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Lecture 07. Vibrational Partition Functions.

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so really talking about vibrational partition functions but let me just say that you did well on
quiz to again very happy with that 69
you got a name out of 107 it's outstanding so be another quiz on Friday and in a week from Friday we have mid-term 1 that's coming right up so we'll have more to say about that but OK so we were talking about this symmetry number on Friday this is a confusing topic but isn't discussed very much in your book volunteers Is this the number of indistinguishable orientations of a molecule period the number of indistinguishable orientations of a molecule so for simple
molecules you don't need any fancy thought process to discover what the symmetry number it's you can just take the molecule can label the Adams turn it around and figure out for yourself how many indistinguishable orientations there are right for example
has sold 3 is the molecule we talked about on Friday now I've taken it and I've labeled the oxygen so we can keep track of them while this figure out what the symmetry number is for S O 3 I think everyone can see that if I take this guy right here and I rotated by 120 degrees dissolved 120 degrees rotation rotates this guy down here this guy goes over here In this case can end up over here right so that's 120 degree rotation right there I just took the molecule in a way her if of I do it again industries the rotate down here so OK is his art so there's free orientation shown on the screen is that it for this molecule what if I take this guy and I flip them over like that what I do and know I flip them over so that the 2 is down here now the 1 is up here but Frieden Mo when it took the guy and I flipped him over like that right with these guys I took this guy I rotated I rotated now I fled all right does this look exactly like this it adopts but the occupants are the same place but if this is not the same as this this or this part I kept the labels on these oxygen so we won't see them the reason I could do this is because this molecule is flat I think you can see if I do this flipping thing with ammonia I'm not going to get something it's indistinguishable because but these 3 guys in the case of ammonia hydrogen is very they're going to be sticking out of the screen or sticking in the screen when I flipped the molecule over it's gonna look totally different all right so the fact that I can flip this guy over and get an indistinguishable orientation it is only because he's flat and then once I got the best I can do with 120 degree rotation business again and so I end up with 1 2
3 4 5 6 indistinguishable orientations for this molecule that are possible I don't think there are more than that all right so you can physically count the cemetery number is just the number of indistinguishable orientations right that's what it is now I have a
little recipe for finding this but I tried to tell you about on Friday and the reason I like this scheme is because it works when the molecule gets bigger and more complicated and you don't want a label every atom and turn it around and flip it over and town town talent like this gives you a faster way to find symmetry number what we do is we identified and holds a symmetric axes what is that we won here as an axis for O 3 that's twofold symmetric but I say it's twofold matter because I can do a 180 degree rotation of this guy and I get the same orientation so it's a twofold symmetry axes then you decide whether this axis contains some plane but does this axis containing airplane if I put in a mere right here does this side of the molecule look just like this side of the molecule no right so this access does not contain a mere playing and then you count the number of these accidents you multiplied by that number and then you multiply by 2 if he answered too was yes How many actors are the ones too 3 copper number of these actors in this case the story but multiplied by the end from the and folder and then you multiply by 2 there
was a mere playing OK so in this case we've got 3
Accies times twofold symmetric Times online because without a multiplied by 2 6 Cemetery number this 6
truancy that 3 steps a symmetry axes
decide whether contains a mere played in this case it doesn't I can't put this bar anywhere along this axis and get a mere .period OK if I if I could do
that then I would be multiplying by 2 and the cemetery number would be 12
OK but students got aluminum chloride but once again I can't because the molecule small I can't put a label it every single
Adam and I can turn it around rotated I can figure out comedies years indistinguishable orientations are I can just do it myself without using any formulas right here is the aluminum
chloride a label all the atoms not by rotate this guy like unless OK the 1 is gonna rotate down here 2 going to be up here the 4th rotate around to where the 3 years the 6 going to rotate up like that so I think you can see if I rotate the molecule like that yes I'm really get this guy right here the ruined see that and their health I take that guy and a rotation like that what happened the 6 is going to rotate down where the 1 is the fighters the rotate up to where the 2 all right but in the 3 in the 4th are not going to rotate
