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Lecture 23. Partition Functions Pt. 1


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Title Lecture 23. Partition Functions Pt. 1
Title of Series Chem 131B: Molecular Structure & Statistical Mechanics
Part Number 23
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 10.5446/18908
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 23. Molecular Structure & Statistical Mechanics -- Partition Functions -- Part 1. 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: 0:02:51 Rotational Spectrum of HCl 0:03:53 Molecular Partition Function 0:22:25 2-Level System Partition Function 0:31:12 Particle in a Box Partition Function 0:38:12 Harmonic Oscillator Partition Function
I remind everybody on today we're going to continue talking about partition functions and hopefully get a little bit more familiar with what this means and how we write them down and will do a few specific examples then next time 1 on Monday we're going to talk about issues like what happens when your system of particles interact and I see if I can get my might not make that noise the pope and maybe not the case Ericsson last couple lectures will learn about what happens when a system of particles interact and then on Wednesday next week the lecture will be given by here today John Mark who is going to show over some examples of partition functions on you'll be really great it's so it's his lecture debut and I will be in Washington DC reviewing an proposals so empty cans an important part of my job but it's about a 3rd of the jobs of the rest of it is managing research group reviewing things like proposals and stuff like that so how that works how people get NIH grants is wrong university submits proposals in reviewers who were or professors review these things and I look at that factors like Is there a well laid-out plan you know that either benchmarks for success if it succeeds is likely to do something important and then everybody has to go to Washington DC and around and discuss these things and give them course so that is what I will be doing next week on Wednesday so that also means that office hours next week canceled because all the traveling Tuesday and Thursday Wednesday the so please
use the Facebook page if you have questions of all the checking that relatively regularly duties will be checking it also and I will have office hours during finals week or examines Friday so there's plenty of time to prepare I don't know exactly when yet but all the posting those later on but I'll definitely have a bunch of office hours during the final weeks of relief funny time to ask questions anybody have any questions for me about stuff before we continue talking about statement OK let's do it right the last time
we ended up talking about the rotational specter of HCL and how we get the intensities of of different peaks and we looked at the relative populations between the grants in the 1st excited states in this rotational spectrum and so we looked at you know the fact that these this relative population just depends on it ended generously in the States and the energy between them and a parameter that's really fundamental to this is the and so that's something that we're meeting coming back to instead that the internal energy that's distributed tells us about which states are accessible and the premier that's really fundamental to that is temperature in some ways what's the really fundamental quantity is beta whatever the t OK so so we saw a specific example of how we get these populations and when to come back to rotational spectra but let's look at how we write down the partition function a more general way so we can write down population of some state in a relative sense a relative to the total number of molecules in the system and that depends on this summer 22 Baidoa which again 1 over Katie's we've got into minus beta Times interview the system and that's divided by the partition function Q and the partition function tells us about how much energy is in different modes of the system so what do I mean by different modes it depends on context it could be we could be talking about different vibrational modes are occasional modes or translation of the molecule bouncing around in a container all of these things could possibly be contained within the partition function electronic states too for that matter most of the time we try to treat all these things separately if it's possible just because it's a pain to have to deal with all these variables simultaneously and usually they don't interact with each other so we try to separate them when can Sony in the previous example we were just talking about the rotational states and we wrote down inserted justified in handling the way the relative population between 2 states that now here's the real definition of the partition function OK so we sum up over all the states the DeGeneres and then we have the minus Bader times the energy of each state and we summon up over estates so again how is that how is all the states defined well for something like an animal or a system that's really easy if we have spin half it's a two-level system only 2 of them there are a no other things that I'd like to little systems particularly if we're talking about electronic spectroscopy a lot of times you only have really low level excited states available on there's a well-defined number of them so in that case we can make that kind of approximation for things like vibrational rotational states we might have to take something looks like an infinite series to to some all states so what you actually do here depends a lot on context and particular mathematics of the situation we have OK so that's how we express the partition function in a general sense and again What this is telling us is it something about how the energy is distributed among different states and more specifically it's telling us something about how many states are accessible To the system at a given temperature all right so let's go back to a rotational specter of HCl and write an expression for it's rotational precious function OK so we need to be energies of all the rotational states In the Genesis I realize my title of the slightest little bit ill-chosen because that's of course entire spectrum of HCl but we get the rotational energy models from its so we the ideas there just the terminology isn't the best