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Lecture 24. Partition Functions Pt. 2


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Title Lecture 24. Partition Functions Pt. 2
Title of Series Chem 131B: Molecular Structure & Statistical Mechanics
Part Number 24
Number of Parts 26
Author Martin, Rachel
License CC Attribution - ShareAlike 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license.
DOI 10.5446/18932
Publisher University of California Irvine (UCI)
Release Date 2013
Language English

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Subject Area Chemistry
Abstract UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 24. Molecular Structure & Statistical Mechanics -- Partition Functions -- Part 2. Instructor: Rachel Martin, Ph.D. Description: Principles of quantum mechanics with application to the elements of atomic structure and energy levels, diatomic molecular spectroscopy and structure determination, and chemical bonding in simple molecules. Index of Topics: 0:02:18 Rotational partition Function 0:08:13 Rotational Temperature 0:12:51 Symmetric Linear Molecule 0:17:01 Rotational Raman for H2 0:23:30 Strokes Lines for H2 0:29:11 The Mean Energy 0:32:42 The Canonical Ensemble
let's said go ahead and get started last week with lots stuff to cover before we do so everybody should have your exams back and um if you have any questions about reading your score is added longing for something like that the deadline for taking care of that is this Friday at midnight so please make sure you look at it and I can have time during finals week to to the the release exams so make sure to get that taking care of I'm also going to be traveling so I might not get to it right away but I will definitely do it the at some point this week on any other questions before we start yes the I'm sorry we assuming that we get done with everything we're going to have a review in The Last Lecture that I'm going to do and then the tears might do another 1 I don't know I'm definitely going to have a bunch of office hours during finals week is I am going to get a chance the canceled this week so I'm going to make up for it by doing some during the finals Jim question to you is the final will be extremely cumulative like every single thing that we've talked about all water so the point the point is you know you're you're getting a chance to redeem yourself if you didn't do so well on either of the 1st exams and so that means that absolutely everything has to be covered so everything's going to be on it and it's going to be really long but that's not yeah that's
how it goes OK I take that piece of paper this is it's good
you want to have a quiz the leading practice for the types of problems that you see on the final about partition functions so we've been talking about partition functions in general and how to deal with different contributions to them and I really recommend reading in the book the whole discussion of translational vibrational partition functions we have gone over that lecture but you know hopefully now we talk about the general stuff more might sink in a little bit better so please do go back review that organ talk about the rotational partition functions in more detail to pick an example to going to buy all of the general sorts of things that we've been talking about apply applied to all the different kinds of partition functions so 1 of the main things that we notice is that at low temperature we have only the really low energy states populated by you know until we get down to Close to 0 we don't have all the population in the ground state even at CDC here even at 1 Calvin we still have to generously winning out over low energy and the most populated state is not the ground state "quotation mark server room temperature which our you know conditions that leave might see some typical experiments at least for talking about things like atmospheric chemistry and then we see lots and lots of different rotational states populated which makes sense because we know that that terror for occasional states don't take very much energy to populate excited states OK so let's talk about when you rivers in particular spent some time with talking about diet molecules so I mentioned before that we can use this approximation for linear Rover we can say that our rotational partition function is approximately equal to Katie over HC times the occasional constant and just for completeness here is the equivalent expression for an Ottoman-era services for something that has 3 different rotational constants a B and C. we're not going to get into where this comes from for the nonlinear River it's a little bit tricky there's a whole little supplemental section in your book of about where it comes from if you're interested govern read it but it's just here for computers at this point of excellent focus on linear reversed and so the approximation that were going users were going to say are right at reasonable temperatures Katie is a lot larger than the separation between states so the energy that we have available to the system is plenty to populate a bunch of states and that enables us to approximate the as an integral and why we wanted to this because it gets just easier to evaluate so you know if if we want to do it numerically the sum isn't necessarily big deal but as far as just being able to do it with paper and pencil in class and get some intuitive insight as to what it looks like it was a very much easier to deal with so that's something that Britain
