 a discussion of the zeroth law and the idea of temperature. In fact, the zeroth law is a funny law of thermodynamics. Unlike the other laws, which give us an equation, the zeroth law does not give us an equation. The first law said that well, D e is an exact differential. The second law will also say that something else, some combination is an exact differential leading to the property entropy, just the way first law led to the property energy. zeroth law does not give an equation for temperature. In fact, the special thing about zeroth law is it gives us a geometric or a state space interpretation of temperature. It gives us some idea about how to measure temperature, but just an idea it does not give a thermodynamic basis for the measurement of temperature. And although we may think otherwise, zeroth law does not tell us what a higher temperature is and what a lower temperature is. Using zeroth law, we have conventionally defined a large number of temperature scales and just by convention for convenience, we have defined larger numbers to mean hotter stuff and cooler numbers to mean cooler stuff. Traditionally, we know that steam is hotter because we have decided to call anything which is which behaves like steam as hot. Steams, flames are hot, whereas something which is frozen like ice is cold. So, hot and cold are relative ideas and the higher temperature and lower temperatures are conventional directions of numbers, higher numbers and lower numbers, which we have associated with it. It is only the second law. In fact, temperature is an involved entity both in zeroth law as well as in the second law. And it is only after going to the second law that thermodynamic temperature will have a proper thermodynamic basis for its measurement. So, temperature is something which pervades two laws, one definition or the basic definition using zeroth law and second one, the proper thermodynamic basis for its measurement in the using the second law. So, just the way the first law helped us define two entities. One, a property of state called energy and two, an interaction called the heat interaction. The second law, when we come to it will allow us to define entropy and also provide a basis for measurement of temperature. Now, notice that although the nomenclature is zeroth law, first law, second law. Historically, the ideas were developed first as those for second law by Carnot, then for first law by Joule and others and finally by zeroth law. But the way we are studying it or developing our exposure of thermodynamics, it is in the another way, first law, zeroth law and then second law. Of course, there is nothing unique about it. This is the way, well, many people and many textbooks develop it and I find that it is the most neat and convenient way of developing things and explaining thermodynamics. So, what is zeroth law? If first law is the statement of systems bounded or interacting across an adiabatic wall, zeroth law is a study of the behavior or generalization of behaviors of systems related by non adiabatic walls. Now, before we do that, let us again look at the consequences. We have using the first law, D e is D q minus D w or if we restrict ourselves, if D e is D u, then we end up with u equals D q minus D w. Now, this u as a property, what does it depend on? Let us look at it. It turns out that if you drive D u as D q minus D w expansion plus D w electric plus so on. You will notice that D w expansion can be written as something like P d. So, you change the volume, you are likely to have a change in u. This is something like E d q. So, you change the amount of charge, you are likely to change D u. So, D u is likely to be a function of some properties of state that D and q and all that depending on the complexity of the system. But then it turns out that without doing any work, you can have all work interactions to be 0. Then even by having some heat type of interaction, you can change the internal energy of a system. So, that means apart from this, u is a function of some other property. What is that property? That is one question. The second thing is on how many properties does u depend on? Why just u? Just the complete state of the system depend on and that brings us to the second state postulate. Although this is a state postulate to a state principle, after studying the second law, we will see a demonstration of why this postulate turns out to be true. And here we come back to our definitions yesterday of a simple system, a complex system and the rudimentary system. The state postulate 2 says that the number of independent intents properties. By intensive, we can say either intensive or specific. That means for a system of fixed mass or a closed system, required define the state of a system equals number of two-way work modes plus 1. This is the state postulate 2. The state postulate 1 was that the state of a system can always be defined in terms of primitive properties. Which primitive properties thermodynamics does not tell us? How many? At that time, we did not know. Now here we have the answer to how many. The answer to how many is find out the number of two-way work modes, add one to that. And that means if we have a simple system, the number of properties needed is number of two-way work mode plus 1, which is 1 plus 1. Because for a simple system, number of two-way work mode is 1, is 2. And in illustration of this is if you have a simple compressible system like a fluid, then the two properties could