Okay we're going to work our way through the language in this problem, and see what its asking of us, and draw some pictures as we go. Tells me what the first ionization energy of this element is, so this is the energy required to remove an electron from an atom in it's ground state. Gives me a value for that. And then it says, a value for the ionization energy in the first excited state. Let's get a picture in our mind of what in the world that is saying. And I want to do my best to draw some pictures here. It's hard to draw circles for me on the screen, but I've got my first, n, this is n equals 1. And we know that there's going to be a s orbital there, with a couple of electrons in there. Then we move out to the n equals two, so we draw a circle here. N equals 2. So it's got an s and a p. It's got electrons in there. We don't know, what the atom is, but we know that it's going to to keep on filling in the sub-shells until it's out. Somewhere out here, and we'll call this shell the outer most shell, it's the n equals some number, I'm going to call it x. And that is, everything's filled up to that level and we've got our ground state. Now, if we were to pull an electron out of this, okay? We know, according to this, to remove that electron away from the atom in its highest energy orbital. The minimum amount of energy required to remove that electron, would equal and we’re going to call this the first ionization energy is 6.84 times 10 to the minus 19 joules. So that's how much energy it's going to take to remove that electron. Now, the next thing it say, this element is in it's first excited state. What in the world does that mean? What we're doing is, we're going to take an electron. Out of this n equals x, and we are going to, instead of pulling it off, we're going to promote it up. 'Kay? To some energy level, that's the next nearest one. So I'm going to call it x plus 1. 'Kay? And from there. If I were to take an electron that was sitting in their, and remove it entirely from the atom, the ionization energy for that transition is given to me, as 2.09 times 10 to the minus 19. 'Kay? So, to remove it from its highest energy but in its ground state, we're pulling it out of here. It takes this much energy. And if we promote the electron from here to its nearest higher energy, right here, then it would only take 2.09 times 10 minus 19 joules. A little less energy to pull it off. So that's the information that they've given me, right up to this period, right here. Now, based upon that information, what I want to have them do, what we need to do is calculate the wavelength of light emitted, when an electron goes from this first excited state to the ground state. So, let me change colors for just a minute here, as we consider what we're going to do next? Now it's saying let's say, there was an electron that got promoted up to this, now that'd be an excited state because the ground state the highest electron ground state would be in this level. I want to take an electron from here and we're going to let it relax to it's ground state, so it's going to go to here. Now, any time an electron loses energy like that, it is going to emit a photon. Okay? So, there's a photon being emitted. And, we want to know the wavelength of this photon that is emitted. So, that's a picture of what's going on, the information that's given to us, and what we're trying to determine? Well, let's see, the first thing that we could do is, we could determine the change in energy of that electron. What is the size of this energy gap? Well, if it takes if we know, how much energy it takes to get, to remove the electron from here. And we know, how much energy it takes to remove the energy from here, if we were to subtract those two, we would know the difference in energy of this transition between those two. So this is going to be equal to and, and electron is relaxing down, so the change in energy is actually going to be final minus initial, so it'll be 2.09 times 10 to the minus 19 joules, minus 6.84, times 10 to the minus 19 joules. And that is equal to 4.75 times 10 to the minus 19 joules. So that's a change in energy of the electron, as it goes from this to this. Well, that has to equal the energy of the photon. It has exactly, the same energy. Only the energy of the photon is the, by the way that's negative, is going to be the absolute value of energy differences of the electron. 'Kay? because the electron is changing in energy, a photon has to have a positive value. So the energy of the photon is 4.75 times 10 to the minus 19 joules. Well, if we know the energy of the photon. We know, that E equals h nu or E equals hc over lambda, we can plug in our information or we could solve for lambda and put in all the other values and obtain the value for that. So what would lambda equal? That's wavelength. It would be hc over E. So we put our numbers in. Okay? We have 6.63 times 10 to the minus 34 joules times seconds for hour h. We have 3 times 10, to the 8 meters per second for our c, so were going to multiply those two there, and then were going to divide by our energy. Okay? Hc over E, and our E was 4.75 times 10 to the minus 19 joules. 'Kay? So our seconds will cancel, our joules will be cancelling and we'll be left with meters. And it wants it actually, in nanometers. So let's calculate it in meters. This would be equal to 4.19 times 10 to the minus 7 meters. And then we can convert that to nanometers, how would we do that? We don't want meters, we want nanometers, and a nanometer is 10 to the minus 9th meters, and so this is going to be 419 nanometers.
