### Video Transcript

For a gas at a constant temperature, if the volume is blank, then the pressure blank. (A) Increased, increases; (B) increased, stays the same; (C) increased, decreases; (D) decreased, stays the same; or (E) decreased, decreases.

In this question, we’re considering a gas, and we’d like to work out what happens to the pressure of the gas if we make some change to its volume. So we’ll consider the gas at two points in time, before and after this change in volume. Before this change, we will say that the gas has a volume of 𝑉 one and a pressure that we will call 𝑃 one. After the change, we will say that the gas has a volume that we will call 𝑉 two and a pressure that we will call 𝑃 two.

In order to work out the relationship between the pressure and volume of a gas, we will use a gas law called Boyle’s law. And Boyle’s law tells us that the pressure of a gas multiplied by the volume of a gas is constant. However, we can only use this law when two conditions are satisfied: one, the temperature of the gas must be constant, and two, the amount of gas stays the same.

Now, the question tells us that the gas is at a constant temperature, so our first condition is satisfied. And the question doesn’t mention anything about adding or removing gas during this process. So we can assume that the amount of gas stays the same. And this means that we can safely apply Boyle’s law to the gas in this question. To answer this question, the first thing we’ll do is use our equation given to us by Boyle’s law to work out the relationship between the pressure and volume of the gas. After this, we can evaluate each of our statements here to work out which one is describing the correct relationship between the pressure and volume.

Starting with our equation for Boyle’s law, we will divide both sides by the volume of the gas 𝑉, where we see that the 𝑉 in the numerator and denominator on the left cancel, leaving us with an expression for the pressure of the gas 𝑃. What we can immediately see here is that the pressure of the gas is inversely proportional to the volume of the gas. And because 𝑐 is a constant, we know that if the volume changes, then the pressure must also change. So we can rule out any statements on the left that say that the pressure will stay the same when the volume changes. These are statements (B) and (D), and we will mark them as incorrect.

To choose the correct answer from the other options, we’ll need to see what happens when we actually change the volume of this gas. If we start with our relationship of 𝑃 is equal to 𝑐 divided by 𝑉 for the gas before we’ve changed its volume and then decrease the volume as in statement (E), for example, such that 𝑉 two is equal to a half of 𝑉 one, then we can use our relationship for the pressure of the gas after change in volume 𝑃 two and substitute in our expression for 𝑉 two, which in this case gives us 𝑃 two is equal to 𝑐 divided by one-half 𝑉 one. Expanding the brackets, we see that 𝑐 divided by one-half 𝑉 one is equal to two 𝑐 divided by 𝑉 one. And we already know that 𝑐 divided by 𝑉 one is equal to 𝑃 one. So in this case, 𝑃 two is equal to two 𝑃 one.

In other words, when we decrease the volume, the pressure actually increases. This disagrees with statement (E) that says when the volume is decreased, the pressure decreases. So we can rule this out and mark it as incorrect. Now we’ve only got two statements left, statement (A) and statement (C). In both of these, the volume is increased. So let’s repeat our process from before, but this time let’s increase the volume. Let’s say that the volume doubles such that 𝑉 two is equal to two 𝑉 one. Substituting this into our relationship for 𝑃 two, we get that 𝑃 two is equal to 𝑐 divided by two 𝑉 one, or equivalently 𝑃 two is equal to a half 𝑐 one divided by 𝑉 one. And once again, we know that 𝑐 one divided by 𝑉 one is equal to 𝑃 one, so 𝑃 two is equal to a half 𝑃 one.

What’s happened here is when the volume has increased, the pressure has decreased. This disagrees with statement (A) that states that when the volume is increased, the pressure increases, so we can mark this as incorrect. And this agrees with our only remaining statement, statement (C), which says that when the volume is increased, the pressure decreases. So we can accept this as a correct answer. For a gas at a constant temperature, if the volume is increased, then the pressure decreases.