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PV diagrams - part 1: Work and isobaric processes

Video transcript

something you see a lot when doing thermodynamics especially problems involving the first law are what are called PV diagrams and now the P stands for pressure and the V stands for volume and this gives you a diagram of what the pressure and volume are at any given instant so what does this mean we'll imagine out a container full of a gas and there's a movable piston on top piston can move up or down changing the amount of volume right this is the volume we're talking about is the volume within here so that movable piston can change that amount of volume and that would change the amount of pressure inside depending on what heat is added how much work is done so say we started with a certain amount of volume right let's say we start with that much volume and the pressure inside is probably not zero if there's any gas inside it can't be zero and so we come over to here let's say we start at this point right here now what do we do I know if I push the piston down my volume decreases and if I pull the piston up my volume increases so if I push the piston down I know volume goes down that means on this graph I'm going that way piston going down means decreasing volume what about piston going up well if the piston goes up then my volume is increasing and I know on my graph I'd better be going to the right now maybe I'm going up and right maybe I'm going down and right all I know is my volume better be increasing so this is increasing volume that's increasing volume that's increasing volume this is not increasing volume so I know if my piston goes up my volume increases I gotta be going to the right word in some way on this graph and if my piston goes down I better be going to the left on this graph somehow now what happens to the pressure you got to know a little more detail about it but just knowing the direction of the piston that lets you know which way you go on this graph so say I've pushed the piston down say I push it down really fast what do you think's going to happen to the pressure pressure is probably going to go up how would I represent that well volumes got to go down pressure would have to go up so I might take a path that looks something like this volumes got to go down to the left pressures got to go up so maybe it does something like that there's really infinitely many ways the gas could get from one state to another it could take any possible range and unless you know the exact details it's hard to say exactly what's going to happen so there's infinitely many possibilities on this diagram you can loop around it's not like a function you can do something like this this gas can take some crazy path through this PV diagram there's infinitely many ways it can take but there are four thermodynamic processes that are most commonly represented on a PV diagram again these are not the only four possibilities these are just the four that are kind of the simplest to deal with mathematically and they're often a good representation and accurate approximation to a lot of processes so the maths good they work pretty well we talked about them a lot the first one is called an isobaric process ISO means constant so whenever you see ISO before something it means constant whatever follows next and this one's isobaric beric will bars that's a unit of pressure so beric is talking about pressure isobaric means constant pressure so how do you represent this on a PV diagram well if you want to maintain constant pressure you can't go up or down because if I were to go up my pressure would be increasing if I were to go down my pressure would be decreasing the only option available is to go along a horizontal line so this would be an isobe well sometimes they're called isobars an isobar for short this is an isobar this is an isobaric expansion if I go to the right because I know volumes increasing and if I go to the left it'd be an isobaric compression because volume would be decreasing but it doesn't have to be in this particular spot it could be anywhere on this PV diagram any horizontal line is going to be an isobar an isobaric process now I bring up the isobaric process first because that allows me to show something important that's true of every process that's just easier to see for the isobaric process in physics the area under the curve often represents something significant and that's going to be true here as well let's try to figure out what the area under this curve represents so first of all to find the area of this rectangle we know it's going to be the height times the width what's the height the height is just the pressure right the value of this pressure over here is going to be the height and the width is the change in volume so if I started with the initial and I end with the final let's say it was the expansion instead of the compression this V final minus V initial this Delta V is going to represent the width of this rectangle so we know area is going to be the value of the pressure times the change in the volume well what does that mean we know that pressure we know the definition of pressure pressure is just the force per area so on this gas the amount of force exerted on it per area and the change in volume what do we know is the volume well how could I represent the volume in here I know this piston has some area so there's some area that this piston has and then there's a certain height this inner cylinder of volume in here has a certain height and then a certain area so we know the volume is just height times area so it'd be height times the area of the piston which of these is changing in this process well the area is not changing if the area of this piston change it either let some of the gas out or it would bust through the sides of the cylinder both of which we're assuming is not happening so I can pull area out of this Delta sign since the area