 So, in this lecture, we are going to discuss the first law of thermodynamics for a system. And as I mentioned before, the laws of thermodynamics are actually made us as statements without any proof, the only proof is that they have not been disproved so far. So, we take absence of proof in this case to be proof of absence. So, based on extensive experiments, Joule and Clausius found out that for a system undergoing a cyclic process, the net heat and network interactions are equal. So, this is what they discovered after going through many experiments where they tried to convert heat to work and so on and wanted to formalize or come up with concepts which will relate the conversion of heat to work. And so mathematically, we may write this statement as cyclic integral delta Q equal to cyclic integral delta W. We say net heat and net work interaction because of the sign convention that we use, things will be taken care of automatically. For instance, in some parts of the work, the system may be receiving heat, but in I mean sorry, in some parts of the cyclic process, the system may be receiving heat and in other parts it may be rejecting heat. Similarly, the system may be producing work in during some parts of the cycle and it may actually be receiving work in other parts of the cycle. So, with the sign convention that we have, the algebraic sum of the heat interaction and algebraic sum of the work interaction comes out to be equal for a system that executes a cyclic process. So, the most important thing here is that this has to be a system and it must execute a cyclic process. Both these things are very, very important. In fact, the cyclic process is of particular significance in thermodynamics because cyclic processes by virtue of the fact that they can run forever can actually result in perpetual motion machines. So, second law or laws of thermodynamics actually take a very close look at systems executing cyclic processes. If it is a non-cyclic processes, then there may not be any restraints or constraints from the laws of thermodynamics. We will see this as we go along as we look at many different examples. So, we can actually rewrite this expression like this cyclic integral delta Q minus delta W equal to 0 and you should know from your high secondary mathematics that if the cyclic integral of a quantity is 0, then we may write this integrand as the differential of quantity like this. So, we may write it as a differential, perfect differential of a quantity. So, we choose to write this as De and we identify E as the total energy of a system and that is a property. So, E is a property and it is the total energy of the system. What we mean by total energy will be clarified in a minute or two. But for now, let us just look at what we have done. So, since this cyclic integral is 0, we said the integrand must be a perfect differential. We wrote it as De and then identified E as a property which is the total energy of the system. Of course, you may be wondering why we chose to write it like this. Why not write it like this? For instance, we could have written this as De equal to delta W minus delta Q also. So, we wrote delta Q minus delta W integral, cyclic integral delta Q minus delta W equal to 0. We could have taken this to the other side and written cyclic integral delta W minus delta Q equal to 0 also. Mathematically, that would have been alright. But for the sign convention that we have adopted here, this is consistent. Why is this consistent? Because when we supply heat to a system, in the absence of any work, supply of heat to a system increases the total energy of the system. Or in the absence of any heat, supply of work to a system which if you recall is negative in sign. If we supply work to a system, that is negative, work done by a system is positive. So, if we supply work to a system, then this entire term becomes positive, resulting in a positive increase in the total energy of the system. For instance, we looked at piston with a finite mass. So, when the piston moved up, we said that the total energy of the piston, meaning the potential energy of the piston increases. So, an agent has done work to increase the potential energy. So, this illustrates that writing it in this manner is not consistent with our sign convention. Writing it in this manner is consistent with our sign convention. Now, let us look at E and see what it comprises of. What is that? In the previous expression, Q is not a property of the system. We discussed this when we talked about properties. W is also not a property of the system, but the difference delta Q minus delta W is a property of the system. That is very important. So, the difference is a property, although each one by itself is not a property of the system. If we integrate the equation, the previous equation or previous expression from 1 to 2, we get the following delta E, which denotes the finite change in total energy between the final state 2 and the initial state 1 is equal to the heat supplied minus the or is equal to the heat interaction minus the work interaction of the system. So, this is first law applied to a non-cyclic process, whereas this is first law applied to a cyclic process. Notice that the change in total energy depends only on the initial and the final system. So, that is actually path independent, but Q and W are both path dependent. So, the difference of Q and W is actually a property, although Q and W themselves are not properties. Now, we have also introduced a few notations here. For instance, we wrote delta, Greek lowercase I mean Greek lowercase D or lowercase delta. We also wrote lowercase D, Roman D here. So, the same convention or the notation that we follow is like this. We use the lowercase D to denote changes, incremental changes in properties. So, we may use DE, we may use DP, DT over T is temperature or D capital U where U is internal energy and so on. So, we use lowercase D to denote incremental change in a property of a system. And we use lowercase delta to denote incremental changes in quantities which are not properties which are path dependent, delta Q, delta W and so on. These are incremental changes, the D and the delta refer to incremental changes. Whereas this capital delta is used to indicate finite changes in properties of the system. Finite change