Lecture 22. The Boltzmann Distribution


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Lecture 22. The Boltzmann Distribution
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Martin, Rachel
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UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 22. Molecular Structure & Statistical Mechanics -- The Boltzmann Distribution. 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: 0:05:23 NMR Population Differences 0:10:34 Statistical Mechanics 0:24:26 Weights of Configurations 0:41:17 Relative Populations 0:43:05 Rotational Spectrum of HCl
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I have announcements to make what is your exams all greater than the consent for scanning on the averages about 64 so great job it's a lot better than last time so I think that people working really
hard and I'm happy that that everybody's going material other announcements next week is last week at a quarter that means on its the last he can summon that you can go to for extra credit so if you want to do it that's last chance and again as a dual week after the summer so that the last 1 and people have been asking me about the final he is definitely cumulative everything is going to be on it's 2 hours so it'll be long on the same that having more questions about stuff like that would just takes the final yes here are the final will be pretty comparable to things you've seen before and don't think there there will be any surprises obviously will be longer because there's more time than it kind and it has to cover everything but otherwise I think it be pretty comparable yes ,comma cheat sheets 1 same as before yes I know it's going to be pretty equally distributed among everything that we've seen this quarter and so I'm going to tell you why that is but so he can is really hard and it takes some time to I understand some of the concepts and not everybody gets it right away I didn't hear that but so what really matters to me at least is what you know at the end and so if you did really well on the final and you have happened to stink it up on 1 of the previous exams then that will be taken into account so that is why the final is completely cumulative there's sort of equal amounts of everything we've learned and I if there's a there's a distribution of what's what's worth how much and that's the default situation but if there's a situation where you know somebody did really really poorly and 1 in the previous exams and much better on the final then that will be taken into account this the nite I pretty much always do this in the town of it usually affects a small but non-zero number of people states so the was mostly in it's not doesn't make such a big difference but sometimes it does so he had a problem with 1 of the previous terms I really encourage you to make sure that you understand what you did wrong and be able to do that on the final write any more questions
OK what's so started talking about statement right so in a way this is a really big departure from the things that we've been doing so so far and in end this quarter he can we talk about some aspects of molecular symmetry and how this relates to spectroscopy and how how spectroscopy works in involves different forms and how we can use that to get structures of molecules and the whole emphasis so far has really been on individual molecules and how we can determine their properties of course we're talking above all techniques usually do spectroscopy we're not looking at a single molecule we have a whole ensemble of them but were during the fact that were fundamentally concerns about properties of 1 molecule whether it's geometry or electronic structure were vibrations invitations and things like that and now with static we're going to make the transition to talking more about ensembles of molecules and properties that have to do with what happens when you get the whole group of molecules together statement is a really important sort of bridging topic because this is what makes the connection between all of these individual properties of molecules from microscopic picture that we've been learning about and the macroscopic properties that you know from day so this is your hopefully a topic that that helps to make some connections between how we know this stuff on my on the microscopic level and how to use it to tell us something about the full properties the things that that we get from
thermodynamics OK so let's start with relating it back to the topic that we've been talking about recently which is an unmarked so in our whole discussion about tomorrow I have mentioned many times but the population differences it is really small when you put yourself on a magnetic field that breaks into generously states in and it induces a population difference between stands that are aligned with the magnetic field verses against the magnetic field and I keep saying that this population differences really small and that's why we need a big 19 of whom I see a stronger signal let's look at what else that depends on and see if we can quantify what energy differences so so far we just know it's
not very big so here is an expression for the actual energy difference so the begins here on members of molecules so this is the ratio number of spins in the state data over a number of spins and state health and it has a pretty simple functional form so we have is it depends on each new survived at its resonant frequencies that the warmer frequency because of our nucleus and then in the denominator there's also means constant and the temperature and so that there is this really simple expression tells us about how it would be more quantitatively what's the population difference between students and the extra Elvis fans are the ones that make up this mechanization vector that's pointing under that we've been talking about manipulating In terms of and so this is why we say that In a normal equilibrium population of spin states most of Boston's aren't giving us thinking we don't have very much in an excess of alpha-beta now we can quantify that and that also gives us maybe another parameter that we can change in order to increase the sensitivity so we know that this depends on the energy difference which of course depends on a lot frequency and there's a dependence