 In this lecture, we are going to connect entropy with the molecular partition function. After having established a relationship between entropy and W, weight of the most probable configuration that is we talked about S is equal to K log W. Now we have to look out for the ways that is how to connect this log W with the molecular partition function that is going to be our next goal. Now let us refer back to our earlier discussion when we talk that instead of W it is better to talk in terms of log W because it will allow us to develop further relations further equations in a more easier way and today the time has come to demonstrate that. Remember that log W is equal to n log n minus summation i n i log n i this we have derived some time ago. Now let us write rewrite in some other way instead of n let me write summation n i because n is equal to summation n i then I have log n and I have minus summation i n i log n i and let us substitute S is equal to K log W. So instead of log W write let us write summation i n i log n minus summation i n i log n i which I can write as K summation i let me write n i log n minus n i log n i let us rewrite this I can write this as minus K summation i n i log n i upon n I have brought the negative sign here because I am keeping this n i into numerator I will do one modification over here I will multiply by n and divide by n here n i upon n and I have log n i upon n what is this n i upon n it is the fractional population. You see the strategy over here we have converted S is equal to K log W into an expression which relates entropy with fractional population. So let us write this as minus n times K summation i this is P i log P i we have connected entropy with fractional population of I H state why we have done this because we already know that a relationship exists between the fractional population of I H state and molecular partition function what is that remember that P i is equal to n i upon n which is equal to exponential minus beta E i upon Q we are moving towards connecting S with Q if P i is this then log P i this is what we are interested in log P i is equal to minus beta E i minus log Q and now I can substitute this P log P i over here let us do that and see what happens. So I have S is equal to minus n times K summation i P i and log P i is minus beta E i minus log Q we can now further work on this this is equal to minus and minus becomes plus. So n times K summation i P i beta E i then plus n times K summation i P i log Q we can further work on this now S is equal to here n K summation P i beta E i let us try to now rewrite P i as n i upon n because that will allow me to further simplify this. So this is going to be n times K summation i instead of P i let me write n i upon n beta E i plus n K log Q summation i P i we can further work on this and and n get cancelled and beta is equal to 1 by K T alright I once again repeat this this n and this n get cancelled beta is equal to 1 over K T. So therefore what I have is 1 by T 1 by T summation i n i E i plus n K log Q I repeat what I have done and and n get cancelled beta is equal to 1 by K T. So what I have is 1 by T and inside we have n i and E i leftover and n K log Q because summation of all the populations fractional populations has to be equal to 1 right. So log Q is constant I take it out of summation. So summation i P i it is the sum of all fractional populations that has to be equal to 1. So I have this expression now we have to now work upon this and let us now see what form it takes remember that this this will be equal to I have summation n i E i this is equal to total energy by T plus n K log Q I will come back to this later on summation n i E i is equal to total energy by T plus n K log Q what we derived was S is equal to E total energy by T plus n K log Q and remember that U minus U 0 is the total energy. So therefore, let us substitute S is equal to U minus U 0 by T plus n K log Q we have now come up with an expression which connects entropy with the molecular partition function. We already have developed equation for U minus U 0 remember that U minus U 0 was equal to minus n by Q del Q del beta at constant volume or remember that we said that we can also write this as minus n del log Q del beta at constant volume. So U minus U 0 will come from here in terms of molecular partition function which can be substituted over here and n K log Q. So this is the relationship of Boltzmann formula for entropy to partition function. So we have discussed relationship between S and W remember that if you want to connect S and W that is S is equal to K log W and if you want to connect entropy with the molecular partition function then the expression to be used is this one. Let us try to apply this formula and calculate the entropy of a collection of n independent harmonic oscillators and further evaluate it using vibrational data for iodine vapors at 25 degree centigrade. So here we are talking about n independent harmonic oscillators. Harmonic oscillators will follow the arrangement of energy levels in a uniform ladder manner. So this is a uniform ladder of energy levels we need to make use of this and we have just derived the expression that S is equal to U minus U 0 by T plus