but the freeing of 4 are going to stay right the freeze stay the front
rotating around like there some sort of rotating around an axis the goes right through the freedom of 4 OK so can everyone see that if I do that rotation and end up with this guy and he's different from this guy and different from this guy but you can see they're all indistinguishable of if I
didn't have the chlorine is
labeled you couldn't tell the difference Kroger OK and then I can with this guy right here rotate them like that again but if I do that by rotates down 6 rotates out for real rotates back sorry for rotates the front 3 rotates the back you get the idea yes 1 of those the work
of the yeah for but now if I take this guy when the brevity and such
what I take this guy and I rotated like I did that guy 2 goes down here 1 goes up there right if I wrote it like that the 1 is going to go Ochsner go down I get this guy right here if I do that there are no other orientation OK and if you don't
believe me mess around with it right for
possible orientations the symmetry number for this molecule must be for right how do we find it with the shorthand method she was an access there's 2 possible symmetry axis is that you can choose and factors for a we years also write your access goes up and down right through the center of the molecule right OK the 1 I'm using goes right through these guys these 2 guys right here these 2 aluminum OK but we can also use won the gold regularly scoring right here right there
are in fact 3 orthogonal axes that we can use for this guy Our ability use
this guy for the sake of argument OK is there symmetry plane along this axis now of course if if I had some artistic talent I could drive properly but Is there is there symmetry plane they're
not How many people think there's a cemetery .period How many people
don't see it thank roads To get the answer is yes there is a right and if I could draw this properly you'd see that decided the molecule looks exactly like this other molecules like just draw properly OK so there's 1 axis this twofold symmetry and the mere plane says that that has to be aghast decide whether it is airplane yes and so we multiplied by 2 students to 4 times 1 see that invited
chosen this axis it would still yes to the Mir playing question but it would still be too right is that the rotating and have to rotate the on only got twofold symmetry there right so would still be 1 axis times twofold symmetry times too and if I chose this axis like that it's still going to be 1 accidents twofold symmetry times too you should get the same answer
every which way you look at it otherwise if these answers are not self-consistent there's a mistake somewhere yes or no no
was only 1 axis like this there's there's too little members is only 1 way to draw on access through those 2 aluminum right but if I wanted jolly access to these 2 chlorine is right here there's only 1 way to do that OK if I won a jolly access to the center of the molecule up and down like this there's really only 1 way to do that but so there's really only 1 of each 1 of these yes there could be more
than 1 year place that's impossible it's like some kind of fun house To me airplanes
no you can't have most 1 but you might have not let's do benzene OK once again we can do the labeling experiment call that 1 ABC D and so on .period I can rotate them down here that's 2 orientations you can't tell the difference right by go all the way around and I flip them over like a pancake and I'd do it again on the Net 12 orientations does everyone see that modified if I just rotated by 60 degrees global boom boom boom I'm going to get sex and then I flip them over like a
pancake and I do it again get 12 right all right heart as hot
as the Panama give this right there is 1 2 3 equivalent Accies that I can drop the department but we won't see that 1 2 3 does this axis containing airplane right there yes OK so the calculation is 3 times twofold symmetric times to 2 dance to support emissaries 12 Bill it will see that I
could also do all the actors right through the center of the molecule coming out of the screen I could use that as an axis to but how many of those Accies are there 1 is a mere plane that it's right in this school plane of the screen it's right there Kerry goes right through cuts the molecule right in half OK airplane and if I do that calculated 1 access times 6 full-time to talk we get the same answer right it's 6 fold symmetric now 1 2 3 4 5 6 4 that access and 6 fold symmetry not twofold currency that OK we're not going to beat us to death anymore but you should be able look at a molecule and figure out what its symmetry numbers any which way you want to label the Adams rotated around in your head cut out the
model OK on the exam it'll be easier if you can do it my way focus now
estimated rotational partition for the Allard 25 degrees and 250 degrees C the value was 10 . 5 9 wave numbers petitioner Const
OK actually pretty big rotational
cost pretty big energy for rotational cost OK here's our choices is the equations that we flashed on the screen on Friday were use this guy I hope you've noticed that this guy and this guy are basically the same equation because if the linear molecule like the center of symmetry Sigma equals 1 OK so this these are just the same equation for goodness sake OK but in this case the molecule center cemetery story is this guy that the rotational temperatures status of our rotation to be overcame being units of
