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At sea level on Earth, the atmospheric air pressure is about 101 kilopascals.
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At the top of Mount Everest, the air pressure is about 34 kilopascals.
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A balloon with a fixed amount of helium gas is taken from sea level up to the top of Mount Everest.
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The temperature of the air is the same at both locations.
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What happens to the size of the balloon?
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Okay, so in this question, we’ve been told that on Earth at sea level the atmospheric air pressure is about 101 kilopascals.
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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.
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In other words, the air pressure at the top of Mount Everest is a lot lower than the air pressure at sea level.
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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.
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We’re also told that the temperature of the air is the same at both locations.
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And what we’ve been asked to do is to find out what happens to the size of the balloon.
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Okay, so let’s draw a diagram representing what’s going on.
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So in this diagram, we’ve drawn sea level here on the left and Mount Everest on the right.
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We’re told that at sea level, the pressure is 101 kilopascals, whereas at the top of Mount Everest the pressure is 34 kilopascals.
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Now, what we’re doing is we’re taking a balloon from sea level up to the top of Mount Everest.
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And what we need to do is to find out whether the balloon gets bigger or smaller, which is it?
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Well, to answer this question, we need to recall something known as Boyle’s law.
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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.
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Now, this condition is really quite important.
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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.
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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.
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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.
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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.
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So we say that 𝑃 sub 𝑠 multiplied by 𝑉 sub 𝑠 is equal to 𝑃 sub 𝐸 multiplied by 𝑉 sub 𝐸.
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Now, we’ve been given values of 𝑃 sub 𝑠 and 𝑃 sub 𝐸.
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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.
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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.
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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.
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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.
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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.
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This way we have a large pressure multiplying a small volume and this is equal to a small pressure multiplying a large volume.
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This is the only way the left-hand side can be equal to the right-hand side.
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Therefore, 𝑉 sub 𝐸 must be larger than 𝑉 sub 𝑠.
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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.
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And this makes sense on some intuitive level.
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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.
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And this is balanced by the pressure of the molecules inside the balloon.
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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.
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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.
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And hence, we have our final answer.
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What happens to the size of the balloon?
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Well, it increases.