 Hello and welcome to lecture 19 of this lecture series on Introduction to Aerospace Propulsion. So, over the last several lectures we have had some discussions on fundamental aspects of thermodynamics like the laws of thermodynamics and properties and so on. And also application of thermodynamic principles to actual applications like in cycle analysis and so on. So, last 2-3 lectures we were discussing about the various power cycles which are commonly used in day to day applications. We had looked at the basic thermodynamic principles behind these cycles and also what are the small modifications that can be made which can lead to improvement in the efficiency of these cycles. So, these were some of the aspects that were discussed in the last few lectures. In today's lecture what we are going to take up is a very different topic altogether. It is not to do with thermodynamic cycle analysis and so on, but what we shall be discussing today are some very basic thermodynamic relations which are commonly used in analysis of engineering systems. And some of these topics are very important in analysis of complex thermodynamic systems. So, let us take a look at what we shall be discussing in today's lecture. We shall be talking about what are known as the Helmholtz and the Gibbs functions. And then we shall be discussing about what are meant by Legendre transformations. And as we apply Legendre transformation to certain equations we get what are known as the thermodynamic potentials. Then we will take up a very important set of relations known as the Maxwell relations which play a very significant role in thermodynamic cycle analysis and otherwise. We will spend some time on discussing about the ideal gas equation of state and also subsequently what is meant by the compressibility factor. We will then take up the other equations of state which are basically modifications of the ideal gas equation states to make it more realistic. And towards the end of the lecture we will be discussing about the Joule-Thompson coefficient or the Joule-Thompson effect which is basically applied for fluid flows through a throttling device or a throttling process. Now, if you recall several lectures earlier we had discussed about what are known as combination properties. Combination properties are those which are basically a set of different properties which put together have certain thermodynamic significance. One such property which we had discussed was known as the enthalpy. Enthalpy was defined as the sum of the internal energy plus the product of pressure and specific volume. So, specific internal energy u plus p v is equal to the specific enthalpy. So, this was the basic definition of enthalpy which is basically a combination property. Today, we will discuss about two more combination properties which are known as the Helmholtz function and the Gibbs function. So, let us take a look at what these functions are and what their significance are. So, we are going to introduce two combination properties. First one is known as the Helmholtz function which we will define or denote by a letter a. So, Helmholtz function is basically an indication of the maximum work that can be extracted from a particular system. We have already discussed about what is the maximum work, what do you mean by useful work, what is known as irreversibility etcetera. And so, based on those definitions or concepts, what we shall define today is what is known as the Helmholtz function. So, Helmholtz function by definition is it basically indicates the maximum work that can be obtained from a system. So, enthalpy is something which we have already discussed. It is a combination property again it indicates the work content or energy content of a system. So, Helmholtz function by definition is u minus T s where u is the internal energy and T s is product of temperature absolute temperature and the entropy. So, this is by definition known as the Helmholtz function. So, what we can immediately see is that Helmholtz function basically gives us an idea of how much is the internal energy, what is the amount of maximum energy that you can get after you deduct the unavailable energy because the product T s is basically the measure of the unavailable energy or the irreversibility which we have defined earlier. So, difference between the internal energy and the unavailable energy is basically the Helmholtz function. And so, it gives us an idea about the maximum work that can be extracted from a particular system. The other combination property is known as the Gibbs function. Now, Gibbs function is again very similar to that of Helmholtz function, but it indicates the maximum useful work that can be obtained from a system. And like Helmholtz function which was just the indication of the maximum work Gibbs function gives an idea of maximum useful work that can be extracted from a system. So, Gibbs function G it is denoted by letter G, G is equal to h minus T s. So, here we are directing the irreversibility or the unavailable energy which is product of absolute temperature T and the entropy s from the enthalpy. So, Gibbs function is less than enthalpy Helmholtz function is less than the internal energy. So, these are two functions which we shall be using in today's discussion. We will also be using making use of the other equations which were known as the T d s equations which we had derived earlier. We will also be using those T d s equations in some of the analysis today. And so, what we will do in fact, the two T d s equations which we had defined earlier and these two equations for the Helmholtz function and the Gibbs function all the four put together are actually known as the Gibbs equation. So, the two T d s equation and the two equations we had defined today for one is for Helmholtz function, the other is for the Gibbs function all of them