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Measurement and Problem Solving

Measurement and Problem Solving
Measurement and Problem Solving www.ThesisScientist.comWhat Is a Measurement • Quantitative = quantity and it must have a unit • Comparison to an agreed upon standard. • accuracy www.ThesisScientist.comScientists have measured the average global temperature rise over the past century to be 0.6 °C • Quantity is 0.6 the unit is°C • Comparison to an agreed upon standard is Celsius temperature scale. • Accuracy: The confidence in the measurement is determined by looking at th’s the last digit, here it is in the 1/10 position, so the certain is plus and minus 1/10 or between 0.5 and 0.7 °C. www.ThesisScientist.comScientific Notation A way of writing large and small numbers. www.ThesisScientist.comBig and Small Numbers • We commonly measure The sun’s objects that are many times diameter is larger or smaller than our 1,392,000,000 m. standard of comparison. • Writing large numbers of zeros is tricky and confusing. An atom’s Not to mention there’s the 8 average diameter is digit limit of your calculator 0.000 000 000 3 m. www.ThesisScientist.comExponents • When the exponent on 10 is positive, it means the number is that many powers of 10 larger. 9 Sun’s diameter = 1.392 x 10 m = 1,392,000,000 m. • When the exponent on 10 is negative, it means the number is that many powers of 10 smaller. 10 Average atom’s diameter = 3 x 10 m = 0.0000000003m www.ThesisScientist.comScientific Notation • To compare numbers written in scientific notation: First compare exponents on 10. If exponents are equal, then compare decimal numbers Exponent 5 2 1.23 x 10 4.56 x 10 8 2 5 1.23 x 10 4.56 x 10 7.89 x 10 10 10 7.89 x 10 1.23 x 10 Decimal part Exponent part www.ThesisScientist.comWriting a Number in Scientific Notation, Continued 12340 1. Locate the decimal point. 12340. 2. Move the decimal point to obtain a number between 1 and 10. 1.234 n 3. Multiply the new number by 10 .  Where n is the number of places you moved the decimal point. 4 1.234 x 10 4. If you moved the decimal point to the left, then n is +; if you moved it to the right, then n is − . 4 1.234 x 10 www.ThesisScientist.comWriting a Number in Scientific Notation, Continued 0.00012340 1. Locate the decimal point. 0.00012340 2. Move the decimal point to obtain a number between 1 and 10. 1.2340 n 3. Multiply the new number by 10 .  Where n is the number of places you moved the decimal point. 4 1.2340 x 10 4. If you moved the decimal point to the left, then n is +; if you moved it to the right, then n is − . 4 1.2340 x 10 www.ThesisScientist.comWriting a Number in Standard Form 6 1.234 x 10 • Since exponent is 6, make the number smaller by moving the decimal point to the left 6 places. When you run out of digits to move around, add zeros. Add a zero in front of the decimal point for decimal numbers. 000 001.234 0.000 001 234 www.ThesisScientist.comPractice—Write the Following in Scientific Notation, Continued 2 0 123.4 = 1.234 x 10 8.0012 = 8.0012 x 10 5 3 145000 = 1.45 x 10 0.00234 = 2.34 x 10 1 2 25.25 = 2.525 x 10 0.0123 = 1.23 x 10 0 6 1.45 = 1.45 x 10 0.000 008706 = 8.706 x 10 www.ThesisScientist.comPractice—Write the Following in Standard Form, Continued 3 0 2.1 x 10 = 2100 4.02 x 10 = 4.02 4 1 9.66 x 10 = 0.000966 3.3 x 10 = 33 2 0 6.04 x 10 = 0.0604 1.2 x 10 = 1.2 www.ThesisScientist.comNumbers 1. Exact 2. Measured: Significant Figures www.ThesisScientist.comExact Numbers vs. Measurements • Exact: Sometimes you can determine an exact value for a quality of an object. A. Often by counting. • Pennies in a pile. B. Sometimes by definition th • 1 ounce is exactly 1/16 of 1 pound. • Measured: Whenever you use an instrument to compare a quality of an object to a standard, there is uncertainty