 This session we are going to study some new things in detail, but before that we let us see what are the outcomes that I expect. First thing is you must be able to evaluate the specific heat of mixture of gases, then you will be able to analyze different processes by first law, then you will be able to define aerobatic elasticity and isothermal elasticity. Now see already in first three or in the last session we have studied the genesis of the first law and certain things about the specific heat and other matters, but thermodynamics is such a science which appears to be simple for the first listening, but if I ask the questions then you will find that how intricate it is and difficult to answer the questions. Now looking at the first law I will ask a simple question first, now this is my say PV diagram and on this PV diagram I draw two curves, I draw two curves where temperature is same, temperature is same, but the masses are different M1 and M2, so this is related to specific heat understanding. Now my simple question is if I want I have two systems for which first is mass M1, second is mass M2 and I have maintained the temperature constant by supplying them heat, then my simple question is whether M1 is greater than less than or equal to M2, you have to justify it by the proper concepts of thermodynamics specific heat, if possible you can use the first law, if it is say you can apply it, second conceptual questions that question that you can under say apply is this is process A to B, if I show it on the PV diagram A to B, I am asking you whether VB is greater than less than or equal to V, now see this appears to be very simple, but if you look at the nature of the process that is shown A to B, you will find that A to B is not passing through say diagram that is sorry instead of PV diagram say it is a PT diagram, I am saying PT diagram, it is not passing through origin, when it is not passing through origin it is not a constant volume process, so you have to use the equation for line y is equal to mx plus c, you have to find out the constant of integration by the given boundary conditions and then you have to evaluate, so try to say give the answers to this particular questions and now we will say see how we will evaluate the specific heat when we have been given a mixture, now you know that whenever we have been given a mixture means there is a gas in which there is gas A and gas B, number of moles are n1 and n2, it has got some say each gas has got its own adiabatic index say gamma 1 and gamma 2, n1 and n2 are the moles, then after mixing them together what happens say logically it is like mixing warm water and cold water you will get some different temperature, so if I mix them together my question is what will be the adiabatic index of the mixture means I have to find out the gamma for the mixture, now for that what we require two things we require one is Cp by Cv is equal to gamma this is the relation see Cp by Cv is always gamma it is the adiabatic index compression index Cp by Cv is never n, n is a very different parameter in thermodynamics Cp by Cv is gamma and another we have Cp minus Cv is equal to R which is always true which is always true whether your gas is monotomic, diatomic or polyatomic, now the question is if the molecular weight of this first gas is m1, second gas is m2, n is the total number of moles n1 plus m2 then what will happen if I mix the gases together then I know that the molecular weight of the mixture will be equal to n1 m1 plus n2 m2 upon n1 plus n2 this is from chemistry, so I use simple Dalton's law and I have done it, now from thermodynamics point of view whatever heat I have supplied to gas A, whatever heat I have supplied to gas B must be equal to total heat supplied to the composite, one thing is there this mixture is non reactive this mixture is non reactive mixture, so what will happen suppose I assume that the specific heat of the mixture is Cv mix then I know that total number of moles are n1 plus n2 then Cv of mixture is I know then temperature rise must be equal to I know n1 Cv1 into T plus n2 Cv2 into T2, so this is my simple equation that is I have written from the simple first law of thermodynamics or it is a simple principle of energy conservation, now one thing I say forgot to tell you about the first law that already we have derived it is the conservation of energy principle because whatever heat we have supplied we have tried to find out where it has gone, so it has gone in expanding the medium and also it has gone in changing the internal energy, so because of this there is a conservation but first law is not complete law for understanding thermodynamics because it never states anything about whether the process is possible or not, so for example suppose I rub my two hands then what will happen my hands will become warm, so as a result there is because of the friction because of the work done there is a heat generation, so you know that moving the rubbing the hands is a mechanical work that I am doing and I am getting heat instead that is I am converting that work into heat and it is becoming warm, now suppose you held you say propose you keep your hands like this say for example and if you supply heat from the bottom with the help of a candle will your hands move automatically never, so this is the second law that we are studying in our other modules, so in the second law we give the direction that where the process will take place, so this is the principle of conservation of energy which every process must follow and also it must obey the second law and then only we say that the particular process is feasible, now by this two relations C p minus upon C v is equal to this and this one we can find that C v of the mixture is going to be r upon gamma minus 1 then C v 1 is equal to r upon gamma 1 minus 1 and say C v 2 is equal to r upon gamma 2 minus gamma 1, why we are doing this we are doing this because we want to bring into the expression for evaluating the specific heat of the mixture, so what I will do now I will write here n 1 plus n 2 into say r upon r upon gamma minus 1 into t which is equal to r upon gamma 1 minus 1 into say C v z is a n 1 into 2 into 2 into 2 into t plus r upon gamma 2 minus 1 into n 2 into t, so if I cancel out the terms I will get n 1 plus n 2 upon gamma minus 1 is equal to n 1 upon gamma 1 minus 1 plus n 2 upon gamma 2 minus 1, this is a very important say relation from which we can get the expression for gamma for the mixture, so this gamma is for gamma of mixture that we can get it, so this is about the specific heats, now if I ask a simple question that we have to apply the first law for various processes, now when I apply the first law for various processes the first process I take for evaluation is say on the p v diagram say constant pressure process either this way or I can go by this way, so 1 2 2 or say here from a to b, now you can see I had drawn this diagram on both that is both diagrams on the p v diagram in the first I go from 1 2 in the second I go from a to b, if I ask you what is this process it is a constant pressure process, the first is expansion second is yes compression because here volume is reduced here volume is increased, now if I draw the isotherm passing through this and passing through this, similarly if I go from this and this you will find that when I go from process 1 to 2 area under the process 1 to 2 is the work done and if this is this direction it is moving it is a clockwise direction, so it is a positive work this is a negative work, now if I apply the first law that is say heat supplied q is equal to w plus delta u and w is p d v plus delta u and you know that p d v plus delta u is always equal to n c v delta t, n c v delta t, so c v is positive delta t is if there is a increase in temperature, if there is increase in temperature it is positive, so if I supply the heat if I get the work what will happen you have to mention that what happens to my say delta u that you can write, now based on this if I ask you a simple question that on the p v diagram I show you two volumes say this is v 1 and this is v 2 and this is my first process I expand this way and this way, so you have to identify which is this process this is isothermal or adiabatic this is isothermal or adiabatic first thing and from that you have to find out in which case we get more work, if it is this process is having a less slope so this is say isothermal and this is adiabatic already you have studied this, so you can see if the same volume that is same expansion is there then area under the isothermal is more than area under the adiabatic is it not, but the problem is in isothermal the pressure is higher than the adiabatic, so if your choice of a device is to take complete benefit of the expansion of the gas then don't go for isothermal go for adiabatic, so these are very complex things that we are going to see in detail in our say next parts now those who are interested in say studying further for them there are certain standard books that you can say refer to that is one is thermodynamics by NaG and second is thermodynamics by Unis-Sengel, now see thermodynamics is