Scientific Measurement (Chapter 3) Dr. Walker

Objectives Use the metric system in measurement and calculations Perform conversions between metric and non-metric units using dimensional analysis Differentiate between accuracy and precision in measurement Use proper scientific notation in measurements Perform calculations using significant figures Determine percent error

Measurement Contains a number AND a unit 39 years old 70 inches tall 200 pounds 37 oC

Metric Units For this class, we will use the metric system for our measurements Units to know: Mass – kilogram Volume – liter Temperature – Kelvin (or celsius) Length – meter Energy -joule

Metric Units Metric prefixes to know Milli – 1/1000 Centi – 1/100 1000 mL = 1 L Centi – 1/100 100 cg = 1 g Think 100 cents = 1 dollar Kilo – 1000 1000 g = 1 kg

Conversion Factors Allow you to convert from one unit to another What you saw on the previous page were all examples 1000 mL = 1 L 100 cg = 1 g 1000 g = 1 kg

Conversion Factors Not all conversion factors involve two metric measurements The following are examples (don’t memorize them) 2.54 cm = 1 in 2.2 lbs = 1 kg 3.79 L = 1 gal 60 seconds = 1 min

An Example Problem My body mass is 97 kg. If 1 kg = 2.2 lbs, what is my weight in pounds? How could I go about solving this problem?

Dimensional Analysis Method of using conversion factors to solve problems T – square method Gift tag Looks like a “gift tag” - who it goes to on top - who is giving the gift on the bottom start to from You typically will not have a unit here. I’ll show you an exception later.

Dimensional Analysis Example My body mass is 97 kg. If 1 kg = 2.2 lbs, what is my weight? Start with the information given Body mass = 95 kg What unit are we coming from? kilograms What unit are we going to – what does the problem ask for? My weight in pounds

Dimensional Analysis Example My body mass is 97 kg. If 1 kg = 2.2 lbs, what is my weight in pounds? We’re going “to” the pound, the unit asked for in the problem, which goes in the top right The initial information (“start”) given is 97 kg. This goes in the top left of the t-square 97 kg 2.2 lbs 1 kg We’re converting “from” kilograms, so it goes in the bottom right

Dimensional Analysis Advantage You don’t have to remember whether you multiply or divide Plug in the numbers into the template and cancel units accordingly

Dimensional Analysis Example My body mass is 97 kg. If 1 kg = 2.2 lbs, what is my weight? After setting up the problem, we treat it like we’re manipulating fractions Top x top, bottom x bottom, then divide accordingly 97 kg 2.2 lbs = 213.4 lbs 1 kg What happens to the kilograms unit? Kilograms are on both the top and the bottom, so the unit cancels. The units are just like numbers this way. Pounds are all that is left.

Another Example I need 15 liters of water for an experiment. If I buy 3 gallons of water at the store, do I have enough? (3.79 L = 1 gallon)

Another Example I need 15 liters of water for an experiment. If I buy 3 gallons of water at the store, do I have enough? (3.79 L = 1 gallon) 3 gallons 3.79 L = 11.37 L 1 gallon I need 15 gallons for the experiment, so I didn’t buy enough water!!

Multistep Example How many seconds are in 1 day? We know… 60 seconds = 1 minute 60 minutes = 1 hour 24 hours = 1 day How do we solve this????

Multistep Example Some problems require more than one conversion to solve Options One Use a different t-square for each conversion Use the answer from the 1st t-square to start the 2nd one Two Use multiple t-squares together to make one big step

Multistep Example How many seconds are in 1 day? 1 day 24 hours 1 day Notice the answer to the first t-square starts the second one 24 hours 60 minutes = 1440 minutes 1 hour

Multistep Example How many seconds are in 1 day? 1440 minutes (Answer!!!) 1 minute

Another way We can do this all in one step Not required 1 day 24 hours minutes seconds 1 day 24 hours 60 minutes 60 seconds = 86,400 seconds (Answer!!!) 1 day 1 hour 1 minute

Uncertainty Precision (left) – reproducibility of measurement How many times you get the same value Accuracy (right) – How close it is to known value

Uncertainty With Data Sets A standard 100 g mass was measured several times, providing measurements of 90.2 g, 90.1 g, and 90.2 g. Is this accurate? Is this precise?

Uncertainty With Data Sets A standard 100 g mass was measured several times, providing measurements of 90.2 g, 90.1 g, and 90.2 g. Is this accurate? NO!! The measurements are NOT close to the 100 g known value Is this precise? YES!! The measurements are all very close TO EACH OTHER!!

Uncertainty With Data Sets A metal with a known density of 3.25 g/mL The density was measured at 3.24 g/mL, 3.23 g/mL, and 3.24 g/mL. Is this accurate? Is this precise?

