Is ENTROPY Really a "Measure of Disorder"? Physics of Entropy EXPLAINED and MADE EASY

hey what's up you lot path here filming today on one of the hottest days in the uk that we've had in a long time it's about 35 degrees celsius here and i am living for it today we're talking about one of my most requested topics we are talking about entropy now entropy is a difficult idea to get your head round so in this video we will be focusing on one specific definition of entropy we won't be looking at all the far-reaching implications of entropy like the heat death of the universe or anything like that instead we'll be focusing on the basics here and trying to get an intuitive sense of what entropy actually is if you can understand that then everything else follows quite nicely in my opinion now you might have heard entropy being described as a measure of disorder or a measure of how much disorder there is in a system but what does this even mean well we'll be delving into that description of entropy by understanding this equation here and don't worry as always you only need to know high school level mathematics to get what's going on if i've made this video correctly so if you enjoyed the video please do hit the thumbs up button and subscribe to my channel for more fun physics content let's get into it let's begin with a fairly abstract idea and bear with me here because we'll see where this is going very shortly let's begin by imagining that we have a box and in this box we have some number of particles let's say we've got three particles in this box it doesn't really matter what those particles are specifically but what actually matters is that they each carry some amount of energy and we're going to assume here that they can only carry specific amounts of energy we'll represent those specific amounts of energy that each particle can carry with these energy levels in this diagram this energy level represents the smallest amount of energy that a particle is allowed to carry this next one represents a slightly larger amount of energy that a particle is about to carry and so on and so forth let's assume for simplicity that this lowest energy level corresponds to a particle carrying an energy e doesn't matter what e is just as long as we know that it's a constant and the next energy level again for simplicity corresponds to a particle carrying an energy to e and the next one is three and four e and five e and so on and of course we can say that there are infinitely many energy levels that our particles can occupy but crucially the particles must have a minimum amount of energy they can carry which is e and they can only carry specific amounts of energy so e to e three e and so on now like i said bear with me here the reason that we're making so many abstractions will become clear shortly and yes this is an abstraction we're not saying that the particles must occupy a certain height within the box in order to have a certain amount of energy we're not saying that particles which contain 2e worth of energy for example are this high off the floor of the box that's not what we're doing here this ladder diagram is just a way for us to represent how much energy each particle has but now let's make a bit more progress let's assume that the total energy of our system of our box is 5e let's also assume that all of that energy is carried by the particles rather than like the walls of the box or something like that what this means is that there is 5e worth of energy distributed over three particles let's call them particle a particle b and particle c so what energy levels can these particles occupy in order for the total energy of the box to be five e well we could start by saying that particle a is in the lowest possible energy level it has an amount of energy e and we could also say for example that particle b is in the lowest possible energy level as well this means that the remaining particle particle c must by necessity be in the 3e energy level because now the particles in the box contain 1e plus 1e plus 3e worth of energy which is 5e in total so this is one possible configuration that our particles could be in in terms of how much energy each of them carries but of course another possible configuration is if particle b for example were to be in the 3e energy level and particles a and c were to be in the lowest energy level and another configuration still is when particle a is in the 3e level and particles b and c are in the lowest possible energy level again the total energy is 5e but we're just changing how it's distributed over the three particles in the box and we can actually arrange these particles in a slightly different way say for example particle a is in the lowest energy level again but this time both particles b and c are in the second lowest energy level 2e because now the total energy of the system is 1e plus 2e plus 2e which is still 5e and equally the particles could be distributed like this or like this and so just as a quick summary we see that there are six different ways in which to arrange our particles so that we've got a box which contains five e worth of energy and there are three particles within that box these six possible ways of arranging our system are known as the six micro states of our system and this total number of microstates that we can have for each system becomes very important when defining entropy and i'm going to talk about one more thing before we finally start saying the magical e word i want to let you guys know that for this video i've done something slightly different than usual i've included in the description of this video or in like a pin comment depending on when i remember to do this a document with a few example questions that you can attempt after watching this video if you don't want to hear about that don't really care then skip to this timestamp here but let me quickly tell you about that document like i said it contains a few questions that you can have a go at and i've tried to make it in such a way that it gives you some useful insight into what entropy actually is i've tried to make the questions relatively tricky but let me know in the comments down below if they're too easy or too hard the document also contains some basic bare-bones solutions right at the end but i also want to make a video walking through each of those questions one by one so check out the link below like i said it's free i wanted to put this first one up so that i could see whether you guys liked it or not and whether you guys even were interested in something like this but in the future i want to start