Lecture 05. The Equipartition Theorum.
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Lecture 05. The Equipartition Theorum.

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5

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27

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CC Attribution  ShareAlike 3.0 Unported:
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2012

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English

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Abstract 
UCI Chem 131C Thermodynamics and Chemical Dynamics (Spring 2012) Lec 05. Thermodynamics and Chemical Dynamics  The Equipartition Theorum  Instructor: Reginald Penner, Ph.D. Description: In Chemistry 131C, students will study how to calculate macroscopic chemical properties of systems. This course will build on the microscopic understanding (Chemical Physics) to reinforce and expand your understanding of the basic thermochemistry concepts from General Chemistry (Physical Chemistry.) We then go on to study how chemical reaction rates are measured and calculated from molecular properties. Topics covered include: Energy, entropy, and the thermodynamic potentials; Chemical equilibrium; and Chemical kinetics. Index of Topics: 0:02:34 In Real Molecules... 0:05:51 Constant Volume Heat Capacity 0:11:37 The Equipartition Theorem 0:39:40 The Translational Energy of a Classical Gas Molecules

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OK so we have a quiz on Friday you
00:10
know and it's just going to be exactly the way it was last Friday the skin transiting to be up there at the front grab when you come in don't sit next to
00:22
anybody we had plenty of room last week right for some reason they gave us this giant of electoral which is really nice what underwrite that I have written it yet but all right at this afternoon when I do that I'm going look of what I've been telling you for the last 3 lectures on related discussion guide to which is posted now on the lectures page of our website and we look at the assigned homework and I'm an abrupt something with 5 questions on it the 1st 2 questions there to be pretty easy the last 3 will be a little bit harder OK just like it was last week it's open book open notes opened anything takes sapped we're not going allout tablet computers
01:12
and regular computers this time OK you can still bring your tablet computer in your laptop if you want but and you can use it after the quiz is over but you can't use it during the is kind a calculator that you want OK employees do not
01:38
use this as a reason to print out the lecture notes because it's very destructive or I
01:49
write anything down you want any formulas put Postit notes in your textbook OK any questions on quiz too you know I tried include all this
02:04
other stuff I tend to merely look at what's in the lectures and try to make sure that I haven't asked you any questions on the quiz that I haven't answered in the lecture from attended to you may also want to look
02:16
at Queen's 2 from last year which I posted on the announcement page because I'm lazy and I'm likely to steal problems of queries too it's just the way I am OK so today we can talk about
02:39
a subject that's hardly discussed
02:41
at all in your book but I think it's really cool and it really helped us obtain an intuitive understanding of how the statistical mechanics stuff right
02:54
real molecules the picture is considerably more complicated than that this is where we started with the statistical mechanics and evenly spaced ladder of states such as that which we would obtained if we had a harmonic oscillator but real don't even vibrate this way even tho we use the harmonic oscillator to approximate vibrations in real molecules but there is an ominous city in real molecules are translation comes pretty close to the harmonic oscillator picture but even in this case that states are not evenly spaced even know I've drawn them evenly spaced years reality they're
03:32
not I'll show you that later on
03:35
and these translational space states are really close together there's many of them between each rotational state of the molecule here's the rotational OK and there's many of these translational states the sandwiched in between each 1 of these rotational states thousands in most cases hundreds of thousands minus rotational states in there many of them In between each 1 of these vibrational states aren't they right so there's a lot of
04:09
complexity in real molecules there really haven't started to talk about it's been discussed in your book and you've been doing some homework problems that relate to this issue but we
04:21
really haven't talked about it in class were going to be lucky in that we can take this problem apart we can treat each of these energetic manifolds write each 1 of these things is what I'm calling the manifold you can treat them separately at the end of the day we can just take the partition function for translation and multiplied by the partition function for rotation we can calculate each 1 of these partition function separately multiplying together to give total partition function for the whole molecule and the reason this
04:53
works is because these degrees of freedom or socalled weakly coupled they don't talk to each other OK multiplying together we
05:05
