 Now, well we have defined temperature. Now let us go further. It turns out that although the idea of ideal gas is good for thermometry, it turns out that in the real world any gases including the famous mixture of gases all surrounding us called air, they behave almost like an ideal gas over a useful range of their state space. Of course, water vapor is another gas it is far from ideal. So, we will not talk about it, but air by itself although it is a mixture or many major components of air like oxygen, nitrogen, argon where helium and hydrogens are generally not abundant in air, but those are good scientific and technical gases industrially important. So, they all behave over a reasonably large range of temperatures and pressures almost like an ideal gas. So, that is why the behavior of ideal gas is of significant interest to us as scientists and engineers and the range is reasonably wide. Remember that as you go to lower and lower pressures the density will generally reduce. So, as you go to lower pressures the behavior will become more and more ideal. As you go to lower temperatures problems arise, but unless you go. So, within 10 or 20 Kelvin of the you know the freezing points or condensing points or boiling points of the corresponding liquids you are reasonably safe and ideal gas behavior is a good approximation. For helium the helium liquefaction temperature is about 4 Kelvin. For hydrogen it is a few tens of Kelvin. For major components of air which is oxygen and nitrogen it is about 80 Kelvin a few degrees of 80 Kelvin. I think 1 is 78 and 1 is 82 something like that. But there are other gases which will liquefy at reasonably high temperatures when compared to the helium and hydrogen and oxygen and nitrogen liquefaction temperatures. Problems arise when you go to high temperatures because although the densities are low the gases themselves may start dissociating. The oxygen instead of a diatomic oxygen molecule will try to dissociate between two monatomic oxygen molecules and so on. So, so long as you are not at a temperature high enough for dissociation to be significant the pressures and the pressures are also not very high ideal gas behavior is a good approximation. Dense physicists and engineers have looked at the behavior of many real gases and seen how good they are as ideal gases. Joule did experiments now Joule was more interested in the thermal or energy aspects interaction aspect than temperature itself unlike Celsius or Fahrenheit. So, Joule worked with ideal gases and Joule realized one more thing. He said that for a gas the internal energy of a gas essentially depends on its temperature and the effect of all other variables like pressure volume seems to be low. So, the he proposed Joule's law. So, just the way we have Boyle's law saying for an ideal gas isotherms are represented by p v equals constant where the constant differs for different isotherms. We have Joule's law which says that the internal energy of a gas of course a system of an system containing an ideal gas of fixed mass. If you change mass naturally the internal energy will change because it is an extensive property larger the system at the same state otherwise same pressure and temperature larger is likely to be the internal energy. So, an ideal gas obeys two laws Boyle's law and Joule's law. The Boyle's law relates p v to temperature at a fixed temperature p v is constant. The Joule's law says u is a function only of temperature of course for systems closed systems with fixed mass. Now the actual relationship between p v and t and the actual relationship between u and t through mass is known as relations of state or equations of state. Remember that ideal gas being a gas or a fluid a system containing an ideal gas will be a simple compressible system. There is only one two way work mode and that is expansion and hence we will require for a system of fixed mass containing a gas or an ideal gas two properties two independent intensive properties to define its state. It could be pressure and specific volume it could be pressure and temperature it could be temperature and specific volume it could be energy and pressure it is left to you. Now pressure and specific volume are convenient both are primitive properties not very difficult to measure but since temperature is an all providing concept pressure and temperature are the more commonly used pair for defining the properties or state of any system not only a gas or an ideal gas. But since pressure and volume are primitive properties we would like to have a relation between pressure and volume and temperature. So temperature as a function of pressure and volume is one of the equations of state and because it does not have any energy related to it we call it the primary equation of state or the equation of state but we also need to know how the specific internal energy is related to pressure, volume or temperature and that relation will be another equation of state. Similarly relation between enthalpy and the two variables which we have selected will also be an equation of state. So equation of state does not mean a relation between p, v and t but by default if we say the equation of state we assume it to be the relationship between pressure, volume and temperature. The p, v, t relation for an ideal gas or the principle equation of state is obtained using the Kelvin definition of ideal gas temperature. We know that the for any state of an ideal gas temperature of the state divided by t ref will be p, v divided by p, v ref. We turn it around and write it as p, v equals p, v ref divided by t ref into t and we divide both sides by the mass of our system giving us p, v equal to p, v I will put ref together into t. Now it turns out that this particular product this particular factor which is the ratio of the product of pressure and specific volume to the Kelvin temperature at the reference state reference is triple point of water is the characteristic of that gas and is known as R the gas constant for that gas and this gives us the first relation which we all know p, v equals R t the so called equation of state of an ideal gas. Now of course this can be written in various different ways for example now again multiply either side by mass and you will get p, v equals m, R, t where m is mass and then chemists look at it and they said they looked at the constitution of the gas for us a gas is just a fluid with different identities like oxygen, nitrogen may be a mixture like air chemists look at what the components are they look at the