# Lecture 06. The Rotational Partition Function.

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the rotational partition function that's the lecture is going to be
about let me just point out that videos of these lectures becoming
available on the lecture page of our website here is the 1st 4 wiry posting these lectures why are we making
videos of them we were trying to disseminate this information is broadly as we can even beyond the classroom but for your purposes this hopefully if you miss a lecture for any reason you can go
back and listen to it if you want to or you can look at lectures before you study for the midterm exam maybe you think you might miss something nice said and it's all going to be there I will be
in expendable component at the end of this quarter all the information the court will be on this website that is the cause of some concern to me actually work OK
we talked on Wednesday about ideal gasses a little bit mainly we were talking about the translational partition function as just review that quickly and then we'll talk about the rotational partition function as we said on Wednesday if we neglect the electronic energy levels the present in a molecule we have 3 manifolds estates with in molecules that determine the thermodynamic properties 3 manifolds of states translation rotation and vibration or if we can neglect the electronic degrees of freedom right and why do
we do that because electronic states are separated by huge amounts of energy 20 thousand wave numbers it's a round number for a huge amounts of energy differences so most of the time we can think of these molecules as existing in a single electronic state we can calculate the thermodynamic properties without considering the possibility that they might be able to access more than 1 electronic state so
we said also that were fortunate we can decouple these various degrees of freedom we can express the toll partition function Freddie molecules of product between them and what we're going to do is we're going to make the assumption that the partition function for the electronic degrees of freedom is just 1 there's only 1 formerly accessible electronic state now we're going focused attention on the translational partition function this is what we did on Wednesday so we said Look the way statistical mechanics contends with translation is to imagine that it occurs within the confines of some volume prelate said that that volume is well-defined geometrically let's say it's a box where can be
one-dimensional boxer two-dimensional boxer three-dimensional box in if that's the case we can use the simple particle in the box equations that we remember from quantum mechanics to
calculate the available translational energies for the molecule OK here's particle particle the box of expression for one-dimensional acts and the 0 . energy is not allowed right here is the lowest allowed value of the energy in years the 2nd you can see the energy spacing increases as a function of quantum number again but so the molecular partition function for Q all that involves we only have to make a substitution for this energy right to use the particle in the box Energy expression is so that's our partition function for translation in 1 dimension which is that the one-dimensional expression for the particle in a box very simple 1 consequence Of this expression tho is that the energy that were in a calculate are very closely spaced all right it we did an example on Wednesday in which we considered a box that was 1 micron in size we talk about the size of a micron a Micron is very small but the
size of a bacteria which is about the smallest thing that you can see in an optical microscope and so
when we did this calculation here's the difference in energy between the ground state in the 1st excited states that difference in energy we can calculate of course is unbelievably small
right 110 million of a wave number we know wave numbers pretty small denomination of energy already right so these translational states are Class II where there's hardly
any energy spacing between different translational states even in a really really tiny box and cost of the box gets bigger that states basing its smaller business you can think about is being sort of an opera limits you would practically encounter you're never going to see a
box at the Micron and size are very rarely and the
states facing gets bigger of course as the entity as you go up and energy the difference between states gets bigger but it stays pretty small and its 10 minus 28 roughly 4 and was 100 between 100 and 101 that states facing still on the Order of Canada minus 28 still really really tiny numbers OK so if state cause I continue we don't have to think about this summation it's completely impractical the spending tens of thousands or millions of states millions of terms the summation we don't want happy evaluated explicitly we don't have to because cause continues we can think about it as an integration this is the integral we have to evaluate this is the trick that we talk about on Wednesday and were done working me evaluating an integral here's the expression that we get for the one-dimensional translation 1 dimensional translational partition function and so this is the calculation that we did Ford Oregon at a middle 1 micron one-dimensional box we calculated 62 thousand 700 thoroughly accessible to translational states and if we did that in 3 dimensions here's the three-dimensional expression for the translational partition function it doesn't fit on the screen I put it on 2 screens and there that's the number write 10 of the 14 translational states at 300 degrees Kelvin so we're
never going to see In a laboratory experiment any evidence quantification of energy for translation 1 were never going to see it right we're never going to have a resolution to tell the
