Math Made Manageable collegechemistry.batcave.net Dimensional Analysis Math Made Manageable collegechemistry.batcave.net

What is Dimensional Analysis? An organized methodology for examining the relationship between measurements (and convert between them) in order to express the same (or related) value differently. Okay, so what is a value? A measurement expressed in terms of a number and a unit. For example: just saying “55” is meaningless. But saying “55 inches” provides us with information.

Objective & Consequences Our original value and our answer are equal. We are only changing the way we express these values. This means that we must multiply by “1” at each step in the process (remember this!). Conversion Factors: reflect some equality. 12 inches = 1 foot 2.54 cm = 1 inch

Rules (and steps) of the Game Start with what you know. A single unit (if possible). At each step: Fill in units first. Canceled Unit: Diagonally Across. New Unit (from a conversion factor): into other spot. Then fill in numbers using the conversion factor. Top value must equal bottom value. Cancel diagonal units. Check remaining units: Is it what we’re looking for? If not  Go back to rule 2 and start another step. Multiply left to right (on both the top and bottom). Divide the top answer by the bottom answer.

A Simple Problem Rule 1: Start with what you know. 2.25 ft Convert 2.25 feet into centimeters (cm). Rule 1: Start with what you know. Here we have only a single unit. Lets place it on the top (i.e. in the numerator) of our first step. 2.25 ft

A Simple Problem Convert 2.25 feet into centimeters (cm). Rule 2: Fill in Units first. Place the unit needing cancelled diagonally across. i.e. on the bottom (i.e. pt 2: in the denominator). 2.25 ft ft

A Simple Problem Rule 2: Fill in Units first. 2.25 ft in ft Convert 2.25 feet into centimeters (cm). Rule 2: Fill in Units first. New unit comes from a conversion factor. Remember 12 in = 1 foot. 2.25 ft in ft

A Simple Problem Convert 2.25 feet into centimeters (cm). Rule 3: Fill in numbers. Numbers also come from the conversion factor. Remember 12 in = 1 foot. Cancel and check remaining units Note that ‘12’ is with ‘inch’ both here… And here! 2.25 ft 12 in 1 ft Inch is not centimeter, so we know we have at least one more step.

A Simple Problem Convert 2.25 feet into centimeters (cm). Repeat rules 2 & 3: Fill in units. Fill in numbers. Cancel and check remaining units. 2.25 ft 12 in 1 ft

A Simple Problem Convert 2.25 feet into centimeters (cm). Repeat rules 2 & 3: Fill in units. Fill in numbers. Cancel and check remaining units. 2.25 ft 12 in cm 1 ft in

A Simple Problem Convert 2.25 feet into centimeters (cm). Repeat rules 2 & 3: Fill in units. Fill in numbers. Cancel and check remaining units. 2.25 ft 12 in 2.54 cm 1 ft 1 in cm is the unit we are looking for. So there are no more steps.

A Simple Problem Convert 2.25 feet into centimeters (cm). Rules 4 & 5: Multiply left to right. On both top and bottom. Divide top answer by bottom answer. 2.25 ft 12 in 2.54 cm 68.58 cm 1 ft 1 in 1 Or just 68.6 cm (remember sig figs!)

A Complex Problem Rule 1: Start with what you know. 40.0 mi 1 hr Convert 40.0 miles per hour into meters per sec. Rule 1: Start with what you know. Here we have a compound starting unit. We avoid this if we can, but in this case it’s not possible Per means divide; hour goes on bottom. 40.0 mi 1 hr

A Complex Problem Rules 2 & 3: Some helpful conversion factors. Convert 40.0 miles per hour into meters per sec. Rules 2 & 3: Some helpful conversion factors. 1 mile = 1.609 km (or 1 km = 0.6214 miles) 1 km = 1000 m (or 1 m = 0.001 km) 1 hr = 60 min and 1 min = 60 sec 40.0 mi 1 hr

A Complex Problem Rules 2 & 3: Some helpful conversion factors. Convert 40.0 miles per hour into meters per sec. Rules 2 & 3: Some helpful conversion factors. 1 mile = 1.609 km (or 1 km = 0.6214 miles) 1 km = 1000 m (or 1 m = 0.001 km) 1 hr = 60 min and 1 min = 60 sec 40.0 mi 1.609 km 1 hr 1 mi Remember to cancel and check remaining units after Each step.

A Complex Problem Rules 2 & 3: Some helpful conversion factors. Convert 40.0 miles per hour into meters per sec. Rules 2 & 3: Some helpful conversion factors. 1 mile = 1.609 km (or 1 km = 0.6214 miles) 1 km = 1000 m (or 1 m = 0.001 km) 1 hr = 60 min and 1 min = 60 sec 40.0 mi 1.609 km 1000 m 1 hr 1 mi 1 km Now lets fill in the rest. But note: hr is on the Bottom, here.

