 Now the zeroth law for some reason it was called zeroth because first and second were already defined people did not want to rename them and it was thought that zeroth law is a more basic law compared to first and second. Finally it turns out that that is not true all three laws have to be and the same have to have the same status. Now the what is zeroth law? The zeroth law is a law which considers the behavior of two systems separated by a non adiabatic partition or wall or boundary whatever you want to say. Then just like first law where Karatheotory said that all adiabatic systems seem to behave in a similar way, generalize that as his statement of the first law. Here also we are going to look at something and generalize it. What we do here is the following. Let us have a system A and a system B and let us say that the wall in between is not adiabatic. First definition a wall which is not adiabatic or a non adiabatic wall we define it as a diathermic wall. So a wall which is not adiabatic and that means a wall or a partition or a boundary or an interface which will allow interactions other than work. That means interactions of the heat kind is called a diathermic wall. So just the way process was short form for a change of state diathermic is a short form for non adiabatic that is it. So diathermic wall allows Q interaction also. Now we simplify the thing by saying and I will simplify the diagram. For the sake of argument let us say that A and B are constrained such that between them only heat interaction takes place. This is for simplicity. This does not distract us from or make us deviate from generality because you know remember that work requires some sort of a displacement. Either a mechanical displacement like movement of a wall or movement of a stirrer or an electrical displacement like movement of electrons and this can always be suppressed by fixing the wall at a place or freezing the stirrer or putting a break on the stirrer or you know opening the electric circuit and things like that. Now what we do is we the observation is this. Remember first law the experiment which is claimed to demonstrate the history of first law is Jules experiment in which he studied the behavior of a system containing some amount of water from a given initial state to a final state and he executed different parts and he found that the energy transfer as dictated by the rising or the lowering of a weight was the same whichever part it took that was the driving force for the first law. So here what we do is we do the following experiment. I am drawing here the state space of A that you can write PAVA if you feel like and this is the state space of B and what we do is we bring state A to some state A1 system A and then we do the following experiment. We have our system B initially at different states B0, B1, B2, B3, B4 whatever we actually said allow it to interact with this system A across a diathermic wall and see whether any heat interaction takes place or does not take place. Of course we can also observe in which direction it takes place but our main thing is whether heat interaction takes place or not. What we find is the following we find that there are some states in the state space of system B. Let me call them B1, B2, B3, B4 and so on. These are states such that if we allow them to interact with system A when it is in state A1 refuse to have any heat interaction even when allowed to have. This is the first part of 0th law. Yes 0th law, one part. There are states B1, B2, etc. Do not queue interaction with system A in state A1. So coming back to the earlier figure, maybe I should put it here. Yes good. Of course that upper part of the figure is not senior but I think you have it this one. So what the experiment we have done is we have system A in a given state A1 and we have experimented with various states of B. 0th law states that first part of 0th law says that you will discover a number of states in the state space of system B, at least one if not more which if you allow them to interact with system A when it is state A1 will refuse to have any heat interaction in spite of being separated by a non-adiabatic boundary. And you can demonstrate it is non-adiabatic because there are other states which will merrily have a heat interaction but there will be a set of states at least one which will refuse to have any heat interaction. This is not a derivation this is a statement say this is of a law such states exist. So this is what I call the existence part. There are states or in mathematical there exist states just the way some theorem says between there exists a value x0 such that f of x0 is 0. So this is the existence part. Now comes the second part which is generally stated as the second sorry 0th law in mini textbook. Hardly any textbook mentions the first part of 0th law but that is a necessary unless you talk of the first law you cannot talk of the first part you cannot talk of the second part. The second part says that this property or actually what I should do is I should let us have some definitions before I come to the second part so that the second part can be shorter definition. Such pairs of states are called isothermal states that means when I say that a1 will refuse to have any interaction with b1 we will say b1 and a1 or a1 and b1 are isothermal. Similarly a1 and b2 in which order you write is immaterial because it is a pair or isothermal pair of states and so on. And another way this is definition a definition b is again just to understand our earlier terminology and put it on a common footing so that we can keep on using the earlier terminology. States a1 and b1 are in thermal equilibrium each other this is nothing but another way of saying a1 b1 are isothermal states and when are the isothermal states when we allow them to interact with each other across a diatomic partition they refuse to have any thermal interaction or heat interaction. Why should they refuse to have a heat interaction because the first part of zeroth law says that such states pairs of states will exist. So with this what we have done is we have looked at the situation which leads us to the idea of zeroth law of thermodynamics no which one. I find certain circular over here how shall I understand that there is heat interaction or there is no heat interaction. If there is a heat interaction there is some interaction some interaction will lead to a change of state which can be observed. What is other than temperature now? We have not yet defined temperature but we have that first state postulate that the state of a system can always be defined using primitive variables. So all that we have to do is keep on observing the primitive variables. If there is a change in any one of them there is a change of state and if there is a change of state