Chapter 3Chapter 3

Scientific MeasurementScientific MeasurementScientific MeasurementScientific Measurement

A. Qualitative Measurements

B. Quantitative Measurements

- give results in descriptive, non-numeric form

- give results in definite form, numbers and units

Examples: Mr. Smith is tall. The room is hot.

Examples: Mr. Smith is 6 feet tall. The room is 70 degrees Fahrenheit.

II. Scientific NotationDef – Scientific notation is simply writing a number as a product of two numbers.

Examples:

12,300,000 = _____________1.23 x 107

0.0000546 = _____________5.46 x 10-5

.

2 parts of a number:

1. Coefficient – a number between 1 and 10

2. Exponent – 10 raised to a power

Note:Positive powers of 10 = large numbers (multiply)

Negative powers of 10 = small numbers (divide)

3.6 x 104 = 3.6 x 10 x 10 x 10 x 10 = 36,000

8.1 x 10-4 = 8.1 ÷ 10 ÷ 10 ÷ 10 ÷ 10 = 0.000 81

1.23 x 107

coefficient exponent

Calculations with Scientific Notation

1. Multiplication and Division

a. Multiplying – MULTIPLY the coefficients and ADD the exponents.

b. Dividing – DIVIDE the coefficients and SUBTRACT the exponents.

Example 1:

(9 x 104) x (2 x 103)

18 x 107 1.8 x 107 x 108

Example 2:

(8 x 104) ÷ (2 x 107)

4 x 10-3

Calculations with Scientific Notation

2. Addition and Subtraction

Rule: First the powers of 10 must be the same!

Then just add or subtract the coefficients and keep the power of 10 the same

Example 1:

(9 x 104) + (2 x 103)

(9 x 104)(0.2 x 104)(0.2 x 104)

9.2 x 104

90 000 2 000

92 000

Calculations with Scientific Notation

2. Addition and Subtraction

Rule: First the powers of 10 must be the same!

Then just add or subtract the coefficients and keep the power of 10 the same

Example 2:

(4 x 108) - (9 x 106)

(4 x 108)(0.09 x 108)(0.09 x 108)

3.91 x 108

Practice Problems

1. 4.5 x 108 2. 3.6 x 10-4

x 2.0 x 105 ÷ 1.2 x 105

3. 2.2 x 105 4. 6.07 x 1012

+ 5.5 x 106 – 4.3 x 1011

5. 6.02 x 1023 6. 7.5 x 10-5

x 2.00 x 10-10 + 9.5 x 10-8

II. Accuracy, Precision, and Error

A. Accuracy – The measure of how close a measurement is to the accepted value.

B. Precision – How close a series of measurements are to each other.

Note: Precision also describes how “exact” a measuring tool allows you to be.

Example: Which is more precise?

19 grams or 18.895 grams

The Dartboard Model

Accuracy =

Precision =

Accuracy =

Precision =

Accuracy =

Precision =

Low

Low

Low

High

High

High

C. ErrorDef – The difference between the experimental

value and the accepted value.

Experimental Value – The value measured in the lab

Accepted Value – The correct value, based on reliable references or calculations

Equation:

[Experimental] – [Accepted] = Error

Sample Problem:

James weighs a metal cylinder and finds it has a mass of 48.34 grams. If the actual mass of the cylinder is exactly 50.00 grams, what is his error?

48.34 – 50.00 = -1.66 grams

D. Percent Error

Def – The absolute value of the error divided

by the accepted value, multiplied by 100- Compares the error to the accepted value,

showing how bad the error really is.

Equation:

% Error = | error | x 100 accepted value

Steve and Martin each order bricks from a lumber yard. Steve orders 500 bricks to build a wall, but the lumber yard only delivers 499. Martin orders 5 bricks, but only recieves 4. Calculate the error and percent error in each order.

