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Lecture 25. Partition Functions.
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Title  Lecture 25. Partition Functions. 
Title of Series  Chem 131B: Molecular Structure & Statistical Mechanics 
Part Number  25 
Number of Parts  26 
Author 
Martin, Rachel

Contributors 
Johns, Gianmarc (Teaching Assistant Chemistry)

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 noncommercial 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  10.5446/18933 
Publisher  University of California Irvine (UCI) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Chemistry 
Abstract  UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 25. Molecular Structure & Statistical Mechanics  Partition Functions  Part 3. 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: 0:01:02 Simulation 0:06:38 Langrange Multipliers: Motivation 0:16:30 Multiple Constraints 0:38:36 Curie's Law 
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00:07
that will be talking about was that NEC is that if you had a group assistance they can occupy different states the 1 that you see the most of what's going to dominate is the 1 that can be achieved in the most number of ways which intuitively sounds reasonable and further always said that if you have a really large system like say other gardeners number of molecules Neuralia see that dominate even more so it's a little abstract so I cooked up the full calculation here and what the calculation does is what we're doing is simple to state system there's 2 states has been tested by sort .period but instead of just looking at 1 point a bunch of times will say we have a box of quarters we shake the box of quotas and would take a peek inside and count the number there I had to tell us you we report the number as a percentage of fat right to expect 50 % heads so
01:03
here a simulation our own so right now I have it set up so that there are 4 quarters in the box and we shake it 4 times so if we look at the results here and we've got it looks like the ones we got to know heads along the way zeropercent heads with a 25 percent heads so that happen once once we got 1 had once we got 50 50 and once we got all tales so we're not seeing what we expect to see like this if you think about the number of ways you can get these things there's only 1 way to get all heads with only 1 way to get all tales right there's for ways to get 1 quarter on heads and for ways to get 1 entails but there and but for a 50 50 Tutu heads to tells the 6 ways to do that you should see the most of that would that so the problem right now and by the way this computers fellow program here it randomly selects between 0 and 1 for each time this isn't just a graphic were actually you doing the experiment and so for the full hour for crops .period flips works in the distribution we expect right now so we call this we say this isn't under sampled system but so we only up 4 times right so what shaking up a bunch more times so let's say we get some programs student myself to shake this Box 3 thousand times and count the number recorded as right so we shake this think 3 thousand times now this be distribution silicon that this makes a lot more sense right with the shape it's calcium right we're seeing a gasoline distribution so now he said that after Intel 3 thousand that is enough time to shake the spot to want to the point where we see we expect to see as you look at these peaks relative to each other but we see the most of the 50 50 Split but there's still a good number that you have 1 head 1 tale on your side that because rose for ways to get that 6 ways to get a 50 50 mix so it makes sense of the pretty close doesn't have that much of a lead 6 before that that you know that's going to dominate and also we still have a pretty fair showing for all tales of all heads right so we're in this experiment 3 thousand times the members small storage we have a bill looks like about 200 times we're in the experiment we got all tales and another to have handed all if you think about that official boxes for quarters got all heads you would be that freaked out by the middle of weird thought that we had a lot for charisma so I'm still a splendid this little bit so will leave the number shakes at 3 thousand so that we know we're sampling system well effort for all of these are times but let's not September quarter solicitors heaquarters in the box by played in they still see gasoline distribution by and it looks like we actually still got a few they're all heads and all tales because that's a little more weird orders on have details but we did shake spent 3 thousand times I but if you notice it does start to tail off by the laughter maybe around 10 % heads and 90 per cent heads rights is starting to get our gas is getting more and more Pete that's a good stride for stride 22 so nicely in here getting more Peter even even more notes telling AFP by around 30 and 70 per cent was shot 220 to decide keeping this to weddings convenient again now look 50 cent heads and tails is really starting to dominate now To get 222 quarters botched shaken up 3 thousand times so now know that got Scottsville statistical significance here and now it's still 1 of by like about 45 and 55 per cent heads I sir please check 2006 that dozens of those abilities on order of 2000 OK so now and this event started getting really simple now rather than settling below 48 or 52 . 