put together are classified as the Gibbs equation. So, two of the equations which we had derived earlier one is d u is equal to T d s minus P d v and d h is equal to T d s plus V d p. So, these are two equations which we had at that time defined as the T d s relation. So, the two other Gibbs equations are a is equal to u minus T s and g is equal to h minus T s which is what we had defined today. So, let us differentiate these two Gibbs equations. If you differentiate we get d a is equal to d u minus T d s minus s d t. Similarly, d g is equal to d h minus T d s minus s d t. So, these are now four differential equations one is in terms of the internal energy one is in terms of the enthalpy and one is in terms of the Helmholtz function and the last one is in terms of the Gibbs function. So, these are the four different differential equations which we have which relates a set of properties of a system set of important properties of a system like the temperature, the pressure, the entropy and the volume and so on. So, these properties are related in terms of certain combination properties like enthalpy Helmholtz function and the Gibbs function. So, what we will discuss next is that it is possible for us to express some of these functions or some of these terms in a in the form of another set of terms by what are known as Legendre transformations that is you can make certain you can map or transform a function which is or a property which is a function of a set of properties to another set of properties by using what are known as the Legendre transformations. So, we will use Legendre transforms to basically transform or map a set of parameters of properties from one set to another which basically helps us in certain analysis which we shall be discussing why we are doing this transformation in little detail little later on that once we do this transformation makes the analysis of thermodynamic systems quite easy. So, let us understand what are known as the Legendre transformations. So, we know that a simple compressible system which is basically a system which is completely defined from the state postulate it says that a simple compressible system can be completely defined characterized by two independent and intensive properties. And so and this is the state postulate and a simple compressible system is one which is devoid of any effects of magnetic effects or gravity and so on or and so what we have seen is that we can actually characterize a simple compressible system completely by using either energy which is internal energy u or entropy s and its volume. So, this is an example of course you can also characterize a simple compressible system by another set of properties just for an example we are going to take up a compressible system simple compressible system which we shall characterized by its energy or entropy and the volume which means that u is a function of entropy s and specific volume v which means we have already seen the T D S equation D u which basically relates the internal energy entropy and the volume. So, D u is equal to T D S minus P D v. So, this is the T D S equation which we have defined such that for constant volume we get T is equal to D u by or del u by del s for constant volume that is if you look at the first T D S equation which is D u is equal to T D S minus P D v this equation is can also be rewritten for constant volume specific volume as T is equal to del u by del s where v is for constant v. Similarly, for constant s we get pressure P is equal to del u by del v at constant s. So, what it means is that if you take up if you look at a basic Gibbs equation one of the Gibbs equations and apply certain boundary conditions you can express one of the properties in terms of the other two like temperature here has been expressed in terms of the energy and entropy P is in pressure is expressed in terms of energy and the volume. Similarly, let us look at what happens if you were to map the equation for entropy in the same fashion. So, entropy as you know s is a function of internal energy and the volume. Let us look at another set of T D S the same equation T D S equation in terms of entropy. So, T D S is equal to D u plus P D v which again by applying similar boundary conditions we get 1 by T is equal to del s by del u at constant v that is if v is a constant we get 1 by T as del s divided by del u at constant v. And similarly, the ratio pressure by temperature P by T is equal to del s by del v at constant u that is energy being constant you can express the properties pressure and temperature in terms of the changes in entropy with reference to the specific volume. The partial derivative of entropy with reference to the partial derivative of the specific volume keeping the energy internal energy a constant. So, what we have basically done is that we can map certain parameters which could be like in the example we have discussed this set of parameters were the first example was energy was expressed or can be expressed in terms of entropy and volume. And we can correspondingly map the properties like temperature or pressure in the form of partial derivatives of some of the other properties like energy del u by T was equal to del u by del s at constant v and so on. So, you can actually map certain properties important properties of a system in terms of partial derivatives of another set of properties of the same system. So, Legendre transformation basically helps us in carrying out this kind of transformation which will basically help us in certain analysis. One example I shall give you is that you are very much interested in determining the change in entropy of a particular process. But as we know entropy is not something which you can very easily measure entropy is a parameter which is very difficult to measure directly. There are no devices which can actually