in the comparison. www.ThesisScientist.comExact Numbers • Exact numbers have an unlimited number of significant figures. • A number whose value is known with complete certainty is exact. From counting individual objects. From definitions. • 1 cm is exactly equal to 0.01 m. • 20 .05 = 1.0000000000000 • 12 inches = 1.000000000000000000000000 ft www.ThesisScientist.com4 fundamental measuring instruments 1. Length 2. Mass 3. Time 4. Temperature www.ThesisScientist.comHow do we make a Measurement • Measurements are written to indicate the uncertainty in the measurement. • The system of writing measurements we use is called significant figures. • When writing measurements, all the digits written are known with certainty except the last one, which is an estimate. 4 45 5.87 .872 2 Estimated Certain www.ThesisScientist.comReading a Measuring Instrument/Device For any Digital Device record ALL the digits www.ThesisScientist.comReading a Measuring Instrument/Device 1. Record all the numbers you can see 2. Make ONE Guess www.ThesisScientist.comSkillbuilder 2.3—Reporting the Right Number of Digits • A thermometer used to measure the temperature of a backyard hot tub is shown to the right. What is the temperature reading to the correct number of digits www.ThesisScientist.comSkillbuilder 2.3—Reporting the Right Number of Digits • A thermometer used to measure the temperature of a backyard hot tub is shown to 103.4 °F the right. What is the temperature reading to the correct number of digits www.ThesisScientist.comWhat is the Length • We can see the markings between 1.61.7cm • We can’t see the markings between the .6.7 • We must guess between .6 .7 • We record 1.67 cm as our measurement www.ThesisScientist.comWhat is the length of the wooden stick 1) 4.5 cm 2) 4.54 cm 3) 4.547 cm www.ThesisScientist.com8.00 cm or 3 (2.2/8) www.ThesisScientist.comCounting Significant Figures • All nonzero digits are significant. 1.5 has 2 significant figures. • Interior zeros are significant. 1.05 has 3 significant figures. • Trailing zeros after a decimal point are significant. 1.050 has 4 significant figures. www.ThesisScientist.comCounting Significant Figures, Continued • Leading zeros are NOT significant.  0.001050 has 4 significant figures. 3 • 1.050 x 10 • Zeros at the end of a number without a written decimal point are NOT significant 2  If 150 has 2 significant figures, then 1.5 x 10 , but if 150 has 3 significant figures, then 1.50 2 x 10 . www.ThesisScientist.comExample 2.4—Determining the Number of Significant Figures in a Number, Continued • How many significant figures are in each of the following numbers 0.0035 2 significant figures—leading zeros are not significant. 1.080 4 significant figures—trailing and interior zeros are significant. 2371 4 significant figures—All digits are significant. 5 2.97 × 10 3 significant figures—Only decimal parts count as significant. 1 dozen = 12 Unlimited significant figures—Definition 100,000 1, no decimal www.ThesisScientist.comDetermine the Number of Significant Figures, 2 2 • 12000 • 0.0012 3 3 • 120. • 0.00120 4 4 • 12.00 • 1201 3 • 1.20 x 10 • 1201000 3 4 www.ThesisScientist.comHow many sig figs •All digits count 6 45.8736 •Leading 0’s don’t 3 .000239 •Trailing 0’s do 5 .00023900 48000. •0’s count in decimal form 5 48000 •0’s don’t count w/o decimal 2 6 3.98210 •All digits count 4 1.00040 •0’s between digits count as well 6 www.ThesisScientist.com as trailing in decimal formRounding • When rounding to the correct number of significant figures, if the number after the place of the last significant figure is: 1. 0 to 4, round down.  Drop all digits after the last significant figure and leave the last significant figure alone. 2. 5 to 9, round up.  