Measurement and Significant Figures .

Measurement andSignificant Figures

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Steps in the Scientific Method

1. Observations- quantitative- qualitative

2. Formulating hypotheses- possible explanation for

the observation3. Performing experiments

- gathering new information to decide

whether the hypothesis is valid

Outcomes Over the Long-Term

Theory (Model)- A set of tested hypotheses that give an overall explanation of some natural phenomenon.

Natural Law- The same observation applies to many different systems

Law vs. Theory

A law summarizes what happens

A theory (model) is an attempt to explain why it happens.

Einstein's theory of gravity describes gravitational forces in terms of the curvature of spacetime caused by the presence of mass

Nature of Measurement

Part 1 - numberPart 2 - scale (unit)

Examples:20 grams

6.63 x 10-34 Joule·seconds

A measurement is a quantitative observation consisting of 2 parts:

The Fundamental SI Units (le Système International, SI)

Physical Quantity Name Abbreviation

Mass kilogram kg

Length meter m

Time second s

Temperature Kelvin K

Electric Current Ampere A

Amount of Substance mole mol

Luminous Intensity candela cd

SI Units

Celsius & Kelvin

SI Prefixes Common to Chemistry

Prefix Unit Abbr. ExponentMega M 106

Kilo k 103

Deci d 10-1

Centi c 10-2

Milli m 10-3

Micro 10-6

Nano n 10-9

Pico p 10-12

Uncertainty in Measurement

A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty. Measurements are performed

with instruments No instrument can read to an infinite number of decimal places

Precision and AccuracyAccuracy refers to the agreement of a particular value with the true value.

Precision refers to the degree of agreement among several measurements made in the same manner.

Neither accurate

nor precise

Precise but not accurate

Precise AND

accurate

Types of Error

Random Error (Indeterminate Error) - measurement has an equal probability of being high or low.

Systematic Error (Determinate Error) - Occurs in the same direction each time (high or low), often resulting from poor technique or incorrect calibration. This can result in measurements that are precise, but not accurate.

Rules for Counting Significant Figures - Details

Nonzero integers always count as significant figures.

3456 has 4 sig figs.


Zeros- Leading zeros do not count as significant figures.

0.0486 has3 sig figs.


Zeros- Captive

zeros always count as significant figures.



ZerosTrailing zeros are significant only if the number contains a decimal point.



Exact numbers have an infinite number of significant figures.

1 inch = 2.54 cm, exactly

Sig Fig Practice #1How many significant figures in each of the following?

1.0070 m 5 sig figs

17.10 kg 4 sig figs

100,890 L 5 sig figs

3.29 x 103 s 3 sig figs

0.0054 cm 2 sig figs

3,200,000 2 sig figs

Rules for Significant Figures in Mathematical Operations

Multiplication and Division: # sig figs in the result equals

the number in the least precise measurement used in the calculation.

6.38 x 2.0 =12.76 13 (2 sig figs)

Sig Fig Practice #2

3.24 m x 7.0 m

Calculation Calculator says: Answer

22.68 m2 23 m2

100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 4.22 g/cm3

0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2

710 m ÷ 3.0 s 236.6666667 m/s 240 m/s

1818.2 lb x 3.23 ft 5872.786 lb·ft 5870 lb·ft

1.030 g ÷ 2.87 mL 2.9561 g/mL 2.96 g/mL

Rules for Significant Figures in Mathematical Operations

Addition and Subtraction: The number of decimal places in the result equals the number of decimal places in the least precise measurement.

