In this video, we’re talking about measuring volumes and in particular of the three different states of matter, solid, liquid, and gas. We’ll be talking about measuring the volume of liquids.
Measuring the volume of a liquid is not the easiest thing because we know if we just have the liquid by itself and no container at all, the liquid will fall to the ground and form a puddle. That’s not very helpful for finding its volume: how much space the liquid takes up. We will need some sort of container for the liquid.
Think for a moment of what kind of container you would design if it was meant to measure the volume of a liquid. Maybe, you would design a collection of identical cups that you can pour the liquid out into. Or maybe taking this idea a step further, you would make one big container that had markings on the side which indicated the volume contained at that level, sort of like a measuring cup.
Well, it turns out that the device used for measuring volumes of liquid is actually pretty similar to the second idea we’ve had. This piece of equipment is called a measuring cylinder or a graduated cylinder. The name “graduated cylinder” comes from the fact that the side of the cylinder is marked out with evenly spaced markings or gradations. It doesn’t have to do with the cylinder being a graduate of any particular thing. Although if it did, that would be pretty impressive.
But anyway, like we said, the cylinder has graduated markings or gradations on its side. And typically, the largest hash marks on the cylinder have a particular volume etched in the glass right next to the marks. For example, this lowest large hash mark might indicate a volume of five milliliters of fluid in a cylinder. And then if we move up one large mark, we see that indicates a volume of 10 milliliters, and then onto 15 milliliters, and so on up the cylinder.
We can see though that along with these large hash marks which have numbers written by them, there are smaller ones in between. Normally, these small hash marks don’t have numbers etched in the glass next to them. But they evenly divide the volume between adjacent large hash marks. Looking at this space between the five- and 10-milliliter marks on our cylinder, we can see there are one, two, three, four small hash marks. That means that those marks evenly divide up this volume from five to 10 milliliters into five even volumes.
In other words, each individual small hash mark indicates a change in volume of one milliliter. Since that’s the smallest change in volume we can measure using these markings on the cylinder, we will call that the resolution of the cylinder. So now that we have an idea for how measuring cylinders work, let’s see how they’re used to actually measure the volume of a liquid.
Let’s say that we put our cylinder under a tap of water and we open the tap. We then fill up this cylinder with some amount of water and turn off the tap. The next step of course is figuring out just how much liquid we have in our cylinder. Finding what this volume is involves making a measurement. And to make the measurement, we’ll need to make sure that we’re in the right position. We want our eye to be on level with the surface of the liquid in our cylinder, in this case water.
This isn’t always the most natural thing because often measuring cylinders set on tabletops which are lower than eye level we’ll need to be very careful and intentional to put our eye at the level where it needs to be to make an accurate measurement. If we don’t, if we have our eye at some other position relative to the liquid surface, say for example that it’s higher than that level, then we risk making an inaccurate measurement of the volume of the fluid.
All that to say, it takes a little bit of extra work to get our eye in a position to make a good measurement. But the reward is that we do get a more accurate reading for the volume of the liquid in the cylinder. Once our eye is in the right position, when we look closely at the surface of the liquid in our cylinder we’ll notice something interesting. To help us see this, let’s consider an expanded view of this section of the cylinder.
When we look really closely at the surface of the liquid, the surprising thing we see is that that surface is not flat. We see that actually there’s a curve to it so that it’s higher at the sides and lower at the middle. Why should this be happening? What we now know to be going on is that there’s an interaction taking place between the liquid in the cylinder and the glass walls of the cylinder. Specifically, the liquid is attracted to the walls. It’s drawn to it by electrostatic force.
This force gets weaker the further away the water is from the walls of the cylinder. That’s why the water near the edges of the cylinder is pulled more strongly, but the water in the middle is not pulled very much at all. This curved shape to the surface of the water is so striking. It even has its own name. It’s called a “meniscus.” And this word comes from a word that means crescent.
