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  O ur original value and our answer are equal.  W e are only changing the way we express these values.  T his means that we must multiply by “1” at each step in the process (remember this!).  C onversion Factors: reflect some equality.  1 2 inches = 1 foot  2.54 cm = 1 inch

Rules (and steps) of the Game 1. Start with what you know. a. A single unit (if possible). 2. At each step: Fill in units first. a. Canceled Unit: Diagonally Across. b. New Unit (from a conversion factor): into other spot. 3. Then fill in numbers using the conversion factor. a. Top value must equal bottom value. b. Cancel diagonal units. c. Check remaining units: Is it what we’re looking for?  If not  Go back to rule 2 and start another step. 4. Multiply left to right (on both the top and bottom). 5. Divide the top answer by the bottom answer.

A Simple Problem  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 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) ft ft

A Simple Problem  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 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 2.25 ft 12 in 1 ft Inch is not centimeter, so we know we have at least one more step. Note that ‘12’ is with ‘inch’ both here… And here!

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 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 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 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 ft 12 in 2.54 cm cm 1 ft 1 in 1 Or just 68.6 cm (remember sig figs!)

A Complex Problem  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 mi 1 hr

A Complex Problem  Convert 40.0 miles per hour into meters per sec.  Rules 2 & 3: Some helpful conversion factors.  1 mile = km (or 1 km = miles)  1 km = 1000 m (or 1 m = km)  1 hr = 60 min and 1 min = 60 sec 40 mi 1 hr

A Complex Problem  Convert 40.0 miles per hour into meters per sec.  Rules 2 & 3: Some helpful conversion factors.  1 mile = km (or 1 km = miles)  1 km = 1000 m (or 1 m = km)  1 hr = 60 min and 1 min = 60 sec 40 mi km 1 hr 1 mi Remember to cancel and check remaining units after Each step.

A Complex Problem  Convert 40.0 miles per hour into meters per sec.  Rules 2 & 3: Some helpful conversion factors.  1 mile = km (or 1 km = miles)  1 km = 1000 m (or 1 m = km)  1 hr = 60 min and 1 min = 60 sec 40 mi 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  Convert 40.0 miles per hour into meters per sec. 40 mi km 1000 m 1 hr 1 min 1 hr 1 mi 1 km 60 min 60 sec  Rules 2 & 3: Some helpful conversion factors.  1 mile = km (or 1 km = miles)  1 km = 1000 m (or 1 m = km)  1 hr = 60 min and 1 min = 60 sec And note: hr is on the top, here.

A Complex Problem  Convert 40.0 miles per hour into meters per sec.  Rules 4 & 5: Remember.  Multiply across the top and bottom first. 40 mi km 1000 m 1 hr 1 min 1 hr 1 mi 1 km 60 min 60 sec  divide the top answer by the bottom answer m 3600 sec Or 17.9 meters / sec THEN

A Complex Problem  Convert 40.0 miles per hour into meters per sec.  Rules 4 & 5: Remember.  We could have used different conversion factors. 40 mi 1 km 1 m 1 hr 1 min 1 hr mi km 60 min 60 sec 40 m sec Still 17.9 meters / sec

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.  Rule 1: Start with the single unit that you know.  Given: 5 ml syrup = 63.5 mg amoxicillin.  Others you should know: 1 fluid oz = 29.6 ml 95 mg Amox  Rules 2 & 3: Fill in units and numbers with conversion factors. 5 ml syrup 63.5 mg Amox 1 fl. oz ml syrup Cancel and Check!

A Dosage Calculation Problem  Convert 95 mg Amoxicillin to ? Fluid oz of syrup. 95 mg Amox 5 ml syrup 63.5 mg Amox 1 fl. oz ml syrup  Rules 4 & 5: Remember.  Multiply across the top and bottom first.  divide the top answer by the bottom answer. 475 fl. oz THEN 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?  Rule 1: Start with the single unit that you know. 175 lb patient  Rules 2 & 3: Fill in units and numbers with conversion factors. 1 kg 2.20 lb 85 mg med 10 kg patient Cancel and Check!  Conversion Factors: 1 kg = 2.20 lbs  10 kg patient mass = 85 mg medicine.  1 tablet = 225 mg medicine. 1 tablet 225 mg med

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