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I've now supplied you with two definitions of the state variable entropy and it's s for entropy the thermodynamic definition said that the change in entropy is equal to the heat added to a system divided by the temperature at which the heat is added so if I obviously if the temperature is changing while we add the heat which is normally the case we're going to do a little bit of calculus and then you can view this as a mathematical or the statistical or the common rhetorical definition of of entropy and this essentially says that entropy is equal to some constant times the natural log of the number of states the system can take on and this is this case when all the states are equally probable which is a pretty good assumption if you have you know stability and a gazillion molecules I can have a gazillion gazillion states you can assume they're all roughly equally likely there's a slightly more involved definition if they had different probabilities but we won't worry about that now so given that we've seen these two definitions it's a good time to introduce you to the second law of thermodynamics second law and that's this it's a pretty simple law but it explains a whole range of phenomena it tells us that the change in entropy for the universe the change in entropy for the universe is when any process is undergone is always greater than or equal to zero so that tells us that whatever when anything ever happens in the universe the net effect is that the there's more entropy in the universe itself and this seems very deep and and it actually is so let's see if we can apply it to see why it explains or why it makes sense relative to some examples so let's say I have two reservoirs that are in contact with each other so I have I have t1 and this is kind of let's call this our hot reservoir and then I have t2 I have t2 a cold call this our cold reservoir well we know from experience what happens if I put a hot cup of water and it has it's sharing a wall with a cold glass of water or cold cube of water what happens well their temperature is equalized if these are the same substance will end up roughly in between if they're the same Fay's so essentially we have a transfer of heat from the hotter substance to the colder substance so we have some heat Q that goes from the hotter substance to the colder substance you don't see an everyday reality heat going from a cold or substance to a hotter Stubbins if I put an ice cube in so let's say some hot tea you don't see the ice cube getting colder and the hot tea getting hotter you see them both getting to some equal temperature which essentially the tea is giving heat to the ice cube now in this situation they're reservoirs so I'm assuming that their temperatures stay constant which is a little bit which would only be the case if they were both infinite which we know doesn't exist in the real world in the real world tyonne's temperatures it gave heat would go down and teas to temperature would go up but let's just see whether the second law of thermodynamics says that this should happen so what's happening here if we call what's what's the net change in entropy 41 so the second law of thermodynamics says that the change in entropy for the universe is greater than zero but in this case that's equal to the change in entropy 41 plus the change in entropy for oh I shouldn't I'm going to call it well let me call it instead of T 1 let me call it let me call it just one for system one that's this hot system up here plus the change in entropy for system two so it's the change in entropy for system one it loses it it loses Q one at a high temperature so it loses so this equals this equals minus the heat given to the system is Q over some hot temperature T one and then we have the heat being added to the system T 2 so plus Q over T 2 this is the change in entropy for the system - right this guy loses the heat and is at temperature 1 which is a higher temperature this guy gains the heat and he it's at a temperature - which is a colder temperature now is this going to be greater than zero let's think about it a little bit if I divide let me rewrite this so let me I could rearrange them so that we can write this as Q over t2 - this one I'm just rearranging it - Q over t1 now which number is bigger t2 or t1 well t1 is bigger right this is bigger bigger now if I have a bigger number than a bigger than this when I use the word bigger you have to compare it to something now 2 1 is bigger than this we have the same number in the numerator in both cases right so if I take let's say 1 over some let's say 1/2 minus 1/3 we're going to be bigger than 0 this is a larger number than this number because this has a bigger denominator you're dividing by a larger number that's a good way to think about it you're dividing this cube by some number here to get something and then you're subtracting that this Q divided by a larger number so this fraction is going to be a smaller absolute number so this is going to be greater than 0 so that tells the second law of thermodynamics it verifies this observation we see in the real world that the temperature will or that heat will flow from the hot body to the cold body now you might say hey Sal I have a I have a case that will show you that you are wrong you could say look if I put an air-conditioner in a room if I put an air-conditioner in a room let's say this is the room and this is outside you'll say look what look what the air-conditioner does the room is already cold and outside is already hot but what the air conditioner does is it makes the cold even colder it makes the hot even hotter it takes some queue and it goes in that direction right it takes heat from the cold room and puts it out into the hot air and you're saying this defies the second law of thermodynamics you have just disproved it you deserve a Nobel Prize and I would say to you you're forgetting one one one small fact one small fact this air-conditioner inside here it has some type of a compressor some type of an engine that's actively doing this it's putting in work to make this happen and this engine right here I'll do it like in magenta it's also expelling some more heat so let's call that Q of the the Q of the engine Q of the engine so if you wanted