### 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in the 1780s in some unpublished documents by a researcher named Jacques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.