Dimensional Analysis and Unit Conversions

Now that we've taken a look at significant figures and how to record measurements, let's take a look at how to do some calculations with these. Specifically, we're going to be taking a look at conversions and then a little bit with density. So we're going to be taking a look at this thing called dimensional analysis, and basically it's a way to convert between one unit and another. What you're going to find is we're going to start with a given. and then we're going to use a conversion factor. We'll talk about those in the next couple slides. And then we plug this in so the units will cancel. I know some of the things that we're going to do in the next couple problems, you may know how to do another way, but I'm trying to show you a method that we will use over and over again throughout the course, this idea of dimensional analysis and getting units to cancel out. So the most common conversions we'll do are metric conversions. And if you'll notice on the left of this slide, table 1.3 gives the metric conversions. But if you'll notice, they like to give them an exponential notation. And I find that's kind of cumbersome to put into a calculator and to work with. So if you'll notice on the right hand side, I've listed in red the six most common metric conversions that you see in chemistry. And so you may want to stop the video at this point and record these into your slot notes. One of the things that I want you to notice is there's a lot of thousands. Okay. And if you'll notice the thousand goes next to the smaller unit. The only one that's not is that centimeters to meters. So really in chemistry, if you're doing a metric conversion, for the most part, it's going to be a factor of a thousand. And if you'll notice in... all of these, the thousand goes next to the smaller of the two. Again, like I said, you may want to pause and go ahead and write these down. We also have some other common conversion factors, which is from the metric to the English unit. And I'm going to be using some of these for the problems. And what you're going to find for the most part, when I give you a problem in this class, I'm going to give you the conversion factor because I'm going to assume that you're going to be able to go look that up or go find the conversion factor, especially when you go to take a midterm and a final exam that's timed. So the first thing we need to do is kind of again familiarize ourselves with the metric system. And so a liter is pretty much about the size of a a quart and a meter is just slightly longer than a yard. Okay. Again, this whole idea of, I used to have to do a lot more with this when I was teaching chemistry 20 years ago, but now I find students are more familiar with the metric system and kind of have an idea of the inner conversion about what the relative sizes are. So now what we're going to do is we're going to take a look at dimensional analysis and we're going to use some of those conversions. Okay so you need to kind of listen to what I'm saying because this is a methodology for solving problems. Like I said some of you could probably do this maybe even in your head this metric conversion. But what I'm trying to do is show you how to do a type of problem that we will use over and over and over again in chemistry. So whenever I run into a problem What I find is I look for the numbers in the unit. Okay, so I've got 83 centimeters. And if I don't know what else to do, I'm going to put a multiplication sign in a division bar. Well, dimensional analysis says let's get rid of the centimeters. The dimension is centimeters, and we're going to analyze it by getting rid of it. So we're going to put it on the bottom. And if you'll notice, our centimeters will cancel out. And now we're looking for meters. So I'm going to put meters on the top. And if you take a look at what you wrote down in your slot notes, there are 100 centimeters in a meter. And so if the 100 is in the bottom, listen to what I'm saying. If the 100 is in the bottom and the reason it's in the bottom is because it goes next to the centimeters. This says. we're going to use division to solve this problem. So we're going to take 83 divided by 100. And we're going to end up with 0.83 meters. If the number's in the bottom, it's division. If the number's in the top, it's going to be multiplication. Notice I didn't say, how do I convert from one unit to another? Well, this time I multiply. This time I divide. Notice the problem is going to tell you which function to use in your calculator. Again, this is that problem solving that I'm trying to show you how to get used to, because we're going to use it with things that are fairly exotic a little bit later on. in the semester. But we're going to default back to this idea, get rid of the units. And if the number's in the bottom, you're going to hit division in your calculator. If the number's in the top, you're going to hit multiplication. So in this problem, we're