 Welcome back to further discussion on mean energies as discussed in the previous lecture mean energy is basically the internal energy per molecule that means, u minus u 0 which we call as the total energy you divide by the number of molecules. So, that becomes the total energy that is divided by number of molecules which is mean energy. And in the previous lecture we have discussed mean energy for translation of a molecule and for rotation of a molecule. Let us go back to the discussion which were which we were having earlier that is the mean energy of a molecule can be obtained from the knowledge of its molecular partition function. And where m is a particular mode which is translational rotational vibrational electron. In the previous lecture we have already discussed the translational we have also addressed the rotational, but we restricted to linear rotor. So, the expression that we got was this that is the mean rotational energy is given by h c b into the expressions mentioned into the numerator and denominator. In these expressions b is the rotational constant beta represents the temperature in the form of 1 by k t. So, a plot of mean rotational energy versus temperature looks like this. Basically this plot is represented as ratio of mean rotational energy with h c b versus t by theta r. So, the behavior of the mean rotational energy versus temperature will look like this. Initially if you see as long as the temperature is less than the rotational temperature the mean energy value is not much, but it starts rising as the temperature increases. Now, this expression that we derived yesterday was based upon explicit summation for the expression of rotational partition function. And we have discussed that when the temperature is high that is when the temperature is higher than the rotational temperature in that case the rotational partition function can be equal to 1 upon sigma h c beta b. This is an expression where temperature is higher than theta r. So, therefore under those conditions your mean rotational energy which will be minus 1 upon q r delta q r by delta beta at constant volume what it will be let us see it is minus 1 over q r 1 over q r from here it will be sigma h c beta b into derivative of q r which is going to be minus 1 over sigma h c beta b whole square into sigma h c b. So, minus and minus becomes plus and 1 sigma h c beta b will cancel with this and what will remain here is sigma h c b will also cancel with sigma h c b. So, what I have is 1 upon beta and beta is equal to k t and that is what is the result here. So, that means, when the temperature is higher than the rotational temperature or at a relatively higher temperature the dependence of mean rotational energy on temperature is linear and you can see here it linearly varies. So, therefore, when to use this expression and when to use this expression will simply depend upon the temperature of interest. If the temperature is not very high then you have to use this expression and if the temperature is high then you have to use you can use this expression. So, remember that this discussion that we have had is basically for a linear rotor. Now, in case you have to deal with the non-linear rotor examples can be water or there are many other molecules. If the rotor is non-linear then you can follow the similar procedure to come up with the energy expression and for simplicity if you calculate the mean rotational energy for a non-linear rotor then mean rotational energy for a non-linear rotor will come out to 3 by 2 k t that I am leaving for you to do this exercise. So, just remember that for a linear rotor the mean rotational energy is given by k t and for a non-linear rotor the value is 3 by 2 k t and you should be able to derive an expression for the mean rotational energy of a non-linear rotor if the temperature is less than rotational temperature. If the temperature is more than rotational temperature then the result is right here. So, I hope that this is clear to you. So, now that we have discussed the translational contribution and rotational contribution the next that we will discuss is vibrational contribution. These general comments we have already discussed earlier. So, therefore, I am not going to discuss these again. Now, let us talk about mean vibrational energy. Mean vibrational energy for a particular mode mean vibrational energy will be equal to minus 1 by q v into del q v over del beta at constant volume. Do not mix this v with this v by this v means vibrational contribution and this v means constant volume. And remember that the overall vibrational partition function is equal to product of the vibrational partition function for each mode of vibration. And here for each mode of vibration we are now calculating the mean energy. Let us proceed towards the derivation. Vibrational partition function is given by 1 divided by 1 minus exponential minus beta h c nu bar. This is a general result with no approximation. The only approximation that is used over here is we have considered it as a harmonic oscillator. So, therefore, now the mean vibrational energy will be equal to minus 1 by q v. This is minus 1 minus exponential minus beta h c nu bar. This is minus 1 over q v. Now, we want to take the derivative of this. The derivative of this is going to be minus 1 divided by 1 minus exponential minus beta h c nu bar square into minus exponential minus beta h c nu bar into minus h c nu bar. We are taking a derivative with respect to beta. Now, you carefully examine what we have 1, 2, 3, 4. 4 negative signs will become positive sign. This will cancel with this. So, what I have is mean vibrational energy is equal to h c nu bar into exponential minus beta h c nu bar divided by x 1 minus exponential minus beta h c nu bar. So, I have got now this as an expression for mean vibrational energy. I can further simplify it. I can multiply the numerator and denominator with exponential beta h c nu bar. So, in that case I will be left out with h c nu bar over exponential beta h c nu bar minus 1. This is the alternate form of what we derived here. I just multiplied the numerator and denominator by exponential beta h c nu bar. So, here what we have is now an expression for the mean vibrational energy which is equal to h c nu bar divided by exponential beta h c nu bar minus 1. There is no approximation involved in it other than considering the vibrational energy level following the harmonic oscillator. Now, based upon this expression we can talk about the dependence of mean vibrational energy upon temperature. Remember that this is a general result neither low temperature nor high temperature approximation has been involved. So, what we have is mean vibrational energy is equal to h c nu bar divided by exponential beta h c nu bar minus 1. So, as I just