difference between 1 state and another winner this close together but it would be practical to even look for that as an experimental outcome it's going to be so hard to make that measure but this equation makes sense this equation right here because with increasing mass the translational states closer together right here our equation for the energy of the states and is in the denominator right so we expect the translational partition function go out with em and it dies In other words if there is a certain amount of thermal energy in the system right is going to be more accessible translational states if the mass of the thing is bigger this goes up Q goes up right proportion of 3 have power In the same thing is true volume this makes sort of intuitive sense the gets bigger these also effectively in the denominator of this expression and so we expect the partition function go up in volume also it's directly
proportional to Boeing looking so
there's some substitutions that we can make we can simplify this expression somewhat in terms of the formal wavelength and we talk about that and then we can calculate the translational energy the internal energy translation by by using this expression for the energy that we derived earlier and basically all we have to do is substitute VII over whatever that Greek symbol is due all right there it is we make that substitution there we make the substitution here we worked through the derivatives and we find that the internal energy the translation is 3 has and T which is just 3 of them where were and as the number of malls if we make an and so a wide but turns into an and this is the internal energy that we had been assuming in the ethnic partition searing calculations that
we were doing on Wednesday anyway wasn't it's just approved that that's right now we
haven't talked about the Selby and 1 really not going to talk about it today but in the future we will show that this equation is true the NLP ages ago that the internal energy policy .period if that's true people's and arty and sell the NLP is justified as an artist so we got an expression from statistical mechanics for the internal energy and for the and in fact we know all of these things because earlier we talk about the entropy this is the Zapotec short equation that we derived last week we've been able using statistical mechanics to calculate all of these thermodynamic parameters including the partition function which is not strictly speaking a thermodynamic parameters but the
connection here with translation is all in the mass of the molecule that's the at tribute to the molecule that were connected to a thermodynamic properties at the beginning of this class we said Look what we want to do with statistical mechanics is connected the properties of the molecule bond distances geometry vibrational frequencies with the thermodynamic properties of a molecule that we can measure always were measuring thermodynamic properties on huge ensembles of molecules so in the case of translation we've done
that right the only connection is in many but that's the only molecular property that couples with this translational partition function the translational partition function really only depends on the matter and with this guy would calculated that and that and that OK so we have established a connection to molecular properties in the renowned properties connection was absent in traditional thermodynamics this begins to fulfill what we set out to do at the beginning of the class just barely
which is barely starting to do that I think we're doing it with so in other words we can measure these properties for molecules that don't store energy any other way they don't rotated on vibrate OK so atomic gasses yeah 1 of the molecule
can rotate them it's got another way to store energy but we have a manifold and rotational states that looks like there I wear this the manifold of rotation state in between 2 vibrational states of molecule but there's a manifold like that here there's another 1 here is another 1 here there's another 1 here and here there's another 1 OK so we're adding a lot of complexity picture now I know you've seen this and I have a whole lecture written that reviews rotational spectroscopy but we've thrown it who knows still on the webpage we're not going to take the time to review all rotational spectroscopy you don't really need to know what for what we're doing but what you do need to know it is you have to understand what the energies of the states are about to be generous these art if remember that the energies are given by this expression each over to why times J and J plus 1 word J is the quantum number that applies here right Jake 0 1 2 3 4 and so on what side a moment of inertia OK so we can calculate these energies I just by substituting in different values of Jr and if we remember that these exports wherever to all why we can just denominators energies in terms of the instead of using other units it's just convenient for us write to the ground state 0 1st excited states going to be 6 12 20 hopefully what you
recall is that these on equally spaced states give rise to perfectly equally spaced rotational lines in a rotational spectrum as everybody remember that remember the reason why where there's a
selection rule the J has to be equal Jr J. filed BJ plus 1 J minus 1 OK so let me just emphasize that when I write B I write it like this right because I'm writing it and where do I really good tools when I evaluate these units of these other yes right ages js but Jules Jewels 2nd 5 kilogram meters of squared OK so and work through those units yes I did jewels rest the stuff canceled OK and I'm comfortable with that but I like being able cults tablets of and jewels because then
I can use my 2 conversion factors and get weighed numbers
art so I'm always doing then I am very rarely doing that's where I write B with a squiggle on and that means B is denominated in terms of wave numbers and then you gotta do conversion here and I just get all messed up when I have to do that so your book will constantly uses equation will cost users equation don't be confused by that your book wants to do everything in terms of weight numbers under do everything in jewels