going to convert 4.59 liters into milliliters. Like the previous problem, we're going to look for the number. So I've got 4.59 liters. multiplication sign, division bar. And I'm going to put liters on the bottom because I want to get rid of liters. And I'm looking for milliliters. So I'm going to put milliliters on the top. And then if you remember that the numbers in red, we found that there's a thousand milliliters in a liter. So because this number is on the top, it's now going to be multiplication instead of division. So 4.59 times a thousand. gives us 4590 milliliters. Notice how the units determine where the number goes, whether it's on the top or on the bottom, and then that's going to tell you whether you hit the multiplication sign or division sign in your calculator. So if you notice in this one we're going to convert the 78.24 milligrams into grams. And so we're going to take the 78.24 milligrams, multiplication sign, division bar. In this case we're going to put milligrams on the bottom, grams on the top, and then if we go back oh we got milligrams and grams. Most common is a thousand and a milligram smaller than a gram. So I'm going to put the thousand in front of the milligrams. And so because it's on the bottom, we're going to do division. And so 78.24 divided by a thousand gives us that 0.07824 grams. Or if we want to record this to make sure we've got the correct number of significant figures, we'd write it as 7.824 times 10 to the minus second in scientific notation. Okay, so now we're going to take a look at the temperature scales or temperature conversions. Fahrenheit basically took water and he boiled it and then he took the, and then what he did is he added salt to it, or actually he froze it and then added salt to it, tried to find the lowest and the highest temperature, and he came up with that scale of 32 to 212. Zero degrees Fahrenheit is pretty much the lowest you can get salt water. before it'll freeze. And so that's not a convenient scale to work with. So Celsius came along and said, well, let's divide it into a scale of 100, which is more convenient. And the zero degrees is where pure water freezes and 100 degrees Celsius is where water boils. Well, and then along came Kelvin and Kelvin says, well, where is absolute zero? And so Kelvin came up with a scale. where the zero on the Kelvin scale is absolute zero. Okay. At the bottom of the slide, you'll notice there are some formulas for converting from Fahrenheit to Celsius and Celsius to Fahrenheit. If we need to do that, basically it's a plug and chug. You put the numbers into the formula and then you can do a little bit of algebra with it. So in this slide, what we're going to do is, or this problem, we're going to convert 172.9 degrees Fahrenheit to degrees Celsius. And like I said on the previous slide, this just becomes a formula problem. So we need the Celsius to Fahrenheit. And so we're going to plug our numbers in, and that's 172.9 minus the 32. And then we're going to multiply by 5 and then divide by 9. and we end up with 78.3 degrees celsius As I mentioned when we were talking about the three different scales, we needed a scale that basically has an absolute because all the other ones are kind of degrees. And so what we find is Kelvin came up with a temperature or estimated a temperature of zero Kelvin. At that point, molecules cease to move and we'll talk a little bit more about that when we take a look at reaction kinetics. But it's been estimated at negative 273.15 degrees Celsius. So this makes our conversion pretty simple in that all we need to do is either add or subtract the 273.15. And the Q trick to this is if you look in the alphabet, C comes before K. And if we were to... put numbers with the letters in the alphabet, C would be smaller than K. Okay, so the K is always going to be bigger than the C. So we don't have to memorize the formula. We just need to remember the 273.15. And so if we've got 212.00 Kelvin, we realize the Celsius has to be smaller. So you're going to subtract the 273.15. and we end up with the negative 61.15 degrees celsius be careful with that negative sometimes you say we have a tendency to drop it off and forget about it if i want to go from celsius to kelvin i need to make the celsius and make it bigger so i'm going to add 273.15 and so i end up with a 320.40 Kelvin. Notice there's no degree sign on the Kelvin because it's an absolute scale. So what we found with problem solving is we need to consider the starting and the ending units. Remember, that's how we get them to cancel out and make a list of equalities or conversion factors that are going to be used. Right now, we've only been using one, but we can add multiple conversion factors into the same problem. And we will a couple chapters down the road. Okay. Make sure your units are written so, again, they cancel out. Once you get the units, then you can put the numbers. And if the numbers are in the top, it's multiplication. The numbers