 Now, it is time to discuss the next contribution to partition function and that is electronic. I have been repeatedly telling that in the k in the order of energies you have the translational rotational vibrational electron. And as we discussed in the previous lecture the vibrational contribution expected at normal temperature is generally not different than 1. And therefore, it is also expected because electronic energy levels are far separated. And therefore, we expect in this case also the value of electronic partition function not to be very much different than 1. However, in this case the degeneracy of the ground state will play a role whether the value will be 1 or equal to degeneracy of the ground state. So, therefore, let us get started in discussing the electronic partition function. The electronic contribution can be obtained directly by summation, we do not need an approximation. What is the reason? Because electronic energy separations from ground state are usually very large. So, if the separations are very large you require exceptionally high temperatures if the transitions are to be brought about by temperature. So, that means, in general you do not need to derive any formula here. One can simply expand this Q e is equal to G 0 plus G 1 exponential minus beta e 1 plus G 2 exponential minus beta e 2 plus so on. We expand, but as commented over here that the electronic energy separation from the ground state are usually very large. If these are very large that means this value is very high and exponential of very high value with a negative sign is going to be close to 0. So, that means, in that case all these upper numbers will be close to 0 and electronic contribution is going to be equal to degeneracy of the ground state. And if degeneracy is 1 then electronic contribution is usually 1 except look at the comment except in the case of atoms or molecules having electronically degenerate ground state that is what I was saying that if there is a ground state degeneracy then the value is going to be G 0 very easy. There are no complicated conversions into some approximation you simply expand this and depending upon the values of the first excited state usually the first excited state itself is very high. Therefore, in that case you will generally have a value of G 0, but in some cases there may be some lower lying first excited state. So, in the case where there is a lower lying first electronically excited state one can generally go up to the first the second term first is the ground state and second is G 1 into exponential minus beta E 1. Let us discuss that with the help of an example as written over here some atoms and molecules have low lying electronically excited states. One such example is nitric oxide NO and I am sure all of you know the electronic configuration according to the molecular orbitals. Here you see a comparison of MOs of N 2 versus NO both are diatomic linear molecules the electronic configuration in the MO of N 2 you see here in 2 S you have 2 in 1 sigma G 2 in the upper level anti-bonding and here 4 in the bonding pi orbital 2 here and the anti-bondings are not there and on the right hand side we see that for NO. I am putting here intentionally 2 different notations because if you follow of some book they will follow this kind of nomenclatures of various MOs and another book may follow this kind of nomenclature for the MOs. So, therefore, I am retaining both one major difference which you note between these 2 MOs for nitrogen you see these atoms energies are represented at the same level because both are it is a homonuclear diatomic molecule. However, when it is a heteronuclear diatomic molecules then oxygen being more electron negative you see the oxygen energy is lower than nitrogen energy. Oxygen has 8 electrons 2 in 1 S 2 in 2 S and 4 in 2 P oxygen is paramagnetic. Nitrogen 7 2 in 1 S 2 in 2 S 4 and 3 in 2 P unpaired and when you fill what you see is the difference between N 2 and NO is there is no electron here and there is 1 electron in the pi star 2 P x orbital in the pi orbital. So, that is what the configuration is pi 1 NO ok. The idea was here to show how the electronic configuration differs for NO from N 2 and eventually the configuration of NO is all these you keep on filling and finally, it is pi 1 and as I said that NO is a system which has low lying electronically excited state. Now, see what happens now that pi 1 electron is there. So, therefore, what happens due to that the orbital angular momentum may to may take two orientations you can note over here with respect to molecular axis one will correspond to clockwise circulation and the second will correspond to anticlockwise circulation around the axis. So, one is clockwise the other is anticlockwise. So, therefore, that leads to when you compare this see you talk about orbital angular momentum. So, therefore, similarly when you talk about spin angular momentum here also you can talk about clockwise and anticlockwise there are two orientations in each case with respect to the molecular axis. That means, there are there is going to be total in all four states the one which is corresponding to spin momenta parallel or angular momentum parallel that is 2 pi 3 by 2 will not go into details of this nomenclature at this point. The other one with 2 pi 1 by 2 with anti-parallel momenta are the lower one. So, parallel versus anti-parallel. So, total there are four you can see you have doubly degenerate here and you have doubly degenerate here and these are low line the upper state is low line. And therefore, at normal temperature you have all the four states accessible. If all these four states are accessible then what will be the expression for partition function? Don't worry about this figure this is essentially the same figure which is shown over here. Somehow it got little different in the in the next one. So, you have 2 and you have 2 and this is separated by some energy which is 121.1 centimeter inverse. So, this is general form that we discussed earlier. The ground state is doubly degenerate you have a value of 2. The first excited state is also doubly degenerate 2 into exponential minus beta e and that e is equal to 121.1 centimeter inverse in this case. So, now depending upon the temperature because see here this one if I write over here this is 2 plus 2 into exponential minus e over k t. So, therefore, both the numbers will matter which numbers the energy separation and the temperature both will matter and one can discuss when the temperature approach is 0 what will be the value when the temperature approach is infinity what will be the value. Generally we have discussed as the temperature increases more and more thermally accessible states are there. And therefore, the