 So let us start with the next topic that is thermometry scales of temperature and equations of state. In the main workshop we will spend the reasonable amount of time on this but here let me quickly go and explain what is meant by thermometry, what is meant by temperature scales and the equations of state. As we go through we will define some useful temperature scales particularly the ideal gas Kelvin scale of temperature and one of the equations of state we will now use is the ideal gas scale of ideal gas equation of, by thermometry we mean the process labeling states with temperature. So assigning any state of a given system with an appropriate temperature is the job of thermometry and what is temperature? It is a label of an isotherm so given a state find out to which isotherm it belongs and the label of the isotherm is the label of that state so that is the temperature of that state. Thermometry is essentially an empirical science in the sense it is experimentally done there is no thermodynamic basis for this temperature and this is because the zeroth law temperature or zeroth law gives us the idea of temperature but it does not tell us which is the best temperature to use or which is the appropriate scale of temperature to use. It does not even tell us which temperature is higher or not. In zeroth law all we can say is given two systems in specified states whether they will interact thermally with each other when allowed to interact across a diatomic wall but even that is a great useful thing. So see how we proceed traditionally the thermometry was done by defining a scale of temperature. A scale of temperature was defining by defining a system a standard system that means specifying a system whose states are labeled this was given the name this standard was given the name thermometer and after that we use zeroth law that means use this system see which of the state of the system is in thermal equilibrium with the system whose temperature is to be measured and just transfer the label because that is what zeroth law does. Then this labeling is done by a procedure which requires what is known as reference states of some system. Let me take one example I will just take two examples one is the so called Celsius scale. The reference system used here I am talking of the historical Celsius scale because today the Celsius scale is defined in terms of the ideal gas scale. The historical Celsius scale used a mercury in glass thermometer. So you had a long capillary in a glass I will just show that part and a bulb I am not showing the glass envelope I am just showing the active part and it contained mercury and it was made up of solid glass so it was a by design it was a rudimentary system you could expand it you could compress it you could charge it discharge it magnetization demagnetization nothing was possible. So because it was a rudimentary system there was only one property which was of significant the scale could be defined by one property and since it was a solid thing with a constant volume the one property which you could obviously see as the state change was the length L of the mercury in the capillary that is the only thing change. So first thing this was rudimentary so this implied one property of any significance and this was length of mercury in the. So we said that look the states are simply labored by the length of mercury and if we set up a scale based on the length that could be a temperature scale. So how do you calibrate it so Celsius used two fixed points he used water as pure as possible for him a system containing water as his system and then he said the normal freezing point of water for the normal melting point of ice was his one fixed point and the normal boiling point the ice point and the normal boiling point steam point was the other reference point. So what he did is on his thing he said look on my thermometer this is the ice point IP and this is the steam point SP and this is these are the sort of fixed points because I can create a system containing water reasonably easily I can put it in the refrigerator freeze it wait till it starts melting and bring this thermometer there and mark the ice point. So long as I do the experiments correctly I have a proper ice point similarly I put it in a boiling channel a so called hypsometer as we say so that it is covered with steam just as the water boils mark that point steam point and then he says that look in between I will provide graduation. So he decided that the ice point will be labored 0 the steam point will be labored 100 and in between he had a uniform scaling put a scale with 0 at ice point 100 at steam point and he said that look suppose I want to measure the temperature of this or label the temperature for this system and if this is my Celsius thermometer I will put it in thermal contact with this now this is not a thermal contact so what I will do is I will put it inside I dare not put it because this is not a thermometer this is a pen but you put it inside so that there is direct contact which is good thermal contact that means you are now asserting yourself that the boundary separating your thermometer and the system is a diathermic boundary remember what we have studied in 0 law then you will find that there will be some interaction and we hope that the interaction is small enough because this body is large the thermometer has a small thing so soon it will reach a stage where it will be in thermal equilibrium with this that means the state