 Now what we will look at is the idea of an equation of state. I am writing it in singular but actually I should write it in plural, equations of state. Now we know that let us consider a system and let us say that the complexity of the system is such that there are n variables or n independent variables which define the state of the system. So there are n properties which define the state. But now there are for example if I take a gas, two properties define its state. It could be pressure volume, now that we are comfortable with temperature, it could be pressure temperature, it could be volume temperature. But then we are all suppose we decide on pressure and volume then we must also know the temperature, it is needed for some calculation. So P and V fixes the state, T depends on the value of P and V. What is the relation? Tomorrow you have internal energy, you have enthalpy, you have entropy, they will all depend on the two variables which are selected. What is the relation? All these relations are known as equations of state. So for example n independent properties and let us say for example a simple system and let us say simply phi 1, phi 2 are the selected pair of property for a system of fixed mass. But other properties for example phi 3, phi 4, whatever you have thermodynamics does not restrict the number of properties in which you will take interest. They will all have to be considered as functions of these, phi 4 will also be a function of phi 1, phi 2. All these relations are known as equations of state. However there is one equation of state which has a special status and that is for a fluid the relation between P, V and T. Example, you take a gas or a fluid or in general any simple compressible system. Two properties define the state. This is what is dictated by thermodynamics but thermodynamics would not say which two properties. So you may select say P, V in which case one equation of state will provide T as a function of P, V. Another equation of state will provide U as a function of P, V. Another equation of state will provide S as a function of P, V and so on. Or you can select P, T in which case one equation of state will provide V, V as a function of for volume as a function of pressure and temperature. Then U as a function of pressure and temperature, S as a function of pressure and temperature and so on. You can go on writing these pairs and all these functions are known as equations of state, all these relations. But when you say the equation of state it usually means a relation between temperature, pressure and volume and that is what we mean by default. So when we talk of an equation of state of a gas you will say P into V minus V is RT, something like that. Or P plus A by V squared into V minus V is RT. Or for a simple ideal gas PV equals RT. We have not yet said what an ideal gas equation of state is but we also know that U also is a function of temperature for an ideal gas. And we know that U is CV into T minus T naught or U minus U naught is CV into T minus T naught where T naught is the reference temperature and U naught is the value of the internal energy at that reference temperature. That also is an equation of state. But when we call the equation of state it means by default the PVT relation. This is just information nothing more than that. Now let us come to the equation of state of ideal gas. Actually the equation of state of an ideal gas is nothing but the Kelvin defining relationship just turned around arbitrary. We have the temperature of an ideal gas divided by the temperature of the same gas at the reference state be equal to PV divided by PV naught. This is the defining relation for Kelvin. Turn this around suppose my system is an ideal gas itself then what will be the relation between Tp and V? It will be this itself. But we usually write it down in this form PV equals PV naught divided by T naught into T. Then we write it as mass of that gas into PV naught divided by T naught into T. And this particular value which is specific to an ideal gas is written down at MRT where R is known as the gas constant for that particular gas. If you change the identity of the gas R is likely to be different. So for oxygen you will have one R, for nitrogen you will have another R, hydrogen a third R, helium a fourth R and air it is a mixture but if you approximate it as an ideal gas which is a good approximation you will have another R. Now at this stage it is not a bad idea to take the students into confidence because they have already come across in physical chemistry PV equals MRT, where N is the number of moles and R is a universal gas constant. And you can link it up in terms of the molecular weight and universal gas constant this can be determined. This make it clear this is not a foray into microscopic thermodynamics or molecular thermodynamics. This is just a link between the universal gas constant and the specific gas constant for any gas. We will not really have much use of a universal gas constant except that from the universal gas constant and molecular weight we will be obtaining the value of the gas constant. So for us PV equals MRT is going to be the equation of state for an ideal gas which can be written down in a simpler format which is PV equals RT. Now here where we are having a forward link ideal gas obeys actually two laws. One is obeys Boyle's law at all points in its state space and this plus definition of the Kelvin scale of temperature gives us the equation of state PV equals RT. Second thing it also obeys Joule's law which was established by Joule using what is known as porous plug experiments or what we can call now is a throttling experiment or free expansion experiment. This indicates that U of an ideal gas is a function only of temperature mind you ideal gas is a fluid although it may be considered ideal. So two properties determine the state since internal energy is a property it should be in general a function of two variables say pressure volume, pressure temperature or volume temperature. What Joule says that if one of the variables