 Welcome back to the next lecture of statistical thermodynamics. Today we are going to talk about residual entropy. Entropy is a very important thermodynamic quantity because the changes in entropy decide or contribute to the spontaneity of a process. The spontaneity of a process usually we talk in terms of changes in Gibbs free energy and we say that if delta G at constant temperature and pressure is negative then the process is spontaneous. And this delta G is a combination of delta H and a combination of T delta S. Therefore, the negative value of delta H and positive value of delta S they contribute to the spontaneity of a process. The second component that is T delta S which is basically the change in entropy suggests that entropy is a very important thermodynamic quantity. So, obviously the question comes that how to determine absolute entropy experimentally. Here we recall the third law of thermodynamics which suggests that entropy of every substance is positive, it may become 0, it may become 0 at absolute 0 and it does become 0 at absolute 0 for perfectly crystalline substances. The catch there is perfectly crystalline substances, what the third law says that the entropy of every substance is positive and it may become 0 at absolute 0 may. That means it does not say that it is 0 at absolute 0, it says it may become 0 at absolute 0 and it does become 0 at absolute 0 for perfectly crystalline substances. That means if a substance is not perfectly crystalline the entropy at absolute 0 may not become 0. Now, let us take a look at some of the comments. First comment is entropies may be calculated from spectroscopic data and they may also be measured experimentally. When we say that entropies may be calculated from spectroscopic data. That means essentially here we are talking about connection of entropy with partition function because partition functions can be determined from spectroscopic data. The second comment is they may also be measured experimentally. When I say they may be measured experimentally then what I mean is we know this expression S at a final temperature is equal to S at initial temperature plus integration T i to T f C p by T d t. According to this expression if I set initial temperature to be 0 that means then I am talking about S 0. If my initial temperature is 0 then S 0 that means then at any general temperature I can write S t is equal to S 0 plus integration 0 to T C p by T d t. So, from the knowledge of S 0 I can experimentally determine the value of entropies and entropy provided I know the heat capacity and heat capacities can be determined calorimetrically. So, this is experimental determination of entropies this is something which you have studied in classical thermodynamics. Calorimeters are required to measure heat capacities and once we know the temperature dependent heat capacity we can easily get the value of the entropy at any temperature T. But this expression that I wrote may not be always true if there is a phase transition that means if I have if I start from solid form at 0 Kelvin then when I heat it A will be solid up to its freezing point then there is an equilibrium. So, A becomes liquid at freezing point and then I can further heat it A let say I can have liquid and any general temperature. The point that I am going to make here is that the comment number 2 says that in many cases there is a good agreement, but in some the experimental value is less than the calculated value. In general there is a good agreement, but that is not necessary sometimes there can be discrepancies also that is what the comment is made over here that, but in some cases the experimental entropy is lesser than the calculated value. Now if it is less then we have to assign a reason for that what could be the possibility. Now look at the next comment one possibility is that the experimental determination failed to take a phase transition into account as I was trying to explain here that suppose if I miss this phase transition then your experimental value of entropy will be lesser than the calculated value. Why? Let us see how to write it for this I will write s t is equal to s 0 plus integration 0 to t f C p of solid by t d t I have covered this part. Now there is a phase transition and phase transition it is delta h fusion by freezing point this is the entropy of phase transition you have studied in classical thermodynamics and then one you further heat this liquid plus integration t f to a general temperature t C p of liquid by t d t. What this discussion suggests is that if by some chance you miss this phase transitions then your experimental value will turn out to be lesser than the theoretical predicted. Remember that the phase transition does not necessarily has to involve conversion of solid to liquid to liquid to gas etcetera. There can be phase transitions within the solid itself for example sulfur orthorhombic monoclinic, carbon diamond form and graphite form both are solid forms. So, if there is a phase transition within a solid and if we somehow ignore that then our experimental value will be lesser than the theoretical value. So, going back to the comments one possibility is that the experimental determination failed to take a phase transition into account. So, what is another possibility? Another possibility is that some disorder is present in the solid even at t equal to 0 by third law I repeat the entropy of all perfectly crystalline substances is 0 at absolute 0, but that is only for perfectly crystalline substances. If the substance is not perfectly crystalline there may be still some disorder present in the solid at t equal to 0 that means then at absolute 0 temperature the entropy will be greater than 0 and that entropy is called residual entropy. The entropy which is still present in a system at absolute 0 is called the residual entropy. Hope this is clear what is the origin and magnitude of residual entropy. Let us consider a crystal which is made up of A B I will talk about A B solid and how this A B is formed it is formed from atoms A and B. In a crystal lattice how these A and B as we are arranged let us take one very simple arrangement of A and B atoms in the crystal lattice. Let us say this one the compound A B is in the lattice arranged or in the in the crystal or in the solids arranged like A B A B A B etcetera it keeps on going in all the possible direction. Suppose A and B are very similar atoms and the overall electric dipole moment of A B is negligible I repeat consider A and B to be very similar atoms such that the dipole moment of A B is negligible that means in this solid form if the arrangement is like you know A B is negligible. A B and somehow it is