Video: Using the Inverse Variation in a Real-World Context

The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

03:13

Video Transcript

The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

So the first thing we look at in this question is the fact that we have volume that actually varies indirectly with pressure. So we therefore we can say that the volume is indirectly or inversely proportional to 𝑝, our pressure.

But this isn’t very useful written like this. So what we do instead is turn it into an equation. We can say that the volume is equal to 𝑘 over 𝑝, where 𝑘 is our proportionality constant. And we write it as 𝑘 over 𝑝 because, as we said, it’s indirectly or inversely proportional.

Now the first thing we do with any problem like this too is we need to find 𝑘. So we need to find our proportionality constant. And to do that, we’re gonna use some information that we’ve got.

Well, we know that when the volume is 1200 cubic centimeters, then the pressure is 200 millimeters of mercury. So we can say that 𝑉 is equal to 1200 and 𝑝 is equal to 200. So what we do now is actually substitute our values for 𝑉 and 𝑝 into the formula we had earlier, which is 𝑉 is equal to 𝑘 over 𝑝. And when we do that, we get 1200 is equal to 𝑘 over 200, because 𝑉 was obviously our 1200 and 𝑝 was our 200.

So now what we do to actually find out what 𝑘 is is we multiply both sides of the equation by 200. And when we do that, we get 240000 is equal to 𝑘. So great! We’ve actually found 𝑘. So therefore, what we can do is actually rewrite our formula. So we’ve now got that 𝑉 is equal to 240000 over 𝑝.

So now we can take a look at the second part of the question, where we actually need to work out, what we’re looking for. So we want to find the volume when the pressure is 300 millimeters of mercury. So therefore, we know that the volume is what we’re trying to find. And our pressure is equal to 300.

So again, we can actually substitute these into our 𝑉 equals 240000 over 𝑝. And when we do that, we get 𝑉 is equal to 240000 over 300. So now in order to actually work this out, we can actually do this kind of in steps.

So the first thing we can do is actually divide it by 100. So as you can see there, we’ve actually divided the numerator and denominator by 100. So now we’ve got 2400 over three. And next, we do 24 divided by three, which gives us eight. And then we have the two zeros that are left over from our 2400. And when we do that, we get the volume is equal to 800 centimeters cubed.

So therefore, we can say that if the volume of a gas at a constant temperature varies indirectly with the pressure of the gas and the volume of the gas is 1200 cubic centimeters when the pressure is 200 millimeters, then the volume when the pressure is 300 millimeters of mercury is going to be 800 centimeters cubed or cubic centimeters.

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