OK so we have the energy level the energy of some little is HCB at times shapeless 1 end remember India in the context of the partition function we wanted to find a ground state have 0 energy just because it makes them happy 0 so we know that there is some 0 . energy it's not actually 0 but we define it that way in this context we also know that the generosity of each level is To date plus 1 and so here's the expression for the partition function so there's not really an upper limit on the number of rotational states you can just put more and more energy into the system in the molecule will rotate faster and faster and as we go up an shapes look closer and closer together it becomes more like a continuum but there's not really an upper limit so we have to to Tucson overall these levels from 0 to infinity and then we plug the expressions for the generosity of the states and for their energies which we we know from having looked at this previously and that can be evaluated numerically pretty straightforwardly using experimental energies so in other words you can counter peaks in the spectrum and you see how many you can realistically and plug and all the energies for these things and calculated value for the partition function and you'll get he'll get a number I want to just mentioned something here which is going to come up again in our example day and that is if you try to to do this with rotational Roman here rather than a purer occasional spectrum we end up with you might count too many states because an invitational Rama you get the same as the configuration of molecule twice served during every rotation so again that the back line will see that more during the year example OK so when we go to evaluate this we can look up the additional concentrates L and it's about 10 . 6 wave numbers and we can plug that into our expressions for the energy and if we take the sum of the 1st 10 terms are looking at the 1st 10 states in the rotational energy let's see what we get for the partition function and if you count the peaks in that spectrum you can see that we have not worried not very many more than 10 so coarse that those a giving your transitions spread states but In other words this approximation is not perfect but you were worth seeing most of the states that are populated if we take the 1st 10 terms OK so I got these numbers out of your book is really straightforward to to calculate the this year just plugging in the energies of the different states so if we evaluate this quantity that were something over for each of these states here's what we get so weary did this we saw that the the relative the population of the 1st excited state to the ground state is about 2 . 7 1 and we know that that's because the generously there are more ways to be in that 1st excited state and then similarly as we get up to the 2nd excited states we have even more ways to do that and at this temperature which is 298 Calvin the generous he still dominating the same thing as we go up today equals 3 there there's still more population in that state and then that starts to level as we get at 2 tables for and so at this point it should be really clear why the relative intensities of the lines in the specter of what the waited Seremban in in the spectrum were looking at the transitions between 1 state another so we can just map the
the height of the intensities under the populations of particular state but it does tell us something about about what's populated and OK so we saw that this starts to turn a word about tables for and the numbers so that the relative population start going down again and then as we get up to J-PAL's we get something that looks like . 0 8 OK so if we had this up the sum of the 1st 50 terms is about 19 . 9 0 2 it was 19 . 9 0 for the the 1st 10 terms so we can see that that there's really not more more than about 10 states populated in the system at room temperature adding another 40 of them doesn't really do much good Of course this is going to change if we change the temperature so if we "quotation mark system down significantly what's going to happen is we're going to get a tighter distribution to there'll be more you more population in the most populated states and things will be a spread out and also b maximum the maximally populated state will have to being lowered if we keep the system up then we're going to get something that's that's much flatter it's much closer to all the states being equally populated OK so this also leads us to an approximation that we can use for these kind of things which is that the rotational partition function approximately equals Katie or HC data and yesterday when we're looking at the side of the spectrum this this quantity was it was just kind of stuck up there that's why it's so it's a reasonable approximation to at the rotational partition function and if you do that you get 19 . 6 in this case so it's not perfect but it does give you something that's in the right ball park and it is very much work yes I know you on it's so it's related to the population but so this is this is universe quantity that were ending up in the air In the partition function so it's it's a relative it's essentially a relative population relative to come here and the ground state that's right yesterday said well you so you so to get the the part of the actual population of the state you have to divide by the partition function which is telling you something about the overall population sees more examples like OK so what this telling what this is telling us is how many states are firmly accessible at a particular temperature and so let's just think about some limiting cases to try to get a sense of us so let's say the temperature is close to 0 we call our system almost all the way down to 0 Calvin there's not very much emotion going on N you know here betas won over Katie and that starts to approach infinity as temperatures close to 0 and so what that tells us is that everything other than the 1st term in the summer is going equals 0 so there are all equal to even the minor sex with X equals infinity except for the first one and that gives us something that's really into it and if we really pull the system down to words close to 0 almost everything is going to be in the ground state level and so on I should point out that a lot of people who were really looking at this from a theoretical physics perspective or unit