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Current time:0:00Total duration:14:04

Video transcript

I'm now going to introduce you to the notion of the efficiency of an engine efficiency efficiency of an engine and it's represented by the Greek letter ADA so you know it sounds like an e it looks like a kind of funny-looking n so Greek ladder ADA this is efficiency and efficiency is applies to engines is kind of similar to the way we use the word in our everyday life if I said how efficient are you with your time I care about what do you do in that hour that I gave you or if I said how efficient are you with your money I'd say how much were you able to buy with that hundred dollars I gave you so if fish'n see with an engine is the same thing what were you able to do with the stuff that I gave you so in an engine in the engine world we define efficiency to be the work you did with the energy that I gave you to do the work essentially and in our carnal world up here this is all the stuff I had done before what was the energy that I had to given you well the energy that I've given you was the energy that came from this first reservoir up here remember when we were moving when we were removing the pebbles from A to B we were moving the pebbles to keep it as a quasi-static process so that we could go back if we had to so and so that the system stayed in equilibrium the whole time if we didn't have this reservoir here the temperature would have gone down because we're expanding the volume and all of that so we had to keep that reservoir there added heat to the system we figured out this multiple times the heat added to the system was q1 it was equivalent to the work we did over that time period so it would've been the whole shaded region not just what's inside the circle but so the heat that we gave you was q1 we laid you later gave some heat back when some work was done for you but what we care about is the work is the heat that we you were given so in this case would be q1 q1 and what was the work that you did well the net work that you did was a shaded area it was the area inside of our Carnot cycle so that's our definition for efficiency it's always going to be a fraction sometimes it's given as a percentage where you just multi-node is point five six you would call that 56% efficient which is essentially saying you're able to transfer 56 percent of the heat energy that you were given and turn that into useful work and so it makes sense at least in my head that that that that that would be the definition for efficiency now let's see if we can play around with this and and and see Wow efficiency would play out with some of the variables we're dealing with in the Carnot cycle so what was the work that we did well we know our definition of internal energy our definition of internal energy this has been more useful than you would have thought for such a simple equation our change in internal energy is the heat applied then the net heat applied to the system let's be careful here let me call it the net heat applied to the system minus the work minus the work done by the system right now when you complete one Carnot cycle when you go from a all the way around back to a and actually I'll make a little side here you could have actually gone the other way around the cycle but if you went over when you go when you go the way we went the first time when you go clockwise your Carnot engine you're doing work and you're transferring heat from t1 to t2 if you went around the other way around the circle you'd be a Carnot essentially would be a Carnot refrigerator where you would have work being done to you and I'll touch on this in a second and you would be transferring energy the other way and this will be important to our proof of why the Carnot engine is the best engine at least theoretically from an efficiency point of view if efficiency is all you cared about but anyway that's what I was talking about so if I go a complete cycle in this PV diagram and end up back at a what's my change in internal energy it's zero my change by internal energy is a state variable so my interchange internal energy or my change in entropy will also be zero that's another state variable when I get from a back to a so over the course of this cycle we know that my change in internal energy is zero what's my what's the net the net heat applied to the system what's the net heat applied to the system well we apply q1 let me do it we applied q1 to the system and then we took out q2 right we gave that to the second reservoir we gave that down there to t2 that second reservoir and then minus work and all of this is equal to zero this is the net I just want to make it clear this is the net heat applied to our system so the work done to this done by the system we just take a W to both sides this equation you get W is equal to Q 1 minus Q 2 so there we have it let's just substitute that back here and instead of W we can write Q 1 minus Q 2 as the numerator and our efficiency definition and then the denominator is still Q 1 and we could do a little bit of math this simplifies this is Q 1 this is the heat we put into it so it's the net heat we applied to the system divided by the heat we put into it so this is equal to Q 1 divided by Q 1 is 1 minus q2 over q1 so once again this is another interesting definition of efficiency they're all algebraic manipulations using the definition of internal energy and whatever else now let's see if we can somehow relate efficiency to our temperatures let me this is q1 right there so what work q2 and q1 what were they what were their absolute values not not looking at the signs of them and we know that it was trans q2 was transferred out of the system so if we said you know q2 in terms of the energy applied to the system it would be a negative number but if we just wanted to know the magnitude of q2 what would be in the magnitudes of q1 well the magnitude of q1 let me draw a new car node I Carnot cycle just for cleanliness I'll draw a little small one over here that's my volume axis that's my pressure axis P V I start here at some state and then I go isothermally what's a good color for an isothermal expansion oh maybe purple it's kind of see an isothermal expansion I'm on a isothermal isothermal year so I go down there and I go