jewels of course for many years this equation OK and
so the rotational temperature here I can calculate what it is 10 . 5 9 convert to jewels divide by K I get 15 . 2 3 degrees Kelvin that's some rotational temperatures pretty cold but actually it's usually colder than that all right but that's critical 15 degrees Kelvin 2 so this temperature is much lower than either 1 of these target temperatures that 25 degrees C not right that's 250 degrees C OK so we expect to have lots of formerly accessible rotational states
don't we where where temperatures are way higher the rotational temperature this molecule by right
15 degrees OK so let's calculate what the partition function is this is our equations Katie be so there's Katie this is just unit conversion B 10 . 5 9 converted to jewels 10 . 5 9 converted jewels only thing that's different is the temperature that partition function I get do those numbers look right what do you think still it about right I mean we we
expect something like 6 significantly bigger than 1 don't we were at temperatures in at 25 deg C were at 298 K the rotation temperatures 15 writes that lots of rotational states are going to be occupied 15 degrees is when we start to have multiple rotational states occupied artwork way above that we have lots of rotational states occupied qualitatively looks about right notice that 250 not 10 times 25 In Calvin units right now it's about a factor too right and that's also what we see here is so we didn't make a mapmaker math mistake OK that was easy 1 on methane but calculated partition function at 298 degrees here in case its rotational constants 5 . 2 4 1 2
wave numbers these rotational constants are often known 2 4 5 6 6 figs right because the spectroscopy you can measure the frequency very very very accurately rate it's not too often in physical chemistry we can measure something with this kind of precision right but with spectroscopy you can measure rotational constants right with very high precision 5
6 figures but we know what to within 1 part in 10 thousand we need the big boy for this all right the other he's a nonlinear molecule methane OK in moment of inertia Vela all the same because of the cemetery of the molecule researchers beat the case we need the cemetery number what is economy going be from methane you think back 4 vertices what's the cemetery around each verdict Vertex story some say 3 4 vertices each of them is threefold symmetric is there a mere plane along those Accies Knoll so what's the cemetery number right 12 OK so we like everything and this is 1 of the 12 of blob of obliging the conversion bubble 36 . 7 1 1 is a partition function I calculated that seem about right yeah it's higher than what we saw for HCl but B is a lot
smaller but these happens
because it was for HCl so at this temperature we expect you to be larger it's about twice as large all right but paying these happens so makes perfect sense OK these rotational petition
pretty easy we just we've only got
these 3 equations in fact it's really only 2 equations but now vibration yes that is the rotational constant
about each of the 3 principal axes of the molecule x y and z so in the case of methane because it's for hydrogen and it's about touches duly symmetric those 3 rotational constants are all going to be the same now if was chloral methane if you had 1 chlorine and 3 hydrogen St then there would be 1 unique rotational
axis that had a different be valued so would be a B squared OK and you have to be told what they are really that you have to be told that's what we needed big equation OK so harmonic oscillator remember way back when we were talking about here is the harmonic potential one-half K X squared or AA-minus sorry are is the equilibrium by distance everyone remembered books along evenly spaced energy levels this is what they are write new plus one-half times new or v rather plus one-half times age knew that's the vibrational quantum number OK we have our normal expression for the partition of onto just like this energy and we got the partition function right
always works this way are and then
all we do is we simplified right but we're always start with this this is always the expression for the partition function the matter whether retired about translation
rotation vibration you can derive the partition
function by just plugging the energy into this expression and simplifying always now to make this a little bit simpler we often neglect is 0 .period energy later on if we calculate the energy using the equations I'm going to show you we can always had net 0 .period energy at the end we know what it is but we can't put it back in if we want to make sure that we don't neglected OK so if we neglect the 0 . energy now the one-half goes away we get this guy right here so if I write that series out but it looks like this the vehicle 0 1 2 3 and so on OK now there is a nice closed form expression for what this series sounds too right it's gonna sound to invest there if that's an acts Of course it's not an it's said age Nova Katie OK so we're going to use the same former substitute age Nova Katie and that's our wrote our of vibrational partition function right there really simple expression How big is it well let's just say a typical the vibrations
