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

2. 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.

3. 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

4. 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.

5. 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

6. 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

7. 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

8. 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 in
1 ft
Inch is not centimeter, so we
know we have at least one
more step.

9. 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 in
1 ft

10. 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 in cm
1 ft in

11. 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 in cm
1 ft in
cm is the unit we are looking for.
So there are
no more steps.

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

13. 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

14. 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 = km (or 1 km = miles)
1 km = 1000 m (or 1 m = km)
1 hr = 60 min and 1 min = 60 sec
40.0 mi
1 hr

15. 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 = km (or 1 km = miles)
1 km = 1000 m (or 1 m = km)
1 hr = 60 min and 1 min = 60 sec
40.0 mi km
1 hr mi
Remember to cancel and
check remaining units after
Each step.

16. 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 = km (or 1 km = miles)
1 km = 1000 m (or 1 m = km)
1 hr = 60 min and 1 min = 60 sec
40.0 mi km 1000 m
1 hr mi km
Now lets fill in the rest.
But note: hr is on the
Bottom, here.

17. 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 = 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.
40.0 mi km m hr min
1 hr mi km min sec

18. 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 km m hr min
1 hr mi km min sec
64360 m
3600 sec
Or 17.9 meters / sec

19. 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 km m hr min
1 hr mi km 60 min 60 sec
40.0 m
2.237 sec
Still 17.9 meters / sec

20. 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

21. 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 ft inch
1 ft
This step only covers one dimension!

22. 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 ft inch 12 inch
1 ft 1 ft
We’d stop here if the problem was
1 ft2 to in2

23. 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 ft inch 12 inch 12 inch
1 ft 1 ft ft
Or 1700 in3 (accounting for sig figs)
1728 in3 1

24. 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.

25. 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

26. 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

27. 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

28. 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

29. A Dosage Calculation Problem
How many tablets for a 175 lb patient?
Conversion Factors: 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

30. 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
tablets
4950
175 lb patient
1 kg
2.20 lb
1 tablet
225 mg med
Answer: 3 tablets