Significant Figures, Measurement, and Calculations in Chemistry

Transcription

1 Instructional Improvement Program Significant Figures, Measurement, and Calculations in Chemistry Carl Hoeger, Ph.D. University of California, San Diego SigFig 2 1

2 Part 2: Calculations in Science: Dimensional Analysis Carl Hoeger, Ph.D. University of California, San Diego SigFig 3 Dimensional Analysis Also known as the Factor-Label Method of problem solving. A quantity in one unit is converted into an equivalent quantity in a different unit. A way to analyze and solve problems by using the units (or dimensions) of the measurement; based on the (correct) assumption that if the units of your answer are right, chances are good that your answer is as well. You will use the units to solve the problem, doing the actual math at the end. Requires the use of conversion factors: equivalence statements that allow us to convert units of one type to another. SigFig-4 2

3 Conversion Factors Derived from equivalence statements: an equality relating one quantity to another. Write in equation form if not already. One foot is 12 inches; there are 15,125 bolts in a Saturn Vue; a mole of carbon atoms weighs grams 1ft = 12 in 15,125 screws = 1 Saturn Vue 1 mol C = g Most useful when expressed as an equivalence ratio. 1ft 12 in 1 mol C g or ; or 12 in 1ft g 1 mol C When 1 by itself appears in an equivalence statement or ratio, that 1 is an exact number. SigFig-5 Equivalence Ratios Let s examine how an equivalence statement is converted into an equivalence ratio: 1 atm = kpa 1 atm = kpa kpa kpa = 1 or 1 atm = kpa 1 atm 1 atm = kpa = 1 1 atm SO, Conversion factors are just a creative way to express 1! SigFig-6 3

4 Conversion Factors (cont) For unit interconversions in the same measurement system (i.e. metric to metric or English to English) conversion factors are defined quantities and therefore have unlimited significant figures. 2 nm = 2 x 10 7 cm; 16 oz = 1 lb For unit interconversions between different systems (i.e. metric to English) cf s are measured values and DO have sig fig limits! EXCEPTION: 1 in = 2.54 cm; this is an EXACT conversion (only one!) Equivalence statements always have the following relationship: big # of a small unit = small # of a big unit 1000 mm = 1 m; x atoms = 1 mol SigFig-7 Conversion Factors (cont) The units in a conversion factor can be treated the same way you normally treat numbers: they can be squared, rooted, canceled with identical units, etc cm = (2.54)3 cm cm 3 Remember to 1 in (1) 3 in 3 = change your 1 in 3 numerical portion accordingly! Note that ONLY the units cancel! 15 in cm3 3 = 246 cm 1 in 3 Thus, conversion factors allow us to convert one measurement in a given set of units into another. SigFig-8 4

5 Conversion Factors (cont) Conversion factors can also be derived from physical properties, chemical measurements, or constants: d Hg = g cm 3 ; heat of vaporization of Hg = kj mol 1 h = J s Conversion factors derived from physical properties or chemical measurements are considered measured quantities and therefore have significant figures limits. It is therefore important to use most precise value you can find when using them and apply sig fig rules as needed: d Hg = g cm 3 (4sf ) vs. d Hg = 13.6 g cm 3 (3sf ) Constants also have significant figure limits, but most constants have values with such high precision that it is rarely necessary to invoke sig fig rules: h = J s, commonly used as just J s SigFig-9 Conversion Factors and Dimensional Analysis: How To Begin by creating a conversion path. What do we have; what do we need; what do we know? Determine what equivalence ratios (conversion factors) are needed. Put conversion string together, adjust and cancel units as called for. Put in actual numerical values. Do the math CAREFULLY! Does the answer make sense? SigFig-10 5

