Ideal gas equation
Voiceover: We just finished talking about Boyle's law, and the experiments that led to the PV part of the ideal gas equation. Now I want to talk about the experiment that led to the V equals T part of the equation. About 100 years after Robert Boyle, there came a French physicist named Jacques Charles. If I didn't look up the pronunciation for this man's name, I probably would have said Jacks Charles, but it's Jacques Charles. This French physicist also liked experimenting with gas, and it actually turns out he was the first person to fill up a hot air balloon with hydrogen gas and fly solo. But in Jacques's 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 gas. It'll be green gas. 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 going to 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. If I show this 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. This is what the piston would look like after the heat was applied. As you heat a system of gas, the volume will also increase. In fact, the volume increases directly with the temperature, or the volume increases proportionally to the increase in temperature. I think I can show this a little bit more clearly if I use a plot of gases increasing with temperature. 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. So at about 300 degrees Celsius, this helium we can see is taking up a volume of about 5 liters right here. As we decrease this temperature, the volume is going to 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 is taking up. 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. This blue line might indicate water vapor, water gas, steam, and this yellow line would indicate hydrogen gas. But all of these gases can be plotted in a straight line. In Y intercept form, that would look like Y equals MX plus B. 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. 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. You can also see that the lines are coming to a stopping point at different places. That's because that all of these gases turn into liquid at different temperatures. They all have different boiling points. 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. The same thing with hydrogen. 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. Charles's 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 zero, and if we move some variables around, we'll see that V divided by T is equal to M. Or, in other words, the quotient of our volume divided by our temperature is constant. It's this same volume as long as the sample size is the same, so the same number of moles, and the pressure doesn't change. This is exactly the concept that we've applied to our ideal gas equation. 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. This is a perfect opportunity to apply Charles's Law. We need to start with V1 over T1 is equal to V2 over T2. 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. 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. Let's convert 25 degrees Celsius to Kelvin. All we would do is take 25 and add 273 which would give us 298 Kelvin. Our initial temperature is 298 Kelvin. We're looking for the final volume so V2. Then our final temperature is 50 degrees Celsius, and we need to convert that to Kelvin so 50 plus 273 is going to give us 323 Kelvin. 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. That's going to allow us to completely cancel out the value on this side, and we'll cancel out our units of Kelvin on this side. What we have is 323 times 4.31 divided by 298, and we're retaining our value, or our unit of liters. That's going to give us a final volume of 4.67 liters. Thanks to Jacques Charles 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.