Add to Watchlist
Lecture 09. Vibrations in Molecules
203 views
Citation of segment
Embed Code
Formal Metadata
Title  Lecture 09. Vibrations in Molecules 
Title of Series  Chem 131B: Molecular Structure & Statistical Mechanics 
Part Number  9 
Number of Parts  26 
Author 
Martin, Rachel

License 
CC Attribution  ShareAlike 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and noncommercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. 
DOI  10.5446/18917 
Publisher  University of California Irvine (UCI) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Chemistry 
Abstract  UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 09. Molecular Structure & Statistical Mechanics  Vibration in Molecules. Instructor: Rachel Martin, Ph.D. Description: Principles of quantum mechanics with application to the elements of atomic structure and energy levels, diatomic molecular spectroscopy and structure determination, and chemical bonding in simple molecules. Index of Topics: 0:00:08 MethaneVibrations 0:11:56 Vibrational Modes 0:13:35 IR Spectrum of Methane 0:15:18 Harmonic Oscillator Energy Levels 0:18:24 Vibrational and Rotational Energy Levels 0:20:37 IR Spectrum of HCl 0:39:28 Force Constants 0:41:55 Anharmonic Potential 
Series
Annotations
Transcript
00:06
so today we're going to finish off our discussion underground examples of vibrations and Holly Thomas molecules and how we can figure out which ones are ironic active by symmetry and then we're going to move on to something cases of more simplified molecules were capital linear molecules where we can calculated using some of some pretty easy calculations like the the bond length the force constant things like that are foreign the bonds involved of course we can do that quality time molecules but we need a computer so In the discussion of which vibrations are firearm active 1 of the things that came up last time is the fact that we don't know anything quantitative about vibrations from that analysis we know where winter whether we can see them in the spectrum but that doesn't tell us what energy they show up that way and it doesn't tell us on anything about how intense they are for them the we need to know some additional information we can get it is the theory so we will get into that today for the cases of atomic molecules then next time on Wednesday going to go over the selection rules in a little bit more detailed were to talk about this in qualitative way we're going to look at how you actually calculated and then that's going to be cared for in the 1st mantra so what CEO Halsey phenomenon that they have any questions before we start yes the materials for the midterm does not cut off everything through you know what we do on on Wednesday is going to be honored OK so let's continue talking about fund vibrations are actually been doing this example of methane which as I said is going to be a little bit harder than probably we're actually can have do in the dance so if you get this year and in shape OK so last time we talked about the the bonding now want to talk about its vibrations and so the question is identified a vibrational modes and determine whether the Roman active and so costly users during the we're going to do this using group theory and so as always when we talk about molecular motion are basis is now a little coordinate system on each added so we need to look at how these things transforming the molecule around OK so methane has 4 Adams and we have to take them all into account because we're talking about these vibrations so firmer basis that we set up always has something to do with the symmetry of what we're actually looking at and so here we are interested in is the displacement of the Adams relative to each other so we need cordon system on each other and so there are 15 little unit actors in our bases here and so if you want to set up the actual matrices for this union 15 by 15 major cities so let's not do that looks so use the shortcut to get the character so that I hope but you know we looked at the water example you get something that's that's 9 by 9 it's useful to do that once you for 1 operation just yourself how it works and then then you're gonna want to see that so that's good OK so 4 the identity operation nothing changes of course and so we get 15 for the character of that so of course are identity matrix just looks like a 15 by 15 matrix with 1 of diagonals and zeros everywhere else now let's look at what happens when we look at the C 3 workstations so this is where it starts to get a little bit more complicated because when we look at C 3 rotations of methane ah basis is not just that the individual bonds anymore like it was we're looking the bonding over there we can use the short cut and it was it was simple enough because we were talking about just the bonds swapping places so here we have these little x y and z axes on each other and when you do 120 irritation X doesn't matter onto wide directly end up with things that are in between and so let's consider this year I'm not going to set up the whole giant matrix the let's consider this sort of piece by piece so we're thinking about this year rendered as our seethrough rotation were holding the molecule by 1 of the ageism and rotating around look the common any age that's on top so for that you know these things to remind him the same and we have to fill in Arbil rotation matrix as we learn before In order to get that and so you plug in the actual Angola were treating 3 and you get that the trace about a 0 and you can do this for the other hydrogen is as well and everything changes places in space and those contribute 0 to trace also and this is something that I if you're not sure about the rotation matrices that's a good thing to do to practice on also recommend that the general form of how you make rotation matrices if you don't remember that might be a good thing to put your she that you have to write down its it's a useful thing in a video of a again the reason that we can't use the the standard shortcut in this case is that for the particular operation that we're doing elements in our bases don't map onto each other if you need a onetoone fashion so we have to do to break it out like this and what we find is that the character of c 3 0
