 What are the most important criterias for a model of an atom to be successful is that it must explain the existence of atomic spectra. Now what is atomic spectra? You see whenever a material absorbs radiation then we find that it only absorbs a very specific certain kind of wavelengths which is known as the absorption spectra. And when that material releases or emits radiation then under analysis we find that it only emits certain specific discrete wavelengths of light which is a characteristic of that material which is known as emission spectra. The emission and the absorption spectra of a material is a characteristic or a signature of that particular material. So any kind of an atomic model that we want to establish must be able to explain why is it that certain kinds of elements have a certain unique wavelength signature while others have different ones. You see you might have seen images of distant heavenly objects, distant galaxies or nebulas where a scientist would claim that gaseous body is composed of this much nitrogen, that much oxygen, that much hydrogen etc etc. How is it that people or scientists on earth are able to make a conclusion about what is the content of the elements present in distant galaxies or gaseous bodies? This is because we are familiar with the unique wavelength signature of various kinds of chemical elements and when we look at the radiation coming from distant objects we can sort of conclude that that object, that heavenly object consists of this much nitrogen or oxygen or hydrogen or sulphur or something else by making a comparison with the known spectra of the known elements in our periodic table. So the question still remains that how can a successful atomic model explain these unique fixed discrete wavelengths corresponding to different kinds of chemical elements. Now in our previous lecture we have talked about the Rutherford model and we saw that the Rutherford model could not explain discrete wavelengths present in the spectra of an element and it was also not able to explain the stability of an atom. Then we talked about the Bohr model. In the Bohr model Niels Bohr suggested that the electron exists in an atom in certain specific orbits only. The electron cannot exist in an atom except in certain specific orbits and the stability condition for those orbits was given by the angular momentum of the electron in that particular orbit is equal to integral multiple of h upon 2 pi. Here the symbols have the usual meaning m being the mass of the electron, v is the velocity of the electron around the orbit, r is the radius of the orbit, h is a Planck's constant, pi is also a constant and n here is an integer that can take the values of 1, 2, 3 and on and on. So the Niels Bohr model of the atom is a kind of a combination of classical ideas of a particle like an electron travelling in a fixed orbit and quantum ideas of there being discrete orbits and a possible in this system and other orbits not being allowed. So the Bohr model is a semi-classical quantum model of an atom and according to the Bohr model we have calculated the total amount of energy that a particular electron in a fixed orbit can have and we found out that the energy of an electron in an orbit is given by minus m e to the power 4 upon 8 epsilon naught square h square times 1 upon n square. You can see that these are all constants mass, electron charge, epsilon naught is a permittivity of free space, Planck's constant these are all constants therefore essentially the energy of the electron in any given orbit depends upon this particular number n. This is a quantum number n associated with different kinds of orbits. Now if you are interested in figuring out how we obtain this particular expression I have derived it in my previous video where we took the total energy to be equal to kinetic energy plus potential energy and then we applied the balancing of forces the electrostatic attraction between the nucleus and the electron and the centrifugal forces and doing necessary substitutions we came up with this particular expression where of course we use the stability condition given by Bohr. So if you are interested in the derivation you can check out the previous video for the detail derivation. Here we are going to start with this particular expression where because these are all constants I can simply write down the energy of any electron in a given orbit to be equal to this value which comes out to be minus 13.6 electron volt. So electron volt is units of energy upon n square which gives us a very simplistic expression for the energy of an electron in a given orbit around a nucleus. In this case we are obviously focusing on the simplest atomic model which is the atomic model corresponding to a hydrogen. So we have a proton here and an electron going around the proton creating the hydrogen atom. So this is the expression for the energies of the different orbits which are possible. So clearly we can see that because n can take values of 1, 2, 3, 4 and nothing other than that we have discrete energy levels. So the Niels Bohr model gives us a prediction of discrete energy levels being there in the hydrogen atom. That means the electron cannot have just any random value of energy around the hydrogen atom. The electron can only exist in certain discrete energy levels. So one energy level corresponding to n is equal to 1, another energy level corresponding to n is equal to 2 etc etc. So for example, so we can say that for n is equal to 1 the energy of the ground state which is the lowest energy state comes