 Good morning all of you and welcome to the second week, the second half day six. Let me start the lecture now, today we have the whole day. So, we will be discussing property relations, but we will also be solving problems on property relations. This is a direct consequence of the first and second law of thermodynamics, particularly the second law of thermodynamics. You can say it is the play on the properties of the property called entropy and many of your ideas regarding entropy etcetera are likely to be more clear when we do this. Now, we are going to look at property relations and we can say property relations or relations between parties of a simple compressible system. Since we are going to have a closed system for our study, we often will work with specific properties. That means, we might as well consider the system to consist of a unit mass of content. The tools which we will use are first law of thermodynamics, second law of thermodynamics, zeroth law will be hidden in the background. We do not have to specifically invoke it, but since we are looking at properties everything is in equilibrium including thermal equilibrium. So, zeroth law is being taken care of. We will look at the what is known as the basic property relation and we will also have this is thermodynamics, but we will also use calculus of exact differentials and relations pertaining to partial differentiation. This part of thermodynamics, I find that number of places, number of universities they either neglect this or it is not simply included in the course content, but we find that it leads including this leads to development of a large number of tools for handling thermodynamic properties. And this allows the students to appreciate for example, questions have already been reasoned. How are steam tables prepared? How many measurements are needed and such stuff? Using this we will be able to show that for a system containing a simple compressible fluid or a fluid system. All that you need to have is the pressure, volume, temperature, data all over the state space and the value of say C p as a function of temperature at one pressure or the value of C v as a function of temperature at one pressure. That means, to measure or to determine steam tables the experimental data needed will be the density of steam or water and steam as a function of pressure and temperature at all pressures and temperature or over the range of pressure and temperature that you are interested in. And then select one pressure say for example, out of that say one atmosphere or one bar or five bar select one pressure and over the range of temperature of interest measure C p as a function of temperature at that fixed pressure. Just this much information is sufficient to determine all other properties like thermal energy, entropy, enthalpy, specific heat at constant pressure, specific heat at constant volume at all pressures and temperatures in that range of measurement. Now, remember in our second law we derived the basic property relation and coming to energy functions that means, something which is related to heat transfer or work transfer. We notice that first law defines energy, but we will not consider energy we will consider the internal energy we will we want only relationship between thermodynamic properties, the relationship between energy and other properties like gravitational potential kinetic that is already known. So, we will say the first law defines you the thermal internal energy that is going to be an important energy function for us. Then second law defines S that is the second relation and then since we have a simple compressible system at rest remember the way we have determined entropy. We said that for a simple compressible system entropy d s is d u plus p d v everything divided by t. We turn it around and write it as t d s equals d u plus p d v. Now, from here the quantitative part begins. Now, let us use the first and second law in its quantitative form. The first law says that d q equals d u plus d w. Notice that this is the property relation in which only properties are present temperature entropy internal energy pressure and volume. This is the statement of the first law where the only property present is d u d q and d w are interactions. And similarly, we have the second law which says the entropy relation t d s is greater than or equal to d q and combine this and this and we will have t d s is greater than or equal to d u plus this is second law. You can say this is second law come first law. In fact, this equation these two equations we have used earlier to show that d w other than p d v work must be the negative work that you must be able to you can do it on the system, but you cannot extract a non p d v or non expansion work out of a simple compressible system. Now, let us see where it leads to particularly we are interested in this property relation because you know almost all important thermodynamic variables are there we have temperature we have entropy we have energy we have pressure we have volume, but we also know that for a simple compressible system because there is only one two way mode of work only two of these can be independent which two that is for us to consider by convenience. We can consider p and v to be independent we can consider t and s to be independent or if you want we can consider u and v to be independent or p and t to be independent the choice is left to us and say suppose we take p and v to be independent. Then we have to consider temperature entropy and internal energy as dependent properties dependent on p and v. So, we can write p d v as it is, but then d u we have to write d u of p and v d s of p and v t of p and v and then we can determine and derive using standard calculus calculus of exact differentials because all these property differentials are exact except that in all this d u and d w these underlined differentials are not exact all other differentials are exact. So, exact differentials they are calculus and related relations between partial derivatives this we will use, but before we do that let us complete our study of energy functions. You can say entropy is also some sort