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