07:15
0 OK so now circuits C 2 so can remember for the methane molecule the C 2 axes are going between the hydrogen carbon bonds so it's so you before we're holding the molecule by 1 hydrogen and now we have to hold it with 2 of them sticking out and 1 is is coming out of your 1 pointing at me so we we look at our C to Accies there then all we have to treat the the common enemy the hydrogen separately so all hydrogen switched position will be so those contribute 0 the character and on the carbon atoms z is dynasty the same an X and Y and contribute minus 1 and so if we and all those out we get minus 1 C 2 OK have about S for so ,comma this one's hardly visualizing methane hold the molecule by the top hydrogen again so that we have the other 3 hydrogen sticking out like this rotate 90 degrees and then reflect through plane perpendicular to the arbitration and when you do that you see that there's the access on the carbon changes sign and everything else did you 0 so we get a minus1 for asked for and again if you have trouble visualizing not you go check out the secular models and and make sure that you can prove it to yourself OK so the last thing that we have to deal with it is our die he drove claims so on these vertical planes we have to look at what the Chatham is doing so for the carbon Excellency stayed the same why changes and the same thing happens for the hydrogen is the in place and then the other ones get reflected and so we have to add up all these contributions to get the characters and we do that we end up with 3 so now we have a reusable representation for devotions of methane so remember we're not just looking at the vibrations yet we have this basis tells us something about the displacement of all the atoms and so we have to reduce our reducible representation and we use the reduction formula to do that and here's what we end up with so we get A1 plus EPlus 31 plus 3 T 2 we had 15 letters in our basis so we should end up with 15 elements here and it might look like we don't have enough until you remember that he is doubly degenerate so that answers to NTU is strictly designed so every she counsels as 3 cemetery species and that representation OK so after we reduce our representation then we have to take out the cemetery species that correspond to translations and rotations because those those don't show up in our vibrational spectra and so if we look at the character table we see that for translations that's going to take care of 1 of the steps that degenerate sets of T 2 so there's an x y and z In there so translation of this molecule 1 NED Accies corresponds to that so that removes 1 of them from consideration OK so how about rotation so RXR wineries E are going to be all degenerate and those fall into the T 1 category and again what is reading this right off the character table so the vibrations are everything that's left over so we have a 1 B and 2 T 2 and so you can look at the character table and decide which ones are a firearm active OK so let's look at what the actual racial modes
11:59
or for methane so there a little bit harder visualize that this is a good picture of of what's going on so we have this this wagon where he has some of the bonds are are moving with respect to the other ones there's a twist the symmetric stretch is easy to see that's the 1 roll the bonds reflecting in an out and then there's the scissors motion so we try to put this in terms of our answer that we got we see that the 1 that some 81 is enacted there's no there's no component of the dipole moments and no component polarize ability that's it corresponding to that and that's fine it just means that we can't see it that vibrational mode doesn't talk doesn't give us anything and spectroscopic methods we find that the ones that are that had symmetry the AR Rahman active and to vibrations are higher and Roman active and so let's take a look at what this looks like in practice so if we look at the IRA spectrum of methane we expect it to have a couple of a few vibrational modes so T 2
13:37
Is that generate so here's our spectrum of of methane and so we can see that we get these peaks you again the group theory analysis just enables us to say that we're there and that there there it doesn't tell us anything about where they are or what the intensities are and if we look at all this fine structure those are the rotation positions so when we excite this molecule to an excited vibrational state there's enough energy in there to excite all the rotational transitions to and so we see all these recreational Leisenring talk about that in more detail in a minute effort ,comma molecules here's the Rollins
14:19