to state and I went down to state B so this is a 2b and we know we were on an isotherm we had this is when q1 was added this is when q1 was added this was an isotherm if your if your temperature didn't change you're in energy didn't change and like I said before if your internal energy is zero then the heat applied to the system is the same as the work you did they cancel each other out that's why you got to zero so this Q one that we applied to the system it must be equal to the work we did and the work we did is just the area under this curve we do this multiple times and why is it the area of the curve because it's a bunch of rectangles of pressure times volume pressure times volume and then you just add up all the rectangles an infinite number of infinitely narrow rectangles and you get the area and what is that and just that should review pressure times volume what's the work right because we're expanding we're expanding the cylinder we're moving up that piston we're doing Force Times distance so the amount q1 is equal to that integral the amount of work we did as well it's equal to the integral from V final let me I shouldn't say V final from VB from our volume at B oh sorry not from our volume at a to our volume at B we're starting here and we're going to our volume at B and we're taking the integral of pressure and I've done this multiple times but I'll do it again pressure times our change so the pressure our height times our change in volume DV we but we go back to our ideal gas formula PV is equal to N R T divide both sides by V and you get P is equal to n R T over V and so you have q1 q1 is equal to the integral of VA from VA to VB of this little thing over here n R T over V DV all of this stuff up here these are all constants remember one on isotherm our temperature isn't changing we could write t1 there because that's our temperature we're at t1 we're on a t1 isotherm right there because we're touching the t1 reservoir but this is all a constant so we can take it out of the equation and then this we've evaluated this multiple time so I won't go into the mechanics of the integral but so this is q1 is equal to our constant terms n R T time's this definite integral all we have left in the integral is 1 over V the antiderivative of that is natural log and then you evaluate it the two boundaries so you get the natural log natural log of VB minus VA which is the same thing as the natural log of VB over VA fair enough this right here is q1 all right now what is q2 q2 was this part of the Carnot cycle q2 is when we went from C to D right we went from C to D so that q2 the magnitude of q2 is the area under this curve right here it's the area under this curve now this is the work done to the system so that's why we subtracted it out when we want to know the net work done by the system and we ended up with this area over here when you subtracted this as well over here on this side but if we just want to know the magnitude of q2 we just take the integral we just take the integral under this under this curve and what's the integral under that curve what's the inner this is the heat out of the system the heat that had to be pushed out of the system as work was done to it so that integral so we could just say the magnitude of q2 q2 I'll do it over here q2 is equal to the same exact logic applies it was just the boundaries are different we're now going from and remember when we cared about the direction I would say I'm going from VC to VD but if I just want to know the absolute value of that area because I just want to know the magnitude I could go from VD I could go from VC to VD and just take the absolute value of it or I could just go from VD to VC and now to get a positive a positive area so let me just do that so this is the integral from V D to V CE remember the cycle we went from VC to VD but I just want the absolute value I want this to be positive so I'm I'm I'm turning it the other way of P DV and we do the exact same math there you get Q 2 Q 2 is equal to n R this time the temperature is T 2 are on a t2 of our times the natural log times the natural log and this time what's it going to be times the natural log of instead of VB over VA it's going to be V CE VC divided by v d vc / a VD now let's use these two pieces of information and substitute them back into that result for efficiency we just got we just learned that you could also write the efficiency of an engine to be equal to 1 minus Q let's let's look back at it 1 minus q2 over q1 1 minus q2 over q1 so let's substitute q2 there and q1 over here and what do you get you get the efficiency of your engine is equal to 1 minus q2 is this expression over here n R t2 times the natural log of VC over VD all of that divided by all of that divided by q1 which is this one over here and our t1 times the natural log of VB over VA now we can do a little bit of canceling obviously the N and the RS cancel that was canceled and now we have these natural logs and all of that but I had a whole video dedicated to show you that VC over VD is equal to VB over VA now if we know that these are equal then this is equal to this so the natural logs of them are equal so we can just divide and what are we left with we're left with the fact that efficiency can also be written as 1 minus t2 over t1 for for a Carnot engine for Carnot engine remember this time you know what we did over here this applied to any engine this was just a little bit of math and the definition of what of what work is and and and well I won't go too much into it right now but this is for a Carnot engine right because we did a little bit of work here that involved the Carnot cycle but this is a pretty important outcome because we're going to show that the Carnot engine in the next video is actually the most efficient engine that can ever be attained and we have to be very careful about that that when we say efficient it means that between two temperature sources you can't get a more efficient engine between those two temperature sources then the Carnot engine I'm not saying it's the best engine or it's a practical engine or you'd want it to power your your lawnmower or your or your jet plane I'm just saying that it's the most efficient engine between these two temperature reservoirs and I'll show you that in the next video