6.8 + 11.934 =18.734 18.7 (3 sig figs)

Sig Fig Practice #3

3.24 m + 7.0 m

Calculation Calculator says: Answer

10.24 m 10.2 m

100.0 g - 23.73 g 76.27 g 76.3 g

0.02 cm + 2.371 cm 2.391 cm 2.39 cm

713.1 L - 3.872 L 709.228 L 709.2 L

1818.2 lb + 3.37 lb 1821.57 lb 1821.6 lb

2.030 mL - 1.870 mL 0.16 mL 0.160 mL

Metric PrefixesKilo- means 1000 of that unit

»1 kilometer (km) =

1000 meters (m)Centi- means 1/100 of that unit

»1 meter (m) = 100

centimeters (cm)

»1 dollar = 100 centsMilli- means 1/1000 of that unit

»1 Liter (L) = 1000

milliliters (mL)

Metric Prefixes

1. 1000 m = 1 ___ a) mm b) km c) dm

2. 0.001 g = 1 ___ a) mg b) kg c) dg

3. 0.1 L = 1 ___ a) mL b) cL c) dL

4. 0.01 m = 1 ___ a) mm b) cm c) dm

Learning Check

Units of Length

? kilometer (km) = 500 meters

(m)

2.5 meter (m) = ? centimeters

(cm)

1 centimeter (cm) = ? millimeter

(mm)

1 nanometer (nm) = 1.0 x 10-9

meter

O—H distance =9.4 x 10-11 m9.4 x 10-9 cm0.094 nm

O—H distance =9.4 x 10-11 m9.4 x 10-9 cm0.094 nm

Learning Check Select the unit you would use to measure 1. Your height

a) millimeters b) meters c) kilometers

2. Your mass a) milligrams b) grams c)

kilograms

3. The distance between two cities a) millimeters b) meters

c) kilometers

4. The width of an arterya) millimeters b) meters

c) kilometers

Conversion Factors

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

How many minutes are in 2.5 hours?

Conversion factor

2.5 hr x 60 min = 150 min 1 hr

cancel

By using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the

numbers!

Steps to Problem SolvingWrite down the given amount. Don’t forget the

units!Multiply by a fraction.Use the fraction as a conversion factor.

Determine if the top or the bottom should be the same unit as the given so that it will cancel.

Put a unit on the opposite side that will be the new unit. If you don’t know a conversion between those units directly, use one that you do know that is a step toward the one you want at the end.

Insert the numbers on the conversion so that the top and the bottom amounts are EQUAL, but in different units.

Multiply and divide the units (Cancel).If the units are not the ones you want for your

answer, make more conversions until you reach that point.

Multiply and divide the numbers. Don’t forget “Please Excuse My Dear Aunt Sally”! (order of operations)

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

You Try This One!

If Jacob stands on Spencer’s shoulders, they are two and a half yards high. How many feet is that?

Learning Check

A rattlesnake is 2.44 m long. How long is the snake in cm?

a) 2440 cmb) 244 cmc) 24.4 cm

Solution

A rattlesnake is 2.44 m long. How long is the snake in cm?b) 244 cm

2.44 m x 100 cm = 244 cm1 m

Learning Check

How many seconds are in 1.4 days?

Unit plan: days hr min seconds

1.4 days x 24 hr x ?? 1 day

Wait a minute!

What is wrong with the following setup?

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

English and Metric Conversions

If you know ONE conversion for each type of measurement, you can convert anything!

»Mass: 454 grams = 1 pound

»Length: 2.54 cm = 1 inch

»Volume: 0.946 L = 1 quart

Learning Check

An adult human has 4.65 L of blood. How many gallons of blood is that?

Unit plan: L qt gallon

Equalities:1 quart = 0.946 L 1 gallon = 4 quarts

Your Setup:

Equalities

State the same measurement in two different units

length

10.0 in.

25.4 cm

Steps to Problem Solving

Read problem Identify data Make a unit plan from the initial unit

to the desired unit Select conversion factors Change initial unit to desired unit Cancel units and check Do math on calculator Give an answer using significant

figures

Dealing with Two Units

If your pace on a treadmill is 65 meters per minute, how many seconds will it take for you to walk a distance of 8450 feet?

What about 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 ENITRE 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

Learning Check

A Nalgene water bottle holds 1000 cm3 of dihydrogen monoxide (DHMO). How many cubic decimeters is that?

Solution

1000 cm3 1 dm 3

10 cm( ) = 1 dm3

So, a dm3 is the same as a Liter !

A cm3 is the same as a milliliter.

Measurement and Significant Figures .

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