 That brings me back to zeroth law. We are going to straighten out our understanding of zeroth law and we will define as we progress what is meant by thermal equilibrium, what is meant by an isothermal state and hence we will finally define what is meant by temperature. Now when you talk of, when you look at almost any text book and unfortunately although the Karatheodor is form of first law which I explained has become a part of good books in thermodynamics. Sears and Salinger and Moran and Shapiro are examples of this. Even Zimansky physics text book is an example of this. For some reason the zeroth law as we are going to formulate, formulation was I think originally proposed by Lansberg, the physicist adopted by many others has not found place in good text book of thermodynamics and everybody assumes that something called thermal equilibrium is known and then provide the statement that if system A happens to be in thermal equilibrium with system B, system B happens to be in thermal equilibrium in system C then A and C are in thermal equilibrium or will be in thermal equilibrium with each other. Weaknesses here apart from defining what thermal equilibrium is that you are talking of systems. Shouldn't we talk about state A can be in any state it feels like, B can be in any state it feels like, C can be in any state it feels like. So rather than systems we should be talking of systems in a given state and second thing is when have A and B brought together you say let they be in thermal equilibrium why should they be in thermal equilibrium. So the fact that a situation exists where they are in thermal equilibrium is not stated it is just assumed. So what we will do is we will take the formulation of zeroth law properly and we will say the zeroth law is our understanding of the behavior of systems by means of systems I mean plural two systems minimal separated by a non-adiabatic. So if first law was our understanding of the behavior of a system bounded by an adiabatic boundary so as it is restricted only to work interaction our zeroth law will tell us what happens when two systems are allowed to interact across a non-adiabatic boundary. So definition one since the word non-adiabatic is going to come quite often we will use a short form word it which is used regularly by all of you non-adiabatic is diathermic or diathermal just the way adiabatic as an adjective applies to mainly a boundary an adiabatic boundary all other definitions to system processes are derivatives from this non-adiabatic also applies to a boundary adiabatic means a boundary is adiabatic if it does not allow anything other than work interaction. Since now we understand what is meant by an heat interaction and adiabatic boundary prevents any type of heat interaction. A diathermic boundary on the other hand allows heat interaction it may allow work interaction also that is not an issue but a diathermic boundary allows heat interaction and a diathermic boundary for us it is simply a non-adiabatic boundary short form of definition. Now we look at this situation I will just show by a not so thick line a diathermic boundary or a diathermic wall or simply a diathermal and I will allow two system a system A and a system B to interact with each other across that diathermic boundary diathermic means heat conducting heat transfer allowing non-adiabatic the basic definition of diathermal means non-adiabatic remember that. Now remember a diathermic does not mean that work interaction is prevented but we also know that for work interaction to take place between A and B suppose they are fluid system the wall separating the two must be allowed to move the piston must be allowed to move if there is a stirrer that stirrer must be allowed to rotate if there is an electrical connection there must be a potential difference and current must be allowed to flow. So we have to take some conscious make some conscious effort to allow work interaction and because of that it is always possible to prevent a work interaction for example if A tries to expand into B by moving this wall I can fix the wall anchor it at a place preventive expansion type of work interaction if there is a stirrer connecting the two I can lock the stirrer up just the way we lock the brakes or the wheel of steering of a car preventing the tau d theta type of interaction if there is an electrical connection I will make it open circuit because unless the circuit is closed there will be no electric current whatever be the potential there will be no electric work interaction. So by this I will consider a restricted place a diathermic wall and we will say just for illustration that interactions between A and B Q only across this wall either this way or this way does not matter just for the sake of argument in simplicity we say that if there is work interaction which comes into the play we restrict it we will say let the piston be fixed. Now this is the basic situation which we are going to look at and it is a bit more complicated because for first law we just had one system it could do only work here now we have two systems there is a partition which allows it but in principle could allow work so we will remove the complication by noting that the work interactions can always be restricted to 0 and which we have done so that we can now study the heat interaction properly without getting disturbed by work interactions. Now we do one thing we I will just re sketch this system A system B diathermic wall and then what I am going to do is I am going to look at the state space of A say xA yA and state space of B say xB yB if you are comfortable write pressure volume pressure volume sigma A whatever and two dimensions I am showing for simplicity and