wanted to OK so here is the relevant in a role that he can use for and so the approximation that we're making is that we can treat these states as basically a continuous variable so we know that's not really true awareness if we get the system really cold or if we have a very light molecules like H 2 Bell make a difference we have to deal with the fact that the states are quantized N. J. is not actually continuous variable but if 4 at close to room temperature we can make the approximation that is and we get a reasonable answer OK so we're just here were just sticking in the sun familiar expression for the partition function and assuming that that we can treat gays a continuous variable and this is something that I was before I went through before but I think I want all fast and get some steps and I got a lot of questions about its renewed again OK so In order to make this easier to deal with we're going to call exercise this term and change our integration variable again just because it makes them happy 0 gives us role that we know how to evaluate and so if we make that substitution for acts as a function we get an expression In terms of an integral there it is easy to handle and so that is where are our approximation for show rotational partition function at reasonably high temperatures comes from and so do I want you to reproduce this duration right now and not really I just want to know where it comes from and be able to to use the approximation of the need to OK so that was something that I got much questions on a thought it was worth just briefly going over it again so this approximation leads us to the concept of a rotational temperature and
that's defined like this so it's called feet and it's just justified as HCB overcame and so for purposes of determining whether we can use that approximation or not whether we can use in in roles of song high temperature in this case means that it's greater than the rotational temperature by a lot and we're going to look at some examples in just a 2nd and we're going to see that for almost all reasonable molecules that they were close to room temperature you have plenty of energy in the system will use this this approximation and so again under these conditions it course given what these you know what the hell this is defined as you can approximate the partition function as approximately equal to the temperature over the rotational temperature and let's look at some actual molecules the stabilizing your book so you have to write down all these numbers it's just sometimes it's really worth looking at actual examples just to get an order of magnitude for how these things look like OK so for each to it's a really light molecule it's for occasional temperature is 88 Calvin so that's actually pretty the for this kind of thing might not seem very warm liquid is about 77 :colon but when you you know what this looks like for the rest of the molecules ring like that HCL into reasonably small molecule but that chlorine atom is a lot heavier than hydrogen the rotational temperature there is about 15 Calvin and then we get to some larger both your molecules for eye to you you have to get really close to absolute 0 for this approximation not to be good and even for C O 2 we have thought about half so each to really has an unusually high rotational temperature and clearly have to be very careful about using this approximation for hydrogen and last year most of it is very high temperature but for the rest of this stuff it's a pretty good approximation of 4 anywhere close to no reasonable room temperature conditions any sort of atmospheric chemistry type study that can at ambient temperature that approximation is going to be just fine so we can also say that molecules that have large moments of inertia or small rotational constants have a really large work ational partition functions and again having a large rotational play partition function tells us that a lot of states Serb-populated room-temperature temperature and that just brings us back to here again what is the partition function telling us it's telling us how many states are populated at a particular temperature OK so now we're going to get into some more detailed specifics for particular thinks so so far this is relatively general it's so it's a useful approximation make a lot of he cabin and actually physics problems are all about knowing you know when we can make some approximations that you know usually that just make them happy 0 make it to do make it easier to deal with follow it's important to know what approximations you're making and why it's OK to do that and that's that's important for general problems now let's talk about some specifics so for some of these molecules we have to be careful not to include too many terms in the song all were looking at at the exact partition function and it if particular let's talk about 2 of d infinity the molecules so when you're molecules that should be the Infiniti age I knew that was that was wrong when I said arts so that so be linear molecules where both ends of the same are so for
this kind of a symmetric linear molecule if we rotated 180 degrees we get the same effect somewhat hits here too so this is the universe is an obvious thing if we rotated 180 degrees we get an indistinguishable configuration and when we start to think about these things we have to go beyond the cemetery arguments that we've looked at previously because are basis is not just the Adams were looking at the overall molecular wave functions and it's not we we have to think about all the more deeply a case of a 2 we really are kind of just looking at the molecular cemetery because were assuming that aaah nuclei are 0 16 items it's by far the most common isotope of of oxygen and 0 16 unanswered spin 0 those so they have no spin and so on why does that matter because if they did have stand we would have to worry about sign changes of the wave function only interchange them a around at example where that does matter in a 2nd but for now all we really have to worry about is the fact that you know what we have only rotate this thing 180 degrees we get the same state and so we were counting are possible configurations will double counted we assume that that's distinguishable so the number of firmly accessible states it is half of what it would be for are 100 nuclear diatonic