 Now look at this for a rudimentary system only one property is needed and the question is which would this property be and for different types of rudimentary system will there be a common property. This question and the earlier question is that you know the thermal energy when people started working with it they got a vague idea of a property which significantly defined the thermal energy but that whole thing could be formalized only when the 0th law of thermodynamics was formalized and that brings us to the proper topic of this lecture and that is the 0th law of thermodynamics. Now again a bit of history the nomenclature 0th law first law second law is rather arbitrary yesterday we saw that historically the second law was thought about first although formalized much later. The second thought process began about the first law but since it got formalized before the second law was formalized it was given the name first law and our counting begins with 1. So, law 1 law 2 much later it was realized that to put temperature on a proper footing you need a law so let us call that the temperature law but then the earlier thought was that temperature must be defined before you define energy or entropy or before you define the first law and the second law and hence they said well you must have a nomenclature known as the 0th law of thermodynamics that is unfortunate because when we started appreciating these two laws particularly when it came to the formulation of karetheodori it turned out that the proper logical order would be first law followed by 0th law followed by second law that is rather unfortunate but that is a matter with numbers and it happened because unlike say Newton's law of motion which came from Newton essentially his own thoughts as published in his principia. So, he put them in a proper order it was a single big body of work that is not true of the laws of thermodynamics laws of thermodynamics have been developed over almost a century or more maybe a map personal feeling is there is no reason why these laws should not be called say the second law may be called the Carnot law the first law may be called the Joule law the 0th law may be called may be Fahrenheit law or something like that in which case the order of the law will be immaterial but anyway let us come to the 0th law of thermodynamics and just the way we said that the first law is our understanding of the way adiabatic systems or adiabatic boundaries or adiabatic partitions behave. The 0th law is our understanding of the characteristic of non adiabatic partitions remember that in thermodynamics we have already used this a and non a work and non work and rather than say non work non work every time we said let us call it heat because that is what historically it is consistent with. So, now let us go to 0th law and let us define we know that a boundary between systems can be adiabatic in which case you will say the interactions of work type only or non adiabatic which means work and or heat that is q. Now again since we are going to discuss the characteristics of non adiabatic boundaries the word non adiabatic non adiabatic will be used again and again and then let us use another word which does not have a negative connotation and which is historically used for these type of boundaries and hence wherever the word non adiabatic comes we will use the equivalent word diathermic. So, a diathermic just like adiabatic is an adjective to be applied to boundaries or partitions or walls or interfaces whatever they are all equivalent a diathermic boundary is a non adiabatic boundary and a diathermic boundary is a boundary which allows the heat type of interaction across it and adiabatic boundary does not allow. So, that means for us now we have classified boundaries as either adiabatic or diathermic, diathermic means non adiabatic and 0th law studies the behavior of systems separated from each other by a diathermic wall. And now let us come to our understanding of 0th law the for the first law we did we thought about an experiment generalization of Joules experiment in which we said that a system was taken from a fixed initial state 1 to a final initial state 2 through any adiabatic process just one system was involved and the process was involved that was adiabatic. Here for the exploration of 0th law we do the following experiment we will only think about this experiment, but you will realize that this is a real life experiment because we have a feel this is the way nature behaves what we do is we take a diathermic boundary or we cannot create a boundary without a system. So, let us say that we to take two systems 1a and 1b may be two cylinder piston arrangements two gases or whatever you have or one copper block and one gas in a cylinder piston arrangement and allow them to interact with each other across a diathermic wall and that means we allow a q type of interaction between them and then we say that so as to restrict ourselves to the proper study of the heat interaction we say we restrict the systems from any W interaction with each other and remember this can be done because one thing which is important is work and heat are interactions those are energies in transit two systems must be involved unless two systems are involved you cannot talk of work you cannot talk of heat when only one system is involved you can only talk of its change of state may be change in pressure change