 So it turns out that the 1D particle in a box model is not just useful as a first example of solving a quantum mechanics problem, but we've seen it can be used to describe the pi electrons in linear conjugated hydrocarbons, at least qualitatively. So the next question is how useful, how quantitatively accurate is that model? So we've seen that for short molecules like ethylene or butadiene, we can use the model to predict that the wavelength of absorption, which is in the ultraviolet for these molecules, gets longer as the molecules get longer. And we saw for molecules of 8 or 10 or perhaps 12 carbons that it's pushing up towards the visible, but we can make even longer molecules still. So I'll put the structure up here of a much longer molecule. So notice, so this is molecules beta-carotene, which is a molecule you might have heard of. That's the compound that gives carrots their characteristic orange color. It's a reddish-orange molecule, at least in large quantities, it looks reddish-orange. So the structure here, notice, is conjugate. I've got double bonds alternating with single, double, single, double, single throughout the whole backbone of this molecule. And it's a much longer backbone than the shorter molecules like hexatronene or octatetrene that we have considered previously. So in particular, if we count the double bonds in this molecule, there's one, two, three, four, five, six, seven, eight, nine, ten, eleven double bonds. So I've got, each of those double bonds has two pi electrons in it. So I've got a total of 22 pi electrons. So that's a much longer molecule, and we can ask ourselves, maybe, to predict what is the wavelength of light absorbed by this molecule to see if we can predict the orange color that it's going to have. So to do that, we can use the particle in a box model. So again, we're treating the pi electrons in this molecule. Each of these pi electrons is confined to a box, so that box is a long, somewhat one-dimensional box, and these conjugated electrons can zip back and forth in this box, and we'll see how well they can be described by this particle in a box model. So again, we have energies, we can predict, if we know these constants m and a, we can predict the energies of these molecules, I'm sorry, these electrons within the molecule. So there's a ground state, E1, E2, E3, and so on. And of course this goes up for a long way. And in order to be able to predict what color of light is absorbed by the beta-carotene molecule, we need to be able to predict which energy levels these electrons actually occupy. Do they occupy the E2 or the E3 or the E1 energy level? So these 22 electrons that live in the beta-carotene backbone, those pi electrons, which energy levels do they occupy? So now we have to think back to the rules you learned in general chemistry with names like the Aufbau Principle and Pauli's Exclusion Rule. So Pauli's Exclusion Rule tells us, for example, that we can't fit more than two electrons into one orbital. So once I've put two electrons into this E1 level, it's full, I can't put any more. Hund's Rule also says things like, I fill singly first, so I put one electron in here and then I put the second one in there before moving up to the next level. But if I have a total of 22 pi electrons and I'm filling them up two electrons per level, I'm going to get all the way up to the 11th energy level. I'll put two electrons in there. Those are my 21st and 22nd electrons. And then the 12th energy level, E sub 12, is empty because I've used up all my electrons climbing this ladder. So that's what we predict is that the lowest 11 energy levels will be full. The 11th energy level is the highest occupied orbital, the HOMO. E12 is the lowest unoccupied level, the LUMO. But when we're thinking about light that will be absorbed by this molecule, in order to absorb a photon of light with energy Hc over lambda, that photon has to have an energy that matches the difference between E11 and E12. There are other excitations I could make. I could lift a molecule from E3 all the way up to E12 if I wanted to. But that's going to be a relatively high energy photon. The lowest energy photon I can find that will be absorbed by this molecule will be one that only lifts a pi electron from the 11th energy level up to the 12th energy level. So we'll predict that the lowest energy photon is going to correspond to the difference in energy between the 12th and the 11th electron, 12th and 11th energy level. And again, the way I got those particular energy levels is by knowing how many electrons were occupying the energy ladder, how many pi electrons occupy this box that we're treating the pi electrons as particles in this box. So we know the expression for how to calculate the energy of each one of these energy levels. It's H squared over 8ma squared times 12 squared, and then H squared over 8ma squared times 11 squared subtracted from the first one. So to proceed any further, we need to know what are these masses and box lengths, the mass. This is the mass of the particle that's confined to the box. So our particle confined to the box is the pi electron, so this is the mass of an electron. And A, the box length, we need to know how long this molecule is, in particular how long the conjugated backbone portion of the molecule is. So we'll come up with just a rough estimate for that, and I'll tell you that the distance between carbons along the backbone of this molecule is about 1 angstrom, or about 10 to the minus 10 meters. So because our backbone has 22 pi bonds in it, so there's, sorry, 22 electrons, 11 pi bonds. So there's 11 of these pi bonds, and there's 10 of the single bonds in between them, because it ends with a pi bond on this side and a pi bond on that side. So the total length of the molecule is going to be those 21 bonds, 11 doubles and 10 single bonds multiplied by 1 angstrom. So that's 21 angstroms roughly, or 2.1 nanometers. And this is a fairly rough calculation. That number is only 1 sig fig, because we'll see when we get to the end that our ultimate answer is not going to be terribly accurate to lots of sig figs, so we're fine with just this rough approximation for the length of the box, but roughly speaking, the backbone of this molecule, the conjugated backbone is 2.1 nanometers long, and that's our value for the box length. So if we go to plug into this difference in energy, we can find Planck's constant squared in the numerator, the denominator, I've got an 8, I've got the mass of an electron, which is 9.1 times 10 to the minus 31 kilograms. My box length is squared, so that's this quantity, 2.1 nanometers, 2.1 times 10 to the minus ninth meters, don't forget to square that. I'm multiplying that by this difference 144 minus 121, so I'm