value of partition function will increase that is a general thumb rule. Let us see what happens we just discussed this that in case of NO the electronic contribution to partition function will be 2 plus 2 into exponential minus beta e. And if you plot this Q e against temperature you expect a value of a total of 4 actually y 4 we are plotting Q e against k t by e. So, therefore, when the temperature is close to 0 in that case. So, you have Q e let us make it more clearer is equal to 2 plus 2 exponential minus e upon k t. As temperature approach is 0 exponential minus e over k t approaches infinity and exponential minus infinity is 0 that means, Q will approach a value of 2 as the temperature approaches 0. And that is what you see over here. And now when the temperature is increasing you can notice here this is an expanded form of this lower temperature region this is how the value of partition function will start increasing. And as the temperature is approaching very high the value is moving towards a saturation that is towards 4. Because when the temperature is infinity when the temperature is infinity then 1 over infinity is 0. So, this will be 1 that means, eventually you should get a value of 4 if the temperature is extremely high. And that is what is commented over here when T is equal to 0 or T approaches 0 Q e is equal to 2 or Q e approaches a value of 2. And when T is very large Q e approaches a value of 4. And at 25 degree centigrade you can put appropriate number that is 298 Kelvin. And you already have an energy value of 121.1 centimeter inverse once you use this 121.1 centimeter inverse and a T value of 298 you will get Q electronic is 2.8. This is the scenario in the cases the molecule having low lying electronics state upper state. But even in that case you see the contribution at 25 degree centigrade is not very large it is simply 2.8. So, therefore, as a general rule let us conclude this discussion. Now, I have Q defined by this general formula and the expansion of this is Q e is equal to G 0 plus G 1 exponential minus e 1 upon k T plus G 2 exponential minus e 2 upon k T plus there can be other terms. Therefore, as I discussed in the beginning usually the energy separations in case of electronic levels very large. If that is very large in general you will have Q e equal to G 0. Exceptions can be there like we took just example of an O where there is a low lying upper electronic state. In that case this could be extended up to this. In general these upper contributions will not be there because those energy levels are far separated and therefore, that occupation is usually very low. So, therefore, again as a thumb rule the electronic contribution to partition function will be usually close to the degeneracy of the ground state and in certain cases this can be including the degeneracy of the first excited state weighted by the exponential factor which involves the energy and the overall value therefore, will be decided both by the energy separation and the temperature. So, by now we have discussed most of the contributions that we need to consider. A system a molecule can have translational contribution, rotational contribution, vibrational contribution and now we have discussed electronic contributions. If you can think of some other energy level then those contributions can also become, but if those energy levels are far separated the contribution is usually going to be 1. The overall molecular partition function is going to be the product of all contributions that is the overall partition function will be equal to translational partition function into rotational partition function into vibrational partition function into electronic partition function. In all these the vibrational partition function can have also contributions due to different normal modes of vibration. So, therefore, partition function which is multiplication of all individual contributions will have maximum contribution from translational followed by rotational followed by vibrational and electronic both these vibrational and electronic are going to be close to 1. So, we are now equipped with the knowledge of molecular partition function, how to calculate, how to evaluate molecular partition function and what it needs is temperature and the respective energy levels information. And where from that information will come will come from the respective spectroscopy, microwave spectroscopy, infrared spectroscopy, electronic spectroscopy. So, once we have the information about the molecular partition function, we have already discussed the connection between the canonical partition function and molecular partition function. And as I discussed earlier that in this lecture series we are mostly concentrating on canonical partition function. So, once we have that connection between the canonical partition function and molecular partition function, then we are ready to connect these further with different thermodynamic quantities. The thermodynamic quantities like entropy, enthalpy, Gibbs free energy, Helmholtz free energy and our main aim is eventually to further connect with equilibrium constant. So, what are the different thermodynamic signatures that we need to know. As I just said when you talk about first law of thermodynamics, its internal energy d u is equal to d q plus d w. Internal energy can be connected with partition function, whether molecular partition function or canonical partition function. Second law entropy s is equal to k log w and s also we have connected with molecular partition function or canonical partition function through u and other terms. Then the third law also deals with entropy. After that the Helmholtz free energy which is a maximum work function that is also connected with the partition function, Gibbs free energy the change in which is a measure of maximum non pressure volume work that can also be connected with partition function right. So, g is connected to h and s. So, if you know g if you know s we have information about h that is also connected with partition function. Once you know g you can talk about delta g you can talk about delta g naught in a chemical thermodynamics. I am sure that you understood the difference between delta g and delta g naught. You can connect both with partition function and with the time in the lectures ahead we are going to connect delta g naught which is connected with equilibrium constant and that we will further connect with the partition function. So, therefore, now with the knowledge gathered on all contributions we will start further connecting these with different thermodynamic signatures and then discuss their applications. We will we will discuss all these issues in the upcoming lectures. Thank you very much.