in the bottom, it's division. So let's take a look at a couple problems. And 63 says, how many milliliters of a soft drink are contained in a 12-ounce can? Notice that black in brackets. Sometimes what I'll do is I'll give you the conversion factor in the problem. And what you'll see is it's always listed in brackets and it's in bold. So this tells us the 29.6 milliliters is equal to one fluid ounce. So we're going to start out with 12 ounces. Multiplication sign, division bar. I'm going to put the ounces on the bottom. We're looking for milliliters. So we get rid of the ounces. So because the numbers on the top it's multiplication, I'm going to take 12 times the 29.6 and I'm going to end up with 355 milliliters or basically a little bit over a third of a liter. Number 69, what many medical laboratory tests are run using five microliters. a blood serum. What is the volume in milliliters? Well, the problem we run into is the, um, we have a conversion from microliters to liters, but we really don't have one directly from microliters to milliliters. And so if I start out with 5.0 microliters, multiplication sign, division bar, I'm going to put the microliters on the bottom. Well, from our chart, We now can find that it's 1.0 times 10 to the minus 6 liters per microliter. But if you notice, this is liters, but we're still looking for milliliters. Well, we've done a liters to milliliters problem already. So if we don't have the final unit we're looking for, what we're going to find is we're just going to add another conversion factor. It doesn't become another problem. We just add another conversion problem. And so we're going to get rid of the liters and we're going to put the thousand milliliters on the top. And if you remember we did that in the problem of converting the 4.59 liters to milliliters. Liters will cancel out. If you notice what we have left is milliliters and that's what we're looking for. So this says take 5 times 1.0 times 10 to the minus 6 times a thousand. and we end up with 5.0 times 10 to the minus third milliliters. So we've done some volume. We can do some mass. A long ton is 2240 pounds and in your slot notes on the page 15 we had some metric and English conversions. And so I've got 2,240 pounds, multiplication sign, division bar. If you'll notice from our conversions, we're going to put 2.2046 pounds. So the pounds will cancel out. Put the kilograms on the top because this is in the bottom. It's going to be a multiple, it's going to, sorry, it's going to be a division problem. And we end up with 1,020 kilograms. Sometimes we have units that are squared and cubed. And once you know the trick, this is pretty simple to do. So both of these are converting miles squared into kilometers. Okay. Or, and, or miles squared into kilometers squared and miles cubed into cubic kilometers or kilometers cubed. And so I would take the 96,981 miles. Notice the miles is squared. So if you'll notice, I square my conversion factor because I've squared the unit. Goes on the bottom because it needs to stay next to the miles. And we end up with kilometers squared. And we end up with 251,180. 80 kilometers squared. So if we estimate the volume of the oceans, we've got it as miles cubed, 330 million miles cubed. Divide that by the 0.62137 cubed is going to give us the kilometers cubed. And that's going to be 1.4 times 10 to the 9th or 1.4, what is that? Thousand billion kilometers cubed. Now what we're going to do is we're going to take a look at density. Okay. Density is mass over volume. So the density of our luma will be 2.70. But not only can density be calculated, it can also be used as a conversion factor. So if you'll notice, we're going to use density as a conversion factor. Gives us the density of mercury. Okay, so we've got 6.0 centimeters cubed, multiplication, sine, division bar. We now know because of dimensional analysis, we got to get rid of the centimeters cubed. We're going to put grams on the top and if you'll notice our conversion factor says 13.5939 goes next to the gram so this is a multiplication problem and we end up with 81.6 grams. I find students sometimes will want to plug this into the density equals mass over volume and sometimes end up with the inverse of the answer just because of the algebra. I find it's easier to use a conversion factor. your use density as a conversion factor. So this one's a little bit different in that we're starting out with grams instead of milliliters. Multiplication sign, division bar. We're going to put the grams in the bottom so they'll cancel out. The 4.39 goes to the grams from the density that's given. centimeters cubed because it's on the bottom it's going to be division and so we end up with 5.1 centimeters cubed this point we're done with section 1.6 i hope you've been working on the problems in your homework quiz okay as you complete the lectures i would work on a couple quiz instead of trying to complete everything and then do all the problems at one time.

