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# PV-diagrams and expansion work

## Video transcript

in the last video we saw that a system could do work by expanding in this in the situation we drew we had the situation where the ceiling was moveable we had this piston and we like in our in our quasi-static process video we had a bunch of pebbles we removed a pebble so the pressure in our system if we assume that it was just so small that the pressure was constant it pushed up on the piston with some force we figured out that that force since pressure is force per area we just multiplied pressure times the area of our piston and we got the amount of force we're applying we apply that and then we multiply that times the distance that we push the piston up and then we get the we get the amount of work that it did by expansion or the expansion work we said well you know we could have rewritten that if we said pressure times our area times our distance we could instead write that as pressure times the area times the distance and the area times the distance is the change in volume and so we came up with a neat little formulation that the work done by a system could be written as the pressure times the change in volume so in this case I wrote the internal energy formula where it's the work done by the system so I did a minus right because when you do work you are giving energy away to someone else so in that situation we did a minus and so we can stead of writing work we could say minus the pressure times the change in volume and remember this is a quasi-static process we're doing it very small increments we're assuming that this change in volume is very small and that the pressure is roughly constant while we're doing this and of course that's not the case right if we did this if this was a large change in volume or if this happened all of a sudden if these were really big pebbles then our pressure will change as we expand so it's hard to say what the pressure times the change in volume is but if we assume things are really really being done in a very very small increments we could say ok let's say the pressure was constant over that small increment and then we can multiply it by the change in volume now let's see how this can relate to some of what we've done before with the PV diagram and so far all we've seen the PV diagram or what I used it for is to kind of help explain the difference between quasi-static processes or to say when macro states are defined but let me now do something more useful with it and this will give you an idea or start giving you an idea of why people who study thermodynamics love these so much so before I did anything when my canister was just here I had all the pebbles on him we were in a state of equilibrium I could describe all of its macro States its pressure its volume its temperature including it I could describe its internal energy as well let me draw it here so let's say I was at this state this was state number one state number one was right there and then let's say I just start removing pebbles remember if I just removed all the pebbles at once the state system is going to go into flux we wouldn't be doing a quasi-static process or a reversible process which isn't always the same thing but for our purposes we wouldn't we wouldn't be in equilibrium the whole time and we'd have to wait to get to equilibrium and at some point we would have some pressure and volume that's down here this is if we weren't doing it as a quasi static process now we are when I showed in the last video we are doing it as a or we're trying to get close to a quasi-static process because we're doing it in small increments and these little pebbles and if these aren't small enough for you could do it in smaller pebbles so we're moving incrementally so for example in that last video we may be moved from there we removed one pebble and we've got right there you move another pebble you go right there you move another pebble you go right there and the benefit of doing these quasi static processes is you really get a path going from one state to the next so if your move if let's say when you remove all but one of the pebbles just you know this describes our path so let's say we are in state two when we've removed all but one let me draw that so state two will look something like this I'll draw it really quick so that's our container that's our piston we only have one pebble left on top and then of course we have the gas now the gas we have a much higher this was let me write this down this is state two and let me write state one was something like this state one the ceiling was lower we had a bunch of pebbles on top of it on top of it and we had a smaller volume and so the gas was bumping into the ceilings and the walls and the floor a lot more I should draw the same number so we had a higher pressure so pressure was high pressure high and volume low now in state two so this is pressure high so this is pressure is this axis this is prep excess volume so we had high pressure low volume and we got to situation after removing all but one pebble we're and we're doing it slowly so we're always in equilibrium so we have a path this is after removing each of the pebbles so that our pressure and volume macrostates are always well-defined but at state two we now have a pressure low pressure low and volume is high the volume is high you can just see that because we kept pushing the piston up slowly slowly trying to maintain ourselves in equilibrium so our macro states are always defined and our pressure is lowered just cuz we're going to be we could have the same number of particles but they're just going to bump into the walls a little bit less because they have a little bit more room to to move around and that's all fair and dandy so this describes the path of the of our of our system as it transitioned or as it experienced this process which was a a quasi-static process so everything was defined at every point now we said that the work done at any given point by the system is the pressure times the change in volume now how does that relate to here change in volume is just a certain distance along this x-axis along whether I should call out the volume axis this is a change in volume right we started off at this volume and let's say when you removed one pebble we got to this volume now we want to multiply that diam as our pressure since we did it over such a small increment and we're over we're so close to equilibrium we could assume that our pressure is roughly constant over that period of time so we could say that this is the pressure over that period of time and so how much work we did it's this pressure over here times this volume which is the area of this rectangle the area of this rectangle right there and for any of you all have seen my calculus videos this should start looking a little bit familiar right and then what what about art when we could take our next pebble well now our pressure is a little bit lower this is our new pressure our pressure is a little bit lower and we multiply that times our new change in volume times this change in volume and we have that increment of work once again is the is the area of this rectangle and if you keep doing that the amount of work we do is essentially the area of all of these rectangles as we remove each pebble and now you might say especially those of you who haven't watched my calculus videos gee you know this might be getting close but these the area of these rectangles isn't exactly the area this curve there's you know