Dimensional Analysis Applying Mathematics to Chemistry Dimensional Analysis or Factor Labeling Method refers to the process of evaluating numbers and units, or dimensions, in solving a problem in order to solve problems in chemistry, both numbers and dimensions need to be considered units can be treated like numbers in calculations therefore mathematical operations (division, multiplication, cancellation) can be performed on units “Equivalence statement” or “cancelling out” e.g. n/n = 1 3PV divided by 3P equals V or: 3PV/3P = V Solve: (5 m)(2 cm)/(2 m) = Solve: (2L)(4g) /(2g) = Solve: (10L)(6mL)/(5mL) =

Dimensional Analysis Applying Mathematics to Chemistry Dimensional Analysis can be used to evaluate numbers and formula using the Conversion Factor 1 m = 100 cm Conversion factor: Divide each side by 100 cm 1 m/100 cm = 100 cm/100 cm = 1 (conversion factor or unity factor) How many meters = 1000 cm? 1000 cm x (1 m/100 cm) = (1000 cm x 1 m)/(100 cm) = 10 m

Dimensional Analysis Applying Mathematics to Chemistry Conversion factors 1 hr = 60 min divide each side by 60 min 1 hr/60 min = 60 min/60 min = 1 conversion factor is a unity factor numerator and denominator contain equal quantities, expressed in different units (hr vs. min) Example Question: how many hours = 90 minutes? Answer Use unity factor: 1 hr/60 min = 1 - 90 min x (1 hr/60 min) = (90 min)x(1 hr)/(60 min) = 90/60 x 1 hr = 1.5 hr

Dimensional Analysis Applying Mathematics to Chemistry Question - Is the answer reasonable? Answer - Yes: 90 min is longer than 1 hr (= 60 min), but less than 2 hrs (= 120 min) Choice of conversion factors 1 hr = 60 min divide each side by 1 hr 1 hr/1 hr = 60 min/1 hr = 1 Alternative conversion factor - 90 min x (60 min/1 hr) = (90 min)x(60 min)/(1 hr) = 90x60 min2/1 hr = 5400 min2/hr Does this look like a reasonable answer?

Dimensional Analysis Applying Mathematics to Chemistry Step 1 What is the problem asking read the problem carefully look up any terms that are unclear if an equation is given, make sure you understand it look for clues: determine, calculate, what mass, what volume know the units associated with the quantities in the problem Step 2 Determine the relationship between the information in the problem and the desired answer 60 min = 1 hr: not given in the problem remember relationships or look them up in a textbook or manual critically examine the problem and use only information that you need Make sure that your equivalence statement is valid 1 ft = 12 in. not: 1 ft = 16 in

Dimensional Analysis Applying Mathematics to Chemistry Step 3 Write a logical equation for solving the problem Derive the conversion factors that are needed to solve the problem Make sure that unneeded units cancel out When a series of conversion factors is needed, map out the correct sequence to follow Conversion of miles to centimeters mi  ft  in  cm 1 mi x (5280 ft/1 mi) x (12 in/1 ft) x (2.54 cm/in) = ? cm Step 4 Check that your answer is reasonable Does the answer have the right magnitude? Does the answer have the correct units? Daily volume of liquid antacid calculated to be 24 L 24 L ~ 24 qts Is this reasonable? Possible error L vs. mL?

Unit Conversion Problem: A car travels 350 miles in 7 hrs Speed: 350 miles/7 hrs = 50 miles/hr In 3 hrs: 3 hrs * 50 mi/hr = 150 mi Question: how many kilometers in 2 hours? Approach: 1 mile = 1.609 km = 1,609 m 1 km = 1 km/(1.609 km/mi) = 0.622 mi Distance per hour: 50 mi/hr = 50 mi/hr * 1.609 km/mi = 80.45 km/hr In 2 hrs: 2 hr * 80.45 km/hr = 160.9 km How many miles is this? 160.9 km = (160.9 km)/(1.609 km/mi) = 100 mi

Dimensional Analysis Applying Mathematics to Chemistry 500 g x (1 kg/1000 g) = 2000 mL x (1 L/1000 mL) = 200 mL x (1 L/1000 mL) = 200 mL x (1 kg/100 mL) x (1000 g/1 kg) = 600 K x (1P/300 K) x (10V/4P) = 100L x (10V/2L) x (10K/5V) =

Dimensional Analysis BBQ Pool Party! You invite 10 people to your Labor day Pool Party, and will serve burgers 1 lb of ground beef will give you 6 burgers Every guest eats 2 burgers How many lbs of beef do you need?

Dimensional Analysis Swimming Pool Problems! OOPS! On Saturday you realize you need to refill you pool! The pool is 8 ft x 20 ft x 5 ft. You fill the pool with a hose that delivers 3 gal/min Can you fill your pool before Monday noon? (How long does to take to fill your pool?) Is the answer reasonable? What is an appropriate unit for the duration? (1 gal = 231 in3)

Dimensional Analysis Can you solve this problem? A barge filled with bricks is floating down the Mississippi river. The barge is 100 ft long and 15 ft wide. The age of the captain is 45. Each brick measures 12 in x 2 in x 6 in. How many bricks can the barge hold?

Dimensional Analysis How many hours are there in a year? Two approaches 1 day = 24 hrs 1 week = 7 days 1 month = 4 weeks 1 year = 12 months 1 year = 365 days Both approaches are correct, but give different answers. Neither approach is accurate, since there are a number of roundings in the conversion factors.