spectrum of of methane and so you notice that the intensities here are higher than the ones on the societies of the Stokes lines versus the anti Stokes alliances is the 1 where the electromagnetic radiation is giving up a quantum of energy to the molecule rather than taking it from the molecules so we can rationalize the intensity is nothing but otherwise these things look kind of complicated and so the rotational find spectrum of different modes are overlapping into really predict what this is going to do you need a computer you can do it you can do it very accurately especially for her 4 molecules better relatively small like this but if you really want to understand the details of what's going on it's best to do a computationally for smaller things like diet atomic molecules we really can do these calculations just with a pencil and paper and so it's more useful to to zoom in and taken a closer look at that OK
15:19
so let's all go back to thinking about what this this looks like in theoretical sense so we're talking about motions of the molecule for a atomic this is like easier to visualize because we just all always happens just this molecule by reading in an hour there's only 1 motion that's going on so the specter of a lot easier to interpret what we can do things a lot more quantitatively without using computational methods and again I don't want to imply that the computational methods art very precisely that you can but incident you really can't you can't just you just can't do it easily in class so we're going to focus on this for purposes of calculating things that are quantitative OK so things to notice about the harmonic oscillator formalism these are all things that they you learned last quarter so we have this system where there is a harmonic potential well of course such an approximation were treating it as a harmonic oscillator that doesn't mean it's it's always so I really like that but for small displacements it works pretty well notice that there is a 0 . energy so the lowest state is not exactly at the the bottom of potential that's that's important for the harmonic oscillator treatment and again remember it's different from the rigid rotor approximation there we were allowed to have a 0 Invitational stay here were not I remember them these things are quantized we have energy increments of HA as we go up and in energy In and of course where looking at the offices of potential energies we can write as as STX wearing solely dependent on of onedimensional system it's only dependent on that interview clear and X equals 0 is 0 that doesn't mean the nuclear touching each other that's its equilibrium in a clear separation and so we measure bond lengths of molecules of course what we're measuring is the equilibrium distance the average distance in reality these things are always moving around by bringing and OK so we can use this to get quantitative information about things like the bond lengths and the force constant which tells us something about how stiff the bonds are and we can calculate a quantitatively using some pretty simple approximations and if you don't remember some of us from last quarter it's useful to go back and review what the harmonic oscillator wave functions look like so they're there for me polynomials you should have seen this last quarter it's useful to go look at a particularly census next lecturing talk about selection rules and might have to to look at different symmetries of different vibrational states OK so here's what that corresponds to when we actually look at the Spectrum so here are agreed lowest energy 2 states Of the harmonic oscillator so where were whereupon quantum number here is is called New typically for the harmonic oscillator where functions so we have new equals 0 and new equals 1 and then all all of these little transitions in between the represented by the red and blue arrows are the rotational transitions so remember vibrational transitions take a lot more energy than rotational ones so when we excited vibrational transitions all the rotational ones come along for the ride and we see them in our spectra and it turns out that kind useful because we can use them to get some valuable information and so were representing the state's indirect rotation as on the state that has won a number of new for the occasional preferred sorry for the vibrational state and quantum number J for the rotational state and if we look at the I spectrum in this case it's HCL we have 2 sides inspector so on this side were going from 0 J 2 1 J minus 1 so we're starting in the lowest energy irrational state taking it up to the 1st excited state and while we're doing that we're going down In rotational transitions on the other side we're going the other way so we're going from 0 2 1 in the vibrational transitions and we are going up in the vibrational restoring the additional state so that that needs to get fixed so let's let's look at that have the correct next once
20:28
also hit that another thing that I want to point out before we move on you know I expected is that sometimes you'll see them plotted with the peaks pointing down some that
20:38