A Complex Problem Rules 2 & 3: Some helpful conversion factors. Convert 40.0 miles per hour into meters per sec. Rules 2 & 3: Some helpful conversion factors. 1 mile = 1.609 km (or 1 km = 0.6214 miles) 1 km = 1000 m (or 1 m = 0.001 km) 1 hr = 60 min and 1 min = 60 sec And note: hr is on the top, here. 40.0 mi 1.609 km 1000 m 1 hr 1 min 1 hr 1 mi 1 km 60 min 60 sec

A Complex Problem Rules 4 & 5: Remember. Convert 40.0 miles per hour into meters per sec. Rules 4 & 5: Remember. Multiply across the top and bottom first. THEN divide the top answer by the bottom answer. 40.0 mi 1.609 km 1000 m 1 hr 1 min 1 hr 1 mi 1 km 60 min 60 sec 64360 m 3600 sec Or 17.9 meters / sec

A Complex Problem Rules 4 & 5: Remember. Convert 40.0 miles per hour into meters per sec. Rules 4 & 5: Remember. We could have used different conversion factors. 40.0 mi 1 km 1 m 1 hr 1 min 1 hr 0.6214 mi 0.001 km 60 min 60 sec 40.0 m 2.237 sec Still 17.9 meters / sec

A Special Case: Higher Orders (area & volume) Convert 1.0 cubic foot (ft3)into cubic inch (in3). Rule 1: Start with what you know. Here we have a higher order starting unit. This means our unit covers multiple dimensions 1 ft  length is 1 dimension 1 ft2  area is 2 dimensions 1 ft 3  volume is 3 dimensions 1.0 ft3

A Special Case: Higher Orders (area & volume) Convert 1.0 cubic foot (ft3)into cubic inch (in3). Rule 1: Start with what you know. Here we have a higher order starting unit. This means our unit covers multiple dimensions 1 ft  length is 1 dimension 1 ft2  area is 2 dimensions 1 ft 3  volume is 3 dimensions Cubed becomes squared… Not fully cancelled! 2 1.0 ft3 12 inch 1 ft This step only covers one dimension!

A Special Case: Higher Orders (area & volume) Convert 1.0 cubic foot (ft3)into cubic inch (in3). Rule 1: Start with what you know. Here we have a higher order starting unit. This means our unit covers multiple dimensions 1 ft  length is 1 dimension 1 ft2  area is 2 dimensions 1 ft 3  volume is 3 dimensions But since we started with ft3, we’ve now cancelled to ft. 2 1.0 ft3 12 inch 12 inch 1 ft 1 ft We’d stop here if the problem was 1 ft2 to in2

A Special Case: Higher Orders (area & volume) Convert 1.0 cubic foot (ft3)into cubic inch (in3). Rule 1: Start with what you know. Here we have a higher order starting unit. This means our unit covers multiple dimensions 1 ft  length is 1 dimension 1 ft2  area is 2 dimensions 1 ft 3  volume is 3 dimensions It takes 3 steps to cancel a cube (or 2 steps to cancel a square) 2 1.0 ft3 12 inch 12 inch 12 inch 1 ft 1 ft 1 ft Or 1700 in3 (accounting for sig figs) 1728 in3 1

A Dosage Calculation Problem A patient requires 95 milligrams of Amoxicillin. The amoxicillin syrup contains 63.5 mg in each 5 ml dose. Your measuring spoons are delineated in fluid ounces. How many fluid ounces will you give. The trick to word problems is learning to parse your information and organize your data.

A Dosage Calculation Problem A patient requires 95 milligrams of Amoxicillin. The amoxicillin syrup contains 63.5 mg in each 5 ml dose. Your measuring spoons are delineated in fluid ounces. How many fluid ounces will you give. What conversion factors do we have? From the problem: 5 ml syrup = 63.5 mg amoxicillin. Others you should know: 1 fluid oz = 29.6 ml

A Dosage Calculation Problem Convert 95 mg Amoxicillin to ? Fluid oz of syrup. Given: 5 ml syrup = 63.5 mg amoxicillin. Others you should know: 1 fluid oz = 29.6 ml Rule 1: Start with the single unit that you know. Rules 2 & 3: Fill in units and numbers with conversion factors. Cancel and Check! 95 mg Amox 5 ml syrup 63.5 mg Amox 1 fl. oz. 29.6 ml syrup

A Dosage Calculation Problem Convert 95 mg Amoxicillin to ? Fluid oz of syrup. Rules 4 & 5: Remember. Multiply across the top and bottom first. THEN divide the top answer by the bottom answer. 95 mg Amox 5 ml syrup 63.5 mg Amox 1 fl. oz. 29.6 ml syrup 475 fl. oz. 1880 Answer: 0.25 fluid ounce

A Dosage Calculation Problem A doctor prescribes 85 mg of medicine for each 10 kg of patient mass. Your patient weighs 175 pounds and the medicine comes in tablets containing 225 mg each. How many tablets will you give. What conversion factors do we have? From the problem: 10 kg patient mass = 85 mg medicine. 1 tablet = 225 mg medicine. Others you should know: 1 kg = 2.20 lbs

A Dosage Calculation Problem How many tablets for a 175 lb patient? Conversion Factors: 1 kg = 2.20 lbs 10 kg patient mass = 85 mg medicine. 1 tablet = 225 mg medicine. Rule 1: Start with the single unit that you know. Rules 2 & 3: Fill in units and numbers with conversion factors. Cancel and Check! 85 mg med 10 kg patient 175 lb patient 1 kg 2.20 lb 1 tablet 225 mg med

A Dosage Calculation Problem How many tablets for a 175 lb patient? Rules 4 & 5: Remember. Multiply across the top and bottom first. THEN divide the top answer by the bottom answer. 85 mg med 10 kg patient 14875 tablets 4950 175 lb patient 1 kg 2.20 lb 1 tablet 225 mg med Answer: 3 tablets