individual molecules they have something like the molecular mass involved and they have something an idea of bulk called the mole saying a mole contains a fixed number of molecules the Avogadro number of molecules. So they rewrite this as number of moles n into a different R into a t this is the number of moles measure of the number of molecules in that and in that case the advantage is that this R now turns out to be independent of the gas and hence it is known as the universal gas constant. Its unit would be unit of p, v that is unit of work like joule divided by unit of temperature that is Kelvin and unit of moles like mole or k mole one should remember that the value of this universal gas constant is 8.314 kilojoule per k mole Kelvin and since the number of moles is the mass divided by the so called molecular weight or molecular mass we end up with since number of moles is m divided by the so called molecular weight which I will write a capital M with a double stem. The gas constant for any particular gas will turn out to be universal gas constant divided by the molecular mass. The units match because the universal gas constant is kilojoules per kilo mole Kelvin and the units of molecular weight or relative molecular mass is kg per k mole. So that gives you the unit of R to be joule or kilojoule per kilogram Kelvin. So this or its variation in terms of p, v equals m, r, t or p, v equals n, r, t where r is universal gas constant are different formats of the same equation of state for an ideal gas. Now if you take any gas actually any fluid the equation of state may not be as simple as p, v equals r, t so temperature could be a general function of p and v or you can write this as p equals r, t by v or v equals r, t by p. These relations may not be as simple as this. We could have a generally v equals some general more complicated function of temperature and pressure or pressure with some general more complicated function of temperature and volume. Similarly any other property for example specific internal energy would also be some general function of temperature and pressure and or you could consider it since we can consider any two variables choices of us we could consider it to be a function of temperature and volume. We could even consider writing this as for example temperature with some general function of pressure and volume and may be even if you consider pressure and volume as the two independent variables you can also be considered to be function of pressure and volume. It turns out that some combinations are more convenient to use than some other combination. Later on we will see for some thermodynamic reasons this particular combination is a very convenient combination to use. Not always for example another property is enthalpy. We have defined enthalpy as u plus p, v where this p, v has nothing to do with the p, v in the ideal gas equation of state. Get rid of that idea. Why this p, v comes up that we will see may be on Friday when we consider open thermodynamic systems. So h can also be considered as a function of either temperature and pressure or it can be considered as a function of temperature and volume or it can be considered a function of pressure and volume. It turns out that when we consider enthalpy after studying the second law it will be clear that considering enthalpy as a function of temperature and pressure is a more convenient way of looking at enthalpy and deriving its properties. So coming back to it the second important thing we want to know is how does u vary with two properties of state and for the time being we will just take it that it is good for us and more convenient for us to consider u to be a function of temperature and volume. And since although we consider u to be a function of temperature and volume remember that u being a component of energy and energy is significant only when its differences are involved. Remember energy is actually defined only as data e. So rather than u we would be interested more in data u or du in a differential form. So that means when we consider variation of u we will write this as du and the variation would be partial of u with respect to t at constant v dt plus partial of u with respect to v at constant t dv. So if you want to consider the variation of u with temperature and volume the two selected independent variables we must know what the values of these two partial derivatives are. These derivatives are important that or at least the first one of these this one it is so important that we have given it a special name. Unfortunately this is a historical name first let us use the symbol. The symbol is cv this does not have a name but later on we will derive a much simpler formula for partial of u with respect to v at constant t plus partial of u this I will leave it as it. This particular derivative given the symbol cv and I would like to continue calling it simply cv cv cv but unfortunately we still continue with its name old historical name specific heat capacity at constant volume. Now this is rather an unfortunate nomenclature although we continue using it because of the following. Let us look at the word everything is per unit mass so the word specific is okay. It is a derivative with volume maintained constant so at constant volume is also okay it is proper. But this heat capacity is unfortunate name because we are not looking at a heat interaction at all we are just looking at relation between properties of a system two neighboring states as how does the u change when at constant volume t changes by a small amount dt that ratio is known as cv. It has absolutely nothing to do with heat interaction if there is a relation with heat interaction that will have to come out of the first law remember I said q equals delta e plus w that is the only relation between q and the rest of our world thermodynamic or otherwise right. So it is only a historically we say historical redundancy or historical take down hand down which forces us to still continue using this particular thing. In a similar fashion remember that if I want to determine the change in internal energy between two states one at t1 v1 another as t2 v2 I must integrate this as a partial differential or partial integral in two variables from state 1 to state 2 across any convenient path. In a similar way let me complete the definition we have said that h is another useful property and it is very convenient for us as we will realize later to consider it as a function of temperature and pressure just the way it was convenient for us to