Chapter 2 Dimensional Analysis

Transcription

1 Chapter 2 Dimensional Analysis 1. Introduction There are many methods one can use to solve mathematical problems in chemistry. Sometimes called "story problems", to the novice they can seem difficult, if not impossible! But with a good "plan of attack" they can be made "child's play". Let's look at four methods of problem solving. 1. Pure Logic This is the most difficult method and definitely not for beginners. By this method you reason out what to do with the numbers according to your knowledge about the quantities involved. 2. Formulas This method is commonly taught in math classes. One memorizes a formula for each type of problem and "plugs in" values to get the answer. Unfortunately there would be too many formulas to memorize to use this method in chemistry. 3. Proportions Some older chemistry books use this method. Proportions can be set up for many of the quantities we study, but they can be confusing to apply. 4. Dimensional Analysis This is by far the most common method used today and will be our primary way of "attacking" chemistry problems throughout the year. In dimensional analysis (sometimes called the factor-label method) you will learn to custom build a formula for each problem as you solve it. 2. Dimensional Analysis Suppose you visited a village deep in Africa and found that instead of money the people traded in various types of animals. You assemble the following trading table: 4 pigs = 9 ducks 1 goat = 3 ducks 2 pigs = 5 chickens Problem How many chickens could you trade for? Solution One could use pure logic but it would be easier with dimensional analysis! If you wish, you could try to figure out the answer to this problem by pure logic; then turn to the net page to see how you can do it using dimensional Dimensional Analysis Chapter 2 1

2 analysis. Two Mathematical Ideas to Apply First ---- If the top of a fraction equals the bottom of a fraction, the value of the fraction is "one". Second ---- If the letter on top of one fraction is the same as the letter on the bottom of another fraction, they cancel. 7 7 = 1 Eamples Each equals "one" if = 3 y = = 1 3y - 2 The "B" c ancels 2 B C 3 A 5 B In dimensional analysis we write each equivalency (such as 4 pigs = 9 ducks) as a fraction. We treat the units (pigs, goats, etc.) as algebraic letters allowing them to cancel. The parts of the problem therefore fit together like pieces of a puzzle. The Problem The Table: How many chickens could you trade for? 4 pigs = 9 ducks 1 goat = 3 ducks 2 pigs = 5 chickens Step 1 We start by listing the given information at the left edge of the page and the unit of what we want to find at the right edge of the page. Thus our task will be to go from goats to chickens. GIVEN FIND chic kens Step 2 We multiply by the fraction, "3 ducks over 1 goat". This is based on one of the equivalencies in our table. Goats are placed on the bottom of the fraction so that they will cancel with the goats given in the problem. Since the fraction we are multiplying by has a value of one, the answer so far (18 ducks) is equal to the original. 2 Chapter 2 Dimensional Analysis 3 ducks = 18 ducks 1 goat Step 3 Similarly, we multiply by another fraction, "4 pigs over 9 ducks". This cancels out the "ducks" and introduces "pigs". 3 ducks 1 goat 4 pigs 9 ducks Step 4 Then we multiply by another fraction, "5 chickens over 2 pigs". This cancels out the "pigs" and introduces "chickens". 3 ducks 4 pigs 5 chickens = chickens 1 goat 9 ducks 2 pigs Each time we have multiplied by a fraction equal to "1" so that the answer will be equal to the we started out with. Each time one unit cancels and another is introduced, until the unit becomes what we are looking for, "chickens". On the Calculator press the keys in the following order: = 20 We have custom built a formula to solve the problem. Writing fractions and canceling

3 units has forced us to arrange the numbers in a particular fashion. Multiplying and dividing the numbers as they are arranged gives the answer: 20 chickens. Use this epanded table to work out the problems below. Choose only the equivalencies you need to write fractions which will cancel units to bring you to the unit of the answer. Problems 4 pigs = 9 ducks 9 goats = 2 horse 1 goat = 3 ducks 5 ducks = 3 geese 2 pigs = 5 chickens 1 cow = 12 geese 20 rabbits = 1 pig 5 horses = 3 bulls 3 goats = 2 sheep 1 bull = 32 pigeons 1. How many goats could be traded for 18 geese? 2. Eight sheep would be worth how many pigs? 3. How many rabbits could you get for 2 horses? 3. Converting Measurements One useful application of dimensional analysis is converting measurements in either the English or Metric system. Most of us can do simple one step conversions, like feet to inches, by pure logic. But when several steps are involved, such as to convert miles to inches, use dimensional analysis. Sample Problem 2 How many meters are in 2.5 yards? Solution To solve this problem we will need to recall certain equivalencies which should be familiar to you. A partial list of these equivalences are on the net page. A more complete list is contained in the appendi of this book. It is best to decide a path through the problem by picking out the appropriate equivalencies in the appropriate order before starting to set up the problem. Since we are starting with yards and have to convert to meters these equivalencies are: 1 yard = 3 feet, 1 foot = ches, 1 inch = 2.54 centimeters, and 1 meter = 100 centimeters. Step 1 Convert the 2.5 yards to feet by multiplying by "3 feet over 1 yard". Hold off on the actual multiplication until the whole problem is set up. 2.5 yd 3 ft 1 yd Step 2 Convert feet to inches by multiplying by "ches over 1 foot". 2.5 yd 3 ft 1 yd 1 ft Step 3 Convert inches to centimeters by multiplying by "2.54 centimeters over 1 inch". 2.5 yd 3 ft 1 yd 1 ft 2.54 cm Step 4 Convert centimeters to meters by multiplying by "1 meter over 100 centimeters". This process has forced us to arrange the numbers in a particular fashion to custom build a formula 1 in Dimensional Analysis Chapter 2 3