even from some other systems that that behave this way in things other than statistical mechanics a lot of people would say that data is really the fundamental parameter rather than temperature because if we think about our standard understanding of how this relates to temperature which is that as we put you know as we have higher temperature that means that we have more entropy more motion things moving around art standard understanding of that means that we can't have a negative temperature that's that seems to be impossible from are intuitive understanding of how molecules work so we have some 0 . energy things cold and absolute 0 and nothing is moving and that's because we're thinking about this in terms of a particular type of physical system that being molecules that are moving around endit other sorts of things we can have negative temperature and it's kind of a bizarre concepts of something that has a negative temperature you know you might think it's really really cold it's not it's really hot so let's think about a system that we have talked about that has a negative temperature so if we think about our anymore system where we have and equilibrium we have spins in the alpha and beta states and there's a little bit of in excess of the office state and then we go to do and inversion recovery experiment we give 182 reports we have harmonization better where there are more stands in the office and state and then we give degree calls and now we have a population inversion so now there are more spins In the beta state then the office state during the time that the system is like that before relaxes backed equilibrium it has a negative temperature there's more there are more spins in the higher energy state than warranted state and on that was 1 of the main reasons well I think why am I Procell got the Nobel Prize for in discovery of some of these and more phenomena because physically that's a really weird set up to be a quick assistant recently some physicists were able to generate the 80 system with negative temperature In terms of actual molecules fold will talk about that more later but when you think about these these kind of of issues and how we can have something where putting more energy into the system reduces the entropy then negative temperature is possible and that's 1 reason why you might want to think about beat as being the fundamental parameter In a thermodynamic sense rather than the temperature itself OK so if we look at the far end of March system it's easy example cause only has 2 levels at least in this been one-half case we have are 2 states alpha and beta and as the temperature gets close to absolute 0 Our partition function the ends up being very close to 1 because everything is in this smaller energy state which is not deterrent OK so we talked about what happens at low temperature we talked about how we can get a negative temperature all will so that what happens when the temperature is high 1 misunderstanding that people have sometimes starting out with this as you might think like OK when temperatures high then you have an excess states in the arab spends in the apostate you don't that's you have to you have to set the system up in a particular way to get that that's not a 180 degree Paul's were we end up with a negative temperature but just heating up the system doesn't do that and doesn't makers have more spins in the excited state so what happens when the temperature is high is that you tend toward equal populations the state so you have plenty of energy there's no reason there's there's less reason why it matters if you're in the war energy state as opposed to hire 1 ends Our partition function tends to yeah just you just get a solution where everything is equally distributed so in this case ah partition function is going to go it's going to end up being too as are temperature goes to infinity OK so
let's look at a some concrete examples of how to write these things down so this is something that you're definitely going to do so I'm still while I'm still working on some practice problems for for this have them up at some point today definitely on I wanted to go beyond the ones that are in your book and give some more examples of the illustrating in some probability ideas so I will definitely have those of later today but I so 1 of the things that you'll need to do is be able to write down petition functions for a fairly simple systems were into some practice problems were you have to use some of infinite series upstart for rotational vibrational states but you know those are a little bit more involved it's good work to see how it works as far as being able to do it on the exam systems like this are more realistic OK so we know our relative or expression for the population of the state ends look at how we write the partition function for the things that are better defined in terms of a small number of states OK so what say we have a two-level system where the lowest state is not degenerate so there's only 1 1 way to get the lower state and the upper state is doubly degenerate and so the 1st thing to do to to be able solve such a problem is you know look at the description words and be able to to write and energy all diagram for that so far are lower state is not generated the upper is doubly degenerate and then it's important to remember that we always define the energy of our ground state 0 in these kinds of problems so it's completely general it doesn't matter what kind of system it is does not all we define 1 0 and we also said that in this case the energy of the 1st excited state workers can call Epsilon and so then we can breakdown partition function and River we have this summer with stick that generous even in front of it and then we have the spokesman distribution looking thing for us the energy of states and so in this particular case we just get 1 plus 2 times even minor steps 1 so this is definitely something that you should know how to do a few have you know that description In the words of some system that contains a small number of states with DeGeneres is given additional write-downs partition function OK so let's talk about some specific contributions to the partition function yes vote due to the