the even with the peaks pointing out it's just a different conventions it doesn't really matter and you know that that's something that you'll see in the literature I it's also important to pay attention to whether spectrum is plotted in frequency units on wave numbers because Of course the energy of the direction of increasing energy is going to go the opposite way depending on on for size frequency units or we like those in the energy will go the opposite way OK so these things also have historical names so if we're going down Invitational states that's called the P branch and if we're going up in the rotational states that's called the ah branch these are just a historical names it's it's useful to know they exist for purposes of reading literature about you for for the most part I'm mostly concerned that you understand that the physical basis for this and what's going on I again here some some examples of spectra here's 1 and wave numbers here's 1 and frequency units you will see both and you should definitely I know how to deal with both like being able to fluently convert between frequency and wave numbers and energy units is definitely something that you should be with you OK so what's what's and look a little bit closer why spectra look the way they do so we have an are branch and a P branch and there's all this rotational fine structure in there and it's a matter because on 1 side were going down a quantum Invitational transitions when we step through all the different transitions that there are relying doesn't represent a state represents a transition between 2 states and if we're looking at you know where going down a rotational transitions depicted on 1 side and going up on the other side but there's no peak in the middle so the fundamental frequency that we're looking at here that's telling us about the energy and going from the 0 2 1 by regional state doesn't have a peak there and the reason for that is the selection rules so we have to have a difference in of irrational state of plus or minus 1 so 0 was not allowed so we don't see a transition year so only 1 that fundamental frequency we have to pick up points in between these 2 sets of vibrational whites OK when I was there to say about that yes at the May Q Branch is the name for the Central Line which is missing as will see next lecture there are sometimes when you do get the will will talk about what situations arise in which there it's called the King Ranch the other thing I want you to mentioned again it look at the intensities here so the lowest energy transition is not the 1 that has the highest intensity some maximum up here and again that's because that generosity when we get to a little bit higher energy states there are more ways for the the system to be in that state and so it's more highly populated if we were to raise the temperature so that more energy is available we would see that those curbs would flatten out we would get the higher energy states more populated and also just the whole thing with would spread out because there's more generously there's more ways to occupy those states OK so we've talked about what the spectral look like and on how to interpret them in a really qualitative way with skin and actually calculating some things from them OK so the fears our spectrum of HCl and this particular 1 is plotted in refrigeration units and we have this character lines in the center that's on either side of the fundamental transition From New equals 0 new equals 1 without a change in occasional state and so that frequency is what's gonna tell us about the energy of vibrational transition that we won it's another thing I didn't bring up before it is if you look at this now but it's blowing out like that you see that the lines splits the nobody know why that is and so were you know we're talking about these bond vibrations there's really only 1 vibration that economic growth so why would have 2 different lines it's like there's very slightly different energy there in the in the bond vibrations yes that some that's a good gas that to the acolytesthe isotope effect so if we have I say deuterium instead of hydrogen there's going to be some you know little natural abundance population different isotopes in the sample and deuterium is going to be a lot heavier than when hydrogen and so will see no difference in the vibrational frequency and a lot of times when we're interested in looking at our vibrational states of molecules 1 way that you can do that is isotopic we label specific things in order to to make the frequency a little bit different if we have time to get into some applications next time I'll show you some examples of our people have been very clever about using isotope labeling to break into generously and look at a different time vibrations in a complicated molecule OK so for now let's on and stick to I'm not very complicated molecule because we can calculate a bunch of stuff about it quite easily OK so I like this picture because it's showing us on the energy level diagram what we what we're seeing in the spectrum OK so we're looking at this transition From New equals 0 2 new equals 1 and then we have the corresponding rotational transitions trying appear so we have to take the average position of these 2 lines and if the center frequency and that's going to tell us the fundamental frequency of that by racial transition which is what what we want now and so we know that in general the fit the spacing between the lines for rotational spectra the same thing here is to be it's 2 times the occasional constant but so that's for the ones that are spaced apart evenly here there's a center went missing so that that center Caroline's is spaced apart by 4 B and so are changing frequency here called Delta is for your age and that's what it is in the works so let's put this in terms of what were actually interested in in finding out so are kinetic
28:28
