Chemistry: Charles's Law

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I'm from Colorado. And in the wintertime in Colorado, it can get to be -20 or -30 degrees Fahrenheit. And one of the things that I've noticed when the temperature gets that cold is that the tires on my car look a little flat. Maybe you've noticed that as well. It turns out that flat tires in cold weather are a manifestation of something known as Charles's Law. Jacques Charles was a French scientist in 1787, and he discovered a direct relationship between the volume of a sample of gas and the temperature of that gas. So here I've plotted somewhat qualitatively the relationship that he discovered. We have plotted over here the volume of a sample. I've got two samples, sample 1 and sample 2. These could be different gases. They could be different amounts of gases. The point is that as you cool them down, if you measure the volume of the gas as a function of the temperature in degrees Celsius, what happens is they eventually get down toward zero.
Let me show you a demonstration of this part of the curve where something can have a really small volume, and rise to a really large volume once the temperature warms up. And to do that, what I have here are some balloons that I inflated previously, and then I cooled down to the temperature of liquid nitrogen, which is about 77 Kelvin. Let me reach in and grab one. It's pretty cold in here. And here we have a purple balloon and it's at 77 Kelvin. But as it warms up, the volume of the gas on the inside of the balloon--remember these started out as inflated balloons. The volume of the balloon is going to increase according to Charles's Law, which is that there is a linear relationship between the volume of the gas on the inside and the temperature. We'll set this thing. It's going to get quite a bit bigger yet. We'll just set it aside. It might explode, but we won't worry about that.
You'll notice that I drew these lines, and they appear to extrapolate to a temperature of -273.15 degrees Celsius. That's the temperature at which the volume of the gas theoretically ought to go to zero. And that inspires a new temperature scale. And this temperature scale we're going to call the Kelvin scale after Lord Kelvin, who suggested it. And what we're going to say is instead of defining the zero of temperature, right now we define the zero of temperature for the Celsius scale to be the temperature at which water freezes. Let's define zero on the Kelvin scale, or zero Kelvins, to be the temperature at which the volume of a gas would go to zero. There's a relationship then between the temperature on the Kelvin scale and the temperature on the Celsius scale, which looks like the temperature on the Kelvin scale is equal to the temperature on the Celsius scale plus 273.15. That means that -273.15 degrees Celsius is equal to zero Kelvin. And notice I didn't say degrees. Kelvin is not associated with degrees. Then zero degrees Celsius is 273.15 Kelvins. And 100 degrees Celsius is 373.15 Kelvins.
Let's get to Charles's Law. In Charles's Law, another one of those laws that we get from Mother Nature, says that while keeping the gas at a constant pressure, if you increase the Kelvin temperature, then the volume will increase proportionally. Another way to state that is that the volume of a sample divided by the temperature, but the temperature in this absolute scale where the volume would go to zero if the temperature went to zero, the volume divided by the temperature is equal to a constant. This is the equation of a straight line. There are a couple of limitations. We have to say that it's for a given amount of gas. And of course when we inflated that balloon by itself, we didn't change the amount of gas on the inside of the balloon. And similarly we had a fixed pressure. In this case it's atmospheric pressure. So our balloon would have followed Charles's Law.
Let's look at the kind of problem that you're probably going to be expected to be able to solve using Charles's Law. The volume of a balloon at 25 degrees Celsius, which is room temperature, is 2.3 liters. So in other words, we just measured the volume of this balloon at room temperature. What is the volume at 50 degrees Celsius assuming that the pressure is constant? So we can apply Charles's Law very straightforwardly from this point. Now a common mistake that is made is to forget that we can't leave things in degrees Celsius. If you made the mistake and said, "Well we've got the temperature 50 here, which is double 25, it must be that the volume doubles," that's incorrect. Because we first have to convert these temperatures into the Kelvin scale. If we do that, 25 degrees Celsius corresponds to 298 Kelvins. And 50 degrees Celsius corresponds to 323 Kelvins. And you'll notice that even though there is a two for one relationship between the Celsius scale temperatures, there's not nearly a two for one between the Kelvin temperatures.
How are we going to use Charles's Law? We'll take . We can, as we saw before, use this. Say that . Rearrange that equation to put it terms of the final volume, which is what we are really interested in. , and let's go ahead and plug in what we've got. []is equal to , which was 2.3 liters. The final temperature is 323 Kelvins. And the initial temperature was 298 Kelvins. Cancel out the Kelvins. That's going to give us something in units of liters. That's exactly what we want. And this turns out to be 2.5 liters. So the balloon is a little bit larger as we warm it up. And that's consistent with our intuitive understanding of Charles's Law, that as you warm things up the volume increases, as you cool things down the volume decreases. And again that's how we explain why tires look a little flat on a cold winter day.
Gases
Gases and Gas Laws
Charles's Law Page [1 of 2]