4 for this problem. At this point you can use your calculator to get the answer: 2.3 meters. 2.5 yd 3 ft 1 yd 1 ft 2.54 cm 1 in 1 m 100 cm = m English and Metric Conversion Table ches = 1 foot 1 kilometer = 1000 meters 3 feet = 1 yard 1 hectometer = 100 meters 16.5 feet = 1 rod 1 dekameter = 10 meters 5280 feet = 1 mile 10 decimeters = 1 meter 40 rods = 1 furlong 100 centimeters = 1 meter 6 feet = 1 fathom 1000 millimeters = 1 meter 1 gallon = 231 inch 3 1 milliliter = 1 centimeter 3 16 ounces = 1 pint 2 pints = 1 quart 1000 milliliters = 1 liter 4 quarts = 1 gallon English to metric conversion: 2.54 centimeters = 1 inch *For the most part, the equivalencies in this table are eact and therefore will not affect the number of significant figures in the answer to the problems in this chapter. The eceptions are marked with a. Problems Use only the equivalencies listed in the above table. 4. How many rods will equal 247 inches? 5. How many centimeters in 34.2 decimeters? 6. A horse race is 55.0 furlongs. How many yards is that? 7. How many ounces are in 1.00 gallon? 8. A "Five K Race" is 5.00 kilometers long. How many miles is that? metric abbreviations km kilomenter hm hectometers dam dekameters m meters dm decimeters cm centimeters mm millimeters ml milliliters L liters Square units are squares with one unit along each side. Thus 1 square foot equals 1 foot times 1 foot. Cubic units are cubes of one unit along each of the three dimensions. Thus 1 cubic foot equals 1 foot times 1 foot times 1 foot. Sample Problem 3 How many square feet in 238 square inches? Solution Write square feet as ft 2 and write square inches as in 2. Remember that units must cancel and that, from algebra, an "X 2 " on top will cancel with two "X's" on the bottom. Therefore: Step 1 Start with 238 square inches. 238 in 2 Step 2 Multiply by "1 foot over ches".? 4 Chapter 2 Dimensional Analysis

5 238 in 2 1 ft Step 3 Multiply by "1 foot over ches", again. 238 in 2 1 ft 1 ft = ft 2 Notice that the same co nversion facto r is u sed twice to cancel th e sq uare u nits. Again we have custom build a formula for the problem. Multiplying and dividing the numbers as they are arranged ( ) gives us the answer: 1.65 square feet. Sample Problem 4 How many square feet are in 3.25 square meters? Solution Remember to plan a path through the problem, before starting to write the set up. We will start with meters, convert to centimeters, then to inches, then to feet multiplying by each conversion factor twice since we are working with square units m cm 100 cm 1 in 1 in 1 ft 1 ft 1 m 1 m 2.54 cm 2.54 cm = 35.0 ft 2 Sample Problem 5 Convert 56.0 cubic inches to cubic centimeters. Solution Use the conversion factor, "2.54 centimeters over 1 inch", three times in cm 2.54 cm 2.54 cm 918 cm 3 = 1 in 1 in 1 in Mied Problems Use only the equivalencies in the table earlier in this chapter. Some of the problems use the same conversion factor more than once, some do not. 9. An area on the ocean bottom is charted as being fathoms 2 in size. How many yards 2 is that? square centimeters is how many square dekameters? 11. How many fathoms are in 63 dekameters? 12. A bo has a volume of 4.6 ft 3. How many gallons does it contain? 13. A cup of coffee is measured to have a volume of 618 milliliters. What is the volume in cubic decimeters? 14. How many rods in 26 miles? 15. Using the defined equality "640 acres = 1 square mile", how many square yards are in 1.00 acre? 16. Hiking trails are often marked on maps in rods. How many kilometers long Dimensional Analysis Chapter 2 5

6 is a trail marked as 82 rods on the map? 17. Coke and Pepsi are now sold in 2.00 liter bottles. What is this volume in quarts? 18. A farm covers a area of two-tenths (0.200) of a square mile. What is the area in square feet? 19. How many liters are contained in a bo measuring 1.2 meters by 2.5 meters by 1.8 meters? (Hint: Find the volume of the bo in cubic meters first. ) 20. What is the volume of an ounce cup in milliliters? 21. How many ounces are in 3.9 in 3? 22. How many hectometers are in 56 furlongs? JPF 9/13/ Chapter 2 Dimensional Analysis