Dimensional analysis
Many calculations in chemistry require that we convert quantities from one set of units to another. Whether we are trying to convert between the British system of units and the SI system of units, or within the SI system of units, we can do this by using conversion factors.
For examples,
- If we wish to convert between meter and inch, we would need the conversion factor 1 meter = 39 inches.
- If we wish to convert between kilogram and pound, we would need the conversion factor 1 kilogram = 2.2 pounds.
- If we wish to convert between liter and milliliter, we would need the conversion factor 1 liter = 1000 milliliter.
The approach to problem solving via inspecting the units of measurements is called dimensional analysis. The basic setup is summarized in the following steps:
- Identify the unit in the given quantity of measurement. This is your information given.
- Identify the unit that you need to present the answer. This is your information sought.
- Figure out the conversion factor to convert the units and write the conversion factor as a fraction.
- Set up the problem as shown below to ensure that units cancel in such a way that you are left with the unit you want.
(information given) x (conversion factor(s)) = information sought
Example: Express 0.5 mg in µg
The process which Inspect the units of the quantities of the calculation.
Step 1: Identify the units in the given quantity of measurement.
information given - the units in the given quantity of measurement is in milligram.
Step 2: Identify the units that you need to present the answer.
information sought.- the units that is required in the answer is microgram.
Step 3: Figure out the conversion factor between milligram, mg, to microgram, µg.
To do that, we go back to the base unit of grams.
1 gram = 1000 milligram ...............................equation (1)
1 gram = 1,000,000 microgram ......................... equation (2)
Since both 1000 milligram and 1,000,000 microgram equal to 1 gram, it follows that
1000 milligram = 1,000,000 microgram ................... equation (3)
which reduces the conversion factor to
1 milligram = 1000 microgram ............................... equation (4)
Equation (4) is the conversion factor for milligram to microgram. We can set up the conversion factor as a fraction. We can either express the fraction with milligram as the numerator and microgram as the denominator (as in (a)) OR express the fraction with microgram as the numerator and milligram as the denominator (as in (b)).
Step 4: Apply the conversion factor to the problem. Follow the basic setup for problem solving.
| information given: | 0.5 mg |
|---|---|
| conversion factor: |
|
| information sought (answer): | ? µg |
(information given) x (conversion factor(s)) = information sought
We are given a quantity of measurement in units of mg. The answer (or information sought) must have units of µg. In solving for this problem, we need to multiply 0.5 mg with the conversion factor to obtain an answer in microgram. Although there are two ways to express the conversion factor between mg to µg, we must select the conversion factor expressed as in (b) and multiply it with 0.5 milligram.
In the above calculation, the units of milligram cancel and the answer is in microgram. Therefore, 0.5 mg = 500 µg.
What would happen if we selected the conversion factor expressed as in (a)? Notice that the units do not cancel out each other. The answer no longer has units of µg, instead the answer has a unit that does not have any physical sense, mg2/µg. The units alone should send you the signal that the calculation shown below is incorrect!
Content suitability
BCIT courses: CHEM 0011
