 Welcome back. Today we are going to discuss a very important thermodynamic quantity that is equilibrium constant. In classical thermodynamics, we have already discussed equilibrium constant at length and connected with the change in standard molar Gibbs energy. In statistical thermodynamics considerations, we are going to connect this equilibrium constant with molecular partition function. The experimental measurement of equilibrium constant can be done in a variety of methods. There can be spectroscopic methods, there can be calorimetric methods, there can be direct methods where a suitable fitting of the model to the experimental data points can give the value of equilibrium constant. Alternatively, we can get the value of equilibrium constant from the value of delta G naught and delta G naught is equal to delta H naught minus T delta S naught. So, therefore, there are variety of methods calorimetric spectroscopic and there may be some other methods which you can use for experimental determination of equilibrium constant. Even when we talk about the spectroscopic determination of equilibrium constant and then those who have consulted the book by Lackowitz and are dealing with the interaction between ligands and biological macromolecules like proteins, then Stern-Walmer kind of analysis can give you the value of quenching constant which under certain circumstances can become equal to equilibrium constant. In this course, we have been talking about the molecular partition function and by now we have discussed all contributions to the molecular partition function. For example, translational contribution, rotational contribution, vibrational contribution, electronic contribution these are the four major contributions that constitute that form the overall molecular partition function and which can be related to various thermodynamic quantities. In today's lecture as I just said we are going to connect the equilibrium constant with the partition function Q. What is the significance of equilibrium constant? Just as a recap to earlier knowledge. Equilibrium constant is a very important thermodynamic quantity because it tells you how much product is formed. Equilibrium constant is the ratio of the concentrations of the product to that of the concentrations of the reactants weighted by the power of their respective stoichiometric numbers. In industry equilibrium constant is very important because as I just said equilibrium constant is directly related to the amount of product formed and therefore in industry they are always in finding out appropriate conditions which can be tuned to achieve a desired value of equilibrium constant and the molecular partition function. Since we have discussed that it is connected to the energy levels of the molecule and therefore that means here also the determination of equilibrium constant by statistical means is in fact spectroscopic method of evaluating the value of equilibrium constant. Let us begin our discussion. There are two things I want to highlight to begin with. One is that we will discuss ideal systems first ideal gases. So, therefore initially I will restrict to gas phase reaction and also the gas phase reactions means I will talk about ideal gases. One ideality we will bring later. Equilibrium constant K is related to delta G naught by this equation delta G naught is equal to minus RT log K. In classical thermodynamics we derived this equation delta G naught is equal to minus RT log K based upon a certain derivation which was delta G is equal to delta G naught plus RT log K. In this equation delta G is the change in Gibbs free energy delta G naught is the change in Gibbs free energy under standard state conditions. Remember difference between delta G and delta G naught it is a common mistake sometimes we do not differentiate between the two, but these two convey entirely different meaning. This delta G is actually the slope delta G delta psi at constant temperature and pressure that means the positive or negative value of this delta G is going to decide in which direction the reaction is spontaneous. At equilibrium delta G is always 0 this slope is 0 and Q is equal to K. So, therefore, when you rearrange that you get delta G naught is equal to minus RT log K. The purpose of this discussion was to again highlight the difference between delta G and delta G naught these two are not the same quantities these two are different quantities. Now, since today's task is to correlate K with Q that means we have to connect this K somehow with the Gibbs free energy or the changes in Gibbs free energy under standard state conditions. From our previous discussion we have derived this equation G is equal to G 0 this is absolute 0 T is equal to 0 is equal to minus n RT log Q by n this we have discussed earlier. And we had also discussed that n number of molecules is number of moles times Avogadro constant. So, therefore, when you convert this upper equation this equation and write n is equal to small n times n a then Q divided by n becomes Q m molar partition function. And throughout you divide by n if this equation is divided throughout by n it results into this equation where n is consumed into per molar quantity. That means what I am writing here is G by n is equal to G m molar quantity. And we have invoked standard state conditions you see throughout this not has been put why I am putting this standard state conditions because the relationship that I am interested in connecting is delta G 0 is equal to minus RT log K. So, therefore, I need to have expressions for Gibbs free energy under standard state conditions ok. So, from here onwards we will talk for the Gibbs free energy under standard state conditions. So, what is a standard state? First of all we should be clear on what is a standard state. We have also discussed or your teachers might have already discussed with you what is a standard state? Usually the answer comes temperature 25 degree centigrade no temperature can be any substance should be in its pure form and pressure should be 1 bar that is the latest recommendations of IUPAC. What is a standard state? The substance should be pure in its pure form pressure should be 1 bar this is very important pressure should be 1 bar temperature can be any there is no fixation of temperature you can have standard state conditions at any temperature. For example, you have nitrogen and 2 the standard state of nitrogen and 2 is gas at 25 degree centigrade. If you reduce the temperature towards 0 Kelvin the standard state will be different. So, therefore, temperature can be any substance should be in its pure form and pressure should be 1 bar. In other words p naught this naught here signifies the standard state conditions that is 1 bar and since here we are dealing with ideal cases ideal gases I will write p v is equal to n r t that is a perfect gas equation or ideal gas equation. So, that means here v I can write this as v by n is equal to r t by p and then after that if I invoke standard state condition then v by n becomes v m naught