molecules and this is dealt with in a general way by defining something called the symmetry number witches given a Sigma and that is the number of indistinguishable orientations of the molecule so you know we're talking about rotation so it's with respect to rotation and in this case the cemetery too and it's pretty hard to the cemetery number being more than that for a linear molecule which is mostly we're going to focus on in this discussion but I do remember in back your mind that you can have symmetric worsened the new spherical writers and so in that case you might have to worry about higher symmetry numbers and so here's what our approximation to the rotational petition to function looks like what we take into account the cemetery number I'm OK so that's what you get when we include the concept of distinguish ability of orientations and again I said that what we're interested in here is the overall molecular wave function and 1 thing I want to point out is that you're rotational wave functions change sign as minus 1 to the J under a seat to workstation and you can go back and look at the if that if that seems a little bit perplexing go back and look at look up with the rotational wave functions are and convince yourself that this is true OK so what see why this matters if we look at a molecule where the Adams on the ends are not spin 0 then this is going to become a lot more important to us and interesting things happen OK so here's the
occasional Roman spectrum for H 2 there's no really lying in the middle because the these researchers who collected the spectrum applied a filter there so it's been filtered out seats the see the army the Roman lines much better so that's why there's this funny little blip in the center of the spectrum instead of being really line they just filtered out so what you notice about this is that there are 2 sets of peaks and once set is much more intense than the other which you know almost looks like you have 2 different species and their right but it's all age 2 so what's going on OK so what's happening here is now we're looking at Adams that are spin 1 half for me from the context of an Aymara we've already learned that protons Austin one-half and you cost were assuming that the both of the the hydrogen is here are protons and not deuterium Merck tritium or some other isotope of hydrogen but given that they are protons there's been 1 half from the islands and if you remember the Pauli principle from 1st quarter that tells us that the overall wave function has changed sign when you interchange the particles and that's just a general thing for her for me answer and so I know that you talked about this last quarter in the context of electrons and electron wave functions with this is true absolutely in general and so if we look at our overall wave function here we have all kinds of things that go into it there's electronic and vibrational and also nuclear spin and rotational components to the wave function here we don't really need to worry about the vibrational and rotational states are sorry I mean the vibrational and electronic states were assuming that were not low enough temperature that things are in the ground state of the on vibrational electronic wave functions and we're just worried about the spin and the rotations OK so why do we get these differences in intensity between these lines it has to do with the selection rules that the specific selection rules for money spectroscopy and how this interacts with the nuclear spin which function so we have seen in a play between 2 completely different parts of the rate of the wave function the rotational part and the nuclear spent part OK so the issue here is that the overall molecular wave function is going to change sign when they rotate the molecule if the spins repaired but not the parallel you look like you don't believe initially supporters so let's look at our hydrogen molecules so here is the case where we have our 2 protons and they have entered peril stands and this species is called her a hydrogen and now if I do see 2 rotation on that here's what I get so it doesn't just walk the positions of the nuclei it does that but it also changes the the sign but the wave function and so now this starts to you know hopefully it should remind you a little bit of these problems that we did we're talking group theory where we had different in orbitals the sway of swapping places may change Sinard didn't depending on the orientation hopefully now there's so you can see a possible use for that in terms of understanding OK so here we have a case where changing needs during C 2 rotation on this thing change is the sign of the spend part of the wave function if we have instead the nuclei idea the nuclear spins parallel to each other that sort of hydrogen and during a C to rotation on that doesn't change there science he did the same thing back and so on when we look at the Parra hydrogen case the fact that the nuclear part changes sign when you rotated it means that only even values and J. allowed and that's because the police collusion principle requires that will be interchange the particles we have to get a change in sign and the spent cartridges sign if there were occasional part change signed to then we would have minus -minus plus and the overall thing wouldn't change sign and that's not allowed so that's why we only see the even cases here for work or hydrogen the opposite is true so the stand part doesn't change sign but we know something has to because we interchange the particles and so that means the recreational park has to change so we've got a good sign change from somewhere in 1 case a comes from the spin part of the wave function In the case it has to come from the rotational apartment so that's why we have these 2 sets of different lines source selection rule is still Delta J-PAL's plus or minus 1 what we're getting even components from 1 source and the odd ones from another 1 OK so we say so we still didn't explain why the intensities are different so we just talked about why we see both sets of peaks spoke like we have more of 1 and the others so let's zoom in on this and look at it a little bit closer Robb