mentioned this is a general result no low or high temperature approximation used over here and I will soon show that at high temperature you can further expand this and get that mean vibrational energy is equal to k t that is a high temperature limit, but if the temperature is not high even if the temperature is high still this expression is equally valid all right. This is the result the exponential beta h c nu bar is equal to exponential h c nu bar by k t. If the temperature is high then h c nu bar by k t is going to be a small quantity and it is like exponential x if x is small then you can expand that 1 plus x minus x square by 2 factorial plus so on keep on going that. Let us follow that mean vibrational energy is equal to h c nu bar divided by now the temperature is high. So, I can expand this 1 plus beta h c nu bar minus so on the squared quantity is divided by factorial 2 etcetera minus 1 which is already there. So, this will give me h c nu bar divided by beta h c nu bar in this h c nu bar and h c nu bar can cancel. So, eventually what we have is mean vibrational energy is equal to k t, but remember that this is a high temperature result or if the bond is weak bond is very weak. So, therefore, when you are not sure whether the temperature is high enough then one can go by the result which we obtained here and if we are sure that the temperature is high enough that many vibrational levels are significantly populated then we can use this result. So, by now what we have discussed is mean translational energy, mean rotational energy, mean vibrational energy. What is left over is to discuss the mean electronic energy, but if we clearly examine all these the order of energy levels as we have discussed repeatedly earlier you have translational then rotational then vibrational then electronic. Translational partition function is quite high rotational partition function is lesser than translational partition function. In fact, many folds lesser than translational partition function and then many folds lesser value is that for the vibrational partition function. For electronic partition function usually it is equal to degeneracy of the ground state this also we have discussed repeatedly. Now, when it comes to the mean energies mean translational energy we have discussed mean rotational energy also expression we have discussed in the previous lecture as well as in this lecture and there you have to be careful about whether you are talking about a linear rotor or you are talking about a non-linear rotor and going beyond the rotational contribution then comes the vibrational. And in vibrational partition function remember that the overall vibrational partition function is the product of partition function for each mode of vibration. Once we have the expression for vibrational partition function it is easy to derive an expression such as this for the mean vibrational energy. Now, when you talk about electronic mean electronic energy in that case also you need to require the expression or the value for the partition function. Let us say I very briefly touch upon electronic contribution electronic contribution to partition function can be obtained directly from this expression q electronic summation j g j exponential minus beta e j where j is a certain electronic state. And we know that electronic energy levels are far separated. So, in general the electronic contribution to the molecular partition function is equal to the degeneracy of the ground state because the excited states are far apart means the difference the distance in energy terms between the ground state and excited state is large. So, therefore, usually at under normal conditions of temperature only the ground state will be significantly populated under that condition or under those circumstances the value of the electronic partition function is approximately same as degeneracy of the ground state. And of course, if the temperature is higher then you can include the higher terms also, but what you will see is that in general the increment by considering the higher states to the value of g 0 are not or is not very high. Therefore, it is ok to assume that the electronic contribution to the molecular partition function is equal to degeneracy of the ground state. And then beyond this if I were to include electronic mean electronic energy this will be equal to minus 1 over q electronic into del q electronic del beta at constant volume. And here you can examine that if the value of q e is constant for example, you know see here this number is a constant value. And next the derivative of a constant with respect to beta is going to be 0. So, that means the mean electronic energy under normal conditions of temperature you do not have to worry about. So, by now we have discussed the mean energies of almost all the modes which we generally consider for a molecule mean translational energy, mean rotational energy, mean vibrational energy and mean electronic energy. Remember that the energy whether we talk in terms of total energy or we talk of internal energy it can be further connected to some other thermodynamic quantities. For example, heat capacity heat capacity at constant volume is the amount of heat required to raise the temperature by 1 unit or 1 Kelvin at constant volume. And what is that heat required to raise the temperature by 1 unit at constant volume q v is simply equal to delta u. So, there you are talking about internal energy and then the temperature derivative of internal energy becomes heat capacity. So, that means once we have the knowledge of the mean energies we can further connect these expressions to obtain expressions for heat capacities whether at constant volume or at constant pressure that is a different matter. But as long as we are talking about internal energy we can connect this to the heat capacity at constant volume. So, remember which result to be used under general conditions of temperature, temperature low temperature let us say and when the temperature is high then you can use the simplified result both for rotational and vibrational contributions. And as we just discussed that for electronic contribution you do not need to worry because in general the value of electronic contribution to partition function is equal to degeneracy of the ground state which is a number and that number is constant. So, after having a knowledge of mean energy for different modes now we are all set to take it forward for the discussion of heat capacities. So, in the lectures ahead we will derive expressions for heat capacities at constant volume and once we have the information about heat capacity at constant volume the heat capacity at constant pressure can easily be obtained from those results. But those we will discuss in the next lecture. Thank you very much.