in temperature change in any property change in its energy. But you cannot talk of an interaction unless a second system is involved and work interaction also means that system a when it does work on b system b accepts that much of work from a it is like a double entry system when a gives up something b must accept something if there is no b to accept a work interaction from a a cannot have a work interaction it is that simple as that. So remember work is a generalization of force into displacement if we prevent any displacement from occurring there will be no work interaction for example if you have a cylinder piston arrangement and if you lock the piston there will be no expansion or compression type of work interaction if you have a stirrer and if you lock the rod of the stirrer there will be no stirrer interaction and if you have an electrical cell with two leads if you make an open circuit there will be no electrical interaction no charging discharging. So it is rather easy for us to prevent work interactions and we will prevent systems a and b by appropriate restrictions from undergoing any work interaction. Now what we do is we do an experiment we do a series of experiment we keep the system a in a state a 1 and then what we do is we try out various states of b and for every state we check any q interaction what we will find and I will be sketching this again and again let us say this is the state space of a and this is the state space of b this is x a y a this is x b y b if you are not comfortable with x and y you can say p a v a p b v b whatever we have a fixed state of a say a 1 and we try different states of b bring it in contact with a across a diathermic partition and check whether there is a heat interaction between them or not that can very easily be checked if there is a hint interaction the state of a will tend to change state of b will also tend to change. So we can detect that there is a heat interaction work interaction has anyway been prevented you will find and this is the first part of 0th law 0th law states that there will be at least one state in the state space of b let me say b 1 which will say that look you are allowing me to have a heat interaction with my neighboring system a but I refuse to have any heat interaction I do not want to have any interaction this is the first part of 0th law not 0th law states there will be at least one. So if you explore enough you may find b 1 b 2 b 3 b 4 b 5 in a large number of cases particularly in our continuum domain you will find a locus of states all of which say that we will not have any interaction of the heat kind with this fixed state of a namely a 1. So let me write the part 1 of 0th law existence in state space of b of states say b 1 b 2 which do not any q interaction with a fixed state of a say a 1 across a diathermic wall remember this is important because a diathermic wall will allow heat interaction but there are some states of b which will say that look we do not want to have a heat interaction in spite of being allowed to have one that means the gate is open but I do not want to go through the gate it is like two tanks connected to each other through a tube you say the fluid may flow but the fluid does not want to flow what do we say in fluid mechanics in fluid mechanics we will say the two tanks will have the same pressure we are going to do something like this here but we will not say what that idea is. Now definition when this happens we say states a 1 and b 1 are isothermal states this is definition of the word isothermal and similarly because we have said not only a 1 and b 1 but a 1 and b 2 also do not have any q interaction so this state a 1 will be isothermal not only with b 1 but also with b 2 also with b 3 b 4 and b 5 so I will say that not only a 1 and b 1 are isothermal even a 1 and b 2 are isothermal and so are a 1 and b 3 so that means the existence part of the zeroth law says that if you have state systems a and b interacting across a diathermic wall then for a fixed state of a say a 1 there are a few states of b say b 1 b 2 b 3 b 4 which are isothermal with a 1 that means even if allowed they will not have any heat interaction with this fixed state of a 1. Now that was the first part the second part and equally important part is what traditionally textbooks talk about as the first law of thermodynamics or the zeroth law of thermodynamics. The second part of zeroth law comes up like this this is the existence part the second part may be called the transitive part what this part says that if you have a diathermic wall and let us say you have a system a with state a 1 and system b with state b 1 and say they are isothermal so q is 0 so that means a 1 and b 1 are isothermal and then you can bring a 1 and try out some other system c you will find at least one system one state in c 1 which will be isothermal with a 1 this is existence this is existence then the second part says that if you bring system a in state a 1 with system c with state c 1 across a diathermic wall you will find that sorry b in its b 1 and c in its c 1 b 1 was isothermal with a 1 c 1 was isothermal with a 1 then the second part of zeroth law says that b 1 and c 1 also b isothermal this is what I call the transitive part so the first part of zeroth law is the existence part the second part of zeroth law is the transitive part and this is the transitive part which has generally been