 Welcome back to the next lecture on vibrational partition function. In the previous lecture we have talked about what is the expression for the vibrational energy and then we discussed about how to set the ground state vibrational energy equal to 0 and then how to account for if we want to overall calculate the internal energy. So, let us take a look at what we have discussed earlier. We talked about this is the overall expression for the vibrational energy levels where V which is the vibrational quantum number is 0, 1, 2 etcetera and E V is equal to V h c nu bar where when you set V equal to 0 energy levels for the ground state are set to 0. So, let us now discuss further Q V is equal to summation V from 0 to infinity you have exponential minus V I will put beta h c nu bar which you can write as is written above or if you do not want you can leave it as such. So, I have now Q V is equal to when V is equal to 0 exponential 0 is equal to 1 plus exponential minus beta h c nu bar plus exponential minus 2 beta h c nu bar plus exponential minus 3 beta h c nu bar plus so on. Now you remember let me just give a background we had E V is equal to V h c nu bar. So, that means E 0 is equal to 0 E 1 is equal to h c nu bar E 2 is equal to 2 h c nu bar E 3 is equal to 3 h c nu bar etcetera, etcetera. And if you look at the difference difference between this and this is h c nu bar different between this and this is h c nu bar difference between this and this is h c nu bar. So, that means difference between the successive energy levels is h c nu bar. That means this difference is h c nu bar each one is h c nu bar. And if you set this h c nu bar if I just write this to be equal to E then you have E 2 E 3 E 4 E etcetera. So, this becomes a uniform ladder of energy levels. Remember the approximation here we are considering that the vibrational excitation is not too large therefore, we are using the harmonic oscillator approximation. So, therefore, this is the same case as that we have discussed earlier when we talked about uniform ladder of energy levels. So, let me come back to this discussion and if you look at the sum it is 1 plus exponential minus beta E plus exponential minus 2 beta E plus exponential minus 3 beta E which can be written as q V is equal to 1 plus exponential minus beta h c nu bar for V equal to 1 for V equal to 2. This second term I can write minus beta h c nu bar square plus exponential minus beta h c nu bar cube plus so on so on. And if I just represent this as x this whole as x then q V is equal to 1 plus x plus x square plus x cube plus so on so on. This is sum of a GP and sum of a GP as we have earlier discussed when we were discussing uniform ladder of energy level it is 1 over 1 minus x beta h c nu bar. This is the expression that you are going to use to calculate or to determine the value of vibrational contribution to partition function. Remember that we have used only one approximation here and that approximation is that we have considered the system to be harmonic oscillator which has led to this uniform ladder of energy level. And we have not used any other approximation. So, for a given vibrational wave number you will have a vibrational partition function. If the vibrational wave number changes you will have another value of vibrational partition function. The overall value of vibrational partition function will be equal to q V 1 into q V 2 into q V 3 etcetera etcetera as we have discussed earlier. So, within the approximation of harmonic oscillator or uniform ladder of energy level we can use this expression exclusively at any temperature. And we can now discuss the conditions you know what happens when the temperature is very high what happens when the temperature is very low can we further modify this expression. So, just a recap as I discussed in the previous slide that you can treat within the harmonic oscillator approximation you can treat the system like a uniform ladder of energy levels which gives rise to equally spaced infinite array of energy levels as shown over here. And a harmonic oscillator has the same spectrum of levels that that is what I have been discussing. And our discussion of expanding this I will just quickly once again write for the benefit of all when V is equal to 0 then it is 1 plus exponential minus beta h c nu bar plus exponential minus beta h c nu bar square plus exponential minus beta h c nu bar cube plus so on so on that is what it leads to. And this leads to this result that is what we discussed in the previous lecture. That means, this expression now we are going to use for calculation for determination of vibrational contribution with this background let us move further. Now there can be different conditions for example, if we consider the vibrational wave number which are very very high that is if beta V h c nu bar is much much higher than 1. Let me make a simple correction over here this V is not there. So, you treat this as 1 over 1 minus exponential minus beta h c nu bar V is not there ok. So, make that correction. So, same thing I will put over here which is equal to 1 over 1 minus exponential minus beta h c nu bar. So, if beta h c nu bar is much much greater than 1 then what happens? If this is much much greater than 1 then exponential minus very high quantity you can ignore that. That means, in that case Q V is going to be approximately equal to 1. That means, we are talking about if the separation is very very high. In that case you will have mostly the contribution due to ground state only that means, when you talk about vibrational levels most of the molecules will be there in the ground state only. And in that case the vibrational contribution is going to be 1. So, when I put this condition that means, I am talking k T is much much less than h c nu bar right beta is equal to 1 over k T. That means, we are talking about here temperatures which are very very low compared to the h c nu bar or k T is much much lower compared to h c nu bar. Example for example, take methane molecule and in the methane molecule the lowest vibrational wave number is 13606 centimeter inverse. And when you use room temperature that is a 25 degree centigrade then the value of beta h c nu bar you can put the number h is blanks constant c is speed of light and nu bar is the wave number. And when you put the numbers it turns out to be 6 and exponential minus 6 is 0.002. So, that means, 1 over 1 minus 0.002 is approximately 1 ok. So, therefore, if the temperature is very low or if the wave numbers are very very large in that case it is understood that the value of vibrational contribution to partition function for that normal mode of vibration is going to be close to 1. And the reason is given over here just in the form of a short calculation. Now, what happens if beta h c nu bar is much much less than 1 or k T is much much higher than beta h c nu bar. That means, we are talking