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### Ch. 2 “Scientific Measurement & Problem Solving”

SAVE PAPER AND INK!!!

If you print out the notes

on PowerPoint,

print "Handouts" instead

of "Slides“ in the print setup.

Also, turn off the backgrounds

(Tools>Options>Print>UNcheck

"Background Printing")!

Types of Observations and Measurements

- We makeQUALITATIVEobservations of reactions — Describes using words

Ex. Odor, color, texture, and physical state.

- We also makeQUANTITATIVE 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?

Accurate & precise!

Accuracy –

how close a measurement

comes to the true value of what is

measured

Precision –

is concerned with the

reproducibility of the measurement

Three targets with three arrows each to shoot.

How do they compare?

Both accurate and precise

Precise but not accurate

Neither accurate nor precise

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!

Le Système international d'unités

- 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

Derived Unitsmade by combining Base Units!

- Volume length cubed m3 cm3
- Density mass/volume g/mL kg/L g/ cm3

g/L

- Speed length /time mi/hr m/s km/hr
- Area length squared m2 cm2

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

Metric System

- An easy way to move within the metric system is by moving the decimal point one place for each “step” desired

Example: change meters to centimeters

1.00 meter = 10.0 decimeters = 100. centimeters

1 kg on earth

0.1 kg on moon

- 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 has a

mass of 1 kg

on earth and

on the moon

Volume – Amount of space occupied by matter

SI derived unit for volume is cubic meter (m3)

m3 = m x m x m

We often use the Liter (L) when

working with liquid volumes!

1 L = 1000 mL= 1000 cm3 = 1 dm3

1 mL = 1 cm3

1 dm3 = 1 L

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.

TEMPERATURE is associated only with

the intensity of heat and is not affected

by the size of the sample.

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.

Heat always spontaneously flows from a

hotter system (higher temp.) to a colder

system (lower temp.).

100 ˚C

373 K

100 K

180˚F

100˚C

32 ˚F

0 ˚C

273 K

Temperature ScalesFahrenheit

Celsius

Kelvin

Boiling point of water

Freezing point of water

Notice that 1 Kelvin = 1 degree Celsius

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

Brick

Styrofoam

- Density is anINTENSIVEproperty of matter.
- Since it is a ratio of mass to volume -does NOT depend on quantity of matter.

The density of 1 g of gold =

The density of 5 kg of gold!

ProblemA 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).

Copper ore

Pure copper metal

SOLUTION

1. 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.

(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)

V

W

K

V

W

K

W

V

K

Solution

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

1)

WATER!!

Denser materials ‘sink’ in

less dense materials!

V

W

Most solids sink in their liquid form.

Can you think of an exception to this?!

K

Finding Volume of an IrregularSolid byWater Displacement

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

Volume of solid

is 8 mL

33 mL

25 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

The number of atoms in 12 g of carbon:

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

The mass of a single carbon atom in grams:

0.0000000000000000000000199

Scientific Notation

6.022 x 1023

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:
- 405789
- 0.003872
- 3000000000
- 2
- 0.478260

move decimal right

Scientific Notation

568.762

0.00000772

n > 0

n < 0

568.762 = 5.68762 x 102

0.00000772 = 7.72 x 10-6

Addition or Subtraction

- Write each quantity with the same exponent n
- Combine N1 and N2
- 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

Calculations

Multiplication

(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

- Multiply N1 and N2
- Add exponentsn1and n2
- Put in proper format, if necessary

Division

8.5 x 104÷ 5.0 x 109 =

(8.5 ÷ 5.0) x 104-9 =

1.7 x 10-5

- Divide N1 and N2
- Subtract exponentsn1and n2
- Put in proper format, if necessary

Significant Figures

- The numbers reported in a measurement are limited by the measuring tool
- Significant Figures in a measurement include all certain digits plus one estimated digit

Significant Figures

- All certain digits plus one estimated digit (used when recording measurements)

Known + Estimated Digits

In 2.85 cm…

- Known digits2and8are 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

Reading a Meterstick

. l2. . . . I . . . . I3 . . . .I . . . . I4. . cm

First digit (known) = 2 2.?? cm

Second digit (known) = 0.8 2.8? cm

Third digit (estimated) between 0.03- 0.05

Length 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 forCounting Significant Figures

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

Number of Significant Figures?

38.15 cm

5.6 mL

65.6 kg

122.55 m

4

2

3

5

Sandwiched Zeros

RULE 2. Zeros between nonzero numbers are significant.

Number of Significant Figures?

50.8 mm

2001 min

.702 mg

400005 m

3

4

3

6

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

1

3

2

4

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

200 L

48,600 mg

25,005,000 kg

2

1

3

5

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

6

3

5

8

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

24 mL

2 significant figures

4 significant figures

3001 g

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

Examples

Round each of the following measurements so they have 3 sig figs:

- 761.50
- 14.334
- 10.44
- 10789
- 8024.50
- 203.514

762

14.3

10.4

10800

8020

204

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 Subtracting

The answer has the same number of decimal places as the measurement with the fewest decimal places.

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

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

26.54

Answer: 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

+

1.1

one significant figure after decimal point

two significant figures after decimal point

90.432

round off to 90.4

round off to 0.79

3.70

-2.9133

0.7867

Significant Figures

Addition or Subtraction (con’t,)

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

round to

3 sig figs

2 sig figs

round to

2 sig figs

Significant Figures

Multiplication or Division

The 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

6.8 ÷ 112.04 = 0.0606926

= 0.061

Learning Check

A. 2.19 X 4.2 =

1) 9 2) 9.2 3) 9.198

B. 4.311 ÷ 0.07 =

1)61.582) 62 3) 60

C. 2.54 X 0.0028 =

0.0105 X 0.060

1) 11.3 2) 11 3) 0.041

= 6.67333 = 6.67

= 7

3

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 considered

to have an infinite number of significant figures)

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

Because 3 is an exact number

On 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

Example:

statement of equality

4 quarters

1 dollar

=

conversion factor

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)

1L

L2

1.63 L x

= 1630 mL

mL

1L

1.63 L x

= 0.001630

1000 mL

Dimensional Analysis

Method of Solving Problems

- Start with the given
- Determine what unit label is needed on the answer
- Add conversion factor(s) & cancel units until you are left with the desired unit label!

How many mL are in 1.63 L?

1 L = 1000 mL

1.9

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

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

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

1000 mL

1 L

= 1.13 x 104 g/L

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 usingDIMENSIONAL

ANALYSIS.

m

x

x

x

343

60 s

1 mi

s

1 hour

= 767

1 min

1609 m

mi

hour

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?

meters to miles

seconds to hours

1 mi = 1609 m

1 min = 60 s

1 hour = 60 min

1.9

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

Known:

100 cm = 1 m

cm3 = cm x cm x cm

m3 = m x m x m

- Convert: 2000 cm3 to m3
- No intermediate needed

OR

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