on the magnetic field in there so we can increase the magnetic fields we can also lower the temperature Of course the problem with doing that in a practical sense is that to really change the Boltzmann populations in and in more simple we have to make it really really cold we have to get down to you know Millie Kellerman to really make a very big difference in the populations so it's typically not the not the most practical thing to do at least in chemistry experiments 1 thing I just wanna mentioned as as a little aside is so people who have done and more in the context of organic chemistry have you heard of other quiet probe for enhancing sensitivity so that's a that's a cool in instrument that could someone have ends a lot of times people who were using more as a technique see this and they say OK we have acquired probe for enhancing sensitivity and they think that what that's doing is calling on the sample in taking the Boltzmann concert at the the the Boltzmann factor it's not what that's actually doing your sample is still at room temperature the electronics of the probe recalled and so that's just reducing thermal noise in the electronics so that's how acquired the works it has nothing to do with this at all it's just increasing the sensitivity of the instrument by reducing thermal noise from random motions electrons in the electronics themselves OK not so important is that regulators one-dimensional because it's so it's something that comes up OK so here's an expression for harmonization vector In terms of Boltzmann constant with all these factors in there so we have and which is the number of stages of course that important if you want a larger signal you can put into the bigger sample or a more concentrated sample we have this factor gamma squared each part squared by time cyclists 1 eye is they the quantum number of the nucleus and then again we have the spokesman constant and the temperature in the denominator so this is kind of our our 1st view of an example of what these population differences look like and I'm going to serve you go back and forth between talking about fundamentals of your probability distributions and things like that and also showing examples of the kinds of things that we've seen before animal tied the 2 together at the end OK so here's what we're really talking about with a statement looks let's make an analogy to part it previous discussions last quarter in this quarter of single molecules so for talking about a single molecules on the microscopic level the thing that tells us everything we need to know about that is a way function
so as we've seen they're all different kinds of wave functions 1st learned about them in a way that General Chemistry In the context of electronic structure but you know we've learned that there are all kinds of vibrational wave functions the anymore wave functions some of which don't even necessarily look like a function in the traditional sense so much but this is the thing that describes what the properties of the molecule look like ended quantum mechanics and spectroscopy that's the quantity that tells us everything we need to know about the system In statistical mechanics the thing that we're interested in what is called the partition function and the partition function applies to an America sought macroscopic ensemble of molecules and it tells us about how the energy is distributed among different degrees of freedom in the whole system but you can't have a partition function from 1 molecule it's something that's that is an ensemble properties however is tied into what the individual molecules are doing by probability concept OK so the partition function is going as high as going to be what tells us the thermodynamic information about an ensemble of molecules In terms of what's going on with the individual molecules so let's look at what we I mean we
talk about an ensemble so the ensemble is a system of and molecules work and is going to be really large in typical examples of look at and it has some total energy which we can call and he again as you've seen in general chemistry and the remote just out of curiosity how many people taken for Mo Udall physics sir engineering who had quite a few but everybody seemed the basics and general chemistry you remember the kinetic molecular theory of gasses and you know how this relates to the ideal gas we know that if we had an ensemble of molecules even know all identical they don't all have identical Connecticut er velocities or anything like that a particular point in time is always a distribution and so on all kinds of these thermodynamic properties that were interested in like and bookie and the temperature and things like that are all dependent on the distribution amenities and distribution doesn't look like a delta function it has some kind of spread out shape ends yet again from the looking at the kinetic molecular theory gasses remember that if we reduce the temperature of our sample we get I shopper so we have you know more of the molecules in the the minimum energy but that the state where more of the molecules in the maximum likelihood state should be be careful where and also we have fewer that have higher energy workers if we make more energy available to the system the distribution not only shifts to higher energy exports spread out we have more diversity of of states going on really look at that in more detail later another piece of information that is important told us is that collisions are important to redistribution amenities so that's how molecules change their state they run into each other they were on the walls of the container and that's how energy gets real redistributed that's important because 1 of the things that we're look at is on a lot of times and statement we make the assumption that if we take a snapshot of the whole huge ensemble and look at the status of all the molecules and distribution of that that that's equivalent to watching a single molecule over a really long time occupying all these different states and seed collisions to be able to redistribute energy OK so let's talk about our ensemble in a little bit more formal terms so we have our system and molecules and the whole thing has energy E but 1 individual molecules doing so stuff is going to move around it's going to change the time but