like a patreon or something so if any of you are interested in supporting the channel that way then i'll make some more documents like this and chuck them on there not quite sure yet i'll keep you guys updated and let me know if you're interested in something like that anyway let's come back to all of these possible microstates of our system as we saw for this particular system with total energy 5e three particles in the box and each of those particles can occupy an energy level of e two e three e and so on there are six possible different microstates the system can occupy and this is important because the total number of microstates that a system can occupy is directly linked to its entropy in fact if we look at the equation that i put up on screen right at the beginning of the video this quantity omega is actually the total number of microstates a system can occupy and s is the entropy of the system k subscript b is known as the boltzmann constant and of course we've got the natural logarithm ln of omega which is the total number of microstates a system can occupy by the way if you're not comfortable with logarithms i'll leave some resources in the description below and this is why the entropy for system depends on how many microstates the system can occupy which brings us around to a rather common description of entropy it's a measure of disorder with this particular system we've seen that there are six different ways to arrange it so its entropy is equal to the boltzmann constant multiplied by the natural logarithm of six but if we think about another system for example where this time the total energy of our system is three e and it's got three particles then there's only one possible microstate the system can occupy one where all of the particles are in the lowest possible energy level because one e plus one e plus one e is equal to three e and there's no other possible combination here because the lowest energy level these particles can occupy is the e energy level and if any of them are in a higher energy level then the total energy exceeds 3e so for this system total energy 3e with three particles the total number of microstates possible is one therefore omega is equal to 1 and its entropy is just kb multiplied by the natural log of 1. incidentally that happens to be 0 but that's another kettle of fish entirely the point is that the first system we looked at which had many possible microstates that it could be arranged in can be thought of as a disordered system the particles can be arranged in lots of different ways and a disordered system has a higher entropy whereas the second system we looked at has a much smaller number of microstates it can be arranged in and this smaller number gives it a smaller entropy this second system is more of an ordered system so hopefully looking at this has given you a direct analog of why entropy is thought of as a measure of disorder it might be a bit of a hand-wavy explanation but again i want to just give you an intuitive sense of what entropy is trying to measure now there is something else i need to mention here if we go back to the system we were looking at earlier total energy 5e three particles in the box and we bring up these diagrams once again six different possible microstates that our system can occupy then when we calculated this system's entropy we made an implicit assumption we assumed that the system is equally likely to be in any one of these microstates because when we calculated its entropy all we said is that there are six possible microstates we didn't say anything about it's more likely to be in this one or it's less likely to be in this one or anything like that this implicit assumption isn't actually very implicit when we think about entropy from first principles which is not what we're doing in this video this assumption is very purposeful the idea that this system could be in any one of these states and it's got an equal chance equal probability of being in any one of these states this assumption known as the fundamental assumption of statistical thermodynamics sometimes called the assumption of equal a priori probability is a rather reasonable one to make for systems that are a isolated from all other systems and b in thermal equilibrium in other words systems that will behave nicely and don't interact with anything else outside of itself again tricky idea to wrap your head around so there'll be some stuff on it in the description below now it's worth noting that this definition of entropy entropy is equal to the boltzmann constant multiplied by the natural log of the number of microstates a system can occupy is not the original formulation of entropy the original formulation of entropy looked much more on a macroscopic scale it looked at the system as a whole and the measurements we could make on that system for example the pressure of that system or the temperature or the volume of that system in other words the macroscopic or large scale properties of that system but it wasn't until much later when we started looking at statistical thermodynamics and studying systems on a much smaller scale though we got this microstate dependent definition of entropy and started understanding it on a more fundamental level and with all of that being said i'm going to end the explanation here if you enjoyed it please do leave a thumbs up and subscribe to my channel for more fun physics content hit that bell button if you want to be notified when i upload and also do hit that link in the description below if you want to try out those questions that i've written for this video there are hopefully some relatively tricky questions that you can have a go at but if they're too easy or too difficult let me know as always if i've made a mistake in this video do tell me as well and we'll try and correct it in the comments as quickly as possible lastly please do check out the two-part mini-series i made recently on one of the coolest theorems in quantum mechanics whilst also talking about quantum operators quantum commutators and expectation values i also have a second channel where i upload some original music planning on putting out some more soon and i have an instagram at path vlogs if you want to follow me on there thank you so much for watching as always and i'll see you really soon

Channel: Parth G

Views: 59,618

Rating: 4.9656291 out of 5

Keywords: entropy, thermodynamics, physics, entropy explained, parth g, microstate, boltzmann entropy, statistical physics, statistical mechanics

Id: mg0hueOyoAw

Length: 11min 13sec (673 seconds)

Published: Tue Aug 18 2020