Steve Martin

499 – 500 = -1 4 – 5 = -1

The error in both orders is the same! In which order is it more significant?

| -1 | 500

x 100 = | -1 | 5

x 100 = 0.2%

499 – 500 = -1

20%

4 – 5 = -1

II. Significant Figures in Measurements

Significant Figures – All the digits that are known in a measurement, plus one estimated digit

How tall is the rectangle on the left according to the ruler shown?

cm1

2

Between 1.6 & 1.7

1.64

Known Digits

Estimated Digit

4

5

What is the volume of water in the graduated cylinder to the left?

mL

Between 4.1 & 4.2

4.13

Known Digits

Estimated Digit

*Note: With all measuring tools, you are expected to estimate one digit beyond what you can actually see marked = significant!

Reporting Measurements in proper Sig Figs

- Measurements must be reported with the correct number of significant figures!

- Significant figures describe the precision with which your measuring tool is calibrated.

0

1

0

1

0

1

0.6 cm 0.62 cm 0.628 cm

Rules for Sig Figs (Rules for Zero’s)

1. All nonzero digits ARE significant.

2. Zero’s appearing between nonzero digits ARE significant.

3. Zeros to the left in front of nonzero digits ARE NOT significant. (They are just placeholders!)

546 = ________ sig figs 13.456 = _________ sig figs

104.3 = ________ sig figs 20.05 = _________ sig figs

0.00224 = ________ sig figs 0.34 = _________ sig figs

35

44

32

Rules for Sig Figs (Rules for Zero’s)

4. Zeros at the end of a number and to the right of a decimal ARE ALWAYS significant.

5. Zeros at the end of a number and LEFT of a decimal are SOMETIMES significant. Unless it is stated that they are specifically measured, we WILL NOT count them as significant.

55.00 = ________ sig figs 320.0 = _________ sig figs

120 = ________ sig figs 33,000 = _________ sig figs

22

44

V. Significant Figures in Calculations

General Rule: Your answer cannot be more precise than the

measurements used to calculate that answer.* You must round your answers to the proper number

of significant figures.

Example: Tennis Ball Drop Times – Averaged

Trial 1 - 0.89 seconds Trial 2 - 0.91 seconds Trial 3 - 1.04 seconds Trial 4 - 0.84 seconds Trial 5 - 0.73 seconds

Average Time - 0.882 seconds 0.88 seconds

1. Rounding – Calculations must be rounded to make them consistent with the measurements from which they were calculated.

How to Round Numbers:

a. First, decide how many significant figures your answer should have.

* Follow the rules for addition/subtraction or multiplication/division below!

b. Once significant figures have been decided, count that many places, starting from the left.

c. Identify the digit immediately to the right of the last significant figure:

i. If it is less than 5 it is dropped.

ii. If it is 5 or greater, you must round up the last sig fig.

Examples: Round the following to 3 sig figs.

a. 13.542 = _________

b. 0.25252 = _________

c. 0.0002398 = _________

d. 12,346 = _________

13.5

0.253

0.000240

12,300

2.40 x 10-4

1.23 x 104

2. Addition and Subtraction Rule:The answer should be rounded to have the same number

of places after the decimal as the measurement with the LEAST number of decimal places.

24.312 0.2332 19.5 0.4+ 4.66 + 0.66257

19.5 2.6 - 4.66 - 1.55

48.472 1.29577

14.84 1.05

48.5

14.8

1.3

1.1

3. Multiplication and Division Rule:Round your answer to have the same number of

significant figures as the measurement with the least number of significant figures.

24.312 0.2332 19.5 0.40x 0.66 x 0.66257

19.5 2.6 ÷ 4.6 ÷ 1.55

312.89544 0.06180453

4.2456… 1.6774…4.2

310 0.062

1.7

VI. International System of Units- Abbreviated SI, after the French name,

Le Systéme International d’Unités

- A revised version of the Metric System.

1. The Metric System

- All metric units are based on the number 10 or multiples of 10.

- Measurements consist of a base unit to which a prefix may be added to make it larger or smaller by a power of 10.