is constant and what I get is f times a over a times the change in the height well the A's cancel a cancels a and I get F times the change in the height but look it this is just force times a distance times the distance by which this height changes so Delta H will be the amount by which this piston goes up or down and we know force times the distance by which you apply that force is just the work so now we know the area under this isobaric process represents the work done either on the gas or by the gas depending on which way you're going so this area is the work this area the value of this area equals the amount of work done on the gas or by the gas how do you figure out which well technically this area represents the work done by the gas because if we're talking about a positive area mathematically that means moving to the right like on a graph and math class the area positive area you're moving to the right so if we want to be particular and precise we'll say that this is a process moving to the right and we know if the volume is going up like this graph is going to the right which means volume is increasing we know the gas is doing work so technically this area is the work done by the gas you can see that as well since this is P Delta V if your Delta V comes out positive pressure is always positive if your Delta V comes out positive the volume is increasing that means work is being done by the gas so you have to be careful if you calculate this P Delta V and you go to your first law equation which remember says Delta U is Q plus W well you can't just plug in the value of P Delta V this is the work done by the gas so you have to plug in negative that value for the work done and also correspondingly if you were to go to the left if you did have a process that went to the left that is to say the volume was decreasing if you find this area and you're careful then you'll get a negative Delta V if you're going leftward because you'll end with a smaller value for the volume than you started with so if you really treat the left one as the final because that's where you end up if you're going left and the right word one is the initial your leftward final point will be smaller than your initial point you will get a negative value here so again you plug in negative of that negative value you'll get your positive work because positive works being done on the gas that sounds very complicated here's what I do quite honestly I just look at the shape I find the area I do the magnitude of the height right the size of it no negatives the size of the width no negatives I multiply the two and then I just look am am I going to the left if I'm going to the left I know my work is positive if I'm going to the right I know my work is negative that I plug into here so I just add the negative sign in makes it easier for me to understand so I said that this works for any process how's that so if I take some random process I'm not going to get a nice rectangle how is this true well if I did take a random process from one point to another say it took this crazy path here even though it's not a perfect rectangle I can break it up into small rectangles so I can take this break this portion up into if I make the rectangles small enough I can approximate any area as the summation of a whole bunch of little rectangles and look at each one of these rectangles well P Delta V that's the area underneath for that one add them all up I get the total area underneath so even though it might be difficult to find this area it's always true that if I could find this area under any process this area does represent the work done and again it's by the gas so in other words using the formula work done by the gas that we had previously equals P times Delta V that works for one small little rectangle and you can add all those up but it won't work for the entire process if you try to use the say initial pressure times the total change in volume that's not going to give you an exact answer that's assuming you have one big rectangle so this formula won't work for the whole process but we do know if you have an isobaric process if it really is an isobaric process then we can rewrite the first law the first law says that Delta U equals Q Plus work done on the gas well we know a formula for the work done by the gas work done by the gas is P Delta V so the work done on the gas is just negative P times Delta V here's a formula for the first law if you happen to have an isobaric process so an isobaric process is pretty nice it gives you an exact way to find the work done since the area underneath is a perfect rectangle but how would you physically set up an isobaric process in the lab we'll imagine this let's say you heat up the cylinder you allow heat to flow in that would tend to increase the pressure so the only way we could maintain constant pressure because an isobaric process maintains constant pressure if I want the pressure to stay the same as heat flows in I better let this piston move upwards while I add heat I can maintain constant pressure in fact you might think kated are you going to do that exactly not so bad just allow the piston to come into equilibrium with whatever atmospheric pressure plus the weight of this piston is so there's a certain pressure down from the outside and then there's the weight of the piston divided by the area gives another pressure this heat will try to make the pressure increase but if you just allow this system to come into equilibrium with the outside pressure the inside pressure is always going to equal the outside pressure because if it's not equal this piston will move up or down accordingly so if this piston can move freely it'll maintain a constant pressure and that would be a way to physically ensure that the pressure remains constant and you have an isobaric process I'll explain the next three thermodynamic processes in the next video
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