out to be if n is equal to 1 I end up getting minus 13.6 electron volt. But essentially minus 13.6 electron volt is the energy of the ground state of an electron in a hydrogen atom. So here I can write this as this being the ground state energy E1 upon n square. So as we go higher up in the quantum number n is equal to 2, 3, 4, 5 the energy also changes according to this particular expression. Now how can this discrete energy level configuration of the hydrogen atom explain the atomic spectra? You see for that we have to understand that why an atom emits or absorbs a radiation in the first place. You see when an atom emits a radiation it emits a radiation because it loses energy. How does it lose energy? Well it loses energy via electronic transitions. What happens in an electric transition? In an electronic transition an electron goes from one energy level to another energy level. An electron goes from a higher energy level to a lower energy level then it releases all the energy which is equal to the difference in those two energy levels in one single photon. And if the reverse happens that means it goes from a lower energy level to a higher energy level then it must require that amount of energy from some incident photon from some external process. So an electronic transition let us suppose from a higher energy level to a lower energy level. This is a Bohr atom with lots of possible orbits according to this particular given condition. So if an electron goes from a higher energy level to a lower energy level then it releases energy in the form of a light photon and that energy of the light photon is exactly equal to the difference in the energies of these two energy levels. And all that energy is emitted in one single photon during that particular transition instead of a gradual release of energy. So this idea of a instant emission of a photon the moment an electron makes a transition leads to the various spectra of chemical elements. You see we can actually come up with an expression for the various wavelengths corresponding to the hydrogen spectra. So for example if let us suppose this is the initial energy level EI and if this is the final energy level EF then any kind of an atomic transition which emits wavelength H nu is basically equal to the initial energy of the initial state and the energy of the final state where energy of these states are given by what this expression. So therefore H nu so this is basically the difference in the energy of both these two levels. So H nu is equal to H let us suppose I call it C upon lambda but what is EI EI is simply equal to E1 upon Ni square minus E1 upon NF square where this is the initial state. So N basically gives us an idea about the state. So the ground state has N is equal to 1, the first excited state has N is equal to 2, the second excited state has N is equal to 3 and on and on. So these are the quantum numbers or we call it the principle quantum number associated with the different possible orbits. So if we simplify this, this simply becomes HC upon lambda is equal to if I take the E out. So if I take it E1 and I say okay let us suppose this is minus E1 then this comes out to be NF square minus Ni square. So this finally becomes or 1 upon lambda is equal to minus E1 upon HC 1 upon NF square minus 1 upon Ni square. So finally we have this particular expression minus E1 upon HC right. What is E1? E1 is nothing but minus 13.6 electron volt, its energy of the ground state. H is a Planck's constant, C is the speed of light. So here these are all constants. So minus E1 upon HC is simply equal to E1 is essentially this particular expression right. So this comes out to be M e to the power 4 upon 8 epsilon naught square CH cube. Now these are all constants. So this gives me a value of 1.097 into 10 to the power 7 per meter. So this value 1.097 into 10 to the power 7 meter is known as the Reiburg constant. So finally this expression then ends up becoming what? We end up getting this particular expression where 1 upon lambda is equal to the Reiburg constant whose value is this times 1 upon the state of the final electron energy level minus the 1 upon Ni which is the initial state quantum number. This is the expression. You see what we have obtained here? We have basically obtained a formula to predict the kind of wavelengths that are going to be emitted by various electronic transitions in the hydrogen atom. And since Nf and Ni these are all integers right, N can take the value of only integers therefore this formula predicts that we only get certain specific wavelengths corresponding to certain specific transitions in the hydrogen atom. We cannot have a continuous distribution of wavelengths being emitted by a particular chemical element instead we have only certain specific wavelengths corresponding to that particular chemical element. So therefore the Bohr model of this fixed energy levels and the photons being emitted because of a transition between these fixed energy levels gives us a very good starting point in explaining the existence of discrete atomic spectra. In fact we can at least look at the atomic spectra of the hydrogen atom okay. So we will see that not only we have discrete wavelengths corresponding to the emission spectra of hydrogen we can also categorize all the wavelengths in series. So there are a few series corresponding to whether the electronic transitions are happening up to ground level or first excited level and on and on. So let's look at that. So here I have drawn these lines which represent the various kinds of energy