- Welcome back. Here we go to our last topic for this section of the class. And it's the topic of density. Now you've probably heard of density before. And it's a little bit about what things float and what things sink. So what I have is little demonstration that I'm going to do for you. And we all know that if you take a glass of water like this and you drink it, or whatever, it tastes yummy. But we all know that if you put ice cubes into water-- so I grabbed a couple of ice cubes there-- they float because that's just what ice cubes do in water. But if you do things just right-- so what I'm going to do is I'm going to get those ice cubes, and I'm going to put ice cubes into this one right here. It's just water. You blow on it. And if you blow on it just right, the ice cubes sink to the bottom instead of floating. Well, how can that be? How can blowing on it make the ice cubes sink to the bottom? And it turns out they can't actually. It turns out it's related to density. But the first thing we want to know is, what is density? If we're saying, hey, these things are related to density, we have to know what density is. And density has a definition. And the easiest way to think about the definition is to actually think about the equation that governs density. And density is mass over volume. It's a relationship between how much stuff is in there, how much does it weigh, it's mass, and how much space does it take up. And if you think of something like Styrofoam, those little Styrofoam peanuts that you get in packaging, or a big Styrofoam if you're opening up your giant widescreen TV thing, huge. They are absolutely huge. But they don't weigh anything. And so, you've got something with a very, very large volume and a very, very small mass. And when you take a small number and divide by a really big number, you get a tiny number. And so the density of Styrofoam is actually very small. However, if you take something like a lead weight, a small lead weight, and try to pick it up, and, oh, gosh, that's really, really heavy because it's got a very large mass and a very small volume. And if you take a large number and divide it by a small number, you get a large number. But the cool thing is, is density is inherent to a subset. So when I've got water here in liquid water, it's got an inherent density. In fact, the inherent density of liquid water is 1.00 grams. That's our mass divided by milliliters. That's our volume. And all that says is if you take 1 milliliter of water, liquid water, it has a mass of one gram. And remember, a milliliter-- oops, my pen is having a hard time here-- a milliliter, 1 milliliter is the same thing as 1 cubic centimeter. A cubic centimeter is when you take a centimeter on each side, and make a cube out of that. And that's actually the definition of a milliliter. So 1.00 grams per milliliter, that's what it is for liquid water. Now it turns out, it does change slightly with temperature but usually on the fourth or fifth digit, so we don't worry about that too much in this class. To three digits, the density of liquid water is 1.00 grams per milliliter. I want you to know this fact because on tests or quizzes I might ask you a problem that involves the density of water and I'm not going to tell you the density of water. So you need to know that it's one, which is pretty easy. In fact, 1.00 to three significant figures. Now why does the ice cube float inside this water? And the ice cubes are going away, but you can see they're still there. Why does the ice cube float? Well it turns out, things float when their density is less than the density of the liquid. So the ice floats because its density is less than that of the liquid. It turns out, when water freezes, it expands a little. So the same amount of water takes up more space. And if you have gram per milliliter, if you're same amount of stuff takes up more space, that makes the density smaller. So the density of ice is slightly less than one. And so it floats in water. Now, how did I get this special ice over here that seems to sink in water? And it turns out, it's not that I did special ice. The ice cubes are the same. They're just from my freezer. What I did is I tricked you because I can do that really easily over the internet. I tricked you. There's not actually water in here. If I take a big whiff-- [COUGHING] It makes me cough a little because what's in there is isopropyl alcohol. Now what you just saw was not a good example of how you're supposed to sniff things in chemistry. I knew exactly what was in my container. And so I did that on purpose for you. Just as a lesson for if you ever do need to smell something in a chemistry lab, you just do this and you just get a little bit towards your face. You waft it is what we tell you. OK. Now isopropyl alcohol is stuff that's in your cabinet. That's why I'm not worried about wearing goggles or anything like that. It is stuff that's in our household. I'm not going to spill it into my eyes. That would be really not good. But it is a just a generic household chemical. So why does that matter? Well turns out, that the density of isopropyl alcohol is 0.786 grams per milliliter. It's significantly smaller than the density of water. And so when you have the ice, which has a density larger than that, you end up having ice sink inside your isopropyl alcohol instead of floating. And so, it wasn't the ice that was different. It wasn't that I blew on it. It was that in this container is isopropyl alcohol, which is less dense than water. And the ice then is more dense than the isopropyl alcohol. And so it sinks. So we have to think about density as this relationship. It's not about how much it weighs. It's not about mass. It's not about volume, how big it is. It's about the combination of the two. It's the density, it's the mass and the volume, the ratio between them. Something extraordinarily heavy can float if the overall density is less than that of water. Consider a giant ship, like an aircraft carrier. That thing's huge, massive, weighs tons and tons and tons. However, because there's so much air inside of it trapped inside, the overall density of the ship is less than that of water, so it floats. But you take a rock, a little tiny pebble, weighs almost nothing. But it