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# Work from expansion

## Video transcript

I've talked a lot about how our change in internal energy of a system can be due to some heat being added to the system or some or some work being added to the system being done to the system and I'm going to write it again the other way just because you see it both ways you could say that the change in internal energy could be the heat added to the system minus the work done by the system so there's two questions that might naturally spring up in your head one is how is heat added to or taken away from a system and how is work done or done done by or done to a system the heat I think is fairly intuitive if I have a and will be a little bit what will be more precise in the future this but I just want to give you the sense of what we're talking about if I have some system here some particles in a and some type of a canister and it's a temperature I don't know let's say it's a temperature T one do you want I'll even give it I'll even give it a minute it's at a I don't know it's at 300 Kelvin 300 Kelvin if I want to add heat to this system what I can do is I can place another system right next to it maybe right next to it you know who knows what size it is and it's got some particles there but it's temperature is much much much higher so this temperature is going to be so this systems temperature let's say t2 is equal to and let's say it's a thousand Kelvin I'm just making up numbers so what's going to happen in this situation you're going to have heat transferred from this second system to the first system so you're going to have heat going into the system now heat and what will work and even internal energy this goes back to our conversation of macro states versus micro States heat is changing the macro state of our systems this system is going to lose temperature the system is going to gain temperature but we know what's happening on a micro level these molecules are going to lose kinetic energy these molecules are going to gain kinetic energy how is that actually happening well we assume that there's some type of a container here maybe it's a solid wall these molecules are going to bump into that wall we're going to make the particles in that wall vibrate and then they're going to make the particles in the the green containers walls vibrate and so when the green containers molecules touch the wall they're going to bounce off with even more kinetic energy with even more velocity because of that vibration in the wall will push them back even further so that's essentially how you get this transfer of kinetic energy or this transfer of heat I think that's really intuitive if we put this next to a cooler a a system with lower temperature we would lose kinetic energy or we would lose heat and there's other ways that we can do it we could compress the content well I don't want to talk about that just now because that'll be touching on work so how can we add or subtract work to a system and this one's a little bit more interesting let's go back to our piston example let's go back to a let me just draw some lines here so I have my container you go it's got a little movable ceiling to it that's our my piston and go back to the example because what we're going to be dealing with especially once I go into the pressure pressure volume diagram the PV diagram that I'm about to go into we want to deal with quasi-static processes processes that are always close enough to equilibrium that we feel okay talking about macro States like pressure and volume remember that if we just did something crazy and the whole system is in flux those macro states aren't defined anymore so let's say we want to do a quasi-static process so I'll have pebbles instead of one big rock I'll draw the pebbles a little bigger this time and I have some pressure so that's my piston and it's it's being kept down by these rocks it's being kept up by the pressure of the gas the gas is bumping into this ceiling it's bumping into everything the pressure every point in the container is the same it's at equilibrium now what happens in that example where I removed I removed one rock from that so let me copy and paste that so if I remove one rock from this thing right here you copy and paste so that's the same thing now let me remove a rock I'll remove this top one was removed what's going to happen well I now have less weight pushing down on the piston and I have a certain amount of pressure pushing up the system it'll vary temporarily go out of equilibrium but we did a very small difference and how much we're pressing down on it so hopefully it won't be a huge change in our equilibrium we'll stay pretty close to it but we know from the previous example instead of this thing flying up it's going just going to shift up a little bit this is just going to shift up a little bit right when we do it is going to be like that right there and let me fill in that part with black because it's not like the space disappeared let me fill that in right there so our thing our little piston will move up a very small amount and what I claim is when this happened when I removed this little pebble from here the system did some work and let's just think about that so work according to the definitions that you learned in first-year physics and when you just using classical or dealing with classical mechanics you learned that work is equal to force times distance so if I'm claiming that when this piston moved up a little bit when I remove that pebble I'm claiming that this system here did some work so I'm claiming that it applied a force to this piston and it applied that force to the piston for some distance for some distance so let's figure out what that is and if we can somehow relate it to other macro properties that we know reasonably well well we know the pressure and the volume right we know the pressure that's being exerted on the piston at least at this point in time and what's pressure pressure pressure is equal to force per area force per area right so the area remember this piston there's some you know you just seeing it from the side but it's a kind of a flat plate or a flat ceiling on top of this thing and now what distance didn't move it let's just call that you know I could blow it up a little bit let me you know it moved it some I didn't draw it too big here some X some distance X right so this change it moved it moved it up some distance X there right so what is the force that it pushed it up well the force we know it's pressure the pressure is force per area so if we want to know the force we have to multiply pressure times area if we multiply both sides of this times area we get force so we're essentially saying the area of this little ceiling to this container right there you know it could be I could draw it with some depth but I think you know what I'm talking about it has some area it's probably the same areas the base of the container right so we could say that the force being applied by our system let me do it in a new color the force is equal to our pressure of the system pressure times the area of the ceiling times the area of the ceiling of our of our container of the piston now that's the force now what's the distance the distance is this x over here the distance is I'll do it in blue it's this change right here I didn't draw it too big but that's that X now let's see if we can relate this somehow let me draw it a little bit bigger so and I'll draw it it I'll try to draw it in three dimensions so let me draw the piston what color did I do it in I did it in that brown color so our piston looks something I'll draw it as a ellipse the piston looks like that and it got pushed up so it got pushed up some distance X let me see how good I can know whoops let me copy and paste that same so the piston gets pushed up some distance X let me draw that it got pushed up some distance X and we're claiming that our so this is the force sorry let me be clear this is the force and this is the distance right so work is equal to our force which is our pressure times our area pressure times our area times the distance right I want to be very clear with that because when I wrote this I said okay the force that we're applying is the pressure we're applying times the area of our cylinder this is the area of our cylinder right here that's the area of our cylinder right there so if you do the pressure the pressure times this area you get the force and then we moved it some distance X right now what is we could rearrange this we could say that the work is equal to our pressure times our area times X what's this what's our this area this area right here times X well that's going to be our change in volume right we've increased this this area times some height is some volume and that's essentially how much our container has changed in volume when we push this piston up the volume of our container has increased you can see that even looking from the side our rectangle got a little bit taller when you look at it with a little bit of depth you see the rectangle did not also didn't get taller we have some surface area surface area times height is volume so this right here this term right here is a change in volume so we can write work now in terms of things that we know we can write work done by our system work done is equal to pressure times our change in volume times change in volume now this has some this has a very interesting reprecussions ear so we could many we can rewrite our internal energy formulas so for example we can write internal change in internal energy is now equal to heat added to the system plus the work plus the work let me say the minus the work done by the system minus the work done by the system well what is the work done by the system what's the pressure of the system times how much the the system expanded right in this case the system is pushing these these these these marbles or these these pieces of sand up it's doing work if we were doing the other way if we're adding the sand and we're pushing down on our little canister we would be worth doing work to the system so this is the situation where I'm doing here where I'm removing the sand and the piston goes up essentially the gas is pushing up on the piston the system is doing the work so if we go back to our little formula that internal energy is heat minus the work done by system so done by done by then we could write this as this is equal to the heat added to the system heat added minus this quantity the pressure of the system times the change in volume it is interesting if the volume is increasing then the system is doing work and this applies we're going to talk a lot more about engines in the future but that's how engines do work they they have a little explosion that goes on inside of a cylinder that pushes up on the piston and then that piston moves a bunch of other stuff that eventually turns wheels so when the volume increases you're actually doing work so I'm going to leave you there in this video in the next video we're going to relate this this new way of writing our internal energy our formula and we're going to relate it to the PV diagram