measure entropy as such. But if you are able to express entropy in terms of parameters which can be easily measured or experimentally like temperature or pressure or volume. Then it makes analysis a lot simpler because now you have entropy which has been expressed in terms of a set of another parameters like temperature or pressure and they can be measured quite easily. Whereas, entropy on the other hand is not something which you can measure very easily and so Legendre transformation helps us in expressing some such parameters in terms of another set of parameters which we can measure easily. So, that is one of the advantages of carrying out this kind of a transformation. So, if you look at the properties of the or the advantages of Legendre transformation. Legendre transformation basically we know that any fundamental expression or relation we need to express it in terms of proper variable. So, that the relation is complete which is why energy will always feature entropy that is if you are expressing energy internal energy as we did in the example which was expressed in terms of entropy s and volume v. But you would rather have temperature as one of the proper variables because basically because entropy is not something that can be easily measured experimentally. Therefore, it is convenient to construct other related quantities in which entropy is a dependent variable instead of an independent variable because if it is a dependent variable then you can express entropy in terms of other parameters which can be measured like temperature and pressure or volume. Whereas, if it happens to be an independent variable then it means that you will have an expression which relates energy in terms of entropy alone and that is something that you cannot measure easily and that is one of the advantages that you would want to carry out the Legendre transformation. Now, let us also take up an example of let us say the Helmholtz function we have already had an example for the energy in terms of entropy and volume. Let us also look at one example which has Helmholtz function as the example. Now, we have already defined the Helmholtz function also known as in some books you will also see Helmholtz function being defined as Helmholtz free energy. So, Helmholtz free energy was defined as a is equal to u minus t s. Now, if you look at a simple compressible system a is equal to u minus t s can be written as if you differentiate d a is equal to d u minus s d t minus p d v and so on. So, we had already differentiated this a few slides earlier. Now, this can be simplified as d a is equal to minus s d t minus p d v which comes from the T D s equation because d a is equal to d u minus t d s minus s d t d u minus T D s is equal to minus p d v from the first T D s equation. Therefore, d a is equal to minus s d t minus p d v. Again if you are about to apply the transformation here we get this equation can be expressed in a different way as entropy s is equal to minus del a by del t at constant v. Similarly, pressure p is equal to minus del a by del v at constant t. Now, what we have here is that entropy can actually be expressed as in terms of partial derivatives of the Helmholtz function or the Helmholtz free energy divided by with reference to temperature at constant specific volume. So, here we can the state function now is much more amenable to experimental manipulation than the internal energy because energy or entropy on the other hand is something that is quite difficult to measure. And once you can express the state function in terms of parameters which can be easily measured that makes thermodynamic analysis a lot simpler because you now have expressed the state variables in terms of parameters which can be easily measured. So, that is why we would like to carry out such transformations for certain parameters which you would like to measure and therefore, you would like to express it in terms of certain functions or parameters which can be which is much more experimentally amenable. And so, we will now define what we are going to refer to as the thermodynamic potential. So, the derivatives that we get from these the transformations are basically referred to as the thermodynamic potential. So, state functions which we obtain as a means of or as an as an outcome of the transformation of the Legendre transformation are known as the thermodynamic potential. So, thermodynamic potential examples of thermodynamic potentials are enthalpy the entropy is also a thermodynamic potential the Gibbs function or Gibbs free energy and the Helmholtz function or Helmholtz free energy. So, state functions which we obtain as a consequence of the Legendre transformation we already seen what is the Legendre transformation which is the set of transformations which we can do to map a set of functions or parameters in terms of other functions or parameters which are much more easier to measure experimentally. So, as you transform or map these parameters from one set of one set to another we can do that by using the Legendre transformation. So, the state functions state variables that you get as a consequence of this Legendre transformation are referred to as the thermodynamic potential examples being enthalpy Gibbs free energy Helmholtz energy and so on. Now, why are they called thermodynamic potential well they are basically called thermodynamic potential because they look similar to or are analogous to the potential energy which is which we come across very frequently in mechanics. So, because of their similarity that is the similarity of these functions the enthalpy entropy free energy and so on. Because of their similarity to the potential energy which we come across in in mechanics these are known as the thermodynamic potential. Because these are potentials which come up in thermodynamic analysis and that is why they