there is an interaction and then we use the work interaction definition to determine whether it is work or heat. So all these tools we have developed so far. So we are not that we have constrained for simplicity otherwise the argument will become longer but the same argument continues. This is only for simplicity that we say that we have constrained the wall is not moving the wall is not electrically conducting. We are saying that it is a isothermal condition. So we can say that some amount of heat is transferring from A to B and the same amount is transferring from A to A we cannot say. No we are saying that we allow A1 and B1 to interact with each other during the interaction for simplicity we have prevented any work interaction taking place. So if there is a change in state of the systems then any interaction which takes place is heat interaction. We have said that since no change in state is taking place there will be no interaction taking place of the heat kind because work has already been done. But the net heat interaction will be 0. We are not we are looking at the gross level so you can always say that look you are saying 0 heat transfer but there may be Q12 equal to minus Q21 with a negative sign. So be it net interaction is 0, net change of state is 0 that is what we are interested. So here isothermal does not mean the constant temperature. We have not defined what is temperature. No we have not defined but isothermal leads to what does the term mean isothermal states here. See we have said here that is what I was looking at. We have what have we done is we have looked at the situation which leads to zeroth law. Then we said the first statement of zeroth law is the discovery that there exist the states B1, B2 which do not have any heat interaction with state A. This is the existence factor. Then the definition is if A1 and B1 refuse to have any heat interaction across a diothermic partition we then define them to be isothermal pair of states. And then just because we have been always talking of thermal equilibrium we say if A1 and B1 or say A1 and B2 any pair of states is an isothermal pair of states. Another way of saying they are isothermal pair of states is to say that they are in thermal equilibrium. So here is a definition of isothermal states that means they must have 0 heat interaction across a diothermic wall. And here is a definition of two states being in thermal equilibrium with each other that means they are isothermal states that means they should be no heat interaction with each other that is all. Remember that we have not yet defined even the concept of temperature at least in this set of lectures. Now we come to the second part, 0th law if the first half was existence the second half is the transitivity property. What is a transitivity property? Transitivity property is something which is something which at the same level for example in mathematics equality is considered a transitive property because if A equals B, B equals C then A equals C. So we can write A equals B equals C in any order that is known as a transitive property. There are properties which are not necessarily transitivity property for example A is the square root of B, B is the square root of C does not mean A is the square root of C. There are properties which are non-transitive, there are properties which are transitivity. So let us see the transitivity property. It says this simply says and this is the statement all of us know as the 0th law of thermodynamics. If A1, B1 are isothermal states and if say B1 and you take some other system C, C1 are isothermal states then this implies that A1, C1 are also isothermal. This is the 0th law all of us have studied in textbooks without really without even talking of the existence but this is the formulation of Landberg and others that this is the way 0th law is properly developed. Now where does temperature come in? Temperature comes in as follows. Now again let us go back to our two systems A and B. This is the state space of A, this is the state space of B and if you want xA, xB, yA, yB. What we have done is we have fixed a state A1 of system A and we have discovered a number of states for simplicity. Let me say now thermodynamics does not say so but let us assume that this is the states of B which are in thermal equilibrium with A1 or which can be in thermal equilibrium with A1. So any one of the states here if you want you can call this B1, B2, B3. These are some properties. If you want you are more comfortable write them P, sorry this is xA, sorry and yeah I think I think this should be xA, yA and this should be xB, yB. I think now it is okay. So this A1 is isothermal with either B1 or B2 or B3. Now what I do is I repeat the experiment by fixing the state of B at say B2 and hunt out in the state space of A whether there are other states and you will find that there are other states in the state space of A which are isothermal with this given state B2 of this. First we did experiment with A fixed at A1. Now we do experiment with B fixed at B2 and maybe you will find a state A2, A3, A4. You can always say that why should there be a continuous locus. There is no reason but we generally find that we have considered the state to be continuous. So in our non quantized thing these happen to be continuous locus. Now the transitivity thing means that A1 and B2 were isothermal. We have discovered that B2 and A4 are also isothermal. So what is the consequence of this? If I were to take two copies of A, one at A1 and another at A4 they would also be isothermal. Extending this we will say that any state of A on this will be isothermal with any state of B on this. Extending it further we would say that take any system which is either A on any one of these states or B on any one of these states. Take some other system which is either A of any one of these states or B of any one of these states. Put them across a diothermic partition. We will discover that they are isothermal. Consequence of 0. These states this set of states of in the state space A is known as an isotherm in the state space of A. So this is an isotherm corresponding to say B2 or B1 and this also is an isotherm. Not just any isotherm these are known because they belong to the same bin. Take any one either from A or B and take any one either from A or B you will find them isotherm. These are known as corresponding isotherms. Now before we go to temperature let us go the absolute basic way. What we are going to do now is let us take a state which is not on these corresponding isotherms in the state space of say A. Repeat the experiment maybe we will find an isotherm here and coming back here we will find an isotherm here. So we had a black pair of corresponding isotherms maybe we had a red pair of corresponding isotherms. Then maybe we