of the thermometer and the state of the water will be two states on the same set of isotopes corresponding isotopes. So they should have the same label they should have that same temperature so find out where that state of the thermometer is suppose it is at L then he said that label the temperature as L minus L ice point here L steam point minus L ice point multiplied by 100 and the he or others started calling this initially as degree centigrade later on degree Celsius and now only Celsius the degree because of the old idea that temperature is the you know idea measure of degree of hotness and all and now we say that there is no such thing as a degree of hotness it is just like a thermo any other thermodynamic property. So remember that the Celsius scale required a definition of mercury in glass thermometer realizing that there is it is simple enough so there is only one property we do not have to worry about it then definition of fixed points for which system as defined definition of fixed points and definition of the reference temperatures reference label 0 here and 100 here. He could have said why 0 and 100 could have said 0 and 1000 actually when he started really doing it he put 100 at ice point and 0 at steam point nothing wrong thermo thermometrically but then it turned out that hotter things have a lower value of temperature cooler things have a higher value of temperature and other scientist told him that look that goes against our idea so he flipped it between 0 and 100 but you look at history he started with 100 at ice point and 0 at steam point and remember that is nothing wrong thermo thermodynamic point and then the third thing was the interpolation law because you have fixed points in this case two fixed points and the state can lie anywhere in between this is the basic idea and if you look at the rumor scale or the Fahrenheit scale the idea was simple define a simple rudimentary system as a thermometer define some reference system in this case water ice steam define some reference points which can be easily replicated in the laboratory define arbitrary labels at the two reference points and then define a interpolating law there is a large amount of arbitrariness is this and it is bound to be so because this is purely experiment we have now a few Celsius Fahrenheit Kelvin a few scales but if everybody insist I can have a scale in my name each one of you can have a scale in your name but then you know talking to each other somebody from Jalandhar will say oh temperature here is very low it is only 1200 degree Jalandhar how do I understand then I tell him no no no it is very hot here you know it is today it was 37 Celsius you say what is Celsius I do not know so some standardization is required and hence maybe after a few arguments people settled on a few cell scales of temperature like Celsius Fahrenheit and later on Kelvin now why did the Kelvin scale and the ideal gas scale came together came up the reason is that we started working scientific situations and industrial situations required that we go much beyond this 0 to 100 range now 0 to 100 range is good enough for day to day work you know typically ambient temperature will not exceed in fact we will never go to 100 ambient temperature will generally be between minus 20 and around 50 okay as you boil something cook something maybe 100, 120 you will notice that already we are going beyond this range if you cook something in a pressure cooker it will be 120 or 110 degrees if you go try free something or go near the poles you will easily find temperatures which are below 0 and people started noticing that a mercury in glass thermometer does not work below something like 39 degree C because mercury freezes does not expand does not remain a liquid anymore similarly as you approach 350 degrees C mercury starts vaporizing thermometer start cracking. So people started using developing ideas for other type of thermometers and the next thing which really extended the range where gas thermometer here the idea was use a simple gas may be air or a pure gas like hydrogen helium oxygen nitrogen fixed mass so measure the pressure measure the volume primitive variables comfort zone and naturally if you plot pressure against volume every isotherm will look in some funny way so have a plot but then you notice here that a gas is not a rudimentary system containing a gas a simple compressible system so people said that look let us try to make it an equivalent rudimentary system so some people what they did was they froze the piston making it a constant volume so volume does not remain a variable so they said will always work at a fixed volume and thus we had the constant volume gas thermometer where only thing which varies is pressure and pressure can be mapped to either Celsius temperature or whatever scale you have so pressure becomes the variable which is mapped on to temperature notice what is done here you have a gas why did we use a gas because if you use a gas like oxygen or nitrogen or even air the range of temperature so range of range in state space where it remains a gas is much wider than that of mercury you can easily go to minus 100 150 degree C or go to 400 500 600 degrees Celsius and there is no problem because that is why gases are attractive then they found that it is difficult to maintain a constant volume as you go to very wide ranges of temperature because the container also expands and contract so they said that look instead of maintaining a constant