 Welcome back to the next lecture of statistical thermodynamics. Today we are going to learn about the molecular partition function. In the previous lecture we have derived expression for Boltzmann distribution and we will move ahead from there. The Boltzmann distribution that is n i upon n where n i is the number of molecules or particles in i th energy state n is the total number of molecules exponential minus beta E i E i is the energy of the i th state and then we have this summation j exponential minus beta E j and that you remember. That came for alpha in the derivation of this expression we have used method of undetermined multipliers and in that method of undetermined multipliers we had to constant alpha and beta alpha we have already accounted for and we will soon find out an expression for the beta. Now, I want to specifically draw your attention to the denominator the denominator which is summation j exponential minus beta i j this is called the molecular partition function. We will name this as small q we will write q is equal to summation j exponential minus beta E j. This j can run from 0 onwards that is 0 1 2 3 4 etcetera. So, let us expand this and see what happens this will be equal to when j is equal to 0 exponential minus beta into 0 is exponential 0. So, first term exponential 0 it is going to be 1 the second term is going to be exponential minus beta E 1. Third term is going to be exponential minus beta E 2 and you keep on going this is the expansion for molecular partition function. Here E 1 E 2 etcetera are first excited state second excited states and so on. Now, suppose that 2 or more states have the same energy. So, if different states have the same energy that forms a level. Therefore, a particular energy level can be 2 fold degenerate if 2 states are having same energy it can be 3 fold degenerate it can be more. So, therefore, in general what we will do is we will write G j fold degenerate C right I will include that G j into my expression now. What I want to say is if 2 states are having same energy then the same term will appear twice. If 3 states have the same energy then the same term will apply thrice. Therefore, the way to incorporate this is that let us include G j exponential minus beta E j and that this G j is the degeneracy of a particular level that is the jth level is G j fold degenerate. So, obviously when you expand this G 0 exponential 0 is going to be 1 then second one G 1 exponential minus beta E 1 plus G 2 exponential minus beta E 2 and then plus will have other terms. So, remember that we started from the ratio n i upon n what is this ratio n i upon n number of molecules in the i th state divide by the total number of molecule. It represents fractional population of i th state and I am going to discuss this in more details soon. So, obviously a question will come in the mind that earlier we have accounted for alpha. So, what is this beta soon we are going to show that this beta is connected with temperature by this expression beta is equal to 1 over k t and this k is Boltzmann constant t is the temperature in Kelvin. Sometimes you even say beta as temperature because we beta is connected with temperature through this term 1 over k t. So, therefore remember that the partition function is given by this expression that is q is equal to summation j G j exponential minus beta E j where beta is equal to 1 over k t. What we need to know is the degeneracy of each energy level. Once we have the values for G once we have the values of E therefore, we can explicitly calculate the value of molecular partition function. Also remember that I am highlighting that this is molecular partition function because we are talking in terms of individual particles we are talking in terms of individual molecules. So, what we have discussed we have started with Boltzmann distribution where n i upon n we refer to as population of the state to be more precise read it as fractional population of the state because it is n i upon n. What is this equal to exponential minus beta E i and this denominator we have just discussed that this is molecular partition function. So, therefore fractional population of any state can be calculated once we have the information about the various energy levels and also we need to know the temperature and the molecular partition function. Reemphasizing on the meaning of each term that we have discussed Boltzmann distribution, fractional population of ith state expressing molecular partition function and also highlighting that the requirement of degeneracy of the ith energy level and that that beta is connected to temperature by 1 by k t how it comes that we will be discussing a bit later. Now, let us discuss what is the interpretation of partition function what does it physically convey. So, some insights into the significance of partition function can be obtained by considering how it depends on temperature. Let us try to understand. Molecular partition function we just discussed is equal to summation j g j exponential minus beta E j or permit me to write in terms of temperature g j exponential minus E j upon k t. Let us expand it what we have now q is equal to put g is equal to 0. So, we have g 0 first term e 0 exponential raise to power 0 is 1, second one we have g 1 exponential minus E 1 upon k t plus g 2 exponential minus E 2 upon k t plus so on keep on going. Now, if you look at the comment that I made that some insights into the significance of a partition function can be obtained by considering how it depends on temperature. Let us concentrate on that we want to talk about what is the effect of temperature. So, when whenever we want to discuss about the effect of temperature the easiest thing is let us first take the extremes. First extreme is when temperature approaches a value of 0 when temperature approaches 0 exponential minus any E upon k into 0 what is this? This is exponential minus infinity this will tend to a value of 0. So, then what will be the value of q all other terms will disappear this term will disappear this term will disappear and all other terms will disappear. So, what will remain is only the first term q is equal to 0 or q approaches 0 when t approaches 0 ok. So, let me write this q approaches a value of g 0 as temperature approaches 0 fine. So, we have talked about the first extreme let us talk about this second extreme. Now, temperature approaches a value of infinity when temperature approaches a value of infinity exponential minus E upon k times infinity this will approach a value of 1 because it is exponential 0 is equal to 1. Therefore, this term is going to be 1 this term is going to be 1 and other all other exponential terms are going to be equal to 1 what will remain is g 0 plus g 1 plus g 2 plus so on. So, in that case your q will become g 0 plus g 1 plus g 2 plus keep on going that means very high value. I want you to appreciate the discussion that when t approaches 0 your partition function approaches a value equal to the degeneracy of the ground state on the other hand when t approaches infinity the value of partition function you see it approaches a very high value. So, therefore, what do we conclude from this kind of result? That means, when the temperature is approaching a value of 0 q is equal to 0 g 0 which is degeneracy of the ground state and when temperature approaches infinity the value of q also increases it increases to a very high value depending upon the temperature. That means, the partition function gives an idea of how many states are