 What we need to do to get rid of this arbitrariness is the following. We must have a formal way of labeling isotopes. In fact, many things have been formalized. Remember the measurement of length. Some king decided that the length of his foot will be the unit, so the foot came up. Then somewhere else he decided that we measure the distance between the equator and the pole and divide it by maybe how many 10 million or 1 million times and say that is going to be our standard measure and so came up the original meter and even these definitions keep on changing. Finally, you have some international society for what standards and measures which comes up and say this is our universal definition of meter, this is our universal definition of a second and this is our definition of mass. In a similar way, we need to formalize the way of labeling isotopes. Again just the way have different units of length and mass and time. We have different ways, but finally we have a standard which puts all these things together. The formal way of labeling isotopes is known as thermometry. Thermometry is nothing but a method of labeling isotopes. So, what is the basic idea in thermometry? The basic idea is define a system preferably rudimentary. Let us discuss this. Why preferably rudimentary? What is the characteristic of a rudimentary system? No two-way mode of work, so only one property of significance and then we say that look let us define that property and use that property, that characteristic as my label of temperature. So just simplicity of course, it is not easy to have a large number of rudimentary systems, but we can always restrict a non rudimentary system to become essentially a rudimentary system. For example, you have a gas in a cylinder piston arrangement, restrain it by say freezing the piston. Then you have a gas at a constant volume and if the volume cannot change, no work of expansion can be done. So, no two-way mode of work it becomes essentially a rudimentary system or you restrict a system. For example, you again have a gas in a cylinder piston arrangement. We know that for a fixed mass pressure and density or a pressure and volume will be two variables which will decide its state and two variables means we will have some confusion in labeling isotherms. So we decide to keep its pressure fixed that means we expose it to a pressure chamber which maintains pressure and we have a leak proof frictionless piston. So although it expands and contracts, the pressure remains the same. So now there is only one variable to worry about and that is the volume. The volume can then be used to label the isotherms of this state. Define a system then define standard states of the system these are known as fixed points then define labels that is temperatures for the standard states that is fixed points. And finally, have an interpolation I will take an illustration and then we will break for T. The illustration is that of thermometry using mercury in glass thermometer. One participant raised an objection to my idea that the mercury in glass thermometer is not a rudimentary system, participant I forget the name and for some reason I have simply deleted it because the mail came after my goading yesterday afternoon that put your queries on the moodle do not send it to me by email I do not want my email blocks email box flooded by queries good you have queries but put them on moodle. So everybody knows what the question is and when I reply to that everybody will know what the answer is and not only me others who understand it can propose an answer and others can comment on it let it be a good discussion forum not by individual emails. So the participant said that look there is a mercury thread there and it can expand and contract. So when it expands and contracts there will be some work done now my question is is there any work done for work done you must have two systems mercury thread movement etcetera is all internal to that system our system the boundary is external boundary of the mercury in glass thermometer whatever happens inside cannot constitute a work interaction only if the external boundary has something happening across it then it can be work interaction or heat interaction. But anyway let us come back to it I will say you have a mercury bulb I will just show a capillary and the only variable of state only property is the length of the thread in the capillary it could be here it could be here it could be here and that will decide the state of the system. So this is the first step we have defined the system the second step standard states here classically we have say the ice point and the steam point the ice point is a system containing water in which both the solid phase and the liquid phase are in equilibrium and the pressure is 1 atmosphere the steam point is again a closed system containing water in which the liquid phase and the vapour phase are in equilibrium and the pressure is 1 bar 1 atmosphere then you say that bring my thermometer this system in thermal equilibrium with the ice point and note the state we label this as isotherm 0 for the steam point we label this as isotherm 100. So, we have defined the temperatures on this we mark an L naught corresponding to 0 and an L 100 corresponding to 100 of these labels. And then we have the interpolation law we assume and we try to prepare the capillary as uniform length as possible we assume and try to make a perfect vacuum above the mercury and then we say that if you have a system which when brought in contact with this mercury in glass thermometer will have an isothermal state of the thermometer which corresponds to the mercury thread being at a length L. Then you say that for that system the temperature will be given by L minus L naught divided by L 100 minus L naught multiplied by 100. Then you say that all this thing was created by a scientist named Celsius. So, in his honor we write the unit at sea and now many of you will be familiar with the idea of temperature as a degree of hotness unfortunate phrase used in many textbooks almost all textbooks that degree is put here. So, that is historical this is in honor of the scientist who created this, but this is an unfortunate historical wastage. So, that becomes the temperature assigned to that just before T we understood the basic idea of thermometer and then looked at the mercury in glass thermometer and took as an illustration the Celsius scale. Notice what we have done we had a system in this case a rudimentary system a evacuated capillary connected to a bulb containing mercury and the only state variable of interest was the length of the thread of mercury in the capillary. We look at