and then we're going to use a conversion factor. We'll talk about those in the next couple slides. And then we plug this in so the units will cancel. I know some of the things that we're going to do in the next couple problems, you may know how to do another way, but I'm trying to show you a method that we will use over and over again throughout the course, this idea of dimensional analysis and getting units to cancel out. So the most common conversions we'll do are metric conversions.

And if you'll notice on the left of this slide, table 1.3 gives the metric conversions. But if you'll notice, they like to give them an exponential notation. And I find that's kind of cumbersome to put into a calculator and to work with. So if you'll notice on the right hand side, I've listed in red the six most common metric conversions that you see in chemistry.

And so you may want to stop the video at this point and record these into your slot notes. One of the things that I want you to notice is there's a lot of thousands. Okay. And if you'll notice the thousand goes next to the smaller unit.

The only one that's not is that centimeters to meters. So really in chemistry, if you're doing a metric conversion, for the most part, it's going to be a factor of a thousand. And if you'll notice in...

all of these, the thousand goes next to the smaller of the two. Again, like I said, you may want to pause and go ahead and write these down. We also have some other common conversion factors, which is from the metric to the English unit. And I'm going to be using some of these for the problems. And what you're going to find for the most part, when I give you a problem in this class, I'm going to give you the conversion factor because I'm going to assume that you're going to be able to go look that up or go find the conversion factor, especially when you go to take a midterm and a final exam that's timed.

So the first thing we need to do is kind of again familiarize ourselves with the metric system. And so a liter is pretty much about the size of a a quart and a meter is just slightly longer than a yard. Okay. Again, this whole idea of, I used to have to do a lot more with this when I was teaching chemistry 20 years ago, but now I find students are more familiar with the metric system and kind of have an idea of the inner conversion about what the relative sizes are. So now what we're going to do is we're going to take a look at dimensional analysis and we're going to use some of those conversions.

Okay so you need to kind of listen to what I'm saying because this is a methodology for solving problems. Like I said some of you could probably do this maybe even in your head this metric conversion. But what I'm trying to do is show you how to do a type of problem that we will use over and over and over again in chemistry. So whenever I run into a problem What I find is I look for the numbers in the unit.

Okay, so I've got 83 centimeters. And if I don't know what else to do, I'm going to put a multiplication sign in a division bar. Well, dimensional analysis says let's get rid of the centimeters. The dimension is centimeters, and we're going to analyze it by getting rid of it.

So we're going to put it on the bottom. And if you'll notice, our centimeters will cancel out. And now we're looking for meters.

So I'm going to put meters on the top. And if you take a look at what you wrote down in your slot notes, there are 100 centimeters in a meter. And so if the 100 is in the bottom, listen to what I'm saying. If the 100 is in the bottom and the reason it's in the bottom is because it goes next to the centimeters.

This says. we're going to use division to solve this problem. So we're going to take 83 divided by 100. And we're going to end up with 0.83 meters.

If the number's in the bottom, it's division. If the number's in the top, it's going to be multiplication. Notice I didn't say, how do I convert from one unit to another? Well, this time I multiply. This time I divide.

Notice the problem is going to tell you which function to use in your calculator. Again, this is that problem solving that I'm trying to show you how to get used to, because we're going to use it with things that are fairly exotic a little bit later on. in the semester.

But we're going to default back to this idea, get rid of the units. And if the number's in the bottom, you're going to hit division in your calculator. If the number's in the top, you're going to hit multiplication.

So in this problem, we're going to convert 4.59 liters into milliliters. Like the previous problem, we're going to look for the number. So I've got 4.59 liters.

multiplication sign, division bar. And I'm going to put liters on the bottom because I want to get rid of liters. And I'm looking for milliliters.

So I'm going to put milliliters on the top. And then if you remember that the numbers in red, we found that there's a thousand milliliters in a liter. So because this number is on the top, it's now going to be multiplication instead of division. So 4.59 times a thousand. gives us 4590 milliliters.

