### Video Transcript

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

On this plot of gas volume against gas temperature, we can count five different lines. There is the black line, the blue line, the purple line, the red line, and the green line. One of these, we’re told, shows how the volume of a gas varies with its temperature when it is kept at constant pressure.

Considering our five lines, we can divide them into two groups. The one group, which is the generally downward-sloping lines, includes the blue and the black curves. The other group, the generally upward-sloping lines, includes the purple, red, and green curves. Now let’s imagine that our answer was among the downward-sloping lines, either the blue or the black line. This would mean that, given some gas, if we increase the gas’s temperature, say by transferring heat energy to it, then the volume of that gas would correspondingly decrease. That’s what both the blue and the black line overall are indicating.

Physically though, we know that it doesn’t make sense for the temperature of a gas to increase and then for its volume to get smaller. In fact, even for a gas held at constant pressure, we would expect the opposite, that as its temperature increases, so does its volume. Since our two downward-sloping lines contradict this conclusion — that as temperature increases, volume will too — we won’t choose either of those as our answer.

For our upward-sloping lines on the other hand, these curves indicate the general trend that as the temperature of a gas increases, so does its volume. The question then becomes “Which of these three lines best represents that change?” There is a law known as Charles’s law, which says that the volume of a gas held at constant pressure is proportional to the temperature of the gas. An equivalent way to write this is to say that the volume 𝑉 equals a constant 𝑘 times the gas temperature 𝑇. Notice that this relationship implies that if we, say, double the volume 𝑉 of a gas — that is, if the volume 𝑉 goes to two times 𝑉 — then for Charles’s law to hold true, the temperature 𝑇 must also double from 𝑇 to two times 𝑇.

This is the meaning of the volume being proportional to the temperature, which implies a straight-line relationship if we plotted it on our graph. So then, considering our three upward-sloping curves, the purple, the red, and the green, which one we want to identify is linear so that if we double the volume, then we also double the temperature 𝑇 of our gas? With that condition in place, we see that only one of these three curves satisfies that requirement. It’s the red line only that demonstrates a linear relationship between volume and temperature. That is, this line is consistent with Charles’s law. For our answer, we choose the red line.