energy use is you were describing with the potential and that's 1 half I made squared and we can write this down in terms of the reduced mass so here a moment of inertia can be written pretty easily in terms of reduced mass because of the diatonic molecule and you know again as we've seen before you know a diet helmet molecule like this where 1 of the items really heavy and the other 1 is light you hopefully singers summer homework problems you end up with the vibration and you're looking like the chlorine is just staying still mobile protons is bouncing in and out OK so we know the spacing between the rotational wines which is the same for the final the fine structure in the vibrational spectrum as it was for the pure recreational spectrum that we looked at and it's for reform the central transition and so we can just plug it in and the change in energy between the states and so know where were interested in getting the change about the change in energy which is just equal to H times the frequency difference so as with any kind of us have to be a problem like this equals each knew that energy difference for the transition is the same as the energy the photon went in and was absorbed to promote that molecule the higher state the only thing that's a little bit tricky here is that we have to remember what the states actually mean so we're not able see a direct line for the transition were interested and we have to infer from the structure of the rest of the rotational said OK so we have the energy therefore I didn't work out the number about but we know what it is it's just H. times at frequencies so we have the microwave value that was calculated for the bundling of HCl so that means like somebody when measured the direct rotational spectrum of this molecule 1 got the bond length which again do you they looked at that and it that's given us . 1 2 7 mm which I looked up to now and calculate the 1 Linkovich seal from the vibrational spectrum and see how well OK so we have our own new our squared equals to age squared overall the energy here and if we plugged in values and sulfur are within the significant figures that we have here given that the numbers that we got the same answer so hopefully this convinces you that you UK users pretty simple analysis to get information about the molecules lease in the case of of diatonic molecules and again obviously this is a lot more useful for larger molecules that that we don't already know everything about the motions of and there we use the same procedure it's just that you need computational methods to do it OK so what's up and continue discussion really look at just had a deal with with different units just as a reminder so again has the same spectrum this time exploded with the peaks going down and it's an wave numbers they're just different conventions and you'll see both in the literature spectroscopy is really the land of thought confusing notation and things being given in different units and drawn different ways that's just because of historical conventions so 1 issue is that chemists and physicists are looking at a lot of the same things and they have different conventions and how they prevent the having presented things on if I were the dictator in the world would be that way but we don't get to pick you know how these things are represented in so that's that's part of it is just learning the conventions that you're going to see in different parts of the field and particularly you were talking about you may be going this summer that he can seminars and seeing people's current research that's part of the language and and part of being able to understand what's what's going on there so let's look at us so we want calculate the bond lengths and you have a spectrum we numbers you want to pick a pair of lines that's relatively close to the center so why do we want to compare reminds close to the center we know that the rotational constant is the same and these lines space fighting To be Invitational fine structure but remember we have centrifugal distortion so when we get the molecule excited to higher and higher rotational states then it doesn't behave as a perfect region bordering stretching more and more and that's the same thing here so if we kick lines corresponding to really high energy recreational states were not going to get the best value for part of rotational constant there and there there will be unnecessary error in the calculation of the violent so we want to pick something from a part of the spectrum where our assumptions are more likely to be correct so take a couple of their close to the sun and so that gives us be in wave numbers so here it has it's only over it so it's an inverse centimeters rather than frequency units with the appropriate conversion and then we can plug all our stuff in and we get something for and again it's going to keep track of units so for the moment of inertia we get kg meters squared which we should and then we can solve for our given that we know of the expression from the notion that I atomic molecules and then last step is check that your answer is in the right units and that it's a reasonable order of magnitude so we did get linked units which is good because we're talking about a bond length and in nearly got something that's on the of strands which is a reasonable value so as far as as calculating these things paying attention to units and whether you got a reasonable order of magnitude is a large part of the battle that will really they long waits and terms of figuring out whether your answer makes sense OK so this is 1 type of information and that we can get from vibrational spectroscopy so we can learn about the links of bonds we can also learn about the force constant of bonds
36:20