calculated separately really get the total partition function of looking for a very different kind of molecule that we care about now we're going to start talking about that later on today but in this lecture were the talk about a shortcut to calculating not disguise but something related to him the heat capacity in the internal the shortcut and it's called the equity partition fear and I think this is discussed right at the end of Chapter 13 but it's discussed very briefly not in enough detail to really understand and then at the end of this
05:44
lecture will start to talk about the
05:46
translational partition function "quotation mark so when your but talks about the EC will petition the concentrates on the internal energy of particularly molecule or the internal energy of a small molecules these 2 variables are difficult to measure directly in the laboratory in other words if you're an experimental physical chemist and you go on to the laboratory to actually
06:13
measure the absolute value of the internal energy of a single molecule or mold molecules are any number of molecules that turns out to be a hard thing to but it's a lot easier to
06:25
measure that the capacity of some volume of molecules to absorb the the heat capacity that Tunisia quantity to measure miracle occurs among the capacity is just a lot of energy material common can store per unit temperature so constant volume he capacity is just a partial derivative of the total energy the system with temperature at constant volume if we can equally well the defined in terms of the average internal energy of a particular molecule OK so hypothetically if the average internal energy of a particular molecule turned out to be Katie over to them that the capacity is just the derivative of Katie over 2 with respect team such a scale all right so if we now the internal energy we can get the capacity and viceversa
07:28
now in I haven't told you why this school year but bear with me this is a plot of the heat capacity as a function of temperature for some generic molecule but what we want appreciate is that as the temperature increases along this axis from left to right the capacity of a molecule the store energy increases in stepwise fashion like this why is that the answer is that at means really low temperatures only the translational states that are available to a molecule are occupied in other words is the only way a molecule can store energy is by
08:12
changing its velocity but it can store more energy by speeding up once he gets
08:21
to a temperature where the thermal energy available at that temperature equals the energy between rotational states of the molecule now the molecule can start to rotate as well as translate right so it's got a new manifold that it can access for storing energy can store energy is translation but also it can store energy in terms of its rotational states because now it is reached the threshold here working the thermal energy available to it is high enough so that can start to no remember these rotational states are much further apart In molecules than the translational states much further apart finally if the temperature is even higher you get to the point where you're exciting many rotations but suddenly you start to have just the threshold of energy you need this excites some vibrations of the molecule now you access the vibrational manifold and you got 3 ways to store energy translation rotation and now vibration OK so let me point out a couple things about this diagram 1st so it's not the slightest bit confusing there's no units here all right but the units for this constant here volume you can best the numbered 7 have K 5 as K 3 have for talking about a single molecule the units are in terms of case before talking about a mall molecules there in units of all are because 1 obbligato number timescales part isn't OK so even though doesn't say what the unit saga that's 3 Absa before talking about more these guys these temperatures here are the characteristic temperatures in your book called status of the invaders of artists the characteristic rotational temperature in the characteristic vibrational temperature the rotational temperature is just b written out in terms of jewels Fred remembers the rotational content for the molecule write out in terms of jewels are right wave numbers and do a conversion to jewels that's what this is always confusing to me divide by KNU units of temperature right because case jewels Calvin but events in jewels somebody get this in terms of Calvin that's what that is right there but likewise that the energy between vibrational states h new if I write that in terms of jewels and divide by K I get that temperature right there so that's the characteristic vibrational and a characteristic rotational temperature we can calculate that these these 2 temperatures for any molecules ones we know be an agent OK so down here we got only translation going on up here we got translation and rotation because the thermal energy is high enough now so that the molecule can rotate as well as translator and finally but here we got all 3 things going on but the capacity of the molecule the store heat increases as it has more channels in which to put the very intuitive idea I think OK so what the heck we petition does is provide a shortcut method for estimating approximately the internal energy in the capacity of any molecule what makes it interesting is it actually works it's so simple
11:56
