Chapter 3Chapter 3

Scientific MeasurementScientific Measurement

Anything in black letters = write it in your notes (‘knowts’)

3.1 – Using & Expressing Measurements3.1 – Using & Expressing Measurements

Measurements without units are useless!

“I walked 5 today.”

“The speed of light is 186,000

“I weigh 890”

“20 of water”

All measurements need units!

We will often work with really large or really small numbers in this class.

872,000,000 grams

0.0000056 moles

= 8.72 x 108 grams

= 5.6 x 10-6 moles

Scientific Notation

6.02 x 1023

coefficient exponent

The coefficient must be a single, nonzero digit, exponent must be an integer.

Standard Notation

To multiply numbers written in scientific notation, multiply the coefficients and add the exponents.

(3 x 104) x (2 x 102) = (3 x 2) x 104+2 = 6 x 106

(2.1 x 103) x (4.0 x 10–7) = (2.1 x 4.0) x 103+(–7) = 8.4 x 10–4

Multiplication and Division

To divide numbers written in scientific notation, divide the coefficients and subtract the bottom

exponent from the top exponent.

2

5

10 x 0.6

10 x 0.3 2510 x 0.6

0.3

310 x 5.0210 x 0.5

Coefficient needs to be between 1 and 10

Addition and Subtraction

When adding or subtracting in Sci. Not., the exponents must be the same.

= (5.4 x 103) + (0.80 x 103)

(5.4 x 103) + (8.0 x 102)

= (5.4 + 0.80) x 103

= 6.2 x 103

Not the same, need to adjust one of the

exponents

Example

Solve each problem and express the answer in scientific notation.

a. (8.0 x 10–2) x (7.0 x 10–5)

b. (7.1 x 10–2) + (5 x 10–3)

a.

Multiply the coefficients and add the exponents.

(8.0 x 10–2) x (7.0 x 10–5)

= (8.0 x 7.0) x 10–2 + (–5)

= 56 x 10–7

= 5.6 x 10–6

b.

Rewrite one of the numbers so that the exponents match. Then add the coefficients

(7.1 x 10–2) + (5 x 10–3)

= (7.1 x 10–2) + (0.5 x 10–2)

= (7.1 + 0.5) x 10–2

= 7.6 x 10–2

Accuracy - closeness of a measurement to the actual or accepted value.

Precision - closeness of repeated measurements to each other

The closeness of a dart to the bull’s-eye corresponds to the degree of accuracy. The closeness of several darts to one another

corresponds to the degree of precision.

Good Accuracy, Good Precision

Poor Accuracy, Good Precision

Poor Accuracy, Poor Precision

Darts on a dartboard illustrate the difference between accuracy and precision.

Accuracy and Precision

Error

Suppose you measured the melting point of a compound to be 78°C

Suppose also, that the actual melting point value (from reference books) is 76°C.

The error in your measurement would be 2°C.

Error is the difference between the actual (accepted) and experimental value

How far off you are in a measurement doesn’t tell you much.

For example, lets say you have $1,000,000 in your checking account. When you balance your checkbook at the end of the month, you find that you are off by $175; error = $175

Now, lets be more realistic, you have $225 in your checking account and after balancing you are off by $175!

In both cases, there is an error of $175.

But in the first, the error is such a small portion of the total that it doesn’t matter as much as the second.

So, instead of error, percent error is more valuable.

Percent error compares the error to the size of the measurements.

ASSIGN: Chapter 3 Worksheet #1

Significant Figures

In any measurement, the last digit is estimated

30.2°CThe 2 is estimated (uncertain) by the experimenter, another person may say 30.1 or 30.3

0.72 cm

9.3 mL

Increasing Precision

The significant figures in a measurement include all of the digits that are known, plus the last digit that is estimated.

Numbers that are NOT significant are called placeholders.

Rules for determining Significant Figures (p. 67)1. Every nonzero digit in a reported measurement is assumed to be significant.

2. Zeros appearing between nonzero digits are significant.

3. Leftmost zeros appearing in front of nonzero digits are not significant. They act as placeholders. By writing the measurements in scientific notation, you can eliminate such placeholding zeros.