are called a thermodynamic potential because of this similarity. And the other aspect of the potential is that each of these potentials provide a complete and equivalent description of the equilibrium states of the system. Because they are all derived from a fundamental relation we have seen the enthalpy is derived from a fundamental relation it relates basically the internal energy and product p v which is a very fundamental all of these three parameters are fundamental parameters. And so similarly the free energy Gibbs free energy or Helmholtz energy they are all derived from certain fundamental relations. And so each of these terms will provide a complete description of the equilibrium states of the system because they all come from a or they are all derived from a set of fundamental properties. Now, so we can actually map or carry out the Legendre transformation for different sets of parameters or state variables and derive the corresponding thermodynamic potentials for these kind of these set of state variables. So, what I will do is I will I have already explained how to carry out this analysis for some examples like energy or entropy or the Helmholtz energy. So, with that background you should be able to now calculate or carry out this transformation for other set of parameters or state variables. So, let us summarize I would summarize the use of Legendre transformation for different state variables and the corresponding thermodynamic potentials. So, in the table that is shown here we have the state variables on the left hand side first column is the second column is are the thermodynamic potentials. So, we have seen that from this set of state variables u and v that is energy and the specific volume we should be able to derive the thermodynamic potential which is entropy using the Legendre transformation. So, this was one example which I had discussed the other example I had discussed was the Helmholtz function that is you can derive the Helmholtz function which is basically the thermodynamic potential and correspondingly we can carry out the Legendre transformation from the state variables for temperature absolute temperature T and the specific volume v. So, carrying using these two state variables you can carry out the Legendre transformation for the Helmholtz function. Similarly, we can carry out the transformations for Gibbs function in terms of the state variables temperature T and pressure p for Gibbs function which is G is equal to h minus d s and for enthalpy the state variables would be entropy s and the pressure p absolute pressure p where enthalpy is u plus p v. So, this is just to summarize how you can express different potential thermodynamic potentials as a consequence of using a set of state variables through the Legendre transformation. So, thermodynamic potential basically helps us in understanding or equating certain potentials which we also see in mechanics and so similarly in thermodynamic analysis we come across certain potentials which are very similar or analogous to the potential energy of mechanics. Now, that we have understood the Legendre transformation and how you can express certain parameters or functions in terms of a set of different set of parameters which are more amenable to measurements like you can express entropy in terms of other set of parameters and so on. So, we shall now discuss about what are known as the Maxwell equations. Maxwell relations or Maxwell equations are basically derived from the Gibbs equations all the 4 Gibbs equations put together. You can carry out the Legendre transformation and derive the Maxwell equation from all these 4 Gibbs equations. So, Maxwell relations basically relate some of the basic properties of a simple compressible system like the pressure, volume, temperature and entropy and we basically get partial derivatives of these properties in terms of the others and this is basically applied for simple compressible systems. But as we will see little later on that we have we can also derive such equations for non-simple systems which involve electric or magnetic force etcetera and get the corresponding Maxwell's relation for those systems. But what we will discuss today is only for a simple compressible system where we express these basic properties like temperature, pressure, specific volume, entropy etcetera in terms of partial derivatives. So, Maxwell relations are basically equations that relate the partial derivative of properties like pressure, specific volume, temperature and entropy of a simple compressible system to each other that is you would have partial derivatives of pressure in terms of temperature and so on. Now, we will make use of the fact that all of these properties are basically because they are thermodynamic properties they are exact differentials. So, these equations will be derived by using the exactness of the differentials of these thermodynamic properties. So, Maxwell relation basically can be obtained by applying these Legendre transformations to the 4 Gibbs equations and what were those 4 Gibbs equations we had already discussed about the T D S equations. There are 2 T D S equations one is in terms of energy u and the other is in terms of enthalpy H. In addition to that the other 2 Gibbs equations one is for the Helmholtz function A and the other is for the Gibbs function G. So, the Gibbs equation can actually be simplified in by using the T D S equations which we get from which we get D A is equal to minus S D T minus P D V. So, that is the third Gibbs equation and D G is equal to minus S D T plus P D V plus D G is equal to minus S D T plus V D P. So, now we have a set of 4 equations all of them put together are known as the Gibbs equation and all these equations if you see carefully they are all of this form which of the general form which is written here as D Z is equal to