of an energy function, but does not have the units of energy. The first energy function we are going to look at is the internal energy or thermal energy. The first two items of this are some sort of a revision because we have already done it see if you look at the first law d q is d u plus d w which is equal to d u plus p d v plus d w other. Now, consider a constant volume process and what we get at constant volume is d u at constant volume plus d w other. This indicates that for a process which is constant volume the change in internal energy will represent or will be equal to the heat transfer provided the non p d v work component be 0. Second warming up we already have defined for convenience another energy function called enthalpy. Enthalpy is defined as u plus p v. Now, if you take a differential you will get d h equals d u plus p d v plus d p. Now, remember our d q is d u plus d w and I will write d w as d u plus p d v the expansion component of work plus d w other. Now, I will write this in terms of this d u plus p d v I will replace by d h. So, this I will get d h there is a minus v d p because d u plus p d v is d h minus v d p plus d w other and that means now look at this d p. If I consider a constant pressure process we get d q at constant pressure is d h at constant pressure at constant pressure this term will be 0 plus d w other. So, this tells us something which we have already realized that the heat transfer during an isobaric that is constant pressure process will be the change in enthalpy of the system provided there is no other component of work simple enough. But now let us define two more functions these are defined because you will notice that u has the same dimensions as p v the dimensions of u which are dimensions of work or dimensions of energy are also the dimensions of p v and they are also the dimensions of T s because we know that dimension of s is dimension of heat divided by dimension of temperature. So, dimension of T s will be the dimension of the heat interaction which is also the dimension of work interaction which is the dimension of energy. So, if u plus p v has been defined as a useful property h maybe you can add u plus T s or u minus T s or u plus T s plus p v or u minus T s plus p v all these combinations. It turns out that of the combinations this combination u minus T s and u minus T s plus p v which is h minus T s are useful combinations they allow us to understand thermodynamics in a different way and there are applications for you know some chemical processes some electrochemical processes which can be understood well by using this and these combinations are so common that just the way we have said that look u plus p v is a common combination and we give it a name enthalpy h u minus T s is also a very useful combination and is given the name Helmholtz function. Similarly, this combination u minus T s plus p v which is also equal to h minus T s is given the name G and it is known as the Gibbs function named after the scientist Helmholtz and Gibbs significant contributors to physical chemistry and to thermodynamics and just the way we have treated energy and enthalpy, but remember in energy and enthalpy we only had to use first law because entropy was not present. So, we had to use first law and that was sufficient. Now, let us try analyzing first the Helmholtz function and then the Gibbs function in a similar fashion. However, since entropy is present apart from first law you will also have to use second law a is defined as u minus T s. So, you take a differential d a equals d u minus T d s minus s d t. Now, d u is d q minus d w. So, write this as d q minus d w minus T d s minus s d t. Now, I will rewrite this as d a plus d w plus s d t d a plus d w plus s d t on one side and I will get here d q minus T d s. Remember here I have replaced d u by d q minus d w. So, where I have used first law. Now, remember that we have said that d s is greater than or equal to d q that is the second law. So, T d s will be greater than or equal to d q or d q will always be less than or equal to T d s giving us this quantity to be always less than or equal to 0. So, here this transformation from this to this is first law and this being less than 0 is second law. So, both the laws have been used and now if this is less than or equal to 0. Let me put it like this d w will be less than or equal to minus d a minus s d t and now consider an isothermal process. So, that this d t can be made put equal to 0 and you will get the work done by a system under an isothermal process is less than or equal to the reduction in the Helmholtz function and look at this sign. Remember in mechanics we have quite often seen that the work done or work which is extracted is reduction in potential energy. So, this is on the right hand side you have something like a reduction in potential on the left hand side is the work done. So, we can say that a is like a potential or a is similar to a potential reduction in a represents maximum work because this is less than or equal to for an isothermal process that is something which we should note. So, that is why the a which is given or is named after Helmholtz we may call it Helmholtz function, but sometimes it is known as Helmholtz potential or Helmholtz free energy. The word free energy is sort of misnomer there is nothing free, but Helmholtz potential is a good enough name instead of Helmholtz function. Now, in a similar way let us look at the Gibbs function. The Gibbs function is defined as u plus T v minus T s which turns out to be equal to h minus T s same treatment differentiate D g is D u plus P d v minus T d s minus S d t. Now, D u is D q minus D w. So, this will be D q minus I will do this thing D w minus P d v I will take this P d v inside plus V d p minus T d s minus S d t. So, I have expanded D u and I have used first law and now I notice that this D w minus P d v is D w other and now what we will do is like the earlier one this D q minus T d s those two terms we will keep on the right hand side everything else we will take on the left hand side. So, on the left hand side we will have D g minus actually here plus because minus D w minus