Carnot efficiency 3: Proving that it is the most efficient

let's say we have two reservoirs let's say up here I have my hot reservoir it's at th for hot and I have my engine it's going to be a Carnot engine because we'll learn that no no engine is better at least from an efficiency point of view we have to be careful when we say better we have our Carnot engine and it takes it operates on heat differentials so it takes some heat from our hot heat source it takes some heat there let's call that q1 and it does some work it does some work works a good thing so I'll make that in green it does some work and then the the surplus energy essentially goes to a cold so the the surplus heat q2 then goes to our cold reservoir do that right there T cold now I made multiple insinuations in the previous video that this is the most efficient that the this is the most efficient engine that can be created between two heat between two reservoirs th and TC now you come along and say no no no no no I know her friend he has invented of a he has invented a new engine that is more efficient than this engine between these same two reservoirs and you go and you you you proceed to draw the same type of diagram for your friend's engine for your friend's engine you say look let me make it clear this is the same reservoir right these same reservoirs we're dealing with she I should probably draw this line all the way because I'm going to do multiple engines here so the same reservoir we're dealing with all right this is all the hot reservoir th th and this is all the cold reservoir I need space for multiple engines that we're going to deal with so your friend has an engine and it'll call it the super engine super engine and your friends claim and I'm your friends claim and I'll show you why your friends claim cannot be true if you believe the second law of thermodynamics so your friends claim they the super engine and they claim that look I can actually take in q1 I can take in that same heat from this heat source up here but I can produce more work than your engine I can produce then your Carnot engine I could produce 1 plus X I don't get to algebraic but let's say you produce W I produce W times 1 plus X of work where X is a positive number so he's saying look X is greater than 0 whatever that number he might feed you is and then the rest of the energy that's left over yet the rest of the energy that's left over is what its q1 minus this so it's q1 minus W times 1 plus X and just to be clear q2 right here I can rewrite that as q1 minus W fair enough so you look at that you come to me with this and I say no no no no this this cannot be true because if this were then we would solve literally all of the world's energy problems and I'm about to show you why we could solve all of the world's energy problems and we would have a perpetual motion machine and and be able to defy all sorts of things if we had this now I had this is my Carnot engine but I could devise a reverse Carnot engine right let me make a reverse Carnot engine so my reverse Carnot engine would look like this Carnot reverse and it's going to do the same thing but in Reverse so instead of taking instead of producing q1 minus W here instead of producing q1 minus W and putting it into TC it could take in q1 I want to make sure I don't run out of space it could take in q1 minus W from our cold source so could take that in or even better let's scale it up a little bit let's take let's say it takes in q1 minus W times 1 plus X so I've just made a slightly larger reverse Carnot engine now if I take in that much and that if I knew in order to do this in Reverse I'm gonna have to take in I'm gonna have to scale up this Carnot engine and reverse everything so an editor instead of producing work I'm not going to need work to go in the other direction and I've scaled it up by one plus X so I'm going to need one plus I'm gonna need my the amount of work here times 1 plus X times 1 plus X and then I'm gonna produce I'm gonna push q1 but I've scaled it up I'm gonna push in q1 times 1 plus X into my hot heat source and once again this isn't defying the laws of thermodynamics I'm picking up some work I need there's work that needs to be done in order to do this but all of a sudden you you you you come to me and say look we this is this is an awesome deal you have this nice engine that works this way my friend has this super engine let's just couple them together let's take the work that he produces right here he produces W times 1 plus X and that just happens to be the amount of work that you need to operate your engine so you just feed that into there so what's the net effect of these two engines what's the net effect of these two engines so if I were to do let me do another let me just go scroll a little bit more actually that might be the best way to do it so let me make sure that we understand that these are the same heat sinks or heat sources that we're using the whole time so that's my hot source my cold source is down here so if I add our two engines together so if I have a you know let me call it a mmm took a new color these colors are getting monotonous nope I wanted to do the rectangle tool there you go all right so I combined these two engines together essentially I just put a big box around