Notice how the units determine where the number goes, whether it's on the top or on the bottom, and then that's going to tell you whether you hit the multiplication sign or division sign in your calculator. So if you notice in this one we're going to convert the 78.24 milligrams into grams. And so we're going to take the 78.24 milligrams, multiplication sign, division bar. In this case we're going to put milligrams on the bottom, grams on the top, and then if we go back oh we got milligrams and grams.

Most common is a thousand and a milligram smaller than a gram. So I'm going to put the thousand in front of the milligrams. And so because it's on the bottom, we're going to do division. And so 78.24 divided by a thousand gives us that 0.07824 grams.

Or if we want to record this to make sure we've got the correct number of significant figures, we'd write it as 7.824 times 10 to the minus second in scientific notation. Okay, so now we're going to take a look at the temperature scales or temperature conversions. Fahrenheit basically took water and he boiled it and then he took the, and then what he did is he added salt to it, or actually he froze it and then added salt to it, tried to find the lowest and the highest temperature, and he came up with that scale of 32 to 212. Zero degrees Fahrenheit is pretty much the lowest you can get salt water.

before it'll freeze. And so that's not a convenient scale to work with. So Celsius came along and said, well, let's divide it into a scale of 100, which is more convenient. And the zero degrees is where pure water freezes and 100 degrees Celsius is where water boils. Well, and then along came Kelvin and Kelvin says, well, where is absolute zero?

And so Kelvin came up with a scale. where the zero on the Kelvin scale is absolute zero. Okay.

At the bottom of the slide, you'll notice there are some formulas for converting from Fahrenheit to Celsius and Celsius to Fahrenheit. If we need to do that, basically it's a plug and chug. You put the numbers into the formula and then you can do a little bit of algebra with it. So in this slide, what we're going to do is, or this problem, we're going to convert 172.9 degrees Fahrenheit to degrees Celsius. And like I said on the previous slide, this just becomes a formula problem.

So we need the Celsius to Fahrenheit. And so we're going to plug our numbers in, and that's 172.9 minus the 32. And then we're going to multiply by 5 and then divide by 9. and we end up with 78.3 degrees celsius As I mentioned when we were talking about the three different scales, we needed a scale that basically has an absolute because all the other ones are kind of degrees. And so what we find is Kelvin came up with a temperature or estimated a temperature of zero Kelvin. At that point, molecules cease to move and we'll talk a little bit more about that when we take a look at reaction kinetics.

But it's been estimated at negative 273.15 degrees Celsius. So this makes our conversion pretty simple in that all we need to do is either add or subtract the 273.15. And the Q trick to this is if you look in the alphabet, C comes before K.

And if we were to... put numbers with the letters in the alphabet, C would be smaller than K. Okay, so the K is always going to be bigger than the C. So we don't have to memorize the formula.

We just need to remember the 273.15. And so if we've got 212.00 Kelvin, we realize the Celsius has to be smaller. So you're going to subtract the 273.15. and we end up with the negative 61.15 degrees celsius be careful with that negative sometimes you say we have a tendency to drop it off and forget about it if i want to go from celsius to kelvin i need to make the celsius and make it bigger so i'm going to add 273.15 and so i end up with a 320.40 Kelvin.

Notice there's no degree sign on the Kelvin because it's an absolute scale. So what we found with problem solving is we need to consider the starting and the ending units. Remember, that's how we get them to cancel out and make a list of equalities or conversion factors that are going to be used.

Right now, we've only been using one, but we can add multiple conversion factors into the same problem. And we will a couple chapters down the road. Okay.

Make sure your units are written so, again, they cancel out. Once you get the units, then you can put the numbers. And if the numbers are in the top, it's multiplication. The numbers in the bottom, it's division. So let's take a look at a couple problems.

And 63 says, how many milliliters of a soft drink are contained in a 12-ounce can? Notice that black in brackets. Sometimes what I'll do is I'll give you the conversion factor in the problem.

And what you'll see is it's always listed in brackets and it's in bold. So this tells us the 29.6 milliliters is equal to one fluid ounce. So we're going to start out with 12 ounces.

