Which of the following is the correct formula for Boyle’s law, where 𝑃 is the pressure of a gas, 𝑉 is the volume, 𝑇 is the temperature, and 𝑘 is a constant? (A) 𝑃 times 𝑇 equals 𝑘, (B) 𝑇 times 𝑉 equals 𝑘, (C) 𝑃 times 𝑉 equals 𝑘, (D) 𝑃 divided by 𝑉 equals 𝑘, (E) 𝑃 equals 𝑘 times 𝑇.
The first thing that we can recall about Boyle’s law is that this law assumes that gas temperature is held constant. This means that among our candidate solutions, if the temperature 𝑇 appears in any of them, we know that candidate can’t be the correct formula for Boyle’s law. This is so because if the temperature 𝑇 does appear in an equation like it does for answer options (A), (B), and (E), then that means the other gas quantity in the equation, whether that be pressure 𝑃 or volume 𝑉, cannot change either. If that were the case, then, for example, for answer choice (A), we would have a constant times a constant being equal to a constant. This is true, but it’s not the correct formula for Boyle’s law.
All three of the equations that involve the temperature 𝑇 since that temperature remains constant can be reduced to simply saying that one constant equals another constant. That’s not the statement of Boyle’s law. So, we’ll eliminate these options. That leaves us with answer choices (C) and (D) remaining. To develop some intuition about which one of these is correct, let’s say that we have a volume of a certain size and inside the volume we have a gas. As the particles of the gas move around and collide with the walls of the container, the net effect of all those collisions is equal to the pressure of the gas.
Now, let’s say that the volume of this container is adjustable, that we can move the lid downwards, say, to decrease the volume occupied by the gas. So with the volume of the gas decreased, what effect do we think this would have on the gas pressure? Since the particles of a gas now occupy a smaller space, we would expect that over some amount of time there are more collisions per unit time between the particles and the walls of the container. This would be true even if the temperature of the gas in the container remains constant. Since at a smaller volume there would be more collisions per unit time between the particles of the gas and the walls of the container, we would expect the pressure to increase.
What we’re finding is that as volume trends one way decreasing, pressure trends the other way. This implies an inverse relationship between gas pressure and gas volume. A mathematically equivalent way to write this is to say the gas pressure 𝑃 is equal to a constant 𝑘 divided by gas volume 𝑉. Notice that if we multiply both sides of this equation by the volume 𝑉, canceling that volume on the right, we find a result that is identical to the solution offered in option (C). This is the answer rather than option (D), which says that if one of these two quantities pressure and volume, say, increases, the other must correspondingly decrease in order for their product to be equal to a constant. Answer choice (C) that 𝑃 times 𝑉 equals a constant is the correct formula for Boyle’s law.