like I always do and then convert later if I need to focus
so let's work the expression fraud the partition for rotational potential to get this energy we know Jason equal to and so on you let me remind you that there can be more than 1 moment of inertia in this expression depending on
How symmetrical molecule is it may have a different moment of inertia in X and Y and OK there can be 3 different moments of inertia and if that's the
case why the overall petition from 2 rotations to to the product of x y and z I will come back to that but for now Q applies for to each 1 of these 3 orthogonal axes x y and z and we can write expression for 1 of them this is just our standard expression for the partition function now at substituting in the energy for it ending dingy the generously the 20 status is which is always equal the 2 J plus 1 you have to know that OK so we ride out this series slipped right up the 1st free future Jade equals 0 that's going to be a wine by Jake was 0 this is just equal 1 if Jezebel the 10 this is going to be sorry and so on and so forth OK and so we can right at the individual terms of this partition function if we want to but however at room temperature how many of these states other than many thermally accessible rotational states are going to be at room temperature what the states facing Water magnitude in rotation 1 2 3 wave numbers paralysis plus say 1 how much thermal energy is there room
temperature units away numbers 207 so there's going to be to first-order hundreds 800 rotational states but that are accessible at room temperature do we wanna write
100 terms in this expression no right room temperature is a temperature that we care a lot about OK so we can treat this expansion here much like we did in the case of translation the state's away further
apart the translational state but there's still qualify continuous OK so we can
instead of writing a summation we can do an integration in much the same way that we did with translation and I'm not going to walk you through this in a brawl but you can see if you wanna go back later on and look at these lights at the end of the day this is what you calculate if you evaluate that integral right the partition function for 1 dimension make a place where on the British Empire State Yuvraj squared we have to customize this expression Ford molecules of different shapes 3 versions linear molecule lacking a center of symmetry any linear molecule the nonlinear molecules they are never going to have to memorize a questions like this are always be available to you you
during your book on the exam but we have to be able to use them to calculate partition functions
OK sort of focused on 1st thing then understand signal we've got to be able to figure out what signal is for any molecule right it's the symmetry factor this a B and C here was a little bit cryptic we haven't seen those before those are just the rotational constants about the 3 major axis of rotation x y and z where we know we won't need to say any more about them but this cemetery factors what is it we do have to I understand that and be
able to figure out what it is for any molecule known principle you can use group theory to do this and I understand Professor Martin beat you over the head with group theory and I'm happy that she did that but we wanted just have sort of an intuitive feel we what it will look at a molecule without doing that group theory on we want we'll figure out what it symmetry of factories OK
what the symmetry number 8 CIA that's the CL that the there are no indistinguishable orientations of this molecule about any axis What is that need well I rotate the molecule about this axis right here the chlorine will move from this side and this side of hydrogen from the side of side and I can tell the difference if I rotated by
180 degrees I can see that it's rotated if I put an
access to the molecule like this and I rotated 180 degrees I can tell it's rotated if I rotate this molecule I can always tell that it's rotated there are no indistinguishable orientations of this molecule about any of its accidents right that means it's Cemetery number is 1 take it failed to mine this is a perfectly symmetrical molecules it's got to indistinguishable orientations about its axis where well I label this chlorine 1 in this chlorine to because otherwise they can't tell them apart the label and they do a 180 degree rotation the 2 goes over here the 1 of of here rotate again to those of a year the 1 over here those are indistinguishable from 1 another In a matter how I put this access to the molecule is only 2 of them the cemetery number is too what is consequences does that happen while the partition functions only half of big for this guy right it has direct consequences for the partition function ammonia 3 indistinguishable orientations what our that will be the axis that matters is the 1 that runs through the center of the nitrogen a three-fold axis of the molecule in a label a hydrogen and I can see that there's 3 indistinguishable orientations that exist around an axis OK and the cemetery number here is story OK what I attaches Egil molecules orientations
military actions relations and this was there's only
1 symmetry axis for this molecule that could where rotation about that axis gives rise to indistinguishable molecule about each symmetry axis in the small only 1 but we're retired workers relies more on 1 take I never learned anything
by actually writing down what the rules are and if I wrote down what the rules are Germany's Cemetery number I would understand right you have that the only way ever learn anything is vitally examples OK federal
