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  • >> 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 ]