M D X plus N D Y. So, this is like a set of exact differentials and if you recall from mathematics which you might have studied earlier on such equations can actually be written in terms of the partial derivatives of these variables here. So, D Z is equal to M D X plus N D Y can be expressed as del m by del y at constant x is equal to del n by del x at y that is these functions that you see here in terms of m and n can be expressed as partial derivatives in this fashion. So, if you have an equation of this form D Z is equal to M D X plus N D Y then it also follows that del m by del y at x is equal to del n by del x at y. Now, since all these equations that is the 4 Gibbs equations are in terms of properties of a system like energy enthalpy the Helmholtz function and the Gibbs function which all of which are properties of a system they should have exact differentials which means all these 4 equations which we had discussed one is in terms of an energy u second is in terms of enthalpy h then the Helmholtz function a and the Gibbs function g all these 4 equations only deal with properties of a system. Therefore, all of these equations should have exact differentials which means all these equations can be expressed in a fashion which was just discussed for a general equation D Z is equal to M D X plus N D Y where m del m del y at x is equal to del n by del x del n by del y at x. So, if we were to express all those 4 equations 4 Gibbs equations and apply the exact differentials for them because all these equations are in terms of properties of a system and if you were to do that then we get a set of 4 relations which are basically known as the Maxwell equations. So, if you take the equations 1 by 1 that is the first T D S equation apply the exact differential property for that second T D S equation and so on then we get the Maxwell relations which are shown here. So, applying this property of or identity of exact differentials to the Gibbs equation then we get from the first and from all the 4 Gibbs equations together we get del T by del V at S is equal to minus del P by del S at V. The second Maxwell relation is del T by del P at S is equal to del V by del S at P. Third Maxwell equation or Maxwell relation is del S by del V at T is equal to del P by del T at V and the last Maxwell relation is del S by del P at T is equal to minus del V by del T at P. So, all these 4 equations are basically known as the Maxwell relations which are basically derived from the Gibbs equations and by applying the exact differentials to all these Gibbs equations. Now, what is the significance of Maxwell relation now that we have defined these 4 equations in terms which are basically partial derivatives of the properties expressed in terms of one another. What is the significance of these Maxwell relations basically Maxwell relations are important thermodynamic relations because they provide a means of measuring changes in entropy using variables which can be easily measured like pressure, volume, temperature etcetera. And the equations that we just discussed or the Maxwell relation which we had discussed is basically limited to simple compressible systems, but we should be able to derive similar relations for non-simple systems which involve electric, magnetic and other effects and so on. And so Maxwell relations which we discussed though were restricted to simple compressible systems which basically do not have effects of electric, magnetic fields and so on. Such equations are also derived for equation or for systems which involve non-simple effects like electric, magnetic fields. So, what we had discussed were so far were firstly the Gibbs and Helmholtz function then we discussed about the Legendre transformation which helps us in expressing one function in terms of set of another functions which can be easily measured and so on. And Maxwell relation is basically a consequence of these transformation applied to the Gibbs equations the 4 Gibbs equations. And what we shall discuss next are the equations of states and we will start our discussion with discussing about the ideal gas equation of state. And then we will see how we can modify this ideal gas equation of state to include real gas effects. Now ideal gas equation of state is something which you are already familiar with and we have used this also in some of the numerical calculations during our tutorials in some of the earlier lectures where we had assumed that air is an ideal gas for the range of temperatures for which we were carrying out the calculations. So, we had defined ideal gas equation there as p v is equal to R T. So, that is the basic ideal gas equation of state which expresses 2 properties which basically relates 3 properties of the ideal gas pressure, volume and temperature. And then there is a constant which is known as the gas constant. Now any equation which basically so if you look at equation of state you can actually have any equation which relates the pressure, temperature and specific volume of a substance is basically known as an equation of state. Now the simplest and the best known equation of state which we are aware of so far is the ideal gas equation of state which is p v is equal to R T. Now in this equation the pressure p is absolute pressure, T is absolute temperature, v is a specific volume and R is the gas constant for the particular gas we are interested in for air we can calculate gas constant by dividing the universal gas constant by the molecular weight of that particular gas. Now the equation of state we had just discussed is meant for an ideal gas. Now real gases can deviate substantially from this ideal gas nature depending upon the pressure and temperature that we are dealing with. So, depending upon that the actual gas behavior can be substantially different from that of an ideal gas. So, is there a way