and it actually works it gives the right answer How does it work consider the classical Hamilton 1
12:11
the harmonic oscillator work here's the way it works you write the classical Hambletonian for the molecule the classical Hambletonian rights of this quantummechanical stuff going on we're going to miss it here then we convert each 1 of these quadratic terms while mission so consider the customers 1 you so won the harmonic oscillator this 2 terms in the Hambletonian a kinetic energy term that's not temperature and a potential energy term that's not volume kinetic energy potential energy the kinetic energy Germans just peace squared over 2 and please the momentum and the man that potential energies onehalf K X squared I just hope slot all right that's the force constant of the bond and that's the displacement of the bond from equilibrium backs AAminus are 0 few OK so there's 2 terms in the classical Hambletonian Beckley petitioned here and says that any quadratic terminus Hambletonian having the form for example 80 squared or B X squared the internal energy of a molecule is Katie over 2 for each such terms How could be that simple Net any
13:38
terminology take the terminal multiplied by Katie over to and the internal energy yes right now gonna work OK
13:52
so the problem of applying the acquis petitions here comes down to writing this classical Hambletonian correctly figuring out how many molds there are right there actually
14:05
participating in the energy storage and then assigning
14:08
each 1 this magic number Katie over to work you've got a mall RT over to I think it's pretty easy no I recall capacity just another U.S. we just said OK so if there is 1 term but there's 1 quadratic term in the classical Hambletonian then this internal energies Katie over to the capacities K over to for a single molecule or or over to promote molecules yes just so that OK so let's calculate something right now all molecules translate in the classical Hambletonian in 3 dimensions for translation is just that P of EC squared the moment Annex squared momentum squared amendments the square divided by the 2 times the mass same for every molecule How many quadratic terms are there 3 1 2 3 at repetition tells us that the translation of the translation is that translation contributes sorry Katie over to to the internal energy of a single molecule 3 because those 3 a quadratic terms OK so the internal energy the single molecules going to be there with the internal energy of a molecules going to be this and he capacities could be derivative of that with respect to you 3 are gratuitously video about the contribution of molecular translation to the heat capacity is 3 are overturned for every
15:49
molecule "quotation mark yeah look at that right 3 are over that's why that's 3 or now is that only approximately correct no that's exactly correct a site sector well wish I'd be in the lecture OK so this is also the toll the capacity from model atomic gasses obviously because a monotonic gas can store energy
16:21
energy any other way it can rotate we can vibrate so this is the whole story for model time aghast like neon or Oregon but it can't do anything all right so this is the capacity of a noble gas for example .period there are
16:41
no box that doesn't happen and that doesn't happen it's just blew OK 4 molecules of more than 1 Adam vibration rotation can also contributed the capacity but vibration doesn't turn on until the temperature approaches the characteristic vibration temperature to things to 4 rotation rotation doesn't turn on what I think turn on
17:06
any doesn't contribute to the capacity so for a
17:10
linear molecule let's say that we're at a temperature that's higher than the rotational characteristic temperature but lower than the vibrational characteristic temperature in other words rotation is turned on but vibration is isn't all right at moderately cold temperature this turns out to be the case let's say Bloomberg hundredweight numbers and thermal energy alright so the Hambletonian for rotation of a linear molecule now this got 2 terms it can rotate accident can rotate and why these are the moments of inertia these are lies so there should be I setbacks and Isobel OK how many quadratic terms of the year 2 could it be that simple rotation about the x axis rotation about the Y axis that's the whole story OK so the internal energy of rotation now is going to be 2 times Katie overturned or formal molecules to times overture In
18:26
that's amazingly simple enough so
18:30
he capacity then the total capacity for the molecule in this temperature range it's got to contributions to its hears the translational contribution 3 or over to that's always the same that's always going to be 3 or overture right is the rotational contribution build right it's too hard to over 2 because the molecules that 2 ways you can rotate they can rotate and acts but it can tumbled working rotated why there is actually 2 coordinates that can the
19:06
provide only to 1 another I couldn't rotate along its axis like this now but can rotate if it's oriented like this you can rotate like that rotate like that right that's the X and Y
19:19