on average there are and sublime molecules in some state epsilon survive How many states are their total that depends on the specific issue wariness more specifics later on but however many of them there are the total energy it's just going to some of the individual energies of course weighted by how many molecules in each state and the total energy that we have can be partitioned among the various states and that's why this thing is called the partition function and for any particular system the lowest energy state is epsilon not and we generally defined that to me 0 and measure the interstates relative to it because that makes stuff easier to deal with Is it really 0 no it's the 0 . energy in the system but we look at the energy of other states relative to OK so what does this look like for some realistic systems In so here are some of the some proteins that have something different conformational states and if you look at these little free energy diagrams associated with them there are different conformational states the proteins can occupy the little local minima and there barriers between them these are some of it this this is just some examples of different states that have different energies that molecules can occupied and as you can imagine for something like a protein that can have serious consequences for the function of the molecules somebody's conformational states are going to be more active others and if you have everything frozen into a low-energy state that is not so active then the molecules immunity is acted as if you have more energy available to to mix among the states here's a here's another picture about showing for some of these particular minimum conformational states which of these confirmations are active-inactive were partially active so this is a good reason why we might wanna know something about the distribution of of energy that's available the system and how the molecules or partition among the different states it's a lot more general unless we can do a lot of things with them With sisters
,comma Texas just 1 practical example OK so if we have ensemble of molecules the lowest energy saves the 0 . energy which Morgan and define as 0 and then the set of populations in each of the states is the number of for each of the state's summed over all the possible states and of course the number of molecules in all the states when we add them all up past the hole in the total number of molecules in the system so if we take Our system of a bunch of molecules and take a snapshot we get an instantaneous picture of what's going on then there are a and sub-zero molecules in the lowest energy state there are and someone in the use of War etc. again what what defines this it's going to be some sort of Boltzmann distribution based on me relative energies of the States but we haven't we haven't quite gotten back to that yet OK so we need to write down our instantaneous configuration and here this is just notation so this is how we write down you know how many molecules are in each of the states and this is an important thing to be able to do because the probabilities of the states are going to meet here are going to come into account so typical case Is that a lot of that a lot of times the lowest energy state is not necessarily the most populated because the DeGeneres Yemen as well so there in many cases it's not degenerate so there's only 1 way to occupy the lowest energy states were as higher energy states are going to have a lot more general
OK so some configurations are a lot more probable than others so as I was just saying if we have all the molecules in the ground state so we make our example really really really cold and we try to put all the molecules in the ground state that means that are total number of molecules and for all in that state and all of our other states have 0 occupancy so there's only 1 way to do that now let's say that we have 2 molecules in the 1st excited state and the rest in the ground state so now we have and minus 2 molecules in the ground state and then we have to to win the next excited state and 0 for everything else let's look at how to write down the number of ways to achieve this configuration see you can intuitively see where this is going right so there's only 1 way to put everything in the ground state it's like you have a whole bunch of panties in a bunch of boxes you could put men and if you have all the pennies in 1 box there's only 1 way to do that but now if few Oregon promote to to the 1st excited states then yeah you're taking 2 pennies under the 1st boxing putting him in the 2nd 1 you can pick any of the detainees you want so there are more ways to do that and so the general expression for this relates to How many choices you have so when you picked the first one out see take you're your 1st Penny in box you have been choices because you could pick absolutely any 1 of them but then when you go to take the 2nd 1 you only have and minus-1 traces left because we're at a point now out so this is how we are he can write these down so there In sub-zero factorial ways to select that and so In general here's a an expression for the number of distinguishable configurations that we can make with some set of objects that can be put in different bands and again this is completely general for probability of sorting anything in any kind of way near here we're talking about the specifics of putting molecules in different excited states but it's completely general OK so another thing to remember With the systems of large molecules is that there really really large we have no other got his number even more molecules moving around the system is fluctuating randomly all the time and it's almost always going to be found in the more likely configurations and so that's why sticking everything at all in the ground state is really really unlikely so when n is large stuff is almost always in the media the more probable configurations right so we can we can look at the weight of the configuration of how likely it is to have to happen by defining the weight so we talked about this on the previous slide now we're just giving a name so this is that the number of ways that we can achieve a particular configuration and of course that's related how likely and so we can we can make some approximations so we