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# Reconciling thermodynamic and state definitions of entropy

## Video transcript

if you followed some of the mathematics and some of the thermodynamic principles in the last several videos what occurs in this video might just blow your mind so not to set expectations too high let's just start off with it let's say I have a container and in that container I have gas particles gas particles inside of that container they're bouncing around like gas particles tend to do creating some pressure on the container the certain volume and let's say I have n particles I have n particles now each of these particles could be in X different states so let me write that down each each particle each particle can be in X different states different states what do I mean by a state well let's say I take particle a let me make particle a a different color particle a it could be down here in this corner and it could be have some velocity like that it could also be in that corner and have a velocity like that those would be two different states it could be up here and have a velocity like that it could be there and have a velocity like that if you were to add up all the different states and there would be a gazillion of them you would get X particles that blue particle kind of X different states you don't know we don't we're just we're just saying look I have this container it's got n particles and we just know that each of them could be in X different states now if each of them can be in X different state how many total configurations are there for the system as a whole well particle a could be in X different states and then particle B could be in X different places so times X if we just had two particles then you would multiply all the different to places where X could be x all the different places where this the red particle could be and then you'd get all the different configurations for the system but we don't have just two particles we have n particles so you'd be multiplying for every particle you'd multiply it times the number of states you could have and you do that a total of n times n times and this is really just combinatorics here you do a 10times this system would have n configurations for example if I had two particles each particle had three different potential states how many different configurations could there be well each could for every three that one particle could have the other one could have three different states so you'd have nine different states you would multiply them you fit another particle with three different states you'd multiply that by three so you have 27 different states here we have n particles each of them can be in Ex different states so the total number of configurations we have for our system x times itself n times is just X to the n so we have X to the n states in our system now let's say that we like thinking about how many states a system can have certain states have less for example if had fewer particles I would have fewer potential states or maybe if I had a smaller container I would also have fewer potential States there would be fewer potential places for our little particles to exist so I want to create some type of state variable that tells me well how many states can my system be in so this is kind of a macro state variable it tells me how many states can my system be in and let's call it s for States s for states for the first time in thermodynamics we're actually using a letter that in some way is related to what we're actually trying to measure s for States and since the states they can grow really large let's say I'd like to take the logarithm of the number of states and it's just how I'm defining my state Val variable I get to define it so I get to put a logarithm out front so let me put a logarithm so in this case would be the logarithm of my number of states so it'd be X to the N where this is number of potential States potential States and you know we need some kind of scaling factor maybe I'll change the unit's eventually so let me put a little constant out front a little constant every every good formula needs a constant to get our units right I'll make that a lowercase okay so that's my definition I call this my state it's state variable if you give me a system I should in theory be able to tell you how many states the system can take on fair enough so let me close that box right there now let's say that I were to take my my box that I had let me copy and paste it I take that box and it just so happens that there was an adjacent box next to it they share this wall they're identical in size although what I just drew isn't identical in size but they're close enough they're identical in size and what I do is I blow away this wall I just evaporated all of a sudden it just disappears so this wall just disappears now what's going to happen well as soon as I blow away this wall this is very much not an isostatic process right all hell's going to break loose I'm going to blow away this wall and you know the particles the particles that we're about to bounce off of the wall are just going to keep going right they're going to keep going until they can maybe bounce off of that wall so very right when I blow away this wall there's no pressure here because these guys have nothing that bounce off to well these guys don't know anything they don't know anything until they come over here and say oh no wall so the pressure is in flux even the volume is in flux as these guys make their way across the entire expanse of the new of the new of the new volume so everything is in flux right and so what's our new volume if we call this volume if we call this volume what's this this is now two times the volume we now have two times the volume let's think about a cup of some of the other state variables we know we know that we know that the pressure is going to go down we can even relate it because we know that our volume is twice it is two times the volume what about the temperature what about the temperature well the temperature change temperature is average kinetic energy right or it's a measure of average kinetic energy so all of these molecules here they have each of them have kinetic energy they could be different amounts of kinetic energy but temperature is a measure of average kinetic energy now if I blow away this wall does that the kinetic energy of these molecules no it doesn't change