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Measurement in Chemistry Factor-Label Method The Factor-Label Method At the conclusion of our time together, you should be able to: 1. Recognize a problem that can be solved with the factor label method 2. Transform a statement of equality into a conversion factor 3. Use the appropriate conversion factor in the correct way so that the labels cancel and the correct conversion is found The Factor label Method A way to solve math problems in chemistry Used to convert km to miles, m to km, mol to g, g to mol, etc. To use this we need: 1) desired quantity 2) given quantity 3) conversion factors Conversion factors are valid relationships or equalities expressed as a fraction and equal to one! Equalities State the same measurement in two different units length 10.0 in. 25.4 cm Conversion Factors Fractions in which the numerator and denominator are EQUAL quantities expressed in different units but always equal to one. You can always multiply any equation by this equality and not change the quantity, just the units. Example: Factors: 1 in. = 2.54 cm 1in. 2.54 cm and 2.54 cm 1 in. For example: 1 km = 0.6 miles the conversion factor is 1 km 0.6 miles or 0.6 miles 1 km Write conversion factors for 1 foot = 12 inches What conversion factors can you think of that involve meters? Conversion Factors Conversion factors for 1 ft = 12 in 1 foot 12 inches or 12 inches 1 foot There are almost an infinite number of conversion factors that include meters: 1000 m 1m 1m , , 1 km 100 cm 1000 mm 1m 1m 0.9144 yards , , 3.28 feet 39.37 inches 1m The Steps to Follow Now we are ready to solve problems using the factor label method. The steps involved are: 1. 2. 3. 4. 5. Write the desired quantity and = Write down the given quantity and put it over 1 Determine what conversion factors you will use to turn the given label into the needed label. Multiply the given quantity by the appropriate conversion factors to eliminate units you don’t want and leave the units you do want Complete the math Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km First write down the desired quantity Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 mi Next, equate desired quantity to the given quantity Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 mi Now we have to choose a conversion factor Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 mi 1 km 0.621 mi 0.621 mi 1 km What conversion factors are possible? Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 mi 1 km 0.621 mi 0.621 mi 1 km Pick the one that will allow you to cancel out miles Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 mi 1 km 0.621 mi 0.621 mi 1 km Pick the one that will allow you to cancel out miles Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 mi 1 km 0.621 mi 0.621 mi 1 km Multiply given quantity by chosen conversion factor Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 mi x 1 km 0.621 mi Multiply given quantity by chosen conversion factor Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 mi x 1 km 0.621 mi Cross out common factors Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 x 1 km 0.621 Cross out common factors Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 x 1 km 0.621 Are the units now correct? Yes – km on both sides! Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 x 1 km 0.621 = 75.68438003 km Now finish the math. The Steps to Follow Now we are ready to solve problems using the factor label method. The steps involved are: 1. 2. 3. 4. Complete the math with no rounding Make certain the sig figs are correct by rounding to the correct number of sig figs at the very end Don’t forget the order of operations when you complete the math Conversion factors do not determine sig. figs.! Factor label Example How many kilometers are in 47.0 miles? (note: 1 km = 0.621 miles) # km = 47.0 x 1 km 0.621 = 75.7 km The final answer is 75.7 km Summary The previous problem was not that hard In other words, you probably could have done it faster using a different method However, for harder problems the factor label method is easiest More Examples 1. You want to convert 100.00 U.S. dollars to Canadian dollars. If the exchange rate is 1 Can$ = 0.65 US$, how much will it cost? # Can$ = 100.00 US$x 1 Can$ = 153.85 Can$ 0.65 US$ The Factor-Label Method Let’s see if you can: 1. Recognize a problem that can be solved with the factor label method 2. Transform a statement of equality into a conversion factor 3. Use the appropriate conversion factor in the correct way so that the labels cancel and the correct conversion is found Learning Check Write conversion factors that relate each of the following pairs of units: 1. Liters and mL 1 Liter = 1000 mL 2. hours and minutes 1 hour = 60 minutes 3. meters and kilometers 1000 meters = 1 kilometer How many minutes are in 2.5 hours? Conversion factor 2.5 hr x 60 min 1 hr = 150 min By using dimensional analysis/factor-label method, the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the numbers! Learning Check You have $7.25 in your pocket in quarters. How many quarters do you have? 7.25 dollars X 4 quarters 1 dollar = 29 quarters Measurement in Chemistry Factor-Label Method Part 2 The Factor-Label Method At the conclusion of our time together, you should be able to: 1. Recognize a problem that can be solved by moving the decimal point. 