Factor Unit Method

Some of you may be worried about the
math that you’ll encounter in a chemistry class. There’s some good news -the Factor Unit Method is a very powerful tool that will help you work
many of the math problems that you encounter. Whenever I see numbers in a
quiz question or an exam question, I say to myself it’s probably a math problem,
and all math problems in our text at least, can be worked in one of two ways. There’s some that we have a formula for- for example density or percentage or
temperature problems, and when you have a formula that’s like a road map – it tells
you how to set up the problem and and where to substitute the numbers. If you
don’t have a formula, then virtually all of the other problems can be worked by
the factor unit method. Let’s look at one and use the factor unit method. This question says “have a paper that’s 11 inches tall. How tall is the paper in
centimeters?” Well that’s a pretty simple problem and I’m sure you could work it a
couple of ways, but what I promise you is that if you use the factor unit method
and let it become a habit, you’ll not only be able to work simple problems but
very difficult problems too. We’re given some additional information – we’re told
that 1.0 inches is equal to 2.54 centimeters. This is an equation. It
represents a truth or a fact, and from a fact we can derive factors that help us
in the factor unit method. The very first step in the factor unit method tells
you where to start, and that’s one of the strengths in that method – says exclude
all the words up here and just look for the number that’s given. Well, step one
we’d write down 11 inches. Step two is to sort of see where you’re going. We’d like
to know centimeters, so we’ll use that unit over on the right hand side. And now coming back to the left, we’ll employ a factor that allows us to
convert inches into centimeters. If I have just one factor, it’s always true
that the unit of the answer will come in the numerator, and the unit of the
denominator will be the same as we start with over here, so that inches cancels
and we’re given units as the product. Let’s substitute in those numbers now
and associated with the unit centimeters is this 2.54 – we’d put that in associated
with inches. 1.0 we’ll place it in. We’d multiply the numerators, divide by the
denominators, the answer that we get is 27.94 centimeters. And we have to keep
in mind significant figures. We’ve got two significant figures over here in a
couple of places, so we’d round this off to two significant figures – 28
centimeters. The strength of this method is that by setting it up so that the
unit comes out right, the number should come out right also. Let’s look at
another problem. In the first problem we looked at an English to metric
conversion. This time let’s look at metric to metric. 28 centimeters is how
many meters. Well here’s something that you probably know or you can find it in
a table – 1 meter is equal to a hundred centimeters. And again up here if I were
to encounter this as a quiz or an exam question, I see a number I’d say hey it’s
most likely a math problem – do I have a formula for it? No, I’m going to use the
factor unit method. Step one, simply write down the number that’s given and the
unit that goes with it. Step 2, we’d like to know meters. Step 3, here’s our factor
unit of the answer in the numerator, unit that we’re given in the denominator
so that centimeters cancel, and now we’ll put in the numbers that are associated
with that with those units from our equation and multiply the numerators, divide by the denominators – we get 0.28 meters. Really pretty easy, but
later on in the text you’ll find that you can work even hard problems that
quickly and that simply. Both of these examples have used just one factor – a
little bit tougher now if we have a problem with two factors. Let’s work one
of those. This question asks how many grams are in 5.0 pounds? We’re given a
couple of equations at the top – these might be available in a table and the text, or
most likely given in a problem. One of those states that 1 gram is equal to 0.035 ounces. The other equation, 16 ounces is equal to 1 pound. And as I look
at all of these numbers I would probably say to myself “hey this looks like a math
problem, do I have a formula for it?” Well no, I don’t know the relationship between
grams and pounds, so I’m going to use the factor unit method. And as I think about
the factor unit method again, I don’t know the relationship between grams and
pounds and it’s not contained up here in these two equations. And so, that’s a clue
for me that I’m going to need at least two factors to work this problem. So, as
we begin, again we’d start with the first step in the factory unit method. The number that’s given – 5.0 pounds. I’ll leave space for a factor, another factor,
and here’s what we’d like to know – grams. Then we can see that our objective is to
convert pounds to grams in these two factors, so I need pounds down here in
the denominator. Going back up to these two equations – here’s pounds, and we know the relationship to ounces, so ounces will go in the numerator. We’d like to
get rid of ounces, so ounces is in the denominator of the next factor, and we’d
like to have grams, so it’s in the numerator. And the first expression tells
us that relationship. So now we’d take those using these units as a guide and
simply put in the numbers that are associated with those units. As we
multiply out, my calculator gave me this answer 2285.7142 grams. Looks like a
terrific answer – got lots of decimal places out there, but of course we know
that the answer is no more significant than the least number of significant
figures back here too in several places. So we need to round that answer off to
two significant figures. Rounding it a little bit we would get 2286. Still too
many digit. Even if we rounded it to 2300, these trailing zeros are
significant, so we have to use the scientific expression. We’ll move the
decimal three places to the left – we correct that by-or balance that by-
indicating three places to the right, 2.3 times 10 to the third grams in scientific notation. Again, the strength of this method is to focus upon
the unit. If we make sure that the unit comes out right, then the numbers going
to be right also.

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