it at all so the temperature is constant so if this is this is t1 then the temperature of this system here is t1 and you might say hey Sal wait that doesn't make sense in the past when the piston when my when my cylinder expanded my temperature went down and the reason why temperature went down in that case is because your molecules were doing work they expanded the the container itself they pushed up the cylinder so they expanded some of their kinetic energy to to do the work in this case I just blew away that wall these guys did no work whatsoever so they didn't have to expend any of their kinetic energy to do any work so their temperature did not change so that's interesting fair enough well in this new world in this new world what happens eventually I get to a situation where my molecules fill the container right we know that from common sense and if you think about it on a micro level why does that happen it's not a mystery you know on this direction things we're bouncing and they keep bouncing but when they go here there used to be a wall and they'll just keep going and then they'll start bouncing here so when you have gazillions of particles doing a gazillion of these bounces eventually they're they're just as likely to be here as they are over there now let's do our computation again in our old situation when we just looked at this each particle could be in one of X places or one of X States now it could be whatwhat's it could be in twice as many states right now each particle each particle could be in two X different states why do I say 2 X because I have twice the area to be in now the states aren't just you know position in space but everything else I have just I have so you know before here I could have had maybe I had a positions in space times B positions or B Momentum's you know where those are all the different Momentum's and that was equal to X now I have 2a positions in volume that I can be and I have twice the volume to deal with so I have 2a positions in volume I can be at but my my my momentum states could are going still be I just have B momentum states so this is equal to 2x I now can be in 2x different states just because I have two times the volume to to travel around in right so how many states are there for the system well each particle can be in two X States so it's 2x times 2x times 2x and I'm going to do that n times n times so my new s so this is you know let's call this s initial so my s final my new way of measuring my state's is going to be equal to that little constant that I threw in there times the natural log times the natural log of the new number of states so what is it it's 2 X to the N power so it's 2 X to the N power so my question to you is what is my change what is my change in s when I blew away the wall when I you know that there was this room here the entire time although these particles really didn't care because this wall was there so what is the change in s when I blew away this wall and just to be clear the temperature didn't change because no kinetic energy was expended and this was all in isolation I should have said it's adiabatic there's no transfer of heat so that's also why the temperature didn't change so what is our change in s our change in s is equal to RS final minus RS initial which is equal to what's our s final it's this expression I write here it is K times the natural log and we can write this as 2 to the N X to the n that's just exponent rules and from that we're going to subtract out our initial our initial s value which was this K natural log of X to the N K natural log of X to the n now we can use our logarithm properties to say well you know you take the logarithm of a minus the logarithm of B you can just divide them so this is equal to K I could factor that out times the logarithm of 2 to the N let me make its upper case n so let me do that this is upper case and I don't want to get confused with moles upper case n is the number of particles we actually have so it's two to the uppercase n times X to the uppercase n divided by X to the uppercase n so we can just these two cancel out so our change in s is equal to K times the natural log of 2 to the n or if we want to use our logarithm properties we could throw that n out front and we could say our change in s our change in s whatever this state variable I've defined and this is a different definition than I did in the last video is equal to is equal to K or do we say n big n the number of molecules we have times my little constant times the natural log of 2 so by blowing away that wall and giving and giving my and giving my molecules twice as much volume to travel around in my change in this little state function I defined is NK the natural log of 2 and what am i what what what really happened I mean it clearly went up right I clearly have a positive value here this is natural log of 2 is a positive value n is a positive value is going to be a very large number than the more molecules we had and I'm assuming my constant I threw on there is a is a positive value but what am I really describing I'm saying that look by blowing away that wall my system can take on more states there's more different things it can do and we'll throw a little word out here it's entropy has gone up and entropy well well actually let's just define s to be the word entropy we'll talk more about the word in the future its entropy has gone up which means the number of states we have has gone up I shouldn't use the word entropy without just saying entropy I'm defining is equal to s but let's just keep it with s s for states the number of states we're dealing with has gone up and it's gone up by this factor actually has gone up by a factor of 2 to the N and that's why it becomes n natural log of 2 fair enough now you're saying ok this is this is nice Sal you have this this this this statistical way or I guess you could this combinatoric way of measuring how many states this system can take on and you looked at the actual molecules you weren't looking at the kind of the macro states and you were able to do that you came up with this macro state that says that's essentially saying how many states can I have but how does that relate to that s that you defined in the previous video remember in the previous video I was looking for a state function that dealt with heat and I defined s or change in s I defined as change in s to be equal to the heat added to the system divided by