Subtitles section Play video Print subtitles >> Bond character is somewhere between that of a single bond and that of a double bond. In fact, it's sort of about 30% double bond and 70% single bond that makeup this picture. So as a result, you have slow rotation about this bond and I'll say slow which means if I put a star next to this ethyl group here, I'll just put sort of a star here to remind us it's special, you do have a dynamic equilibrium where you swap positions right. So in this case, the start ethyl group assists to the carbonyl. In this case, the start ethyl group is trans to the carbonyl but this equilibrium is slow on what we'll say the NMR time scale. What I really want to do in today's talk is to give us more of a feeling of what's slow, what's fast both in time and also in energy. So when I saw this is about 30% double bond character, I want you to get a feeling for what that means in kilocalories per mole and how I know that number. Then I also want us to get a feeling for what happens as we cross from the slow regime to the fast regime so we'll get sort of one equation, a couple of calibration points on energy and time out of today's talk. Now, if you think about this, let's take a simpler situation. The case that people invariably use for didactic purposes is something with singlets. The case that sort of was the classic was dimethylformamide just because it's easy to think about and easy to simulate. So here with methyl groups, of course, you have singlets and just like the ethyl groups for one of them is more down field; the one that's CIS to the carbonyl is more down field; the one that's trans from the carbonyl is more up field. You have the same with the methyl groups here. I just want to, we'll call these HA and HB. So you have some equilibrium here we can call this KA and KB. In the case of a perfectly symmetrical molecule, the rate constant is going to be the same but if these were two groups say of disparate size like a tert butyl group and a methyl group, then the rate one way would be faster than the other because you would have an equilibrium constant that wasn't 1. In other words, you'd have an equilibrium constant where if you had a bulkier group, say the bulkier group would spend more time here and the less bulky group would spend more time here or vice versa. All right. So let's just imagine a little thought experiment for this situation. So, the situation we just saw was one where you have 2 singlets so this is just my old drawing of an H1 NMR spectrum. At some condition where you're slow, you're going to see 2 sets of peaks. For the same window if it were very fast, you'd see 1 peak and I'm just drawing the same spectral window and sort of plotting this out so I guess I'll kind of put these on the scale, on the same scale. So we can say this is fast and, of course, what's the best way to take something that's slow and make it fast in the laboratory? >> Heat it up. >> Heat it up. So this is cold and this is hot. All right somewhere between that point of cold and hot you hit a middle point which is called medium, yep, medium, okay. You hit a middle point, medium, where you're at what's called coalescence. Now at coalescence what's happening is each of these peaks is getting broader and broader until they merge. This is the uncertainty principle at work. Remember we talked about line width and I said that if you were able to measure the velocity for infinitely long, in other words, if there were never any spin flip, any relaxation, any swapping of population between alpha and beta states, your lines would be infinitely sharp, but I said when we were talking about the uncertainty principle your lines are typically about a hertz wide or a little less than hertz wide because your relaxation time is on the order of a couple of seconds. In other words, you cannot get your lines infinitely sharp because you're only literally measuring the velocity for finite amount of time. As you heat things up, what's happening is you're flipping faster and faster and so your lines are broadening out. So, what's really happening at coalescence so let's just go back to the equation I presented by the uncertainty principle, which is if you look at your line, in theory there's some exact position of the line. In other words in theory if you could make that measurement infinitely, your line would have this position but what you're getting is signal out here because you're not measuring that line with infinite time. You're not able to because of relaxation and so you have a certain width and that's the value when we talked about the uncertainty principle we called Delta nu, right? Delta nu basically is the half line width at half height. In other words, it is the level where you're sort of within these arrow bars so you're plus or minus Delta nu of that theoretical central value and we said from the uncertainty principle Delta nu times the time that's the lifetime so it's not really a half life it's almost like a half life, it's an eth life, you know, 1 over 2.3 instead of 1 over 2.0. Delta nu times time is equal to 1 over root 2 pi. In other words, if you're able to make that measurement for a second, then Delta nu is going to be .22 hertz or the line width at half height, the full line width is going to be .44 hertz. If on the other hand you're only able to have that lifetime be say a tenth of a second, so I'll say t equals .1 second, now we get to Delta nu is equal to 2.2 hertz. In other words, now that line has become 4.4 hertz