4 were never getting that and and this thing is totally random like it's not so funny math going on it's just selecting between 0 and 1 of a bunch of time and as you can see where not even seeing those less likely configurations so there's so now like the number of waging a 50 per cent is on the order of thousands of workers is 1 way to get all heads so I can see that and if you think about that if you show a boxer 2014 got all headed totally freak out ahead of time but so just enacted that I set this thing to go up to 3 thousand so it goes a little more so if we're talking about ensemble of quarters of the at 3 thousand sounded a lot like that a lot of quotas but talk about molecules that's like nothing because of Agadez number we have had 20 0 stood at 3 thousand to get on the order of a mole so if you dominates this much for a twostate system which is 3 thousand cause you can imagine if you have a whole mold molecules have the most configuration has the most number of ways will really start to dominate so that's why did this demonstration is our identity an incredulous person if you tell me this is going down and went well really it's still could happen but sure enough if he actually won the calculation you really don't see those those unlikely configurations happy so hopefully you believe in that
06:33
that sort of thing now look good to the
06:36
regularly scheduled programming
06:39
so .period about which sold this is thus leaving the Lagrange multipliers so the strategy with Lagrange multipliers is that we want to maximize the number of ways like we just prove politically for the experiment that the configuration which shows the highest number of ways to get something is 1 contaminated so if you so that the functional and maximize right so that the formula for figuring out number of ways of doing something is right there factorial isn't it the thing is factorial is a really difficult to deal with and mathematically so there's a check would take a lot of it and the reason we take a lot of it the logarithmic functions something ,comma tirelessly increasing seen the plot along functionary does that it doesn't suddenly drop a star going crazy at the right it just keeps steadily increasing increasing more slowly but nonetheless always increase therefore if you take the loss of some function W wherever that what function has a maximum it's going to be the same as the maximum for a simple equation with the factorial there so that's why we're allowed to take a lot of it because the maximum in the same place for both function and the reason we would take a lot of there's 1 more step will trick always talked about sterling's approximation and what Sterling's approximation allows us to do is we can get rid of those factorial because like this of the really difficult to deal with mathematically so that's why we're actually met maximizing the lot of W and not just studying is because it allows us this trick to get rid of factorial on hand and the maximum we find is going to be the same maximum that we would have found had we done it the hard way so it's it's a nice little to do that and this is how the this is not a partition function gets to ride and so when looking for a global maximum we have constraints rightly so With the coin flip there was no like energetic and constrained type thing but for molecules weaved were posing to constraints the wonder is that of the total number of molecules never changes and the other is that a lot the total energy the system never changes so because of that if we add up the number in each state we should get the same number every time I if it is you have you know 3 more in 1 state than you had before those 3 have become out of another 1 right just producing molecules out of thin and say we will lose any models the same thing happens for the energy of the Arab all their energies on if he had the energy of each individual molecules the total needs to remain the same alltime so are constricts assistance were at maximizing a function with 2 constraints the best way to do it is with the method of of Lagrange multipliers so here's how we that we do not want was so the basic tricked out with this if you want to maximize the simple onedimensional function you take a derivative of it and figure out where that function has 0 derivative right that's where to their maximum and it's an extreme on Soviet however of a function of multiple variables then you're going to do the same approach for you take a partial derivative with respect to all the variables and add them up and there should be 0 and that's where Maxim I'm but nowhere but that's that's just tell before global maximum more interested in the Maxim with respect to some constraints so the way you do that but these 3 equations at the bottom of the just for arbitrary variables x 1 to win the but is the way I prefer to think about it I like to move the term with the land and from there to the other side and think of it as the derivative of the function is equal to limit the time of derivative of the constraints and that just resonates more with me but it's the exact same thing on the rationale behind this is that it has to do I will continue to much because we just want use we don't need approval grant buyers works will that the mathematicians but I basically if you had like a normal vector coming off of both surfaces the point of that that satisfies these conditions is when the parallel to each other so collectors Apollo to each other but not necessarily the same length that the land is what you multiply the 1 by a factor to making the same line so in and then that's where these derivative derivatives are equivalent to the equivalent in likened the angle but to make them the same length as the multiplier and that's what makes them altogether so the the lenders ,comma ties everything together and and we can actually use it added greatly to help solve these things so you make a lot more sense sensitive example so 1st example let's say we won't make a play area for popping but we only have 40 we have 40 feet defense material so I'm we want to give the popular most Member space to play bikers were nice people so are we want the most area with a set perimeter so we can only we can only do so much with the primitive and another thing