4. Zeros at the end of a number and to the right of a decimal point are always significant.

5. Zeros at the rightmost end of a measurement that lie to the left of an understood decimal point are not significant if they serve as placeholders to show the magnitude of the number.

5 (continued). If such zeros were known measured values, then they would be significant. Writing the value in scientific notation makes it clear that these zeros are significant.

6. There are two situations in which numbers have an unlimited number of significant figures. The first involves counting. A number that is counted is exact.

6 (continued). The second situation involves exactly defined quantities such as those found within a system of measurement.

HOLY SMOKES!!

A shorter method for determining which numbers are significant in a measurement.

1.All nonzero numbers are significant.

2.Zeros are also significant if they are NOT placeholder zeros.

3.Counted and exact numbers have an unlimited number of sig figs.

The Tischer Method!™

A placeholder is a zero that ‘holds place’; it is only there to show how big or small a number is.

Underline the zeros that are placeholders

100 1.00 0.23

0.0034 1.01 1005.4

0.10 100.0 54.0

A placeholder is a zero that ‘holds place’; it is only there to show how big or small a number is.

Zeros to the right of a number & after the decimal are more than placeholders, they are part of the measurement and are significant.

1. All nonzero numbers are significant.

2. Zeros are also significant if they are NOT placeholders.

3. Counted and exact numbers have an unlimited number of sig. figs.

How many significant digits are in the following measurements?

a) 150.31 grams b) 10.03 mL

c) 0.045 cm d) 4.00 lbs

e) 0.01040 m f) 100.10 cm

g) 100 grams h) 1.00 x 102 grams

i) 11 cars j) 2 molecules

“Box and Dot” Method for Counting Sig Figs

1. Draw a box around all digits from the 1st nonzero digit on left to last nonzero digit on the right.

2. If a dot is present, draw a box around any trailing zeros.

3. Any boxed digit is significant.

SOURCE: JCE Vol 86 No 8 (Aug ‘09) W. Kirk Stephenson

Another Shorter Method

An answer can’t be more accurate than the measurements it was

calculated from

Rules for Add/Subtracting Sig FigsThe answer to an +/- calculation should be rounded to the same number of decimal places as the measurement with the least number of decimal places.

3.2 cm

2.05 cm

+ 1.1 cm 3.15 cm

32.10 g

+ 5.0012 g 37.1012 g

37.10 g

Rules for Mult/Division Sig Figs The answer to a x/÷ calculation should be rounded to the same number of sig figs as the measurement with the least number of sig figs.

2.0 cm

x 1.89 cm 2 3.78 cm

3.8 cm2

8.19

8.1

g

mL 1.01111111... g mL

1.0 g mL

Always round your final answer off to the correct number of significant digits.

ASSIGN: Chapter 3 Worksheet #2

3.2 – Units of Measurement3.2 – Units of Measurement

SI Base Units (page 74)Quantity SI base

unitSymbol

Length meter m

Mass kilogram kg

Temperature kelvin K

Time second s

Amount of substance

mole mol

Luminous intensity

candela cd

Electric current

ampere A

SI – International System of Units

We will use all of these in this class

How is Mass different from Amount of Substance?

Commonly Used Metric Prefixes (page 75)Prefix Symbol Meaning Factor

mega M 1 million times larger than the base 106

kilo k 1000 times larger than the base 103

BASE Base Unit (meter, second, gram, etc)

deci d 10 times smaller than the base 10-1

centi c 100 times smaller than the base 10-2

milli m 1000 times smaller than the base 10-3

micro μ 1 million times smaller than the base 10-6

nano n 1 billion times smaller than the base 10-9

pico p 1 trillion times smaller than the base 10-12

Volume - Amount of space occupied by an object (remember?)

1 L = 1000 mL 1 mL = 1 cm3

Normal units used for volume:

Solids – m3 or cm3

Liquids & Gases – liters (L) or milliliters (mL)

The volume of a material changes with temperature, especially for gases.

Mass - Measure of inertia (remember?)