Drop all digits after the last significant figure and increase the last significant figure by one. www.ThesisScientist.comExamples of Rounding For example you want a 4 Sig Fig number 0 is dropped, it is 5 4965.03 4965 780,582 780,600 8 is dropped, it is 5; Note you must include the 0’s 5 is dropped it is = 5; note you 1999.5 2000. need a 4 Sig Fig www.ThesisScientist.comMultiplication and Division with Significant Figures • When multiplying or dividing measurements with significant figures, the result has the same number of significant figures as the measurement with the fewest number of significant figures. 5.02 × 89,665 × 0.10 = 45.0118 = 45 3 sig. figs. 5 sig. figs. 2 sig. figs. 2 sig. figs. 5.892 ÷ 6.10 = 0.96590 = 0.966 4 sig. figs. 3 sig. figs. 3 sig. figs. www.ThesisScientist.comDetermine the Correct Number of Significant Figures for Each Calculation and 1. 1.01 × 0.12 × 53.51 ÷ 96 = 0.067556 = 0.068 7 is in place Result should 3 sf 2 sf 4 sf 2 sf of last sig. fig., have 2 sf. number after is 5 or greater, so round up. 2. 56.55 × 0.920 ÷ 34.2585 = 1.51863 = 1.52 1 is in place 4 sf Result should 3 sf 6 sf of last sig. fig., have 3 sf. number after is 5 or greater, so round up. www.ThesisScientist.comAddition/Subtraction 25.5 32.72 320 +34.270 0.0049 + 12.5 59.770 32.7151 332.5 59.8 32.72 330 www.ThesisScientist.comAddition and Subtraction with Significant Figures • When adding or subtracting measurements with significant figures, the result has the same number of decimal places as the measurement with the fewest number of decimal places. 5.74 + 0.823 + 2.651 = 9.214 = 9.21 2 dec. pl. 3 dec. pl. 3 dec. pl. 2 dec. pl. 4.8 3.965 = 0.835 = 0.8 1 dec. pl 3 dec. pl. 1 dec. pl. www.ThesisScientist.comDetermine the Correct Number of Significant Figures for Each Calculation and Round and Report the Result, Continued 1. 0.987 + 125.1 – 1.22 = 124.867 = 124.9 8 is in place 3 dp Result should 1 dp 2 dp of last sig. fig., have 1 dp. number after is 5 or greater, so round up. 2. 0.764 – 3.449 – 5.98 = 8.664 = 8.66 6 is in place Result should 3 dp 3 dp 2 dp of last sig. fig., have 2 dp. number after is 4 or less, so round down. www.ThesisScientist.comAddition and Subtraction Look for the .71 .56 + . 1 5 3 = . 7 1 3 last 82000 82000 + 5.32 = 82005.32 important digit .1 10.0 9.8742 = .12580 0 10 – 9.8742 = .12580 www.ThesisScientist.comBoth Multiplication/Division and Addition/Subtraction with Significant Figures • When doing different kinds of operations with measurements with significant figures, evaluate the significant figures in the intermediate answer, then do the remaining steps. • Follow the standard order of operations. n  Please Excuse My Dear Aunt Sally.    3.489 × (5.67 – 2.3) = 2 dp 1 dp 3.489 × 3.37 = 12 4 sf 1 dp 2 sf 2 sf www.ThesisScientist.comExample 1.6—Perform the Following Calculations to the Correct Number of Significant Figures, Continued a) 1.100.51204.00153.4555 0.65219 0.652 0.355 b) 105.1 100.5820 4.8730  4.9 c) 4.5623.99870452.6755 452.33 52.79904 53 d) 14.840.558.02 0.142 0.1 www.ThesisScientist.comBasic Units of Measure www.ThesisScientist.comUnits • Units tell the standard quantity to which we are comparing the measured property.  Without an associated unit, a measurement is without meaning. • Scientists use a set of standard units for comparing all our measurements.  