this is a video of ink separating from water there's nothing physically impossible about this situation this is a thing that could actually happen but still we never see it happen and by the time I was done with the first sentence of this video you'd probably already figured out that I just revers the footage in this case I reversed a clip of the ink machine from my last video but why do you know that why do humans have an intuition for what direction time should move here's another example two rolling steel balls colliding with each other I'm playing the same clip forwards and backwards here which do you think is playing forwards and which do you think is playing backwards this one is a lot [Music] harder entropy is a complicated thing and if you take nothing else from this video remember this example with the colliding Steel walls in this interaction where two particles Collide there is nothing physically preventing both of these interactions from existing in fact if you played this clip one way then froze time reached into the system reversed the momentum vectors of both balls and click play again it would play out precisely in Reverse but let's look at a more complicated system this time we have 10 particles interacting five black and five Chrome it looks pretty normal then all of a sudden all the black and chrome balls end up on opposite opposite sides that seems really weird and if you're thinking that's weird because this clip was reversed you'd be right this clip was reversed this is actually the start of the video playing forwards now I started the balls in two distinct groups and then agitated them so that they would randomize pretty quickly the order of the starting condition Fades into this disordered mess but what about this clip now there are balls bouncing all over the place just like before and then look at this all the colored balls end up on one side again this clip is not reversed but it took me hours of work to create I started the balls in a random configuration then I mixed and mixed and mixed and mixed until they all ended up separated again I had to wait for luck to be on my side we're going to come back to this example in a little bit with a more detailed analysis but remember the condition with all the balls on separated sides isn't impossible to generate by random chance it's just really really unlikely it's much more probable for a system to to evolve from ordered to disordered than for an equivalent system to evolve from disordered to ordered when we make the leap to systems with large numbers of particles like the 10 trillion trillion water molecules visible in this example the unlikely options like all of the ink and water separating from each other are so unfathomably unlikely we can reliably say that they never ever happen entropy is one of the hardest to understand Concepts in physics because there's nothing like it that we interact with in the macroscopic world the the second law of Thermodynamics that an isolated system's entropy will always increase is less of a physical law and more of a statistical one like you won't win the lottery every day for the rest of your life if anybody watching thinks that it's possible like not even likely but possible for you to win the lottery every day for the rest of your life then you're not going to enjoy thermodynamics in a few minutes after we have a working definition of entropy hopefully you'll see that this becomes a rather Circ argument because the most likely outcomes are defined to have higher entropy so I'm not kidding about the statistics you can define a useful definition of entropy by flipping coins and that definition is going to end up being completely equivalent to the example with the balls and even equivalent to the example of ink mixing with water it's going to take some doing but I hope that by the end of this video you have an appreciation for why entropy has to increase how it can do work and why it has some very strange but first we're going to take a detour for 5 minutes and 27 seconds and flip some coins and then we'll come back to reality and we'll look at the ink tube in an actual demonstration of Osmosis so imagine that we have whoa we don't need that many up imagine that we have four coins and we flip each of them Tails heads heads heads so we got one tails and three heads that's not the most likely option but it is a likely option when we flip four coins like this There are 16 wait well I don't have 16 fingers two to the four There are 16 possible outcomes and each of those possible outcomes is equally likely notice that there's only one way to get all tails and one way to get all heads there are four outcomes where one coin is Tails and three coins are heads there are four outcomes where one coin is heads and three coins are tails and there are a whopping six different ways to flip four coins and have two of them come up heads when you flip four coins the most likely outcome is that two of them will be heads and two of them will be tails you don't know which two of the coins will be heads and tails but that doesn't matter you have a six out of 16 chance of selecting one of these states I'm holding off the rigor here for a few minutes thermodynamic and the audience don't get mad I'll fix it but for now let's say that the entropy of this system is four the likelihood out of 16 of getting three heads and one Tails if I