the arbitrariness this was selected in an arbitrary fashion the standard states or fixed points are selected ice point and steam point in some arbitrary fashion y 0 y 100 well Celsius thought it to be so and a very simple law of interpolation. So, there is a certain amount of arbitrariness in this and you will notice that the mercury in glass thermometer leads us not only to the Celsius scale, but Fahrenheit also defined a corresponding scale where he took the lower fixed point and called it 0 we call it now 0 degree Fahrenheit that was a mixture of I think the lowest freezing mixture of water and ammonium chloride eutectic mixture. Instead of steam point he took the upper point as his own body temperature named it as 96 degree Fahrenheit and we ended up with the Fahrenheit scale. So, this is somewhat arbitrary that is one the second thing is as science grew technology grew we started using systems at temperatures much higher than 100 degree C much lower than 0 degree C and we started noticing that the mercury in glass thermometer cannot be used at temperature below something like minus 40 degree C because mercury freezes and you cannot go to very high temperatures because mercury starts boiling at around 350 or 360 degree C. So, that becomes a limit. So, scientist and engineer started looking at different thermometric possibilities. So, the mercury in glass thermometer gave way to what is known as the gas thermometer and the idea here was not to use a rudimentary system, but a gas in a cylinder piston arrangement and its pressure and its volume are the two variables and hence any property of this system and mind you pressure and volume are primitive variables. So, any property of this system including the temperature would be a function of these two variables. So, we now have to define temperature as a function of pressure and volume well the two variables are not going to be very helpful to us. So, there are two possibilities one is a constant volume gas thermometer in which we freeze the piston in which case V is kept constant and temperature would then be a function only of pressure and the second one is the constant pressure gas thermometer in which the pressure is maintained constant and then the temperature is a function only of volume. After trying both finally, scientist and engineer zeroed in on the constant pressure gas thermometer as the one which is easy to implement and gives you reproducible and consistent results, because whatever experimental scheme you devise unless it is neat precise and reproducible it is not of much use. While exploring gas thermometers the following thing was noticed constant pressure gas thermometer means that pressure was constant and the volume now is a measure of temperature. Questions still remain which gas how much 1 kg 1 gram and which pressure you use a different gas you use a different quantity of the gas you get a different you use a different constant pressure you may end up with different scales of temperature. But anyway the scales of temperature were defined by using some fixed points the ice point and the steam point were good enough for very basic purposes and temperature was defined as a function of volume using some interpolation law. But then came a scientist named Boyle and he explored the state space of many gases and he came up with an interesting observation. He says that isotherms in the state space of any gas are approximated by tangular hyperbolas. What does this mean? This meant that I forgot to write on the PV plane. So, if you take the PV coordinate and you take a gas our system is a gas and you determine a set of states which are isothermal to each other you will get a curve like this. You take a different set of isothermal curves you get a set like this. Third set you will get like this, fourth set you may get like this. Each one of these is represented by P is approximately not exactly proportional to 1 over V that means P into V is approximately a constant and different constants for different isotherms. So, he came up with an idea that look if this is approximately constant then perhaps that constant which is different from different isotherms the value of that constant itself can be used as the label of a temperature. Then exploring further he realized that approximation is better low densities for any gas and this he proposed as a law which we today know as Boyle's law. Boyle's law says that for a system containing a gas low densities that means low pressures and high volumes isotherms are represented by PV equals constant and different constants for different isotherms and that brings us now this for real gases is an approximation but then he came up with the idea which we today call an ideal gas. Any gas will obey this law approximately the approximation becomes better and better at low densities but then we define an ideal gas as a gas which obeys Boyle's law all over its state that is our definition of an ideal gas. That means for an ideal gas on the PV plane each isotherm is exact rectangular hyperbola represented by PV equals and this is now used to define what we call the ideal gas scales of temperature. Notice the plural here indicating that the ideal gas scales of temperatures are not unique the basic idea is like this. Here we take as our system or thermometer a system containing an ideal gas. Now an ideal gas is a gas about which we can think about but we can get a very good approximation of it by using a gas like helium or hydrogen at very low pressure. Make a measurement of its pressure make a measurement of its volume then we know that the PV product is going to represent temperature. So we need a mapping between the PV product and temperature. Now this mapping makes the scales a bit arbitrary. We can define a fixed point and we can define a mapping and we can define an ideal gas scale of temperature. Anybody is free to do that and many people have defined you know the ideal gas for example there is a Rankine scale of ideal gas temperature but we will look at one temperature as the standardized ideal gas scale of temperature and that is the ideal gas Kelvin scale of temperature. For this the thermometer is a closed system containing an ideal gas in a cylinder piston arrangement and we have a proper measurement of the pressure and the volume. So that for any state of this the PV product can be measured and determined. So this is our thermometer that is step 1. We are going to have one fixed point. The fixed point which was realized only in the last 100 years is the triple point of water. Triple point of water we will see tomorrow when we study properties of water and steam is the state where all the