volume why not maintain a constant pressure then the volume of that chamber becomes a measure of the temperature again arbitrary mapping as was done by Celsius and we ended up with a constant volume gas thermometer. So then of course there were physicists who started mapping these isotherms in detail and we came up with a physicist engineer called Boyd Boyd discovered that isotherms of a gas are approximately of the type P into V is constant given isotherm he plotted an isotherm that is found out a set of isothermal states given isothermal state he found that the equation is approximately P V equals constant and then he found that this approximation becomes better and better as the pressure is low and the volume is large so when the density of the gas is low he found that this is a very good approximation. So he proposed what is known as Boyle's law, Boyle's law says that P V of a gas is constant for a set of isothermal and then he says this is approximately true for many gases and then of course physicists do not like approximations they want abstractions exactness so that they could understand the basic idea. So physicists came up with an idea that look let us consider an ideal gas, the ideal gas is defined as a gas which obeys Boyle's law all over state space that means whatever be the pressure whatever be the volume it will obey Boyle's law that means find out an isotherm for a given isotherm P V will be a constant yes historically we do not know when we present it this is the way we present it. Historically he did experiments he came up with exactly this fact no other scientist say let us you know most gases behave like Boyle's gases approximately the approximation is good at lower pressure lower temperatures how low how high pressure and volume that depends on the individual gas for hydrogen helium the range is very wide for water vapor and other heavy gases the range is very low but then you know physicists being mathematically oriented tend to work with exact thing though approximate so they said let us assume that there is a gas which obeys Boyle's law exactly and see what are the consequences and then they started calling it first they call it a Boyle gas but then they said an ideal gas because then it gives an idea that there is some idealization approximation the other term sometimes used is a perfect gas we will use ideal gas or a perfect gas interchangeably there is nothing special. Now remember that this constant will be different for different isotopes and then the idea came that suppose I have an ideal gas that means if I do experiments with a real gas with low pressures and large volume so low densities and then I can label the temperature using the PV product because the PV product represents a temperature so why not use the PV product as a major of temperature this idea led to what are known as ideal gas scales of temperature I am calling scales because you have decided to use an ideal gas as a thermometer but again you have to define reference states you have to define reference temperatures you have to define a law of interpolation and depending on what you do you will have different scales of ideal gas each one of them can have different scales of ideal gas but again this will lead to confusion so standardization the Kelvin scale of ideal gas temperature IGT is ideal gas temperature here the reference system is water the reference state is triple point why triple point at triple point all three phases coexist because it is a pure substance pure chemical the phase rule says that both pressure and temperature have to be fixed at some values you do not have a choice saturation temperature may differ with pressure but when it is triple point it is a fixed point in the state space so you do not have to specify temperature you do not have to specify pressure and the reference value is temperature of the triple point was defined as 273.16 Kelvin the reason for that we will see in a few minutes so ideal gas or approximated by a real gas was used as the system for measurement the reference temperature was triple point of water and the T of triple point was defined as 273.16 Kelvin the law of interpolation was very simple PV by PV at triple point that is the reference state was temperature I will put this divided by temperature so this was defined temperature of the triple point was defined this had to be measured and this had to be measured this had to be measured once for your thermometer this had to be measured whenever suppose I want to measure the temperature of this take your ideal gas system bring it in thermal contact with this adjust the pressure and volume so that there is zero interaction so thermal equilibrium is established find out the PV product you have already determined PV product at the triple point by creating a system with water at triple point so you have measured PV at triple point already you have measured PV now T triple point is defined so you compute out this temperature this was the method for determining the temperature on the ideal gas Kelvin scale now why 273.16 Kelvin the reason is whenever we do something new we must know how it is related to something old which we have been traditionally using and we want that relation to be as simple as possible so that conversion or movement from one to the other is easy not very complicated so till the Kelvin scale came up which is based on just one fixed point the Celsius scale was the standard scale you know the old CGS method of units