know that yeah get into the approximations a minute a so just stare quick sort of practice exercise if we look at 20 identical objects with 6 different states they can be and they have the following configurations if you have a calculator work this out see see what you get what you will learn is that the the number is surprisingly large so even for you know we only have 20 things which you can imagine that the cluster of 20 molecules is really unrealistically small you know again in the case of molecules were talking about other divers number there's even more things going on but here we only have 20 molecules and we have 6 books column vibrational states so we have softened the ground state and maybe find excited states so you know very small system In the chemical sensors and we get something like I'm quite 3 times tended to eat configurations for this really unrealistically tiny system so this is something to remember when we're talking about petition functions and as you developing an intuitive sense for which states are going to be more populated than others at 1st you know we look at the intensities of spectra like when we talked about rotational spectra and why the intensities that keeps look the way they do you know 1st it's kind of surprising that the ground state isn't the most populated but we start to think about numbers like these and realize that there is only 1 way to get the ground state whereas the higher energy state somewhat more generous then we start to see why the lowest energy states are not most populated OK so now we can now we can start to think about using some approximations so we have this expression for W it turns out taking the natural log of it is useful because we can rearrange some stuff just using the properties of of natural logs and we can write this thing In a little bit different form that enables us to use Sterling's approximation and this is a really nice thing to be able to do because taking factorial is of huge numbers is the difficulty of its computationally intensive when we start to talk about realistic systems and it provides lots of opportunities for for making mistakes so it's useful believes these approximations so Sterling's approximation is just natural log of X Factor oil is approximately equal to excellent X minus X and so we can use that to get an approximate expression for the weight of our
configuration so again it's just simplifies things makes our lives easier and in even the actual systems that were generally talking about there are so many different confirmations and ways to achieve them that this is a find enough approximation OK so now the next thing that we want to do it is try to find what's the dominant configuration so we send the ground state is not the most populated because although it has the lowest energy it's unlikely because there's only 1 way to get it so how do we find the predominant configuration so to do that we're gonna want to maximize the weight so we want maximum likelihood of being in particular confirmation and so we're going to do that by varying and survived a number of molecules in that in that state and and look for it the 1st derivative of Wg 0 and so we can write down some expressions like it so for example we know that if we add up all the total numbers of molecules we have to get capital and which is the total number that we started with and if we add up the numbers of molecules and each state times the energy about that state then we better get the total energy mn we would like to be able to maximize our configuration so unfortunately we can't accept that equal to 0 and all for it that would be nice and convenient but it doesn't work that way because the populations aren't independent so if we take you know if we take care of a molecule from 1 state that means we have to put it into another state To do that they'll have to go somewhere we have a limited number of states for molecules to the and so the and eyes started not independent and we have to worry about that so you know said we just were just going to use variable and variation method to to maximize OK so what we end up with is on this solution where we have fun NO Iowa again is going to have a visa exponential weights and this constant Bader determines the most probable populations and Bader equals 1 over constant times the temperature and I know I need to give you on some practice problems for this material that exploring that these kind of things is something that you're going the he did doing homework seeking see how it works but so when it comes down to is pretty simple and beautiful so we have this this parameter Bolton's constant that comes out of of looking at these kinds of probabilities and it also gives us another way to understand what the temperature means so 1 thing that that comes up you know when people take thermal for the 1st time is give people say Oh entropy is really confusing you know this is something that that you ends up suddenly mysteriously take and on thinks so I think that entropy is actually pretty intuitive yeah that's just talking about the numbers of ways to get different confirmation the thing that's confusing as temperature you have is this year with what does it actually mean and what is it looked like on microscopic level compared to our sort of everyday understanding of temperature so it does fall under this discussion of what's the at the dominant configuration of the states OK so the only confirmations that allowed are ones that are consistent with having constant total energy so you can't have you know configurations that don't yeah you can have configurations that don't conserve energy in the same system so that's another way of saying that if you add up the number of molecules in each state times the energy that state you have to get the total energy and that and the populations are not independent and these are constraints on the system and so then we go to minimize our were going to maximize our weight and get the dominant configuration Kantor said that's 0 because you we said they're not independent and so we need to use variation methods like that for example you can use Lagrange multipliers which means you want to multiply here constrict each constrained by a constant and added To the whole thing and then treat your variables as independent a