

Example Convert 5.2 cm to mm

• Known: 100 cm = 1 m1000 mm = 1 m

• MUST use m as an intermediate

cmmcm 52

10012.5

Example

Convert 0.020 kg to mg

• Known: 1 kg = 1000 g1000 mg = 1 g

• Must use g as an intermediate

kggkg 000,20

11000020.0

Advanced Conversions

• A more difficult type of conversion deals w/units that are fractions themselves

• Be sure convert one unit at a time; don’t try to do both at once

• Setup your work the exact same way

When unit labels are fractions (or ratios), unzip them!

11.3 g/mL can be written as 11.3 g 1 mL

OR 1 mL 11.3 g

Ex. Convert 11.3 g/mL to g/L

11.3 g 1 mL = 1.13 x 104 g/L

1000 mL

PROBLEM: Mercury (Hg) has a density of 13.6 g/cm3. What is the mass of 95 cm3 of Hg in grams?

Solve the problem using DIMENSIONAL ANALYSIS.

95 cm3 • 13.6 gcm3 = 1.3 x 103 g

The speed of sound in air is about 343 m/s. What is this speed in miles per hour?

What is the given? What do you have to convert?

1 mi = 1609 m 1 min = 60 s 1 hour = 60 min

343 ms x 1 mi

1609 m 60 s

1 minx 60 min

1 hourx = 767 mi

meters to miles seconds to hours

Advanced Conversions

• Another difficult type of conversion deals with squared or cubed units

• Be sure to square or cube the conversion factor you are using to cancel all the units

• If you tend to forget to square or cube the number in the conversion factor, try rewriting the conversion factor instead of just using the exponent

Square and Cubic units• Use the conversion factors you already

know, but when you square or cube the unit, don’t forget to cube the number also!

• Best way: Square or cube the ENTIRE conversion factor

• Example: Convert 4.3 cm3 to mm3

4.3 cm3 10 mm 3 1 cm ( ) =

4.3 cm3 103 mm3

13 cm3

= 4300 mm3

Example

• Convert: 2000 cm3 to m3

• No intermediate needed

Known:100 cm = 1 m cm3 = cm x cm x cmm3 = m x m x m

3002.0100

12000 mcm

mcmcmcm

3 002.0100

12000 mcm

Ch. 2 “Scientific Measurement & Problem Solving”

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