so here it is these lines are there is also a large peak labeled and to that's from some nitrogen that's in the sample weakening nor are the ones that were interested in are the ones that are more to the left of the spectrum and so now we want to explain these intensity differences so what's going on there and the reason we have these intensity arises because this is if we look at the statistics of this process there were 3 times as waste as many ways to get on Jr as the so we can look at best and for completeness and this cannot tell you the the rules for figuring out the intensity of odds even lines for the nucleus with a particular value and so these are the rules for half integral spin and for integral spent so that's just so you know how to deal with that in cases where you don't have you there's been 0 costing one-half but for now we're only going to talk in detail about the it's been one-half case OK so where this comes from is if you have this things if you have this thing where nuclear spends there parent there's only 1 way to do that it's analogous to a single at state while we're talking about electronic wave functions it's not a single state exactly in minutes it's the nuclear spin on 1 of us but there's only 1 way to get it and then there are 3 ways to get the parallel case which is analogous to a triple state and so here is the picture of what those look like so here the dark blue 1 the the paired case the single at state and the other 3 the components of the troubled state and if this is so yet this is a little sketchy here you don't quite remember go back and check out Chapter 4 that's where that's where the stuff is 1st introduced there's a resembling the explanation of your book OK so that's why we have 3 times as much intensity in the on state says the even ones so we see where they both come from and how we get the differences intensive question all yes of those at those are the 3 ways to add up the stands to get the report state go go back and check out Chapter 4 there's a there's a pretty good description of it in there OK so another interesting thing to know about this is that also and hydrogen into converted very slowly so you have your whole bunch of hydrogen stored somewhere you're going to have a mixture of both of these things and you can just convert from 1 to the other very readily and so that's why we see both in the spectrum another interesting consequence of this is that the author of hydrogen that can't exist in even states can't exist in DJ equals 0 recreational state and so even really really low temperatures it rotates and Wales itself off and that this is a problem for hydrogen storage so soaking 1 have a bunch of hydrogen and keep it around at low temperature you have to get it into the compare hydrogen state people also use Parra hydrogen for doing and more experiments in which they can increase the polarization of the sample very specifically and introduce Parra hydrogen into organic molecules when I have time to get into that it's a it's a new research area OK so we looked at the rotational statistics for this and by the way this is all described in Chapter 10 I just felt that it was too soon to talk about it and because we haven't really gotten into the statistics of how we get different populations so soaking worried more about then the nuclear statistics additional spectroscopy that is in Chapter 10 fights Let's look at the partition function but this thing so if we want to ride out the the full partition function we have to take into account that a quarter of the state's come from even J and three-quarters of them come from bondage I but think and so that is specific to the case of what happens if both in Europe Adams Spain 1 have a are excellent said move on and start talking about how can we get bulk properties from the partition function so I said that this is the partition function and that the Boltzmann distribution is the connection between the microscopic treatments of systems and the macroscopic treatments like we know about from
thermodynamics start to look at how that occurs so if we take something like the mean energy so far where it works time we've just been talking about the partition function as you were summing up over all the energy the particles and tells us which how many states are populated it also contains the information that we need to get things like energy and so if you assume that you have a system with an independent particles the mean energy of a particle is just going to be the total energy divided by the number of particles of course we know that they don't all have the same energy the some Boltzmann distribution and how wide it isn't all that depends on the temperature but the mean value is given by US end that's a be this kind of asylum and so if we use the Boltzmann expression for the population of a particular state L we get something that looks like mess and then of course we would like to rearrange thus so that it's only a function of Hugh because it would be nice to have the partition function or the the mean energy just as a function of the commission function and so we can do that we can write like this and you notice that this thing ends up containing a essentially a time derivative because bidders whatever Katie and so we do that here's what our some ends up looking like and then we can evaluate this to get minus 1 over Q debater and so you can kind of start to see why some people who work with static Michael lot particularly in systems other than physical and chemical systems consider the bidding to be the natural parameter rather than temperature for experimentalists temperatures pretty natural because we can measure it but