 So as one interesting example of how we can apply the particle in a box model to some real world molecules, let's consider the example of dye molecules or molecules that take on very distinctive colors. So we've seen that not just linear hydrocarbons, but also conjugated molecules that can be thought of as two dimensional boxes can be treated with a particle in a box model. So a benzene molecule for example, we can treat fairly well with the two dimensional where we say the electron, the pi electrons anyway are confined to a box that approximates the size of this benzene molecule. So just like we saw for the small 1D particle in a box molecules for when the box is relatively small just a few carbon atoms on a side, the energy levels for excitations of this molecule turn out to be in the ultraviolet. They're higher energy than the visible portion of the spectrum and if I were to make that molecule bigger by adding more benzene molecules, benzene rings onto the molecule, I can increase the wavelength of light that that molecule absorbs and by making it large enough I can move it into the visible portion of the spectrum and then by changing the size of the molecule I can affect what colors of light it absorbs and what colors it doesn't absorb and therefore what color the molecule appears to be. So as one particularly interesting example of that, I'll bring up a molecule in a picture over here that you're likely familiar with from previous chemistry courses. If you've done acid-base titrations it's very likely you've used phenolphthalein as an indicator and this is a molecule that in its acidic form with the two protons here takes on this form and then if we remove those protons as you typically do, remove two protons as you do if you titrate with hydroxide for example, you can form this more basic form of the molecule and so the molecules are distinct and distinct structure in their acidic and basic forms and what's interesting about them from the point of view of the particle in a box is here we see for example a conjugated portion of the molecule, a benzene ring not conjugated with the rest of the molecule, so this molecule is going to absorb in the ultraviolet portion of the spectrum just like benzene does, absorbs light in the ultraviolet, the boxes are not large enough for those pi conjugated electrons to absorb light in the visible portion of the spectrum, so a solution containing this molecule is clear, it's colorless because it doesn't absorb any visible light. On the other hand when you titrate with hydroxide, when you consume some of these protons and generate the more basic form of this molecule then we can see if I draw a box around this portion of the molecule, this portion is now all conjugated because we have alternating double single double single double bonds and it's planar, so the two dimensional box in which these pi conjugated electrons are confined within is larger than it used to be and it's pushed just large enough now that absorbs in the highest energy portion of the visible spectrum, so it absorbs a little bit of the violet portion of the spectrum and as a result the color of this molecule ends up looking pink or the color that it reflects is a little bit tinged toward the red side of the spectrum because it absorbs a little bit on the violet side of the spectrum, so the reason phenolphthalein is such a good acid-base indicator is because it undergoes a color change when it changes from acidic to its basic form and the reason for that is essentially because the size of the box that the conjugated electrons are confined to changes and as we've seen from the energy levels of those particle in a box energy levels, changing the size of the box changes the energy levels and changes the wavelength of the light that gets absorbed, so here's a great example of how a chemical reaction can change the way a molecule behaves as a particle in a box. As a different example, I can clear this picture away and bring up another picture with a list of molecules, so let me take away these decorations. We can talk about this collection of molecules over on the right hand side and these are all dye molecules, these are molecules that could be used to dye clothes for example and the particular details of the molecules are not important other than the fact that the molecules have different colors as are indicated by their names and you can see that each of these molecules behaves a lot like a two-dimensional particle in a box. I have a large conjugated section of the molecule, within this portion of the molecule there's pi-conjugated electrons that can move freely within the molecule and then that shape of this box determines the wavelengths of light that are absorbed and consequently the wavelengths of light that are not absorbed and why this molecule looks blue. Other molecules with different sized boxes absorb different frequencies of light and molecules like this, that black 29 for example we have a mix of some small conjugated sections of the molecule, some larger sections here and some more moderate sections here so this molecule absorbs light via electrons in several different sized boxes so it absorbs lots of different colors of light by absorbing almost all the light that's shined on it it ends up with a relatively black color. So here's again another example of how chemistry can be used by changing the size of the box that these planar conjugated electrons are confined to. We can change the energy levels directly and therefore change the color of the molecules so that's a big part of what for example a dichemist does is adjusts the properties, synthesizes different molecules in order to make them different colors. That's not to say that you need to constantly be thinking about quantum mechanics if you do the organic chemistry of preparing these molecules as a materials chemist or a dichemist but it is helpful to be able to think about what should happen as you increase the size of a molecule, you should redshift the color of the light that gets absorbed or equivalently if you change the substituents on these molecules if you add electron withdrawing substituents or electron donating substituents you can change the number of molecules that occupy these energy levels and consequently also change those energy levels. So at a qualitative level at least it's useful to be able to think about the particle in a box model when you're thinking about what color these molecules are. So it also illustrates that even a qualitative understanding is good enough if we were to try to quantitatively predict exactly what color this molecule would be. It's not exactly a rectangular box, it's not quite true that the electrons feel no difference in energy anywhere they go in this molecule. They might have a different energy near this carbonyl group than far away from it on a carbon atom or in the middle of one of these six member rings and so on. Particle in a box model is not an exact representation of this molecule but it's certainly good enough to give us some qualitative idea of how these molecules behave.