is that we've decided we're making this thing rectangular by the way that we define the area right so the area is just like the ones with the exams wine and the perimeter is locked in at 40 but to get the primary just add the 2 EC unit to exercise into seismic program so we have a functional will maximize with respect to a constraint on so we apply method of Lagrange multipliers so the 1st thing we want to do is take all this partial derivatives so take a partial derivatives of function and partial derivatives of the constraints and an ICC functions work with so it would do all this and I now remember what I said that of the derivative the function with respect to stop some variable is equal to land at times a derivative of the constraints with respect to the same variables so you can have Y equals to end in exit was 2 and so good that there so we have our wine accident in I X Maxson why Mexico to and so that we can plug this into the constraint you put that in the constraint to invariable left his lenders can solve Philander so when that we figure out is 5 and we conclude that back into the 2 win and we see that exit life equals 10 feet so the biggest area we can get to the the the dog with 40 the defense material is a 10 by 10 square which apply could've you know problematic gas that without doing with large multipliers but don't overlook this example when you're studying because it's nice to look at something really simple and just focus on the technique so well didn't don't dismiss the sort of thing that can be really useful study tools and say 10 by 10 square factory into a little more complicated so now we have something called a parabola Lloyd and that's that yellow surface the tournament UCI colors with salubrious style but somehow the probability is is that that don't look and think and then that we have a constraint of this size the plane the Blue Plains so the idea we would we would find the highest point on this function so that effort that's why the yellow function that constraint the 1 sex is the Blue plane so if we wanted them the just the global maximum of that that probably we just walk to the top of the mountain where they're right but the problem at hand is that we have to walk in such a way that we're touching both the a plane in the probable Lloyd and then find highest point right so looks like it's somewhere somewhere off to the left but so just go looking at it that we note can be somewhere around America let sellers
14:43
amassed so the 1st thing we want to do is take the 2 derivatives of the function and so we have because there with the chain rule that slated 2 out from the official at the channel and that saw that for X and Y will then now we have the global maximum right so the top of the mountain is at the . 3 6 but that's not what we want right we want that the top of a mountain with the assumption that we have to be touching the plane as well so that means we need to take a derivatives of the constraints as well so we take the constraints it's just negative 1 in 1 7 remember it's going to be the derivative of the function equals land has driven the constraint so we set that up and as you can see from this other both now expressed in terms of when sources they're both in terms of land or we can take the 1 on the left multiplied by negative 1 and then the both the quarter Lander and industry Palander the equal to each other right so we can eliminate Linda express this way similar looking at that we can rearrange that and say that Expos Y equals 9 right so we know Expos why has steeple 9 and that gets us almost the whole way there but if we look back at a constraint we realize that why my sex equals 0 the all Y equals acts so if Y equals X and Y plus X equals 9 then they both happen before . 5 around the room good so far the fucking going fast that they have not yet given yes yeah very good so I know what's next all right so and so those 2 examples were both the method of Lagrange multipliers given that you have 1 function maximize 1 constraint on but for a partition function room we have 2 constraints you have the total number has remained the same the total energy has remained the same so you can do the level of multipliers with as many constraints as you want but before you add the harder it gets to do and it doesn't really lend itself well to lecture because it's 1 of those derivations it takes so long by and you forget we're doing in the beginning the thing but works well for reading and is actually really did want any your chapter that further information section of the chapter on the Bolton distribution so check that out it's it's a good read but I like it and I hope so but this is this is the process he would do what I can actually do this is what the process and it's in the book on where we have all these constraints and and you see that you subtract them will unite Alekperov said people but we are subtracting right but in this case it is subjected to constraints and each 1 has its own Lagrange multipliers labor land and abated and that made constant actually does end up being the very constant that arm the familiar with striker won over Katie so that's all say that actually driving the partition function but want to understand how where it comes from so that you know that you to the miners they times the energy that's that's it all comes out of this I sort hope OPM is what ever is the largest normal vector Boeing in the bottom left that is the function the extremes of summarily farmers put out after L but he added don't think too hard about the thing say about the power of actresses were mostly interested in applying this thing but I just have a look at myself as I was curious and and a little bit about and that's what it boils down to but keep the focus on applying for special for for this class but I've got to carries abided if only to have