P d v I will put it as D w other and then on this side I will have minus V d p plus S d t the term which will remain on the right hand side will now be simply D q minus T d s which we know from the second law of thermodynamics is less than or equal to 0 this is second law. If first law was used here the second law has been used here. So, if the right hand side must be less than or equal to 0 again using that inequality we have now. So, D w other must be less than or equal to minus D g plus V d p minus S d t just transpose these terms and now consider a constant a pressure like constant pressure and constant temperature a process in which there is either a phase change or a chemical reaction chemical or electrochemical something and in that case we will get D w that is the work done other than P d v work bring a constant pressure come constant temperature that is isobaric isothermal process is less than or equal to minus D g for that same isothermal isobaric process. So, what does this tell us this tells us that g is like a potential reduction in which represents maximum non expansion work for a constant P constant T process and that is why just the way the Helmholtz function is sometimes named Helmholtz potential the Gibbs function is often named Gibbs potential or Gibbs free energy and those of you who might have studied the some physical chemistry in detail or those of you who have some chemical engineering background or those of you who have studied reactions of interest to mechanical engineers and that is typically our combustion reaction you would have noticed that it is the Gibbs energy difference which will tell us a lot many things about chemical reactions. Now, after having seen the basic ideas of these four functions we will come to the part where we will now start deriving a large number of property differences for that we will have to look at four functions four energy functions u h a and g these will in turn be considered as functions of any two properties. Although we have a choice of any two properties the question we will ask is for each of these four is there a natural pair of properties natural or most convenient there is nothing natural about it actually one should use the word most convenient pair of properties pair because we are looking at a simple system. So, two properties would define the state of a closed system of such a closed system. So, we will consider any one of these properties phi as a function of some two properties x y the first question is which are likely to be or which is the pair which is the most convenient to consider as an independent pair of variables for these and then we will also notice that since phi x y are properties their differentials are exact differentials that means integration of the differential of any of the properties along any curve would be independent of the path taken by that curve will depend only on the initial and final state and when we do this we will use some ideas from differential calculus and partial derivatives. For example, when phi is a function of x and y with some minor requirement as continuity and continuous derivatives we can write the differential of phi as d phi is partial of phi with respect to x constant y dx plus partial of phi with respect to y at constant x dy you will find this relation and all its properties in any good book on calculus my favorite is Thomas or the current editions are by Thomas and Finney, but I am sure there are many many other books will do this, but out there they will not be writing this subscript y and x because in mathematics when you consider partial differentiation when you say x and y partial differentiation with respect to x would automatically mean that y is maintained constant, but in thermodynamics it is not so when I say the variation of temperature with pressure the question that arises is what is it that is being maintained constant it could be variation of temperature with pressure when volume is maintained constant in which case I am considering this will be of useful if I am considering temperature as a function of p and v that means I have selected p and v as my pair of independent variables or it could even be variation of temperature with pressure at constant entropy which would mean that I am considering pressure and entropy temperature as a function of pressure and entropy and pressure and entropy as my independent variables so that is why just writing partial of t with respect to p will leave this question open and that is why we will be very particular in writing this the variable or variables which are maintained constant as subscripts for that partial derivative, but our main thing is this if phi is a continuous function then the differential of phi can be written down like this and we use the inverse of this that means if we come across a relation if we come across a relation where d phi is m d x plus n d y where in the thermodynamic language phi x y are properties then we must have m equal to partial of phi with respect to x at constant y and n equal to partial of phi with respect to y at constant x and not only that under some minor conditions of continuity etcetera we know that when you take the second cross derivative of a function say suppose you take the second derivative of phi it does not matter whether you first do it with y and then do it with x like this this is equal to the second derivative second cross derivative when you first differentiate with x and then differentiate with y and because of this now remember this can be written down as partial with respect to x at constant y of d phi by d y I will write it as d first let me take the first derivative here is partial of phi with respect to y at constant x second derivative here is partial of this with respect to x at constant y and this will be equal to the first derivative here is partial of phi with respect to x at constant y and the second differentiation is partial of this with respect to y at constant x but what is this partial of phi with respect to y at constant x is n so this will be partial of n with respect to x at constant y that should be equal to partial of m with respect to y at