 Let us continue our discussion on contributions to equilibrium constant. In the previous lecture, we discussed how to get an expression for the number of molecules in the products or in the reactants. Reaction that we are talking about is our reactants in equilibrium with products and we have come up with an expression something like this ok. So, our main aim is to get N p divided by N r. Now, number of products and number of reactant molecules our system was like this, this was for reactants and this was for products products and this difference is delta E 0. This is what we are talking about and then we have various energy levels all these and that these also has various energy levels and we are talking about their distribution. So, let us talk about summation I n I for reactants. This I am talking about for reactants, this is equal to number of moles of reactants ok. Remember I am talking about reactants and this is equal to N by Q into summation. Let me put r because I am talking about reactants I will just put r for reactants exponential minus beta E I am just putting r you can put small r capital R whatever. What are these E r? This energy E r or you can read this as E r whatever way you say r I am reserving for reactants. So, that means when I talk about this and I measure the energy of any of the level of reactants that is with reference to the 0 point energy of the reactants. This one gives me N r is equal to N pi Q into this one is nothing, but the partition function when you only consider the reactant molecules. What we have here? We have this N is the total number of molecules which are combined in the combined reactants and product sector. This is the combined for the combined system the molecular partition function. This is the molecular partition function which is only applicable to the molecules which are in the reactant side ok. Let us mark this as our equation number 1. Now let us come towards the product side. Now let us talk about products that means here I will use the same expression and I now I will change to product is equal to N product and this is equal to N upon Q into I am writing for product exponential minus beta E p prime. Notice that I have put prime over here I did not put prime over here there is a reason why I did not put prime over there and I am putting prime over here. The reason for that is when you look at the products the 0 point energy level of product is above the 0 point energy of the reactants by delta E naught and these energy levels are with reference to this 0 point energy level. Once again these energy levels of the products are with reference to 0 point energy of the product which is shown over here. So, that means when I consider the combined system this E p prime is equal to E p plus delta E 0 because for the combined system the 0 point energy level is at this R level. Let us relook at what we should write now N p is equal to N upon Q into summation for products exponential minus beta that is what I am saying E p plus delta E 0 which is equal to N by Q into exponential of course summation p minus beta E p into exponential minus beta delta E 0 and this is equal to N upon Q and if you recognize this this is Q of the products into exponential minus beta delta E 0. We have this expression for N p and we have this expression for N R. Now you can take the ratio K is equal to N p by N R. So, N by Q N by Q will cancel out you have Q p by Q R into exponential minus beta delta E 0. I will elaborate on this beta del E 0 is equal to del E 0 by K T and if I want to express per mole then I will multiply by N A. So, K times N A will become R if I am expressing per molecule then delta E 0 by K T is enough and if I am expressing per mole then I will multiply a numerator and denominator by N A. So, then this per molecule energy will become per mole and K will become R. So, what we have done is by the arguments of the number of molecules which are occupying the energy levels corresponding to reactants and the number of molecules which are occupying the energy level corresponding to product we have come up with an expression which is same as that we get from the derived expression. But what is the key result of this and what are the key points that we should remember? One is that the value of K is dependent upon N p and N R and we have discussed that N p or N R is going to depend upon the population of each state. When I talk about population of each state that means we are talking about entropy and when I am talking about what is lying about what where product is lying above reactants 0 their 0 point energies are separated with a positive number that means we are talking about an endothermic reaction. So, basically K which decides delta G naught delta G naught is minus RT log K it incorporates what we derived what we discussed in chemical thermodynamics that delta G naught is a combination of delta H naught and delta S naught. Let us now take a very specialized specific example. In that example let us consider that the reactant has only one ground state available the other upper excited states are so far apart and the product has a uniform ladder of energy levels. The system that we are discussing here is R in equilibrium with B. We have reactants which has only ground state available that means can I write Q R which is equal to summation generally we write like this J G J exponential minus beta E J only ground state is there degeneracy is 1 that means the value is equal to 1 because E 0 E for J is equal to 0 the energy level is 0 therefore exponential 0 is equal to 1 for Q reactant we know, but for product what is given to us that it has a large number of evenly closed spaced levels. Careful examination says that this is a uniform ladder of energy levels that means when I talk about Q P is equal to summation J exponential minus beta E J there here E J is equal to J times E because this each energy level is separated by E and J can start from 1 to 0 1 2 etcetera because the 0 of this is lying above that of reactant by delta E 0. We have already derived this expression we are not going to get into the details of that Q P this is a uniform ladder of energy levels which is 1 plus exponential minus beta E plus exponential minus beta E square plus so on sum of a GP that means my Q product is equal to 1 over 1 minus exponential minus beta we have derived this kind of expression earlier. I have an expression for Q products and I have an expression for Q reactant do not get confused by this rotation this actually means Q reactant and this actually means Q of product and we know their difference in 0 point energy levels is delta E naught and since