Sometimes it requires more than one conversion factor to solve a problem using Dimensional
Analysis. The basic approach is the same. Please watch our first Unit Conversion video
for the introduction to this method of problem solving. Remember the general strategy is
to look at the units you start with, look at the units you want to end up with, and
use one or more conversion factors to get there.
starting unit x conversion factor = ending unit
The conversion factor looks like this: ending unit over starting unit.
So the starting units cancel, leaving the ending unit.
Here are some more complicated problems, that require at least a couple of conversion factors:
Here’s our first example. Convert 24.5 g/cm3 to kilograms per liter.
Remember, put what you know on the left, and what you want to end up with on the right.
Leave lots of room for your work.
24.5 g/cm3 =some number of kg/liter
We know we want to go from g to kg, and from cm3 to liters. We can look up those conversion
factors and then we’ll have to think about the right way to use them.
There are 1000g in 1 kg. So you could write 1000g over 1kg - that’s a fraction equal to 1.
You could also write 1kg over 1000 g. That’s also a fraction equal to 1.
and there are 1000 cm3 in 1 liter. Again, you can write this as a fraction equal to
1. 1000 cm3 / 1L = 1, and 1 L/ 1000 cm3 equals 1.
Here’s where we have to be really careful. Which way do we write these conversion factors
so the units cancel in our problem?
24.5 g/cm3 (1kg/1000 g) g has to go on the bottom to cancel.
24.5 g/cm3 (1kg/1000g) (1000 cm3 / 1L) next, cm3 has to go on top. Check to make sure all
your units cancel, and you will be left with kg/L .
Multiply all the way across on the top, and multiply all the way across on the bottom.
24.5 kg/L
Here’s another example: You exhale 20.0 mL of CO2 with every breath. If you breathe
15 times per minute, how many liters of CO2 do you produce each month? Assume 30 days
per month. We’re starting with 20.0 mL/breath and we
want to end up with L/month. That’s going to require a lot of conversion factors, so
leave a big space. 20.0 mL/1 breath (15 breaths/1 minute)
(60 minutes/1 hr) ( 24hr/1 day) ( 30 day/1 month)( 1L/1000 mL) = some number of L/month
Make sure your units cancel. Multiply the top all the way across, and multiply
the bottom all the way across. Don’t forget to check your significant figures.
There was a measurement of 3 significant figures in the problem: 20.0 mL, so we’ll state
our answer with 3 significant figures.
12,960,000/1000 = 12,960 - let’s put this in scientific notation to easily see 3 significant
figures. Count over 1, 2, 3, 4 decimal places. 1.296 x 10^4 L /month
Now round to 3 significant figures - 1.30 x 10^4 L /month