flip all these coins again now we got two heads and two taals which conveniently is the same thing that I wrote In The Script the entropy of this system has changed the entropy of this system is now six because there are six out of 16 possibilities that give us two heads and two tails entropy is not a conserved quantity there's nothing wrong with me re flipping the coins and saying oh the entropy is different now you can keep flipping the coins and keep updating the system and the entropy will keep changing so now let's upgrade to 10 coins this means I need to expand my spreadsheet because now there are 1,24 different outcomes for this experiment but only 10 possibilities for the sum we can't flip 10 coins and get more than 10 heads so now our distribution looks like this surprisingly the most likely outcome for flipping 10 coins is seeing five of them turn up heads in fact there are 252 different ways for that to happen but what about a situation where only one coin is heads of course because any of the 10 coins could be that one there are 10 different ways for that to happen if we've got 10 coins this one could be heads or this one could be heads or this one could be heads there are 10 different ways for that to happen another way to think about it is permutation so instead of flipping them you can go well you know that one could be heads or that one could be heads and the rearranging definition is going to be the one that's a little more relevant for mixing but mathematically they're equivalent you can just keep flipping coins entropy is commonly referred to as disorder or Randomness but these terms are pretty biased by our human intuition for what it means for something to be sorted or ordered out of these 1024 possible Arrangements of coin flips which of these feels more likely to you all tals or heads Heads Tails heads Tails heads heads Heads Tails Tails I'll give you an intermission to think about it that's a trick question both of these patterns are equally likely you could flip coins for a very long time and not flip 10 coins and get a string that was heads Heads Tails heads Tails heads Tails Tails heads Heads but ask yourself did you notice that during the break I actually flipped all four of these coins while flipping 10 coins and getting all tals and getting this particular sequence are equally likely getting all Tales is significantly less likely than getting a sequence that sort of looks like this one humans like to see patterns and that has led us to this definition of entropy where stuff like this is said to be disordered or to have a lot of entropy the way we quantify this is to add up the results in this particular case we have a sequence with six heads and Four Tails there are 210 different ways that we could have flipped 10 coins and gotten six heads and four tails but there is only one way to flip 10 coins and get all Tails this pattern has more entropy because there are more ways to make a similar pattern so what's the scoop with the second law of Thermodynamics if entropy is just this unconserved quantity and it's free to go up and down and do whatever it wants why do we see the entropy in closed systems or the entropy of the entire universe going up all the time why doesn't it ever go down let's look at a very skewed version of the coin example if we have a system of two heads and eight tails there are 45 ways to achieve that outcome out of that big list of 1,024 options 45 of them have two heads and eight tails so I just ref flipped all the coins and I could have gotten the state where I flipped 10 coins and got all Tails which would have had our fake entry measurement of one out of 1,024 but I didn't I got a state with uh let's see four heads and six tails if I had flipped 10 tails in a row the entropy of the system would have decreased like there's nothing wrong with that it's totally legal I can physically flip a coin and get a tail so I might as well be able to do it 10 times in a row but it's not very likely it's significantly more likely to have gotten a setup like this with four five or six heads in the sequence because there are combined 672 ways to achieve one of those outcomes let's return to the balls for a minute we can map this system directly onto our coin example if we have a chrome ball on the left side that's like flipping a coin and having it come up heads if we have a chrome ball on the right side that's like flipping a coin and having it come up Tails if we reverse these definitions for the marbles that I've Sharpie black then this Arrangement is equivalent to flipping 10 coins having seven of them come up tails and having three of them come up heads if they separate completely like this that's like flipping 10 heads or if they separate completely like this that's like flipping 10 Tails as these balls bounced around I filmed the whole thing and then I fed 24 minutes of marble footage into a script I wrote that took most of a day to track all the balls and calculate How likely various configurations were for each frame it finds all the balls figures out which color is switch figured out which side of the line they're on and does the math to calculate a score that Maps the current configuration of balls onto the coin flipping model even though I was standing there watching the whole time I only actually saw the balls sort once while watching live but it actually happened six