rotation we're talking about so the total incapacity is just that some of these 2 things fivehour over 2 there's no vibrational contribution because we way below the characteristic vibrational temperature we said We're in this range here were way below the vibrational the temperature were vibration would turn on OK we're up here right that's 5 over to yes that's just what we calculated right so it looks like this plot probably applies to linear molecule because it wasn't a linear molecule this would be 5 are over to it would be you know that too would be a
20:08
lot so it wasn't a linear molecule they can rotate in all 3 dimensions x y and z right so that too would be on 3 now for a nonlinear molecule yes this is just what I said x y and z 3 Katie over 2 or 3 RT over to the but that would be the told the capacity 3 article was a nonlinear molecules this plot that I stole up Wikipedia obviously applies to a linear molecule the translation rotation but no 1 at higher temperatures were start to excite not only rotation translation but also vibration as we said earlier the classical Hambletonian by vibration actually contains 2 terms a little bit more then for translation or rotation because even for a single mode there's 2 terms in the classical Hambletonian victory potential energy in the kinetic energy all right and we're going to start these guys over here 3 and minus 5 A 3 and minus 6 vibrational modes per molecule right depending on where the molecules linear nonlinear part of it linear it's going to be free and minus5 so following through with the predictions of the ethnic partition Fareham we're going to get for each molecule to over to per mile all from all molecules through to watch over to promote true Katie over 2 Kermode right because the classical Hambletonian contains 2 quadratic terms for each mode so socalled directly interferes there we've got for nonlinear molecule we've got rotation right 3 4 x y and a hand with got a contribution from vibration which is going to be the 3 and minus5 for linear molecules with 3 and minus 6 for nonlinear molecules that's translations rotation and vibration and so the total is going to be that the group on what should be 1 of them so 3 minus 5 is nonlinear molecules 3 in my size my nonlinear mode said so right sorry yes the success story which between nonlinear OK so for example for a diatonic molecule that to be free of minus 6 what to keep the applicant so if in fact that was a 6 this would be 3 are over to for the translation the diatonic molecules they can rotate and X and Y so the 2 over to Ford's rotation and then it would be 3 times to for the number of Adams minus 6 times are alright so this would be 5 are plus 6 months later this year OK that's what we would have gotten for in nonlinear molecules 7 have are let's do some examples we UCI competitions some fear and to estimate the constant volume already capacity of bite methane and benzene at 25 degrees OK so the 1st thing that you want figure out here this is where you are on
24:48
this plot in other words how
24:50
many terms are there going to be yearly capacity expansion is translation only contributing translation and rotation for translation rotation and vibration the way that you figure that out is 1st of all if you're at a temperature of near room temperature what we know about whether the vibrant with
25:12
whether the rotations of the molecule are going to be excited or not How much thermal energy is
25:22
there room temperature what's Katie In any units that you wanna use question How much thermal energy is at room temperature and wave numbers right 207 are 200 roughly by 200 and what is the energy spacing for rotation of moderately seismological number energy spacing for rotation
25:57
Watson B Is it a thousand wave number anybody wanna 400 knowledge to hunt a handful write 1 3 1 of a small number right for rotational states only way numbers are therefore vibrations round numbers all
26:36
stretching frequency member that from organic chemistry big blob over on the left hand side you spectrum those entities anybody remember 3 thousand wave numbers but in a to stretch OK so order magnitude a
26:54
thousand wave numbers I'd still wanted to for rotation of thousand for vibration OK so qualitatively this is going to help us figure out where we are on this diagram right in case a bite to both of them vibrators are heavy on they is a big molecule 126 grams from all and so on that's a pretty low frequency but that's a pretty low energy rather 200 wave numbers all right but In the range where we sort of expected it to be a writer thousand wave
27:28
numbers is 2 1 6 Fig that's how much it on energy there is no vibration OK so what do we do
27:35
we have to think about whether this thing is going to be vibrating at 25 degrees C while at 25 deg C we got to underweight members of thermal energy we now wanted to is enough to excite rotation so this baby is rotating but we don't have to worry about that Soward were definitely here were just not sure where we are here are we up here In other words is that vibration excited or we down here is that vibration not excited why we're ready concluded that 25 deg C we got 200 wave numbers so were Klaus Will the vibration of eye to contribute to the heat capacity while we're not sure we can calculate the 2 characteristic temperature from his 240 weighed numbers which is a convert to 242 jewels divide by K we get 308 Calvin but that's a little bit higher because that's not 200 wave numbers 214 so that's why that's 308 OK and so were right here right this line turns out to be 308 my number just below that OK so this state this vibrational mode of the ITU is significantly turned on but it's starting to contribute to the heat capacity we can there assume we're down here were summarized here and we always make the hightemperature assumptions but it's starting to contribute were going to use less To calculate are ACM petitions Thira he capacity confident and knowledge of overestimated a little bit