thermally accessible at a given temperature when t approaches 0 only ground state is thermally accessible when t approaches infinity infinite number of energy states are thermally accessible that is what is highlighted in this slide that as t approaches 0 we just discuss that q approaches a value of g 0 g 0 is degeneracy of the ground state and when t approaches infinity q value also approaches infinity that means very high number many energy states are thermally accessible. So, in some idealized cases the molecule may have only a finite number of states then the upper limit of q is equal to the number of states. Suppose in some idealized states idealized cases you have only 2 states available you have only 3 states available you have only 10 states available then the limit of q will be equal to the number of states to conclude this that partition function how do we interpret partition function gives an idea of how many energy states how many energy levels are thermally accessible at a given temperature. Let us take certain examples consider only the spin energy levels of a radical in a magnetic field spin energy levels that means in this case there are going to be only 2 states with the spin quantum number plus half and minus half. Therefore, how do we interpret that that the partition function of such a system is expected to rise to a value of 2 because there are only 2 states available as the temperature increases towards a value of infinity. Now, let us consider another system in which there are only 2 states available look at this the ground state energy equal to 0 because you know we have discussed in the beginning we will start with the ground state energy equal to 0 and then if there is some 0 point energy that number can be added if we want to calculate total energy of the system. We will address that a bit later the second energy state is at a separation of e how do we write the partition function for this system q is equal to 1 plus exponential minus beta e q which is equal to summation j g j exponential minus beta e j there are only 2 states the degeneracy of the ground state is 1 and energy is 0. So, 1 into exponential 0 is equal to 1 plus degeneracy of the second state is also 1 or second level is also 1 and exponential here the energy is e that is the expression for the partition function and that is what is written over here and expressing beta in terms of temperature you have 1 plus exponential minus e upon kt we should be able to conveniently write the expression for molecular partition function depending upon how many states are there in a system. Here we are talking about some idealized case in which there are only 2 states and in that case in that system the partition function is given by this expression. Let us have a little bit more discussion on this. So, we have these 2 states we have this expression for the molecular partition function we can have some discussion on the effect of temperature. If you plot molecular partition function versus temperature look at this left hand side figure the value when the temperature you know is 0 the value starts from 1 and then sharply rises starts from 1 which is that 1 this is 1 it sharply rises and then if you look at figure which is given on the right hand side it rises towards a maximum value of 2 when temperature approaches infinity it is exponential minus e upon infinity exponential 0 that means this part also becomes equal to 1. So, 1 plus 1 2 the partition function value starts from 1 and then rises as the temperature becomes very high it goes towards a value of 2. So, look at the comment the partition function for a 2 level system as a function of temperature is shown in this figures. The 2 graphs differ in the scale note here 0 2 about 1 which is shown in a expanded form over here the idea is to show that how the value of partition function will sharply rise and when k t upon epsilon becomes relatively very high the value then moves towards a value of 2 partition function a value of 2. Initially very fast and then a slow approach towards a value of 2 as the temperature approaches a value of infinity all right. Now, let us talk about fractional population of each state p i is equal to exponential minus beta e i upon q we have discussed this. So, therefore, we can write p 0 fractional population of the ground state equal to exponential 0 which is 1 upon q and p 1 which is equal to exponential minus beta e this is the energy upon q. So, p 0 is 1 upon q q is 1 plus exponential minus e upon k t or minus beta e and p 1 is given in this form. If I write this as exponential minus e upon k t upon 1 plus exponential minus e upon k t and this equal to 1 over 1 plus exponential minus e upon k t when t approaches infinity then this becomes exponential 0 that means exponential 0 is equal to 1. So, in that case this is going to be let us say first p 0 this is 1 over 1 plus 1 0.5 p 0 is 0.5 in the second case p 1 is also going to be 1 over 2 which is 0.5 as temperature approaches infinity as temperature approaches 0 you will see that when temperature approaches 0 then in that case what you have p 1 will turn out to be 0 p 0 will turn out to be 1. But what is important to note in this discussion is follow usually a common error is to suppose that once we increase the temperature and then all the molecules should be pushed to the upper state that is a common error that is a common error is to suppose that all the molecules in the system will be found in the upper energy state when t is equal to infinity that is the common assumption. However, if you look at this result what does this result say that when t approaches infinity each state is equally populated here we have 2 states and p 0 is also equal to 0.5 that is the fractional occupancy of ground state is 0.5 and fractional occupancy of the first state first excited state is also 0.5 that is what is written over here. However, please note that as t approaches infinity the populations of states become equal. So, we can generalize that the same conclusion is true for multi level systems here we talk about 2 level systems, but the conclusion can be extended to multi level systems that is as t approaches infinity all states become equally populated. Remember that it is not that when t approaches infinity all the molecules from the ground state are you know populated in the excited state no when t approaches infinity all states become equally populated. So, therefore, the take home lesson from this lecture is that the molecular partition function gives an indication of the average number of states that are thermally accessible to a molecule at the temperature of the system. What are the other key points at t equal to 0 only ground level is accessible and g is equal to q is equal to g 0 that is molecular partition function is equal to degeneracy. At very high temperature virtually all states are accessible and q is correspondingly very large that also we discussed high temperature means this k t is much higher than the energy of that state and q has a very high value and low temperature means k t is much much lesser than energy and q is close to g 0. Therefore, let us remember what is the expression for molecular partition function how to interpret the molecular partition function and how to express the fractional population of a state in terms of molecular partition function. We will discuss further how to express molecular partition function for states or for the number of states which are more than 2 in the next lecture. Thank you.