or is there a method of accounting for this deviation of actual gas from the ideal gas behavior. So, what we will do now is to define one parameter which can partly account for this deviation and this factor we shall define as what is known as the compressibility factor. And so compressibility factor is basically defined as the ratio p v by R T, Z is defined as p v by R T and this is basically known as the compressibility factor. This is primarily meant to account for the deviation of real gas effects from the ideal gas behavior. So, for an ideal gas obviously Z is will be equal to 1 and for a real gas Z may be greater than 1 or it may be less than 1 depending upon the pressure and temperature at which it is operating. So, the further away Z is from unity whether it is less than 1 or greater than 1 the more is the deviation of the gas from the ideal gas behavior that is if Z is much less than 1 or Z is much greater than 1. It means that that particular gas is deviating from the ideal gas behavior by that much amount, because for an ideal gas Z is equal to 1. So, compressibility factor is one parameter which can probably account for some of these real gas effects. And this is basically because real gases behave differently at different pressures and temperature and so which means that how do you account for this deviation if you have just one parameter. In this case what we do is that we normalize the pressures and temperatures with reference to the critical pressure and temperature of that particular gas. And so if you let us say normalize pressure with reference to the critical pressure we get a normalized pressure which we shall now define as what is known as the reduced pressure. Similarly, temperature divided by temperature at the critical state is defined as the reduced temperature. So, when we reduce the temperatures and pressures that is when we normalize pressures and temperatures then the behavior of different gases because now they have all been normalized is basically the same is more or less the same. I will show you one example where we will see that though they are not exactly the same they can the behavior of different gases can be assumed to be falling the same trend. So, in order to account for the different behavior of the gases at different pressures and temperatures we normalize them. So, the reduced pressure P r is equal to P divided by critical pressure and T r is equal to T divided by the critical temperature. Now, if you do that basically what happens is that the compressibility factor or the z factor will now be approximately the same for all the gases at the same reduced temperatures and pressures. And this is basically known as the principle of corresponding states that is once you normalize the pressure and temperature of a particular gas with reference to the critical values then the compressibility factor for all the gases for the same set of reduced temperatures and pressures will be approximately the same this is basically the principle of corresponding state. So, we can now express the compressibility factor in terms of reduced pressures and reduced temperatures for a variety of gases and then we can what we get is the generalized compressibility chart. So, if you look at the generalized compressibility chart which are basically meant to be used for all set of gases all gases and then we shall be able to make certain conclusions based on the generalized compressibility chart. So, I have one example here of a compressibility chart which is plotted in terms of set of gases for methane, ethylene, methane and so on the several gases which have been shown here. On the y axis we have the compressibility factor z which is p v by r t x axis is reduced pressure and these graphs have been plotted for different reduced temperatures. So, what we can see is temperature ratio temperature reduced temperature varying from 1 all the way up to 2 and so on. You will find in certain books these charts given for many more reduced temperatures all the way up to 5 or 6 and the, but the basic message is the same that we shall discuss now. What you see here is that as the reduced pressures are lower as it approaches 0 then compressibility factor approaches 1 which means that as for very low pressures irrespective of what the temperature ratio is the compressibility factor approaches 1 which means all the gases will approach ideal gas behavior. And for certain reduced pressures you can say that you can see that the behavior and temperature ratios the behavior is substantially different from that of ideal gas behavior. For example, for a reduced pressure of 1 and reduced temperature of 1 the compressibility factor is 0.2 which means that it is substantially different from 1 it is substantially away from 1 which means that these the gases at these reduced pressures and temperatures will deviate substantially from ideal gas behavior. And so based on the general compressibility chart or generalized compressibility chart we can derive the following conclusions. At very low pressures which we have seen where P r is much much less than 1 or is that as it approaches 1 gas behavior is very similar to that of the ideal gas and regardless of temperature because all the temperature curves were merging towards z equal to 1. At high temperatures T r greater than 2 ideal gas behavior can be assumed regardless of the pressure which you can see here in this diagram that for as you increase the reduced temperature at 2 or beyond that it is the curve is more or less flat which means that regardless of pressure the gas behavior is like that of an ideal gas which occurs for reduced temperatures of 2 and beyond that. And the deviation of a gas from the ideal gas behavior is highest in the vicinity of the critical point that is around critical point when let us say pressure reduce pressure is 1 