2000 wave numbers no that's actually pretty energetic vibration but 2000 wave numbers I think you'll
agree right that's not as energetic as of proton vibrating let's go lower energy demand OK so that's what the frequency turns out to be in that number and I calculate which you is I get almost exactly 1 for the partition function right at room temperature now the partition function is 1 . 0 0 0 0 6 4 OK so this is totally different than what we saw with translation and vibration tight translation and rotation with translation we had millions of accessible States did we with a partition function that in 1 dimension of little 1
micron box was 67 thousand remember that make a three-dimensional botching got millions but accessible
states with rotation we get 20 30 40 rotational states that are accessible at room temperature with vibration we get wild but what is that number Maine it means that molecules are almost always in the ground
vibrational level at room temperature but it is unusual for the 6 them vibrational excited states in a molecule to be excited at room temperature but that would only happen if he hadn't really heavy atoms like trying to
right fired on as of vibrational energy of 214 wave numbers are much thermal energy is there room temperature 200 OK so I been is excited right but my goodness and units to 126 grams per mole Adams bowling balls right there they're barely moving on very low energy doesn't take much energy to excite vibration at room temperature by golly it is excited all right but With the exception OK so at room temperature just 1 vibrational status formerly accessible in general there's exceptions but in general write very different from translation rotation now withstood calculation we try to have a molecule chlorine dioxide has 3 vibrational modes of frequencies of 450 945 1100 wave numbers what the partition function at 298 degrees Kelvin
OK How much thermal energy is there room temperature to 100 wave numbers what is the lowest
energy vibration here 450 OK so we want to keep that in mind because what we expect the partition function to be here just looking at this problem before we calculate anything we expecting petition number partition function of 10 100 about 1 should be close to 1 1 and change right something
right 1 . 0 something maybe right is that what our intuition is telling us yeah because
the thermal imagers way lower than that OK the Stewart there's a molecule but How many vibrational modes does it have 3 ah but it's good question asked that I say anything about the generously in these 3 states here I didn't say anything about it but we know that it's a nonlinear molecule slid 3 minus 6 In his 3 3 Austria's 9 ministries so that only 3 the vibrational states in this molecule of I give you free energies you know they're all 90 generous right OK so 3 of 6 3 and sold the partition functions just going to be the at times that times that but the end what's the partition function with that guy right I'd just take my expression for the partition function but played in my beta my age my see all my constant that's a temperature that were talk money get 1 . 1 2 special higher than we were
expecting we expected 1 and change but 1 . 0 something a little
higher than we expected OK if I do the same calculation for a 945 1100 I should get smaller numbers yet smaller than that OK so the overall partition function is that times that times that for that right now do these numbers all make sense while "quotation mark 1st of all they're all 1 change secondly that was bigger than that which is bigger than that right at the end the season in the right order that's a dead giveaway that we made some kind mistake and so qualitatively yes we can live with that number it seems about right it's a little higher than what we were expecting maybe but just barely OK now if we make it really hot this is 200 thousand degrees this is 100 thousand degrees right there you can in principle have lots of vibrational accessible states in the increase squads linearly with temperature Of course the molecule falls apart way before you can do this right up there is no such thing as a bond dissociation energy that is large enough to allow you to see this but if you could if the molecules didn't fall apart at these this is what you would see right this is for 1 thousand wave number mode this is for 100 wave number mold the vibrational temperatures are 144 K and 1400 agrees K OK we can always calculate vibrational temperatures just age new but I guess it would fall apart that's only about 1 electron volts for most molecules OK so 298 degrees Kelvin is a high temperature of a translation of a high temperature for rotation but low temperature for vibration we wanna keep that
in mind right what the translational states lots of rotational states but in general very few vibrational states are thermally accessible at temperatures near room temperature what about the