if you look at the the Mafia how things these things come out data sir makes a lot more sense are excellent so I go back to our 2 mobile system from the animal or case and again look at how the partition function depends on temperature so we write this down before got sort of an intuitive sense of what the partition function does which I'm hopefully everybody remembered on the quiz and we can look at this and say now we know the mean energy at some temperature and if we play again what the particular partition function is here we can evaluate it and get get an expression In here were just simplifying that and in general we have to remember a couple of things 1 is that you have an energy of the ground state that you have to put back in there that's what this epsilon GS is because remember in the partition function a lot of times we just say or write the energy the ground 0 and if we're just talking about the partition function that's totally fine if we want to relate that back to measurable quantities that we can get from experiments then we have to stick that back in there so we need to pay attention to the energy the ground state and then also noticed that OMB I replaced the DE here with a partial derivative because when start talking about the many Amex temperature might not be the only parameter that's important there might be pressure and volume and other things and so we have to take the Winthrop expresses this is a partial derivatives keeping things constant OK so we can also say DX over Abraxas DL acts and write down our expression for the on the it's like OK so that's the important point with that being able to relate this to the 1st of all the boat qualities we've also kind of danced around relating it to entropy but we haven't quite gotten there yet I also want to point out a little bit of an analogy to quantum mechanics so we said that in quantum mechanics were talking about a particular molecules or atoms and everything about the system is described by the shortening your equation In the end of particularly particularly the Hambletonian and in stop were talking about the partition function so instead of having the Hambletonian describing the energy and everything about the system being contained in the wave functions In here we have and average energy of the system and then everything about that system is contained in the petition function so you again it's not it's not exactly the same but we can see the analogy and so the last thing that we need to get to is talking about a little bit more specific kind of thought experiment for working with patrician functions so we've seen how to get the mean energy this and now we need to talk about it a little bit more precise definition of an ensemble so we've talked about just as we have a collection of and Adams are molecules and we've been pretty imprecise about what's what's going on with that how we define that let's be a little bit more careful about it now so an ensemble is on imaginary system were we have some kind of a closed system of particles in its replicated in times their identical and all of our end replicates are informal contact with each other so what does that mean so there are different variations of this this the Micro canonical ensemble where the number of molecules the volume and the energy on common the 1 that we're going to worry about this quarter Is the canonical ensemble so in this case we have the number of molecules the volume and the temperature in common so why is that important it's easier to manipulate so the Mica canonical ensemble is may be easier to think about theoretically as it has this natural variable of energy but if warranted try to connect this to experiments that you can actually do the temperature is something that's easier to actually I control and then there's also the grand canonical ensemble worry of chemical potential volume interpreter in common and this 1 can exchange particles as well as energy from 1 replica of the system to the other and then we're not going to talk about a whole lot right now I just want to introduce witnesses that it looks familiar to you later when it comes up maybe next quarter or the next time you see it
OK so for the canonical ensemble just like we've seen for our discussions of partition functions for our particular set of molecules some configurations are more probable than others so we're not going to have all the energy in 1 system just like 4 a set of of molecules Moroccan have everything in the ground state and this side and with the toll they it is the number of Little closed systems within within our system and in the thermodynamic lament that's the the ball case where we just have a huge amount of these things then and goes to infinity In that case a single configuration dominates and we can write the configurations and the weights of the configurations exactly like we did 4 the partition function and the most probable configuration just like any other cases the 1 you get were you maximize the weight subject to the constraint of having constant energy and that is something that John Mark is going to go over with you on Wednesday we bend over working on the lecture for that I think it's going to be really good is going to go through some examples and show you some practical things and so what you get out of this is the canonical distribution that also has a canonical partition function associated with that and so all of this is analogous to to the little Q partition function that we've been talking about but the importance here and this will be continued next time is that what we set this up as a canonical ensemble we lose the constraint that all the molecules have to be independent and so this is going
to be important because you can use it to look at condensed phases and real gasses not just ideal gasses OK think we're about this time but I will see you all on Friday
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