back at about so Of that we've got to have a serious an example of the plots of the partition functions so if we're interested in knowing the average speed of light I have been adamant these particular noble gasses but if you planted them they would look like this and if you notice that the helium 1 is a lot faster than everyone else right and if you notice that the the correlation here's the lighter it is the faster it's gone right the reason for this is that temperature is tied to the kinetic energy of these gas molecules flying around in 3 dimensions right so if the kinetic energy is with relates to temperature but kinetic energies to pieces it's man onehalf and the square writes of his mask and lost both tied into it so basically if these are it used for gasses are all the same temperature than all but the same kinetic energy so it's all the same Connecticut Energy 1 has less mass and it's going to go faster rate is onehalf and the square so that's why you see it's just kind of like qualitatively so let's talk about finding the average kinetic energy of the gas molecule so there's a equation for Connecticut Energy the Peachtree headed on but down here we have a distribution here and this distribution of represents the probability of finding a molecule in a particular energetic state so it's a population right so basically you have to stay here interested in over all the possible states and I so it's kind of like taking a fractured right it doesn't look immediately like a fraction of an integral on the bottom line but if you remember an integral is a way of approximating a summation rights if you have if you have a summation and you're adding all these things together and integral is is a way of approximating that right and you can think of you know have area the convert under the curve and you divide all opened about 2 little boxes and defined area of each box and add them up that this nation right so the functional give you the height of the box C have the function times that the change in the next corner which is your DX right so it's right there when you when you take an integral you're you're really just taking a bunch of areas and adding together such immediately look like we had the son of a bunch of different possibilities on the bottom and 1 possible only on the topic that's what this is but we can simplify little further so that function on the bottom there's just been a goal of gasoline which is a very well known integral but so there is the role of 50 CDA there the your constant so in this case are constant is just 1 half but they and right so then that's that's how we go from the equation left the 1 on the right is that we just performed an
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integration and it's a constant Sonakul out front so this is gonna represent a population of more concise way that analysts say so interested in average kinetic energy so you think out we need average velocity to get there because we know the masses can be a constant of whatever we're dealing with but we need the average velocity to figure out where average kinetic energies but if you take the average velocity 1 dimension and were just in the extraction you get equal probability of going right and left right so whatever distributions of speedy have going right you can have that same distribution going left said the average of all this can work out 0 I superior that so instead really take off from the start of the average velocity with the average of the square the velocity which is called the variance and as it should be familiar with variance in if you're not beautifully familiar with the screwed variants which is on standard deviation I think people against the government after this attacks so so working for the expectation value of the velocity squared now it's important notice about this integral here is you've seen this type of treatment before we need to acquire mechanics but because ,comma mechanics you wave function when you multiply by its Kunkel complex conjugate gives you a probability distribution right that's that's what away function is is we look apply like by complex conjugate it you the probability distribution of where the article could be ready so for interested the average position we took up position operator which in 1 dimension is just acts and he say which that in between the wave function and its complex country right so he essentially multiplying the thing were interested in by the probability density were doing the exact same thing here and it's just that we start with a probability density so we don't need to multiply anything by complex conjugate with the exact same process and integrated overall space that you're interested in so that's what we're doing and it's essentially a weighted average rate so when you add still going back to Holland in a growth is really the limit of the sum it's kind of the same thing in that this variable we multiplied by whatever that function is at that point and then you keep adding together so we have is a weighted average will do it sample later were we don't need an integral because there's a small enough amount of states that that we can actually just take a standard weighted average so keep that in mind OK so that's why I was so that the integral where we do and Smeltz taking so if you notice of the squared isn't even functions and think a gas the new functions it even times even as even so trip with even functions is you don't need to integrate the entire thing right you can integrate only to the right of 0 and then just double right has whatever is on the right side the same on the left and property of even functions and also what we're talking about that let's think about remember I said with the velocity just qualitatively Mckeighan added Flossie couldn't go right or left it will understand mathematically if