Video: Selecting the Graph That Matches the Pressure-Volume Product of an Ideal Gas against Volume

Which graph could represent the result of pressure times volume, ๐‘ƒ๐‘‰, for an ideal gas versus its volume at constant temperature? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E

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Video Transcript

Which graph could represent the result of pressure times volume, ๐‘ƒ๐‘‰, for an ideal gas versus its volume at constant temperature?

We want to construct a graph of the pressure times volume versus the volume for an ideal gas, which means that the pressure times volume should go on the ๐‘ฆ-axis and the volume should go on the ๐‘ฅ-axis. Since weโ€™re interested in the behavior of an ideal gas, we can use the ideal gas law to create an expression that we can graph. We want an expression where the pressure times volume will be on one side of the equation, where we can graph it as ๐‘ฆ, and the volume should be on the other side of the equation, where we can graph it as ๐‘ฅ.

Just looking at the ideal gas law, we already have the pressure times volume on one side of the equation. But the volume isnโ€™t present on the other side of the equation. This may seem a little strange at first. But all this means is that the pressure times volume and the volume are independent of each other. Before we graph, letโ€™s take a look at the right-hand side of the equation.

The question tells us that weโ€™re at a constant temperature. So ๐‘‡ is a constant. ๐‘… is the gas constant. So thatโ€™s a constant as well. ๐‘› is the amount of the gas that we have in moles. Assuming we have a sample of gas in a container, this would be constant too. Since everything on the right-hand side of the equation was a constant, we can combine them into one constant, which Iโ€™ll call ๐ถ. Now, this looks quite a bit like Boyleโ€™s law, which tells us that the pressure and the volume are inversely proportional to one another. Or you can express it as the pressure times the volume is equal to a constant.

Now, weโ€™re ready to graph. If youโ€™re a little confused about what ๐‘ƒ๐‘‰ equals ๐ถ should look like, you can imagine graphing something where ๐‘ฆ is also equal to a constant, such as ๐‘ฆ equals five. This would be a straight horizontal line, which matches answer choice B, which is the correct graph to represent the pressure times volume of an ideal gas versus its volume.

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