the temperature that the system that the heat was added at so let's see if we can if we can come up if we can see whether these two things make the same you know if these things are somehow related so let's go back to our system and go to a PV diagram and see if we can do anything useful with that so PV diagram alright okay so this is pressure this is volume now when we start it off before we blew away the wall we had some pressure and some volume so this is v1 and then we blew the way the wall and we got to actually let me do it a little bit differently no I wanted to I don't want that to be just right there let me make it right there so that is that is our v1 this is our original state that we're in so state initial or however we wanted that's not initial pressure then we blew away the wall and our volume doubled right so this is we call this - v1 our volume doubled our pressure would have gone down and we're here right that's our state - that's this scenario right here after we blew away the wall now what we did was not a quasi-static process I can't draw the path here because right when I blew away the wall all hell broke loose and things like pressure and volume weren't well-defined eventually got back to an equilibrium where this filled the container and nothing else was in flux and we could go back to here and we could say okay now the pressure in the volume is this but we don't know what happened in between that so if we wanted to figure out our Q over T or the heat into the system we learned in the last video the heat added to the system is equal to the work done by the system we'd be at a loss because the work done by the system is a the area under some curve but there's no curve to speak of here because our system wasn't defined while it was in while all the hell had broken loose so what can we do well remember this is a state function this is a state function so it's and this is a state function and I showed that in the last video so it shouldn't be dependent on how we got from there to there right so this change in entropy or actually let me be careful with my words this change in s the change in s so this you know s 2 minus s 1 should be independent of the process that that got me from s 1 to s 2 so this is independent of whatever crazy path I mean I could have taken some crazy quasi static path like that right so I can any path that goes from this s 1 to this s 2 will have the same heat going into the system or should have the same or let me take that any any any system that goes from s 1 to s 2 regardless of its path will have the same change in entropy because there are there same change in s because there s was something here and it's something different over here and you just take the difference between the two so what's the system that we know that can do that well we can go on we can say that let us say that we did a isothermal and we know that these are on the same isotherm right we know that the temperature didn't change I told you that because no kinetic energy was expended and none of the particles did any work so we can say we can think of a theoretical process in which you know instead of doing something like that instead of doing something like that we could have had a situation where we started off with our original container with our molecules in it we could have put a reservoir here that's equivalent to the temperature that we're at and then this could have been a piston that was maybe we were pushing on it with you know some rocks that are pushing in the leftward direction and we slowly you slowly move remove the rocks so that these gases could push the piston and do some work and fill this in higher volume or twice the volume and then the temperature would have been kept constant by this heat reservoir so this type of a process is kind of a sideways version of what I've done in the Carnot diagrams that would be described like this you'd go from this state to that state and it would be a quasi-static process along an isotherm so it would look like that so you could have a curve there now for that process what is the area under the curve what is the area under the curve there well the area under the curve is just the integral and we've done this multiple times from our initial volume to our second volume which is twice it of P times our change in volume right P is our height times our little changes in volume give us each rectangle and then the integral is just the sum along all of these so we get our that's essentially the work that this this system does right and the work that this system does since we are on an isotherm it is equal to the heat added to the system because our internal energy didn't change so what is this we've done this multiple times but I'll redo it so this is equal to the integral of V 1 to 2 V 1 PV equals NRT right NRT so P is equal to NRT over V n R T over V DV and the T is T 1 now all of this is happening along an isotherm so all of these terms are constant so this is equal to the integral from V 1 to 2 V 1 of n R T 1 times 1 over V DV I've done this integral multiple times and so this is equal to I'll skip a couple of steps here because I've done it in several videos already the natural log of 2 v1 over v1 write the antiderivative this is the natural log take the natural log of that minus the lateral log of that which is equal to the natural log of 2 v1 over v1 which is just the same thing as n R t1 times the natural log of 2 interesting now let's add one little one little interesting thing to this to this equation so this is NRT but if I wanted to write in terms of the number of molecules and is the number of moles so I could rewrite this as I could rewrite n as the number of molecules we have divided by six times ten to the 23rd power right that's what n can be written as so if we do it that way then what is our what is remember all of this we were trying to find the amount of work done by our system right but if we do it this way this equation will turn into so let's see the work done by our system this is our quasi-static process it's going from the same state that's going from this state to that state but it's doing it in a quasi-static way so that we can get an area under the curve so the work done by this system is equal to I'll just write it n times R over six times ten to the 23rd times T one natural log of two fair enough let's make this into some new constant for convenience let me call it a lowercase