which you choose is temperature it will not depend on pressure or volume but of course if you say PV then it will depend only on the PV product but that means it depends on P and depends on V. But if you say U will be a function of T and some other parameter that other parameter will not have an effect it will be a function only of temperature. Now this Joule's law is nothing very special although initially it was experimentally derived later on using property relations as dictated by thermodynamics we will be able to show that if the equation of state of a gas is of the type PV equals RT will be able to show that the internal energy will have to be a function only of temperature. This is one of the funny things thermodynamics does thermodynamics never dictates what the value of a particular property for a particular system should be. It allows us a large freedom you can have any property any density any pressure any volume but then it says that if PV varies like this if PV equals RT then U must be a function only of temperature. It puts some restrictive relations between property if it says that if PV varies like this you should vary like this if PV varies like this something else should vary like this all sorts of funny relations it will give but it will give only relations it will dictate relations between properties it will not dictate properties by themselves remember that and this is a forward link we will be able to later on show this. We looked at the ideal gas and we noted that it obeys Boyle's law at all points in its state space and although we write it as PV equals RT we should realize the following the Boyle's law really says that PV will be some function of temperature only because PV product represents temperature. So this is Boyle's law writing this as RT with R as a short form from here to here it is the Kelvin scale definition be clear in the components it is the ideal gas the Boyle's law applicability gives you this Kelvin scale definition gives us this and of course the Joule's law is U only of T. Now let us take one step further and let us say a system which is simple compressive usually this is known as a fluid system many of the problems which we come across in thermodynamics will be based on such fluid systems what we generally call a simple compressible substance but we will call it a simple compressible system first two properties will define the ideal state the most common properties which we have is PV T so PV or VT or PT choice is ours and of course here we are talking of a fixed mass out of this since temporary temperature is of interest these two are the common ones rather than P and D although thermodynamically nothing wrong in using P and D but when it comes to the most easily variable easily measurable properties perhaps the most easily measurable is temperature after all that Joule's law we agree that the most easily measurable so long as you do not go to very odd temperatures very easily measurable is temperature and that simple contraption so crude so simple mercury in glass thermometer when it comes to anything between 0 and 100 degree C for a mercury glass thermometer it is Kish Jarkipatti you make it long enough you are able to measure even up to 0.1 or 0.05 C and if you really use that Beckmann thermometer you can even measure differences to 0.01 degree C although we say it is archaic it is one of the excellent and very simple devices to measure thermometer all of us have it but in a narrow range that is our clinical thermometer that is something say from 32 to 42 Fahrenheit or Celsius but usually it is calibrated in Fahrenheit and that is because tradition no doctor will talk of a temperature other than Fahrenheit at least in this part of the world so you go there and say my temperature is 310 Kelvin we will send you to the mad house than to anywhere else so suppose you select any one of these pairs and again the second easy only measurable is the pressure rather than the volume because the container could be of an arbitrary shape the measuring the volume may not be easy measuring the volume of a medicine in a syringe is easy because it is a very calibrated cylinder with graduations but measuring the volume of water in this bottle is not very easy because the shape is rather arbitrary it is approximately cylindrical but when it is like this it is not easy to measure the volume of a but pressure is the next easy to measure after a temperature so that is why the pressure temperature combination is a very common combination and whenever you go to a tabulation of properties it is usually be based on pressure and temperature rather than volume and temperature or pressure and volume. Now if you take pressure and temperature or volume and temperature say the next first thing happens is in this case pressure as a function of volume and temperature will be the major equation of state in this case volume as a function of pressure and temperature will be the primary equation of state then internal energy or thermal energy as a function of volume and temperature internal energy as a function of pressure and temperature such specifications we will have to provide we have defined H to be U plus Q that is the secondary property which we have defined we will find its usefulness later that also we will have to be provided as a function of volume and temperature and pressure and temperature it turns out that when you consider the variation of energy or thermal energy with temperature this turns out to be we will soon see why a more convenient way of representing internal energy whereas when it comes to enthalpy it turns out that the natural independent variables for enthalpy are pressure and temperature and of course primarily thermodynamic quantity of interest is the energy. Now for many fluids particularly gaseous systems I am not talking of an ideal gas mind if you plot U as a function of T U will be a function of T and something else usually we take it as V and you will find that as temperature increases U increases with temperature and also as volume so this will