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Chemistry: Dimensional Analysis

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About this Lesson

  • Type: Video Tutorial
  • Length: 11:22
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 122 MB
  • Posted: 01/28/2009

This lesson is part of the following series:

Chemistry: Full Course (303 lessons, $198.00)
Chemistry Review (25 lessons, $49.50)
Chemistry: Introduction to Matter and Measurement (13 lessons, $14.85)
Chemistry: Scientific Measurement (5 lessons, $7.92)

In this lesson, you will learn how to convert measurements from one unit to another. You will learn how to create a conversion factor, which is always a factor of one that is constructed from a known quality. When creating a conversion factor, you will want to make sure to choose one that allows you to cancel out unwanted units. Prof. Yee also discusses significant figures when using conversion factors. Since conversion factors are considered exact, they have an infinite number of significant figures that are not limited. The data will limit the number of significant figures that you will use in an answer. Objects that are counted are also considered exact with unlimited significant figures. You will learn how to link multiple conversion factors, which is sometimes necessary when converting from one unit to another. Prof. Yee reminds you to make sure you always ""take the units along for the ride"" to ensure that you are finding your answer in the proper units. Finally, you will learn the proper conversion factors for converting both from degrees Farenheit to degrees Celsius and from degrees Celsius to degrees Kelvin.

Taught by Professor Yee, this lesson was selected from a broader, comprehensive course, Chemistry. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/chemistry. The full course covers atoms, molecules and ions, stoichiometry, reactions in aqueous solutions, gases, thermochemistry, Modern Atomic Theory, electron configurations, periodicity, chemical bonding, molecular geometry, bonding theory, oxidation-reduction reactions, condensed phases, solution properties, kinetics, acids and bases, organic reactions, thermodynamics, nuclear chemistry, metals, nonmetals, biochemistry, organic chemistry, and more."

Gordon Yee is an associate professor of chemistry at Virginia Tech in Blacksburg, VA. He received his Ph.D. from Stanford University and completed postdoctoral work at DuPont. A widely published author, Professor Yee studies molecule-based magnetism.

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Thinkwell
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An Introduction to Matter and Measurement
Scientific Measurement
Dimensional Analysis Page [1 of 3]

Suppose I was going on a trip to Japan and I wanted to know how much my US dollars were going to buy in Japan.
What I want to do is look up how many yen I get, the yen is the Japanese unit of currency, so I’d look up, maybe on
the Internet, how many yen I get for each US dollar. And then, using that as a conversion factor, I’d figure out how
many yen I could get for my US dollar.

So suppose that I look it up and it turns out that 1 US dollar, abbreviated USD, is equal to 106.823 Japanese yen.
That’s a relationship between dollars and yen. And the beauty is that you can take those sorts of relationships and
turn them into factors of one, or unity factors, or whatever you want to call it. The point is that they are going to be
dimensionless numbers overall, so you can multiply anything by one and you can use that to use your conversions.

So, if we take this relationship and we divide through by 106.823 yen, so divide both sides by that number, then we
would get a relationship. So ignore my hands for a moment, and we would get a relationship that looks like what’s left
over here. In other words, 1 dollar divided by 106.823 yen is equal to 1. And then if we did it the other way around, if
we divided both sides of this equation by 1 dollar instead, we would end up with an equation that looks like that, that
106.823 yen divided by 1 dollar is equal to 1.

What are we doing? Well, we’re taking the units along for the ride, and that’s the key. We always want to take the
units along for the ride. And if we take the units along for the ride, what we get are a bunch of things that actually
amount to 1, and we can multiply by one of these factors, or one of these factors, without penalty. In other words, we
can do it and we affect a conversion, but we haven’t changed the value of the number.