Looking at the crescent shape of this surface, we can see why that’s so. The fact that liquids in measuring cylinders form a meniscus impacts how we measure the volume of those liquids. The horizontal level we pick at which to measure the liquid volume is where the meniscus has a flat section to it, in this case the bottom of the meniscus.
The reason for this choice is fairly practical. When we think about it, there’re really only two locations on the surface of this water that we could reliably pick out by eye: one is at the flat part of the meniscus where we’ve identified already and the other is at the very top of the meniscus. By picking the flat part of the meniscus as our measurement point, we make a more accurate estimation as to the volume of the liquid in the cylinder.
Notice too that whether we measure at the top or at the bottom of the meniscus that will make a difference in the value we report for the volume of this liquid. The difference between these two points comes out to one milliliter of volume. Now notice we’ve been saying that we measure at the flat part of the meniscus. That’s important to remember because sometimes the meniscus points in the opposite direction. That is, for some liquids, the meniscus points upward rather than downward.
This happens when the liquid rather than being attracted to the walls of the cylinder is repelled from it, which means that the liquid over here and over here in this case not only has been pulled downward by gravity, but it’s also being pushed away from the walls. And that gives this meniscus curve its shape.
So sometimes, we’ll be working with a liquid like water, where the meniscus points downward. We can call that a concave meniscus. In other times, we’ll have another liquid such as mercury, where the meniscus points the opposite way, upward. We call that a convex meniscus. But here is the great thing about measuring this volume: in either case, whether the meniscus points upward or downward, we still measure the volume at the flat part of the meniscus; that is, its level at the center of the cylinder.
This means we will report the volume of this water as 22 milliliters. But we’ll report the volume of this other liquid — we could say it’s mercury — as 23 milliliters. And that all has to do with the fact that we measure the meniscus at its flat section, regardless of which way it curves. Knowing all this, let’s get some practice measuring volumes of liquids through a couple of examples.
The diagram shows a measuring cylinder with liquid in it. What is the volume of the liquid?
Taking a look at this diagram, we see it shows us a graduated or measuring cylinder, which measures the volume of liquids in units of milliliters. We notice further that the large hash marks on the side of the cylinder are marked out in units of five milliliters. So it goes five, 10, 15, 20, and so on. To figure out the volume of the liquid in our cylinder, our first step will be to put our eye on the level of the surface of the liquid.
When we do this, we get an up-close view of this section of the cylinder. Looking closely at this part of the cylinder, we notice in our expanded view that the liquid surface curves upward. This tells us that whatever this liquid is, the glass walls of the cylinder repel it. And more than that, we know that the volume of the liquid is measured at the flat part of this curve, the flat part of the meniscus.
So the question then becomes “what does this volume correspond to?” It’s two small hash marks above the larger hash mark of 45 milliliters. To find out, we can recall that the small hash marks on our cylinder evenly divide up the volume between the large marked ones. So for example, if we were to count the number of small hash marks that appear between the 10-milliliter mark and the 15-milliliter mark, that would show us how much volume change each small hash mark corresponds to.
Let’s do that now. Let’s count the number of hash marks that appear in between these two larger labeled ones. So if we start just above 10 milliliters, we count one, two, three, four small hash marks, then five, a medium-sized one, then six, seven, eight, and then nine hash marks. And the next hash mark is the large labeled one, 15 milliliters. This means that the volume between 10 and 15 milliliters on this cylinder is divided up into 10 even volumes.
That tells us that each small hash mark corresponds to a change in volume of one-half of a milliliter. We can write it this way: we could say that Δ𝑣, the smallest measurable change in volume according to the markings on the cylinder, is equal to one-half or 0.5 milliliters. Now that we know this fact, we can go back to our up-close view and figure out which volume this second hash mark above 45 milliliters corresponds to.
Beginning at 45 milliliters, when we move up one hash mark, that means we’re now at a volume of 45.5 milliliters since each small hash mark corresponds to a volume of Δ𝑣. But we move up a second one. That means that our volume now is 46.0 milliliters. That’s the volume at the measurement of the flat part of the meniscus, the top part of its curve in this case.