sinks right to the bottom of water because it's lots of mass in a small space. So it's got a high density, higher than water. So that's what we're going to be thinking about in terms of density. Is that it's always that relationship between mass and volume, not just mass, not just volume. In this class, we're primarily going to use units of grams per milliliter. Although any mass unit and any volume unit can work for a density, we're going to stick mostly with grams per milliliter unless we're doing conversion problems just for fun. So let's just do a couple example problems just to get used to using density. "Find the density of a 7.22 milliliter sample of metal if it has a mass of 23.88 grams." What are we going to do? Well, if we're saying we're going to find the density, one of the first things you do in this class, when you run across something that has an equation associated with it, it's never a bad idea to write down that equation. So any time we see density, it's a great idea to just write down that we know density is m over v our shorthand for mass over volume. And so if we want to know density, we need to know mass and we need to know volume somehow. Luckily in this problem, they're given to us. We just have to know our units well enough. Mass. Well, we've got two numbers in here. We've got a milliliter sample and we've got grams of our sample. Which one corresponds to mass? And that is our grams, 23.88 grams of our sample. And we're going to divide by the volume, which is 7.22 milliliters. And so you plug that into your calculator and you get a whole 3.31 grams per milliliter. So this is a unit that's got two parts to it. It's not just grams, it's not just milliliters, it's grams over milliliters. And we need to make sure we're using that unit any time we have density. Significant figures. We do significant figures when we do density. This top one has four significant figures. This bottom one has three significant figures. It's division, which means our answer can only have three significant figures. And we get 3.31 grams per milliliter as the density. But any of those variables we can solve for in density. We can solve for d, m, or v as long as we have the other two. So in this next problem, we're finding the volume of a piece of wood when we know the density and we know that it weighs 18.3 grams. And we know that grams is a measure of mass. So what I encourage you to do is always start with your equation. I know we're solving for volume. But the chances of you making an algebra error if you try to write down the algebra after doing it in your head is much, much higher than if you write down the correct equation then do the algebra on paper. So if I'm solving for volume, one of the things I'm going to show you incorrectly first, is I see people all the time. They want to say, oh, let me just take m and then I get v equals d over m. That's not OK. Because what do we have? We have volume in the denominator not in the numerator. I'm going to delete all these guys here because they are not correct. What do we need to do? Any time what we're solving for is in the denominator, the first thing we need to do is get it out of the denominator. How do we get volume out of the denominator? We multiply both sides by v. One on top, one on bottom cancels out. And we get m equals d times v. Now we can solve for volume by dividing both sides by d. And we get v equals m over d. Now that we have that equation, we can plug in the numbers from the problem. What's our mass? Our mass is in units of grams. We have 18-- sorry. 18.3 grams. And our density, 0.24 grams per milliliter. Notice that as I'm writing out these things, I'm writing out units on every single step. I'm not just writing 18.7 or 18.3 divided by 0.24. I'm writing out units in every single step. I expect you to be doing that whenever you submit work to me. There is going to be units on every single step of your problem. From the very beginning to the very end, you're going to show how units cancel. I've got grams here, grams canceling there. Now what my units end up being is 1 over, 1 over milliliters, which we talked about before. 1 over 1 over something is just that something. So we get milliliters out of that, which is what we want. We wanted a unit of volume. So it's good that we got a unit of milliliters. That's also a good thing to double-check. When you get to the end of your problem and you check what units you got, it hopefully matches the units of what you're expecting. Quite often, I see people calculating volumes and they get units of grams. Or they're calculating mass and they get units of grams per milliliter. That's a density. So making sure that the units that you get are the units that you want is also a good check on your work. So this is just a plug into a calculator. 18.3 divided by 0.24. And we get 76.25 milliliters. What are our sig figs? We've got three here. We've got two here. So we can only take two down here. And we get 76 milliliters as our answer. That's the volume of that piece of wood. So those are simple problems using density, where we're just plugging in some known values from problem statement. Sometimes in lab, we don't have the volume of something needed to find the volume of something. And for simple objects, it works really well. So if you have a cube, and there's x length on each side, it's just x times x times x is going to be the volume of the cube, or x cubed. So if it was a 1-centimeter cube, it would be 1 times 1 times 1. And it'd be 1 cubic centimeter. So as example, 1 centimeter times 1 centimeter times 1 centimeter is equal to 1 cubic centimeter, or centimeter cubed is how we would say that. For a cylinder, there's an equation that governs this. You may or may not remember that it's the area of the circle times the height. That's the volume. We're not really going to use that too much in this class. Or a sphere, if you remembers, 4/3 pi r cubed is the volume of the sphere. Again, we're not going to really use that too much in this class. So there's ways of calculating the volume of regular objects. But it gets a little bit harder if you've got something shaped like that. How do you calculate the volume of something like that? This isn't a moot question. If you had a gold nugget back