to figure out the total entropy created total entropy for the universe it would be the entropy of the cold room it'd be the entropy of the cold room entropy of the cold room plus the change in entropy for outside I'll call it outside maybe I'll call this for the room right so you might say okay this change in entropy for the room it's giving away heat let's say the room is roughly at a constant temperature for that one millisecond we're looking at it it's giving away some Q at some temperature t1 and then and it's that's a minus and then this the outside is gaining some heat at some temperature t2 and so you let me to say hey this number right here this number right here is a smaller number than this one right because this the denominator is higher so if you just look at this this would be negative entropy and you'd say hey this defies the second law of thermodynamics no but what you have to throw in here is another notion you have to throw in here the notion that the outside is also getting this heat from the engine so it's also getting this heat from the engine over the outside temperature and this term I can guarantee you I'm not giving you numbers right now will make this whole expression positive this this this term will turn the total net entropy to the universe to be positive now let's think a little bit about what entropy is and what entropy isn't in terms of words so when you take a a intro chemistry class the teacher often says entropy entropy equals disorder which is not incorrect it it is disorder but you have to be very careful what we mean by disorder because the very next example that's often given is that you know they'll say look a clean room let's say your bedroom is clean and then it becomes dirty and they'll say look the universe became more disordered the dirty room is is is is it has more disorder than the cleanroom and this is not a case of entropy increase so this is not a good example not a good example why is that because clean and dirty are just states of the room remember entropy entropy is a macrostate variable it's a macrostate variable it's something you use to describe a system where you're not in the mood to sit there and tell me what exactly every particle is doing and this is a macro variable that actually tells me how much how much time would it take for me to tell you what every what every particle is doing it actually tells you how many states there are how much information I would have to give you to tell you the exact state now when you have a clean room in a dirty room these are two different states of the same room if the room is if the room has the same temperature it has the same number of molecules in it and everything then they have the same entropy so clean to dirty it's not more entropy now if you had a nap for example I can have a dirty cold room dirty cold room and let's say I were to go into that room and and you know I work really hard to clean it up and by doing so I had a lot of heat to the system and my sweat molecules drop all over the place and so there's just more stuff in that room it's all warmed up to me so - a hot clean room clean room with with with sweat in it with sweat so it's got more stuff in here it can be configured in more ways and because it's hot every molecule in the room can take on more states right because the average kinetic energy is up so they can kind of explore the the spaces of how many kinetic energies they can have there's no there's more potential energies that each molecule can take on this is actually an increase in entropy from a dirty cold room to a hot clean room and this actually goes well with what we're now into a room and I start cleaning it I am putting heat into the room and the universe is becoming more I guess we could say it's the entropy is increasing so where where does the term disorder apply where does the term to sort apply well let's take a situation where I take a ball I take the ball and it falls to the ground and then it hits the ground and there should have been a question that you've been asking all the time since the first law of thermodynamics once the ball hits the ground so the ball hits the ground right when it got thrown up it had some potential artery to the top then that gall gets turned into kinetic energy hits the ground then it stops and so your obvious question said what happened to all of that energy right a lot the conservation of energy where did all of it go ahead all that kinetic energy right before the ground that it stopped right it seems like it disappeared but it didn't disappear so when the ball was falling it had a bunch of Mahlon these everything had a little bit of heat but let's say you know the ground was reasonably ordered it was reasonably ordered they were probably vibrating the ground molecules were vibrating with some with some kinetic energy and potential energies and then our ball molecules were also vibrating a little bit but most of their motion was downwards right most of the ball molecules motion was downwards now when it hits the ground what happens let me see the interface of the ball so the ball the ball molecules at the front of the ball are going to look like that there's a bunch of them it's a solid it'll maybe be some type of lattice and then it hits the ground and when it hits the ground so the ground is another solid like that all right we're looking at the Mac of the microstate what's going to happen these guys are going to rub up against these guys and they're going to transfer their what was downward kinetic energy and a very ordered downward kinetic energy they're going to transfer it to these two these ground particles and they're going to bump into the ground particles and so when this guy bumps into that guy he might start moving in that direction this guy will start oscillating in that direction and go back and forth like that that guy might bounce off of this guy and go in that direction and bump into that guy and go into that direction and then because that guy bumped here this guy bumps here and because this guy bumps here this guy bumps over there and so what you have is what you what was relatively ordered motion especially from the balls point of view when it starts rubbing up against these molecules of the ground it starts making the kinetic energy or