Name Section Date Dimensional Analysis Solving chemistry problems that involve dimensional analysis can seem complex and confusing. But, when broken down into smaller parts, these problems make me sense. Dimensional analysis is imptant to understand f many reasons other than chemistry. Many of the products we buy are made in countries that use the metric system. F example, suppose that we buy an entertainment center online. The entertainment center arrives from England and requires assembly. The instructions specify inserting screws at certain distances from the edges. All of the measurements are in centis s. But we are used to making measurements in inches yards. How would we make these conversions? Dimensional analysis is the key. Multiplying Fractions When multiplying fractions, multiply all of the numbers in the numerats first, followed by multiplying all of the numbers in the denominats. The last step is to divide the numerat product by the denominat product. Example What is the product of 3 and 5 2? The expression could be written in one of two ways: 3 2 5 2 3 5 We then obtain the result as follows: 2 2 2 = = = 0.33 35 35 5 Whenever the same number occurs in both the numerat and denominat, they cancel. 3 = = 0.20 3 5 5 If the number we are analyzing is a whole number, remember that this really means that the number is over (the whole number in the numerat and the number in the denominat). F example, the number 4 really means 4. We do not actually write these numbers as fractions, but it is imptant to remember that the whole number acts like it is a numerat in the product. Wld of Chemistry MC6 Copyright Houghton Mifflin Company.
Example 2 What is 4 of 5? Remember, the number 5 really means 5. 5 5 5 = = =.25 4 4 4 We also can multiply several fractions together at once using the same guidelines. We can cancel out numbers to simplify the expression (but it is not necessary). Practice Problem Set. What is the product of 4 7 and 2 3? 4 6 5 2465 240 2 = = = 0.29 3 5 7 8 3578 840 2. What is 4 0 of 60? 3. Evaluate the following expressions: 2 a. = 352 9 3 7 = 405 b. 3 c. = 2 Multiplying Units When multiplying units, use the same principle that you use f multiplying fractions. If one unit is in the numerat, and the identical unit is in the denominat, they cancel each other out. Any remaining units are used in the answer. centi centi = = You also can multiply several units together at once using the same principle as f fractions containing numbers. Wld of Chemistry MC7 Copyright Houghton Mifflin Company.
centi kilo second centi mega kilo = mega second It is very imptant to note that if a unit appears once in the numerat but me than once in the denominat, we can only cancel one of the unit expressions in the denominat. Think of this concept in terms of fractions. If the number 4 is in the numerat, and two 4s are in the denominat of different fractions, we only cancel out one of the 4s on the bottom, not both. 4 3 3 3 = = = 0.5 5 4 4 5 4 20 Example 3 Evaluate the unit expression below: 2 2 kilogram second = second kilog ram The squared fact is equivalent to multiplying the fraction by itself. kilog ram kilog ram second = second second ki log ram ki log ram Now we can evaluate the expression by canceling out units. kilog ram kilog ram second second Practice Problem Set 2 second kilog ram kilogram = second. Evaluate the following unit expressions: grams milligrams kilog ram = ki log ram gram a. b. mole grams liter mole gram = c. ml cm 3 mg = ml cm 2 2 kg g m d. = m kg g Wld of Chemistry MC8 Copyright Houghton Mifflin Company.
e. g ft ml = ft 2 ml g 2. Fill in the missing unit expression in each problem below: pint? liter a. cup = liter cups? gallon b. moles liter? moles = liter gram? second Perfming ConversionsDimensional Analysis Now that we know how to evaluate number and unit expressions separately, we just combine the two steps to do conversions! We use the same principles as in the previous sections, just make sure that the final answer contains both a number and a unit. Example 4 One side of a backyard fence measures 32 inches in length. How many feet does this represent? ( ft = 2 in.) To convert from inches to feet, we are going to have to use the conversion fact given in the problem. We have to understand one key point about conversion facts. When the facts are given to us, we can write them as a ratio to help us solve our problem. The equivalence statement ft = 2 in. can be written two different ways: ft 2 in. 2 in. ft This does not change the meaning of the conversion fact. There are always 2 inches in foot, which is expressed in both ratios above. We set up our expression so that our units cancel, and we are left with feet (since this is what the problem is asking f). 32 in. ft 32 ft = = 26.0 ft 2 in. 2 Wld of Chemistry MC9 Copyright Houghton Mifflin Company.
Example 5 A golfer putted a golf ball 7.8 ft across a green. How many inches does this represent? We use the equivalence statement as we did in Example 4, except we use the ratio that causes our units to cancel so that we are left with inches. 2 in. 7.82 in. (7.8 ft ) = = 94 in. ft Opposite ratio than in Example 4 Example 6 How many centis are in 5.0 inches? (2.54 cm = in.) Again, to convert from inches to centis, we are going to have to use the conversion fact given to us in the problem. The equivalence statement 2.54 cm = in. can be written two different ways: 2.54 cm in. in. 2.54 cm We set up our expression so that our units cancel, and we are left with centis. 5.0 in 2.54 cm in 5.02.54 cm = = 38. cm Sometimes perfming conversions requires us to use me than one conversion fact. We can solve these problems as separate steps all in one step. Example 7 An iron sample has a mass of 3.50 lb. What is the mass of this sample in grams? (2.2046 lb = kg) ( kg = 000 g) The conversion facts can be written in the following ways: 2.2046 lb kg kg 2.2046 lb 000 g kg kg 000 g We can first convert our mass sample from pounds to kilograms and then perfm a second step to convert from kilograms to grams. Step : 3.50 lb kg 3.50 kg = =.58 kg 2.2046 lb 2.2046 Wld of Chemistry MC20 Copyright Houghton Mifflin Company.
Step 2:.58 kg 000 g.58000 g = = 580 g kg We also could solve this problem in one step by setting up an expression such that all of the units cancel except f grams. 3.50 lb Practice Problem Set 3 kg 000 g 2.2046 lb kg 3.50000 g = = 580 g 2.2046. A dining room table measures 6.0 ft in length. How many inches does this represent? ( ft = 2 in.) 2. How many cups are in a 64-oz. pitcher of lemonade? (8 fluid ounces = cup) 3. Perfm the following conversions: Useful conversion facts: kg = 000 g g = 000 mg ft = 2 in. in = 2.54 cm km = 000 m hr = 3600 s mi =.6093 km a. 0.28 kg = g = mg b..5 ft = in. = cm c. 6.00 m/s = km/hr = mi/hr 4. Baking soda and vinegar are mixed in a balloon. A gas is produced and the balloon expands to a volume of 2.00 L. What is the volume of the balloon in cm 3? ( L = dm 3 ; dm = 0 cm) Wld of Chemistry MC2 Copyright Houghton Mifflin Company.