clarity it is a complex system it could be more than two dimensions. What I am going to do is do this experiment by some interaction or by prior manipulation I am going to put A bring A to some state A1 let us say let the initial state of A B A1 I am going to experiment with various states of B and every time I am going to see as I bring it in contact with A which is at this fixed state A1 is there a heat interaction all work interactions have been prevented or restricted so if there is a heat interaction the state A1 will change because its energy is changing similarly if B is in some initial state B0 if there is an interaction even its state will be changing so it can be detected easily whether a heat interaction is taking place or not I am trying to hunt out situations where in spite of being allowed to have heat interaction across the diatomic wall heat interaction does not take place the first part of 0th law tells us that in the state space of this system B there exist states which will be such that when allowed to have across a diatomic wall heat interaction with a fixed state of A say A1 they say I do not want to transfer any heat from A or to A this is the existence part of 0th law which is not mentioned in very clearly mentioned in any of the books the first part of 0th law is existence so what I will do is I will erase all this and say that because of existence there will be some states let me call them B1, B2, B3, B4 a number of such states do unique state there could be a number of such states now thermodynamics does not say so but when we look at our systems which are non quantized continuous continuum type of systems we find that these states form a continuous subset that means in two dimensional space a curve in three dimensional space a surface and so on a restricted set of states in the state space of system B the characteristic is you have A at your fixed state of A1 and you have B in any one of these states B1, B2, B3, B4, B5 bring them together across a diatomic wall when you say diatomic wall you are allowing heat transfer but A1, A in A1 and B in B1 or any one of these says I do not want to have any heat transfer and this is allowed because 0th law says such states exist that is the first part of 0th law okay now a definition when you have two systems in their states say A1 and B1 which have this characteristic that when allowed to interact across a diatomic partition they say I do not want to interact heat that means this is A1 this is B1 etc even if allowed Q is 0 such states are known or are defined as isothermal states short form so all I say instead of long winded that state B1 it is such that when allowed to interact with system A with state A1 across a diatomic wall it refuses to have any heat interaction even when it is allowed all this is shortened by saying states A1 and B1 are isothermal states this is the definition of isothermal states. So two states of two systems A1 of A and B1 of B are isothermal means this bring them in contact with each other across a diatomic wall they will shake hands but say no heat transfer this is the definition of what we mean by isothermal states and the using this definition the first part of 0th law 0.1 is the existence of isothermal states so you can say after this definition in the state space of B there will be some states there will be a number of states which will be isothermal with a specified state of 1 now what we will do is just to clarify I will do it on the next page this is A so xA yA this is B so this is xB yB we have A1 and we have let me first draw this and let us say we have B1 B2 B3 now we did the experiment by fixing the state of A1 A2 A1 then we found this let us repeat the experiment the other way round let us say that we have fixed the state of B to B1 I keep that fixed and I hunt out in the state space of A I am sure I will find a number of states A1 will be one of them but there will be more than A1 so I will end up with a locus of state something like this may be A2 A3 A4 A5 so this is states isothermal with A1 these are states of A these are states of B isothermal B1 such states will also exist now I put the question A1 is isothermal with B1 also with B2 also with B3 B1 is isothermal with A3 A4 A5 A2 along with A1 will A4 and B2 be isothermal we say yes because you are familiar with the second law second part of the zeroth law that is what all books say we have not yet said but that is the second part of zeroth law the second part of zeroth law says that the property of two states being isothermal with each other it is a reciprocal relation A is isothermal A1 is isothermal with B1 B1 is also isothermal with B1 but the second part says that this property of isothermality of a pair of states is a transitive property which is what we know traditionally as the zeroth law of thermodynamics. So the second part 0.2 is the transitive property which says that if A1 and B1 are isothermal states or simply isothermal means what is isothermal that big explanation which we will not talk about anymore we know what isothermal means and if B1 and some C1 are isothermal then A1 and C1 will be isothermal this is like saying if A1 equals B1 and B1 equals C1 then A1 equals C1 the transitive property in mathematics this is the transitive property or equality in thermodynamics because of this I will sketch it here again I will show y A x A y B x B and we had we had a B1 here and we had an A1 here we had A2 A3 A4 and we had B2 B3 B4 B5 initially we started by saying A1 and B1 are isothermal A1 and anything on B of isothermal then we said B1 and anything here is isothermal then I ask the question is A3 and B2 isothermal and we said yes because of the transitive property it goes