we just part of the instead of the square these and other functions yet at times even if you take that and really 0 so the math works out as well it matters are intuition and in that year average velocity is going to be 0 which is 1 reason why things like variances standard deviations are so important because sometimes you can get more information from that and can from an average averages just kind of go to things it because it's easier to understand and so if we saw this in but also well known integral but it simplifies down to 1 over be on so the average velocity only depends on 1 over beta and so we're out to get the average kinetic energy rights that's what happened the squared and if we had the average value of any of the square meters put that right so the average the ad bridge of the square of velocity is the size have Connecticut Energy works out to be 1 onehalf Katie I'd say that's it that's nice simple just works up onehalf Katie but since not too many experiments are in that infinitely long too of infinitely small diameter ranks as so we have we had molecules going 1 dimension out we need worry about the rest of the 2 dimensions right but this is an ideal gas so we can work on the assumption that the velocity in the x y and z are not correlated with each other in anyway so that's true it we can treat as a linear system meaning we just add a month so if you're interested near average kinetic energy in all 3 dimensions you just added added up 4 x y and z so instead of onehalf Katie history has Katie was multiplied by to account for all 3 dimensions such I figure the average of velocity of an ideal gas molecule but again so but as as we showed the average velocity the average speed is due to come the harder it gets right so this is just this just shows a few Balton distributions for different temperatures and you see that as a gets hotter in 2003 Southeast average is near the center of the distribution is much further to the right and it's more spread out and it spreads like that because you know more more states are energetically accessible by right so another example is a cool so for this 1 lets say um where exposing some power magnetic substance to a magnetic field right arm so you have all these magnetic dipole the complained in any way right when you put a magnetic field on they try to align with the magnetic field so then there's a statement heeding magnet makes its makes it lose its magnetic position and why so everything about the situation of like a permanent magnet like some refrigerator that's probably fairminded magnetic but will go there anyway but if the if you've got a field that's trying to align all the all the dappled in 1 direction then he it's just ,comma somebody walk around to stick him all over the place so the ideas he soso Richard Fineman 1 of my favorite that scientists in history is to talk about how well Adams jiggle rights if you have any substance the molecules the ads all jiggling around in the harder it gets the more the jiggling and so the same thing happens in in in a substance that so far there has a magnetic titles with the dappled kicked around as the atoms get kicked around so the harder it is the more you can these things around randomly so you've got to see the things you've got a magnetic field that's trying to align the dog calls it 1 way and then you've got he just kicking and kicking around randomly so that's why you can make a magnet news he loses magnet isolation by heating it up because you are answer with
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the with look of permanent magnet on the fridge What is exposed to field long enough and alQaida just get stuck that way but if he heated up you newcomer giving their duty to go around all over the place and then the randomly 1 yes I you will have to be strong that's a good question I don't know about this I found to be honest I don't know I'm not going there giant taken into some mature the look that up on blood I know you know with the superconductors that they can have electromagnet often have to run really cold but I don't know about just calling permanent magnet and that increases that the the field I never heard of this but that that doesn't mean and thus exist so we're approximate the system as a as a yearly collect a twostate system right so we've got these die they .period any which way but Wall will just say and it's a good approximation of what to say that all these titles are either pointed with the fields against the field right so benefits with the field will call parallel efforts against the field will climb into crowds so which 1 he thinks higher energy parallel apparent get it apparel is 1 thing about potential energy like Soviet potential energy is pushing on it wants to pop over right but once it's part of its effort to do anything else right itself like dropping something potential right the fury fell on the floor careful from self since then but so let's see here so the energy of for the ground state recall that negative you not be nice and then if it's antiparallel then it's going to be the same unit but positive so therefore the energy difference is going to be to not be enough I'm so if we want so there's a major difference to which I find the partition function and remembrance of the partition function was in the previous example that were on the bottom we had that integral of all the possibilities before this particular system is only 2 possibilities so we don't need an integral to do it right only need is that is awaited summer Of these these different states and and the states that that we add up and if you take 1 thing at a lecture today don't forget this it's gonna be the did generously that state times each to the minus beta times the energy so that's how we got 1 and even minus to Baden you not being alive is because I am so the ground state