29:15
but because this isn't a perfect science were not going to get the capacity exactly 2 3 6 flags were shooting for 1 stickfigure OK
29:24
so we're going say yes that noticed turned not because we're somewhere on this rising portion of this curve the molecule starting to store energy in its vibrational modes as well as rotation and so on we've got translational contribution to the capacity we've got to rotational degrees of freedom because it's a linear molecule and we've got 1 vibrational mode or 1 hour we're going to include the whole we don't sweat it out so it's 5 or over twoplus hours 7 hours over 2 actually this is 3 . 5 are right and the actual he capacity of ID is 3 . 4 so we overestimated
30:09
the capacity slightly but not by that we had a really good job of guessing what would they I just
30:17
using ethnic partition fear I
30:21
like simple intuitive things like this that allow you to get the right answer right we're always looking for better chemical integration so that we can an order of magnitude level figure out what's going on right that's the real challenge but later on we can calculate this the 3 6 figs if we want but we want to get the aren't we want have an intuitive understanding of how big
30:45
that number is so we can get that with this petitions there notably left out this motive we left out the vibrational mode would be way too low 2 . 5 0 OK so that's a justification for including it even if we're not all the way turned on here were here were not all the way up here alright but we're going to include it anyway and most of the time that's going to get as close to the right answer but this guy obviously more complicated here all the vibrational frequencies that applied and methane 1367 1582 and so on I do these vibrations contribute to the capacity at 2 AM at room temperature what you think
31:47
1367 is going to be storing energy for you at 25 degrees Kelvin where the thermal energy is how many with numbers 200 you get
31:57
200 wave numbers of thermal energy right and the lowest vibrational energy level the methane is 1367 I was at 13 13 67 going to be storing energy for no
32:19
200 wave numbers 1300 wave numbers you need 1300 wave numbers of thermal energy before this thing gets turned on right not sure a plugin telling of plugin take the lost 1 of these number take a 1367 plugin converted to Jules divide by Kate 1367 lost 1 1966 degrees Kelvin that's how hard it would have to be for this lowest mold to get turned off right so at 298 K it's not on you see 1 just the energies too high for this vibration It's not getting excited at 298 good OK so were not up here I would assume none of these guys is turned on we are right here the words rotations are turned on but no vibrations are turned off but that's the hard part of this little calculation is figuring out what you include include the vibrations or not you're almost always including translation and rotation it was year really low temperature but you're including translation and rotation but the question is usually concerns vibrations which ones are turned on which ones aren't some might be turned on others are not OK so he capacities going to be translation plus rotation this is not a linear molecules so that the story and awarded include no vibrational modes because all at energies are too high so we're going to predict the capacity is 3 or are and the reality is for methane it's 3 . 2 OK so the reality is that these vibrational modes are contributing a little bit we neglected them but in reality they're contributing a little bit we miss that but we get awful close we get the right answer to 1 6 Fig a benzene His got all Of these different states pay attention to this column is the actual frequency 1100 3 thousand 1300 location we include any of these in our His benzene and be storing energy in any of these notes right here the all too high How much thermal
35:04
energy is there room temperature 200 never forget that number all right what about here here's
35:12
another mode the ch out of playing away 600 wave numbers it's a little bit higher than 200 but not way higher right 3 thousand forget it 1 thousand forget it 900 very high 6 84 another 1 and so that's a little lower than all the rest 1400 1100 but when we see what happens if we include 2 moats but will include this guy 684 and will include this guy 651 all the rest of them were much higher the still calls we get so let's go with 651 684 way Members only translation rotation nonlinear molecule so that the 3 if was landing area would be at write to dance to because and I would have to tell you there's there's no way you could know each 1 of these knowledge is doubly degenerate right so there isn't 1 651 wave number modus too right yet include that each the generously corresponds to another way the molecule can store the energy right so there's really for years so it's for so 7 hour would be the total the capacity that we asked me if we include 2 miles and forget all the rest and you can see we don't quite get once they think of accuracy the actually capacity is 8 . 8 so that means benzene can use