constant x note this link this link we will be using quite often certain surprising and major relations will be derived using this. So, for us to remember are these important from partial differentiation the first important relation is this the second important relation is this and the third important relation is this second and third go together because second defines what is n and m now let us begin our exploration of our properties first let us start our exploration with internal energy and we again start with our basic property relation T ds is du plus p dv and just because we do not want to write big capital letters we will consider 1 kg of a mass of the system to be 1 kg and hence we will write hence forth all our property relations using specific value does not matter we can continue with capital letters and extensive properties but in textbooks we will find almost everything done in terms of intensive properties but they are equivalent. So, from this property relation we will notice that the differential du can be written down as T ds minus p dv this is nothing but our basic property relation transposed but now look at this is exactly like our remember u s v are properties. So, compare this with our relation on the previous page we have here du equals T ds plus minus p into dv and here we have a property d phi equals m dx plus n dy. So, we can write m and n as partial derivatives of u. So, this gives us T must be partial of u with respect to s at constant v and minus p or I will write p must equal minus partial of u with respect to. Now, notice what happens here we have partial derivative of internal energy with respect to entropy at constant volume related to another property temperature. Similarly, partial derivative of u with respect to volume at constant entropy is related to a sort of unrelated variable pressure but before that you notice from here what we see the question we asked sometime ago was is there a natural pair or a most convenient pair when we differentiate a property. And here we will notice that if you consider u as the property since its differential using the property relation can be directly in terms of ds and dv s and v turn out to be the natural variable. So, the most convenient variables for our property the thermal internal energy u there is something more important to this remember when we started our study of thermodynamics we said that any system is defined by defining its boundaries. So, the first property we should define for any system is its volume any thermodynamic system should have a property known as volume the geometric volume a primitive property. But then the moment we say it is a thermodynamic system and apply first law and second law to it first law will say that it will have an internal energy u second law will say it has a property called s. So, in a way you can say that the for a simple compressible system or for a simple system compressible or otherwise the three inherent properties from the thermodynamic point of view would be v the extent u the energy and s the entropy. So, this way you can say that this is the basic thermodynamic relation for any simple system and for such a system then you can say that this relation is the thermodynamic definition of temperature and this relation is the thermodynamic definition of pressure related purely to the three basic thermodynamic or absolutely fundamental thermodynamic properties volume energy and entropy. So, after this discussion now we come to the next level remember t now happens to be the first partial derivative of u with respect to s volume maintained constant and minus p is the first partial derivative of u with respect to v when entropy is maintained constant. It is now time for us to apply the third relation and see what we get now what we have to do is take this first derivative represented by t differentiate it with the second derivative when the first derivative is maintained constant partial of t with respect to v at constant s will equal here the minus sign is with p. So, I will get minus partial of p with respect to v sorry with respect to s at constant you will find a relation which we could never have imagined relation between t v and s relation between p s and v let me write this equation as m 1 will come to this later. Now, let us treat in this similar fashion the next energy function enthalpy we have will work in terms of specific enthalpy this is u plus p v and if you expand you will get d h is d u plus p d v plus v d p this is definition of h this is expansion in differential, but now from our basic property relation you will notice that d u plus p d v is v d p sorry d u plus p d v is t d s. So, this become t d s plus v d p here d u plus p d v is replaced by t d s using the basic property relation. So, this tells us that s and p are the most convenient variables for enthalpy and now look at it d h is t d s v d p. So, let us use the calculus of partial derivatives and exact differential and you will get one relation temperature equal to partial of h with respect to s at constant p and specific volume equal to partial of h with respect to p and cross derivative partial of t with respect to p at constant s must equal partial of v with respect to s at constant. Let me call this equation m 2. Notice that absolutely funny relation between derivative of temperature with respect to pressure at constant entropy and derivative of volume with respect to entropy at constant pressure. Something like this which we were always saying that thermodynamics puts restrictions on properties or relates properties to each other in some way, but it will never dictate what a value of a particular property will be. It will never say this should be the volume or this should be the entropy or this should be the pressure. It will say that if the volume changes with entropy at constant pressure this way temperature should vary with pressure at constant entropy in the same way at that point that is all it says. So, it just relates variation in properties to each other. So, this way we have taken care of and derived some important relations using the internal energy and the entropy as basic properties.