# Measurement and Significant Figures

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Measurement and Significant Figures

Steps in the Scientific Method
1. Observations - quantitative - qualitative 2. Formulating hypotheses - possible explanation for the observation 3. 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

A law summarizes what happens
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 2 - scale (unit)
A measurement is a quantitative observation consisting of 2 parts: Part 1 - number Part 2 - scale (unit) Examples: 20 grams 6.63 x Joule·seconds

The Fundamental SI Units (le Système International, SI)

SI Units

Celsius & Kelvin

SI Prefixes Common to Chemistry
Unit Abbr. Exponent Mega 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 Accuracy
Accuracy 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.

Rules for Counting Significant Figures - Details
Zeros - Leading zeros do not count as significant figures. has 3 sig figs.

Rules for Counting Significant Figures - Details
Zeros - Captive zeros always count as significant figures. 16.07 has 4 sig figs.

Rules for Counting Significant Figures - Details
Zeros Trailing zeros are significant only if the number contains a decimal point. 9.300 has 4 sig figs.

Rules for Counting Significant Figures - Details
Exact numbers have an infinite number of significant figures. 1 inch = cm, exactly

Sig Fig Practice #1 1.0070 m  5 sig figs 17.10 kg  4 sig figs
How many significant figures in each of the following? m  5 sig figs 17.10 kg  4 sig figs 100,890 L  5 sig figs 3.29 x 103 s  3 sig figs 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 Calculation Calculator says: Answer 3.24 m x 7.0 m
100.0 g ÷ 23.7 cm3 g/cm3 4.22 g/cm3 0.02 cm x cm cm2 0.05 cm2 710 m ÷ 3.0 s m/s 240 m/s lb x 3.23 ft lb·ft 5870 lb·ft 1.030 g ÷ 2.87 mL 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. =  18.7 (3 sig figs)

Sig Fig Practice #3 Calculation Calculator says: Answer 3.24 m + 7.0 m
100.0 g g 76.27 g 76.3 g 0.02 cm cm 2.391 cm 2.39 cm 713.1 L L L 709.2 L lb lb lb lb 2.030 mL mL 0.16 mL 0.160 mL

Metric Prefixes 1 kilometer (km) = 1000 meters (m)
Kilo- means 1000 of that unit 1 kilometer (km) = meters (m) Centi- means 1/100 of that unit 1 meter (m) = 100 centimeters (cm) 1 dollar = 100 cents Milli- means 1/1000 of that unit 1 Liter (L) = milliliters (mL)

Metric Prefixes

Metric Prefixes

Learning Check 1. 1000 m = 1 ___ a) mm b) km c) dm
g = 1 ___ a) mg b) kg c) dg L = 1 ___ a) mL b) cL c) dL m = 1 ___ a) mm b) cm c) dm

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 m 9.4 x 10-9 cm 0.094 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 artery

Conversion Factors Fractions in which the numerator and denominator are EQUAL quantities expressed in different units Example: in. = 2.54 cm Factors: 1 in and cm 2.54 cm in.

Learning Check 1. Liters and mL 2. Hours and minutes
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 min = 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 Solving
Write 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 = 29 quarters X
You have \$7.25 in your pocket in quarters. How many quarters do you have? 7.25 dollars quarters 1 dollar = 29 quarters X

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) cm b) 244 cm c) 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 cm = 244 cm 1 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 min x 60 sec 24 hr hr 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: cm = 1 inch Volume: 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 = L 1 gallon = 4 quarts Your Setup:

Equalities length 10.0 in. 25.4 cm
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 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 cm mm 3 1 cm 4.3 cm mm3 13 cm3 = = 4300 mm3

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

So, a dm3 is the same as a Liter ! A cm3 is the same as a milliliter.
Solution ( ) 1000 cm dm 3 10 cm = 1 dm3 So, a dm3 is the same as a Liter ! A cm3 is the same as a milliliter.