36:58
some of these modes it's sort of a thousand wave numbers they can contribute a little bit to the capacity even way at work way lower temperature where we're at we get redder thermal energy of 200 wave numbers we should be turning aside things aren't we get to a thousand but the they get take a little bit even at
37:19
this low temperature the contribute a little bit that's that's the difference between the 7 that we calculated the 8 . 8 the benzene actually has yes we should have but which go back and work write these guys worse item wave of about yeah you know who who's going to know but you can't expect it In reality if we include the nite and we've never Mulroney to close to the right answer about how would we know that
37:59
so this is a very approximate Park all right in and in this particular case it wouldn't be exactly clear which most to choosing which ones to not choose ready to even be more complicated than menace bye bye and many for simpler molecules are in most cases it's fairly obvious which modes to include which ones not to include now the way most people treat at the petitions here is they just include all the the at partition fearing gives you the high temperature limit for the heat capacity OK and you can see where was Frank include all
38:37
modes in the molecules to be really happy before can access all these modes in the case of benzene OK I definitely the quiz question on this coming year for Friday OK now With that intuition we need to be able to calculate exactly how big each 1 of these guys it's I know you've been doing that already but you've been doing the homework on the discussion but now we're going to talk about lecture just briefly today we're gonna start by talking about translation here's a molecule that's the translational energy of afford moving in threedimensional this is my when lower In of and the quantummechanical gas energies are given by the article about miles here's the classical but if we know the velocity and x y and z we can calculate the kinetic energy OK but quite mechanically we use the particle in the box right are the dimensions of the box L X L why that should be not acts bells even to be not sorry why and these are the quantum numbers for each of those dimensions but remember this With that probably from all quarters OK so these are what those wave functions look like forget mistakes in 3 dimensions all right blast past so now really concentrate attention on ideal model atomic gas is just for the moment such gasses of no internal energy informal rotation of operational discernment as ground electronic data system used to be considered in our analysis of 1 of the tacit assumptions we've been making for the last 20
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minutes is that the electronic states the molecule are not contributing anything to the heat capacity because we're only occupying a single electronic state some molecules that would be a bad assumption but there's relatively few where you've got lowlying electronic states that contribute to the heat capacity we talk about
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1 and a right but there's very few examples like that OK because these various energy manifolds rotation vibration translation to be separated the solution among atomic gas translational energy will also provide us with a general expression of the translational energy of any gas no matter how many Adams advance all we need to know is how big it is OK so consider 1st of all model Tommy gas in 1 dimension we've only got a onedimensional term here we've only got a quantum number for acts and add dimension for acts the molecular partition function is just wouldn't have applied this energy into the expression of the partition function but I'm that fully get there no vintages a very closely spaced consider for example if I put this Oregon and in a 1 micron box 1 micron how big is 1 micron "quotation mark term under 6 meters right how because of red blood cells but the smallest thing that you can see in an optical microscope anybody know you got an
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optical microscope with the by the world's best Zeitz optical microscope you pay 12 thousand dollars for it's got objectives like beer cans on it right you look through it what's the smallest thing that you can see what's the smaller size the thing that you can see only people had microbiology class come on you guys microbiology people should know the answer to this the public is the bacteria about a
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micron Tennessee bacteria yet just barely yes the right 1 micron but an optical microscope you conceal 1
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micron object right as a matter of what you pay for it because you can't see more than a fraction of a wavelength of light what's the weather like the green light half a micron right turns out that's the smallest
