Just wanted to do a little introduction or some practice on unit conversion, since unit
conversion is typically one of the simplest mistakes to make on a test problem or homework problem
and it's something you just need to practice and get used to and comfortable with so that you
don't make those mistakes. So, I have three examples I want to go through, and just kind
of talk about a set-up. Once you get comfortable doing this set-up, and more of it becomes
automatic in your head, you don't necessarily need to use this strategy, but at least
to start, and this is the way that's suggested, and that is,
if we are going, in this case, from say one cup and we want to go to a certain amount of moles
of water. We have to be able to figure out what conversion factors we need, and what
those are so we can convert from that one cup of water to those moles of water. So what
we typically do is draw something that looks like this, with the line in the middle and
units on both the top and the bottom. And then, start putting our conversion factors
in between. So, you have to have an approach, and I just happen to know that we can go from
a volume to a mass using density, and to go from mass to moles using the molecular weight.
So, I'll go ahead and show you how to go about that, and just in terms of the setup, how
you would approach it. There are a couple different conversion factors you could use
for density. The one I can recall off the top of my head is that there is 1 gram of water
per 1 milliliter of water. So that makes it easy to go from there. And then, you could
memorize or look up some other volume conversions; for instance, if you cook a lot, hopefully you'll
know that, in terms of cups and milliliters, there are 240 milliliters per cup.
And so it's important that you notice which side the top or bottom that we're putting these
units. We want the "cup" on the bottom, so that we can cross those off. And the same
things here: we want the "milliliters of water" on the top so we can cross those off. So now,
we have "grams of water". To go from grams to moles, we can use the molecular weight.
And since we want grams to cancel, we're going to put "grams" on the bottom. So how many
grams of water are there per mole of water? Well, you could look that up or calculate
it, and it should be 18 grams. So now we can cross out our "grams of water" and "grams of water" and we're left
with moles. The only thing you have left to do is make these multiplications, where we
have 240 over 18, and that is going to give you the amount of moles of water. It comes
out to about 13.3 moles of water.
So let's try another one. Here's a question for you: something I recall from when I was younger
was whether or not you think you can out run an alligator that could go up to 20 miles
per hour if you can cover the 40 yard dash in 5 seconds. Now, there are obviously a lot
faster people out there, but relatively, five seconds for forty yards isn't too far off.
So the way we would start this is write our forty yards in five seconds, because that's
our velocity that we want to compare with the velocity of an alligator. We're going
to continue this line across, and we want to get to miles per hour. So now this is going to
help us see what conversion factors we're going to use.
I'm going to go ahead and tell you to pause at this point and try to fill this in and calculate what your
speed in miles per hour is.
Now, you could either choose to go with yards or seconds
to start with. It doesn't really matter. First thing that comes to my mind is how to get
rid of the seconds.
We know that there's 3600 seconds in one hour. That's going to cross those off. We have our
hours and now we have to go from yards to miles. You could probably look something up,
or if you know the conversion between yards to miles, you could use that. I do know
that there are 3 feet per 1 yard, and there are 5280 feet per one mile. So now you can see
now that the feet are cancelling, the yards are cancelling, and we have our miles per
hour. So the only thing left to do is to multiply our top numbers divided by our bottom numbers,
and you should see that this comes out to 16.4 miles per hour, meaning that you'd probably
be lunch.
The final question, which is something that you may come across in your travels, or if
you're curious. The question asks whether or not it's cheaper to drive in the U.S. with an average
20 miles to the gallon right now, or in Germany, where the average is about 6 Liters per 100 kilometers
and that's how they report it over there. So there's a couple different conversions you have to make,
and some information that you're not given that you have to be resourceful enough to
look up. So how do we go about this?
Well we need to find some way to compare these two to each other, so let's do it two ways:
let's compare how much it costs to drive a mile in each country, and also how much it
would cost per gallon of gasoline.
Alright, so let's start with the U.S., since it's already in the units we were talking
about. If we go 20 miles to 1 gallon of gasoline, what unit conversion would get us towards
miles to dollars? Well, we have to convert the dollars to gallons--so we'll say that
one gallon probably costs us 3 dollars. So this conversion should yield about 6.67 miles
to the dollar, which is also going to be about 15 cents per mile. And we already have our
price per gallon, since that's what we've looked up.
Now we just have to convert Germany's system over. So let's start with the six Liters per
100 kilometers. I'll draw my single-dimensional equation, and we want to get to dollars per mile.
So let's start with the kilometers. We know that there's 1.6 kilometers per 1 mile. That'll
give us our mile and cancel out kilometers. Now all we have to do is get to our prices.
we know that their price of gasoline, if you look it up right now, is about 1 Euro per Liter, and
there's about 1.5 dollars per 1 Euro. These may not be that accurate depending on when
you look at this, but if we do this calculation, then we'll see it's about 14 cents per mile.
That's really not that different from what we see here.
Well, per mile, it's not--but if we wanted to check to see what it was on a volume basis,
1 Euro over 1 Liter using our dimensional equation, 3.78 Liters per 1 gallon
and then again the price conversion $1.50 per Euro, if you calculate this out, you'll see that the
price of gas is about $5.70/gallon. Although distance-wise it doesn't cost less, the reason
for that is that the cars in Germany are just getting better gas mileage on average than the cars
over here in the U.S.