A four-metre cubed volume of gas is at a pressure of 1000 pascals. The gas is allowed to expand at a constant temperature until its pressure is half of the value before expansion began. How many times greater is the volume of a gas after its expansion?
Okay, so, what we have here is a volume of gas which initially we’ve been told is four metres cubed. And this gas initially is at a pressure of 1000 pascals. As well as this, we can choose to say that the gas is at some temperature, which we’ll call 𝑇. We don’t know the value of 𝑇, but this will become important in a second.
What we’ve been told that then happens is that the gas is allowed to expand. So, over time we can see that the sphere of gas expands. It gets larger. And it does this until the pressure is half of the value before the expansion. So, we can say now that the pressure is 500 pascals. This is half of what the value was before, which was 1000 pascals. Now what we need to do is to find out how many times greater the volume of the gas is after the expansion?
So, let’s say that the volume of the gas after the expansion is 𝑉. And as well as this, we know that the gas is allowed to expand at a constant temperature. So, the temperature of the gas is still 𝑇, whatever that 𝑇 may be. Now at this point, what we can say is that the volume of the gas after expansion 𝑉 is equal to 𝑛 times the volume before the expansion, which was four metres cubed, where 𝑛 is the number of times greater than the volume 𝑉 is compared to the volume four metres cubed.
Now that might sound a bit complicated. So, a simpler way to think about this is that 𝑉 is 𝑛 times four metres cubed. And what we want to find out in this question is the value of 𝑛, how many times the initial volume four metres cubed is 𝑉.
Now to figure this out, we can recall something known as Boyle’s law. Boyle’s law tells us that the pressure of an ideal gas, which is what we assume this to be, is inversely proportional to the volume of the gas, provided that the temperature of the gas is constant.
In other words, as the pressure of a gas gets larger, for example, the volume gets smaller. But this is only true if the temperature is kept constant, which in this case, it is. We’ve been told that the gas is allowed to expand at a constant temperature. And since the condition for 𝑇 being constant has been fulfilled by our gas, we can, therefore, apply 𝑃 is proportional to one over 𝑉 to our gas.
Well, what we can choose to say is that the pressure of the gas is equal to some proportionality constant 𝑘 divided by the volume. Now essentially what we’ve done here is change this proportionality into any equality. And we do this by putting in a proportionality constant 𝑘. Now we don’t know what this proportionality constant is but that’s not going to matter.
Because then, what we can do is to rearrange this equation so that the pressure of the gas multiplied by the volume of the gas is equal to the proportionality constant. And because this proportionality constant is indeed constant, this must mean that whatever the value of 𝑃 and whatever the value of 𝑉 they must always multiply together to give this same value. And of course, we must remember that this is only true for our gas because 𝑇 is constant. That’s why this whole equation applies.
But anyways, so, what we can, therefore, say is that the pressure of the gas initially, which was 1000 pascals, multiplied by the volume of the gas, which is four metres cubed, must be equal to the pressure of the gas after the expansion, which is 500 pascals, multiplied by the volume of the gas 𝑉.
This is because, regardless of what the values of 𝑃 and 𝑉 are individually, the product of 𝑃 and 𝑉 at any one time must be the same as the product of 𝑃 and 𝑉 at any other time. And that product in both cases must be equal to 𝑘. And so, essentially, what we’re saying is that 𝑘 is equal to 𝑘, since both the left-hand side and right-hand side are equal to 𝑘.
So, if we wanted to, we could calculate the value of 𝑘, but that’s not what we’re trying to do, really. We could also, if we wanted to calculate the value of 𝑉, that’s the final volume of the gas after the expansion. However, that’s also not what we’re trying to do. Remember, what we’re trying to do is to calculate how many times larger 𝑉 is compared to the initial volume four metres cubed. In other words, we’re trying to calculate 𝑛.
And if we divide this equation by four metres cubed on both sides, then what we see is that four metres cubed cancels on the right. And so, 𝑛 is equal to 𝑉 divided by four metres cubed. Hence, what we need to do is to take this equation and rearrange it so that we have 𝑉 over four metres cubed on one side. And since that will be equal to 𝑛, whatever is on the other side must also be equal to 𝑛.
We can do this by dividing both sides of the equation by four metres cubed and 500 pascal. This means that on the left-hand side, the four metres cubed in the numerator cancels with the four metres cubed in the denominator. And on the right-hand side, the 500 pascals in the numerator cancels with the 500 pascals in the denominator. So, what we’re left with is on the left-hand side we’ve got 1000 pascals divided by 500 pascals. And on the right-hand side, we have 𝑉 divided by four metres cubed. But then, we said that this whole thing was equal to 𝑛. And so, we can substitute 𝑛 to the right-hand side of the equation.
Now coming back to the left-hand side of the equation, we see that the unit of pascals cancels in the numerator and denominator. And so, all we have left is 1000 divided by 500. And hence, our final answer is simply going to be a number without any unit, which is perfect. Because a number is exactly what we’re trying to find. This number is equal to the number of times the final volume 𝑉 is greater than the initial volume four metres cubed.
And it’s at this point that we can see that the value four metres cubed wasn’t even useful in our calculations. We could’ve equally called this value 𝑉 one, or the initial value of the volume, and still been able to calculate 𝑛. Because all that mattered here was the initial pressure, and the final pressure, and the ratio between these two.
So, now we can see that 1000 divided by 500 is equal to two. In other words, the volume of the gas after its expansion is two times greater than the volume before the expansion. We could use this answer to work out the final volume of the gas, but we haven’t been asked to do this. We found what we need to already. And so, we’ve got our final answer to this question.