Okay, so we just finished talking about Boyle's Law and the experiments that led to the PV part of the ideal gas equation. And now I want to talk about the experiments that led to the V equals T part of the equation. So about a hundred years after Robert Boyle, there came a French physicist named Jacques Charlet.
And if I didn't look up the pronunciation for this man's name, I probably would have said Jacques Charlet, but it's Jacques Charlet. And this French physicist also liked experimenting with gas and actually turns out he was the first person to fill up a hot air balloon with hydrogen gas and fly solo. But in Jacques'experiments with gas and temperature he found that if you heat a gas in a closed container, say like a piston, so I've got a piston here and I'll fill it with with gas, it'll be green gas, and this piston will be under constant pressure because As the atmosphere is pushing down on top of the piston, then the pressure of the gas pushing up is gonna equal the atmosphere.
But under constant pressure, with the same amount of particles, as you heat this piston, so let me apply some heat here, and what we'll see is that the volume of the gas will also increase. So if I show the same piston after the heat was applied, we'd see that the gas was taking up more volume. Even though there's the same number of particles here. We still have six green particles of gas So this is what the piston would look like after the heat was applied and so as you heat a system of gas the volume Will also increase and in fact the volume increases directly with it with the temperature or the volume increases Proportionally to the increase in temperature And I think I can show this a little bit more clearly if I use a plot of gases increasing with temperature And so this is what a plot of volume expansion would look like for four different gases as we're increasing the temperature.
This pink gas would be helium. And so at about 300 degrees Celsius, this helium we can see is taking up a volume of about five liters right here. And as we decrease this temperature, the volume is gonna decrease proportionally. This straight line is showing this down to at zero degrees Celsius.
We've got just a little over three liters that this helium's taking up. And then we've got this green gas, and this might be methane. And we're seeing the same thing. As we increase the temperature, we're increasing proportionally the volume that the methane's taking up.
And this blue line might indicate water vapor, water gas, steam. And this yellow line would indicate would indicate hydrogen gas. But all of these gases can be plotted in a straight line.
So in y-intercept form, that would look like y equals mx plus b. And if we substitute the values that we're using in this graph, our y is our volume, so we would see that y is equal to v, and our x is our temperature. So if we fill that all the way in here, we'd have v is equal to mt plus b.
Now if you're wondering why the slopes are different, it's because the different gas samples in this example would have different number of moles, and you can also see that the lines are coming to a stopping point at different places, and that's because that all of these gases turn into liquid at different temperatures. They all have different boiling points. So with methane, the boiling point would be about negative 100 degrees Celsius, but we could kind of extrapolate this line down.
With water vapor, the boiling point is 100 degrees Celsius, so that's kind of why this straight line stopped. But we can extrapolate this line all the way down as well. And the same thing with hydrogen.
And if we extrapolate these values out to find their y-intercepts or their b-values, we would see something really interesting. And that's that all of them have a volume of zero at the exact same temperature, which is negative. 273.15 degrees Celsius, which is also zero Kelvin.
And so Charlet'Law is actually another proof that zero Kelvin is absolute zero because we can't have a negative volume for gas. All of these gases have to take up some volume. So the lowest temperature that we could theoretically achieve for any of these gases is negative 273.15 degrees Celsius or zero Kelvin. Now if we take our equation, which is v equals mt, and now we don't need the b because our y intercept is 0, and if we move some variables around, we'll see that v divided by t is equal to m. In other words, the quotient of our volume divided by our temperature is constant.
It's the same volume as long as the sample size is the same, so the same number of moles and the pressure doesn't change. And this is exactly the concept that we've applied to our ideal gas equation. So let's try to use this concept in a problem. If the volume of a piston filled with gas is 4.31 liters at 25 degrees Celsius, Then what is the volume of the gas after it's heated to 50 degrees Celsius, assuming that the system doesn't experience a change in pressure?
What we're looking at is a change in volume related to a change in temperature, assuming constant pressure and assuming a closed system with constant moles. So this is a perfect opportunity to apply Charlet'Law. So we need to start with V1 over T1. is equal to V2 over T2.
And again, we're just saying that the initial quotient of the volume and temperature is equal to the final quotient of the volume and temperature, because volume divided by temperature is constant. So our initial volume is 4.31 liters. And our initial temperature is 25 degrees Celsius, but when we're using the ideal gas law, we really need to be operating in Kelvin, because Kelvin allows us to not use negative values for temperature.
So let's convert 25 degrees Celsius to Kelvin. And all we would do is take 25 and add 273, which would give us 298 Kelvin. So our initial temperature is 298 Kelvin.
And we're looking for the final volume, so V2. And then our final temperature is 50 degrees Celsius. And we need to convert that to Kelvin, so 50 plus 273 is gonna give us 323 Kelvin.
And so that's the value that we'll input for our final temperature. Oop, I noticed that I put T1 here, that's actually T2. Our final temperature is 323 Kelvin. So to continue solving this, we need to multiply both sides by 323 Kelvin to isolate.
our final volume. So times 323 Kelvin, and that's gonna allow us to completely cancel out the value on this side, and we'll cancel out our units of Kelvin on this side. And so what we have is 323 times 4.31 divided by 298, and we're retaining our unit of liters.
and that's going to give us a final volume of 4.67 liters. And so thanks to Jacques Charlet, we know that if we're looking at a closed system under constant pressure, then we can predict the change in volume related to the change in temperature, or vice versa.