in a property is denoted using capital delta. So, capital delta E is E2 minus E1. So, capital delta P would be P2 minus P1, capital delta T would be T2 minus T1 and so on. So, that is the notation that we will follow. What is that? There is no such thing as delta Q or delta W because delta denotes finite change in the property of a system between say initial final and initial states and Q is path dependent. So, an expression such as delta Q is actually meaningless which is why we have written Q12 meaning heat interaction between in this process between states 1 and 2 or W1 to 2 along this process. So, it is very important to keep this notation in mind because the notation also helps you understand what is being talked about what is being written and so on. Now, let us take a look at the total energy. So, E has been identified as the total energy of the system and it is equal to the sum of the energy stored by the system in different modes. It could be internal energy, potential energy, kinetic energy and so on. There could be many forms in which system can store energy and E is the total energy and it is a sum of all the forms, energy stored in all the forms by the system. The first one which is the internal energy denoted by U is the energy possessed by the molecules that comprise the system in the form of translational rotational kinetic energy. So, if you take any system it consists of molecules. If you take air it consists of oxygen and nitrogen molecules. So, these molecules can move in all three coordinate directions. They can also rotate and about their own axis. So, there is energy stored in all these forms. So, this and other modes such as latent heat all these together constitute the internal energy of the system. So, the kinetic energy of the molecules is taken into account in the form of internal energy of the entire system itself. Now, potential energy is of the system is recognized like this. So, in the entire system as we mentioned before we may have a piston cylinder mechanism with some gas inside. Let us say the gas is our system. We need to start accounting for energy stored in the system in the form of potential energy when the system as a whole is made to undergo a change in elevation. Otherwise, there is no change in potential energy of the system. So, we need not worry about potential energy at all. So, kinetic energy again when the system as a whole starts to move with a certain velocity then the system has a kinetic energy and we need to keep track of the kinetic energy to account for any changes in kinetic energy. Notice that when we say kinetic energy this is kinetic energy of the system as a whole not kinetic energy of the molecules that comprises system that part has already been accounted for through the internal energy. So, this is kinetic energy of the system as a whole. So, the system as a whole starts moving like this that is kinetic energy. So, the total energy E may be written as U plus P plus Ke plus other things if they are relevant. If there is no change in potential energy, if there is no change in kinetic energy then we simply neglect these things and say that the total energy of the system is just equal to the internal energy. Now, if you look at the modes in which the system can store energy the internal energy mode may be categorized as a disordered mode because it is associated with the molecular motion which is always random. So, that is a disordered mode of energy storage. On the other hand, the potential and kinetic energy system may be considered to be ordered modes because the potential energy means what the energy the system possesses as a whole. So, I am lifting the entire system. So, the entire system has potential energy. So, this is an ordered form or the entire system starts moving like this. So, this is kinetic energy of the system. So, that may be considered to be an ordered mode whereas, the internal energy because it is associated with the motion kinetic energy of the molecules which are in random motion we consider that to be disordered. The importance of this distinction a distinction comes from here. When we supply heat to a system it can cause a change only in the internal energy of the system. When we supply heat to the system it only can change the internal energy of the system meaning the molecules start moving with higher kinetic energy when we supply heat to the system. So, it causes a change in the internal energy which is a disordered mode. So, because we are supplying heat to increase the energy of the system in that to increase the energy of a disordered mode of the system only a part of the heat may be converted to work. Because it is disordered not all of it can be converted to work that is the reason why we have this constraint. You may recall that we mentioned in the beginning that when we burn a certain amount of fuel we know the calorific value of the fuel. So, we know more or less the amount of heat that we are supplying. So, when the engine eventually produces work our expectation is that the amount of work produced should be very close to the calorific value or of the fuel that we have burned. But it never comes out to be the case and now we know why because when we supply heat to a system we are actually supplying energy to the disordered mode in the system and because it is disordered not all of it can be recovered in the form of work. So, heat can access only the internal energy more whereas work transfer can access all the other modes including the internal energy mode we will demonstrate this also. So, that is the difference between work transfer to a system and heat transfer to a system or heat transfer to a system can access only the disordered energy storage mode in the system whereas work transfer can access the disordered mode as well as ordered modes like kinetic energy and potential energy. Let us look at this through an example. Let us say that we have a closed rigid vessel like this which contains a certain amount of a working substance and we take this to be our system. Now what we are going to do is we are going to transfer a certain amount of energy say 100 joules in one of the following ways. The first way is to actually raise the vessel through a height of 1 meter we take g to be 10 meter per second square so that transfers 100 joules of