Video Transcript
A gas with the volume of three cubic meters is at a pressure of 500 pascals. The gas is compressed at a constant temperature until it is at a pressure of 1500 pascals. What is the volume of the gas after it is compressed?
In this question, we have a gas, and this gas is in some sort of container. The question tells us that this gas is then compressed. And what we have to do in this question is find the volume of the gas after the compression. Let’s first take a look at the gas before the compression. The question tells us two things about the gas before the compression. It tells us that it has a volume of three meters cubed, and we will call this volume 𝑉 sub one. The question also tells us that the gas is at a pressure of 500 pascals, and we will call this pressure 𝑃 sub one. After the compression, we are told that the gas is at a pressure of 1500 pascals, and we will call this pressure 𝑃 sub two. And the volume of the gas after the compression is unknown, so we will call this 𝑉 sub two.
Now, apart from this information about the pressure and volume of the gas before and after the compression, the question also tells us a couple of other things. The first is that the amount of gas stays the same. The question tells us that the gas is compressed, and we’re not told that any other gas is added or taken away during this compression. If we think in terms of particles, even though the volume of gas is reduced, the number of particles hasn’t. What’s actually happened is that these particles have ended up, on average, closer together. When the gas is compressed and the particles move closer together, we might be concerned about the temperature of the gas changing due to an increased number of collisions.
However, the question tells us the second piece of information that the temperature of the gas is constant, and this is highlighted in the question text where it says the gas is compressed at a constant temperature. So, we have to work out how the volume of the gas changes with pressure when the amount of gas stays the same and its temperature is constant. To do this, we can use Boyle’s law, which tells us that the pressure of a gas multiplied by the volume of a gas is constant, but only when the amount of gas stays the same and the temperature of the gas is constant. We can apply Boyle’s law to this question before and after the compression.
Before the compression, the pressure of the gas multiplied by the volume of the gas is equal to some constant 𝑐. And after the compression, the pressure of the gas multiplied by the volume of the gas is just equal to that constant as well, 𝑐. Putting these together, the pressure of the gas multiplied by the volume of the gas before the compression is equal to the pressure of the gas multiplied by the volume of the gas after the compression. And we know the pressure and volume of the gas before the compression and the pressure of the gas after the compression. So the only unknown in this equation is the volume of the gas after the compression, 𝑉 two.
Therefore, if we want to work out 𝑉 two, all we have to do is rearrange this equation to make 𝑉 two the subject. Starting with our equation, we can divide both sides by 𝑃 two, where we see that the 𝑃 two in the numerator and the denominator cancel on the right. And this just leaves us with 𝑉 two. Writing this a little bit more neatly, the volume of the gas after the compression, 𝑉 two, is equal to the pressure of the gas before the compression, 𝑃 one, divided by the pressure of the gas after the compression, 𝑃 two, multiplied by the volume of the gas before the compression, 𝑉 one. Substituting our known values for 𝑃 one, 𝑃 two, and 𝑉 one into this equation, we get 𝑉 two is equal to 500 pascals divided by 1500 pascals multiplied by three meters cubed.
And before we go any further, we can quickly look at the units and see that the pascals in the numerator and the denominator of this fraction will cancel. So, the units of our final calculation will just be meters cubed, and this is exactly what we would expect for a volume. So, we’re good to continue. The next thing we can notice is that 500 divided by 1500 simplifies to just one divided by three. Therefore, our calculated value of 𝑉 two will just be one divided by three multiplied by three meters cubed, which is exactly one meter cubed. So, the volume of the gas after it is compressed, 𝑉 two, is exactly one meter cubed.
