Dimensional Analysis 4

Uploaded by AlanEarhartChemistry on 19.12.2011


This is an example of a volume conversion.
We’re converting 348 cubic inches to cubic centimeters.
Volume conversions throw something new into the mix.

Inches cubed... centimeters cubed... What’s the relationship?
We know this- 1 inch is defined to be 2.54 centimeters.

Well, let’s try this and see what it gets us.
I’ll start with the 348 cubic inches. The given information.
And then I have my conversion- inches and centimeters.
So I think opposite units. The inches go into the denominator...
The centimeters go into the numerator.
And then when you multiply it out... oh, wait...
Let’s check out units.
That’s ugly! That’s not what I want!
Inches cubed x centimeters over inches.
Unit-wise that’s inches squared centimeters. That’s not cubic cm.
That doesn’t work.

Four cubed over four is four squared.
OK, that doesn’t work. When you do a volume conversion,
you have three dimensions to worry about. Not just one.

It’s not just inches to centimeters.
There are three dimensions- inches cubed to centimeters cubed.

Wait... there we go... There’s my conversion, right?
Well, the left side in correct, but the right side is not correct.

It’s really (1 inch)^3.
In math, if you do something to one side, you have to
do the same thing to the other side. It’s not just cubing the unit,
it’s cubing (2.54 centimeters).
That’s the correct relationship.
Now, since the 1 inch equals 2.54 centimeters is
an exact relationship, don’t cube 2.54 and then round it.
It’s exact, just leave it as 2.54 cubed.
Or, if you do cube it, write out all of the decimal places. Don’t round it.
Let’ go back and fix this.
This is one way to work the problem,
it’s cubic inches (you have three dimensions to worry about). So I have
three inches units to convert to three centimeters units.
When we multiply this, don’t cube the 348. It’s already been done.
It’s 348 x 2.54 x 2.54 x 2.54.
And that gives us 5702.7 cubic centimeters.
Now the units are correct and that’s what the problem asked for.
That’s good. Let’s check significant figures.
348 has three. The 2.54...
That does not have three.
That’s exact.
Now it would have affected the problem either way.
At least it would not have affected this problem.
But it’s still an exact conversion. So the final answer
has three significant figures because of the 348.
So 5702.7...
So I’m going to round off... oh, 0. You end up with 5700 cubic cm.
You need to ask yourself how many significant figures are in 5700.
There are only two the way it’s written.
This is one of those rare examples in which you must write
the answer in scientific notation. In order to write 5700
with three significant figures (it was rounded properly),
you must use scientific notation. 5.70 x 10^3.
Now that’s just one way to work the problem.
There is another way to work it.
Still start the same way.
348 cubic inches. The difference is in how you write the conversion.
Instead of writing all three of the out,
you are cubing it.
You get the same answer because it’s 348 x 2.54^3.

Which way should you work it?
The top way or the bottom way? They are both good.
What I like about the top way is that it explicitly shows me the map.
I don’t have to wonder how many 2.54’s there are.
There are three. It is a little more writing, though.
The one on the bottom is not as much writing, but a common mistake
I see in volume conversions is that people will take the 348
and multiply it by 2.54 and hit the equal sign.
It’s 2.54 cubed.
So watch your area conversions- that’s a square unit.
And watch your volume conversion, that’s a cubed unit.