So today we're going to move on to another type of spectroscopy, called infrared spectroscopy. And if you took chemistry in school, I think you have done some applications of infrared spectroscopy. So what we're going to do today, is we're gonna get more involved in the theory behind the spectroscopy. And then we're gonna to move on to look at how it's useful in in analyzing various types of molecules. So, as i say here in the first line of the slide, is that the atoms in a molecule, you don't think of them as fixed, that they're constantly, constantly moving. And this, this should be automated to move but, yeah there we go. As we can see here, we have some simple molecules. And molecules, the atoms in them, are constantly vibrating, if you like, moving, back and forth. So, when you think of a bond length, when you talk about a bond length in a molecule that you measure, you're really talking about an average value, because the bond is stretching and contracting, like you see on the slide here. So if you have a simple molecule, a diatomic simple molecule like, say, H-Cl, then the only motions that the atoms can do in there is they can move back and forth. They can have what they call a stretching of the bond, a lengthening and compressing. So when you get more complex molecules, like you get in biological systems. Then you have other types of modes. And likely two of the molecules here on the right, you can see that the stretching there of all the bonds. And also, on the bottom, you can see bending of the atoms. So, they're stretching, and there's bending going on. And when you measure an infrared spectrum, you're measuring the energy of these movements really. So the vibrational motions, like these. The energy needed to excite them, from their ground state or their excited state, which is what all spectroscopy is about. That energy that you need for that, comes within the infrared region of the electromagnetic spectrum. So, if you remember again, support the electromagnetic spectrum from the first day. You have the high energy regions, the gamma rays and the X rays. Then you come along to what we talked about the last day, the UV and then you move into the visible region. And then lower energy from that, you have the infrared region. So this is the region of the electromagnetic spectrum, that you observe vibrations of molecules. And of course in practical applications as we move to later on, what you're trying to do, you're trying to gain from spectroscopy, you're just trying to get some information about the structure, the geometric structure and sometimes maybe the electronic structure of the molecule. So that's the main points on that slide there. So now we're gonna move into the, the theory behind it, because you need to have an appreciation of that as well as just being able to assign assign spectra. So, the simple model that you have for the movements of the atoms In a molecule, is that you treat a molecule as what they call a simple harmonic oscillator. So, it's like you have two atoms and then they're fixed by a spring. And basically they, the atom movements are, for diatomic anyways, are just for stretching and the compression of all that strength. So what you have is something like that there. So there you have Simple diatomic system. You have one mass m2, m1. And then, it's joined by a spring and from that spring, you're going to have what we call the equilibrium bond length, the position which is say, at its lowest energy. And that's the stable one, but you can imagine with a spring, we can pull it and we're exerting force on it. But if we let it go, it will go back to it's equilibrium value. We can push it together and if we let it go again it will come back to it's equilibrium value. So you imagine bond lengths are moving back and forth between the compression and lengthening, but they have an equilibrium bond length which is at lowest energy. So there's a very simple law that relates the energy, the force that you exert on such a system. And this is known as Hooke's Law, and some of you have done Physics might have come across this. But the force, the F(r) here as a function of r, which r is the distance, is given by this here. It's minus k times r minus re and r, so you lengthen that spring then r is the r that you get when you lengthen it. And re is just what we call the equilibrium value, the one, the lost energy bond length. So F(r) = minus k. You need to remember the minus, because what you have to remember is if you're forcing it, If you are pulling the atoms apart, increasing the distance. Therefore the force is actually against it, so the force if you compress it the same way the force is actually against you. So that is why you have this minus sign here. So F(r), is equal to minus k or min re. So r is the actual length as I said, re is the equilibrium length and then we have [COUGH] this k here. So what we're saying here is that F(r) is proportional to r- re, and then you have a proportionality constant and that proportionality constant is k. And we call that the force, the force constant, and if you like k is a kind of a measure of how difficult it is to do that, or the strength of the bond between the two atoms. So the force constant, the way you measure it in Newtons at N, Newtons per meter because force is Newtons, and then if you divide the cross by R, or minus R E as in distance. SI units we like to keep it in meters, so it's Newtons divided by meters or Newtons to the meters, -1. So here you have again this analogy between the spring. If you have a large force constan,t then you have a strong spring, or strong bond. If you have a weak force constant, small force constant. Then you have a relatively at least, weak spring. Okay. Okay, so let's just talk again, it's best just to talk about the simple system of diatomic to get these theoretical ideas In your heads. So we're talking about diatomic molecules. So we've already said that the strong bonds have large force constants. So how do we know that? So here we just show. For some very simple diatomic molecules, hydrogen, HCl, chlorine, and nitrogen. And here on this column here, we have the force constants and then here we have the bond energy that you measure for that bond. So here we have all of these above. The nitrogen are single bonds. So what you can see is that the force constant is directly proportional to the bond energy. The bond energy's the amount of energy you need to break that bond. So, as you can see, the force constant varies with that. And then you can see you have nitrogen here, which has a triple bond, and you should know that if a triple bond is going to be a very strong bond system between the two nitrogen atoms. And you can see that's reflected in the bond energy, the amount of energy required to break the bond, and then that's reflected in the large forced constant for that bond. Right, so we talked about the force. We like to talk about, in spectroscopy we like to talk about the energy. So you want to know what's the energy of the spring or the bond? And the term we use for energy, we use v. And it's a function, talking about diatomics again. It's a function of the distance. So it's v of r, and that's equal to. Half k or enter or K into r minus r e squared and you can remember that for any of you that have a mathematics knowledge, you can also know that f of r Is related to the energy by (-dv4)/(dr), or the force is equal to the negative of the change of the energy as a function of distance. That's from basic, basic physics. And then from that, if you integrate across each side, what you'll get is that the, I try to write it in here, so we're taking the integral of each side so we get that the integral of F(r) Is equal to -V(r), okay? That's just some simple algebra. Or we could say, let's make it a bit easier, and we can just bring the minus sign The minus sign over there. Now we know that F(r), we've already saw it on the last slide. F(r) is equal to -k(r-re). So we've got to integrate that, and then we've got to change the sign. So for those of you who know a bit of calculus. If you integrate r(re), you increase the power by two, and you divide by the new power. So you get minus a 1/2 k, or n or r- re, V squared. And of course we know there's a negative sign outside it, so therefore v of r is equal to plus a half k into r minus r [INAUDIBLE]. So that's just, if you have a bit of mathematics it's nice to be able to do that, but that's where this equation comes from. Otherwise you just have to try and remember that equation. So another thing is that this equation then, if you plotted it out and here it's shown here. If you pull out it gives you a parabola. All right, it's like a half KX squared, it gives you a parabola like we've shown here. And so what it means is here you have the equilibrium value of that spring or bond down at the bottom. That's the minimum energy. And then, as you stretch it, you increase the energy. So the energy goes up. And as you compress it. If you push it together too much. Then the energy goes up as well. And that kind of simple plot is a very good approximation, especially near the equilibrium value for how the energy at the bond changes as a function of its distance.