be at say V1 at a different volume you may have say at V2 at another volume you may have a variation because U is a function of both temperature and volume this is the way it will be depicted it turns out that the variation of U with temperature is more significant than the variation of U with volume for many cases and hence of the two derivatives dU by dt at constant V and dU by dV at constant T of these two derivatives this turns out to be more significant than the other two other one it is so significant that it has been given a special name definition Cv the symbol given is Cv and that is partial of U with temperature at constant volume since we are looking at a fluid system two variables so that is we have selected them to be T angry if it is a complex system say an electrolyte then may be temperature volume and may be electric field this would be the three variables and then Cv would be dU by dt at constant V and any other properties which also needs to be maintained constant but that is for more complicated system generally in a first course in thermodynamics we will not worry about such system unfortunately the traditional name for this is specific heat at constant volume specific is okay constant volume is okay but the word heat is not okay that is because Cv has no direct relation to this is definition what is Cv it is known as the specific heat at constant volume by definition it is the way thermal energy varies this is specific thermal energy varies with temperature when volume is maintained constant it is a partial derivative of U with respect to T at constant volume heat is not a property it is an interaction whereas this being a relationship between properties a derived relation with property Cv is a property it is a derived property it requires two states because dU by dt means you take two neighboring states having the same volume but different temperatures one and two let the different temperatures T1 and T2 let the internal energies be U1 and U2 and U2 minus U1 divided by T2 minus T1 is a good approximation to Cv take the limit as 2 tends to 1 and this is the limiting value units of this are joule per kilogram or it could be joule per kilogram degree C and here we can either use Kelvin or degree C because a temperature difference is involved it is dt whereas earlier I should have written down here R what are the units of R units of R are joule per kilogram Kelvin can I write this K as degree C no it is temperature itself remember that this K is because T itself is represented here whereas what is represented here is temperature difference and when it comes to temperature difference because of our definition current definition of the Celsius scale temperature difference in Kelvin is the same thing as temperature difference in Celsius your question madam. So this will be joule per kg degree C or joule per kg Kelvin instead of joule it could be kilo joule mega joule milli joule depending on the value that is only a matter of convenience similarly in a similar fashion when it comes to enthalpy is this is a convenient representation and hence there are two variables dH by dt two derivative partial derivative at constant P and dH by dt at constant T it turns out that this is more significant than this and hence this is given the name Cp and again unfortunately the name is specific heat at constant pressure we continue with the name knowing fully well that it has nothing to do with heat it is just a relationship between property dH is equal to dq is it not if pressure is constant dH equals dq no I will show that it is not you are we have to apply first law from this definition we cannot say that wait a few minutes we will come to second problem now all this from here for gaseous system this is a general gaseous system now let us go back to ideal gas for an ideal gas u is a function only of t that is joules law right now H by definition now remember all this is for an ideal gas H by definition is u plus PV and PV equals RT boils law so ideal gas relation is used again by joules law u is a function of t and this is only RT so that means H for an ideal gas is only a function of temperature derivation from joules law and ideal gas law and because u is a function only of temperature and H is a function only of temperature for an ideal gas Cv will turn out to be simply du by dt there is no need of a partial derivative and this will be at most a function of temperature similarly because H is a function only of temperature Cp will turn out to be notice I am not using triple equal to I am using only a double equal to it turns out to be equal to dh by dt which could in general be a function only of temperature because H is a function of temperature derivative will also be a function of now the next thing look at this we have come to the conclusion that H equals u plus RT differentiate this and you will get dh by dt is du by dt plus R and this gives you the relation Cp equals Cv plus R and again mind you and this is a funny thing Cp in general could be a function of temperature Cv in general for an ideal gas could also be a function of temperature but R will not be a function of temperature so although Cp and Cv vary with temperature the difference will be a constant value because the difference will have to be R notice that and quite often Cv with t may be small in which case we have the approximation of an ideal gas with constant specific because of this for u2 minus u1 where 1 and 2 are different states will simply be integral Cv dt from state 1 to state 2 do not have to ask what the pressures and volumes of the two states are similarly H2 minus H1 this will equal Cv into t2 minus t1 this will equal Cp into t2 minus t1 only when Cp Cv are constant mind you the difference always has to be constant equal to R so if one of them is a constant the other one automatically has to be a constant you cannot say Cv is a function of temperature but Cp is constant that you cannot say if Cv is a function of temperature Cp has to be Cv plus R so a corresponding shifted function of temperature will be Cp.