we always call 0 because of we can establish energy scale anyway would like and then later if you need to correct for 0 .period energy you can always add it back in sales call the ground state 0 energy so we have 0 energy and ended the genocide both of the states is only 1 of each state rights of the digestive systems 1 for both of these so you can have the the 0 is our ground states that's what we have 1 and then you go ahead into the minus Each into the miners better time 0 seems there the unit have 1 times to the minus Bader times that energy but that the manager we've raised which is up to you not be not so this is our partition function now from the partition function we can figure out the population of each state summarize said it's a fraction right you have that you have so many you have that the 1 particular state you're interested in over all the possible states silly had so for the ground state the 1st term was just 1 c have won of acute gives you the population the ground state and opportunities stood in the population in that the excited state that's going to be the 2nd term so into the minus 2 beta you not be and then you divide that by armed the partition function so those are populations so for interested In the average magnetic moment the way we're going to do that it's going to be a weighted average again so the weighted average is going to be the magnetic moment times its population in that particular state and so this is a weighted average of the had discussions since you've seen this with some of the absolute number of people in the room all different ages and what the average age you'd add up the number of people there than silica 5 of the 12 year olds and 20 20 year olds right you have 5 times the 12 and then you add that to the government 20 the Herald stands at 20 and then divided by the total number of people that's a weighted average of the same thing here so if you've got yeah I know so for the 1st 1 for the magnetic moment of pointing with its that positive you not use just multiplied by the population that's in that state which would be the peanut butter and then at the state .period against it were kind that negative New not then you multiply that by the population that states that the P 1 and that all the way to the right we've just expanded that so if you wanna get fancy we can realize that this is the definition of the hyperbolic tangent and and you can write it that way but but that's not as productive instead were given that we're gonna make approximation here but an approximation is OK but they have to be justifiable rights so where going to justify this it is let's say that we have a weak magnetic fields for temperature right so rare dominating forces we have the magnetic field Tiger won the lineup evenly and then you've got the heat has just taken everything all over the place randomly so I was saying that he is winning basically right so and you can do that with just a really weak magnetic field or you can do it with really high heat right away so the important thing is that the ratio is much less than 1 so if this is much less than 1 then I this strategy were is doing tell series centered at 0 so if you remember what Taylor series the idea is that you tell a series of centered somewhere right and whatever sounded the closer you stated that point the more accurate detail a series is right and if you tell a series of interviews stable across the net .period Senior Challenge series approximation can be pretty good with just a few turn so that's that's coming approach that we're taking here is that we can spend Stuart Taylor series and only has a few of the terms because we're so close to 0 and was centered at 0 so article that said that the McLaren series a 4 7 0 so but looking at this 1st expressed in parentheses around this came from the numerator that faction of the this faction the bottom left here times the Nunavut that's what we're working
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with finding approximation for so on the Taylor series forever for even the ax is just 1 plus X was X squared over to was Cubillas over 6 races it follows that pattern but we're just going to take the 1st 2 terms in college good enough writers were really close to 0 and so we're going to do that and so 1st 2 terms this measure we take that exponential function right there it would just be 1 the mind is so we'll call the entire thing that to minus 2 beta not being not that entire thing contacts so they are not the 1st teachers would be 1 plus acts on our case 1 minus 2 major new not been so their our 1st 2 terms in the series and we're stop at 2 terms so that the distribute that minus sign in there then you get 1 ones the ones go away and you just left with 2 beta you not be right set approximation the numerator nothing that denominator will the same 10 cents thing take the 1st 2 terms so you can have 1 minus 2 patently not be not this time the ones positive to get 1 plus 1 equals 2 and then you subtract the 2 data not be enough I'm so now we can make even 1 more approximation because up top we said that made at times New not not riverbed is just 1 of Katie is really small it's much much less than 1 so that's true that was subtracted really tiny number from 2 well really care at 1 . 9 9 9 9 9 something or if it's too late which is considered to opposites it's so close to 2 soaring approximated being and then if you write that out with that with this faction this is what you get to the expectation value for the magnetic moment for this particular substance in wheat field at high temperature is just that you not squared Be not over take 4 times they have real estate so this is not a securities law this is known as Terri's Law and has a plot of it and that you know at high temperature populations tend equalized in this country confirms what we were thinking about just kind of a qualitatively rates its temperatures just kicking stifled tackles all over the place in a magnetic field is too weak to try and keep a maligned inaccuracy much of the net magnetic moment since we have the questions on the register blown through this stuff from unwanted fast and Israeli only back up will we get to the end with time left plenty of someone rehash of from the Northeast to attend