Uncertainty With Data Sets A metal with a known density of 3.25 g/mL The density was measured at 3.24 g/mL, 3.23 g/mL, and 3.24 g/mL. Is this accurate? YES!! The numbers are all very close the known value Is this precise? YES!! The numbers are all very close TO EACH OTHER

Scientific Notation Used for really big or really small numbers Examples Avogadro’s Number – number of atoms in a mole (you’ll learn this later) 6.022 x 1023 Mass of an electron 9.1 x 10-28 g

Rules One number to the left of the decimal Regular Scientific Move decimal until the above rule is followed Exponent equals the number of places moved Regular # > 1, exponent > 0 Regular # < 1, exponent < 0

Examples To scientific notation From scientific notation 9.2 x 108 356000 0.00026 217 From scientific notation 9.2 x 108 4.28 x 102 5.632 x 10-3

Examples To scientific notation From scientific notation Notice we didn’t put the starting or ending zeros in the number From scientific notation 9.2 x 108 = 920,000,000 4.28 x 102 = 428 5.632 x 10-3 = 0.005632

Significant Figures Significant figures show how many numbers you may count in a measurement Many measurements give you a decimal to several places --- can you really trust it? Examples Room measurement = 11’4” x 15’2” In decimal form = 11.33 x 15.17 Square footage = 171.8761 ft2 Can we really get a measurement to THAT degree of accuracy?

The Rules 1) All nonzeros are SIGNIFICANT! 45 in = 2 sig figs 2369 ft = 4 sig figs 115 m = 3 sig figs

The Rules 2) Sandwich zeroes (zeroes between non-zeros) are SIGNIFICANT! 104 m = 3 sig figs 2008 ft = 4 sig figs 302 s = 3 sig figs

The Rules 3) Trailing zeros do not count…unless they’re behind a decimal 4500 ft = 2 sig figs 2000 yds = 1 sig fig 3570 L = 3 sig figs

The Rules 3) Trailing zeros do not count…unless they’re behind a decimal 25.0 m = 3 sig figs 10.0 m = 3 sig figs 10.50 m = 4 sig figs 2.0 x 103 m = 2 sig figs Gives the number 2000 2 sig figs instead of one! 2.00 x 103 m = 3 sig figs Gives the number 2000 3 sig figs instead of one!

The Rules 4) Leading zeros do not count! 0.0056 m = 2 sig figs 0.0748 L = 3 sig figs Notice the sig figs do not start until you hit a nonzero number! 0.0560 L = 3 sig figs Notice the combination of rules 3 and 4 The sig figs don’t start until the 5 The decimal makes the last zero count!

Additional Examples 2307 in = ____ sig figs 2.50 L = ____ sig figs 0.0078 g = ____ sig figs 170 in = ____ sig figs 0.08650 J = ____ sig figs 4.550 x 106 s = ____ sig figs

Additional Examples 2307 in = 4 sig figs 2.50 L = 3 sig figs 0.0078 g = 2 sig figs 170 in = 2 sig figs 0.08650 J = 4 sig figs 4.550 x 106 = 4 sig figs Number is 4,550,000 Scientific notation designates an extra sig fig

Rules For Operations Multiplication/Division Based upon lowest number of significant figures 15 x 25 = 375 Round answer to 2 sig figs – 375 rounds to 380!! 2 2 Answer can only have 2 sig figs !!! Pay attention to first digit you got rid of – standard rounding rules apply!!

Rules For Operations 25.0 + 2.658 27.658 Addition/Subtraction Based on least significant place value Since there is no sig fig in the Hundredths or thousandths place In the top number, you can’t use them in the answer!!! You must round your answer to the tenths column 27.658 becomes 27.7!! 25.0 + 2.658 27.658

More Practice - Operations 24 m x 58 m = _____ m2 56 L + 2.35 L = _____ L 250 ft x 175 ft = ______ ft2 8432 m2 / 12.5 m = ______ m 14.2 g + 8.73 g + 0.912 g = ______ g 22.4 m x 11.3 m x 5.2 m = ______ m3 20.6 oC – 0.312 oC = _____ oC

More Practice - Operations 24 m x 58 m = 1392 1400 m2 56 L + 2.35 L = 58.35 58 L 250 ft x 175 ft = 43750 44000 ft2 8432 m2 / 12.5 m = 674.56 675 m 14.2 g + 8.73 g + 0.912 g = 23.842 23.8 g 22.4 m x 11.3 m x 5.2 m = 1316.224 1300 m3 20.6 oC – 0.312 oC = 20.288 20.3 oC

Percent Error The bars in the numerator refer to absolute value % Error = Accepted – Experimental ------------------------------- x 100 Accepted The bars in the numerator refer to absolute value All % error values are POSITIVE

Example A standard 100 g mass was measured at 90.2 g. What is the percent error?

Example A standard 100 g mass was measured at 90.2 g. What is the percent error? % Error = 100 – 90.2 ------------------ x 100 = 9.8 % 100

Example I was pulled by a police officer who measured my speed at 68 mph with a calibrated radar gun. My speedometer read 55 mph. What was the % error of my speedometer?

Example I was pulled by a police officer who measured my speed at 68 mph with a calibrated radar gun. My speedometer read 55 mph. What was the % error of my speedometer? % Error = 68 - 55 ------------ x 100 = 19.1 % 68

Terms To Know Measurement Accuracy Precision Accepted value Experimental value Significant figures Liter Kilogram Joule Meter

Required Skills Converting to and from scientific notation Determining accuracy and precision Calculating percent error Determining significant figures Making calculations with significant figures Solve conversion problems using dimensional analysis (t-squares)