and the reduced temperature is 1 that is at the critical point temperatures and pressures the deviation of the gas from the ideal gas behavior is the highest which also is seen clearly from this generalized compressibility chart. So, these are some observations which can may which we can derive when we look at the compressibility chart that based on the compressibility chart you can actually see how an actual gas is with reference to an ideal gas in terms of the reduced temperatures and pressures and finally expressed in terms of the compressibility factor. Now what we had discussed now was the equation of state as applied for an ideal gas now there are there are certain limitations as we have already seen that it is only for a certain set of temperatures and pressure ranges that the behavior of a certain gas can be approximated to that of an ideal gas, but what about the other temperatures and temperatures pressures. So, the ideal gas equation of state has lot of limitations and therefore, over the years they have been several other forms or modifications of the ideal gas equation of state which are meant to account for some of these deficiencies of the equation of state. So, there has several equations which have been proposed some of them in fact there are several of them one of the earliest one is known as the van der Waals equation it is the one of the earliest equation for a long time that was used then the B T Bridgeman and equation is one of the most popular equations and the one of the most recent and most accurate equation is the Benedict Ben Rubin equation. We will only discuss the first two equations in little detail because the third equation is very complicated and I would limit my discussion to the van der Waals equation and the B T Bridgeman equation because these are the ones which have been used popularly. So, van der Waals equation basically includes the effect of intermolecular attractions which is expressed in terms of a by v square where a is a constant and also the volume occupied by the gas themselves that is there is certain volume associated with the gas molecules. So, van der Waals equation takes into account these two effects one is the intermolecular attraction forces and also the volume of the gas. So, the van der Waals equation of state is p plus a by v square and v minus b where b is the volume of the molecules themselves. So, that has been reduced from the volume itself. So, p plus a by v square into v minus b is equal to R T. So, here these two constants need to be determined from the critical point data of that particular gas. So, this is the van der Waals equation and the second equation which is more popularly used is the B T Bridgeman equation and this is expressed in terms of five experimentally determined constants and it is basically the equation is p is equal to R U T by v bar square into 1 minus c by v by v bar t cube multiplied by v bar plus b minus a by v bar square where a and b are expressed again in terms of constants. And so you can see that this equation is fairly quite complicated as compared to the simple ideal gas equation of state. However, this equation has been popularly used in getting a much more better estimate of real gas behavior as compared to the ideal gas assumption. And basically the equation of state can be expressed as a series. So, p v is equal to R T can be also expressed as p is equal to R T by v plus a T by v square plus b T by v cube and so on. So, this is a series it can basically be expressed as a series and so this and similar other equations are basically referred to as the varial equations of state. And these coefficients that you get here a of t b of t and so on are simply functions of temperatures and these are referred to as the varial coefficients. So, there are several other sets of equations which are basically expressing the ideal gas equation in terms of a series and these are known as the varial equations of state. And so these coefficients basically need to be determined either experimentally or theoretically from statistical mechanics. And as the pressure approaches 0 you can see that the varial coefficients will vanish and the equation will reduce to that of an ideal gas equation of state. So, it is only accounting for as you deviate from the ideal gas behavior which could be at different higher pressures and so on. Then some of these equations will actually show up lot of differences as compared to the ideal gas equation of state. And so similarly in literature you would find that there are many more forms of the equation of state. And many of them are very complicated and involve lot of other parameters and constants and coefficients which need to be determined experimentally. And some of these are used in modeling of real gas effects especially when they involve certain combustion of gases etcetera. The combustion products no longer can be treated as an ideal gas. And so you need set of equations which can model these effects in an accurate manner. And it is in these applications that many of these equations are used. But for simple thermodynamic analysis it is always a practice to assume air or even the combustion products as an ideal gas and carry out analysis based on that assumption. Now, we will now discuss another effect which is different from what we have already been discussing about. It is basically to do with flow passing through a constriction or a restricted passage or during a throttling process. Now, if you recall during our discussion on steady flow energy equation applied to such processes we had derived that or we had understood that basically for throttling processes the enthalpy remains a constant. At that point I also made a statement that during a throttling process temperature may either be constant it may increase or decrease and it depends upon certain set of