reverse B A then A B then B A and let us repeat in all direction. Look at the switch over here it is perfectly crystalline substances this is perfectly ordered we are talking about scenario at 0 Kelvin perfectly ordered here the arrangement is not perfectly ordered there is a configurational disorder configurational disorder. Why this configurational disorder arises is that A B has negligible dipole moment that means whether in the crystal you have A B A B A B A B A B kind of arrangement or you have A B A B A B A B and one or two molecules or some several molecules adopt B A kind of orientation then the energy the minimum energy of the crystal remains same it does not change. The minimum energy in the minimum energy conformation you can either have A B kind of arrangement or you have B A kind of arrangement it does not make any difference and this leads to configurational disorder because at absolute 0 you can have A B A B A B A B kind of arrangement or you can have A B A B B A B A A B B A kind of arrangement also so that is a disorder and this disorder leads to residual entropy there is still some entropy due to this configurational disorder at absolute 0 that is called the residual entropy. So, I hope the origin and magnitude of residual entropy is clear to you we will take an example. Let us take the example of carbon monoxide in the solid form at absolute 0 carbon and oxygen atoms do not have a very large you know electro negativity difference in other words the dipole moment of carbon monoxide is negligible and in this if you look at the arrangements you have C O and O C C O C O C C O C it is not perfectly crystalline at absolute 0 because there is some configurational disorder, but this configurational disorder does not change the energy of the crystal that means at absolute 0 the minimum energy conformation can have either carbon monoxide arranged like C O or like O C it does not make any difference to possible orientations of carbon monoxide which gives rise to the same minimum energy in the crystal. Look at the comments now the molecular dipole moment of carbon monoxide is very small we just discussed either of two orientations possible with virtually the same energy at absolute 0. So therefore, W is equal to 2 for one molecule of carbon monoxide and if there are n molecules then W is equal to 2 raise to the power n once you substitute W is equal to 2 raise to the power n which is substituted here it becomes n k log 2 and n k log 2 is n r log 2 because capital N is equal to small n number of moles into Avogadro constant and then n times n a times k is equal to k times n a is equal to r n r. So that is why you have n r log 2 over here now you substitute the numbers n is equal to 1 r is equal to 8.3145 joules per kelvin per mole and log 2 whatever the value is when you solve it comes out to 5.76 joules per kelvin per mole that is even when the temperature is absolute 0. This is the residual entropy this is an example of residual entropy where even at absolute 0 there is still some disorder. The experimental value is 5 joules per kelvin per mole. So there is a good agreement between the experimental value and the calculated value. I hope with this example the meaning of residual entropy is relatively clearer. Now let us take an example of another molecule HCl hydrogen and chlorine the electronegativity difference between hydrogen and chlorine is used highly polar molecule dipole moment is high that means in the crystal if I have this HCl HCl HCl and then I have HCl ClH HCl this arrangement will have different energy than this arrangement because of a bigger electronegativity difference between the constituent atoms. Therefore there is only one minimum energy confirmation possible only one I repeatedly say only one minimum energy confirmation possible and that is shown over here. If at all HCl transforms or rotates in ClH then that arrangement will give different minimum energy different energy because of interaction energy will be different between within the adjacent molecules. So this is perfectly ordered systems what does that mean when I say perfectly ordered system that means the weight of a configuration is equal to 1. There is only one way you can arrange HCl to get minimum energy confirmation then our usual S is equal to k log w if w is equal to 1 then S is equal to k log 1 this is equal to 0. What we have is for the perfectly ordered systems there is no residual entropy this is the value at absolute 0. Now we come back to our original definition of third law of thermodynamics it said that entropy of each substance is positive which may become 0 at absolute 0 and it does become 0 at absolute 0 for perfectly crystallized. Substances and here is the example of perfectly crystalline substances this is the example of perfectly crystalline substances. So the comments which are there in the background here is the same that I discussed that it is the large electronegativity difference between H and Cl which accounts for this observation for which accounts for this fact. So at t equal to 0 Kelvin then w is equal to 1. So S is equal to k log 1 which is equal to 0 that means the residual entropy of perfectly crystalline substances is equal to 0. Now if we want to use this formula S is equal to S is equal to k log w if I want to use this formula then I need to know the weight of a configuration most probable configuration. And when we set temperature equal to 0 when we move towards 0 that means we are moving towards the alignment of the atoms of the molecule or alignment of the molecules in a sort of ordered manner. This order may or may not be perfect because as we just discussed the case of carbon monoxide and HCl in some cases different orientation may not change the energy. And in some cases different orientation changes the energy and therefore there can be more than one minimum energy conformation or only one minimum energy conformation. If there is only one minimum energy conformation at absolute 0 then w is equal to 1 then S 0 is equal to 0. To theoretically calculate the value of residual entropy therefore you need to know how many minimum energy conformation are possible at absolute 0. We will take some examples and discuss ahead that how this entropy residual entropy depends upon the minimum energy conformation and how to also calculate the weight of the configuration which is in the most probable form. But the take home lesson from this lecture is the residual entropy is the entropy of a system which is still present at absolute 0 and that entropy can come by virtue of configurational disorder. We will move ahead by taking more examples but that will be done in the next lecture. Thank you very much.