do take another thing here and maybe we will find another set here blue here we will have blue here. We can map the state space of A and B and mark out these corresponding isotherms. Remember what is the advantage of this if when we have this map we can do the following that will be possible only after we do the second law. I want to impress on you that at this stage there is no way we can order these isotherms. But when you do the experiment naturally it will follow that way. That is after second law. Zero-claw does not say so. Zero-claw does not say so. It does not say so but your experiment should always fall in the same line. It may be from zero-claw it is okay. It may be from second law it is inconsistent. I have done it deliberately for you to ask that way. But what I am saying is your experiment cannot give you different order of lines. Based on what thermodynamics? Based on which thermodynamics? Your experiment itself cannot give you different directions. No but suppose I were to see I have blue, black, green and red here. If I were to put blue, black, green, red you would have asked me why are you putting them in the same order. I have no answer to that just now. Your experiment gave it. I mean your experiment cannot give you this order as you have drawn out. At this stage I do not know. I am just saying that. You need not know the second law but your experiment will not give you lines like this. But why are you sure that your experiment does not give you so? Zero-claw does not say so. I know. I do not know now. But I know still this will not be so. Yes but we will come to that when we come to second law. That is all I am telling you. Agreed. At this stage I am doing this deliberately for people to ask a question so that I can answer that at this stage we cannot do anything more than this. I cannot assert that they have to be in a particular order. You need not assert but I am saying naturally it has to happen because. I will demonstrate that it has to happen and I will demonstrate that I was wrong here only after discussing second law. That is fine. Because the same thing has happened in one year I have put them in order and some students say sir why should they be in order? There is no answer. And I put them not in order and some students say why are they not in order? There is no answer. At this stage there is no answer. The problem is when we learn from the basics we should not go for a priori assumption. Yes and I am not making a priori assumption. That is making a priori assumption. Without a priori assumption you should start. I will come to that when we come to second law. In 0th law I cannot say. I will only say that a distinct isotherm here is a distinct isotherm there. The only thing we can argue is if an isotherm is corresponding isotherms are black and if there is a distinct green isotherm here which does not interact with this black isotherm that means it is distinct then on the other set also it has to be distinct. And if a black and green isotherm they intersect at some point then we have made some mistake the same color should be there for both. Only one you do. So see a distinct isotherm means if they interact they have a heat interaction and if they say that more interaction more exploration makes the two isotherm meet then there is something wrong in some experiment which we have done. Now what is the consequence of this? Consequence of this is the following. Suppose we are given a state system A and system B system A in some state say A0 system B in some state say B0 and we are asked to predict whether there will be an interaction between them if when they are allowed to interact across a diatomic partition heat interaction between them. All that you do is find out whether where A0 lies which is the isotherm on which it lies. Find out for B0 which is the isotherm on which it lies. If you find that they lie on the corresponding isotherms we can say that there will be no heat interaction. If they do not lie on the corresponding isotherms for example if A0 is here and B0 is here we will say there will be no heat interaction because both belong to the black isotherm. But if A0 lies on red isotherm and B0 lies on the green isotherm we will say yes if there is a diatomic partition there will be some heat interaction. At this stage we can only say there will be some heat interaction we will not be able to say in which direction it takes place. Now you will say that look it is possible that my system is here state is here then we will say that look I have not done experiments in detail maybe I should do some more experiment and find out this isotherm do some more experiments and find out the corresponding isotherm here and see whether the given state lies on that. In principle you can do more detailed experiments and map this out in detail. Now the question arises is how do we label this isotherm. Here I have given them in colors but there is over the sensitivity of the eye is not very good what I call green you may call something different you know there are various level of color sensitivities with all of us. I can use lines of different thickness different dot lengths but it becomes confusing and hence we give these things some labels we have started giving them labels and these labels are what we call temperature. So the next definition is temperature is nothing but label on corresponding isotherm this is the basic definition of temperature. So here we say that look my system A and my system B in the specified states will not interact thermally with each other across a diothermic partition because they belong to the same set of corresponding isotherms namely black. We will say now in short that look the temperature of system A is black temperature of system B is also black since they are at the same temperature their isothermal states belong to the same corresponding isotherms and hence no heat interact long winded way of thing but this is now I hope we understand what is meant by temperature and heat interaction. But this labels could be colors labels could be thickness of lines design of lines dotted lines and they are not useful if you want to go details we need labels why not numbers and that brings us to what is known as the experimental science of thermometry and the idea of what is a thermometer. Thermometry is the process by which isotherms in the state space of any given system can be labeled usually quantitative the advantage of quantitative labeling is that we can have precision hopefully to the extent possible. But there is also a trap here in an assumption here that when we put numbers the numbers are already in order so by default we are putting temperatures also in order we are doing it by thermometer not by any principle of thermodynamics I think t time is t ready so we will break.