and dimensioning in physics so we were interested in checking how the temperature on the Kelvin scale compares with temperature on the Celsius scale it turns out that let us take ice point let us take triple point and let us take steam point here this is defined to be 273.16 this is defined why is it defined we do not know on the Celsius scale this is defined at 0 degrees C this is defined as 100 degrees C it turns out that the triple point is approximately 0.01 degrees I am not underlining it because it is approximate if you measure the ice point on the Kelvin scale and if you measure the steam point on the Kelvin scale it turns out that the ice point is 273.15 K the steam point is 373.15 K and you will notice that the difference in temperature between the steam point and ice point is 100 units on the Kelvin scale and also by definition is 100 units on the Celsius scale the 273.16 reference temperature was extracted to make the difference between steam point and ice point to be almost exactly 100 units on the Kelvin scale so that the conversion from Celsius to Kelvin could be very easy just add to 73.15 to Celsius and you will get Kelvin or subtract to 73.15 from Kelvin and you get Celsius. Now for a few years this two definitions the old definition of Celsius and the new definition of Kelvin they went together but then the standardization people realize that temperature is a unique number so there should not be two independent temperature scales with some conversion factor between them then somebody measures on a Celsius scale somebody measures on a Kelvin scale and conversion is not exact we have a trouble as to arguing out which one is right so this is the old definition of Kelvin Celsius scale today's definition is the standardized Celsius scale is defined as T on the Kelvin scale minus 273.15 on this standardization the ice point on the Celsius scale is 0.01 C we do not use the degree anymore but some physicists and many of us keep on using degree because using C is confusing C is used for Coulomb as well as for Celsius. Now this turns out to be 100 Celsius almost exactly but not really exactly but the it is so good that we do not have to worry about how many decimals to use after that. Similarly 0 Celsius is the ice point not by definition but it turns out to be so the only certain temperatures today are the triple point temperature which is defined as 273.16 K as definition of the Kelvin scale and by the related definition of the Celsius scale it is 0.01 Celsius and remember the triple point of water the temperature at the triple point of water is the only temperature today which will never be measured because it is defined to be 0.01 C or 273.16 Kelvin. If tomorrow some person comes to you and say look I have made a very accurate thermometer and I measured the temperature of the triple point of water and I found it to be exactly 0.01 Celsius on my thermometer it must be a good thermometer all that you say is you have never measured the temperature all that is your thermometer is calibrated properly because whatever you say temperature of triple point of water will be 0.01 Celsius by definition you do not have to measure it. If you measure it to be 0.02 Celsius or anything else than 0.01 there is something wrong with your thermometer do not talk to me about it. So remember that this temperature triple point of water is the only temperature in our life which need not be measured because it is defined to be so. So somebody asks you what is the mating point of sulfur somebody will tell you a value how do you know you will say that fellow has done an experiment it is published in this journal this paper and all that. What is the triple point temperature on the Celsius scale the answer should be 0.01 Celsius why the reason should be it is defined to be so remember this. Now with this we come to the next if we have defined the ideal gas temperature scale using I am using now reference instead of triple point as T by T reference then I can just turn it around and now we get PV equal to PV ref by T ref into T and I can now because there is a total volume here I can write it in terms of the mass and specific volume. So this becomes PV by T at the reference point by T by T. This in terms of the definition of the ideal gas scale of temperature gives the ideal gas equation of state this is nothing but the definition of ideal gas temperature worked around. Now depending on what your gas is depending on the identity of the gas this pressure volume pressure into volume specific volume divided by temperature at the reference stage turns out to be a characteristic of that particular gas and we write it as PV equals MRT where R is nothing but PV by T at the reference state which is known as the gas constant for that gas. At this stage you can instead of mass you can use bowls and then talk about go into physical chemistry and bring up the universal gas constant let us not spend time about it that you know very well how to do that is done reasonably well in all textbooks. Now this equation one which is a consequence of Boyle's law definition of an ideal gas which obeys Boyle's law all over the state space definition of ideal gas scales of temperature followed by definition of the ideal gas Kelvin scale all these relate finally link us to this ideal gas equation of state. Since R depends on the individual gas sometimes it is also called specific gas constant as opposed to the universal gas constant and of course at this stage you