Weight - Force of gravity on a mass; measured in pounds (lbs) or Newtons.

Weight can change with location, mass does not

Energy – Ability to do work or produce heat.

Normal units used for energy:

SI – joule (J)

non-SI – calorie (cal)

How many joules are in a kilojoule?

How many calories are in a kilocalorie?

1 cal = 4.184 J

Temperature – measure of how cold or hot an object is.

Temperature – measure of the average kinetic energy of molecules.

Normal units used for temp:

SI – kelvin (K)

non-SI – celsius (°C) or Fahrenheit (°F)

K = °C + 273

yucky!

Celsius

Kelvin

100 divisions

100 divisions

100°CBoiling point

of water373.15 K

0°CFreezing point

of water273.15 K

Density is an intensive property

mass

volumeDensity =

Normal units for density:

g/cm3, g/mL, g/L

Densities of Some Common Materials

Solids and Liquids Gases

MaterialDensity at

20°C (g/cm3)Material

Density at 20°C (g/L)

Gold 19.3 Chlorine 2.95

Mercury 13.6 Carbon dioxide 1.83

Lead 11.3 Argon 1.66

Aluminum 2.70 Oxygen 1.33

Table sugar 1.59 Air 1.20

Corn syrup 1.35–1.38 Nitrogen 1.17

Water (4°C) 1.000 Neon 0.84

Corn oil 0.922 Ammonia 0.718

Ice (0°C) 0.917 Methane 0.665

Ethanol 0.789 Helium 0.166

Gasoline 0.66–0.69 Hydrogen 0.084

ASSIGN:

Read 3.2

Lesson Check 3.2; #23-35 (page 82)

3.3 – Solving Conversion Problems3.3 – Solving Conversion Problems

12 inches = ______ foot

1 minute = _____ seconds

10 cents = _______ dime

24 hours = ______ day

1 km = ________meters

4.184 Joules = _______ calorie

Fill in the blanks…

These can all be used as conversion factors

Conversion Factors

Any measurement that equals another measurement is a conversion factor

How to use conversion factors…to convert measurements…

1. Place the measurement you want to convert over 1.

2. Multiply by a conversion factor to cancel units.

Example 1

How many seconds are in 8 hours?

1.Start with given information.

2.Use conversion factors to cancel units until desired unit is left.

60 min = 1 hr; 60 sec = 1 min

Multiply across the top,

divide along the bottom of the conversion factors.

hours 8 x hour 1

minutes 60 xminute 1

seconds 60

seconds 28,800

Example 2

How many km are in 25 miles?

1.Start with given information.

2.Use conversion factors to cancel units until desired unit is left.

1.6 km = 1 mile

Multiply across the top,

divide along the bottom of the conversion factors.

miles 25 mile 1

km 1.6 x km 40

kmx 10 0.4 1

5.26 mumu = 8 nunu;

1.76 nunu = 12 fufu;

0.826 fufu = 1000 bubu.

1. How many mumus are in 10 bubus?

Example 3 – TRY IT!

2. How many fufus are in 26 mumus?

Example 4 – TRY IT!

5.26 mumu = 8 nunu;

1.76 nunu = 12 fufu;

0.826 fufu = 1000 bubu.

A student converted 10.1 feet as follows:

How many significant digits should the answer have?Why not 1?

ASSIGN:

Chapter 3 Worksheet 3

Conversion Factor QuizConversion Factor Quiz

1 horsepower = 745.7 W

2.54 cm = 1 inch

Show your WORK for credit.

LEFT: 1. Convert 6.5 horsepower into watts

RIGHT: 1. Convert 100 watts into horsepower

LEFT: 2. How many cm are in 12 miles?

RIGHT: 2. How many cm are in 20 miles

Practice With ConversionsPractice With Conversions

1 horsepower = 745.7 W

2.54 cm = 1 inch

Show your WORK for credit.

LEFT: 1. Convert 6.5 horsepower into watts

RIGHT: 1. Convert 100 watts into horsepower

LEFT: 2. How many cm are in 12 miles?

RIGHT: 2. How many cm are in 20 miles