different times for a total of a whopping 56 frames of video let me say that again in 24 minutes of bouncing the balls were completely sorted in either direction for less than two seconds this is not a very likely outcome and the unlikeliness of that outcome is extremely well predicted by the coin flipping model here's the distribution of coin flip probabilities and here's the data from the ball tracker almost identical one of these plots was generated by adding up all the possible ways to flip 10 coins and the other was generated by watching balls bounce for half an hour and analyzing their motion and they're the same math is so satisfying when it works so now let's zoom out what if we were flipping 100 coins or 1,000 coins or 10,000 coins I had to simulate these instead of solving for every possible solution in Excel but as you can see the larger the system gets the tighter the distribution of probable outcomes if we have a system of 10,000 coins and 3,000 of them are heads the next time we flip all those coins we are basically guaranteed to have a near 50/50 ratio of heads and tails it's so unfathomably unlikely to get 3,000 heads and 7,000 tails that it just won't happen this tube of water a real system of particles in the real world has about 10 to the 25 particles in it these kind of huge numbers make probabilities into certainties I mean that's why the second law of Thermodynamics is referred to as a law there's no thermodynamics police force like literally there is no fundamental force of physics that causes systems to increase in entropy but you will never witness a large system break the second law of Thermodynamics it sounds really circular when stated like this because I'm basically telling you that entropy increases because the high entropy states are most probable when the definition of a high entropy state is that it's very likely to happen but that's basically it I mean that's why it's confusing using it's just human nature that these very probable States also tend to look messy and disordered the crazy thing to bring this back to reality is that energy which is pretty tangible thing and entropy which is pretty obscure are Loosely interchangeable you can actually store useful energy that can do work in the real world in a system that is ordered like the order itself sort of stores energy okay now back to the real world this is a tube filled with water if I pour water in one side the water inside sloshes back and forth until the level on both sides is the same this is because gravity pulls on the water in this tube and gravity pulls on the water in this tube equally if at any moment the water level in one side is higher there's more total Force there and it falls back down and equilibrates this is another tube filled with water but it's a little more complicated on the left side I've just filled up the tube with tap water but on the right I filled the tube up with water mixed with sugar in between at the bottom here I've inserted a membrane that lets Water Pass from one side to the other but not the sugar if you want to you can think of this just like the marbles one side has Chrome balls one side has a mixture of black and chrome the Chrome ones are allowed to go from side to side but the black ones are trapped on the right in the regular tube of water gravity pulled down and the water ended up equalizing so the water on both sides was at the same height in this case you can see that I started with more water on the clear side than the sugar side but because the sugar water is denser and heavier I think the two sides were actually close to balanced at the start that means that it's pretty weird that at first the sugar water side goes up despite being heavier for some reason we're moving water through the membrane from left to right even once the water level on the right gets higher than the left the water keeps pumping this system is doing work it's lifting water against the force of gravity and the only thing driving this process is the fact that the water over here is pure and the water over here is mixed with sugar this process is called osmosis and it's weird like really really weird so where's the entropy as I complained in the last video water is clear which means you can't actually see what every water molecule in the Tomb is doing so let's repeat this experiment at marble scale and then we'll do some math to try to quantify what we just saw on the left we have a pure substance just water but on the right we have a very concentrated solution lots of solute ink water whatever and less water unfortunately I don't actually have a color sensitive marble filter so I'm going to be doing this by hand and speeding up the footage later but I swear I'm trying to agitate the marbles as uniformly and unbiasedly as possible as I shake the border the marbles move around every time that a chrome marble passes the center line I let it through and every time a black marble tries to cross the center line I turn it around as though it hit a wall you probably can't can't notice it in real time because I'm using a very small enclosure here but there are on average actually more balls on this side at any given time I reran the same tracker script from before threw out any frames obscured by my hand and found that the average distribution wasn't 10 balls on both sides it was more like 10.7 over here and 9.3 over here