I have both these Q's and I know delta E naught now I can write an expression for the equilibrium constant what is that equilibrium constant it is equilibrium constant will be equal to Q P over Q R into exponential minus delta E naught by RT. We have just discussed that Q reactant is equal to 1 we have also discussed that Q product is equal to 1 over 1 minus exponential minus beta E. Now suppose if the temperature is high or if energy levels are very close compared to this temperature this spacing is very small or the temperature is high then I can write Q P is equal to 1 over 1 minus I can since this is going to be small if temperature is high then 1 minus beta E plus so on. This we have discussed earlier this is equal to 1 upon beta E or is equal to k T upon E this is a high temperature result we have Q P we have Q R substitute in this Q P is k T upon E so here we have k T upon E Q R is 1 so do not need to worry into exponential minus delta E naught by RT. For this specific example the expression for equilibrium constant is k T by E into exponential minus delta E naught by RT ok we have derived some expression is this expression telling us anything qualitatively or quantitatively. Well this particular expression only applies to the given condition if the system given to you if the system changes remember that the corresponding expressions final expressions are also going to change this particular expression which applies to this system now let us see what qualitative information we can derive from it. In industry how to optimize a process so that you get a higher value of equilibrium constant or a higher value of yield is very very important. Now how to improve the value of equilibrium constant how to increase the value of equilibrium constant or how to decrease the value of equilibrium constant an equally valid question will depend upon various factors. Number one is the reaction exothermic or endothermic number two what is the stoichiometry of the reaction that is the value of delta N G we are talking about gas phase reaction. The exothermicity of the reaction endothermicity of the reaction is going to guide us whether the temperature should be increased or temperature should be lowered. Similarly the number of moles participating in the reactants and the product will also tell us whether the pressure should be increased or pressure should be decreased. And accordingly suppose if the reaction is endothermic you remember leach et alier principle what is leach et alier principle leach et alier principle is that if a reaction at equilibrium is disturbed by an external factor then the reaction will shift in such a direction. So, as to undo the effect of the externally applied disturbance for an endothermic reaction according to leach et alier principle if you increase the temperature the value of equilibrium constant also increases. For an exothermic reaction on the other hand earlier I talked about endothermic reaction that is increase in temperature will increase the value of equilibrium constant. For an exothermic reaction an increase in temperature leads to reduction in the value of equilibrium constant. Similarly you can talk about the effect of pressure total pressure is increased at equilibrium then also the reaction composition at equilibrium will shift according to leach et alier principle. That means it will move towards the direction where the pressure is lower the leach et alier principle should also be applicable over here. That means the discussion that we had should also be extendable to alteration in temperature and alteration in pressure. The treatment that we are covering over here here in the expressions for example, if you look at the expression for the equilibrium constant it has Boltzmann constant it has absolute temperature it has energy that energy is the spacing between the adjacent energy levels of the product it has gas constant it has temperature and it has the difference in the 0 point energy levels. So, we should be able to address the effect of temperature over here. Obviously when you are going to alter the temperature the population of the states here corresponding to P is going to change. What is the prediction here? This one where the 0 point energy level of the product is higher than the 0 point energy level of the reactant that means we are talking about an endothermic process. If I increase suppose the reactant has some population and if I increase the temperature then the population of upper states is going to increase. If the population of upper states that is the products is going to increase that is going to raise the value of equilibrium constant, but is the same answer we can get from this expression that we can try to obtain by comparing this factor versus this factor. The increase in temperature or the decrease in temperature is going to lead to what? Before I start discussing that I want to reiterate that in statistical thermodynamics what we have discussed is discussed equilibrium constant in terms of population of different energy levels. When I talk about population of different energy levels that means I am also simultaneously addressing the entropy. And if at the same time I am talking about the delta E 0 this delta E 0 which is the difference in the 0 point energy levels that means here I am talking about exothermicity or endothermicity ok. So, if delta E 0 is positive the situation looks like this. If delta 0 is d 0 is negative you should be able to redraw the similar figure if delta E 0 is negative. We can address the exothermic reactions we can address the endothermic reactions. So, in the next lecture what we are going to discuss is that when we increase the temperature or when we decrease the temperature then which factor becomes dominant and in which direction the equilibrium constant will move. Once again obtaining an optimum value of equilibrium constant is very very important in this to be. And that is why the reaction conditions need to be modified. So, that you can achieve the desired results. Here by using the concept of statistical thermodynamics we are trying to explain that what is the effect of disturbance of external factors on the equilibrium constant. One such external factor that we are discussing at present is temperature. Therefore, the effect of temperature on any equilibrium becomes very important because depending upon exothermicity or endothermicity the value of equilibrium constant will become higher or lower. More qualitative and quantitative insights into equilibrium constants are still possible from the derived expressions, but those we will discuss in the next lecture. Thank you very much.