2. Use the appropriate conversion factor in the correct way so that the labels cancel and the correct conversion is found with two changes of labels or labels that are squared or cubed. Dealing with Two Units Convert 55.00 km/h to m/s 55.00 km x 1000 m x 1 h___ = h 1 km 3600 s 15.28 m/s A patient requires injection of 0.012 g of a pain killer available in a 15 mg/mL solution. How many milliliters should be administered? When you see a number with two units like 15 mg/mL, it can be used as a conversion factor. What it really says is that 1 ml of the solution contains 15 mg of the drug. ? mL = 0.012 g of drug 0.012 g drug mg drug mL soln 3 mg drug 10 1 mL soln ? mL = 0.012 g of drug 1 g drug 15 mg drug ( = 0.80 mL soln )( ) Dealing with Two Units, Your Turn If your pace on a treadmill is 65 meters per minute, how many seconds will it take for you to walk a distance of 8450 feet? 1 meter = 3.28 feet # s = 8450 ft 2380 seconds x1m 3.28 ft x 1 min 65 m x 60 s 1 min What about Square and Cubic units? Use the conversion factors you already know, but when you square or cube the unit, don’t forget to cube the number also! Best way: Square or cube the Entire conversion factor Example: Convert 4.3 cm3 to mm3 ( ) 4.3 cm3 10 mm 1 cm 3 = 4.3 cm3 103 mm3 13 cm3 = 4300 mm3 Learning Check A Nalgene water bottle holds 1000 cm3 of dihydrogen monoxide (DHMO). How many cubic decimeters is that? Solution 1000 cm3 ( ) 1 dm 10 cm 3 = 1 dm3 So, a dm3 is the same as a Liter! A cm3 is the same as a milliliter. Converting Metric to Metric A rattlesnake is 2.44 m long. How long is the snake in cm? a) 2440 cm b) 244 cm c) 24.4 cm Solution A rattlesnake is 2.44 m long. How long is the snake in cm? b) 244 cm 2.44 m x 100 cm 1m = 244 cm Converting Units of Length Made Easy 0.5 kilometer (km) = 500 meters (m) 2.5 meter (m) = 250 centimeters (cm) 1 centimeter (cm) = 10 millimeter (mm) 1 nanometer (nm) = 1.0 x 10-9 meter O—H distance = 9.4 x 10-11 m 9.4 x 10-9 cm 0.094 nm An Easier Way A rattlesnake is 2.44 m long. How long is the snake in cm? G _ _ M _ _ k h da _ d c m _ _ μ _ _ n 1. A move from 1 meter to centimeters is two places right 2. Move the decimal place of the number two places right 3. 244 cm Another Example: How many millimeters are there in 4.5 cm? G _ _ M _ _ k h da _ d c m _ _ μ _ _ n 1. A move from cm to mm is one place right 2. Move the decimal place of the number one place right 3. 45 mm Another Example: How many kilometers are there in 4.5 cm? G _ _ M _ _ k h da _ d c m _ _ μ _ _ n 1. A move from cm to km is five places left 2. Move the decimal place of the number five places left 3. 0.000 045 km The Factor-Label Method Let’s see if you can: 1. Recognize a problem that can be solved by moving the decimal point. 2. Use the appropriate conversion factor in the correct way so that the labels cancel and the correct conversion is found with two changes of labels or labels that are squared or cubed. Learning Check: 2 kilometers is the same as how many millimeters G _ _ M _ _ k h da _ d c m _ _ μ _ _ n 1. A move from km to mm is six places right 2. Move the decimal place of the number six places right 3. 2 000 000 mm, 2 x 106 mm Metric Conversions #1: Write 550 mm as meters. G _ _ M _ _ k h da _ d c m _ _ μ _ _ n 1. A move from mm to m is 3 places left 2. Move the decimal place of the number 3 places left 3. 0.55 m Learning Check A person’s blood contains 185 mg of cholesterol per deciliter of blood. How many grams of cholesterol are there in 1 liter of this blood? A. 0.0185 g B. 0.185 g C. 1.85 g D. 18.5 g E. 1850 g English and Metric Conversions If you know ONE conversion for each type of measurement, you can convert anything! You must use these conversions: Mass: 454 grams = 1 pound Length: 2.54 cm = 1 inch Volume: 0.946 L = 1 quart Learning Check An adult human has 4.65 L of blood. How many gallons of blood is that? Unit plan: L Equalities: qt gallon 1 quart = 0.946 L 1 gallon = 4 quarts Your Setup: gal = 4.65 L x 1 quart x 1 gallon 1 0.946 L 4 quarts = 1.23 gallons Exit Quiz There are 12 inches in a foot, 0.394 inches in a centimeter, and 3 feet in a yard. How many centimeters are in 1.000 yard? # cm = 1 yd x 3 ft x 12 in x 1 cm = 91.37 cm 1 yd 1 ft 0.394 in Exit Quiz #6 on WS Change 9.4 miles to km (1 mile = 1.6 km) # km = 9.4 mi x 1.6 km 1 mi = 15 km Exit Quiz With a U.S. dollar you can buy 1.1 Euros, 130 Yen, or 25 Rubles. How many Yen can you buy with one Ruble? # Yen = 1 Ruble x 1 US $ x 130 Yen = 5.2 Yen 25 Rubles 1 US $ Exit Quiz Calculate how many feet are in 1 meter. (use 1 cm = 0.394 in) # ft= 1 mx 100 cm x 0.394 in x 1 ft = 3.28 ft 1m 1 cm 12 in