like this A1 and B1 are isothermal B1 and A3 are isothermal so A1 and A3 are isothermal that means if I have A on one side and a copy of A on the other side one side at state A1 another of the copy of A on the other side at state A2 they will also be isothermal and since A1 and B2 are isothermal B1 B1 and B2 are isothermal and hence any one of this and any one of this are isothermal so these two sets of states are isothermal with each other although they are in two different state spaces so these states together form a set of isothermal states because of this transitive property we now have a set of isothermal states not only in the state space of A but also in the state space of B and if we make a system C and play with it we will have a corresponding set there. This set of states in B you know which are isothermal with a given set of A is known as an isotherm this also is an isotherm but because they are in two different state spaces we say that these are corresponding isotherms so you take any state any pair of states from corresponding isotherms may be belonging to same state same system or different systems they will be isothermal with each other that means bring a diothermic wall allow them to interact they will say well we do not want to have any heat interaction. Now where do we bring in temperature for that we will extend this let us work with two systems A and B and let me say that we have already done this experiment and we have this set of black states here and black states here which are corresponding isotherms now let me repeat the experiment with say a different state of A it is possible that if that state is not on this isotherm I will get a different set of isothermal states in B and if I fix one state out there I will get a different set of isothermal states in A so the black isotherms are corresponding to each other the red isotherms is a different set of isotherms again corresponding with each other and let me again do repeat the experiment with say a green set of states it is possible that I will find a green pair of corresponding isotherms and if I start with a blue set I may find a blue set of isotherms now after this then what have we achieved let us say we have mapped these isotherms with different colors we will knowing that between A and B if you have a blue state here and a blue state there they are on corresponding isotherms same thing with black same thing with red same thing with green and you can have shades of gray pink yellow orange whatever you feel like fill up all these what have we achieved we have achieved the ability to answer the following question given an arbitrary set arbitrary state of system A and an arbitrary state of system B say A 50 and B 50 or A K and B J the question is if we allow them to interact across a diathermic wall will there be a heat interaction between them or not if we have to answer this question what do we do we go to the state space of A look at where A K exists suppose A K exists on this blue isotherm we go to the state space of B find out where B J exists if it happens to exist on the blue isotherm we will say no heat interaction will take place because they are isothermal states being on the same corresponding isotherm in this particular case blue isotherms but if it is not on blue we will say no one is blue one is orange so there is likely to be a heat interaction because you have a diathermic wall heat interaction will take place at what rate we do not know in thermodynamics but we will say 0th law says because they are not on corresponding isotherm 0th law does not dictate q equals 0 q can be nonzero how to know whether the heat interaction is there or not okay the moment there is a heat interaction first law comes into operation q equals delta e plus w we have say already prevented w so the moment there is a q there is a delta e the moment there is a delta e the state will change so that tells us whether there is a heat interaction or not if only internal energy change the internal energy we can change only by measuring the temperature so but without the knowledge of temperature I will take a concrete example because this is a typical mental state we have come into picture let us take two simple fluids okay one containing say oxygen one containing say nitrogen let us say so of this let us say this is state system one this is system two v1 and p1 are the two variables out of which v1 is fixed here v2 and p2 are the two variables out of which v2 is fixed now my question is if I allow this system to have because v1 is fixed I allow this my only variable is p1 I will say that look at this value of fixed v1 p1 is 2.4 bar and at this value of v1 I will put numbers here say 1.2 meter cube and here I will say 1.9 meter cube these are fixed values and if you want m1 kg could be say 0.9 kg m2 kg could be say 1.2 kg in fact m1 should be this m2 should be this okay and then I say p2 is 3.3 bar I bring them together if there is no interaction anyway volume cannot change energy will change and since only pressure can now vary pressure will vary and I will note that pressure is changing but if I find that this 0.9 kg of oxygen at 1.2 meter cube and 2.4 bar where 1.2 meter cube is fixed and 1.2 kg of nitrogen at a fixed volume of 1.9 meter cube and p2 of 3.3 bar if I allow them across a diatomic wall and I find that after 1 minute 10 minutes however long I wait no interaction so this remains 2.4 bar that remains 3.3 bar I will say no interaction because no state change state can change only with interaction I have prevented work being done so that is no interaction only interaction will be indicated by change in pressure which is not getting indicated so then I will conclude that these are