Video: Charles’ Law

In this lesson, we will learn how to use the formula 𝑉/𝑇 = constant (Charles’ Law) to calculate the volume or temperature of a gas that is heated or cooled at a constant pressure.

12:53

Video Transcript

In this video, our topic is Charles’ law. This law tells us about the behavior of gases when the pressure of that gas is held constant. In our diagram, showing a gas at two different states, this one and this one, the piston pushing down on the gas exerts the same pressure in each case. But looking at these two states, we can see that the space the gas takes up, as well as the temperature of the gas, is not constant. Charles’ law connects these properties of a gas.

To understand this law better, we can consider the fact that it’s an experimental gas law. That means it’s a mathematical relationship that was derived from experimental observations. Charles’s law was first developed in the 1780s in some unpublished documents by a researcher named Jacques Charles. Around that time, scientists were experimenting to better understand the behavior of gases, their pressure, their volume, their temperature, and so on. We can imagine an experiment involving an enclosed gas, which is under some amount of pressure. In our case, that pressure is provided by the weight pushing down on the gas. Now, if this top plate is able to move up and down within this container, while not letting any gas through, and we keep the weight on this plate the same. Then that means the gas will always be subject to the same amount of pressure. That is, its pressure will be constant.

Under these conditions, we can make a measurement of the temperature of this gas as well as its volume, the space it takes up. And then, let’s say we plot that point on our graph over to the right. So we have a temperature of whatever value this is and a volume of whatever value this is. Let’s say that we then start to raise the temperature of this gas by applying some heat to it. If we do that, we’ll observe that as the gas heats up, its volume increases. And if at one moment in time we make a measurement of the volume and the temperature, we would have another data point to plot on our graph. After plotting that point, if we let the temperature of the gas continue to increase, that is, continue to heat it, then we’ll observe the gas volume continue to grow as well.

From this, we can get a third data point to plot. We can recall that as all this goes on, the pressure the gas is under is the same. We have the same weight pushing down on it all throughout. Charles’s law then is a description of the volume that gas takes on, compared to its temperature, when the pressure of that gas is held constant. If we draw a line of best fit through our data points, we can see what this relationship looks like. When gas pressure is constant, then the gas volume, we can call it 𝑉, is directly proportional to the gas temperature. We can call that 𝑇.

Now, when we say that one quantity is directly proportional to another, what we’re saying is that if we were to double the second quantity, in this case 𝑇, that would mean we double the first one as well. Looking on our graph, if we were to consider the temperature and volume of our first data point, then if we were to double the gas temperature, increase it to up to this value. Then direct proportionality tells us that doubling the temperature implies a doubling of the volume. Or let’s say we go the opposite way. Let’s say that, instead of doubling our temperature, we cut it in half, to this value here. Once again, a direct proportionality means that we cut the volume in half as well. That point then would still lie along our line right about here. So that’s what we mean when we say that 𝑉 is directly proportional to 𝑇. Gas volume changes by whatever factor gas temperature changes.

Now, from a mathematical perspective, there’s another equivalent way to write this expression, 𝑉 is proportional to 𝑇. It’s equivalent to say that 𝑉 is equal to a constant, we’ll call that constant 𝑘, times 𝑇. Both of these ways of writing the relationship mean the same thing. But it’s this way that we see more often. If we look up Charles’s law in our textbook or online, we’ll often see it written as 𝑉 is equal to 𝑘 times 𝑇.

Before we can put this law to use, though, there are a couple of important things to realize about it. Within this equation, there are two assumptions, one of which we’ve named so far and one of which we haven’t. The first assumption is that gas pressure is held constant. In our experiment, we saw this going on because we had a constant weight pushing down on the gas. The second assumption, though, has to do with the temperature and, in particular, the units of temperature that we use. Some fairly common temperature scales include the Fahrenheit scale, the Celsius scale, and the kelvin scale. Well, it’s the Kelvin scale that’s accepted as the way to express temperatures in the SI system. It’s important then if we want to use Charles’ law to make sure that the temperatures we’re working with are on that scale, they’re expressed in Kelvin.

When we use Charles’s law, often it’s to compare the volume and temperature of a gas at one point, say, initially, with the volume and temperature of that same gas later on. Looking back at our graph, let’s say that this data point represents our initial gas state. And this represents its final state. Since each one of these points has an associated gas volume and temperature, we can make up variables for those. Let’s say that the gas volume of this first point is 𝑉 one and the temperature is 𝑇 one. And then, at our final point, we’ll say the gas volume is 𝑉 two and the temperature is 𝑇 two. And we could write these variables out as coordinate pairs, 𝑇 one, 𝑉 one and 𝑇 two, 𝑉 two.

To see how these two points compare with one another, let’s clear a bit of space on screen. And then, we’ll take Charles’ law. And we’ll rearrange it a bit. The way it’s written now, this law says that volume, a variable, is equal to 𝑘, a constant, times 𝑇, another variable. But let’s rearrange this equation so that the constant value 𝑘 is on one side by itself. To do that, we can divide both sides of the equation by the temperature 𝑇, meaning that that term cancels out on the right-hand side. In this rearranged form of Charles’s law, we have 𝑉 divided by 𝑇 is equal to a constant.

But notice that this volume and temperature could be any volume and temperature pair for a given gas held at constant pressure. In other words, it could be this volume temperature pair or this one or any other along this line. This means that we can take our two points, 𝑇 one, 𝑉 one and 𝑇 two, 𝑉 two, and use them in this rearranged expression. Because this expression is true for any volume and temperature pair on our curve, that means it must be true for this one. And then, by the same argument, it must also be true for this one, 𝑉 two and 𝑇 two.

