Question Video: Relating the Volume and Temperature of a Gas at Constant Pressure Physics • Second Year of Secondary School

Video Transcript

Which of the following formulas correctly relates the volume and temperature of a gas that is kept at a constant pressure? 𝑉 represents the volume of the gas and 𝑇 represents the temperature. (A) 𝑉 times 𝑇 equals a constant. (B) 𝑉 squared times 𝑇 equals a constant. (C) 𝑉 times 𝑇 squared equals a constant. (D) 𝑉 divided by 𝑇 equals a constant. (E) One divided by the quantity 𝑉 times 𝑇 squared equals a constant.

We’re thinking here of a gas whose volume and temperature can change but whose pressure remains constant. If 𝑉 represents volume and 𝑇 represents temperature, we’re saying that both of these quantities may increase and decrease.

There is a law that relates these quantities of a gas when that gas is held under constant pressure. But even if we can’t recall that law, we can still use some physical intuition about our gas to narrow down our answer options. Say, for example, that we heat this gas, increasing its temperature. That will increase the average kinetic energy of the particles of the gas and tend to make the gas expand or increase in volume. This is true even when the gas is held at a constant pressure. We know then that when the temperature of a gas held at constant pressure increases, its volume should increase too.

Notice now among our answer options that the right-hand side of all of these options is a constant. For our first three answer choices, if the temperature 𝑇 is increased, then for 𝑉 times 𝑇 to equal a constant or 𝑉 squared times 𝑇 or 𝑉 times 𝑇 squared, that must mean that as temperature increases, volume decreases. We’ve seen though that physically that’s not what happens. Rather, they both increase together. Therefore, we can cross out answer options (A), (B), and (C).

And even though it’s harder to tell right away, answer option (E) has the same problem. For one over the quantity 𝑉 times 𝑇 squared to be equal to a constant, that would mean that as, say, the temperature 𝑇 increased, the volume 𝑉 would have to decrease. Otherwise, the left-hand side of this expression would not equal a constant. Again though, we’ve seen that both 𝑉 and 𝑇 increase together. Therefore, option (E) won’t be our final answer either.

Finally, let’s look at answer choice (D). Let’s imagine that the temperature 𝑇 increases, say by a factor of two. We can say then that the temperature 𝑇, whatever it was, goes to two times that temperature. In order for 𝑉 divided by 𝑇 to be a constant, as claimed by this answer choice, the volume 𝑉 would also need to double, say to two 𝑉, if the temperature 𝑇 doubles. If 𝑉 then increases by the same factor that 𝑇 did, then it’s indeed the case that 𝑉 divided by 𝑇 equals a constant.

Option (D) then is our answer. The formula 𝑉 divided by 𝑇 being equal to a constant correctly relates the volume and temperature of a gas that is kept at a constant pressure.