be for example, pressure and volume or pressure and specific volume, if it is a constant mass system, closed system. So, pressure and specific volume would be the two independent intensive properties. What happens if you have a complex system? The number of properties needed will be greater than 2. Again this is n 2 w plus 1. Since n theta 2 w is greater than 1, this will be greater than 2. Depending on the level of complexity, it could be 3, 4 and so on. Most of our systems or almost all systems which we are going to work with are going to be simple systems of some kind or the other, usually simple compressible systems. Now what about a rudimentary system? For a rudimentary system, since the number of two-way work modes is 0, the number of properties needed are 1. Although it says 1, we do not know which one, but if we define more than one properties, it means that all the other properties will depend on one single property. So, you define one single property that will define its state and that will define all the other properties which way you may need to define. Just the way for a simple compressible system, we have said it is a simple system. So, two properties are needed. So, a simple compressible system, again two properties are needed. If we select the two properties to be say pressure and volume, then all other properties which can be extracted or which can be defined like temperature, internal energy, entropy will all depend on these two properties, pressure and specific volume. Again thermodynamics does not say that you use pressure and specific volume as the two properties. We can use pressure and say enthalpy as the two properties, then specific volume, internal energy, temperature should be extractable from the values of these two properties. This question still remains. Is there a significant property if there is a unique property? That brings us back to our mainstream discussion on the zeroth law of thermodynamics and let us now define a word, again a definition. Just the way we began the discussion of the first law of thermodynamics by defining an adjective adiabatic. Here we define a word diathermic. Diathermic is defined again as an adjective. It means not adiabatic and this implies heat interactions possible. So an adiabatic wall or an adiabatic boundary allows only work interaction, prevents all heat interaction because heat interaction is defined as a non-work interaction. First law, a diathermic wall, a diathermic partition, a diathermic boundary, a diathermic interface means a non-adiabatic wall boundary partition interface means work interaction may be possible but apart from that heat interaction is also possible. And our zeroth law, just the way our first law was a generalization of the behavior of adiabatic systems. The zeroth law is based on a study of behavior of systems separated by a diathermic boundary. Now the traditional view of zeroth law, if you see most of the textbooks and even some very good textbooks are not exceptions to this. Zeroth law talks about thermal equilibrium without really defining what thermal equilibrium is. It says that if two systems A and B are in thermal equilibrium with each other and if A is A can independently be in thermal equilibrium with a third system C, then B and C will also be in thermal equilibrium with each other. So well this in a way is a part of zeroth law of thermodynamics but let us go back and look at the proper way zeroth law of thermodynamics is to be studied and appreciated. Actually this traditional statement of zeroth law of thermodynamics has two weaknesses. Number one, we have not defined what thermal equilibrium is and two, this is only a part of the zeroth law of thermodynamics and perhaps a secondary part. The primary part of zeroth law of thermodynamics has not been mentioned at all and maybe that is the reason why we have not defined thermal equilibrium. Zeroth law has two parts, one part the first part is what I will call the existence part and second part is what I call the transitivity part that is if A is in thermal equilibrium with B, B is in thermal equilibrium with C then A will be in thermal equilibrium with C. Let us see how this is developed. Again let me emphasize that zeroth law is our study and appreciation of the way systems behave when they are separated or they are allowed to interact with each other across a diathermic wall. Again for first law we said let us study an adiabatic system between two specified states 1 and 2 and let us look at different adiabatic processes joining the two states from 1 to 2. Here we look at two systems let us say system A and system B separated by a diathermic partition. A diathermic wall allows heat transfer to take place, it allows also work transfer to take place. So what we will do is to simplify the argument we know work transfer is a primitive. Some sort of a generalized force and generalized displacement is involved. For example, if the work transfer possible is expansion of a piston we can prevent that interaction by freezing or sealing the piston. If the work transfer is the stirrer tau d theta kind we can lock the stirrer so that stirrer work is prevented. If it is the electric kind EDQ we can open the circuit so that no current flows no charges either provided or extracted it means electrical work is prevented. By restricting such a system we can set up the two systems and set up the diathermic wall in such a way that we look at only the heat interaction that takes place