3 main base units:

a. Length = meter (m)

b. Volume = liter (L)

c. Mass = gram (g)

Metric Prefixes: (Memorize!)PrefixPrefix SymbolSymbol ValueValue Power of 10Power of 10

giga- G 1,000,000,000x larger 109

mega- M 1,000,000x larger 106

kilo- k 1,000x larger 103

hecto- h 100x larger 102

deka- da 10x larger 101

deci- d 10x smaller (1/10th or 0.1) 10-1

centi- c 100x smaller (1/100th or 0.01) 10-2

milli- m 1000x smaller (1/1000th or 0.001) 10-3

micro- 1,000,000x smaller(1/1,000,000th or 0.000 001) 10-6

nano- n 1,000,000,000x smaller(1/1,000,000,000th or 0.000 000 001) 10-9

pico- p 1,000,000,000,000x smaller(1/1,000,000,000,000th or 0.000 000 000 001) 10-12

Examples:

1 kilogram = 103 grams (1000x larger)

1 millimeter = 10-3 meter (1000x smaller)

1 hectoliter = 102 liters (100x larger)

1 m

10 meters

101 m = 1 dekameter = 1 dam

1 meter stick

10 birds = 1 meterEach bird = 1/10th of a meter (10-1 m)

1 bird = 1 decimeter (1 dm)

Example:

2. Measurements

A. Length – measure of linear distance

- Unit = meters (m)

Associations:

1 meter 1 yard (39.5 inches)

1 inch = 2.54 centimeters

1 mile = 1.6 kilometers

1 kilometer = 0.6 miles ( 10 km = 6 mi )

B. Volume – the amount of space an object takes up

- Units = cubic meters (m3)

liters (L)

SI Unit

Non-SI Unit

Ways to Measure Volume

1. Geometric Solid – (cube, rectangular solid, pyramid, cylinder, sphere, etc…)

- use an equation to derive the volume from length measurements

Example 1: Rectangular Solid

volume = length x width x height

volume = meters x meters x meters = m3

volume = cm x cm x cm = cm3

Kleenex Box = 22 cm x 11 cm x 5 cm = 1210 cm3

Example 2: Cylindrical Solid (area of base x height)

area of circle = r2

volume = r2h

* r = radius (1/2 the diameter)

Petri Dish

diameter = 9.5 cm

radius = 4.75cm

height = 2.0 cm

volume = 141.8 cm3

2. Liquids – pour the liquid into a measured container (graduated cylinder)

- read the level of the meniscus in liters or mL

Note: Liquids are usually measured in liters or mL, while solids are usually measured in m3 or cm3.

Associations:

1 liter 1 quart (4 cups)

1 gallon 4 liters

1 mL 20 drops

1 mL = 1 cm3 = 1 cm x 1cm x 1 cm

1 L = 1 dm3 = 10cm x 10cm x 10cm

1 L = 1000 mL

3. Irregular shaped solids – Water displacement!

- Dunk an object in water and the water level goes up!

4

5

4

5

Volume

_______

Volume

_______

4.13 mL

4.78 mL

- Simply subtract the final and initial volumes to get difference – Indirect method!

C. Mass – measures the quantity of matter an object contains

- SI unit – kilogram (kg) – standard!

- Base unit = gram

Weight – measures the pull of gravity on a given mass

- Weight changes with location.(On moon = 1/6th earth)

- An object can be weightless, but never massless.

1 kg = mass of 1 L of water at 4 degrees Celsius

1 kg = 2.2 lbs

1 ounce 28 grams

1 kg = 1000 grams

1 penny 3 grams

VII. Temperature

Definition – a measure of the average kinetic energy of particles in matter.

- Kinetic energy is energy of motion, therefore, temperature actually measures the speed of particles.

Wikipedia.org – Temperature (click here)

- Temperature also describes the direction of heat flow – from hotter to colder

a. When you hold an ice cube it feels cold= heat flowing from hand to ice

b. When you hold an cup of coffee it feels hot= heat flowing from coffee to hand

Temperature Scales:

a. Celsius scale – based on the boiling and freezing points of water

b. Kelvin scale – based on absolute zero, the lowest temperature theoretically possible.

* At absolute zero, all particle motion stops!

Fahrenheit = FCelsius = C Kelvin = K

Freezing point of water

Boiling point of water

Absolute Zero

-273

0

100

-460

32

212 373

273

0

The End