there are two opposing forces at play first and more obviously is the effective pressure collisions between balls are trying trying to evenly distribute all of the balls spreading them out so it would be 50/50 there' be 10 on one side and 10 on the other but we also have this entropic effect where the Chrome balls want to spread evenly to both sides and the black ones can't if I made the bounding box really really big so there were no collisions between the balls you could imagine that on average the 15 Chrome marbles would spread evenly on both sides and the five black marbles would still all be on one side in reality we get something somewhere in between these two extremes weirdly enough then the this means that Chrome marbles concentrating on one side is actually the most probable outcome so what does it mean to be a mixture well if you have a mixture of two particles they're free to move around and swap places with each other so we have to consider all possible permutations of a set of particles that kind of sounds familiar so let's do the same math as before but this time let's put different amounts of solute on both sides of the system and solve for the entropy on both sides independent ently on the left we start with one black marble and nine Chrome marbles then on this side of the membrane we have 10 different ways to arrange our particles on the other side we have six Chrome Marbles and four black Marbles and there are a lot of ways we could mix these particles together 210 different ways in fact now we've got this membrane between these two groups that the Chrome balls can pass and the black ones cannot so what happens if we take a chrome ball from either side and cross the membrane if a single Chrome ball passes from right to left we would add a tiny bit of entropy to this side because now there are 11 places to put our solute particle but we've massively decreased the entropy on the right from 210 down to 126 there are many fewer ways to rearrange the system after you remove a single Chrome ball this change has made the entire system more ordered we're sorting the balls and it results in a total decrease in entropy but what about the other way if we take a chrome ball from the left and move it to the right we slightly decrease our options on the left down to nine from 10 but there are now Oodles of arrangements for balls on the right we started at 210 possible arrangements and the addition of one more ball takes that all the way up to 330 so at this point through excruciating counting we have determined that the system with more mixed marbles has more entropy it's more likely but what in the real world actually makes this happen why why does the system evolve it happens because of random fluctuations thermal fluctuations when you get down to the scale of atoms and molecules everything's vibrating and zipping around with thermal energy all the time just like these marbles because everything's hot and if it's hotter it actually randomizes better so in the example with 20 marbles we have the two competing forces pressure was trying to make them 10 and 10 but entropy was trying to significantly skew that result to one side if the marbles were bouncing faster in a wide open space that let them really zip around and randomize we would be biasing that system towards the more entropic result higher temperature actually allows entropy to do more work on a system more temperature results in more atomic scale jiggling which results in more randomization and eventually more energy output from the randomization of the system because the hotter it gets the more you randomize it in the case of the water column in the osmosis experiment the energy was used to push water up against the force of gravity to make some very Hasty approximations this sugar water has a density of about 1.3 G per CC and it went up by about a foot and a half this tube is about a/ an inch in diameter so it took about 170 Mill of energy to lift this water this high and we did this at room temperature about 300 Kelvin that means that in the process of adding pure water to the sugar water that was already here we created about 6 * 10-4 Jew per Kelvin of entropy that entropy Jewels per Kelvin implies that if the water temperature were higher the same amount of added disorder could actually do more work if I heated this system up the water would actually lift up higher not because of the density or gravity or anything like that but because the molecules near the membrane at the bottom would be shaking faster and randomizing more now before we wrap up this video I need to come clean to all the thermodynamics in the audience and say that the numbers that I've been using as entropy the likelihood if we've got you know 1,24 options to flip all these coins the numbers that I've been calling entropy are actually multiplicities of the system every possible arrangement of this system is known as a micro state of the system and the number of possible Arrangements that exist without flipping any coins is known as the multiplicity of the system so the number that I've been reporting as entropy is actually the multiplicity of the system in order to get the real calculable entropy of a system you need to find the macro state of the system or at least the multiplicity of that macro State the likelihood that you get a similar outcome you need to take the natural log of