consider u to be a function of temperature and volume and again since u is defined as u plus h is defined as u plus pv only the changes in enthalpy are important the absolute values are not of much significance or any significance. So only differences and changes are important hence we will be interested in looking at dh and dh can be expanded as partial of h with respect to t at constant pressure multiplied by dt plus partial of h with respect to p at constant temperature dp a relation very similar to the earlier relation this one here we had dt and dv here we have dt and dp for h again we do not have any specific name for this but this has a very specific name first the symbol is cp. So this is written as cp dt plus partial of h with respect to p at constant t dp and this cp unfortunately still carries with it the old nomenclature specific heat capacity at constant pressure again we have absolutely no objection to specific in fact at constant pressure has to be mentioned because that what is maintained constant when we differentiate h with respect to t but this is the unfortunate part of this nomenclature and unfortunate because the name specific heat capacity and the definition of specific heat and even the heat interaction in many school books on heat and even college books on heat and thermodynamics relate q directly to cp and cv and some temperature differences leading to a lot of confusion. I wish all of us should remember that the only relation between the heat interaction q and the rest of the world is through one and only one equation and that is q equals let me again write it here q equals delta e plus w any other relation must be a relation derived out of this using the specification of the situation at hand or may be after making appropriate assumptions. Now all this is for any fluid again this is any fluid or any gas or any liquid that is why I am using the word fluid. Now let us consider the special case of an ideal gas. Now since an ideal gas is a gas and a gas is any fluid what is written on this sheet is applicable what is written on this sheet is also applicable. However, the ideal gas obeys Boyle's law it obeys Joule's law and Boyle's law plus definition of the Kelvin scale of temperature gives us the equation of state pv equals RT and Joule's law says u is a function only of temperature you can say that this means u is a function of temperature and volume, but volume has no effect or u is a function of temperature and pressure, but pressure makes no difference in mind you this is the special case of an ideal gas. Now look at the consequences u is a function only of temperature. So the first consequence is what about H? H is defined as u plus pv u is a function only of temperature Joule's law pv is nothing but RT which is only a function of temperature and that means for an ideal gas H is a function only of temperature. Second one because of this and because of this because of this we have partial of u with respect to v at constant temperature is 0 and because of this we have partial of H with respect to p at constant temperature to be 0 and hence du becomes partial of u with respect to t at constant v dt plus partial of u with respect to v at constant t dv I will again write here ideal gas otherwise you may take it out of context. This is 0 because it is an ideal gas so this simply becomes cv into dt in a similar fashion you can show that dH will be partial of H with respect to t at constant p dt plus partial of H with respect to p at constant t dp this is 0 because it is an ideal gas so this will simply become cp dt and these two relations let me call this 1 and let me call this 2 1 and 2 give us that u2 minus u1 the internal change difference in specific internal energy between two states will simply be integral cv dt 1 to p and similarly H2 minus H1 will be integral cp dt between the same two states 1 and 2 and of course you can extend this if cv and cp are constants then you can take them out of the integral sign and then you can write this as cv into t2 minus t1 and cp into t2 minus t1 and of course based on this now you can derive certain other things for example you can always show that cp minus cv equals r you can you should also notice that cv is defined as partial of u with respect to t since u is a function only of t this partial derivative can also be written down for an ideal gas as an ordinary derivative since u is a function of t the cv will also be a function of t in a similar fashion cp will also be a function of t so you should remember that here cp is a function only of t cv is a function only of t but r will not be a function of t so for a general ideal gas we define ratio of specific heats gamma it turns out that this is a good short form which we need to use later on we will find out what its significance is when we consider if we study kinetic theory it will give you some physical significance of that and it will be a good short form for many of our exercises particularly in compressible flow you will use the symbol gamma dozens of times in your derivations and exercises but in the general case this turns out to be a function of t now again we have a still special case over a small range of temperature we have ideal gas with constant specific heats remember that when you write this it should be always constant specific heats because the two together will be functions of temperature if one of them is constant because r is constant other one will have to be also independent of temperature so for an ideal gas con with constant specific heats you have the following things u2 minus u1 will simply be cv into t2 minus t1 h2 minus h1 will be cp into t2 minus t1 and you will have gamma is cp by cv will be a constant and because of this we can define certain stuff by just working around that cp will turn out to be gamma by gamma minus 1 into r and cv will turn out to be 1 over gamma minus 1 into r these are useful because quite often the characteristic of an ideal gas particularly with constant specific heats is defined in terms of the ratio of specific heats gamma and the molecular weight for example I can say assume air to be an ideal gas with molecular weight of say 29 or 29.2 or something and gamma of 1.4 or consider hydrogen to be a gas with molecular weight of 2 kg per k mole with gamma of 1.4 or I can say helium ideal gas with molecular weight 4 kg per k mole and a gamma of 1.67 so using the molecular weight one obtains r and then using this value of r and gamma you can obtain cp and cv and use it for solving our exercises.