should also or characteristic there are characteristic Gauss gas constant specific gas constant particular gas constant sometimes they even write RH2 meaning it is for hydrogen RO2 for oxygen etc that is routine part of thermodynamics let us not spend time on it but this still does not tell us how is the energy of a gas related to its state. So remember that this is a consequence of Boyle then ideal gas then ideal gas scale of temperature then the Kelvin scale of temperature remember based on ideal gas isotherms you can have any temperature Kelvin is just one of them then simultaneously because Boyle discovered that well there is something special about gases if isotherms turns out to be rectangular hyperbola is a system mathematicians jump at it oh it is so simple so they will crank out all sorts of things. Joule did experiments with gases his famous porous plugs experiments or free expansion experiment whatever you call them and he discovered that u of an ideal gas is a function only of temperature because he worked with q equals data u plus w so by measuring data w and q he could determine data u and he found that look if the initial state and the final state of an ideal a system containing an ideal gas are at the same temperature same temperature we know okay then there is no data u because he suppressed all other data is and then he found that q was equal to w data u was then q minus w you found q was equal to w so this is known as Joule's law so an ideal gas consist of two laws which it obeys one the ideal gas equation pv equals mrt or written in specific form p small v is rt and the second one Joule's law which says that the thermal energy of a system containing an ideal gas is a function only of its temperature or specific thermal energy is a function only of temperature and these 1 and 2 together make up the ideal gas equations of state an equation of state is in principle any relation between properties various properties of state relation between pressure volume and temperature is one such equation of state and sometimes it is called the primary equation of state because temperature is related to primitive variables whereas this is a thermal equation of state which relates thermal energy to temperature we see Charles law is an empirical law which was discovered independently but when you take Boyle's law and Joule's law together and particularly when you define the Kelvin scale in terms of this Charles law gets subsumed into it we do not have to consider it as a separate thing what Charles law says that I do not know whether it is at a constant pressure the volume of a gas varies linearly with temperature that was the earlier idea and based on that earlier ideal gas temperature scales were also based on the 2 fixed points if you go through the old books on thermodynamics they will find ideal gas scales of temperature based on 2 fixed points because earlier because of the standardization of the Celsius scale the ice point and steam points were the standard 2 fixed points so they defined it and then they found that all temperatures tend to converge at some minus 273.15 Celsius then say why not define that at 0 so that is the historical development now we say that look this puts the whole thing on a simple straightforward framework and today's definition of the Kelvin scale is exactly as I have defined here today's definition of Celsius scale it simply temperature on the Kelvin scale minus 273.15 unfortunately what happens is because of this we just do not have to talk about Charles law it becomes part of the intermediate history of thermodynamics now because of this we now come to a stage where u is a function of t but how does it vary for an ideal gas again experimental evidence tells us that although it is a general function of t as it decided by Joule we found that ut is roughly a approximately linear function of temperature over a narrow range you can fit a polynomial if you feel like slightly varying but it is approximately linear and when something is approximately linear the slope of that is of important because that decides the direction so we define the slope as Cv at this stage it should immediately be pointed out that for a non-ideal gas u will not just be u of t this may not work but since a gas is a simple system containing a gas is simple system any property will be a function of two variables because state will be defined by two variables so we must write for a non-ideal gas u is either function of temperature and volume or a function of temperature and pressure it turns out that this is more convenient to use this is less convenient to use although there is nothing wrong in using that and now we come to the definitions for any fluid fluid means gas liquid whatever if we consider u to be a function of temperature and volume in the general case then we define this coefficient partial of u now I have to use partial derivative because there are two variables with t at constant v is called the specific heat at and here we come to a funny thing this is purely a historical name out of these five words at constant volume is perfect because it is a derivative partial derivative at constant volume specific is also okay because this u is internal energy or thermal energy per unit mass but heat is that is purely historical because as we have defined it it has absolutely nothing to do with heat interaction and when only properties are involved what is a thing which is an interaction doing out there it is a