between the two systems and the experiment we do is the following. It is an exploratory experiment and we do the experiment as follows. Just for illustration and simplicity I consider the two systems to be simple systems but that is just because I want to show them on plain paper so that two properties. Let us say this is system A state space of system A this is state space of system B and may be I have x A and y A as the two properties if you want write P A and V A and this is x B and y B are the two properties which define the state of system B. Why two well simple illustration if you take complex system draw it in three dimensions or four dimensions etcetera. Now the exploration which we are going to do is as follows. We are going to take our system A in a state here say A 1 and what we are going to do is we are going to initially bring state B system B in different states and allow those states to interact across a diothermic partition with the fixed state of this system A at A 1 and you will find that for most of these experiments there will be a heat interaction between A and B A remaining at A 1 and B 1 of those but the first part the existent part says that in the state space of B there is at least one state but in general more than one state not have any heat interaction with the fixed state of system A when allowed to have heat interaction across a diothermic partition. So this is the existent for a fixed state A 1 of A. So that means if I take A 1 A in A 1 and B in B 1 allow them to interact across a diothermic partition they will say we will not interact there is no need for us to interact and this can be checked out that the state of A 1 will not change state of B in B 1 will not change that means there is no interaction we have already prevented work interaction we have allowed only heat interaction but even that does not take place. So the first part of zeroth law is that in the state space of the system B there exist states at least one which will not have any heat interaction with a fixed state of system A namely A 1. So this is the existent part. Then again definition the pair of states A 1 B 1 etcetera A 1 B 1 and also A 1 B 2 and A 1 B 2. And so on the states which do not have any thermal interaction or heat interaction with a fixed state of A across a diothermic partition are called isothermal states that is definition and this is A part of the definition. These definitions are only to put our earlier terms which are used regularly like isothermal equilibrium on proper footing. So if I say the state A 1 of A and B 1 of B are isothermal state means that they will not have any heat interaction even when allowed to have a heat interaction across a diothermic separating wall. And A 1 and B 1 are in thermal equilibrium each other. So this way we have defined what we mean by a pair of isothermal states and we have defined what is meant by two states being in thermal equilibrium with each other. So two states of two systems A and B namely A 1 and B 1 if they are in thermal equilibrium with each other that means they are isothermal states and that means by definition they will not have any heat interaction across a diothermic wall. So this is the first part of the zeroth law of thermodynamics existence of isothermal state and hence consequence definition of isothermal states and thermal equilibrium. Now the second part the second part is the transitivity part but for the transitivity part let us extend this. If we extend our experiment again state space of A state space of B we did an experiment with A 1 and we found out that there are a number of states say B 1, B 2, B 3, B 4, B 5, B 6. We did the experiment by fixing A at the state A 1. We can repeat the experiment by fixing the state of B at say B 1 and explore various states in the state space of A and we will find perhaps that we will have a set of states A 2, A 3, A 4 all in thermal equilibrium with B 1 or any of them isothermal with respect to B 1. So we have found A 1, B 1 is isothermal, A 1, B 6 is isothermal, A 1, B 2 is isothermal. We have also found B 1, A 2 is isothermal, B 1, A 3 is isothermal and so on. Again extending our definition this set of states with our continuum and continuous variation of states properties we will find generally that these are all on a locus. This is supposed to be an isotherm in the state space of B and this set of states is known as or is called an isotherm in state space of A. Now we will link the two isotherms by using the second part that is the transitivity part, transitivity parts of isothermal states. This is what we all know. We say that let A 1, B 1 be a pair of isothermal states and let us explore the same one and let A 1 and say some C 1 third system be also a pair of isothermal states. Then the transitivity property implies that if you bring B 1 and C 1 together across a diathermic part they will not interact they will also be isothermal states and that means let us go back to our experiment. We have said that look with respect to A 1 any one of these is an isothermal state with respect to B 1 any one of these is also an isothermal state. Now what we will do is we will create a replica of system A that means a perfect copy of system A. One we will bring at A 1 another we will bring at A 3 and we know A 1 and B 1 are isothermal states. We also know that A 3 and B 1 are isothermal states. So if we bring A 1 and A 3 two copies of the same system at two different states together across a diathermic