K so the work we did is equal to the number of particles we had times some new constant will call that Bolton is constant so it's really just eight divided by that times T 1 times the natural log of 2 fair enough now that's only in this situation the other situation did no work right so I can't talk about this this system doing any work but this system did do some work and since it did it along an isotherm Delta the change in internal energy is equal to zero so the change in internal energy which is equal to the heat applied to the system minus the work done by the system this is going to be equal to 0 since our temperature didn't change so the work is going to be equal to the heat added to the system so the heat added to the system by our by our little our little reservoir there is going to be so the heat is going to be the number of particles we had times Boltzmann's constant times our temperature that we're on the isotherm times the natural log of two and all this is by product of the fact that we doubled our volume now in the last video I defined change in s as equal to Q divided by the heat added divided by the temperature at which I'm adding it so for this system this quasi-static system what was the change in s how much did our s term our s state change by so change in s is equal to heat added divided by our temperature our temperatures t1 so that's equal to this thing and K t1 times the natural log of 2 all of that over t1 so our Delta add these cancel out and our change in our s quantity is equal to is equal to n K times the natural log of 2 now you should be starting to an experience an aha moment when we defined in the previous video we were just playing with thermodynamics and we said gee we'd really like to have a state variable that deals with heat and we just made up this thing right here it's a change in that state variable is equal to the heat applied to the system divided by the temperature at which the heat was applied and when we when we use that definition the change in our S value from this position to this position for a quasi-static process ended up being this NK natural log of 2 now this is a state function state variable it's not dependent on the path so any process that gets from that gets from here that gets from this point to that point has to have the same change in s so the Delta s for any process is going to be equal to that same value which was n in this case K times the natural log of to any system by our definition right it's a state verbal I don't care whether there you know it disappeared or the path with some crazy path it's the state's only function of that and of that our change in s so given that even this system we said that this system that we started the video out with it started off at this same v1 and it got to the same v2 so by the definition of the previous video by this definition it's change in s it's change in s is going to be the number of molecules times some constant times the natural log of 2 now that's the same exact result we got when we thought about it from a statistical point of view when we were saying how many more different states can this system take on and with what's what's what's mind-blowing here is that what we started off with was just kind of a nice you know macro state in our little Carnot engine world that we didn't really know what it meant but we got the same exact result then when we try to do it from a measuring the number of states a system could take on so all of this has been a long 2 video winded version of an introduction to entropy entropy entropy and in thermodynamics a change in entropy entropy is s where I think of it as four states in the thermodynamic or Carnot cycle or Carnot engine world is defined as the change in entropy is defined as the heat added to system divided by the temperature at which it was added now in our in our statistical mechanics world we can define entropy we define entropy as some constant and let's it's especially convenient if this is Boltzmann's constant some constant times times the natural log of the number of states we have sometimes it's written as Omega sometimes other things this times the number of states we have and what we just showed in this videos these are equivalent definitions or at least for that one case I just showed you this is equivalent definitions when we used the number of states for this how much did it increase we got this result we got this result and then when we use the thermodynamic definition of it we got that same result and if we assume that this constant is the same as that constant if they're both Boltzmann Boltzmann's constant both 1.3 times 10 to - 23 then our definitions are equivalent and so the intuition of entropy we were in the last one we're kind of struggling with it we just defined it this way boron what does that really mean what change in entropy means is how many more states can the system take on you know sometimes when you learn it in your in your high school chemistry class I'll call it disorder and it is disorder but I don't want to you know I don't want you to think that it's somehow you know a messy room has higher entropy than a clean room which some people sometimes use as an example that's not the case what you should say is is that a stadium full of people has more states than a stadium without people in it that has more entropy or actually I should even be careful there let me say a stadium with let's say a stadium at a high temperature has more entropy than mire than the inside of my refrigerator that the particles in that stadium have more potential states than the particles in my refrigerator now I'm going to leave you there and we're going to we're going to take our definitions here which I think are pretty profound this and this is the same definition and we're going to apply that to talk about the the second law of thermodynamics and actually just a little aside I write Omega here but in our example this was this was 2 to the N and so that's why it's simplified this was the the magnet or no sorry this was X the first time and then the second time it was this was X to the end the first time then the second time when we double the size of our room or our volume it was X to the N times 2 to the N I just want to make sure you realize how what this Omega relate is to relate relative to what I just went through anyway see in the next video
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