use of the year of the this is not the only 1 of the this will work on the 1 you get the but was so so where did the work in the equation of Jean-Paul Studio Plus 1 come from it's further it's for occasional system of 1 or more of the linear molecule in this 1 did I say it's a rotational partition function at all it's just really general right so I just told you the 2 generously of the bottom state is 1 and the generosity of the upper state is too I How do I know that who knows who knows who knows what even bananas it's it's just very very general so you know don't don't get it don't get excited about using the specific rules for certain things if you're not looking at that situation it's an easy mistake to make that and that's why we're going to go through a few examples of different kinds of of partition functions but on this particular case you know you don't even know what it is so that so how we got injured and receive the states is is unknowable OK so here is a general partition function for or at least the year the energies involved in it OK so if we In us if we sum up all the energies of the partition function is with respect to the exponential so world we're going to multiply those altogether what's what's going on here with these different contributions OK so we're looking at all the degrees of freedom With respect to motion that are molecule can have plus also the electronic transitions that's not really emotion exactly but it's convenient to counted anyway because it's something that we often have to worry about 4 molecules OK so we have translational component and a rotational component and the vibrational component and also an electronic ones N I mentioned this before it's it's worth bringing up again because it really makes your life easier if it is at all possible we only want to look at 1 of these things that at a time because usually they don't interact and so it just makes the math 1 easier for only adding up 1 set of degrees of freedom at once also in context we usually only care about 1 of them once because we're looking at a particular type of spectroscopy your word were analyzing some experimental data in which we would be unlikely to have all these things going on at the same time here is an exception if we have AT system were more than the ground electronic state is excited so we have a lot of electronic transitions going on of course we know from looking at the electronic spectroscopy that when you excite various electronic excited states then of course all the vibrational excited that the vibrational transitions get excited as well so no 1 hour molecule gets promoted to excited electronic state that induces a bunch of vibrations and so if we have more than the ground electronic state populated than we can't separate those 2 for most molecules that we're going to be looking at at room temperature only the ground electronic state part is populated so that's a pretty good approximation for most of the things that we look at and we can generally always suffer seperate the translational Invitational states OK so another thing that's important about this general kind of partition function is that it's usually not possible to solve analytically so this is where Mathematica as your friend you know you're going to need a lot of numerical solutions the examples that we're doing classical or going to involve things where the some approximation we can make but if you really get into doing statement you you do this in grad school 1 of the things you'll see it is In when you start getting into more advanced problems theirs you know there's always some little trick that enables you to make some approximation it turns out that's kind of how you solve every problem so there's the of approximation that you can make or you just brute force to it with numerical methods OK so now that we've seen how we write these things down in general and we've talked about how we want to keep them separated if we can just look at some specific examples so the ones that that I'm going through today or in your at least if I get as far as I
think I'm going to get the sum at the end the not but what see how would you OK so these are examples that you've seen before in different contexts which is nice because you know that the basic story we can just talk about how the how it relates to the partition function as the 1st woman to look at is the electronic states have you do with the particle box so this should be really familiar from last quarter I know that they spent a bunch of time on this so you know the energy levels for the army and values of the particle in the box and you are all for these new know what they are the only thing that's different is as always where are you were going to find me the aloft where to put everything in terms of lower energy state and so we can write to us if we define me the other 1 for the energy for any enables 1 as epsilon we can write the epsilon level and is in squared minus 1 times Epsilon OK so now a look at the translational partition function for article and so we got and squared minus 1 times Epsilon we're willing to approximate that as something from 0 2 but even license they'd Absalon DNA and so were and were not taking role said that the sum here because that's easier to deal with just mathematically and so on we can rearrange just fervor the sake of convenience just make easier to do yes sir they want to go I want you view of things like this it was yeah hang hang with bcm rearranging stuff so that to make it easier to do if it doesn't give a damn that's sight there may be entire writes Mark but what's what's OK so where rearranging us to look at it in terms of access so this is a one-dimensional system by definition right a it's a particle one-dimensional box and so on where integrating this over and over and With respect to acts and so if we evaluate the and role we get an expression for ah translational partition function as a function of X and yet another and now that I'm looking at this and I think that the answer quarries question I shouldn't get so many steps in the beginning so I should written up the song you know from equals 1 new Infiniti and show that were transferring this to an integral In terms of acts from the Infiniti EX but hopefully the theater makes sense OK so Hughes are partition function for the translational part of the year the