Multiplication sign, division bar. I'm going to put the ounces on the bottom. We're looking for milliliters. So we get rid of the ounces. So because the numbers on the top it's multiplication, I'm going to take 12 times the 29.6 and I'm going to end up with 355 milliliters or basically a little bit over a third of a liter.

Number 69, what many medical laboratory tests are run using five microliters. a blood serum. What is the volume in milliliters? Well, the problem we run into is the, um, we have a conversion from microliters to liters, but we really don't have one directly from microliters to milliliters.

And so if I start out with 5.0 microliters, multiplication sign, division bar, I'm going to put the microliters on the bottom. Well, from our chart, We now can find that it's 1.0 times 10 to the minus 6 liters per microliter. But if you notice, this is liters, but we're still looking for milliliters.

Well, we've done a liters to milliliters problem already. So if we don't have the final unit we're looking for, what we're going to find is we're just going to add another conversion factor. It doesn't become another problem. We just add another conversion problem. And so we're going to get rid of the liters and we're going to put the thousand milliliters on the top.

And if you remember we did that in the problem of converting the 4.59 liters to milliliters. Liters will cancel out. If you notice what we have left is milliliters and that's what we're looking for. So this says take 5 times 1.0 times 10 to the minus 6 times a thousand. and we end up with 5.0 times 10 to the minus third milliliters.

So we've done some volume. We can do some mass. A long ton is 2240 pounds and in your slot notes on the page 15 we had some metric and English conversions.

And so I've got 2,240 pounds, multiplication sign, division bar. If you'll notice from our conversions, we're going to put 2.2046 pounds. So the pounds will cancel out.

Put the kilograms on the top because this is in the bottom. It's going to be a multiple, it's going to, sorry, it's going to be a division problem. And we end up with 1,020 kilograms. Sometimes we have units that are squared and cubed. And once you know the trick, this is pretty simple to do.

So both of these are converting miles squared into kilometers. Okay. Or, and, or miles squared into kilometers squared and miles cubed into cubic kilometers or kilometers cubed.

And so I would take the 96,981 miles. Notice the miles is squared. So if you'll notice, I square my conversion factor because I've squared the unit.

Goes on the bottom because it needs to stay next to the miles. And we end up with kilometers squared. And we end up with 251,180. 80 kilometers squared.

So if we estimate the volume of the oceans, we've got it as miles cubed, 330 million miles cubed. Divide that by the 0.62137 cubed is going to give us the kilometers cubed. And that's going to be 1.4 times 10 to the 9th or 1.4, what is that?

Thousand billion kilometers cubed. Now what we're going to do is we're going to take a look at density. Okay. Density is mass over volume. So the density of our luma will be 2.70.

But not only can density be calculated, it can also be used as a conversion factor. So if you'll notice, we're going to use density as a conversion factor. Gives us the density of mercury.

Okay, so we've got 6.0 centimeters cubed, multiplication, sine, division bar. We now know because of dimensional analysis, we got to get rid of the centimeters cubed. We're going to put grams on the top and if you'll notice our conversion factor says 13.5939 goes next to the gram so this is a multiplication problem and we end up with 81.6 grams. I find students sometimes will want to plug this into the density equals mass over volume and sometimes end up with the inverse of the answer just because of the algebra.

I find it's easier to use a conversion factor. your use density as a conversion factor. So this one's a little bit different in that we're starting out with grams instead of milliliters.

Multiplication sign, division bar. We're going to put the grams in the bottom so they'll cancel out. The 4.39 goes to the grams from the density that's given. centimeters cubed because it's on the bottom it's going to be division and so we end up with 5.1 centimeters cubed this point we're done with section 1.6 i hope you've been working on the problems in your homework quiz okay as you complete the lectures i would work on a couple quiz instead of trying to complete everything and then do all the problems at one time.

Transcript for:Dimensional Analysis and Unit Conversions

Transcript for:
Dimensional Analysis and Unit Conversions