isothermal states but if I find that soon after allowing that interaction may be the pressure of one is changing pressure of the other is also changing then I say ah something is happening and since I have prevented work from being exchanged some heat interaction must be taking place that is the way we hunt out isothermal states why solve it in case of here we have taken the example of oxygen is gases where the pressure and where may change if we take both to be like metals or blocks. So in that case the question is how to identify or if you are let us not look at system are you going to say that you are looking at a system which is a simple system rudimentary system or complex system if you are saying that the solid is a rudimentary system even then there will be rudimentary system means no two way mode of work okay. But whatever are the other single way mode of work I can prevent but there will be one state variable may be the length of the solid may be the twist in the solid that one variable of state will be there that will change if there is a heat interaction yeah however may however micro yes whatever be your system there has to be one variable and that is why I emphasize that remember it is important to know that even if there are no two way modes of work a thermodynamic system will require one property to determine its states. So you cannot say that I have a system which has no property by which I can determine its state zero property system does not exist the simplest system is one property system. So there will be some length some color some funny property which we do not know that will change properties are not changing no sir p1 and p2 that is when there is no change in the values we say that there is no interaction how do we define that it is an isothermal state isothermal means no our definition if there is no interaction that means they are isothermal states isothermal means we used to say that the temperature is remaining I have not yet used the word temperature no then for me isothermal means no interaction like this no heat interaction I do not want to use the word temperature till I define it so do not use the word temperature for me let us say this isothermal means this that across a diothermic wall no interaction no heat interaction work interactions have been prevented by holding it or preventing them isothermal states means only that we have not yet defined temperature we will do it but as far as we consider the graphs no sir if we say isobar it is about a curve having constant pressure line isotherms it is a line having constant temperature that is your definition we will we will define temperature the other way we will soon do it sir yes sir what is the harm in studying this law directly from point number 2 means what is the significance of its first part first 0.1 you mean yeah because we you cannot talk of isothermal states unless they exist so the first part you may consider is a minor part but it is necessary because I say I started doing this experiment which was that experiment okay I said that look start with A1 and try to hunt out in the state space of B whether such states exist the first question is why should I hunt you are asking me to hunt for a cat in this room first demonstrate that there is a cat in this room so that is why that first part is necessary okay you know your father tells you as a kid they go to the car car ke front seat pe aach ka akbar rakhae leke aau he goes there find there is no akbar in the car anywhere and he comes like sir father are you sure there was an akbar there so we were first then the smart kid if this happens twice or thrice we will ask your father sir are you sure there is a newspaper there father says yes I know there is a newspaper there going so when we hunt for an isothermal state 0.1 talks about existence of an isothermal state because unless 0.1 exists we cannot continue formally with 0th law and if we cannot continue formally with 0th law we cannot define that property with you call it temperature right we will start calling it soon. So coming back to our last slide this slide the advantage of this was the ability to determine whether system A given in a state A1 and system B given in a state B something when allowed to interact across a diatomic wall will there be a heat interaction or not and all that we have to decide is to which class of isotherms that system state of A belongs and which class of isotherms or which isotherms state of B belong. If both of them belong to blue we say no interaction if both of them belong to black we will say no interaction if both of them belong to orange we will say no interaction but if one isotherm is blue another isotherm is orange we will say yes there is likely to be an interaction. Now the basic idea of temperature is something which we have used. We have said one set of isotherm as blue, another as black, third one as orange, fourth one as pink, fifth one as yellow. What are these labels? These labels in thermodynamics we call temperature. That is our definition of temperature. We must provide labels to isotherms and these labels are known as, so instead of say as that lady said a system or a state of A belonging to the blue isotherm and a system, a state of B belonging to the blue isotherm will not exchange heat when allowed to exchange heat across a diathremic partition because they belong to the same blue isotherm. We say they have the same temperature. So the labels, what is the temperature of the state of A blue? What is the temperature of the state of B orange? Temperatures are different, so there will be heat transfer. But if the temperature of the state of A is orange, state of B is also orange, there will be no heat