about high temperature now when the temperature is very high then I have q v is equal to 1 over 1 minus exponential minus beta h c nu bar this is very small right. And if you treat this beta h c nu bar as x 1 over exponential minus x if x is very very small then you can expand this 1 minus with this exponential minus x in this form 1 minus x plus x square minus x cube by 3 factorial etcetera etcetera it will come the square terms cube terms because it is anyway small the square of small is further small cube of small is further small. So, all these higher values can be ignored. So, what you are left with 1 minus 1 minus beta h c nu bar. So, in that case what I have is here 1 over 1 and 1 get cancelled I have beta h c nu bar only and beta is equal to 1 over k T. So, when you put beta is 1 over k T you have vibrational contribution as k T over h c nu bar I want to highlight 2 results over here. One is this one where we wrote q v is equal to 1 over 1 minus exponential minus beta h c nu bar this can be used at any temperature there is no need to worry about whether the temperature is low temperature is high wave number is low wave number is high nothing to worry about you can go ahead and use this expression. However, when the temperature is high or when the temperature is low you need to worry about if the temperature is low you cannot ignore if you cannot ignore that and anyway we have shown if the temperature is low or if nu bar is very very high then q v is approximately coming to be 1. However, if the temperature is high compared to energy separation in that case we show that q v can be approximated by k T over h c nu bar when we can use this result and when we can use this result. This result can be used anyway as we just discussed at any temperature, but this result when can be use in order to use that we will come to that what temperature can be used over here, but you see if I write q v is equal to 1 over 1 minus exponential minus beta h c nu bar. If the temperature is very very low for your simplicity I will write this also 1 minus exponential minus h c nu bar by. So, if the temperature you see is very very low when the temperature is very very low then exponential minus very high number can be ignored in that case q v is approximated by a value of 1. I hope it is clear temperature low this number is high exponential minus high number can be ignored. So, q v is approximately equal to 1 if the temperature is high if the temperature is high what did we have we had the result we just derived which is k T over h c nu bar. So, k T over h c nu bar that means it is when k T is much much higher than h c nu bar. This is what was the condition that is k T is much much higher than let me correct this k T is much much higher than h c nu bar all right. So, look at the comment the vibrational partition function of a molecule in the harmonic approximation note that partition function is linearly proportional to temperature when the temperature is high. When temperature is high you use this result and this result basically you know q v is directly proportional to temperature k is constant h is constant c is constant nu bar is constant for a given mode and then it is almost linearly dependent on temperature right this is expressed in terms of k T over h c nu. But then again you know bringing back the same issue that at what temperature or under what conditions of temperature we can use this approximation for that it is a good idea to introduce a temperature which is called characteristic vibrational temperature. How to introduce characteristic vibrational temperature we had q v is equal to k T over h c nu bar. So, you can mathematically show whatever I am going to discuss that now characteristic vibrational temperature is introduced by writing this equation k times theta v is equal to h c nu bar where theta v is the characteristic vibrational temperature. That means alright theta v is equal to h c nu bar by k we can calculate this you know you can work out the units of this it will turn out to be temperature you calculate this theta v and on the other hand we have let us say experimental temperature experimental temperature which is T. Now if temperature T experimental temperature is higher than theta v then you can use this approximation. So, that is the utility of characteristic vibrational temperature that is it is the temperature above which this approximation is valid. And since we have defined the characteristic vibrational temperature theta v as h c nu bar by k then I can write for h c nu bar by k I can write theta v that means the vibrational partition function then becomes T by theta v. So, now we have three forms of vibrational contribution to partition functions one form is let me write down again v dot q v is equal to 1 over 1 minus exponential minus beta h c nu bar. Second form we write 1 over beta h c nu bar or is equal to k T over h c nu bar. Third form we had basically a modified form of this which is T by theta v and we have discussed in details under what conditions to use this under what conditions to use any of these. This one can be used at any temperature at any wave number. The only approximation in deriving this was we have used harmonic oscillator approximation that is the vibrational is not too much vibrational excitation is not too much. And then when the temperature is high or weak bonds in that case we got this expression which can be used. However, in order to use this expression what you need is to find out the characteristic vibrational temperature and see if the experimental temperature is much higher than the characteristic vibrational temperature then only you can use this approximation. And then by using the definition of characteristic vibrational temperature we define that q v is also equal to T by theta v. So, therefore, these are the different means these are the different ways of evaluating of calculating or of measuring the vibrational contribution to partition function. So, what we have discussed in this lecture and the previous lecture is how to calculate how to measure the value of vibrational contribution to partition function. First we discussed that there can be more than one normal modes of vibration. The overall value of vibrational contribution to partition function will be equal to multiplication of each mode of vibration. Then how to evaluate each normal mode of vibrations contribution to partition function you can use any of these three equations depending upon the temperature. So, now we are equipped with evaluating the partition function for translational degree of freedom, for vibrational degree of freedom and for vibrational degree of freedom. We will make the things more clear by solving numerical problems for each case before we switch over to electronic contribution to overall partition function. And that we will do in the next lecture. Thank you very much.