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'Kay, a nice example of the application of the particle and one dimensional

box model is the analysis of the electronic spectra of, of polyenes.

Now polyenes are, as you should know from general chemistry, are conjugated pi

systems, and they have alternating carbon

carbon single, and carbon carbon double bonds.

And [COUGH] let's take a, one of the simplest examples of a a polyene.

So a simple example would be butadiene, and

butadiene, CH2, CH, CH, double bond CH, CH2.

Now this contains, again from your general chemistry, this contains

four pi electrons or one pi electron for each carbon atom in the molecule.

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So, even though it's not not exactly linear in shape,

what you can assume is that these pi electrons, they, they,

they move a, move along the molecule like the particle in

the, in the one-dimensional box that we were, were talking about.

And you again, you assume that the potential energy along

the chain is constant, but rises sharply at the the ends.

So what do you call this?

You call this the free, free electron molecule.

And, we can use the particle in a, in a

box model to calculate the the energies for, for the system.

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So what we're saying here is, if we go

back to our butadiene molecule, if we assume it's

a linear shape, we can say that this is

our, our, our, our particle in a box model.

And here we go from, from 0 to L, so we can work out this distance 0 to L.

And if you work out a number of single and double bonds in that molecule,

you can find, that for butadiene it's about 556 picometers.

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So we've worked out our L.

So, to understand the the, the spectra, what you get in when you absorb

radiation, is you get a transition from, from one energy level to the other.

So we can draw out our energy levels here for

the for the particle in the box, which is light.

So here we have a qualitative energy level guy, and here we have E.

And here we have our different energy levels, for our particle in the box.

So here we have n equals 1, n equals 2, n equals 3, n equals

4 and, and so far on, we could keep going, and now we can know,

let's put them in a, in a different color here, we know the

energies of these from our particle in a box presentation.

So they're h squared over 8 mL squared, for n equals 1.

What you then have, you have 4h squared, all over 8 mL squared for n equals 2.

And then you have 9h squared, all over 8 mL squared, for n equals 3.

Now, so what happens when you irradiate this at the right frequency?

So you have h new radiation frequency

coming in here, is you cause an excitation.

Now let's first of all, we have 4 pi electrons,

so we have to fill these electron energy levels, levels up.

So we can fill them up as so.

So with 4 electrons, and as fill, you

learned again in general chemistry, you put 2

electrons into each, and you have you have

opposite spins for the, for the two electrons.

So that would be your four pi electrons in the, in the butadiene molecule.

Now, when you come along with the appropriate radiation, and you

can match this gap here, n equals 2 to n equals 3,

what you get is you get a transition of this electron

from the n equals two level to the n equals 3 level.

So if we go over here, we again have our, sketch out our energy levels.

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So, we can easily work out then the energy

difference, or the energy corresponding to this radiation because

we can simply clarify, classify this as the energy

of this level minus the energy of this level.

So it's 9h squared over 8 mL squared, minus 4h squared over 8 mL squared.

So we can write that down.

So we can say, delta E, for this transition

here, is equal to 9 minus 4, brackets

h squared all over 8 mL squared.

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And now your mass here, we're talking

about electrons, so it's the mass of electron,

and the mass of electron is 9.11 by 10 to the minus 31, that's in kilograms.

And then you need to multiply that by the length, length of the box in this case,

the length of the butadiene molecule, and we said

that up here at the start that's 556 picometers.

Picometers is about 10 to minus 12, so we'd like to keep it in SI

units, so it's 556 by 10 to the minus 12, and that's in meters,

and that of course, it's all squared, so that's all squared.

So if you plug that into your into your calculator, you

should get a value of 9.74 by 10 to the minus 19.

And that's given in, in joules.

And then, if we wanted to get the wavelength of that, we could say, lambda

is equal to Planck's constant times the speed of light, divided by this delta E.

And that should come out 203 nanometers.

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So you could move on to some other systems, and

the next system you could try, say, would be hexatriene.

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So here we have a structure, something like this.

So you have CH2 double bond CH, CH,

other double bond CH, CH double bond CH2.

And, again, you would approximate that

as linear going from 0 To L, in this case

your L is going to be 837 peak measures.

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So you could keep going on like that, you could get, and what you will

find is, as you keep increasing the length of the polyene, from 0 to L, as

that gets bigger, then this wavelength value here

is going to shift up to higher and

higher values until eventually you come up to

I think beta carotene, which has 22 atoms.

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So if we talked about beta carotene, that's got

22 atoms, so that's an extended polyene chain

and the L is approximately 2900 picometers.

So, what you find, this is the pigment that's present in, in carrots.

So as you, you increase that L, what you find is this wavelength here will shift

to higher and higher values and eventually will

shift into the visible regions of the spectrum.

because visible region of spectrum, in terms of wavelength,

goes from about, about 400 up to, up to 700.

So once you get into that, then

you're absorbing radiation in the visible region of

the spectrum, like you were for beta carotene, and you get a, you get a color.

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