and p instead of p I write p naught what it suggests is that when you are dealing with the gaseous systems and treating the gases at ideal gases then you can always calculate the volume by using r t by p naught volume need not be given to you should be able to calculate from the knowledge of gas constant and from the knowledge of temperature p naught is anyway 1 bar. So, therefore, under standard state conditions it is easier to get v m naught no data needs to be given it is very important to understand the concept of standard states I hope this description of standard state is clear to you. Now, let us express the molar Gibbs function under standard state conditions for any species j any species j. So, g m naught minus g m naught at 0 is equal to minus r t log q j m naught by n a m stands for molar j stands for species j naught stands for standard state conditions n a is the Avogadro constant. And as mentioned over here q j m is the standard state molar partition function of species j now consider any reaction any gas phase reaction one such example is a moles of a reacting with b moles of b to form c moles of c and d moles of d this is the expression that we are going to use this one this expression allows me to express molar Gibbs function of j under standard state conditions in terms of molar Gibbs function of j at absolute 0 and the molecular partition function what I am interested in is knowing delta g naught. And some books will write a subscript r you can write r means reaction. So, reaction Gibbs free energy change under standard state conditions can be obtained from summation j nu j g m j under standard state condition what it means if I expand this for this reaction let me put a partition over here that means for this reaction the way I will expand this expression is in this let me explain various notations r stands for reaction this symbol stands for standard state condition nu j is the stoichiometric number which is positive for products and negative for reactants I repeat nu j is the stoichiometric number which is positive for products and negative for reactants. Use this positive for products that means what I will write is equal to I open this summation this will be c times g m naught c plus d times g m naught d minus a times g m naught a minus b times g m naught b. When I said positive for products and negative for reactants that stoichiometric number it is becoming clearly here I am writing c and d both as positive numbers they are products a and b they are negative numbers here written over here these are the reactants ok. What I will do is now let us go step by step I am interested in finding this expression summation j nu j I will retain nu j over here and in bracket I will use this expression over here this is equal to g m naught j at absolute zero minus r t log q j m naught by n a we have simply substituted for g m naught this means I can now write delta r g naught is equal to summation j nu j g m naught j at absolute zero minus r t is common constant it comes out. So, this one j nu j log q j m naught by n a try to understand it what is simply done is we use this equation which says that reaction Gibbs free energy under standard state conditions can be obtained from the standard state Gibbs function of the products and reactants weighted by their stoichiometric number and what I did was then I retain nu j of this and for g m j I substituted this and we arrive at the next equation which is equal to delta r g naught is equal to this summation at absolute zero minus r t into this expression nu j log q j m naught by n a we will continue from here. So, what we have is delta r g naught is equal to summation j nu j g m j zero this is my first expression from the previous equation this one minus r t log minus r t summation j nu j log q m j naught by n a purely general equation this is what we had minus r t nu j log q j m naught by n a ok. Now, if you very carefully look at this is also delta g naught at absolute zero. So, that means I have under general conditions delta g naught is equal to this is delta g naught for j at absolute zero minus r t and then allow me to write next expression like this what I have done is I have taken this stoichiometric number as power permitted mathematically ok. Now, we need to write something for this consider g is equal to h minus t s is equal to h is equal to u plus p v minus t s and since we are dealing with ideal gases which is u plus n r t minus t s and does not this equation says g at absolute zero is equal to u at absolute zero because when t is equal to zero then g becomes u. Now, I can write next one delta r g naught is equal to delta u I will say j at zero minus r t summation when you open the summation this is log a plus log b plus log b plus log c right and when you say log a plus log b plus log c etcetera that will be equal to log a into b into c etcetera etcetera I can convert this summation into product. So, I can write this as a product that means this will be minus r t log product of q j m naught by n a it is the power nu j. So, m and j are getting interchange do not worry about that. So, log product of all this is equal to log a plus log b plus log c etcetera. So, I have used that mathematical transformation I hope by now it is clear how we got this expression what we will do is that instead of delta u we will write this as delta e zero just a symbol to describe that this is the difference in the zero point energies instead of delta u u shows internal energy we will just write rewrite in the term delta e naught ok. So, the new expression that we get now is delta r g naught is equal to delta e naught minus r t log pi j q j m naught by n a raise to the power stoichiometric number. So, by now we have an expression for delta g naught and we have some right hand side which is expressed in terms of change in zero point energies and the molecular partition functions that means the right hand side of this equation can now spectroscopically be determined. So, what we have other equations that we have with us are delta g naught is equal to minus r t log k this expression this equation we have derived when you are learning in classical thermodynamics r is a matter of convenience usually r is written for reaction. Now you see this delta g naught and this delta g naught can be equated and once you have these two equations we can come up with an expression for equilibrium constant. So, you might have noted the strategy since we wanted to develop a connection between equilibrium constant and molecular partition function and we realize that the equilibrium constant can easily be expressed in terms of changes in standard molar Gibbs energy. So, we decided to express Gibbs energy in terms of molecular partition function then through a series of steps we have come up to this expression where the right hand side can be spectroscopically determined. The next step will be to equate these two and then come up with an expression for equilibrium constant, but that we will discuss in the next lecture. Thank you very much.