multiplying that by 23, and when I do that calculation, plug those numbers into the calculator, what I find is 9 10 to the minus 19th joules, I get 3.2 times 10 to the minus 19th joules, is the difference in energy between these two energy levels in particular. I'm less interested in the energy of that gap or the energy of the photon than I am in the wavelength of the photon, so if I rearrange that equation to solve for the wavelength of the photon, again using Planck's constant, the speed of light, 3 times 10 to the 8th meters per second, divide that by the energy that I've just obtained, I suppose it's worth the second to talk about units. In this first calculation I had joules squared, so one of those joules turns into the joule that we have for our answer. The units that remain behind, there was a kilogram meter squared in the denominator, second squared in the numerator, that kilogram meter squared under second squared, one joule is a kilogram meters squared per second squared, so that composite kilogram meter squared, second squared which turns into a joule in the denominator, which cancels one of the joules in the numerator. In the second expression, I've got a Jewel on top, jewel on bottom, seconds on top, seconds in the denominator on top, so all I'm left with at the end is meters, which is good because those are the units that a wavelength should have. And when I put those numbers into the calculator, I find 6.3 times 10 to the minus 7 meters for 630 nanometers. So our prediction is the beta-carotene molecule, since it has electrons in its 11th level, but not the 12th level, if it behaves as a particle in a box. Light of wavelength 630 nanometers is required to lift those pi electrons from the 11th level up to the 12th level, so we predict it absorbs light at 630 nanometers. Well, let me point out now that answer is not 100% correct, so here we begin to see the limitations of the particle in a box model. We predicted 630 nanometers. We didn't actually deserve that many sig figs because the model is only an approximate model. In the real world, beta-carotene absorbs at about 4 to 500 nanometers, so we can see that this roughly 20-30% error in the calculation, that's because the beta-carotene molecule is actually not perfectly a particle in a box. The potential energy is not zero along the backbone. There's a slight preference for being some places along the backbone than others. It's not perfectly one-dimensional. The box has some two-dimensional or three-dimensional character to it, so the particle in a box model isn't perfect, but it's good enough to give at least a rough estimate. If we want to think a little further about the color of beta-carotene, if the molecule actually absorbs light of 400 to 500 nanometers on the, roughly speaking, the visible part of the spectrum ranges from about 350 nanometers on the violet end up to about 700 nanometers on the red end, depending on how good your eyes are on either end of the spectrum. If it's absorbing light in the 4 to 500 nanometer portion of the spectrum, that's in the green-blue violet portion of the spectrum, 400 to 500 nanometers, so beta-carotene or containing beta-carotene or other plants that might contain beta-carotene. When white light gets shined on them, they absorb the light that's in the green-blue violet part of the spectrum. They don't absorb the light that's in the red-orange-yellow part of the spectrum, so the part of the light that gets reflected back to our eyes is in this orange side of the spectrum. So that's the reason that the molecule appears, reddish-orange or carats appear orange is because those colors are not absorbed by the molecule and those are the colors that remain to be detected by our eyes. So summing up what we've done, we can use the particle in a box model for real conjugated molecules to make predictions about what colors of light they'll absorb. The answers aren't perfect, so as long as we understand that they're going to have some error, we can at least make a rough prediction of what colors of light are going to be absorbed. It's more useful perhaps in the qualitative style of predictions, saying that beta-carotene is going to absorb at a much longer wavelength than hexatriene or octatetrene. So oh, we're going to do one more thing actually before we stop. Once we've calculated the energies, the difference in energy between energy level 11 and energy 12, I can do one more calculation relatively easily, which is to ask myself, what is the probability that a molecule, that a pi electron in this molecule will occupy the 12th energy level relative to the 11th energy level? That takes us back to the Boltzmann distribution, that ratio of two probabilities, the relative occupation of the 12th level compared to the 11th level, is this e to the minus difference in energy between states relative to kT. So we've already calculated delta E, that was 10 to the minus 19th joules, kT. If we take Boltzmann's constant, and let's say we're interested in doing this calculation at room temperature at 298 Kelvin. If I plug in Boltzmann's constant and 298 Kelvin into this expression, I find that kT is equal to a different very small number of joules, 4.1 times 10 to the minus 21 joules. And without even carrying this calculation further, getting a numerical answer, we can already see that the energy difference, so this gap between energy levels, 10 to the minus 19th joules or so, is about 100 times larger than the value of kT. So when I calculate this ratio, this number is going to be somewhere up around 100, so e to the minus 100 or so, that's going to be a ridiculously small number. So that ratio is very small, that means that the probability of occupying the 12th energy level is very small compared to the 11th energy level. So that's a good illustration of why it was okay to be using the Aufbau principle when we put the electrons into these levels. It's not actually true that the electrons only ever occupy level 11 and never occupy level 12. What is true is that the probability that they will occupy level 12 is much, much smaller than the probability that they occupy level 11. So this type of calculation, Boltzmann's distribution, that tells us the probability of occupying energy 12 is so much lower than energy 11 because this gap in energy is so much larger than kT. That's essentially the origin of the Aufbau principle that tells us it's okay to fill from the bottom up and never skip a step because the probability of an electron occupying a level higher than it should is quite small. So we're going to continue next and explore this ratio of delta e compared to kT a little bit more and see what that can tell us about energy levels that are occupied in molecules other than just beta care team.