00:01
Okay, so for this problem, a gas expands in two different ways.
00:07
First, it's iso -baric, and then it's adiabatic.
00:13
And so we basically want to analyze this situation by drawing a pv diagram, getting the work, computing the temperature.
00:21
Let's see, when did i actually ask for the temperature? final temperature.
00:27
So we just need the final temperature, and then the heat flowing.
00:32
Into or out of the gas.
00:35
So let's start this one out by drawing our pv diagram.
00:40
So here is p, here's v, and then first it expands isobarically.
00:47
So we have a constant pressure.
00:48
So here's our v not.
00:50
And then i'm actually going to, let's see, erase this.
00:56
Actually, no, i'm going to label this p, p, not, because that's exactly what it is, and then relabel the axis.
01:02
So here's p.
01:03
And then this is my axis label oh right out of space okay and next it expands isobarically to four v not all right and this is like gonna be really not to scale unless i erase this v not so here we go here we go i can extend this axis and then double this distance let's put that right there.
01:42
And if a gas is allowed to expand without any heat added, the pressure on the container is going to decrease.
01:51
There's going to be less collisions kind of per second because there's the container is bigger, which is one way to think about it.
01:57
Or you can kind of remember that this is what the adiabatic curves look like from studying the chapter.
02:04
Great.
02:05
And now we have the pv diagram and we can go ahead and address be the work done equals question mark.
02:16
So the work is going to, of course, be the area under this curve.
02:25
So to find this area, we need to find the square rectangular part is kind of easy.
02:33
It's just going to be v .0 times p.
02:35
Knot.
02:35
But this shape, we have to use the formula for it, which is given, i think it's, it's around equation 19 .24.
02:46
I don't think it's exactly it.
02:47
But you get an equation, but in the equation, you end up needing to find this final pressure.
02:54
So in preparation for using this formula that i haven't actually written out yet, i just want to find this final pressure.
03:01
And then i'm going to label each point.
03:04
So here's your p3.
03:07
Oops, not p3.
03:09
I think i'm just going to label the points as 1, 2, and 3.
03:13
So if i have a subscript, that's what they refer to.
03:17
Here's one, here's two, and then there's three.
03:21
Great.
03:22
So p3, we want to find first.
03:28
And so to do that, you need equation 19 .24.
03:34
And if you do some a little bit of algebra, you get that p3 is the initial pressure divided by 2v0 over 4 .0.
03:44
4v0, where i'm using the equation that's where you have like an initial and final pressure for an adiabatic process, an initial and final volume, the initial volume being 2b not, the final volume being 4 be not.
04:00
And then of course that simplifies, but you could write it this way and you can actually sort of see where i got this number.
04:11
And so you get this nice simple equation for that.
04:14
And i guess like the problem stated that there was a given heat capacity so i'm just going to go ahead and it's not in terms of gamma so i just want to put it in terms of cv instead of gamma so as a reminder gamma is equal to 1 plus r divided by cv where r is the gas constant and so then just i'll go ahead and write that explicitly.
04:46
So you have your one -half to the one plus r divided by cv.
04:55
Okay.
04:57
And now i did that a little calculation...