So let me express this by doing a conversion. Suppose I have 50 bucks and I want to know how many yen that’s
going to amount to. You take, what we’re trying to find out is the number of yen in 50 dollars, so let’s put that over on
the left. And the answer has to be in yen. So, overall, the units have to end up being yen. We take 50 dollars, which
is the number we’re trying to convert, i.e., it’s the data. And we include the units, and then we take a conversion
factor, and the conversion factor could be either this guy or this guy, and we have to decide. Which of these are we
going to use? We can multiply by either one with impunity, because, again, they amount to a factor of one when you
include the units along for the ride.

Well, which one of these are we going to choose? And the answer is we want to choose the one, which allows us to
cancel out the units that we don’t want. So we don’t want dollars. So that means we want to choose the conversion
factor that has dollars in the denominator, and then we’ll have yen in the numerator, because the dollars are going to
cancel out. And we can actually go ahead and cancel out the units. So we cancel out dollars with dollars, and then
we multiply 50.00 times 106.823. And then we have four significant figures here in original number, so the overall
conversion is the number of yen in 50.00 dollars is 5,341 yen. That’s all there is to conversion factors, with the
exception that you might have to use more than one conversion factor, and I’ll ultimately show you that.

Now, there are all kinds of different conversion factors. There are all these that you are familiar with. One foot is
equal to 12 inches, one pound is equal to 16 ounces, blah, blah, blah. And one thing to understand is that these are
considered exact. And being exact, what’s implied is that there are infinitely many significant figures. So
1.000000000000 feet is equal to 12.00000000 inches. In other words, these sorts of conversion factors, and any
other that you might write or that you are given, do not limit the number of significant figures. It’s the data that limit the
number of significant figures. And so, going back briefly, it wasn’t that 1 dollar limited our significant figures, because
this is a conversion factor, so these numbers are considered exact. It’s this number here, the data, that limited our
number of significant figures, and we wanted to limit it to four, because there are four here, and so there’s our answer.

Now, there is something else that can be exact, and that is things that are accounted. In other words, a family has
two children. That’s considered to be an exact number, 2.0000000. So, if you wanted to ask how many errors are in
the family, how many errors are on the children, it would be 2 per children times 2 children exactly. So you don’t
worry about limiting significant figures with counting things.

All right, so suppose we wanted to calculate the flow rate of Niagara Falls. And we know, in gallons, that it’s 1.5 times
105 gallons per second and we want it in liters per second, so it’s a nice metric unit. And we have the identity that 1
gallon is equal to 3.7854 liters. So we can remember, make two different factors of one from this relationship, and the
one that we’re going to want is the one that allows us to cancel out gallons, because gallons isn’t our data. And we
want to cancel out the gallons and what we want is we want liters left over, because the answer has to be in liters per
second, which is the rate. And think about this though, that seconds, which is in the answer, and seconds, which is in
the data that we’re starting with, that’s going to go along for the ride and nothing’s going to happen to it. In other
words, we’re sort of halfway there already. We have seconds in the denominator of our units in both the answer and
the piece of data that we’re starting with, so that’s okay. All we have to do is correct for the gallons. So we’ll use a
unit of one, which has gallons in the denominator, so that we can cancel out the gallons.

And then we’ll use this,
liters, 3.7854 liters in the numerator. And again, gallons cancel out, and then we multiply 1.5 times 105 times 3.7854
liters. And then two significant figures here, because 1.5 has two significant figures. And so the rate of water going
over Niagara Falls in metric units is 5.7 times 105 liters per second.

Now, I was mentioning that seconds is going along for the ride here, but what happens if we’re trying to convert both
the numerator of our unit and the denominator of our unit? For instance, a car is traveling 75 miles per hour, what is
its velocity in meters per second? So we have to convert miles into meters, and we have to convert hours into
seconds. And for this problem we’re going to need a whole bunch of different conversion factors. We have that one
mile is equal to 5,280 feet and blah, blah, blah, and you can read all of these, and they’re probably mostly familiar to
you. The one that might not be familiar is one inch is equal to 2.54 centimeters, and that’s the unit that relates length
in the British unit to length in the metric system. And this is actually, again, an identity. It means that one inch is
exactly 2.54 centimeters.