relation between properties so remember that it is CV historically we will continue to call it specific heat at constant volume but by definition it has nothing directly to do with the heat interaction this is first definition the second definition which we will use is that of enthalpy this is a short form H is defined as u plus pv which gives us H to be equal to u plus pv this CV is dependent on two variables in for a general gas for general gas so if there is a term du by del u by del t there must be a term del u by del v also yeah there is a term if necessary we will use it but the term which is partial of u with respect to t at constant volume that term is defined at CV okay but rather in normal thermodynamics we never use that term del u by del v because most of the time we get away by assuming our gases to be ideal there are exercises here in which for example for van der Waal gas we will show that it is a function of both temperature and volume we will take care of that because u is a function only of temperature see it should be very careful for an ideal gas del u by del v is 0 provided the derivative is at constant temperature because in thermodynamics you can have any two variables I can consider u and say pressure u and p as variables then del u by del v at constant p need not be 0 because we are not asserting that temperature is uniform. So this is the second definition and this is just a short form similarly this is not a short form but this is a derived property and the third definition is Cp is defined as partial of specific enthalpy with respect to temperature at constant pressure this is called specific heat at constant pressure other names for this are isobaric heat capacity and for CV it is isochoric or isometric heat capacity those are the other names and if you want to be more specific this should be specific heat capacity at constant volume and specific heat capacity at constant pressure. So remember these three definitions and do not read in them anything other than definitions enthalpy is u plus pv CV is partial of u with respect to t at constant volume why constant volume because it is convenient later on we will see to consider u is a function of t and v and Cp is partial of enthalpy with temperature at constant pressure that means here enthalpy is conveniently considered a function of t and p again because we will notice that this is more convenient than considering enthalpy as a function of. Now with this we notice that this is generally true for any gas but for an ideal gas u is a function of t pv which is RT is also some function of t hence CV becomes du by dt Cp becomes dh by dt and h is u plus pv and since u is only a function of t pv is also a function of t h also becomes a function of t and hence Cp is only dh by dt actually I should write this first and then this again further approximation and of course you can then show that Cp minus CV is R because dh by dt will be du by dt plus R du by dt is CV further often it is assumed CV and Cp are constant thus simplifying things further but up to here these relations are only for an ideal gas we may be on very slippery ground the moment we go away from an ideal gas for a real gas these definitions apply and for a real gas because pv need not be a function only of temperature u need not be a function only of temperature hence h need not be a function only of temperature and hence CV and Cp also need not be constant or need not only be functions of temperature later on we will show that for a slightly more complicated gas like Van der Waals gas u is a function of temperature and volume h is also a function of temperature and something else volume or pressure whatever it is but CV is a function only of temperature although u is a function of temperature and volume CV is a function only of temperature it does not depend on volume but Cp is a function of temperature and either volume or pressure so as you go away from ideal gases things start becoming more and more complicated however you should also impress on students that unless you go to very high pressures as long as you are at lower pressures and unless you consider a process in which the temperature varies very widely over more than a few hundreds of degrees C or there is a change of phase the ideal gas approximation or even the ideal gas approximation with constant specific is a good one provided you use the appropriately averaged values of CV and Cp that is what you should tell the students because that is really true in fact unless you use esoteric gases like ammonia and things like that or water vapor you use gases like oxygen nitrogen even dry air few hundred degrees C from say minus 100, 150 to 400, 500 degrees C you can get away with assuming them to be an ideal gas with constant specific heat but if you are doing experiments between 50 to 150 you will use one value of CV and Cp you are doing experiments between 250 and 500 you will use a slightly different value of CV and Cp the appropriate averages over those ranges and you can get away with reasonably good predictions. Now at the end of this the last part of this is exercises so I will ask you to look up your exercise sheet, page 3 why specific heat at constant pressure is greater than specific heat at constant volume okay for an ideal gas using we have derived Cp equals Cv plus R and since R is a product of pressure and volume and temperature as we defined is always a positive number R is a positive number so Cp is greater than Cp but