so the force constant is telling us something about how floppy your house stiff that bond so we're going with the harmonic oscillator approximation were just assuming that we have wheels frames in between our Adams and they're bouncing back and forth and of course you can have a really soft spring or you can have a very stiff spray and forced us the molecules with tells us about that and so that has to do with the vibrational frequency so noticed that in order to if the bond lengths we didn't actually use anything that had to be done from the vibrational spectra were just using the fine structure of the rotational transitions which we could have gotten for microwave spectroscopy by microwave spectroscopy is not typically use very much because we can get all this information from something like I OK so Oh is our frequency ingredients which we have our center frequency that we measured office spectrum in perks and we have to converted and that equals the square root of failure over the reduced mass so what's came here that's the springs constant so just were using hooks loss from introductory physics since we're making this assumption that molecule behaves like a simple spring and that's the the force constants that were interested in getting so if we plan on our values again remembering to pay attention to units 1 of the things that that's that could cause somebody to slip up here is that you have here your mass in atomic mass units you probably produced nasty and then I when we have our work force constant that needs to be in kilograms because we will end up with something in a new leaders and so you need a conversion factor from kilograms to India and so we get a force concentrates Cl of almost 500 kilometres so just because I imagine people are going to be concerned about this on the exam I will give you things like the convergence between kg and 2 in the USO unit conversions and physical constants that you might need like that I'll give you on equations I'm not going to give you because you have you have a cheat sheet OK so we learned something about our force constant the bond that tells us whether it's stiffer floppy but since we only talked about HCl so far we haven't really put it in context we don't know what that means Solis just look at this for a few different diet comic molecules so if we look at HF here vibrational frequency In parts it's force constant is 970 Newtons per meter so we can see that HCL is a lot flop eared NHS which makes sense if we think about it yet again with our intuition from general chemistry so we know that you know fluorine is a much smaller molecule has a smaller electron cloud it's more like a negative those electrons are held more tightly to the nucleus and so we expect it to have a stiffer bond when it's making it a bond with hydrogen than HCL if we keep going down bromine is a little bit less different by HP is floppy ears still there then the HCl and for HIV have a much more dramatic effect so as we're going down the Holden's we see that the force constant the bonds is getting smaller Cook corresponding to those electrons being held less tightly and making a oneyear bond so looking in the other direction you here were comparing going on holiday tunes were everything is a single bonds if look at carbonmonoxide so that's a triple bond between 2 Adams the growth relatively small we get a very large force constant for that so that that CEO proposed 1 is very stiff and the nitrogen monoxide triple 1 is a little bit less staff so again knowing these exact numbers is an allimportant except that it gives you some insight as to into what we're talking about when we look at that force constant relates back to 2 things that we know from general chemistry OK so the last thing that I would like to do today is to talk about what happens when we have an harmonic potentials so so far we've just assumed that were looking at harmonic potentials everything is ideal we can treat everything is a perfect harmonic oscillator what happens if we can't so far we haven't and harmonic potential looks like the US and our harmonic approximation isn't perfect that means that we have to have a correction term 2 Our of vibrational state and so here's the equation for the Morse potential which is a commonly used potential and so we just end up having the add some correction terms to account for the fact that are potential is not a perfect harmonic oscillator as far as what you need to be able to do with this right now I just want you to know that it's there that in some cases we are dealing with molecules that are going to behave as a perfect harmonic oscillators and in that case there are things that we can do it and there corrections that can be made for the animosity about potential and in computational chemistry is 1 of the important things in various problems is coming up with potentials that accurately represent the physical reality so not just for vibrational states but for all kinds of processes that happen atoms and molecules spectroscopic weight if you can come up with a potential that describes the wheel what these with the energy differences between these states look like that so large a part of being and will solve problems OK so we're going to quit therefore today on see should be able to do a lot of practice problems that because Beyond that we've gone through has these things qualitatively right we still have 5 minutes please pay attention for a 2nd so it should not have the information they do most of the practice problems the online next time we're gonna talk about how you calculate the selection rules give in figure out whether particular transitions are going to happen or not and for Cleveland and sometimes people talk about applications anybody have any more questions right now or I will see 1 wanted
00:00
Computer animation
07:14
Computer animation
13:35
Computer animation
15:18
Computer animation
20:37
Computer animation
36:19
Computer animation