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Measurement and Significant Figures www.lab-initio.com
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 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 - number Part 1 - number Part 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)
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. 0.0486 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 = 2.54 cm, exactly
Sig Fig Practice #1 How 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 10 3 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. # 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 m 2 23 m 2 100.0 g ÷ 23.7 cm 3 4.219409283 g/cm 3 4.22 g/cm 3 0.02 cm x 2.371 cm 0.04742 cm 2 0.05 cm 2 710 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 Prefixes Kilo- 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 cents Milli- means 1/1000 of that unit »1 Liter (L) = 1000 milliliters (mL)
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 m 9.4 x 10 -9 cm 0.094 nm O—H distance = 9.4 x 10 -11 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) milligramsb) grams c) kilograms 3. The distance between two cities a) millimetersb) meters c) kilometers 4. The width of an artery a) millimetersb) 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 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 You have $7.25 in your pocket in quarters. How many quarters do you have? 7.25 dollars 4 quarters 1 dollar 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 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 100 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 60 min x 60 sec 24 hr 1 hr 1 min 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 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 Read problem Identify data Identify data Make a unit plan from the initial unit to the desired unit Make a unit plan from the initial unit to the desired unit Select conversion factors Select conversion factors Change initial unit to desired unit Change initial unit to desired unit Cancel units and check Cancel units and check Do math on calculator Do math on calculator Give an answer using significant figures 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 cm 3 to mm 3 4.3 cm 3 10 mm 3 1 cm 1 cm ( ) = 4.3 cm 3 10 3 mm 3 1 3 cm 3 1 3 cm 3 = 4300 mm 3
Learning Check A Nalgene water bottle holds 1000 cm 3 of dihydrogen monoxide (DHMO). How many cubic decimeters is that?
Solution 1000 cm 3 1 dm 3 10 cm 10 cm ( ) = 1 dm 3 So, a dm 3 is the same as a Liter ! A cm 3 is the same as a milliliter.