energy while we got this nice expression that we derive chemistries least intuitive equation all all we have to do is plug in our petition function here and here we can calculate the energy directly from that OK and so you can do that simple fired there's your equation 13 . 3 9 that's the energy for N molecules that's the vibrational but but I this equation 0 . energy so we can add and are we can calculate the energy using in this equation and then added h over to to make that 0 .period corrections if I about it but if there are multiple states we have a term like this for every state In other words more than 1 h new energy is added make sense at temperatures over so high above above a long weekend if the temperature is much higher than that of characteristic vibrational temperature we can approximate the number 1st to terms that fine if its series even series would like that take the 1st 2 terms 1 minus 1 is they're all H Ugandans of age news we just end up with big and over Bader and as the number of molecules so the total energy just and Katie and the reason that's interesting is because that's what we were calculating what we did the act partition Theron revolt the classical Hambletonian for vibration and had 2 terms insults to K 2 times Katie over 2 yes Katie promoted remember this OK so the equity the ethnic partition theorem tells us that we've got party in terms of energy mode all our resources in terms of the capacity right so we've got 3 or over 2 or 5 or over 2 7 hour to remember yeah well it turns up this was right right it's the high temperature limit if you take a partition function you look at the high temperature limit you get the the extra partition predictions right for their
contribution to the heat capacity in the contribution to the internal energy OK here's a midterm
exam question from a couple years ago they have 1 like this on year on this exam almost everyone got a but a a is easy what is the answer to a it chlorine oxide molecules called for freeze Calvin hommage vibrational energy does retain at that frigid temperatures you think
that's what they said how much energy is a molecule retain at 4 degrees Kelvin How much vibrational energy does it retained With physical chemist
yes page How many age
news are there 1 2
3 like this molecule has 3 vibrational modes you can't stock is 0 .period energy out of any 1 of them even at 4 degrees
Kelvin the molecule is going to keep bending a little bit it's gonna keep asymmetrically vibrating it's going to keep symmetrically vibrating even at 4 degrees Kelvin right OK so the answer is
0 .period energy of 680 0 . energy free 30 in Israel .period engine 973 calculate that you get 991 wave numbers it contains 991 wave numbers of energy even that 1 merely Calvin right you
can't remove that energy from molecule "quotation mark mechanic says you can't do it you guys have had 20 weeks of quantum mechanics but is everyone see when
talking about here he can't remove the 0 . energy from any mode and and half of stuff you know the How can I convert that Jewell Stoddard of the OK now 1 wall of steel to always worn a one-liter container 2 thousand degrees Kelvin what fraction these molecules of the 680 wave number vibration mode excited right going out you have something very similar to this to answer here's what the equations pager that exam looks like OK equations are intentionally disorganized on this page the purpose being that you have to fish out the right wine and it will not be obvious which 1 2 euros so if you don't know what you're doing you could be in trouble you need this guy and this guy that's the petition to adjust arrive rightful vibration are and that is the normal equation that's the Bolton distribution all right in the denominator the Bolton distribution lot is the partition function we just arrived this form of the partition function for that equations were that in for that right there OK and what we what we want and little and over and what little over began we wanna know what fraction of the molecule OK now here's the confusing part am I going to use in this partition function the overall partition function for the molecule in other words to 2 Q multiplied together all a Mike images used to 680 because I'm only asked about the 680 plate number mode a massive is the 680 wave never mode I'm excited but you see that conundrum Rick everyone see that the difficulty in the calculation here is what he did use for this Q down here you don't use the overall partition function for the molecule which has 3 terms Senate or you just can't use it to 680
president the Pierce here's how you
make the decision right if you are asked to calculate a fraction of the molecules for which the 680 mode is excited it's in its 1st vibration excited vibrational level and the other 2 modes are anything then he only used to 680 down here but In other words if you're if the if the identities of the 2 mile doesn't specified they can be anything all we want to know is is the 680 excited or not right then you just put Q 680 down here but in half on the other hand you're asked but said the draft I want to see it the 680 mode excited and these are the 2 are not
I want to know what fraction the molecules have 680 excited and the other 2 modes I'm excited so you've
specified what the other 2 modes are doing a that's a different story that include all 3 it was so if you specify all 3 you need all 3 in the denominator if you specify what I'm allowed only 1 and a lot 1 is doing and I'm terrible at the other 2 you only use the 1 that you care about I don't think a textbook but does a good job of explaining this