particle box and remember that beta equals 1 over Katie and so we can write or partition function as X over capital where Landers defined as this collection of stop notice the mass of the particles in there and it has dimensions of life and if you check out the translational partition functions that are in your book I really recommend doing the reading for 4 these for for these topics definitely before coming to class on Monday at this it's particularly Benson book in this chapter there are a lot of examples are a lot of stuff going on it's useful to look at them I so this this land as is quantity that's going to be important for translational partition functions in general and 1 of the things in your book is an extension of it 2 3 dimensions and if you work all these things out you'll notice that it has dimensions of late it's related to the debris wavelength and so what that means is the partition function increases with the length of the box and the massive particles which should be consistent with your intuition about how this works so you have your system with the particle the box and you know that it's it's it behaves more in quantum like way for a much smaller particles and 4 smaller areas of confinement whereas when you get to a longer one-dimensional boxers heavier particle that it behaves more like a classical system and the levels were closer together looks more like a continuum you get the same kind answer for the 1st doing this in terms of partition function so remember getting a larger quantity for the partition function means that more levels are accessible to the system at a given temperature so it looks more classical for heavier things and larger boxes OK so that's 1 example the translational partition function it doesn't have to be for something like a part of boxy can do this for just particles moving around in a flask of it's a little bit more boring for the classical systems because there's not much of a year there's not much conversation going on in almost all the levels were equally populated in
that case we can also do this for something that looks like vibrational spectroscopy so we can write down a harmonic oscillator partition function so here's our potential for a perfect harmonic oscillators or potential looks like a problem and then we have all of these vibrational States harmonic oscillator wave functions which are for me ,comma meals they're equally spaced and the harmonic oscillator levels are also known to generate just like the ones for the width for the particle box and we know that the separation between the levels is we can :colon epsilon we know that it's new and we can define the lower energy 1 as being 0 so the 1st excited states at 1 the 2nd 1 is to epsilon etc and so we can start to write down expression for the partition function devastating so we have our energies in terms of epsilon spacing between them and we know they're all on degenerate so that takes care of all of that term and we can start to and these things up and we can observed that this starts to look like an infinite series that we know when it converges to and so we can use this expression as the partition function for the home of the harmonic oscillator OK so patently know that again this is a case where you know you write serve at how this is going and then use some approximations that you know or hear this is approximation if you haven't affirms that the series converges to that it's just that you know recognizing what the demand comes out as if you get if you get we understand that you get more experienced during these things were different systems OK so weekend play around with this little more and so I look at whatever if it's series converges to and so we have our partition function for the harmonic oscillator ends we can use this to to get some relative populations so the fraction of molecules in some particular level with energy that's 1 again we get this buyout taking you're taking you to the miners Beta and Epsilon Over Q that the partition function and we can write out what this is and again we get very pretty intuitive result that as we decrease the temperature only lowest energy occupied and it's kind of nice to look at some of these systems where the state's Romonda generous because that gives us a really good intuitive feel for you know how things are just depending on the energy of course we get into things where there is DeGeneres that often wins and so at high temperature again our petition function goes to infinity In this case here we have this problem that's going up to infinity there's an infinite number of vibrational states that are excited Of course in a real molecule that's not a realistic approximation right because in that case we would have lost potential were eventually if you put enough library vibrational energy molecules in a vibrant itself apart but in this idealized system we have an infinite number of levels and the populations are going to tend to be useful the high temperature so so far we've looked at some examples for various systems that were familiar with let's go back to our a translational kind of problem and something that we've seen for a general chemistry OK so certain ballooned into this but you know here I thought I found some some actual examples for right so we talked about the Maxwell Holtzman distribution of molecular speeds for noble gasses sofa for ideal gasses and at some given temperature if we look at the the speed distribution Of these said these Adams we see that the helium the 1 that's the lightest has a really broad distribution there's all kinds of different speeds going on in there and so it's relatively flat whereas as you know on the heaviest 1 not only has a much lower average speed but it has a lot near distribution of different velocities that it can have and you know we think about that that landed parameter that has dimensions of wavelength we can also get some relationships between that and the speeds of molecules so this goes back the kinetic molecular theory of gasses
and so we can think about the effects of having heavier particles as being sort of analogous to taking the same particle and looking at a different temperatures so having heavier particles is that it is going to to look somewhere in terms of how it behaves as taking the same kind of gas and cooling it down so In on the same the actual details of that for next time I just wanted to to introduce a give people time to think about it on later today I will have some practice problems posted and I will see you on Monday ahead of the weekend
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