So we can easily compare our results. • Each of the units is defined as precisely as possible. www.ThesisScientist.comThe Standard Units • Scientists generally report results in an agreed upon International System. • The SI System Aka Système International Quantity Unit Symbol Length meter m Mass kilogram Kg/g Volume liter L Time second s Temperature kelvin K www.ThesisScientist.comVolume 3  1 mL = 1 cm www.ThesisScientist.comRelated Units in the SI System • All units in the SI system are related to the standard unit by a power of 10. • The power of 10 is indicated by a prefix. www.ThesisScientist.comCommon Prefixes in the SI System Decimal Prefix Symbol Power of 10 Equivalent 6 mega M 1,000,000 Base x 10 3 kilo k 1,000 Base x 10 1 deci d 0.1 Base x 10 2 centi c 0.01 Base x 10 3 milli m 0.001 Base x 10 6 microm 0.000 001 Base x 10 9 nano n 0.000 000 001 Base x 10 12 Pico p 0.000 000 000 001 Base x 10 www.ThesisScientist.comMeasurements and SI M 1,000,000 gram g k 1,000 meter m d 0.1 liter L c 0.01 seconds s m 0.001 Kelvin K m 0.000 001 n 0.000 000 001 p 0.000 000 000 001 www.ThesisScientist.comMeasurements and SI liter L m 0.001 (m = .001)L mL = .001L or 1000 mL = L www.ThesisScientist.comPrefixes Used to Modify Standard Unit 3 • kilo = 1000 times base unit = 10 3 k=1000 or k = 10 3 1 kg = 10 g 9 • nano = 10 times the base unit 9 n=.000000001 or n = 10 9 1 nL = 10 L www.ThesisScientist.comUnits • Always write every number with its associated unit. • Always include units in your calculations. You can do the same kind of operations on units as you can with numbers. 2 • cm × cm = cm • cm + cm = 2cm • cm ÷ cm = 1 Using units as a guide to problem solving is called dimensional analysis. www.ThesisScientist.comProblem Solving and Dimensional Analysis, Continued • Arrange conversion factors so the starting unit cancels.  Arrange conversion factor so the starting unit is on the bottom of the conversion factor. • May string conversion factors.  So we do not need to know every relationship, as long as we can find something else the starting and desired units are related to : desired unit start unit desired unit start unit related unit desired unit start unit desired unit start unit related unit www.ThesisScientist.comProblem Solving and Dimensional Analysis • Many problems in chemistry involve using relationships to convert one unit of measurement to another. • Conversion factors are relationships between two units.  May be exact or measured.  Both parts of the conversion factor have the same number of significant figures. • Conversion factors generated from equivalence statements. 2.54cm 1in  e.g., 1 inch = 2.54 cm can give or 2.54cm 1in www.ThesisScientist.comSystematic Approach 1. Write down the given amount and unit. 2. Write down what you want to find and unit. 3. Write down needed conversion factors or equations. a. Write down equivalence statements for each relationship. b. Change equivalence statements to conversion factors with starting unit on the bottom. www.ThesisScientist.comSystematic Approach, Continued 4. Design a solution map for the problem.  Order conversions to cancel previous units or arrange equation so the find amount is isolated. 5. Apply the steps in the solution map.  Check that units cancel properly.  Multiply terms across the top and divide by each bottom term. 6. Determine the number of significant figures to report and round. 7. Check the answer to see if it is reasonable.  Correct size and unit. www.ThesisScientist.comConversion Factors • Convert inches into centimeters. 1. Find relationship equivalence: 1 in = 2.54 cm 2. Write solution map. in cm 3. Change equivalence into conversion factors with starting units on the bottom. 2.54 cm 1 in www.ThesisScientist.comExample 2.8: • Convert 7.8 km to miles www.ThesisScientist.comExample: Convert 7.8 km to miles. • Write down the given quantity and its units. Given: 7.8 km www.ThesisScientist.comInformation Example: Given: 7.8 km Convert 7.8 km to miles. • Write down the quantity to find and/or its units. Find: miles www.ThesisScientist.comInformation Example: Given: 7.8 km Convert 7.8 km to miles. Find: mi • Collect needed conversion factors: 1 km = 0.6214 mile www.ThesisScientist.comInformation Example: Given: 7.8 km Convert 7.8 km to miles. Find: mi Conversion Factor: 1 km = 0.6214 mile • Write