n K log Q. What we need here is U minus U 0 what is U minus U 0 is equal to minus n by Q tell Q del beta at constant volume. So therefore, what kind of information I require? I require an expression for partition function which we have already discussed for a uniform ladder of energy levels. The partition function for a uniform ladder of energy levels was 1 over 1 minus exponential minus beta that we have derived earlier. Therefore, U minus U 0 is going to be minus n divided by Q which is 1 minus exponential minus beta e into del Q by del beta at constant volume. So we need to take derivative of this. This is going to be minus 1 over 1 minus exponential minus beta e whole square into minus exponential minus beta e and into I have minus e minus n by Q into del Q by del beta at constant volume. So what I have now U minus U 0 is equal to 1 negative second third fourth fine. So I have n from here I have e from here exponential minus beta e and I have 1 minus exponential minus beta e from here. This is the expression for U minus U 0 U minus U 0 by t that is what we are interested is equal to n what is 1 by t because beta is equal to 1 over k t. So you use 1 over t is equal to k times beta. So 1 over t I will use as k times beta k times beta I have e I have exponential minus beta e and then I have 1 minus exponential minus beta e and then the second term is n k log Q. Log Q you can write from here if Q is 1 over 1 minus exponential minus beta e then log Q is going to be minus 1 minus exponential beta e. So once you have this this log Q can be substituted here and U minus U 0 by t we already have n k beta e exponential minus beta e over 1 minus exponential minus beta e substitute in this overall equation and you will get this result. Here in order to get this result from this I multiply a numerator and denominator by exponential beta e. So therefore the transformed result is going to be s is equal to n k beta e over exponential beta e minus 1 minus log 1 minus exponential beta. So therefore you can essentially here express entropy as a function of temperature from the knowledge of the energy separation for the harmonic oscillators. This is the expression which connects the entropy of a collection of and independent harmonic oscillators with the temperature. The question is for iodine vapor. So we need the data for iodine. We have derived this expression s is equal to n k beta e over exponential beta e minus 1 minus log into exponential 1 minus exponential beta. And the system under consideration is uniform ladder of energy levels. The separation between simultaneous energy states or energy levels is e. The data given now for iodine is beta e is 1.036 and molar entropy once you substitute this number over here the molar entropy comes out to be 8.38 joules per Kelvin per mole. This is at 25 degree centigrade. But the expression this expression permits you to connect the entropy with the temperature here. And let us take a look at now the plot over here. The plot is entropy s by n k you can take n k to the other side and plot s by n k versus k t by e. This is one way of plotting. Essentially you are plotting entropy versus temperature in some other form. And from the plot you see here when temperature approach is 0 the entropy also approaches 0. And as the temperature increases the entropy also increases in a manner which is without limit. So this particular example is exclusively for harmonic oscillator that is for a system which has a uniform ladder of energy levels. And this expression allows us to interpret or discuss entropy as a function of temperature. Suppose if you have to deal with different systems where it is not a uniform ladder of energy levels. And then you have to express entropy as a function of temperature. How will you do that? Let us have a discussion on that s is equal to u minus u naught by t plus n k log q. This expression that we have obtained is actually obtained based upon this expression. And in order to obtain this expression we need to have information on u minus u 0 by t. And u minus u 0 by t is also expressed in terms of partition function. Let me complete this expression. This will be 1 by t. And u minus u 0 is what? Is minus n by q del q del beta at constant volume plus n k log q. So essentially what we need is the information on molecular partition function. As I just discussed in this example, here we dealt with uniform ladder of energy levels. You can come across systems where there is doubly degenerate ground state, non-degenerate first excited state, doubly degenerate second excited state, triply degenerate next excited state. You should be able to write an expression for the molecular partition function. Once you know the expression for molecular partition function, substitute over here and obtain an expression for the entropy. So, what we discussed in this lecture is that in order to obtain entropy, you need either an information on w weight of a configuration or you need information on molecular partition function. Once we have information on w and q, we can easily write an expression for the entropy of the system. We will solve more numerical problems and discuss more applications on entropy in the next lecture. Thank you very much.