fine .period so for 1 moment
on wanted to buy was side OK case this is the equation right here a plug in the numbers this is what the denominator is equal to right here I just can't do that said this is right here resigned just putting the dean unjust and they export rather right it's minus 4 . 4 8 9 0 0 8 6 and someone I plug that in here and here I get 23 . 7 per cent all the molecules will have the 1st excited state excited OK at 2 thousand degrees Kelvin but does anyone have
any intuition about whether this number makes sense this I don't
right .period to 24 per cent 2 thousand degrees Kelvin art in order to figure it out we need to know what the characteristic vibrational temperature is at 680 right so we can calculate that 978 now we compare that with 2 thousand his 2000 yes OK so we expect there to be an appreciable population in this
vibrational states right where half of the way right at 2
thousand degrees but we put more energy into the molecule the necessary but be necessary to populate the state's we expect the state to have appreciable the talk about 2 thousand degrees in 2002 4 they're not is how much thermal energy it would take to start to populate this vibrational
level but at that temperature
we would expect the vibrational level to just start getting populated where were at 2 thousand degrees right so we expect there to be an appreciable population by 24 per cent could be right doesn't sound unrealistic and once you're
talking about look him now what we
did like everybody remembered by the way with these numbers up here the scripts in front of the element the body remember what those are I didn't use them here but now I'm using them what are those what is that 35 right there yeah it's the atomic mass on about an isotopic links Europe of chlorine
oxide right the 16 is the mass of the oxygen in atomic mass unit right 35 mask the chlorine heartily convert that 35 into the actual massacre chlorine and how to do that what I want another mass of a single isotopic leap year "quotation mark scoring 35 what I did 1 of the units about 35 GM
promote right so that we know how much wine scoring 35 ways I just take 35 divide by other gutters number right yeah that's what
you do OK get free vibration modes Bloomberg all damages so what fraction these molecules have the the 680 excited the 330 excited but not the 1973 now organize specify all 3 a lot 1 quantum of energy in that I want quantum of energy in that guy and I want that guy to be In its ground state but I can ask that question ratifies the question that way I'm just and after both Frasier better use the overall vibrational partition function down here Sudan the man that energy right there is the total energy write to tricky things but you regularly use the overall vibrational partition function because I'm I'm telling you I want all 3 of these things to be a certain way any any which way anyway I ask you for them to be about was that you and I was applied and it would still be the same I'm still telling you how I want those 3 states to be configured and then on energy is the toll vibrational excitation energy in the molecule the toll right I said 1 331 680 Adam together if I said to me it would be to 687 to 330 by the energy here is the tall vibrational
energy for all of the different expectations that you want consider the see I think this is rather confusing concern and interest to the honest 2 1 what if there was
the generously of yeah so 680 was doubly degenerate the B 2 in these terms to the singly degenerate terms multiplied together for success OK and we did because ulation smaller number blood of yesterday makes sense because it's smaller than it was before to be a lot smaller because on our target was a tiny subset 1st of all these 2 things excited the editing work insisting that the unexcited so there's a tiny fraction of molecules are gonna meet the specifications
OK part on Wednesday
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Metadaten

Formale Metadaten

Titel Lecture 07. Vibrational Partition Functions.
Serientitel Chemistry 131C: Thermodynamics and Chemical Dynamics
Teil 07
Anzahl der Teile 27
Autor Penner, Reginald
Lizenz CC-Namensnennung - Weitergabe unter gleichen Bedingungen 3.0 Unported:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen und nicht-kommerziellen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben.
DOI 10.5446/18941
Herausgeber University of California Irvine (UCI)
Erscheinungsjahr 2012
Sprache Englisch

Technische Metadaten

Dauer 49:06

Inhaltliche Metadaten

Fachgebiet Chemie
Abstract UCI Chem 131C Thermodynamics and Chemical Dynamics (Spring 2012) Lec 07. Thermodynamics and Chemical Dynamics -- Vibrational Partition Functions -- Instructor: Reginald Penner, Ph.D. 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 thermo-chemistry 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: 0:00:41 The Symmetry Number 0:07:09 Aluminum Chloride Atoms 0:13:03 Example: Benzene 0:15:43 Rotational Partition Function of HCl 0:19:12 Rotational Partition Function of Methane 0:22:02 Vibrational States 0:33:24 What About Vibrational Energy? 0:36:13 Vibrational Modes

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