this is only for an ideal gas for real gases or real fluids when we derive property relations in our chapter or item 10 we will be deriving a general relation between Cp and Cv more complicated range. Any practical evidence of Cp and Cv? Critical evidence means Cp and Cv have been measured for large number of materials Cp can sometime come very near Cv but Cp has never been shown to be less than Cv and we can even show it that for all stable mechanical materials Cp is greater than or equal to Cv that can be shown thermodynamically after we derive property relations using second law of thermodynamics. No but that only tell you that H is greater than U but Cp means variation of H with temperature and Cv means variation of U with temperature H is higher than U does not mean that the slope of H is higher than the slope of U got it H can be higher than U but at some stage U can be varying sharply H can be varying shallowly. So when they have to be later you cannot ever overcome and become lesser is it not? What? The slope of U and slope of H. No, no, no. If they do not cross the slopes have to be. No, no, no that is not true for example if I consider temperature U may vary like this H may vary like this it is U plus Pv yeah so this is plus Pv this is H this is U but it is possible that at some stage the slope of this is lower than the slope of this at that point at some other point see in principle I can show you such cases but his question is right just now we can demonstrate Cp is greater than Cv only for an ideal gas but later on there are exercises here for example just to satisfy your curiosity come to page 11 property relation PR3 using the results of exercises PR1 and PR2 show that Cp minus Cv is all this thing for a system of simple compressible system. The volume system can remove the constraint system will move. So when we supply the same amount of it there is expansion. So expansion qualitatively is that is the work is done by the system. So it consumes certain amount of heat supply that is a qualitative thing I am not sure all everybody will be satisfied it perhaps that is acceptable but what you see here on page 11 that exercise is a more rigorous way of doing the same thing and there are many other exercises there but the result of PR3 it is something which is used to demonstrate that Cp is greater than or equal to Cv for stable systems because the mechanical stability also comes in there. So now we come to end of 7. So I recommend that you look at exercises F1 we will not solve any of these exercises but I will comment on a few of these. All these exercises are based on the working fluid essentially being an ideal gas I think each and everything is an ideal gas there is no non ideal behavior and also I think in most of these you can assume that the ideal gas is an ideal gas with constant specific heats. The recommendation here is the following always begin with never write an equation other than this for your first law always begin with Q equals delta E plus W. The next step should be write delta E is delta U plus other components then see by reading through the statements see whether there are any references to other components I do not think except perhaps for 1.7 depending on how you define the system there will be no reference to any other component but do not directly write Q equals delta U plus W you may get into trouble. Write this read through the statement very properly and then make the assumption if appropriate that delta E equals delta E again do not jump to the conclusion that W is integral PDV always write so first thing is always write delta E is delta U plus delta E other summed over appropriate. If you want to neglect this see that neglecting that is appropriate under the circumstances similarly W you write W expansion plus W stirrer plus W electric plus something else there are problems here where there are stirrers there are electrical work transfers all those things are there so be careful do not simply say W is integral PDV you are likely to get into trouble then depending on what the process specifies for example 1.2 says that it is a rigid container rigid container means it is a constant volume process do not jump to the conclusion constant volume means DV is 0 W is 0 W expansion will be 0 but there may be other components of work for example here there is a stirrer there so there will be a stirrer work involved do certain assumptions based on the specified thing and then assume as appropriate then use constraints as specified and obtain a final form of first law. The first law should always be this when you say first law Q equals delta E plus W that is first law nothing else if there is anything else that has to be derived or a special case of this in particular students have a habit and I am sure a few of you also have a habit you read here 1.2 oh rigid container so Q must be equal to MCV delta T do not ever make that mistake if I see a student thing I have scans of students this thing if I see a student writing straight away the first thing it is a constant volume process hence Q equals MCV delta T I put a red line after that I said 0 I am not going to look at your answer book and return it I have to do it once after that for the rest of his life the student will start with Q equals delta E plus W and I have appropriate evaluation scheme the moment he writes close thermodynamic system hence first law is Q equals delta E plus W he gets that one mark which is associated and he may get