transfer. Now basically zeroth law ends here by saying number one isothermal states exist. There are corresponding isotherms in any pair of systems you take and the property of being isothermal is transitive which is very useful for us. So we do not have to worry about point to point. We can talk of a set of isothermal states and the corresponding set of isothermal states and it is a good idea to label isotherms. So all that we have to do is look at the state of A, find out the label of the isothermal surface or locus it exists on. Find out the label of the state of B. If both the labels are the same, we say no heat interaction. If the labels are different, we will say yes there will be heat interaction. These labels we say temperature. So now we can say that if temperature of the system A is equal to the temperature of system B, there will be no heat interaction. If they are different, there will be a heat interaction. Now the basic idea I will leave you in a confused state and the confusion will be cleared only after the second law of thermodynamics. The basic zeroth law scheme ends here by labeling isotherms. How to label them? Zeroth law does not say. That is part of thermometry. That is by convention we do it in a particular way. Zeroth law does not dictate. Zeroth law gives us the idea of temperature. Zeroth law does not tell us how to label them. Whether to label them by colors or label them by names of fruits or names of animals. One could have a leopard isotherm and a giraffe isotherm and a jackal isotherm, a cat isotherm and a dog isotherm. It is perfectly alright. And zeroth law in particular, get this clear, does not tell us which temperature is higher temperature and which temperature is lower temperature. That is not the. Zeroth law only talks of the existence of a useful property because each state will have a label. It depends on that state which isothermic belongs. So it is a property. So zeroth law tells us that temperature is a useful property. And what is temperature? Labels provided to isotones. That is where zeroth law ends. This is the thermodynamic definition of temperature. So temperature remember is only a label provided to isotones. Nothing more, nothing less. And zeroth law by itself does not tell us what those labels should be. Whether they should be numbers, they should be names, they should be symbols, whatever. And in particular it does not tell us, it does not provide us a idea of a higher temperature and a lower temperature. That is left to the second law. And zeroth law is only yes or no. If two states are isothermal, that means their temperatures are the same, there will be no heat interaction. But if the temperatures are not the same, that means they are not on corresponding isotones, there will be a heat interaction. Which way it will be? Zeroth law does not say. How to label temperatures? They say corresponding isotones should have the same labels. That is the only restriction. Without that or more than that zeroth law does not tell us anything. You would have noticed that we have not used the phrase thermal equilibrium. We used isotermal states and all that. So now let us just for completeness define and I am doing it late just to emphasize that thermal equilibrium is nothing new. It is another definition, another short form. When two systems across a diathermic partition are allowed to interact and if A1 and B1, if they are isothermal, then we say that A1 and B1 are in same thing. Both of them mean the same thing. Now we have to extend this a bit because quite often we come across a system is in thermal equilibrium. We are not talking of two systems here. We can define the temperature of a system when it is in thermal equilibrium. Such statements you would have come across. In that case, the only meaning of that is suppose you have a system and this is in thermal equilibrium. A system in thermal equilibrium only means that if we imagine a partition across the system and say system X and let us say one part is X1 and another part is X2 and we say if this happens to be suddenly put a diathermic partition, will get separated into two parts, will the two parts be in thermal equilibrium with each other or not? If the situation in the system was such that well whichever way we put the partition, the two parts will be in thermal equilibrium with each other, then we say that the system is in thermal equilibrium period. No other system is needed. So a system in thermal equilibrium essentially means a system throughout which the temperature is uniform and that means the system can be described by a single value of temperature. But this is nothing telling nothing more because we have defined thermodynamic equilibrium as unique values of all properties. So thermodynamic equilibrium means unique value of all properties including temperature and unique value of temperature means whichever part of the system you take temperature is the same that means whichever partition, whichever way you partition the system the temperature on either side of that partition would be the same. So the two parts of the system will be isothermal, so they will be in thermal equilibrium, so the system it said to be in thermal equilibrium. This is nothing great, I am just giving you the meaning of terms generally used in books on thermodynamics. And now you would agree that the whole book on thermodynamics can be rewritten without using the word thermal equilibrium. Because we have understood what is meant by isothermal states, we have understood zeroth law. You know this is the situation in thermodynamics because the same concept is known by many different names for historical reasons. So all that we have to do is be comfortable with that, okay.