 Welcome back. We will now move towards the second part of zeroth law. Now, although in the first part we say at least one state exists between two systems such that the pair would be isothermal. In general, we find that for many systems, let us say this is system A and this is system B. This is the state space of A. Let us represent it by simply xA, yA and that of B by xB, yB. And we have seen that the first part says that for a fixed state of A1, there is at least one state in the state space of B, say B1, which is isothermal integrated. Now, we find that in general for a large number of systems, if we keep our hunt on, we will find that for the same fixed state A1 of A, there are a number of states in the state space of B, say B1, B2, B3, B4, B5, which are isothermal with this fixed state A1 of system A. And with our assumptions that properties change continuously, usually these states will form a continuous locus in the state space of B. And we would say that any state on this locus will be isothermal with state A1 of system A. Let us now do the experiment in the other way. Let us say that instead of fixing state A1 of system A, let us fixed state of B to B1. Now, B1 and A1 are isothermal. Let us hunt whether there are other states in the state space of A1, which are isothermal with the state B1 of system B. And we will find that yes, perhaps there are more states A2, A3, A4, A5. And these states also will form usually a continuous locus and all these states are isothermal B1. Now, let us ask ourselves a question. Now, A1 and B1 are isothermal, A1 and B5 are isothermal, B1 and A3 are isothermal. But suppose I take A3 and B5, will they be isothermal? This brings us to the second part of the zeroth law of thermodynamics. The first part was about existence of isothermal states. Second part what we can say is the transitive property of isothermal states. What this means is if say state A1 of system A and state B1 of system B are isothermal and we have seen that state B1 and say state A3 are isothermal, then this implies that states A1 and A3 are also isothermal. This is the second part and which we call the transitive property. Let us see in some detail what this transitive property means. Now, we have seen that this is the state space of A, this is the state space of B. Here we have a set of isothermal states, A1 is one of them, A3 is one of them. All are isothermal with respect to state B1 of system B and state B1 of system B is a state of isothermal states in which there is a A B5 also. Now, the same experiment which we have done taking system A on one side and system B on one side. We can also repeat by taking system A on one side and a replica of system A that means an exact duplicate, an exact copy equivalent to its originally non-respect. We call it a replica and let us call it another copy of A. So, what we mean by that is the following that if system A in its state A1 and system B in its state B1 are isothermal, no heat transfer across a diatomic wall and system B in its state B1 and system A in its state A3 are also isothermal, then this implies that if I bring a system A in its state A1 and a replica of system A in its state A3 are also isothermal. So, this pair is also isothermal because A1 and B1 are isothermal, B1 and A3 are isothermal. So, A1 and A3 are isothermal. Now, this means that there is something common between states A1 state B1 and state A3. In the sense you take any two of these three they will be a pair of isothermal states. Now, we know that A1 and B5 are also a pair of isothermal states. So, that way not only A1 and B1, A1 and B5 but by the second part of the zeroth law of thermodynamics B1 and A3, A3 and B5 any combination is a pair of isothermal states. Extending this argument further we will say that you take say we had an A2 here, we had an A4 here, we had an A5 here and here we had B4 and here we had B2 and B3. It turns out that by extending this you take any state of A on this set, this isotherm and any state of B on this isotherm put all these states together and just pick up any two of them one of A, one of B or two of A using a replica of A or even two of B using a replica of B all these states in pairs will be isotherms. So, what happens is now we call this isotherm in the state space of A and this isotherm in the state space of B. We call these two isotherms corresponding isotherms. What do we mean by that? Let us have a set of isothermal states or sets on an isothermal set in the state space of A. Let us have a set of states of B on an isotherm put all those states together pick up any two they will be isothermal. Now, let us extend this experiment. Again let me draw a state space of A, let me call it YAXA and let us have a state space of YBXB and let us say that we have discovered a set of corresponding isotherms in the state space of A and B. So, these you see black lines are the two corresponding isotherms. Now, I may repeat the experiment by using another state of A say let me draw it by a red point here and let me call that A11 and let me hunt out by experiment an isothermal state here, let it be B11. Again repeating the experiment, we will find that in the state space of B there is a set of states each one of them isothermal with a given state A11 of A and again then fixing B11 of B hunting out in the state space of A, we may find an isotherm corresponding to B11. So, now we have one pair of corresponding isotherms the black one in A and the black one is B and the red one in A and the red one in B. We can repeat the experiment for example, we can take another state in A and find out an isothermal state in B and extend in the experiment we may find green isotherm in the state space of B and the corresponding green isotherm in the state space of A. Again we are exploring in detail we may get a pink isotherm here and a pink isotherm there and of course if you have the patience you could you know extrapolate this and do exhaustive experiments. Now what is the use of all this? The use of all this is to answer the following question the question is let us say given state say let me call this A star of A and B star of B. Then the question is are A star and B star isotherm and how do we answer the question if we have mapped the state space of A and the state space of B very nicely then all that we do is locate A star in the state space of A. It is possible that A star happens to lie on the black isotherm and it is possible that B star happens to lie not on the black isotherm but some other isotherm say red isotherm. If that is the case then we say those two are not isothermal states because they do not belong to corresponding isotherms. Whereas if we find that both A star and B star say belong to the same isotherm say red then we will say that they are isothermal states. Thank you.