their movement go in all sorts of random directions right this guy is going to make this guy go like that and that guy go like that and so when the movement is no longer ordered forget it if I have a lot of molecules let me do it in a different color if I have a lot of molecules and they're all moving in the exact same direction then my microstate looks like my macro state the whole thing moves in that direction now if I have a bunch of molecules and they're all moving in random directions my ball as a whole will be stationary I could have the exact same amount of kinetic energy at the molecular level but they're all going to be bouncing into each other and in this case we described the kinetic energy as internal energy or we described it as temperature or you know where temperature is the average kinetic energy so in this case when we talk about the world is becoming more disordered you think about the order of the maybe the velocities or the energies of the molecules before they were reasonably ordered the molecules they might have been vibrating a little bit but they were mainly going down in the ball but when they bump into the ground all of a sudden they start vibrating in random directions a little bit more and they make the ground vibrate in more random directions so it makes all at the microstate everything became just that much more disordered now there's an interesting question here there is some probability you know you might think look you know this ball came down and hit the ground why doesn't the ball just you know isn't there some probability that if I have a ground if like that these molecules just rearrange themselves and just the right way to just pop these two just hit these these ball molecules in just the right way there's some probability just from the random movement that on its some second all of the ground molecules just hit the ball molecules just right to send the ball back up and the answer is yes there's actually some infinitesimally small chance that that happens that you could have a ball that's sitting on the ground and this is interesting you can have a ball that's sitting on the ground and while you're looking you'll probably have to wait a few gazillion years for it to happen if it happens at all it could just randomly pop up and that's some red so there's some random very small chance that these molecules just randomly vibrate in just the right way to be ordered for a second and then the ball will pop up but this does not the probability of this happening relative to everything else is essentially zero so when people talk about order and disorder the disorder is increasing because now these molecules are going in more random directions and they can take on more potential states and we saw that here and you know it on some level entropy seemed something kind of magical but on some level it seemed something it seems relatively common sense in that video it to video I think it was the last video I had the case where I had a bunch of molecules and then I had this extra space here and then I removed the wall and we saw that these molecules the these molecules will I mean you know we know there's there's always some molecules are bouncing off this wall before because we probably had some pressure associated with it and then as soon as we remove that wall the molecule that would have bounced there just keeps going just keeps going there's nothing to stop it from there in that direction there's a lot of stuff it could bump into other molecules and it could bump into these walls but in this direction the odds of it bumping into everything is especially for these leading molecules is essentially zero so it's going to expand to fill the container so that's kind of common sense but the neat thing is that the second law of thermodynamics as we saw in that video also says that this will happen that the molecules will all expand to fill the container and that the odds of this happening are very low that they all come back they all come back and go into an ordered State now there is some chance just from the random movements once they feel that they all just happen to come back here but it's a very very small probability and even more and I want to make this very clear s is a macro state macro state we never talk about the entropy for an individual molecule if we know what an individual molecule is doing we shouldn't be worried about entropy we should be worrying about the system as a whole the system as a whole so even if we're looking at this system even if we're looking at the system if we're not looking directly at the molecules we won't even know that this hat we won't even know that this actually happened all we can do is look at the statistical properties of the molecules how many molecules they are what their temperature is all their macro dynamics their pressure and say you know what a box that has these molecules has more state than a smaller box in the box when we had the wall there even if by chance all of the molecules happen to be collecting over there we wouldn't know that that happened because we're not looking at the microstates and that's a really important thing to consider when someone says that a dirty room is has a higher entropy than a than a clean room they're looking at the microstate and entropy essentially is a macro state variable you could just say that a room has a certain um amount of entropy so entropy is associated with the room with the room and it's only useful when you really don't know exactly what's going on in the room you just have a general sense of how much stuff there's in the room what's the temperature of the room what's the pressure in the room just the general macro properties and then entropy will essentially tell us how many possible microstates that macro system can actually have or how much information and there's there's a notion of information entropy how much information would I have to get you give you to tell you exactly where every what the exact microstate is of a system at that point in time well anyway hopefully you found this discussion a little bit useful and clears up some misconceptions about entropy and gives you a little bit more intuition about what it actually is see in the next video

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