energy to the system. In the other one we cause the vessel to move with the speed equal to square root of 20 so that half MV square becomes equal to 100 joules. In the other case we are going to transfer work in the form of paddle work and in the last case we will transfer the 100 joules as heat to the system and let us see what happens in each one of this case so that we can understand the difference between heat transfer and work transfer. So, first case we lift the entire vessel through a height of 1 meter. So, basically what happens is we put in work causing a change in potential energy of the system. So, the work that we put in causes a change in potential energy of the system. So, now the system is at a higher elevation that is 1 meter. Now we can actually use the system to produce some work for example, we could connect it to another weight and allow the system to gradually come down and the weight that is connected to this will then be lifted up. So, that is useful work if you remember that is the definition of work that we gave in the beginning. If the work is done by a system if the combined interaction of the system with the surroundings is the raising of weight. So, I can easily imagine the system coming down and raising a weight. So, we can realize the potential energy change in the form of work and then I can actually now the system is back at the ground level where it was before now I can again put in work raise its height and convert that to work and keep going like that. So, whatever work I put in is translated into change in potential energy and the change in potential energy is translated entirely into work that comes out of the system assuming everything is ideal. Assuming that the mechanism that we use for converting the PE to work is ideal that there are no losses we recover all of the potential energy change in the form of work and this can execute a cyclic process it can keep doing this again and again this is work transfer. So, all the work can be transferred through this system to something else. So, this is basically transmission of work in the same way. So, here we are accessing the potential energy or ordered mode one of the ordered modes of energy storage of the system. So, here we are accessing the other another ordered mode of energy storage of the system namely kinetic energy. So, we for instance you know we can imagine connecting up the system up to let us say catapult and then launching it with the velocity of square root of 20 meter per second. So, that transfers 100 joules of energy to the system. So, we put in work which is then converted to a change in kinetic energy of the system. Now, we can imagine compressing a spring using the system. So, the system is moving like this we have a spring like this. So, the spring is compressed. So, all the kinetic energy that the imported to the system will now be stored as spring energy which can then be connected to another mechanism to do work or we can connect the moving system itself to another mass or something else and use the kinetic energy of the system to raise the weight. But the important point is the kinetic energy can be entirely converted to work. We put in some we put in 100 joules of work it is converted into 100 joules of kinetic energy change, which can then be converted to 100 joules of work. So, all the 100 joules that we put in can be realized in the form of work. So, this is also transmission of work whatever we put in access an ordered mode it is simply transmitted entirely to do work. Now, in the third case we access the disordered mode of energy storage system namely molecular motion or internal energy. So, we have a paddle wheel arrangement here mass is connected to a pulley which turns a paddle wheel. So, what we do initially is we disconnect the mass from the pulley and we use an external agent to raise the mass to a certain height just like what we did here. So, that is putting in work into the mass. So, that is transferring work to the mass. Now, we connect the mass to the pulley and we allow the mass to slowly come down. So, as the mass slowly comes down the paddle wheel turns and so, whatever work we put in here is converted to internal energy of the system. Notice that our system is the same throughout. So, whatever work we put in is converted into internal energy of the system. Now, if we connect it to a reservoir and make sure that none of the heat is lost to the surroundings process can be executed in an isothermal fashion. So, that the temperature of the system remains the same and all the heat is then transferred to a reservoir which we connect like this. So, this is a reservoir. So, if I connect a reservoir I can get all the heat from the system while its temperature remains constant. So, what we have done here is we have put in work into the mass as it descends slowly we converted to internal energy and then we can then convert it to heat. In fact, in principle all the work can actually be converted to heat even in this case because work is an ordered form but we transfer the energy to a disordered form which is internal energy which can then be converted to work. Note that here we have conversion because we are putting in work and we are taking it out in the form of heat. Here we put in work we are taking it out in the form of work and same here. So, this is these two cases are merely transmission of work whereas here we actually have conversion of work to heat. In principle all the work can be converted to heat that is nothing in the laws of thermodynamics that prevents that from or that prohibits that from happening. So, we convert the work to heat and then if you want to repeat the process we disconnect the mass from the pulley and then connect it to a device which can elevate it to this height and then connect it again and then repeat the process. So, the process can be repeated continuously. So, this can be run as a cyclic process conversion of work to heat we can run as a cyclic process all this we can run as cyclic processes. All the three examples that we showed so far can be run as a cyclic process. Now, if I look at this situation where I am giving 100 joules of heat to the system. Now the system expands and in doing so it performs some work. So, I give some heat there is a change in internal energy of the system