a solution map for converting the units: km mi 0.6214 mi 1 km www.ThesisScientist.comInformation Example: Given: 7.8 km 2 significant figures Convert 7.8 km to Find: mi miles. Conversion Factor:1 km = 0.6214 mile 0.6214 mi Solution Map: km  mi 1 km • Apply the solution map: 0.6214 mile 7.8 km mi 1 km = 4.84692 mi • Significant figures and round: 2 significant figures = 4.8 mi www.ThesisScientist.comPractice—Convert 30.0 g to Ounces (1 oz. = 28.35 g) www.ThesisScientist.comConvert 30.0 g to Ounces • Write down the Given Given: 30.0 g 3 sig figs quantity and its unit. • Write down the quantity Find: oz. you want to Find and unit. • Write down the appropriate Conversion 1 oz = 28.35 g Conversion Factors. Factor: g oz • Write a Solution Map. Solution 1 oz Map: 28.35 g • Follow the solution map to Solution: 1 oz 30.0 g 1.05820 oz Solve the problem. 28.35 g • Significant figures and Round: = 1.06 oz 3 sig figs round. • Check. Check: Units and magnitude are www.ThesisScientist.com correct.Example 2.10: • An Italian recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use www.ThesisScientist.comAn Italian recipe for making creamy pasta sauce calls for 0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use • Write down the given quantity and its units. Given: 0.75 L www.ThesisScientist.comAn Italian recipe for making Information creamy pasta sauce calls for Given: 0.75 L 0.75 L of cream. Your measuring cup measures only in cups. How many cups should you use • Write down the quantity to find and/or its units. Find: cups www.ThesisScientist.comAn Italian recipe for making Information creamy pasta sauce calls for Given: 0.75 L 0.75 L of cream. Your Find: cu measuring cup measures only in cups. How many cups should you use • Collect needed conversion factors: 4 cu = 1 qt 1.057 qt = 1 L www.ThesisScientist.comInformation An Italian recipe for making Given: 0.75 L creamy pasta sauce calls for Find: cu 0.75 L of cream. Your Conversion Factors: measuring cup measures only 4 cu = 1 qt; in cups. How many cups 1.057 qt = 1 L should you use • Write a solution map for converting the units: L qt cu 4 cu 1.057 qt 1 qt 1 L www.ThesisScientist.comInformation An Italian recipe for making 2 significant figures Given: 0.75 L creamy pasta sauce calls for Find: cu 0.75 L of cream. Your Conversion Factors: measuring cup measures only 4 cu = 1 qt; 1.057 qt = 1 L in cups. How many cups Solution Map: L  qt  cu should you use 1.057 qt 4 cu 1 qt 1 L • Apply the solution map: 1.057 qt 4 cu 0.75 L 1 L 1 qt = 3.171 cu • Significant figures and round: 2 significant figures = 3.2 cu www.ThesisScientist.comExample 2.12: 2 • A circle has an area of 2,659 cm . What is the area in square meters www.ThesisScientist.comExample: A circle has an area of 2 2,659 cm . What is the area in square meters • Write down the given quantity and its units. 2 Given: 2,659 cm www.ThesisScientist.comInformation Example: 2 Given: 2,659 cm A circle has an area of 2 2,659 cm . What is the area in square meters • Write down the quantity to find and/or its units. 2 Find: m www.ThesisScientist.comInformation Example: 2 Given: 2,659 cm A circle has an area of 2 2 Find: m 2,659 cm . What is the area in square meters • Collect needed conversion factors: 1 cm = 0.01m www.ThesisScientist.comInformation Example: 2 Given: 2,659 cm A circle has an area of 2 Find: m 2 2,659 cm . What is the Conversion Factor: area in square meters 1 cm = 0.01 m • Write a solution map for converting the units: 2 2 cm m 2 0.01 m   1 cm  www.ThesisScientist.comInformation Example: 2 Given: 2,659 cm 4 significant A circle has an area of 2 figures Find: m 2 2,659 cm . What is the Conversion Factor:1 cm = 0.01 m area in square meters 2 2 2 0.01 m  Solution Map: cm m  1 cm  • Apply the solution map: 4 2 110 m 2 2 2,659 cm m 2 1 cm 2 = 0.265900 m • Significant figures and round: 4 significant figures 2 = 0.2659 m www.ThesisScientist.com3 3 Practice—Convert 30.0 cm to ft 2 (1 cm = 1 x 10 m) (in class) www.ThesisScientist.com3 3 Convert 30.0 cm to m 3 1. Write down the Given Given: 30.0 cm 3 sig figs quantity and its unit. 3 2. Write down the quantity Find: m you want to Find and unit. 3. Write down the appropriate Conversion 3 (1 cm = 0.01 m) Conversion Factors. Factor: 3 3 4. Write a Solution Map. Solution cm m 3 Map: 0.01 m   1 cm  5. Follow the solution map to Solution: Solve the problem. 