leader but the cemetery number Will 12 hour wife 12 where there are a 4 threefold accidents 4 where that the Ajax's starts at the 1 of these points and goes to the center of the triangle across from it at 1 2 3 4 vertices undetected he drained parts of the studies for taxis that go right through the center so for this guide the right therefore this guy right through here for this like there and I you can imagine that if you rotate about an axis that goes through this triangle here there are 3 equivalent orientations around each 1 of those accidents but the other way to think about it is that there are 6 twofold actually How yeah iterative drop and accessed through the center of having a hard time yet .period so mentioned act goes to the center of this edge in the center of this said see annexes the goes like that on access the ghost in the center of this said to the center of this said those edge centers are twofold accidents Fertitta dreams all right and the 6 of those guys by each 1 of those two-fold axis is twofold symmetry 6 times to his 12 or 3 times forced 12 I trolled the same symmetry elements independent of which 1 of those 2
sets of actors I choose I can't use To get 24 that's the wrong number on and counted the same symmetry elements twice if I do that write to you choose 1 set of equivalent
vaccines you choose to go From the vertex the center of the triangle or from the center of this adds to the center of that edge over there the senator said that the you use the edges on use the vertices can't use involved we should get the same answer so it's a nice way to check whether your cemetery numbers correct should get the same answer no matter which set of vertices would set axioms you elect to you for these other geometries but likewise 24 24 like 60 that 6 CQ can get those numbers that the times 3 it's also 4 times 6 that 6 times for also 3 times 8 that's 12 times 5 it's also 20 times 3 In other words there is different sets of equivalent Accies for each 1 of these highly symmetrical
geometries probably more than 2 but soon example what is the cemetery
number of linear molecule 2 right all it also vessel wrong 6 wife
becomes not like ammonia so the 1st
question ask is as is SO 3 flats or is it triggered by caramel Turns out flat 24 electrons in a sense of larger electrons fear as is no longer the sole no long hairs on it's going to flat but if it's flat and how does this work while there's 2 ways to get this number right there is the symmetry axis for the molecule right the oxygen and sulfur and there's 3 adults I think you'll agree that each 1 of those Accies has to equivalent orientations this is the 1 that makes that is absolutely clear Rice you've got to 2 and ingenue or 6 total the cemetery number but the other way to think about this is what if you had an access that goes right to the sulfur perpendicular to the screen right such an access would have threefold symmetry would I can rotate this oxygen down to their desire to nobody here is likely not be here 3 different ways to look at that all right but I have to consider that access not only in this direction but also in in this direction this to sees the coincident with 1 another running in different directions why is that true for this molecule because there's a plane cemetery in the screen for this molecule all right any symmetry axis that as a plane symmetry and
used white OK with ammonia that plane symmetry doesn't exist along a symmetry axis does it the molecules get this triangular shape if I put a mere plane along an axis I can't reflect the ammonia it is stay symmetrical at the ammonia as get this triangular shape about the plane Maryland write
these oxygen would be like coming out of the screen writes all morning has a cemetery number 3 SO 3 at the cemetery number of 6 because it's black it's more symmetrical that's the reason OK if you understood what I just said you can tell me what the cemetery number is of this highly symmetrical locked the door molecule 24 How did I get that there is there are 6 accidents was this a least 3 ways to do this problem but withstood the simplest way wife Chen for aid but I don't use each 1 of these axes twice because of eye drops there is symmetry of the molecule but the mere right here right OK if I put a right here this looks exactly the same as that OK so use that access White this to access there to force 6 Accies and around each axis this for equivalent orientation there's a least 2 other sets symmetry axis of the same answer when I could drop a symmetry axis through the bisecting this fluorine sulfur flooring bond but what is twofold symmetry there but there 12 of those guys if you look carefully to build a molecule and you look at it How about this guy aluminum chloride these are chlorine OK if I put an axis through here and I at the center of that access is the molecule to be symmetrical about it it right or put an access through here that compelled the same symmetry elements to waste all right there is the site of the molecule look the same as the site of the molecules along an axis yes OK so it's to times 2 is for the rest of the cemetery number for aluminum chloride for write to move in this direction 2 in this direction to in this direction or you'd rather too in this direction and 2 in that direction In this case is only 2 equivalent exit that never seen it explained this way in
any textbook but this will algorithm that we just described I think always works and fun sample were just doesn't work tried out yourself and see what you think
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 Title Lecture 06. The Rotational Partition Function. Title of Series Chemistry 131C: Thermodynamics and Chemical Dynamics Part Number 6 Number of Parts 27 Author Penner, Reginald License CC Attribution - ShareAlike 3.0 Unported:You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. DOI 10.5446/18940 Publisher University of California Irvine (UCI) Release Date 2012 Language English

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