that probability and then you need to slap a boltzman's constant out front this means that in the marbles doing osmosis example where we were calculating the entropy of two different systems that were sort of part of the same system and then adding those entropies together we actually needed to calculate you know the likelihood of each of these systems as a multiplicity then take the log then add those logged values together which complicates the process a little bit but it doesn't change any of the results that I talked about it's a little less cut and dry as a demonstration because you've got to do more math but if you do this math for a very large system you can basically predict the future if you can find the highest entropy state for a system you can reliably say the system is going to approach this state given enough temperature and enough time you don't need to know how every individual particle is going to bounce you don't need to be able to observe every interaction between water molecules and ink or sugar or something at a membrane uh and you know solve for all that physics you just need to be able to take gross probabilities and you can predict reliably the evolution of the entire system it actually gets more accurate the bigger the system is and for ridiculously huge systems like I don't know a cup of water that you're stirring some sugar into you can be darn sure that it's going to find its highest entropy state if you give it enough time and enough temperature simply because that's the most probable thing for it to do so now you like me are probably wondering how this script got so bloated and where's the ink machine again so I'm sorry but entropy is cool actually it it's not cool it sort of relies on heat in the last video I built this contraption that mixes ink and water together in a clear tube and it looks really cool that was kind of the point I spent a whole video talking about how it lets us observe fluid dynamics in a pipe but I didn't talk about the most interesting part the title on the front here I wrote entropy of mixing and that is the increase in entropy that results from combining ink and water in this way and the coolest part of this entire project is that it happens in Reverse I'm actually separating the ink and water so that we can mix them again that's what's happening inside this filter cartridge at an extremely high pressure ink and water are going in at about 100 PSI and pure water is coming out the other side so how does this work well in this demo I did earlier we took pure water and mixed it with sugar water and we saw that through the process of osmosis this system was actually able to do work by increasing entropy we were mixing two substances together creating entropy and from that actually extracting energy in the form of pressurized water that could lift to a higher height Against Gravity if in this example we were able to put like High Press air or something into this tube we could actually back drive this entire system we could force water and sugar to separate through this membrane giving us pure water on the other side the side effects here is that we can't actually get rid of the sugar so the the mixture in this tube would get progressively more concentrated we'd effectively be squeezing pure water out of like syrup and when we do this we would be sorting the system we would be decreasing its entropy putting all the water on one side and all the sugar on the other side and we can't do that unless we expend energy from outside the system and that's exactly what this pump is doing inside the ink machine it's injecting useful energy or thermodynamic free energy into the ink water filter system to sort it more mechanistically this big beefy pump is just shoving an ink water mixture at high pressure into this filter the water goes through and what's left over concentrates into almost pure ink so we're basically squeezing water out of ink in this machine so that the pure water and the leftover ink can get separately reinjected into the tube as we just spent way too long calculating we have to expend energy to do this by separating the ink and water we actually reducing the entropy of the system and this costs energy in this case we can measure the energy being used by the pump very directly and that's quite a lot this is nominally a 100 gallon per day Ro filter as purchased so very approximately we're looking at about 5 jewles of energy to extract 1 cubic cenm of water from Ink and in bigger units that's about 1.4 kilowatt hours of energy per cubic meter of water of course there isn't a cubic meter of water in here but you can imagine that it takes that much power to cycle it until we get a cubic meter through this isn't really a filtration process yes technically we're forcing a mixture through a filter of sorts where pores don't let the solute through but it's literally at a molecular scale and that means we have to use more complicated physics the physics of Osmosis is thermodynamics entropy and at the end of the day statistics there's absolutely nothing there is no law of physics preventing me from picking this up and shaking it until all the ink separates at one end and all the water separates at the other end but am I going to be waiting like a really long time for that to happen yeah in a system of 10 to the whatever many particles it's just not going to happen [Music]