6 3 110 m 1 3 5 3 3.0010 cm 310 m 3 1 cm 3 −5 3 6. Significant figures and Round: 30.0 cm = 3.00 x 10 m round. 3 sig figs Units and magnitude are 7. Check. Check: www.ThesisScientist.com correct.Density • Inverse relationship between mass and volume. 3 • Solids = g/cm 3  1 cm = 1 mL Mass Density • Liquids = g/mL Volume • Gases = g/L • Volume of a solid can be determined by water displacement—Archimedes Principle. • Density : solids liquids gases  Except ice is less dense than liquid water www.ThesisScientist.comPlatinum has become a popular metal for fine jewelry and costs more than gold. A man gives a woman an engagement ring and tells her that it is made of platinum. Noting that the ring felt a little light, the woman decides to perform a test to determine the ring’s density before giving him an answer about marriage. She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds that the 3 ring displaces 0.556 cm of water. Is the ring made of platinum 3 (Density Pt = 21.4 g/cm ) www.ThesisScientist.comData: She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds that the 3 ring displaces 0.556 cm of water. Is the ring made 3 of platinum (Density Pt = 21.4 g/cm ) Given: Mass = 5.84 grams 3 Volume = 0.556 cm 3 Find: Density in grams/cm Equation: m  d V m, V d m Solution Map:  d V m and V d www.ThesisScientist.comApply the Solution Map: m  d V 5.84 g g  10.5 3 3 cm 0.556 cm 3 3 Since 10.5 g/cm 21.4 g/cm , the ring cannot be platinum. www.ThesisScientist.comPractice—What Is the Density of Metal if a 100.0 g Sample Added to a Cylinder of Water Causes the Water Level to Rise from 25.0 mL to 37.8 mL www.ThesisScientist.comFind Density of Metal if 100.0 g Displaces Water from 25.0 to 37.8 mL 1. Write down the Given Given: m =100.0 g 3 sig figs quantity and its unit. displaces 25.0 to 37.8 mL 3 Find: 2. Write down the quantity you d, g/cm want to Find and unit. m CF 3. Write down the appropriate 3 d  1 mL = 1 cm Equation: Conv. Factor and Equation. V 4. Write a Solution Map. Solution m, V d Map: m d  V 3 5. Follow the solution map to Solution: 1 cm 3 12.8 mL12.8 cm Solve the problem. V = 37.825.0 1 mL = 12.8 mL 100.0 g 3 d 7.8125 g/cm 3 12.8 cm 3 3 6. Significant figures and round. Round: 7.8125 g/cm = 7.81 g/cm 3 significant figures Units and magnitude are 7. Check. www.TCheck hesisScien :tist.com correct.Density as a Conversion Factor • Can use density as a conversion factor between mass and volume Density of H O = 1 g/mL \ 1 g H O = 1 mL H O 2 2 2 3 3 Density of Pb = 11.3 g/cm\ 11.3 g Pb = 1 cm Pb 3 • How much does 4.0 cm of lead weigh 11.3 g Pb 3 4.0 cm Pb 45 g Pb x = 3 1 cm Pb www.ThesisScientist.comMeasurement and Problem Solving: Density as a Conversion Factor • The gasoline in an automobile gas tank has a mass of 60.0 kg 3 and a density of 0.752 g/cm . What is the volume • Given: 60.0 kg Solution Map: 3 • Find: Volume in cm 3 kg g cm • Conversion factors: 3 1000 g 1 cm 3  0.752 g/cm 1 kg 0.752 g  1000 grams = 1 kg www.ThesisScientist.comMeasurement and Problem Solving: Density as a Conversion Factor, Continued Solution Map: 3 kg g cm 3 1000 g 1 cm 1 kg 0.752 g 3 1000 g 1 cm 4 3 60.0 kg 7.9810 cm 1 kg 0.752 g www.ThesisScientist.com
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