we just got through watching electrons that were in an excited state fall down to a lower State and as they did so they admitted energy and we observed that energy and one of the big things we observed was we didn't observe a rainbow instead we observed only select discret colors being admitted for example here's our hydrogen that we observed we only saw four discrete colors if when we looked at Mercury we specifically really only saw those greens and those Blues there right we didn't observe rainbows so what could we conclude from this experiment well if we weren't observing a whole range of colors then our electrons were only making certain set transitions otherwise we would have seen a rainbow everybody write down Lucky Charms we didn't see a rainbow of fruity flavors being displayed for us we didn't see those electrons crashing into any energy State we saw them only move between certain set States so electrons so electrons can have only certain set energies inside an atom then we saw that different elements have different colors what could we conclude from that well if the different elements had different colors that means that the allowed energies for electrons are different in different atoms so what was it we were observing we were observing these electrons moving between different allowed energy states inside an atom as they fell down we were seeing the colors they were admitting now it's very very easy in the laboratory to actually measure the wavelength of light for some really cheap defraction grading that's labeled we're talking an instrument that costs in its crudest crudest form 2550 bucks to get a really cheap um grade school level of a wavelength so measuring the wavelength of an electron emits as it Transit i s is really easy why is that important well we just learned that wavelength is related to frequency which is related to energy so when we measure the wavelength emitted as an electron moves from one energy state to another energy State what that's a measure of is it's a measure of the difference in energy between two energy states to clean up my figure here if I know that as an electron moves between the third energy level and the second energy level if it emits 656 nanometers of a photon or that's the wavelength of the emitted Photon then I can calculate the frequency and I can calculate the energy of that emitted Photon which represents the energy differ between the second and the third energy level well if I accumulate enough of these differences then I can figure out the energy values for the actual energy levels so this is a huge breakthrough using really crude instruments we're able to actually calculate the allowed energy of an electron in an atom and to do that we're going to use what's known as the renberg formula he's the individual that first did the math and first made these observations and his formula says that the inverse technology is great until it's not that the inverse the inverse of the wavelength is equal to the ryzen BG constant and that's a constant that is specific for the hydrogen atom notice it has that little subh there in other words for this calculations that he is doing and presenting to us are only going to ascertain to a hydrogen atom so for the hydrogen atom the inverse of the ID wavelength is equal to a constant times the quantity of the inverse of n^2 final minus the inverse of n^2 initial where n is an element of the positive integers where N is a positive integer notice your book or some textbooks depending on the One You're referencing uses m for nfal I like to use ninal instead of a random m in this course nearly all the calculations we're going to be doing are going to deal with a hydrogen atom if we were dealing with helium or calcium or oxygen then the value of my constant here would be different also notice that this equation solves for for the inverse of wavelength it doesn't solve for wavelength it solves for the inverse of wavelength let's look at a quick example what is the wavelength emitted when an electron falls from the fifth energy level to the fourth energy level of a hydrogen like atom so we have one over the wavelength equals our RH constant times 1 over n^2 final minus 1 over n sared initial well our constant is equal to 1.097 time 102 inverse nanometers times our final state right we're falling to the fourth so our fourth energy level is our final so N squared is 4 minus our fifth energy level so 5 squar and that gives us 2.46 8 25 * 10 to the -4 inverse nanometers is equal to the inverse of our wavelength we don't want an inverse wavelength what's an inverse wavelength what we want is the actual value of the wavelength well that's easy enough to solve for what we do is we raise both sides to the negative one yes and that gives us wavelength is equal to 4.0 51 * 10 3 nanom ERS two things to watch for on an exam first remember your equation solves for the inverse of the wavelength therefore after you punch your equation into your calculator you're going to have to raise to the negative 1 power don't forget that that is the most common thing I see when students email me for help about homework problems they forgot to take the inverse take that multiplicative inverse the other thing they often forget is remember your constants in inverse nanometers so your wavelength is in nanometers that this equation gives you not meters which creates a problem when we try to answer Part B what is the energy associated with the emitted wavelength well we know that energy equals Plank's constant times frequency and we know that speed of light equals wavelength times frequency well we've got a wavelength right but we need frequency for our Energy Formula so let's solve our speed of light equation for frequency so we rearrange and we get frequency equals the speed of light divided by wavelength now I substitute in for my frequency and I have energy equals Plank's constant times the speed of light divided by our wavelength remember our units for this equation has to be meters for our wavelength right now we have our wavelength in nanometers so we're going to have to do a quick conver version so that's our wavelength in meters so now we can come over here and plug and chug our way through the rest of the problem we've got Plank's constant times our speed of light 2998 10 8 m/s divided by our wavelength which we just said was 4.51 * 10 -6 meters right we just converted our nanometers from the last answer into our meters there punch that into our calculator and we discover that our energy is 4.94 * 10 to the -20th Jews however we still have one last thing to do and that has to do with a sign convention when it comes to energy our electron is falling and our electron are falling emits energy right it's going from the fifth energy level to the fourth whenever an electron Falls it emits energy energy admitted or released is all always referred to as negative energy absorbed by a system is always viewed as positive in chemistry and physics plus or minus when it comes to energy tells you the direction the energy is going whether it's going into the thing we're studying or whether it's going out of the thing we're studying and in this case it's going out of our electron so to make this answer 100% correct it really needs to be negative our energy should be the energy equals - 4.94 * 1020th now we just took the long way to answering that problem there's actually a much quicker way to do that and that's through a quick modification of our ryzen Berg formula because we don't really care about inverse wavelengths we said that was awkward the only reason you're taught taught about it is it'll save you some time on a few homework problems plus it's the way ryzen Berg initially did it most of the time as chemists or PE chemists were so much more interested in the actual energy value so let's modify this formula for energy well we just said that energy is equal to Plank's constant times frequency and that if you rearrange your speed of light formula to solve for frequency in terms of wave lengths you get C divided by wavelength so if we come up here to our energy Pro thing and substitute just like we did in the last problem we know that energy equals Plank's constant times the speed of light divided by wavelength well we've got an inverse or dividing by wavelength right here in Our Risen Verve formula right so what we can do is we can take the speed of light and the Plank's constant and multiply both sides of our reisenberg equation by that now we do have to do a quick modification there we are going to have to adjust our units because of that nanometers but after we adjust our units and multiply by planks constant in the speed of light we wind up with a brand new version of reisenberg formula based on energy it says that the energy of an admitted electron is equal to the opposite of renberg's constant times the quantity 1 over n^2 final minus 1 over n^2 initial however this time we've changed the unit units for our constant our constant is now 2.78 * 108 jewles so we have the constant in energy same formula we just multiplied both sides by Plank's constant and the speed of light and modified for nanometers so what is the difference between the energy of the fourth energy level and the third energy level of a hydrogen like atom to do this type of problem we can do it much faster the one the way we just did it by using this new constant plug that into our calculator and we get a 1.59 * 10 to the19 jewles chemistry is easy life is hard why doesn't everybody write down Frosted Flakes everybody write write down Frosted Flakes we'll stick with breakfast cereal for this lecture