6 DA Example #1 A ruler is 12.0 inches long. How long is it in meters? (1 inch = 2.54 cm) Set up a conversion path, putting needed equivalence statements in it: Put conversion string together: in? cm m 1 in = 2.54 cm 100 cm= 1 m 12 in 2.54 cm 1 in 1 m 100 cm = m = 0.30 m (2sf ) Answer has 2 sf; all conversions here are exact SigFig-11 DA Example #2 Jules Verne wrote a book 20,000 leagues under the sea. How far is this in feet? Much more difficult; need a lot of uncommon conversions : 1 league = 3 nautical miles; 1 nautical mile = 10 cable lengths; 1 cable length =100 fathoms; 6 ft = 1 fathom Set up a conversion path, putting needed equivalence ratios in it: league? feet Put conversion string together: 1:3 1:10 nautical mile 6:1 fathom 100:1 cable length SigFig-12 6

7 3 nautical mi leagues 1 league 10 cable lengths 1 nautical mi 100 fathoms 6 ft 1 cable lengths 1 fathom = ft = ft (1sf ) SigFig-13 DA Example #3 A more common problem seen in chemistry is illustrated here: A molar solution of HCl (hydrochloric acid) has a density of 1.17; how many ml do you need to measure out to ensure that the volume you have contains 7.0 g of HCl? Here you are actually faced with TWO problems: a value with no given units (1.17) and too much information (0.032 molar)! Approach: what do you need? Volume of HCl that contains 7.0 g of HCl what do you know? Density of solution = 1.17 g/ml (!); molar =?? Set up a conversion path, starting this time with what you need: g HCl Put conversion string together: 1.17 g/ml ml HCl 1 ml HCl soln 7.0 g HCl needed 1.17 g HCl = 5.98 ml = 6.0 ml (2sf ) SigFigs limited by starting amt SigFig-14 7

8 Multistep Problems: More Complex DA Usually involves complex units: values naturally expressed as a ratio, where units may be understood : Gas mileage; speed; density; molar mass STRATEGY: Solve problems by breaking the solution into steps. Convert complex units, using dimensional analysis. Many complex tasks in daily life are handled by breaking them down into manageable parts; Consider steps in cleaning a car: I. vacuum the inside II. wash the exterior III. dry the exterior IV. apply a coat of wax SigFig-15 Multistep Problem Example You are driving your sports car to Tahoe at an average speed of 68 mph. After 250. minutes you decide to stop for lunch. How many kilometers have you driven in this time and how much money have you spent on gas to drive this distance? Your car gets 32 mpg and gas costs 82 cents per liter. There may be more than one way to approach problems like these. Here is one solution path for this problem: 1. Determine time in hours spent driving; 2. Determine miles driven; 3. Determine kilometers driven; 4. Determine money spent. Unusual Conversions needed: 1 mi = 5280 ft; 1 in = 2.54 cm; 1 gal = L SigFig-16 8

9 1. Convert minutes to hours: Multistep Solution min 1 h 60 min = 4.17 h (3 sf ) 2. Determine miles driven: 4.17 h 68 mi 1 h = 284 mi (3 sf ; only 2 sf allowed) SigFig Convert miles to kilometers: Multistep Solution ft 284 mi 12 in 2.54 cm 1 mi 1 ft 1 in 1 m 100 cm 1 km 1000 m = 457 km = 460 km (2 sf ) 4. Determine money spent: 284 mi 1 gal 32 mi L 1 gal $ L = $27.62 = $28 (2 sf ) SigFig-18 9

10 Summary Errors occur during scientific measurements, regardless of how careful one is! Significant Figures allow us to convey the precision of our measurements to our audience. Units are as important as the numerical values are. Equivalence ratios (conversion factors) allow us to relate a value in one unit set to another. Dimensional Analysis is a way of using one or more equivalence ratio(s) to solve problems. SigFig-19 Questions? Comments? This presentation has been brought to you through the generous support of the University of California, San Diego s Instructional Improvement Program. If you have questions, comments, or wish to use these presentations in your University s courses, please feel free to send an to Dr. Carl Hoeger at chemcook@gmail.com This is episode Dimensional Analysis ; if you have specific comments or questions regarding this episode please note this in your SigFig-20 10