proposed and written in almost all textbooks of thermodynamics as the zeroth law most of the textbooks will say and many of the good textbooks which I referred to are also not exception to that they will say that zeroth law says that if systems a and b are in thermal equilibrium with each other and systems b and c are also in thermal equilibrium with each other then systems a and c will also be in thermal equilibrium with each other the problem here is they talk of systems not states of the system the more important problem is they do not define what is thermal equilibrium is and here we have you know expounded zeroth law without really talking about thermal equilibrium and now we define our thermal equilibrium like this we have said that the existence part of the first law that between two systems any two systems a and b for a fixed state of the system which is a1 system a at a1 there will be some states of b1 which refuse to have any heat interaction and we call these states isothermal with this state of a1 another way of saying that a1 and b1 are isothermal states is that the same thing is saying that a 1 and b 1 are in thermal equilibrium with each other it is the same meaning of being isothermal states. So, the words isothermal the words being in thermal equilibrium with each other mean the same thing. And both mean the same thing that if two states which are isothermal or two states which are in thermal equilibrium with each other are allowed to interact with each other across a diatomic partition they will refuse to have any heat interaction that is all that means. Now, let us proceed where does this lead us to? This leads us to some interesting stuff and what is the use of this? The use of this is to determine whether given two systems in two states system A in state A 1 and system B in some state B j when they are allowed to interact with each other across a diatomic partition whether there will be a heat exchange or not. So, for that we first have to generalize this. Let us say that we now restrict ourselves to two systems system A and system B and I am showing state space of A and state space of B. Let us say that we have a we have already seen that we have a fixed state of A 1 and in the state space of B we have a number of states say B 1, B 2, B 4, B 6 which are isothermal with A 1. We can do the other way we can fix a state here of B 1 of B let us say B 3 it is isothermal definitely with A 1, I may explore and find out that there are number of states here A 2, A 3, A 4, A 5 which are isothermal with it. And then by our transition part of the zeroth law if A 1 and B 3 are isothermal B 3 and A 4 are isothermal then A 1 and A 4 are also isothermal that means if I make a copy of the system A 1 in state A 1 and another in state A 4 and allow them to interact with each other across a diathermic wall there will be no heat interaction. And by extending this you will now say that not only is this state isothermal with a given state here. And not only are these states isothermal with any given state here this set of states of A and this set of states of B you take any pair 1 of A 1 of B or 2 of B or 2 of A they will all be isothermal states. So, this set of states I will call this as states of corresponding isotherms. So, we come to the following definition a set of isothermal states isothermal with respect to one state of some other system is known as an isotherm in state space. And an isotherm in state space of A will have a corresponding isotherm in the state space of B. And if you take a third system a corresponding isotherm in the state space of C. Now, let us do a different experiment let us say that we will take a state of A not on this isotherm some other isotherm. Let me take now a red colored pen and let us say that out here I select another state of A let me call that A 11. And I do an experiment with the state space of B and I may find that there is a B 11 here and a B 12 here and a B 13 here and so on. May be I will be able to hunt out an isotherm here which is corresponding to this state of A. And if I fix an iso a point here in the state space of B and work with various states of A I may find that there are a number of states here which are corresponding to which are isothermal with this. So, now here I have another pair corresponding isotherms. Now, I have a black pair of corresponding isotherms and I have a red pair of corresponding isotherms. Now, that means I can extend this and then I can have a green pair of corresponding isotherms and a blue pair of corresponding isotherms and a yellow pair of corresponding isotherms. What is the use of this? The use of this is to be able to answer after mapping these isotherms the following question. Let me go back to my given a state of A say A x and let us say given a state of B say B x. We are asked the question if I allow them to interact with each other across a diatomic wall non-adiabatic wall will there be a heat interaction or not. Then what do I do? I look at the state space of A find out to which isotherm A x belongs may be A x belongs to the green isotherm. Then I find out from the state space of B to which isotherm does the B x belong. It is possible that B x belongs also to the green isotherm. If it belongs to the green isotherm we will say that there will be no heat interaction between state A x of A and B x of B if allowed to interact across a diatomic wall. But if A x