further mark but if he starts with the wrong foot I say go back home I do not want maybe our students are good students and they become cranky but they finally at this rate when it comes to end semester the evaluating their answer books then becomes a pleasure because nicely solved everything similarly there are processes here where the process is constant pressure again for constant pressure do not jump to the conclusion that Q equals MCP delta T this is this must be a derived relation it may turn out to be similarly another issue with sometimes you write constant volume process so Q must be equal to delta U but this assumes two things it assumes that delta E equals delta U which may or may not be true it also assumes that W is only W expansion and because of constant volume it is 0 it also assumes that there are no other modes of work similarly it is improper to straight away write Q equals delta H for a constant pressure process Q equals delta H for a constant pressure process only when delta E equals delta U and the work transfer is only by the expansion mode but if you have a stirrer if you have an electrical work input then Q is not equal to delta I and hence Q is not equal to MCP delta T if you write MCP delta T there is an additional assumption involved and that is apart from delta E equals delta U and W is W expansion for an isobaric process it also assumes that the there is no change of phase hence delta H can be written down as integral MCP delta T then you assume that CP is a constant and hence you can write it as MCP delta T and I will leave it as an exercise to you to create exercises remember that we are going to have always Q equals delta E plus W and let us say that we restrict ourselves to simple situation so this is delta U plus delta E kinetic plus delta E potential let us say others we neglect plus here we say we just put delta W expansion plus W stirrer plus say W electrical there could be others but we neglect there are 1, 2, 1, 2, 3, 4, 5, 6, 7 you can create combinations and then say the process is constant volume or the process is constant pressure or the process is isothermal if you want or the process is adiabatic that means Q is 0 indirectly and you can create problems the more combinations you use and more complicated problems you create it is good for you and it is good for yours too again I would like you to look at exercise 1.6 F 1.6 this is a generalization of that mental set which leads to the following hilarious thing you ask a student hey what is an isentropic process and he says PV raise to gamma is constant can I ask him that look what does it have to do with an isentropic process then he starts I can derive it serve for an isentropic then he starts derivation and then I said but here you are assuming this here you are assuming that but I just asked for what is an isentropic process and you simply came up with PV raise to gamma is constant now here you will notice in F 1.6 we are deriving a similar relation PV raise to k is constant and you will notice that there is a parameter C there and the relation which you will determine in terms of Cp Cv and C4k will show you that if C is 0 that means if DQ is 0 it is an adiabatic process then you will end up with PV raise to gamma is constant notice that no second law is involved and from analyzing this problem in detail you will notice that for PV raise to gamma is constant you have to have the following things you have to have an ideal gas you have to have at constant specific heats in that process delta E should be replaceable by delta U so there should be no change in other components of energy DW should be DW expansion there should be no other component of work and after all that it has to be a quasi static process so that we can integrate it over the process otherwise it ends up only with that differential part we are unable to integrate and then after this you will realize that PV raise to gamma is constant even for an ideal gas with all this has nothing to do with second law nothing to do with entropy and hence nothing to do with it being an isentropic process it may turn out to be an isentropic process but that is what we will discover after second law but for basic derivation we do not have to assume it to be an isentropic process there are other combinations with some electrical work of charging discharging and all in particular look at problems 110 and 11 these are the problems in which the process is suspected to be non quasi static in one case in the second case the process is not necessarily quasi static so be careful while solving this but when you properly solve it you will notice that it is still possible for you to determine the net work the expansion work and all that but this work will not be from first principle this work interaction will be derived using first law because whether the process is quasi static or whether the process is non quasi static the first law in the form that q equals delta e plus w would still be applicable so if you are able to evaluate q or if q is specified say 0 for example if it is a diabetic if you are able to evaluate delta e evaluation of delta e requires only the end state then you are able to determine w that is indirect computation of w that is what I mean meant when I said that if a process is non quasi static then you may not be able to evaluate w as an integral of dw over that process but you may be able to compute that interaction out using some other method and that is using first law.