wall we will know that they are also isothermal states. So that gives us the importance of the word isotherm in this and isotherm in this state space. Extending this if we create a replica of B you will find for example that B 1 and B 5 are also isothermal states. Not only that extending this argument we will see that you bring system A at any one of these and let it interact with any one of the systems B at any one of these states B 1, B 2, B 3. They will be in thermal equilibrium with each other. So this isotherm and this isotherm this and this are now known as corresponding isotherms in the state space of A and state space of B respectively. Now what is the importance of this? The importance of this is by mapping isotherms we can determine whether Q interaction will take place or not between two systems at which are in specified states. For example let us now extend the experiment like this. We have system A and we have system B. Again the two properties of A let them be x A and y A. Two properties of B let them be x A and y A. Two properties of B let them be x B and y B. And we have already seen that we have already done one experiment. We will say this had A 1 etcetera on it and this we had B 1 etcetera on it. Then let us repeat the experiment and select some other state of A say A 11. And we will find that in the state space of B we will have a set of states may be B 11 on them but others 11, 12, 13 etcetera which are all isothermal with respect to the A 11 state of system A. And then we select one of these and find out the corresponding isotherm here like this. And then we repeat the experiment may be with another state of A say A 21. And we find that in the state space of B there is another set of states may be with B 21 and other states on it which are isothermal with this state of A. And then fixed one of these and we will find that there is a corresponding isotherm here. So we will have a black isotherm here in state space of A, black isotherm in the state space of B, may be a blue isotherm in the state space of A, blue isotherm in the state space of B, green isotherm in the state space of A, green isotherm in the state space of B. And may be if you want even a red isotherm in the state space of A and the red isotherm in the state space of B. Now what does this help us to determine? It helps us to determine, suppose we are given answer to the question which is as follows. Let us say we are given a state A naught of A and state B naught of D and we have to answer the question if I allow state A naught and state B naught to interact with each other across a diathermic wall, will there be a heat interaction or not? All that we have to do is find out which is the isotherm or what is the color of the isotherm to which A naught belongs. Suppose it belongs to the blue isotherm here, we find out the isotherm or the color of the isotherm to which B naught belongs. If it belongs to green then we will say yes there will be a heat interaction, but if instead of green it belongs to blue then we will say no there will not be any heat interaction. If it belongs to any other isotherm other than green other than blue which is the isotherm of A naught we will say that a heat interaction will take place. That is all we can do here, we cannot determine whether the heat interaction will be from A to B or from B to A. Now if this is understood then the next definition, definition of temperature. Now remember what we did we said that given a state A naught of A and given a state B naught of B, we can say that if they are allowed to interact across a diathermic wall then heat interaction will take place if they do not belong to corresponding isotherms, but if they belong to corresponding isotherms say blue and blue or green and green and black and black or red and red then no interaction will take place. To simplify this we define a temperature, the definition of temperature is simply temperature is nothing but a label provided to corresponding isotherms in the state space of various system. In our illustrative case we have just considered two systems A and B, but we can extend it, we can have a system C, we can map out the isotherms just the way we have done this and then we will have a map of state space of C with appropriate isotherms the location of red, blue, green, black, brown what you have. So these labels provided are called temperatures and remember that temperatures are nothing but labels one for every isotherm and what is the advantage? After defining this temperature we will say that A naught and B naught will not have a heat interaction if the temperature of A naught and temperature of B naught are the same. What does it mean? That means the isotherm label for state A naught is the same that means the states A naught and belong to this pair of corresponding isotherms in the respective state spaces and hence there will be no heat interaction. Now remember that labeling these isotherms is rather arbitrary. I have labeled it as black, blue, green, red because some colors were possible. For example, I could have defined an yellow isotherm. I do not think you are able to see it. I could have defined a pink isotherm, but now this is arbitrary. I could have named them after stars for example or after famous cricketers for example. So we could have a Gavaskar isotherm, we could have a Tendulkar isotherm, we could have a Merriman isotherm and so on. This arbitrariness will not work. So we come to the science of thermometry.