parameters. So, we will now see what is it that causes change in temperature it could either decrease or increase depending upon the pressure. So, we will now see how we can estimate or determine changes in temperature during a throttling process. Now, during a throttling process there is definitely a pressure drop associated with flow where it passes through devices like valves or capillary tubes and so on. So, enthalpy of a fluid remains a constant which is basically an outcome of the energy equation. Now, the temperature may remain same it may increase or it could decrease depending upon the throttling process. So, the behavior of fluids during such flows is described by what is known as the Joules-Thompson coefficient. So, let us look at what is been by the Joules coefficient or Joules-Thompson coefficient. Joules-Thompson coefficient mu is basically equal to partial derivative of temperature with pressure for constant enthalpy. So, del T by del P at constant H is Joules-Thompson coefficient. So, it follows from this that if mu is less than 0 the temperature will increase. If mu is equal to 0 temperature remains constant and if mu is greater than 0 the temperature decreases. So, if you can calculate this gradient of temperature with pressure during a constant enthalpy process then depending upon the gradient you can determine whether temperature will increase remain constant or decrease. So, let us look at one example here we have flow passing through a porous plug here the initial condition is P 1 and T 1 which is fixed and as you change the porosity the downstream conditions can change which means if you start at the initial state here exit states which are shown here by these black dots can be defined or can be vary depending upon the porosity here. So, you may end up getting different exit states on the constant enthalpy line. Now, it means that there is certain point on the enthalpy equal to constant line where the slope is 0 or where you have Joules-Thompson coefficient equal to 0. So, the line that passes through all these points is known as the inversion line and the temperature at a point where a constant enthalpy line intersects the inversion line is called the inversion temperature. So, the slopes of h equal to constant line are negative at states to the right of the inversion line and positive to the left of the inversion line. So, that will be clear from this sketch I have shown here illustratively that on this constant enthalpy lines plotted on the temperature pressure plot all these lines are enthalpy equal to constant. So, at the point where slope is equal to 0 if you join all those points you get the inversion line and this temperature is known as the inversion temperature. So, on left of this inversion line you have mu is greater than 0 which means temperature decreases in this zone and if you are operating in the other zone where mu is less than 0 the temperature should increase which means that as you start throttling from here if you are operating on the positive slope line then you end up getting reduced temperature and if you end up operating on the right hand side of the inversion line the temperature after the throttling process will increase. So, on a constant enthalpy line it is necessary if your aim is to reduce temperature then it is necessary that you operate on the right hand side where the slope is positive and if you are operating on the right hand side you end up getting an increase in temperature. So, this is basically the significance of the Joule-Thompson effect or Joule-Thompson coefficient which basically helps us in understanding that as you throttle it is not necessary that temperature will always decrease because throttling devices are very commonly used in refrigeration cycles and air conditioning systems and so on. And so if you need throttling to be effective you need to make sure that on the constant enthalpy line you are operating towards the left of the inversion line where the Joule-Thompson coefficient is greater than 0 which means that you will end up getting reduced temperature but if you are operating on the right hand side the temperature will increase. So, that was about the Joule-Thompson coefficient which describes throttling process in much more detail. So, let us recap what we had discussed in today's lecture we are coming towards the end of this lecture and in today's lecture we had discussed about two new functions the Helmholtz and the Gibbs functions which are combination properties like enthalpy. We had discussed about Legendre transformations which help us in mapping set of parameters to another set of parameters then as a consequence of Legendre transformations we discussed about thermodynamic potentials and then the Maxwell relations which form the basic tool for analysis of different thermodynamic systems. We discussed about the ideal gas equation of state and the real gas effects and how to account for real gas effects in terms of the compressibility factor and also we discussed about the other equations of state like the van der Waals equation and so on. And subsequently we discussed about the Joule Thomson coefficient which is basically applicable for throttling processes. So, that brings us to the end of this lecture and in the next lecture what we shall do is we will have a tutorial session we shall be solving numerical problems related to gas power cycles from auto diesel and dual cycles. We will also be solving some problem from Brayton cycle and variance of Brayton cycle like Brayton cycle with regeneration reheating and so on. And then we will be discussing about the vapor power cycle basically the Rankine cycle and possibly we will also solve some problems based on today's lecture that is the thermodynamic property relation. So, we shall take up these during the next lecture which would be lecture 20.