 Welcome back. Let us continue where we left off. We demonstrated that the efficiency of a reversible 2T heat engine is a function only of the temperatures between which it works and nothing else. So this is a purely thermodynamic relationship. It does not depend on the properties of any material, solid, liquid or vapor. This is a basic thermodynamic relationship. No more details are available. We only know that it can only be a function of T1 and T2. What type of function that we do not know? In fact, this function we can define to set up a set of thermodynamic temperature scales. Let us see how. We know that the efficiency of any heat engine, 2T heat engine in particular is defined as the work done divided by the heat absorbed. If we apply the first law, the work done can be represented as Q1 minus Q2, heat absorbed minus heat ejected. So this becomes Q1 minus Q2 by Q1 which equals 1 minus Q2 by Q1. Now that means if we consider this to be a reversible engine, then this will be W divided by Q1 for a reversible engine. This will be Q1 minus Q2 by Q1 for a reversible engine and this will be 1 minus Q2 by Q1 for a reversible engine. And now since this is a function only of T1 and T2, that also means that Q1 by Q2 or Q2 by Q1, this ratio depends only on T1 and T2. In mathematical terms we can write either Q2 by Q1 is a function of T1 and T2 or let us take the inverse of this and write that Q1 by Q2 for a reversible engine is a function only of T1 and T2 and of nothing else. This is a very important result. Now what type of function can F be? Remember thermodynamics puts some restrictions but gives us lot of options. So here it says that look some function of T1 and T2, thermodynamics itself will put some restriction on F but will not dictate too many details of F. Let us look at what are the possible restrictions and hence what are the possible form or a convenient form for F of T1 and T2. Let us consider the following situation. Let us say that we have a reservoir say T1 and let us say we have another reservoir at say T3 and we have a reservoir at some intermediate temperature say T2 and let us say that the three temperature levels are such that T1 is greater than T2 is greater than T3. Now let us consider three different reversible engines. One is a reversible engine let me call it 1, 2 which works between T1 and T2. Let us have another reversible heat engine let me call it RE23 which works between T2 and T3. This provides work W12, this provides work W23 and let us have a third engine RE13 which works directly between 1 and 3. The reservoir at T1 and the reservoir at T3 and let it provide work W13. Now let us adjust the working of the engines in such a way that T1 provides heat Q1 to this engine RE12. Let T1 also provide the same amount of heat Q1 another stream of heat to RE13. Let RE12 reject an amount Q2 to T2 then let us adjust RE23 in such a way that it absorbs the same amount Q2 from T2. Let RE23 reject heat Q3 to T3. Now let me ask this question this Q3 prime rejected by RE13 to T3. Will it be equal to this Q3 rejected by RE23 to T3 or not. And we will realize that RE13 is a reversible engine working between T1 and T3. What does the combination of RE12 and RE23 do? Because T2 absorbs Q2 from RE12 and provides Q2 to RE23. T2 does not really come into the picture it just absorbs heat and rejects it. So all it does is it anchors the temperature between RE12 and RE23 to this temperature T2 and that means if you consider this system which consists of RE23 and RE12 together it is another 2T engine working between T1 and T3. And that means that if the 2 reversible heat engines this combination and RE13 are working between the same 2 temperatures T1 and T3 they must have the same efficiency. And that means if they are absorbing the same amount of heat they must be producing the same amount of work and they must also be rejecting the same amount of heat to the reservoir at T3. This implies that our Q3 must be equal to Q3. And an implication of this is that if I write Q1 by Q3 as the combination of ratios Q1 by Q2 by Q3. This is the heat absorbed to heat rejected ratio of this reversible engine RE13. So this should be the unknown function F of T1 T3. Q1 by Q2 that is the ratio of heat absorbed and rejected by this reversible engine RE12. So this must equal F of T1 T2. And Q2 by Q3 is the ratio between heat absorbed and heat rejected for this reversible engine RE23. And hence this must equal F of T2 T3. That means this relation forces this relation this into this on our currently unspecified function F. And this is what thermodynamics has done for us. Initially it says some function F of T1 and T2 but thermodynamics itself has restricted the type of function F to this type of function. So any function F which satisfies this is acceptable. Now again the question what type or form can F. If F of T1 T3 needs to be equal to F of T1 T2 into F of T2 T3 then one simple possibility is to consider F of T1 T2 of this kind. One simple possibility is to consider F of X, Y to be the ratio of some function G of X to the same function of Y. If we have something like this then naturally this is satisfied because if we assume this then what happens is this becomes G of T1 by G of T3 equal to G of T1 by G of T2 into G of T2 divided by G of T3. If this is there then this and that means this is satisfied. So that means we now have this function F of say T1 T2 defined in terms of another function where G is a function only of one temperature. Now the next question is what about the function G. Remember that we define our F in such a way that Q1 by Q2 for a reversible to T he can give was F of T1 and T2 and now we know that this has to be of the type some function G of T1 and G of T2. Now if we define G then we get a relation between function of temperature and the thermal energy or heat interaction between a 2T heat engine reversible and its reservoirs and that can be used to define scales of temperature. Then we can define scales of temperature which depend only on this relation and that means which depend only on the corollary of Carnot theorem and that means they will not depend on the material properties of any material that is used to develop and set up a reversible engine. So it will depend only on the principles of thermodynamics and hence these scales of temperature notice the plural here because if we define a different type of G we will get a different temperature scales. These scales of temperatures are known as thermodynamic temperature. Why thermodynamic? Thermodynamic because they do not depend on the property of any particular material that way they are distinguished from our temperature scales like the Celsius scale or the ideal gas Kelvin scale will depend on the property of some material. The Celsius scale initially dependent on the property of glass and mercury and the ideal gas temperature scales depend on a hypothetical ideal gas and an ideal gas can be approximated by a real gas which is at a sufficiently low density. Unlike those temperature scales which we may call empirical temperature scales because they depend on the property of some material. If you use this scheme we will be able to define temperature scales which depend only on the principles of thermodynamics. Hence these will be called thermodynamic temperature scales. We will soon define one particular thermodynamic temperature scale and that will be the thermodynamic Kelvin scale of temperature. Thank you.