it's a little inexact it's a little area here and what I would say is well if you if you're worried about that what you should do is use smaller increments of volume and if you want to have smaller changes in volume along each step what you do is you remove even smaller pebbles and this goes back to trying to get to that ideal quasi-static process so if you did that eventually the rectangular Delta V's would get smaller and smaller and smaller and the rectangles would get thinner and thinner thinner you'd have to do it over more and more steps but eventually you'll get to a point if you assume really small changes in Delta in our Delta V and calculus world that infinitely small changes you write it as d-v so if you if you take a sum of all the pressures times the DVS you get the area under this curve so the way to think about it when you're looking at this PV diagram if someone says you're going from this point this pressure in this volume to this pressure in this volume and they say how much work did you did you say okay well I just have to figure out the area under this curve if you wanted to know the real math behind it if you could get your pressure as a function of volume if you don't know any if you haven't watched the calculus videos you can ignore this little aside I'm going to do here but we're essentially this is this curve right here you could if you could write it this way let's say you could write pressure as a function of volume right and when you're in algebra you learn a curve is you know Y is a function of X but here Y is pressure and X is volume so it's pressure as a function of volume so the area under this curve is the integral of the pressure as a function of volume that's the height at any point right x our very small change in volume so times our very small change in volume and you take the sum from our starting volume so the volume initial - volume final and we'll do this in the future when we actually especially when we start touching on entropy but this is a neat result if you even if you don't know the calculus or if this confuses you if you've never seen it integral before you could ignore it but you could look at this intuitively you say the work I did is the area under this curve now let me ask you one more thing let's say some work is being done to the system so we start adding some marbles back right so let's say actually let's say we're going from this direction let's say we start at state 2 and we go in that direction so direction matters so let's say we go in that direction right there so I should put some arrows and I'm overloading this picture so much actually let me just do a new picture that's probably the best thing to do so this pressure volume I'm actually going to do - nope let me just do pressure volume I'm going to do two graphs here all right so in the first one is pressure volume pressure volume we started here at 1 and we went here to 2 so our system was essentially pushed up on the piston and it could be a curve or line I'm not going to get too particular right now but it was going in this direction and so we can say that the work done was the pressure times the increase in volume at any moment so the work done was the area under this curve the work done was the area under that curve now if we started at position 2 if we started at position 2 and we go to position 1 2 - 1 now what's happening now we're compressing so if we're going in that direction you might say okay maybe the work done by the system is still the area under the curve well you'd be closed because what's happening now working out compressing the system we're adding the marbles back we're putting energy into the system so if we do that remember your work done by the system was pressure times an increase in volume now it's going to be your pressure times a decrease in volume so when you go back in this direction when you go back in this direction the area is not the work done by the system it's the area it's the work done to the system and maybe I'll do that in a different color so blue for or green for work done to the system now let me throw you another little interesting idea and this is actually a key idea it's good to get the intuition here so let me just draw a very simple PV diagram again nope use the wrong tool let me use another PV so let's say we start at some state here state one and I do I do something you know I'm in a quasi-static process and it you know it's doing something weird and I get to state two here I get to state two here and it's going in this direction right so my volume is increasing so in this situation what is the work done by the system easy enough it's the area under this curve it's the area under this curve now let's say that I keep doing some type of I do some type of quasi static process but it takes a different path I'm doing something else other than adding the marbles directly back so my new path looks something like this my new path looks something like this to get back to state one to get back to state one so these arrows are going back so now what is the work done to the system well my volume is decreasing so it's the area under the second curve the area under the second curve is the work done to the system so if I want to know what the network the system did going from state one to state two and then going back to state one remember this the pressure and volume diagram what is it well the work that the system did was this whole area under this brown curve and then it had some work done to it which is the area under this magenta curve so the network it did is essentially the white the whole area - this red area so the network it did would be essentially just the area inside this loop the area inside that loop and hopefully you don't have to know calculus to do this although calculus you would actually use to compute to actually compute these areas but I just want to give you that intuition that the that the area inside this closed loop is actually the amount of work that our system has done when it's going and it what's important is that the direction that it's going so it increased volume then decreased volume so it's kind of this this clockwise motion this is the work that our system has done which I don't know to me is a pretty interesting thing and later we can use this notion to come up with some other ideas behind our state variables I'll make one little aside here remember our state our state variable pressure volume we did set up to it that we went back to that state that stayed the same and I want to say another thing for our purposes when we're dealing with ideal gases that they where the internal energy is essentially the kinetic energy of the system if we go and do all sorts of crazy stuff and come back our internal energy hasn't changed so the internal energy is always going to be the same at this point so if I said I did all of this stuff and came back here what is my change in internal energy it's zero change is zero now if I said I went from here to here I would have a different internal energy and my change would be something real but since this is a state function it doesn't care how I got there if I took all these loops and got back there just says look if I'm at this point in the PV diagram my internal energy is the same thing so if I get if I start at this point and I finish again at this point I have had no change in internal energy and we'll talk more about that in the next video but I just wanted to leave you there and get you to this intuition behind the areas under the curves in the PV diagram