them they're both operating between these two heat sources these two reservoirs so I call this the you know your super engine plus my reverse Carnot engine so what's happening now what's the net what's the net heat that's being that's being taken in or put out of here so we have we have q1 so in this direction we have let me say we have q1 - W one plus x but in this direction we have q1 so this is in this direction we could rewrite this I want to make sure you're clear on the algebra this could be rewritten as what is q1 times 1 plus X times or minus W times 1 plus X right now in these and if you compare these terms this is the same as this term this term is bigger than this term right this term is clearly bigger because we're multiplying it by something larger than 1 it's bigger than this term so this upward if we combine these two the upward movement from my reverse or the amount of heat I'm taking up from my reverse Carnot is going to be greater than the amount of heat being put in by your friends super engine and we can actually calculate them out we can just take this amount minus that Mount and that's the net upward movement so the net upward movement from our cold reservoir is what it's this value minus this value so minus Q 1 minus W 1 plus X if we take a - we're going to subtract it so it's minus and a plus these cancel out this minus cancels out with so this first term could be rewritten as Q 1 plus Q 1 X right we could rewrite it that way so this cancels out with that and so the net upward movement when we combine the two engines is Q 1 times X fair enough now what about the work transfer well whatever work this guy produces is exactly the amount of work that I need so there is no net no outside work has to be done on the system it just works this guy produces work this guy uses the work now what's the net heat transferred up to our hot reservoir to our hot reservoir what's the amount of heat what's the difference between these two and this is clearly a larger number than this one so we have the upward movement dominates so what's this minus that so this can be rewritten as Q 1 plus Q 1 X right I just distributed the Q 1 we're going to subtract that out minus Q 1 and it's with Q 1 X so the net when we combine the two engines is q1x so what's happening here I have no external energy or work has to be expended into this system and it's just it's just taking heat from a cold body and it's moving it to a warm body and it does this indefinitely it'll do this as much as I want to I can just build a bigger one it'll do it I don't even a larger and larger scale so if you think about it I mean I could I could I could I could heat my house with ice by just making the ice colder I could I could you know I could create steam from from things that are arbitrarily cold this is just this goes against the second law of thermodynamics this is you know the the net entropy in this world is going down because what's happening here this is just a straight-up transfer of q1x from a cold body to a hot body so what's the net change in entropy here the change in entropy of the universe well the hot body the hot body is gaining some heat so it's q1x over over the temperature of the hot body and then the cold body is losing the same amount so it's minus q1 x over the cold body now this is a bigger number this is a bigger number then this is right because this is a the denominator is smaller this is a cold body it's temperature in Kelvin will be a smaller number so this is going to be less than zero which the second law of thermodynamics tells us cannot be the entropy cannot shrink in the universe this whole thing is just a it's an independent system and the entropy is shrinking and we can make the entropy shrink arbitrarily if we just scale up our X's enough so this is why the Carnot engine is the most efficient engine possible because if anyone claimed to have a more efficient engine you could couple it with a reverse Carnot engine and then create this perpetual reverse I guess you could call it a perpetual refrigeration machine that just out of out of the blue creates anti entropy from anywhere and it would be this perpetual energy source that creates energy out of nothing and so this is just something that cannot be done in our world especially if you believe the second thermodynamics so the most efficient engine is the Carnot engine where its efficiency is described as one minus the temperature of the cold body divided by the hot body so if I have two temperature reservoirs let's say that my hot one is at 500 Kelvin and my cold one is at 300 Kelvin and I have some engine that takes heat from there and transfers there and does some work the most efficient engine if I were to remove all the friction and the engine the most efficient the highest efficiency I could get would be 1 minus 300 Kelvin over 500 Kelvin which is 1 minus 3/5 which is 2/5 2/5 which is equal to 0.4 which is equal to 40% so if someone tells you that they made an engine that takes that that operates between a reservoir that's 500 Kelvin and 300 Kelvin and they say oh I've achieved 41 percent efficiency I've really polished the thing well you know that they are lying so anyway hopefully you found that reasonably interesting and and I'll see you in the next video