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

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

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

Â systems, and they have alternating carbon

Â carbon single, and carbon carbon double bonds.

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

Â So a simple example would be butadiene, and

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

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

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

Â 1:04

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

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

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

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

Â And you again, you assume that the potential energy along

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

Â So what do you call this?

Â You call this the free, free electron molecule.

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

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

Â 1:44

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

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

Â a linear shape, we can say that this is

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

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

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

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

Â 2:19

So we've worked out our L.

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

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

Â So we can draw out our energy levels here for

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

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

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

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

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

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

Â energies of these from our particle in a box presentation.

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

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

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

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

Â So you have h new radiation frequency

Â coming in here, is you cause an excitation.

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

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

Â So we can fill them up as so.

Â So with 4 electrons, and as fill, you

Â learned again in general chemistry, you put 2

Â electrons into each, and you have you have

Â opposite spins for the, for the two electrons.

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

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

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

Â what you get is you get a transition of this electron

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

Â 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

Â difference, or the energy corresponding to this radiation because

Â we can simply clarify, classify this as the energy

Â of this level minus the energy of this level.

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

Â So we can write that down.

Â So we can say, delta E, for this transition

Â here, is equal to 9 minus 4, brackets

Â h squared all over 8 mL squared.

Â 6:09

And now your mass here, we're talking

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

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

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

Â the length of the butadiene molecule, and we said

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

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

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

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

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

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

Â And that's given in, in joules.

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

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

Â And that should come out 203 nanometers.

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So you could move on to some other systems, and

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

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Â So here we have a structure, something like this.

Â So you have CH2 double bond CH, CH,

Â other double bond CH, CH double bond CH2.

Â And, again, you would approximate that

Â as linear going from 0 To L, in this case

Â your L is going to be 837 peak measures.

Â 9:10

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

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

Â that gets bigger, then this wavelength value here

Â is going to shift up to higher and

Â higher values until eventually you come up to

Â I think beta carotene, which has 22 atoms.

Â 9:33

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

Â 22 atoms, so that's an extended polyene chain

Â and the L is approximately 2900 picometers.

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

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

Â to higher and higher values and eventually will

Â shift into the visible regions of the spectrum.

Â because visible region of spectrum, in terms of wavelength,

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

Â So once you get into that, then

Â you're absorbing radiation in the visible region of

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

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Â