Hey, everybody. So let's get started with our problem here. We have an ideal gas in a sealed container. We're told what the initial volume, pressure, and temperature are, and we're told that the pressure and the volume are both going to double. Let me just go ahead and draw out a sketch of what's going on here. So imagine that I have this kind of sealed container like this, but I have a sort of plunger that I can move up and down. This thing can sort of go up and down like this. So what happens here is that this plunger, imagine it's kind of like halfway through this container, which means that the gas is sort of contained inside of just this area over here. This is the gas. We have an initial volume, pressure, and temperature. Now, what happens here is I'm going to move the plunger up so that it goes like this. So that's basically the top of the container. So now, all the gas has now effectively sort of doubled in size. Right? The container is now sort of bigger than it was before. This is what's going on in this problem here. We have a final volume, final pressure, and final temperature. What we're trying to find in this first part here is what the final temperature is.
We've got an ideal gas. We're going to change some of the characteristics or variables like pressure and volume. We want to figure out what's the final temperature. In order to do that, we're just going to stick to the steps here. The first step we're going to do is write out our changing ideal gas equation. So I'm going to do pinitialvinitialninitialTinitial = pfinalvfinalnfinalTfinal. What we're looking for is Tfinal.
Now, we want to cancel out any constant variables because they'll just cancel off from both sides of the equation. Pressure and volume clearly don't remain constant because we're both told that they both double. However, this is a sealed container. So if it's a sealed container, what happens is that nothing is allowed to go in or come out. So, the number of moles, the amount of gas inside the container does actually remain constant. Alright. So now that that step's done, we're going to go ahead and solve for our target variable. Tfinal equals pfinalvfinalTinitialpinitialvinitial.
Since the pressure and volume both double, I can rewrite the initial pressure of the final pressures as multiples of the initial. It means that pfinal, if it's just double the initial pressure, I can write this as 2pinitial. I can do the same exact thing for the volume. The volume also doubles, so vfinal is just 2vinitial. Then, Tfinal just becomes 4Tinitial. Whatever you do to one side, you have to do to the other. As both pinitial and vinitial are multiplied by 2, Tfinal has to increase by 4 times.
The initial temperature, which is 40 degrees Celsius, is first converted to Kelvin (adding 273 to it) giving 313 Kelvin. So, Tfinal just becomes 4 × 313, yielding a final temperature of 1252 Kelvin.
Now with part b, we are supposed to figure out how many moles of gas there are. Remember the moles of gas is just n. There are different ways I can do this. If I'm looking for n, that was the only variable that I canceled out of my equation. I can just go ahead and use pv = nRT to solve for this. It doesn’t matter whether you use the initials or the finals, as n remains constant. So, your pinitialvinitial divided by RTinitial equals n. The initial pressure is 0.15 atmospheres, converting it with 1 atmosphere = 1.01 × 105 Pascals gives 1.515 × 104 Pascals. The initial volume of 2.8 liters, converted to cubic meters (1 liter = 0.001 cubic meters), gives 0.0028 cubic meters. Working this out gives approximately 0.016 moles, which is your final answer. Let me know if you guys have any questions.