wide at half height. [ Pause ] So what you were really seeing at coalescence when I have this sketch of this broadened hump like this, what you're really seeing is 2 fat lawrenciums [phonetic] that are adding up underneath there. So let me make this sort of with dotted lines to show a fat lawrencium like so and if the separation of the lines here, right, if this separation here if we call this Delta nu lines for want of a better term, in other words, the separation of the lines in hertz, at this point at coalescence then now each of these lines is fattened out so that its Delta nu, not the Delta nu of lines, but this distance here, this Delta nu is now, so if this is what we call Delta nu of lines that this Delta nu is now half of the Delta nu of lines. Does that make sense? In other words, each of the lines is broadened out so that it's half width at half height is halfway across and that is when your lines are going to be coalesced where you're no longer going to see a distinct line on the left, line on the right. If they're broadened anymore, they're going to be merged together and eventually you just have a single peak and you're at this situation here, but right now when they're broadened out they're broadened out to a point where they have merged together and so at that point Delta nu of the lines the separation of the lines times tau, which is now going to be our lifetime at coalescence is equal to 2 over root 2 pi. This is just the equation that we have over there except now because we have the difference each of these is fattened out halfway. If we have 2 of them, it's going to now be 2. So, what this boils down to then is a simple equation that tau when you just work this out is equal to 0.450 over the Delta nu of the lines. In other words, the lifetime at coalescence. [ Writing on board ] Is equal to .450 divided by the separation of the lines at a lower temperature. [ Writing on board ] Does that make sense? [Inaudible question] The .54 is simply what happens if I take in my calculator 2 divided by root 2 divided by 3.1415 and then I put that in the numerator and put the Delta nu lines in the denominator. [Inaudible question] Well, you mean the 2? Well, I'm saying because here our Delta nu is half of the separation. [Inaudible question] Within what you can measure it's exactly. Let me show you. Maybe the best way is for me to show you how things look as you vary here. So, basically if you go any, coming to, if it was more, you'd start to pull in and you'd start to pull together. If it's less, you'll see a dimple in the middle and let me show you what this can best be pictured as and this is just a simulation, this is from a chapter undynamic NMR spectroscopy in a book, let's see which book? This may be a book on dynamic NMR spectroscopy. So this is an old, just an old drawing of a simulation of what you would expect and it's really based on dimethylformamide and it's actually probably based on why DMF on a 60 megahertz spectrometer or something like this. So their simulation is as follows, and the reason I say it's a 60 megahertz spectrometer is the lines in this simulation are Delta nu lines is equal to 20 hertz. In other words, out of 60 megahertz NMR spectrometer that would be about 3/10 of a PPM, which is pretty reasonable. Now, on a 500 megahertz spectrometer that would be .04 PPM. So anyway for their little simulation, they're saying imagine that you have 2 lines, those 2 singlets, and they're separated by 20 hertz. Imagine that you have a T2, that's a relaxation time, of .5 seconds. In other words, imagine that the native lifetime for this molecule due to relaxation was half a second. In other words, that your lines are about .9 hertz width, full width at half height, not half width but full width at half height. So that's how your normal spectrum would look. Now, imagine that you start to heat this sample up so that you have rotation between the 2. So you have the 2 flipping back and forth. So, imagine here, for example, that K for, you know, our equilibrium this is our like dimethylformamide spectrum where we can call this A and B or star. [ Pause ] Imagine now that our rate constant, oh, yeah, imagine that our rate constant was 5.0 per second. If your rate constant is 5 per second, then your lifetime is 1 over K, right? So your lifetime at this point is 200 milliseconds, it's .2 seconds. [ Pause ] So your lines have met because they're not staying in the CIS or trans state as long. You can think of this as we started, here we're told and here we're starting to heat the sample up. Here they actually have a very slow K, K is equal to in this case .1 per second. In other words, it's a 10-second lifetime. Do not swap at any appreciable rate. As you heat up the sample in this simulation, they go to K equals 5 and K equals 10 per second so your lifetime is now .1 seconds and finally you get to a point and you notice so here you are and now your line width is still less than half the distance between them. So you still see this dimple here, this is it. Now you can kind of see right at this point now they are coalesced together. So, right at K is equal to 44.4 per second they're now coalesced together. Then as you heat the sample up more, as you get hotter and hotter, as you get faster and faster, now we get to K equals 100 so now your lifetime is 10 milliseconds and then you get to K equals 500 so your lifetime is now 2 milliseconds and finally you get to K equals 10,000. So now your lifetime is a tenth of a millisecond. [ Pause ]