Unit Conversions Example #4 – Compound Units

This example involves converting between two
compound units that are both made up of several other units. The question is “Convert 8.31 Newton metres
per Mole Kelvin to foot pounds-force per pound-mole Rankin”. This question is based on the units used to
measure the ideal gas constant, so it is actually based on a real-life example, believe it or
not. Given the correct conversion factors, you
should be able to answer this question without even knowing what these numbers and symbols
mean. We’ll start as we always do by setting up
the original amount as a fraction. Write it as 8.31 Newtons times metres over
moles times Kelvins. Now look at what units the answer is supposed
to be in. We need to change Newtons to pounds-force,
metres to feet, moles to pound-moles and Kelvins to Rankins. Let’s start with the Newtons to pounds-force
conversion. We can use the given information that 1 Newton
=.2248 pounds-force. We need a conversion factor as a fraction
with Newtons on the bottom because Newtons is on the top in the original amount. So starting with the original amount, we need
to multiply by the conversion factor here: .2248 pounds-force over 1 Newton. Because this will make the Newtons on the
top in the original cancel with the Newtons on the bottom in the conversion factor. Now we could do this operation now; do the
8.31 times .2248 and see what we get, but it is actually easier to do this question
all in one step by multiplying by the other three conversion factors at the same time. So let’s move on to the next conversion factor
and you’ll see what I mean in the end. Now to change metres to feet, we have the
given information that 1 metre=3.28 feet and we need to make that into a conversion
factor. We need metres to be on the bottom and feet
to be on the top, so you multiply by 3.28 feet over 1 metre, and I’ll add that to the
sort of multiplication train that we have going here. Now we do the same thing with Kelvins to Rankins
using the information 1 Kelvin=1.8 Rankins. However, this time I need Kelvins to be on
the top to cancel with the Kelvins that were on the bottom here in the original. Again, I just put that next to my other multiplications. Finally, to change moles to pound-moles, I
need to put moles on the top to make it cancel with moles on the bottom in the original and
pound-moles on the bottom. So, I’m going need to multiply by 1 mole over
453.59 pound-moles. Now we have this long line of multiplication
that we’re going to do all at once, but it’s best at this point to check to make sure all
of our old units are going to cancel. So you can see that Newton on top here is
going to cancel with Newtons on the bottom here, metres here will cancel with metres
down here, Kelvins on the bottom here is going to cancel with Kelvins on the top in this
one and moles is going to cancel with moles on the top over here. So that only leaves pounds-force, feet, Rankins
and pound-moles in the right places for the answer. And just so you can see how all the units
are cancelling out, I’ll put it on a new line, so you can see all the cancellations. Now that I’m confident that I’m going to end
up with all the right units for the answer, I’ll put down the units for the answer first. And now all that’s left is to do the multiplications
with the numbers. After working through the multiplication of
the fractions, we end up with 8.31 times .2248 times 3.28 on the top and 453.59 times 1.8
on the bottom. So using my calculator that gives me 6.127
for the to pand 816.462 for the bottom. Take the top and divide by the bottom and
that gives you the final answer of 0.0075 pounds-force feet per pound-mole Rankin. And that is the final answer, so I’ll right
down my final statement, confidently stating that 8.31 Newton-metres per Mole-Kelvin is
the same as 0.0075 foot pounds-force per pound-mole Rankin. Or if you want you can write this as scientific
notation as 7.5 times 10 to the power of negative 3 foot pounds-force per pound-mole Rankin. Now is there any real way to check this answer?
It’s definitely a little hard to tell just by looking whether the answer makes sense. These aren’t… we normally wouldn’t use this
kind of compound unit in real life. This is kind of an extreme example… so really
all you can do is check over all the steps, make sure all the units are cancelled out
right and check your calculator work. And… I can tell you this is the right answer,
but if you want to check it for yourself, it’s always a good idea to make sure.

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