Video: Understanding Boyle’s Law

At sea level on Earth, the atmospheric air pressure is about 101 kPa. At the top of Mount Everest, the air pressure is about 34 kPa. A balloon with a fixed amount of helium gas is taken from sea level up to the top of Mount Everest. The temperature of the air is the same at both locations. What happens to the size of the balloon?

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Video Transcript

At sea level on Earth, the atmospheric air pressure is about 101 kilopascals. At the top of Mount Everest, the air pressure is about 34 kilopascals. A balloon with a fixed amount of helium gas is taken from sea level up to the top of Mount Everest. The temperature of the air is the same at both locations. What happens to the size of the balloon?

Okay, so in this question, we’ve been told that on Earth at sea level the atmospheric air pressure is about 101 kilopascals. Now, at the top of Mount Everest, which is the highest mountain above sea level on Earth, we’re told that the air pressure is about 34 kilopascals. In other words, the air pressure at the top of Mount Everest is a lot lower than the air pressure at sea level.

Okay, so we’re told that a balloon with a fixed amount of helium gas is taken from sea level up to the top of Mount Everest. We’re also told that the temperature of the air is the same at both locations. And what we’ve been asked to do is to find out what happens to the size of the balloon. Okay, so let’s draw a diagram representing what’s going on.

So in this diagram, we’ve drawn sea level here on the left and Mount Everest on the right. We’re told that at sea level, the pressure is 101 kilopascals, whereas at the top of Mount Everest the pressure is 34 kilopascals. Now, what we’re doing is we’re taking a balloon from sea level up to the top of Mount Everest. And what we need to do is to find out whether the balloon gets bigger or smaller, which is it? Well, to answer this question, we need to recall something known as Boyle’s law.

Boyle’s law tells us that the pressure of a gas multiplied by the volume of a gas is equal to a constant if the temperature of the gas is a constant. Now, this condition is really quite important. Lucky for us, we’ve been told that the temperature at sea level, which we’ll call 𝑇 sub 𝑠, is equal to the temperature at the top of Mount Everest, which we’ll call 𝑇 sub 𝐸. And therefore, this condition satisfied that if 𝑇 is equal to constant bit, which means that the product of pressure and volume must stay constant.

In other words, if we say that at sea level, the pressure of the gas is 𝑃 sub 𝑠 and the volume of the gas is 𝑉 sub 𝑠 and at the top of Everest the pressure is 𝑃 sub 𝐸 and the volume is 𝑉 sub 𝐸, then we say that for sea level 𝑃 sub 𝑠 multiplied by 𝑉 sub 𝑠 is equal to constant and for Everest 𝑃 sub 𝐸 multiplied by 𝑉 sub 𝐸 is also a constant. But more specifically, because 𝑃 multiplied by 𝑉 must be the same anywhere and it must be the same constant, we can also say that 𝑃 sub 𝑠 multiplied by 𝑉 sub 𝑠 is equal to 𝑃 sub 𝐸 multiplied by 𝑉 sub 𝐸 because these constants on the right-hand side of each equation are the same constant. So we say that 𝑃 sub 𝑠 multiplied by 𝑉 sub 𝑠 is equal to 𝑃 sub 𝐸 multiplied by 𝑉 sub 𝐸.

Now, we’ve been given values of 𝑃 sub 𝑠 and 𝑃 sub 𝐸. In this case, we’re not trying to work out exactly what happens to the volume of the balloon because we don’t know the initial volume of the balloon. But we know that the pressure at sea level 𝑃 sub 𝑠 is much larger than the pressure of the top of Everest 𝑃 sub 𝐸. In other words, what we have is a large pressure multiplied by some volume is equal to a small pressure multiplied by some other volume.

But if the product 𝑃 sub 𝑠 multiplied by 𝑉 sub 𝑠 must be equal to 𝑃 sub 𝐸 multiplied by 𝑉 sub 𝐸, we need to have 𝑉 sub 𝑠 being much smaller than 𝑉 sub 𝐸. And the reason is the following: in order for the left-hand side 𝑃 sub 𝑠 times 𝑉 sub 𝑠 to be equal to 𝑃 sub 𝐸 times 𝑉 sub 𝐸 and for the pressure 𝑃 sub 𝑠 to be larger than 𝑃 sub 𝐸, we need to have 𝑉 sub 𝑠 being smaller than 𝑉 sub 𝐸. This way we have a large pressure multiplying a small volume and this is equal to a small pressure multiplying a large volume. This is the only way the left-hand side can be equal to the right-hand side. Therefore, 𝑉 sub 𝐸 must be larger than 𝑉 sub 𝑠.

In other words, the volume of the balloon at the top of Mount Everest must be larger than the volume of the balloon at sea level. And this makes sense on some intuitive level. If we consider the balloon at sea level, then what we have is a very large air pressure — lots of molecules around the balloon exerting a pressure inwards onto the balloon. And this is balanced by the pressure of the molecules inside the balloon.

But let’s say the balloon has a certain volume here due to the pressure of the molecules pushing in, whereas at the top of Mount Everest the balloon becomes a lot larger because the air pressure is much lower there’s a lot fewer molecules pushing inward on the balloon. So because we have a fixed amount of gas inside the balloon, the molecules that are pushing outwards from inside the balloon will expand the volume of the balloon.

And hence, we have our final answer. What happens to the size of the balloon? Well, it increases.

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