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talks fast so I don't know I'm am
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so no what we're doing example ,comma near and dear to my heart protein folding example right so we talk about our home so we talked about like toy problems silicon ,comma character toy problem would will be like a particle in a boxer harmonic oscillator rate should now about for protein folding right so proteins the size of the body on the ribosome as they come off the riders only fold also sorts of different ways because you know that the different amino acid attracted to each other in different ways and so they can fold up into the shape and really consistently Andes and that's gives their America functions is the structure so it's really fascinating area of research so you can call it that this is the talk of a protein folding so we've got 6 beads like a string 6 beads organizers these beans are all attracted to each other so sailor general's forces beads all attracted to each other but and they can then ran different ways and and possibly stick to each other so idyllic for partition function for this thing and forgot that populations of these different states so as as an approximation let's say that these things can only snap into place at right angles right so only bending this thing can do is at right angles is 21 different ways of you can configure and I think Dr. Martin for drawing every single 1 of the things that must have taken forever so you have all these states here and so this 21 different ways you can bend this thing at right angles without having anything touch if you allowed 2 of these beads to touch those 11 ways to do it and if you allow To context to happen this for ways do it right so the user are microstate so I think this fall is due at 11 ways to do it 21 ways to do it when we're talking about the start the the generously good figure so we're talking about the DeGeneres CDs different states because energetically reinstated the same by the the same state energetically Whirley showing 3 different energetic states but the 1st so that they all have different agendas is more than 1 way to do it so now let's try and figure out which is which 1 do you think is the highest energy state we think is the lowest energy state however the bottom although the 1 with the Formica states you think that's the highest lowest energy well it's good and again we're thinking of potential energy thing here these things are attracted to each other than collapsed together if we look at the 1 but the former states it's already collapsed radiate collapsing further than that and so that's going to be a problem that can be like a ground state and then suddenly 1 contact would be the 1st excited state issued call and then the highest energy state is the 1 with the 21 microstates that will do that so let's try
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and find a partition function for this thing up as a 1st of all this is the energy we're talking about right the energetically they're all the same state is is there's only 3 different energetic states with they habit the did to generously and were looking at it again ,comma linear relationship between energy and number of contacts so the last context there are the higher energy you are so that's why we just come with a linear relationship here if you look at the coefficient funny not because 0 1 2 so but there's a partition function so looking at the ground status for different ways to make their ground state so the genesis for and where was called about energy level 0 so then you have the 2 0 which is 1 services for writers to generously 4 times in a 0 is 1 for the 2nd 1 is 11 ways to do it I and our function is the energy for that particular state is he not so it's easy to the minus the nite Tasmania and the last 1 is 21 ways to do it and energy but determined was that to unite sounds can be 21 times the to the minors how to unite times better all right so members for its petition functions of these smaller systems that we can actually write out a return partition function that typically things it too complicated for this attitude to get them although we can have this toy problems like we do ,comma mechanics to look at that and you can have a partition function of so it was with that we determined the average kinetic energy of the gas molecules and talk about translational energy right which is a really good way to describe energy those atomic gas right because Adams a considered points is as far as the mechanics of for the most part so I'm all your interest in his translational energy but you can have a gas like beginning can rotate rights then you have rotational partition functions as well so there's all kinds is British branches and come up with that the
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pocket into college and so on equals 1 over Katie I prefer always used beta and Dr. Mott was saying this that makes experimentalist tend to favor 1 over Katie because temperatures on can actually you know to temperatures of more natural you know if you're running experiments and theory people tend to favor beta Smith dash available arm so the ad and I think I just I have cultivated every time it's equivalent any really questions as we have almost a little over 5 minutes if there's anything you like me to do again but not in color today it's up to you france and the place if