Okay, so we want a rate and we want it to end up to be in meters per second. And we start with our data, which is 75
miles per hour, and then somehow we have to get rid of miles and turn it all into meters. Well, we have something, a
conversion factor involving miles and feet, and we want to use it with miles in the denominator and feet in the
numerator. So there’s one unit of one. And then we have to get rid of the feet, because we want to get it into meters,
so that’s 12 inches over one foot. That’s another unit of one. And then now we have inches, but we want to get into
centimeters, so we’ll use our 2.54 centimeters over one inch. And now, we have to get into meters, and so we’ve got
centimeters in the numerator. So we put centimeters in the denominator and put a meter in the numerator. And there
are 100 centimeters in one meter, so that’s another unit of one. And now we have to deal with the second, excuse
me, with the hours. We want to get rid of hours, hours in the denominator here, so we’ll put hours in the numerator
and minutes in the denominator. And then we don’t want minutes, we want seconds, so we have put minutes in the
numerator, seconds in the denominator. And they key is that all these units are going to cancel out. So miles cancels
out with miles, feet cancels out with feet, inches cancels out with inches, centimeters cancels out with centimeters,
hours cancels out with hours. Denominator, numerator. Minutes cancels out with minutes.

What’s left over? Well, all
these things have canceled out. We have meters here and seconds, and that’s exactly what we want.

So you multiply 75 times 5,280 times 12 times 2.54 divided by 100 divided by 60 and divided by 60 again, and we get
34. And we report it to significant figures, because we started at 75 miles per hour. So 75 miles per hour corresponds
to 34 meters per second.

Now, the final conversion sequence that I want to tell you about is converting from degrees Celsius to degrees
Fahrenheit, and very often in textbooks you’ll see expressions that look like the [Blue’s] conversion sequences here,
where whatever the temperature is in degrees Fahrenheit, you subtract 32, multiply by five -ninths and that gives you
the temperature in degrees Celsius. Alternatively, if you take the temperature in degrees Celsius, multiply it by ninefifths
and add 32, you get the temperature in degrees Fahrenheit. And the key is these two expressions just tell you
that 32 degrees Fahrenheit, which is where water freezes, is equal to 0 degrees Celsius, and 100 degrees Celsius,
which is where water boils, is equal to 212 degrees Fahrenheit.

Well, what’s wrong with these expressions? The answer is they aren’t good about keeping track of units. The units
are implied, but I’m, in a second, going to show you exactly what they should read.

Again, the key is to bring the units
along for a ride. Trust me, I do this myself. When I’m doing these sorts of conversion factors in my laboratory, I keep
the units. I drag the units along and make sure they all cancel out and make sure everything works out right. That’s
how you don’t make a mistake.

So what should these temperature conversion equations say? They should say the temperature in degrees
Fahrenheit minus 32 degrees Fahrenheit, so this whole expression is in degrees Fahrenheit. And then multiply by a
conversion factor fi ve degrees Celsius for every nine degrees Fahrenheit. And then the units are going to work out to
give temperature in degrees Celsius. And similarly there’s an expression that relates temperature in degrees Celsius
to temperature in degrees Fahrenheit, where now the units all work out. So these things are actually going to be true
to their units.

And the final one I’m going to mention here is that we’re going to ultimately learn something called the Kelvin scale,
and the Kelvin scale is related to the Celsius scale. The size of the degree is exactly the same, so one Kelvin divided
by one degree Celsius. And then there’s an additive constant, 273.15 Kelvins, and that’s how we get from the Celsius
scale to the Kelvin Scale. If you’re not familiar with it yet, don’t worry about it. It’s going to come up later.

So remember, always include the units when you’re asked to work a problem. Figure out what the units are supposed
to be in the answer. Figure out what units you’ve got to start with, what your data look like. And then arrange a series
or unit factors, units of one, that allow you to get all of the units to cancel out to give you the units that you want in
your answer, and you’ll be safe and you’ll get these problems correct.

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