 So we've seen that the big particle partition function or big system partition function if we have a bunch of identical and independent particles can be written as the partition function for each individual particle raised to the nth power if we have n of them. That derivation that we talked about in the previous lecture made an important assumption that we need to spend a little bit of time talking about now and that assumption is that the particles were distinguishable from one another that I can tell the difference between them even though I've been calling them identical particles I assume that I could distinguish between them in some way and when I made that assumption is when we said if the total energy can be broken down into the sum of energy for part a and energy for part b I've just put labels on the individual parts of the system so I'm assuming somehow I can tell part a apart from part b if I'm talking about two identical molecules I'm assuming I can tell the difference between molecule one and molecule two and there's some circumstances where that's perfectly reasonable. So for example let's suppose I have let's stick with the case where I have only two subsystems or two particles maybe those particles are in completely separate containers I have a container containing this molecule and across the room I have a container containing molecule number two they might be identical molecules but I can tell which one is molecule a and which one is molecule b I can walk to one side of the room make a measurement of the energy and then I know I'm talking about the energy of particle a walk to the other side of the room make a measurement of this particle and I know I'm talking about particle b so those are identical particles chemically identical but distinguishable because I know which one I'm talking about I can put a label on them there's other circumstances as well I don't have to put them on opposite sides of a room in separate boxes if I just physically localize them in some way if I have a surface and I have molecule a stuck to the surface over here molecule b stuck to the surface over there they don't have to be on opposite sides of room they only have to be some short distance apart but as long as they're not moving if they're stuck where they are I can probe the system and make a measurement of this particle particle a measure this particle particle b and I can tell the difference sometimes the label is something we call more literally a label maybe particle a contains a different isotope than particle b so particle a has a carbon 13 isotope and particle b has a carbon 12 isotope so I know when I'm making a measurement if I'm measuring the energy of the particle with the C 13 that's the one I'm calling a C 12 is the one I'm calling be so point of these illustrations is to illustrate that there are plenty of cases where the particles are in fact distinguishable from one another identical particles that have some difference either some geometric or spatial difference between each other that allows me to tell the difference between them and in that case this expression is perfectly fine but more often we have identical particles that we really can't tell the difference between for example if I were to talk about the box containing a bunch of molecules avogadro's number of molecules in a container of a gas then I don't really have any way of distinguishing one of the molecules of the that gas from another if I make a measurement of one molecule I don't know if it's the same molecule or a different molecule as I make a measurement of sometime later because they're not confined to different regions or localized to some region in space and they don't have name tags to separate them from one another so turns out this is not the right result when I have indistinguishable particles and the reason that's true let's take a simple example with just n equals 2 so if I have two identical particles if I were to write the full system partition function as the individual molecule partition function squared then that looks like something like the partition functions remember are the sum of the Boltzmann factor so I've got a bunch of different energy levels let's go ahead and label those energy levels one two three four five and so on the identical particles have the same list of energy levels somewhere I'm paying attention to energy level I and some different energy level J each of these and so I haven't yet squared this term but each of these single molecule partition functions is a sum of a bunch of partition some of a bunch of Boltzmann factors somewhere in there there's a e to the minus E I over KT somewhere in there also there's the e to the minus EJ over KT for some particular I and some particular J when I square it I multiply by the same thing when I complete that square when I when I multiply them out there's going to be one term that looks like e to the minus EI times e to the minus EJ there's going to be one term that looks like e to the minus EJ times e to the minus EI and of course there's going to be a I term and lots of other terms that I haven't written down but the point is the square of the individual partition functions includes this term with EI and EJ in this order and it also includes a mirror image term with a J and then an I in the opposite order if the particles are distinguishable then both those terms deserve to be in the partition function that's like saying system A molecule A has energy I and molecule B has energy J or EJ this term says system A has energy EJ and molecule B has energy E sub I so I can tell the difference between those A and I and B and J is different than A and J and B and I those are two different microstates of the system I could make measurements and tell them apart but if the particles are indistinguishable then I can't tell the difference between whether molecule A is in this state and molecule B is in that state or vice versa because the molecules don't have name tags all I know is that one of the molecules is in state I and one of the molecules is in state J and I can't tell by definition because they're indistinguishable I can't tell the difference between them and what that means is I've over counted the number of microstates this microstate and this microstate are the same as each other I don't get to count them twice so I've counted each one of these states exactly twice as often as I should have because there's two different ways two different orders in which I could have written these two terms or another way of saying that is the number of permutations of I and J because there's two of them is two factorial permutations for the more general case if I write Q to the N so not just one partition function times another but I'm gonna have a whole long list of these N of them multiplied together there's gonna be N different permutations of each one of these terms and if the particles are indistinguishable I've over counted by the N factorial different ways of reshuffling those terms that look different algebraically but are in fact exactly the same if the particles are indistinguishable so actually if we do have indistinguishable particles the right way to calculate the partition function for the whole system is the single molecule partition function raise the N power and then divide that by N factorial to fix this over counting so when we have indistinguishable particles that's what we need to do when we have distinguishable particles we don't need to worry about dividing by the N factorial so we need to calculate the partition function differently with the shortcut depending on whether the particles are distinguishable or indistinguishable and as one last note I'll point out a common source of confusion which is this apparent similarity between the words identical and indistinguishable so make sure to keep these two terms in mind and they have the meaning that they had when we were talking about these examples here when I say two particles are identical in this context what I mean is that they're chemically identical they're the same molecule with the same components the same atoms that make up the molecule and so on so chemically identical molecules can be distinguishable if they're in different places and if they're going to remain in those different places or if they have different isotopic labels for example when I say indistinguishable I mean something even stronger than saying they're they're identical particles are indistinguishable if not only are they the same molecule but there's no physical way I can tell them apart so make sure and keep those two points in common two points in mind so now we know how to calculate the full system partition function for independent subsystems or for identical particles that are either distinguishable or indistinguishable the next step is to figure out how to actually use these partition functions to do something useful and calculate some properties of the system that we're interested in