levels corresponding to the various orbits in which the electron is present around the hydrogen atom. So let's suppose the lowest line this represents n is equal to 1 the ground state energy level then again here we have n is equal to 2 and then n is equal to 3 and then n is equal to 4 and n is equal to 5 and n is equal to 6 and on and on and theoretically speaking we can go up to n is equal to infinity okay we can go up to n is equal to infinity. Now what are the various kinds of transitions possible and the wavelengths associated with them the most simplest of them is of course if a electron from the first excited state jumps to the ground state. So if I have this particular transition then for n is equal to 2 to 1 then this is particular wavelength which is known as the hydrogen alpha for the Lyman series. So all the wavelengths that fall in this category of transitions up to the ground state are known as Lyman series. So in Lyman series what we have is the final state nf is equal to 1 so that corresponds to wavelengths for this series being 1 upon lambda is equal to r here we have 1 minus 1 upon n square so ni square is n square where n can be 2, 3, 4 and on and on. So you can have transitions from here to here we can have transitions from n is equal to 4 to 1 we can have transitions from n is equal to 5 to 1 we can have transitions from n is equal to 6 to 1 and on and on. So these wavelengths corresponding to transitions from all the states up to n is equal to 1 represent what is known as the Lyman series. Similarly we can also look at the transitions up to n is equal to 2 so if I have a transition from let us suppose n is equal to 3 to n is equal to 2 this would represent the hydrogen alpha for the Barmer series. If I have a transition from n is equal to 4 to n is equal to 2 this would represent the hydrogen beta for the Barmer series and then on and on from 5 to 2, 6 to 2 and on and on. So this is the Barmer series. So for the Barmer series what we have for the Barmer series nf is equal to 2 and therefore all the wavelengths associated with are equal to 1 upon 2 square minus 1 upon n square where n can take the values of 3, 5, 3, 4, 5, 6 and on and on. Now let me rub this portion to look at the other series possible. So next we have what is called the passion series corresponding to all the transitions up to n is equal to 3. So nf is equal to 3 thus the wavelengths corresponding to that fall under this particular formula 1 upon 3 square minus 1 upon n square where n can take the values of 4, 5, 6 and on and on. So these are the transitions from let us suppose n is equal to 4 to 3, 5 to 3, 6 to 3 and on and on. So this is the passion series. Next we have what is called the bracket series for the transitions up to the level n is equal to 4. So the wavelengths corresponding to that are 1 upon 4 square minus 1 upon n square where n can take the values of 5, 6 and on and on. So this is the passion series and here we have the bracket series up to transitions from 5 to 4, 6 to 4 and other levels. So this is the bracket series. And lastly not exactly lastly but I think these are more than sufficient as far as the knowledge is concerned and you have the fun series where the final state is 5. So the wavelengths corresponding to that are r 1 upon 5 square minus 1 upon n square where n can be 6, 7 and on and on. So this is the fun series, 6 to 5 transitions, 7 to 5 transitions and on and on. So there you have it. The wavelengths corresponding to the various transitions between the discrete or the fixed energy levels of a hydrogen atom result in these specific wavelengths or these series of specific wavelengths that we see when we look at the emission spectra of a hydrogen. And if we look at the emission spectra of a hydrogen we will see exactly the same thing thereby validating this idea of the Niels Bohr model of the atom. So the Niels Bohr model of the atom was quite successful at least for explaining the stability of the atom and the atomic spectra corresponding to single electron atoms like hydrogen. However, it still had deficiencies. It was still not able to explain for example multi-electron atoms or complex systems. That is because the Bohr model is as I said initially a semi-quantum classical model of an atom where we had these electron particles travelling in fixed orbits where the orbits are have energies which are discrete. So it is a semi-classical quantum model. Now to get a much more comprehensive and a detailed and an accurate idea of the various kinds of spectra of not just hydrogen but also much more complex atomic configurations we will need to abandon these Bohr model or any kind of a classical assumption completely. And later on we will see when we study Schrodinger's picture of what happens around the nucleus of a hydrogen atom. We do not really have an electron going around in a fixed orbit but instead we have an electron cloud of probability of the electron around the hydrogen atom as well as other kinds of multi-electron systems. And that kind of a advanced quantum mechanical model will give us a much more accurate and a detailed description of the actual atomic spectra that is seen in nature. But as a starting point the Niels Bohr model was indeed very much successful in explaining something very very crucial at least in part because at least for hydrogen atom which is the atomic spectra and therefore sets the stage for other kinds of ideas to come forward. So that is all for today I am Divya Jyothidas and this is for the love of physics. Thank you so much take care bye bye.