belongs to the green isotherm and B x belongs to say the other isotherm say red isotherm then definitely there will be a heat interaction between the two. So, this means that after mapping corresponding isotherms in the state space of two systems we gain the ability to decide whether state A in a given state and system A in a given state and system B in its given state will interact across a diatomic partition by the process of heat interaction or not. If they belong to the same pair of corresponding isotherms no interaction will take place because they are isothermal states. In fact we call them isothermal states because no interaction will take place. If they belong to two different isotherms then naturally a heat interaction will take place. So, at this stage I hope you have understood the meaning of number one diatomic partition. Second first part of zeroth law that is existence of isothermal states the definition of isothermal states then the definition of isotherms the transitive part of the zeroth law the pairs of corresponding isotherms different pairs of corresponding isotherms and hence our ability to decide whether a heat interaction will take place or not across a diatomic partition. Now remember we have here as an illustration a black isotherm here and a corresponding black isotherm here a green isotherm here and a red isotherm here and a corresponding red isotherm here. Now I think we do not have enough patience to keep patience to keep on sketching the green isotherm the yellow isotherm the blue isotherm and the violet isotherm. And we can do that in as much detail as we want and we will come to a stage where we will be able to ask and answer this question whether a heat interaction will take place or not. So, that means the labels on isotherms are very important the nomenclature of isotherms is important. So, important that we give a special name to labels on corresponding isotherms what are these labels called the labels go by the name temperature what I said earlier that in our thermodynamic language this black isotherm and this black isotherm which are corresponding isotherms and are labelled as black isotherms have the same temperature this red isotherm and this corresponding red isotherms also have the same temperature that is the basic meaning of the thermodynamic term temperature. Temperature is nothing but labels on corresponding isotherms. So, when we assign a state temperature that means we find out to which isotherm it belongs to and provide it a corresponding label. So, we can say here that any state on this black set has a temperature which I may call black any state here or any state here has a temperature which is red. If two states have the same temperature no heat interaction if two states have different temperature a heat interaction is will take place across a diothermic partition. Now the basic idea of zeroth law ends here the basic idea of zeroth law ends here it ends up with the idea of temperature as labels on isotherms the requirement that corresponding isotherms must have the same label. So, must have the same temperature. So, after all this and all understanding all that we say that two systems a and b in states a 1 and b 1 will have across a diothermic partition heat interaction if their temperatures are different if their temperatures are same they will not have any heat interaction. This is the basic understanding of temperature, but now here we notice and we will see we appreciate the strength of zeroth law it has given us a proper foundation to understand temperature that is the strength. The weakness is the following it does not tell us which is the so called higher temperature which is the so called lower temperature. It just tells us whether two temperatures are sort of equal or not if they are unequal zeroth law does not say this is higher and this is lower zeroth law says that is not my job. It does not even say anything about the direction of heat transfer that is still a question mark. If I have two systems again a and b states a 1 and b 1 say a 1 has a temperature black, a 2 has a temperature green. I continue without using numbers because numbers automatically give us an order which is sort of arbitrary zeroth law says that there will be heat transfer it does not say whether it will be from a 1 to b 1 or a 2 b or b 2 a that it does not say and zeroth law does not say whether black is a higher temperature than red or red is a higher temperature than black. Now, that brings us to a problem I can explore state space of various systems and label the corresponding isotherms I like colors. So, I will label them as black, blue, green, red, violet I can have light gray, dark gray whatever I can do. Then my friend again explores those systems, but he is an animal lover he may call corresponding pair of isotherms the dog isotherm some other corresponding pairs as the cat isotherm leopard isotherm and peacock isotherms and so on. One of you may be very fonds of flowers and you may have a rose isotherm and a chamele isotherm and things like that. So, there will be absolute arbitrariness and total chaos and tomorrow somebody will say that look I have a system which is on the dog isotherm will it have a heat transfer with your system and then I say I do not understand what diga isotherm means I understand what is a chamele isotherm means. So, I cannot give you an answer. So, this arbitrariness is of no use.