because heat can access only the internal energy of the system or the disordered mode and that is converted to displacement work. So, here the piston moves up and we have displacement work. Now, the piston goes up to a certain height let us say the piston goes up to here. Now, you may those of you who have studied thermodynamics may even argue that as long as we ensure that the process is isothermal all the heat can actually be converted to work. So, none of the heat that we put in actually results in changing the internal energy of the system it can be converted to work and that is true. So, we can convert all of it to work. Now, the question then comes is as we saw in all these cases we are actually able to run all these processes as cyclic processes. We are able to bring the system back to its initial state and repeat them as cyclic processes what do we do in this case? How do we accomplish that in this case? If I give another 100 joules of heat the piston will move up even more and again the same amount of work will be there. But that is not a cyclic process if I want to run it as a cyclic process I must bring it back to the initial state or the state it was in before I started the process. So, I need to bring it back like this. If I want to do that then I need to put in work which means that if I want to run this as a cycle I put in a certain amount of heat it causes a certain change in internal energy which then gives me a certain amount of work. But if I want to run it as a cyclic process I need to put and some put back in some amount of work. So, the network that I get from the cycle is actually less than the amount of heat that I am putting in. So, in this case if I want to run this as a cyclic process I cannot convert all of the heat into work. I have to use I have to put in some amount of work to get the system back into its initial state. If I do not run it as a cyclic process then I can convert all of the heat into work. The constraint comes only because you want to run this as a cyclic process. So, as I said before cyclic processes have the potential to be perpetual processes. So, there are additional constraints on cyclic processes from the laws of thermodynamics. So, if you want to run this as a cyclic process that means I have to put in some work and the network that I get from the cycle is less than the heat that I am supplying or the net heat that I am I am sorry is less than the heat that I am supplying to the system. The net heat and the net work will be equal first law says that we have already seen that. But the net work that I get from the system will be less than the amount of heat that I am putting in. So, in this case W net will be less than the heat that I am supplying. So, if I summarize this we can say that energy supplied in the form of heat cannot be entirely converted into work. It is important to note that the emphasis here is on engines that accomplish the conversion continuously by means of a cyclic process. So, this is true only for cyclic processes. So, cyclic processes are very special. So, we cannot convert all of the heat to work if you try to run an engine in a cyclic process. Otherwise, it is eminently possible. Now, we saw this example before where we showed that depending on where I draw the system boundary, whether I draw the system boundary like this or whether I draw the system boundary like this the displacement work is different. In this case the displacement work is integral 1 to 2 PDV where P is the system pressure and this pressure we showed was equal to the pressure of the atmosphere plus the pressure due to the force exerted by the external agent plus the pressure due to the weight of the piston. Whereas, if I draw the system boundary like this then the displacement work is the system pressure that we mentioned just now minus P piston. So, this is the pressure of the gas minus P piston. So, we mentioned in passing that although the displacement work is different it is accounted for properly in the first law. What do we mean by this? That is what we are going to look at now. So, if you look at this system the total energy of the system is nothing but the internal energy. There is no change in potential energy of the system. Remember, there is no change in potential energy of the system because we are not lifting the entire piston cylinder mechanism. So, delta PE is 0 delta KE is also 0. So, the total energy of the system is just internal energy. Whereas, in this case the total energy of the system is the internal energy of the gas which is over here plus the potential energy of the piston. We need to take into account potential energy of the piston because it changes during the process the piston rises during the process. So, this changes during the process. So, we need to take that into account. So, in this case there is no change in potential energy of the system. So, you can see that these two expressions work together so that we may account for something as a work transfer or as energy transfer. In this case, this is accounted for as energy transfer change in potential energy of the piston is accounted for as energy transfer. Here the change in potential energy of the piston is because of the work transfer from the system to its surroundings. So, we can account for things properly. Once you draw the system boundary, what crosses the system boundary is work or heat. What is inside the system is accounted for through our total energy term. So, we have to be very careful about the system that we define where the system boundary is because the system boundary determines the nature of the interaction. We have already seen that something may be classified as a work interaction or heat interaction depending on the boundary. We also said that the work interaction magnitude and sign may become different depending on where the system boundary is. Now, we are saying depending on where the system boundary is something may be work transfer and same thing may be classified as energy transfer depending upon where we draw the system boundary. So, it is very important to draw the system first. That is why we spend so much time discussing the definition of the system and all the subtleties associated with defining the system. So, once you define the system, then all these aspects become very clear.