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dimension you're going to see does matter how much money you pay for your microscope right if you if you don't pay enough you will see that OK 1 micron is the tiny box it's the smallest thing that you could possibly see in an optical microscopes were not given the molecule very far removed not only that were only considering its motion in 1 right not wines eh OK for these energies are very closely spaced wearing a woman so I I don't think he are what the debt with the state's spacing between the ground translational energy level in the 1st excited translational level calculate that final what is Page it's a kg now right so when you're doing this on the quiz on Friday and make sure that you use kilograms OK that's 10 minus 6 meters right Palace squared OK and we've got any equals to to square his former minus 1 history not that the energy we get 2 . 4 8 instead of minus 30 jewels big energy
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a small 2 notes right it's jewels right it always looks small right if that was bigger be tenor minus 20 10 minus 18 still seems like a small number right we converted to
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wave numbers we know that small all right 1 . 2 5 to 10 minus7 wave numbers Tiny a unit of energy right 1 wave number to get the molecule the rotate an a minus 7 a case of the states they think is really really tiny by user largescale years fought to 2 . 4 against a minus 30 it's right there all right is as I increased the quantum number I'm looking at the states facing higher higher but what happens there'd be the state's giving a closer OK
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so the states are requires I continuous there a tremendous density of states there's so
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close together but the almost a continuous distribution by and since that's the case we can evaluate we can turn this summation into an integral when integrate over all the state's Zero to Infinity I just move that guy into the integral organ integrate across all of the state's ends of X order usable trick because the interval right and we plug everything and were dismissed substitute Alpha for everything here except for and because enters the integration variable OK and when we do that and we evaluate with the integrated with that which was equal to this is the expression that we get after integration we find out that the partition function for 1 direct 1 dimension is just the dimension divided by age rule 2 .period overpaid right now we can
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calculate exactly what the partition function is a onedimensional as many 6 figures we want this to the
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translational punch partition function 1 dimension for any molecule any molecule all we need to know is it's
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OK calculator the petition which 1 dimension phenomenon confined to 1 micron onedimensional boxer's trainer degrees Kelvin bird OK 10 minus 6 1 over Katie Wright we have to not take 300 we have to know In units of kg kilograms In my emphasizing this point enough we want the massive 1 and so we divide by so why look at the periodic table it's 39 . 9
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4 8 grants from all but 1 down kilograms we're not going to forget
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that already get none of the above when the right answers actually right there OK and so will we calculate this number at 62 thousand
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this 62 thousand translational states and this 1 micron box amazing right there are in principle 62 thousand
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formerly accessible translational states at room temperature in this 1 micron box that's a large look what whatever was the threedimensional box while we just have to keep the same expression we just arrived we Cubitt so that's not a square root anymore it's 3 halves and now we have include L X L Y L z get a cube that H as well OK with Cuba everything because the translational partition function overall is just accidents Q Y Prince accusing the OK and so that this is just the volume obviously and so if I just moved things around that the expression debt and is an even simpler expression if I substitute something called the thermal wavelength and you've seen all this
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already people looking at the homework
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problems we talked about the thermal wavelength already right this is just a way parameterize this equation a little more conveniently because if we win if we define wavelength like this way than that translational partition function just that really really easy to remember OK and so on we can calculate the energy from this by using the equation that we derive the least intuitive equation in chemistry OK she was just this so we implied that in 4 key you where did we do that there we get it right there there's queue of pesos it's just some like you over these the debate the over the formal way like you With a former Waverly cubicles that alright in your view trust me right that's what we get for the internal energy for us should be a mole this should be the average internal energy because we got case here unless we use an equal other goggles number OK so the bottom line is this is 3 have 1 what we knew that the Mac with partitions there it's 3 has are
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Teva translation now we proved that that's exactly right OK what is that 100
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slides triggered foresee 1 Friday