And now, look at this. If we focus on these last two terms to the far right, we can see a relationship between these two points along our curve. The initial gas volume divided by the initial gas temperature, whatever those values are, is equal to the final volume divided by the final temperature. This relationship has a lot of practical use because if we know three of these four variables, then we can use this relationship to solve for the fourth. For example, let’s say we have a scenario where we know the initial volume and temperature of a gas. And then, say that the temperature of the gas changes to some new final value. And we know that as well. Knowing these three values, we can then solve for the fourth one, the unknown volume 𝑉 two. And of course, it doesn’t have to be 𝑉 two that we always solve for. So long as we know any three of these four variables, we can solve for the fourth.

Knowing all this about Charles’s law, let’s get a bit of practice with these ideas through an example.

Which of the lines on the graph shows how the volume of a gas varies with its temperature when it is kept at a constant pressure?

Alright, looking at this graph, we see that it shows us the volume of a gas against its temperature. There are five different lines on the graph, the black one, the blue one, the pink one, the orange one, and the green one. And we want to figure out which of these five lines correctly shows the relationship between gas volume and temperature when its pressure is kept constant. One thing we can notice early on is that these five lines generally divide into two different types. For one type of line, as the temperature of the gas goes up as it increases, the volume of the gas decreases. That’s true for both the black line as well as the blue line. They show gas volume decreasing as gas temperature increases.

Another thing about these two lines is that they don’t go through the origin. In other words, they say that when gas temperature is zero, the gas volume is not zero. It’s some positive value. So that’s one type of line. The other type is represented by the pink curve, the orange curve, and the green one. For those three lines, when gas temperature increases, volume increases as well. And along with that, all three of them pass through the origin. We want to pick which of the five lines correctly shows the relationship between volume and temperature when pressure is constant. By thinking about these physical parameters, temperature and volume, we can start to eliminate a few of these lines from contention.

If we think, for example, about varying the temperature of a gas, it makes sense that as temperature decreases, the volume of a gas, the space it takes up, would decrease as well. That’s because lower temperatures mean lower average speeds of the molecules in the gas. And if those molecules are under constant pressure, like we’re assuming they are, then the space they take up would get smaller and smaller as their speed decreases. Based on this physical reasoning, we would expect that as our temperature gets smaller and smaller, the volume of our gas would too. But in the case of the black and the blue lines, we see the opposite is happening. As temperature decreases and approaches zero, volume goes up.

For this reason, we can eliminate these two lines from consideration. They won’t represent this relationship between volume and temperature when pressure is constant. That leaves us with these other three lines, the pink line, the orange line, and the green line. As we saw each one of these goes to zero volume when the temperature is zero and they all show an overall relationship, whereas temperature increases, volume does as well. So, so far, each one of these three lines demonstrates behavior that makes sense physically. We would expect a gas being heated at constant pressure to behave this way.

To figure out which of these three lines is correct, we’ll need to dig a bit deeper. We can recall a law named Charles’s law. In the case of a gas at a constant pressure, like we have here, this law says that gas volume, 𝑉, is directly proportional to gas temperature, 𝑇. When one variable is directly proportional to a second variable, that means if we multiply the second variable by some factor, say, a factor of two or a factor of one-half or a factor of four. Then the first variable responds in the same way, either by doubling or by being cut in half or multiplied by four. Said another way, by whatever multiple our second variable changes, then the first one must change by that same multiple. That’s what direct proportionality means.

Now, looking back over at our graph, notice that we have some tick marks, marked out on the horizontal and vertical axes. We can use these marks to help us explore the directly proportional relationship between volume and temperature that Charles’s law holds. Here’s how we can do it. Let’s say that we pick a point that all three of these curves pass through. Let’s say we pick the point right here. Now, at that point, the gas has a certain temperature, whatever this temperature is. And it has a certain volume, whatever this volume is. Charles’s law tells us that if we were to double the temperature of our gas, that would involve increasing its temperature from right here to up to here, the fourth tick mark. Then the law says because 𝑉 is directly proportional to 𝑇, the gas volume would also double. In other words, according to Charles’ law, if our gas temperature were to double, from here to here, then our gas volume would double as well, from here to here.

If we find out where this new double data point would lie, we could trace a horizontal line over from our volume and then a vertical line up from our new temperature. And the point where they intersect is where our gas will now be. We can see that, of our three lines, the pink, the orange, and the green, only the orange line passes through this point. So it’s that line, and only that line, that follows the relationship described in Charles’s law. Only for the orange line does gas volume double when gas temperature is doubled. This then is our answer choice. It’s the orange line that shows how the volume of a gas varies with its temperature when it’s at a constant pressure.

Let’s summarize now what we’ve learned about Charles’s law. In this lesson, we saw that Charles’s law is an experimental gas law that relates gas volume to gas temperature. We saw that there are several different but equivalent ways of expressing the law mathematically. The law says that when a gas pressure is constant, then the volume of that gas is directly proportional to its temperature. Or equivalently, the volume of the gas is equal to a constant, we called it 𝑘 times temperature.

Or another way of saying it, for two different volume and temperature pairs for a given gas held at constant pressure, the ratio of the initial volume to its initial temperature, 𝑉 one to 𝑇 one, is equal to the ratio at the final volume, 𝑉 two, to final temperature, 𝑇 two. And lastly, we saw that two assumptions are involved in the Charles’ law equation. The first is that gas pressure is held constant. And the second is that temperature is measured on the kelvin temperature scale. Keeping these two assumptions in mind, Charles’ law is a useful relationship between gas volume and gas temperature.

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