in the olden days and you wanted to determine whether it's gold, one of the ways they did that is by determining density. But a nugget has a shape of a nugget. It doesn't have a something that's easily measured in terms of volume. So how do you measure something when it has an odd shape? How do you measure its volume? Well, one very common way is by using what's called the method of displacement. And the method of displacement is relatively simple in theory. You take something like a rock here or a cylinder, and you're trying to find the volume of it. Now a cylinder, you say, well we had an equation there. But that's for a perfect cylinder. What if there is a scratch in it? What if it looks like this one and there's like dents and different shapes on the edges. Then that equation doesn't work anymore. So what we do is we take a graduated cylinder. And it's got all these markings on it, which tell us exactly the volume that is present-- whoops. Uh-oh. My computer's having a hard day. There we go-- the volume that's present at any given point. And so we can measure the volume. And so we fill up the graduated cylinder part way. We measure the initial volume. Here it was 35.5 milliliters. We then put our item into the graduated cylinder. Measure the new volume. And that tells us what the volume of our object was when we subtract the two of them. So we'd here, in this case, we take 45 minus 35.5 and we'd get 9.5 milliliters. Remembering that when we subtract, our rule for significant figures is number of digits after the decimal. I kept one digit after the decimal. So even though those two numbers that we started with had three significant figures to begin with, I ended up with only two significant figures because of the subtraction rules versus the multiplication rules. So that's a really common way that chemists will use in order to calculate the volume or determine the volume of irregular objects. And once you have that, you can then use that volume in equations where you're trying to find density by taking the mass of the object. So here is just a thought problem to think about. We've got room temperature. Water has a density of 1.00 grams per cubic centimeter. Remember, a cubic centimeter and a milliliter are the same thing. So I could also write grams per milliliter. It would be exactly the same. "But as salt solution has a density of 1.2 grams per cubic centimeter." So its density is slightly higher. So what I want to ask without doing any calculations, what do you think is heavier, 20 milliliters of water or 22 milliliters of salt solution, or are they the same? You thought I froze. Yeah, you thought it was buffering. No. I was just pausing. OK. So which one's going to be heavier? Well, at first glance, you might be like, well, if I have 20 milliliters of each, isn't it going to be the same? But think about it. If you had 20 milliliters of lead weight and 20 milliliters of Styrofoam, are they the same? No. Because they have different densities. And the same here. The thing that has a lower density means that it's going to be lighter if you have the same volume. Imagine, these are grams per cubic centimeter. What if I had 1 cubic centimeter? Then that 1 cubic centimeter, 1 milliliter, would weigh 1 gram for a water solution. And a salt solution would weigh 1.2 grams. So same volume, more mass. And so the 20 milliliters of salt solution is what we think would be heavier. In this case, it would be more massive, if you want to call it that. So we can think about problems like that as well. Lastly in this part on density, we're going to talk about something called specific gravity. Then what is specific gravity? It's a way of talking about density where we remove the units. How can you just remove units? Well, you remove units by dividing by them. So specific gravity has a definition, which is the density of your sample divided by the density of water. Now this seems OK. But the funny part is, what's the density of water? And we said to three significant figures. It's 1.00 grams per milliliter. So to find a specific gravity, you're taking the density your sample and you're dividing by one. And when you divide by one, what do you get? You get that same number back. So the specific gravity of something with a density of 1.2 grams per milliliter is 1.2 grams per milliliter divided by 1.0 grams per milliliter, is equal to 1.2. And what you end up with is you end up with the density again but without any units. Now it turns out, if you need more digits, four significant figures, five significant figures, the density of water is not one. And so we actually end up with a slight change in our specific gravity versus our density. But in this class, we never actually end up using a density more than three digits. And so, what we end up with is our density back without units. But we have to understand where that came from. This is actually quite commonly used clinically. If you've ever taken a urine test, they take your urine. And one of the things they do is they run a specific gravity test. And what that does, very quickly, is it tells them whether you are hydrated or not hydrated. If you are dehydrated, there can be lots of stuff dissolved in your urine, and you're going to have a high specific gravity for your urine. And they'll be able to tell that fairly quickly. So you need to know how to use specific gravity. If I gave you the density of-- I'm sorry, the specific gravity is something and say, hey, the specific gravity is 3.5, you'd have to be able to manipulate this equation to find the density of the sample by multiplying out by the density of water. And you'd end up in that case getting 3.5 grams per milliliter. But the neat thing about specific gravity, it's one of the very few numbers in chemistry that has no units, no units at all. It's kind of weird but that's just how it is. And I think that was the point of making something called specific gravity. It's the way of talking about density without having to worry about what units it's in. All right. That is, I believe, our discussion of density and specific gravity, and the end of this section of our chapter. We're going to move on next, and we're going to talk about energy, which is a fun chapter. Thanks so much for listening. Bye.