So by getting our eye in the right position level with the surface of our liquid in the cylinder and then measuring at the flat part of the meniscus of the curve of the liquid, we found its volume. It’s 46.0 milliliters. Now, let’s look at a second example of measuring liquid volumes.
Victoria uses two measuring cylinders to find the volume of a liquid. She fills the first cylinder up to the top and then pours the rest of the liquid into the second cylinder, as shown in the diagram. She determines that the total volume of the liquid is 90 milliliters. Which of the following statements explains why this answer is incorrect? a) Victoria is reading from the top of the meniscus of the liquid in the first cylinder, rather than the bottom. b) Victoria is reading from the bottom of the meniscus of the liquid in the first cylinder, rather than the top. c) Victoria has not accounted for the liquid about the 50-milliliter mark on the first cylinder. The actual volume of the liquid is greater than 90 milliliters.
Okay, so taking a look at our diagram, we see the two measuring cylinders: the one on the left which is filled up to the top and the one on the right which is partially filled. In the problem statement, we’re told that Victoria claims that the total volume of this liquid is 90 milliliters. But we’re also told that that measurement is incorrect. These three choices a), b), and c) are all options for explaining just why it is that that value is incorrect.
Taking a look at these choices in order, we see that answer option a) claims that the problem is that Victoria is reading from the top of the meniscus of the liquid in the first cylinder rather than the bottom. Interestingly, if we look at answer option b) then, we see that this is very similar to a). This says that the problem is that Victoria is reading from the bottom of the meniscus of the liquid in the first cylinder, rather than the top.
So answer options a) and b) both boil down basically to this. One says that Victoria is reading the volume here. But she should be reading it here, whereas the other says the reverse.
Let’s take a look for a moment at these two measuring cylinders. We see that the cylinders are identical, that they both measure a maximum volume of 50 milliliters. Above that maximum level, neither cylinder has any marks or gradations on it. When it comes to the level of the liquid in cylinder one, this means that whether Victoria measures the level here at the top of the meniscus or here at the bottom, in either case there won’t be a corresponding value for the volume of that liquid. That’s because either level is above the maximum level marked out on the cylinder, 50 milliliters.
This tells us that neither answer option a) nor option b) can really explain why it is that her reading is incorrect. The issue is not where she measures the meniscus on the liquid in cylinder one, but rather it’s that the liquid in this cylinder is above the maximum measurable value. This leads us to answer option c). This says that Victoria has not accounted for the liquid above the 15-milliliter mark on the first cylinder. And that’s true.
Recall that Victoria reported the overall liquid volume as 90 milliliters. To get that, we would add together the 50 milliliters marked out here from cylinder one to the 40 milliliters of liquid in cylinder two. But notice that this leaves out some of the liquid, all of the liquid above that maximum 50-milliliter mark in cylinder one.
Answer option c) goes on to say the actual volume of the liquid is greater than 90 milliliters. And we see that’s true as well. The liquid volume is 90 milliliters plus however much is contained here. So we pick answer option c) as the accurate explanation for why it is that Victoria’s reading is incorrect, that the liquid volume is not 90 milliliters, but rather is more.
Let’s now summarize what we’ve learned about measuring volumes of liquids. We’ve seen in this section that liquid volume is measured in a measuring — also called a graduated — cylinder. And we’ve seen that those cylinders look like this, with volume markings etched into the side of them. Secondly, we saw that liquid in a cylinder forms a curved surface. That’s called a meniscus. For a liquid like water, that curved surface looks like this.
And then, finally, we saw that to measure liquid volume accurately, we measure at the flat part of the meniscus. So in the case of a meniscus that curves down like this, we would measure here, whereas in the case of a meniscus that curves upward, we would measure here. In both cases, we measure the flat part of the meniscus. This then is our process for measuring the volume of a liquid.