case of seen Lagrange mold or Lagrange multipliers and this kind of methodology before OK quite a few but not everybody or it'll definitely come up with some practice problems protests operates this is ah so what I'm what I've done here is unjust putting in and that is multiplying a constant times each of these constraints that being you know having add up the total number of that add up the number of molecules each stating that the total number and add up the energy of the molecules in all the states and get the total energy and I'm putting in these constants alpha and beta related to those and then just putting that into my arms original equation that I'm trying to maximize and then I conclude these things independent since my constraints are here and so now given these constraints Alford made it came from the limitations on our system that we know from just physical common sense now that these constraints are in here now my populations are all independent and can do second-set deal and W he called a 0 and so on this
expression Is that equal to 0 when and II has its most probable values and so I get this expression for L & W and notice I change the index on the sunny for differentiating Ellen W. just To avoid confusion and so then if we do that if we differentiated here's what we got and so we can take a derivative of the 1st term and here's what we end up with so far and when we get to the of his 2nd term I know I'm going through this quickly basically at this point I just want to get to the result and you can see essentially how we get there and then you're going to do a little bit of practice with us homework problems and we can go back and talk about it people are confused OK so so I changed the index on the summation so as to not get the differentiation variable and I used with the with or something over and so it is thought we get this situation where if I was not the whole day dmgt will 0 and does equals 1 and so we get this expression in terms of direct delta function and what falls out of the whole thing is with us so we had to take the derivative the 1st from the second-term separately and so we get there -minus Ellen and survived plus 1 plus Ellen and plus 1 and so that this gives us -minus natural log of ends of Iowa and which comes back to these things that we've been looking at that we get as a bowls and distribution OK so now let's put this back in In terms of our constraints so we've got this thing that came out of this Ellen and survival and and then we will we had adding constraints of their constants associated with you have to add up the total number of molecules to begin again and having add up all the energies to the total energy and so that's where we get these parameters if we look at it was also invaded means then we see that traded determines the most probable populations and this comes out in terms of temperature so again that was fast on if you've seen certain Lagrange multipliers before or if you haven't hopefully at least you get an idea work comes from I don't expect everybody to get all the details right away I just kind wonderful once introduced you will have some practice on this for the whole work the really important thing to take away from the right now is the result and that this is where the Boltzmann distribution comes from OK so here are relative populations if you have if you want to know How much of the the molecules present during stay Iverson stated we take the error ratio leave these Boltzmann factors and so what we can see right away and that the take-home message is that the relative populations of 2 states falls off exponentially with the energy difference and so for example we can go back to looking at rotational states he has we talked about in the early part of the quarter with rotational spectroscopy and actually come up with a quantitative expression for the relative populations of the J. equals 1 and equals 0 rotational states evasive of HCL at 25 degrees and so this is going to be based on In a recent falls off exponentially with the energy differences but it's also going to be based on there generosity so further the on the ground state there's no generously rate cuts there's only 1 way to do that for urging equals 1 we have 3 different values events Sunday we have minus 1 0 and plus 1 and so it's too generously is free so and there are 3 times as many ways to get that 1st excited state as the ground state economic cost more and there and so again here's what year the spectrum looks like being so weird know that the ground state the most populated the energy of a level with quantum number J is HC times rotational constant times Jason J. plus 1 so the difference between these 2 states is going to be to H C indeed and we can look at the value of the rotational concentrates Cl and so that 298 Calvin we have to make sure With these things and Calvin so that our units workout we get about 270 wave numbers for this factor Katie or C and so then To get our relative populations we have to stick this factor generously in front of it so we have 3 over 1 4 the difference and generous and then it falls off exponentially as the energy difference with come between the 2 states and so the relative populations here are described by this quantity and we did the factor that we have about 2 . 7 times more intensity In his 1st excited state on the ground state which you know it's not exactly that tracks pretty recently with the generously so we see that we have 2 factors here when determining the relative populations of states 1 as the energy difference and that's important but that generously In ways he is even more important because we have a lot of fun of molecules and the probable confirmations are much more likely to be occupied OK
so again you know where to start to get into the suffering to talk about it more renounces practice problems the take-home message so far is you should be able to look at differences in states you should know how the relative populations depend on the generosity and also the and the differences between them if you can't reproduce his durations right now that's alright wingers have some more opportunities to practice on the main thing is just knowing the results all regret therefore today and I will see next time


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