three phases of water whether it is solid the ice liquid that is water and gaseous that is vapor or steam they exist together in equilibrium at the same pressure and same temperature. And since water has just one component it is a single chemical substance but three phases are together the phase rule of physical chemistry dictates that we have no choice on pressure we have no choice on temperature. If you decide on water the triple point will have a fixed pressure and a fixed temperature. What we do is we say let this pressure be whatever it is I think it is 0.006 bar or something like that we will find it tomorrow in team table. This is something which we define and for the Kelvin scale we define this to be 273.16 Kelvin. Of course immediately many of you will come up with a question why 273.16 we will see why 273.16. Kelvin later. So our thermometer is a closed system containing an ideal gas our fixed point is the triple point of water. Our temperature defined for the Kelvin scale at the triple point is 273.16 Kelvin. Note that we have now decided not to use the word degree because we know temperature as is nothing like a degree of hotness we appreciate temperature for what it properly is. So this is the third item. The fourth item for defining the Kelvin scale of temperature is the interpolation rule. What we do is the following say this is the system whose temperature is to be measured. Let us say this is our reference system containing water at triple point. We do the experiment as follows. We take our system the thermometer and bring it in thermal equilibrium with our system containing water at triple point. Measure its pressure and temperature P triple point V triple point and determine the PV product at triple point. Then of course for this we will have to experiment with its state space and keep on measuring P and V. Then let me call not at triple point but as reference because tomorrow we are going to use TPTP in a very specific meaning. Then we bring our so called thermometric system and adjust its state in such a way that it remains in thermal equilibrium with the system whose temperature is to be measured. We measured its pressure we measure the volume and determine the PV product. After doing this thermometric measurement the defining relation for temperature is T by T ref is PV by PV ref. So, here PV is measured here PV ref is measured here. T ref is defined to be 273.16 k and PV is measured here. T now is the temperature of on the ideal gas Kelvin. This is the experiment and this is the definition of the ideal gas Kelvin scale temperature and its measurement for a system. Now a few things first thing why this 273.16 k and why this particular ratio type of definition? Why not linear interpolation? Why only one fixed point? It turns out that because of Boyle's law just one fixed point is sufficient that simple algebra. The ratio and 273.16 k comes from the fact that if we define our Kelvin scale like this then it turns out that the Kelvin and Celsius scales become very nicely related to each other and that was necessary because the Celsius scale was reasonably well established by the time the Kelvin scale was proposed and standardized. And hence it would always be better if the shift from Kelvin to Celsius is a very simple shift. So, remember the Celsius scale was defined in such a way that for ice point it was 0 degrees C, steam point it was 100 degrees C. These were definitions. If you take triple point of water and measure its temperature it turned out to be approximately 0.01 degrees C. This is on the Celsius scale. On the Kelvin scale the triple point of water was defined as 273.16 Kelvin. Define the ice point now turns out to be almost exactly 273.15 Kelvin and the steam point was 273.15 Kelvin. Now see the difference on the Celsius scale the difference in temperature between ice point and steam point is 100 minus 0 that is 100 degree Celsius. On the Kelvin scale because of the definition of the Kelvin scale again on that scale the temperature difference between the ice point and steam point again turns out to be 100 Kelvin. That means temperature differences between the Kelvin and the Celsius scale are aligned. 100 degrees C temperature difference and 100 Kelvin temperature difference are almost exactly the same. That means in magnitude the degree of temperature on the Celsius scale is equal in magnitude. To degree of temperature difference on the Kelvin scale it is for this alignment that we have 273.16 K defined as the triple point of water and because of this it turns out that the temperature of any system in degree Celsius is temperature of the same system on the Kelvin scale minus 273.15. And now since we cannot have two standards of temperature then there will be confusion. So today the definition of Celsius scale is this. The historical definition of Celsius scale is what we have been looking at earlier. But today the definition of the Celsius scale is measure the temperature on the Kelvin scale subtract 273.15. Because of that what happens is on the Celsius scale 0.01 degree C is the exact value of the triple point of water. The ice point is 0 degrees C approximately but within very very good approximation I think third or fourth or fifth place of decimal. And the same thing is true of the steam point it is also 100 degrees C may be to 4 or 5 significant figures but still approximately. The only exact temperature by definitions today are the temperatures of the triple point of water which is 273.16 Kelvin on the ideal gas Kelvin scale and 0.01 degree C on the Celsius scale. And this is one temperature the triple point of water which will never be measured it is by definition 273.16 K. Now let us turn this around. Now we know how to measure the temperature on the Kelvin scale using an ideal gas. But let us turn our attention to the ideal gas as a system itself. So let us take a system ideal gas. And what is our definition of an ideal gas? An ideal gas is a gas which obeys Boyle's law all over its state space. Just now we have seen that the defining relation for the temperature of an ideal gas is this. Now let us say we have measured the temperature and now we are ready to turn this around. If we turn it around we will get pV equals pV by T at the reference state multiplied by T for an ideal gas. Now here the volume on the right hand side will depend on the mass being an extensive property. So we rewrite the right hand side as m into pV by T at the reference state which is the state which is in equilibrium with the system containing water at triple point multiplied by T. Now remember temperature, pressure, volume at the reference state for a gas each one is a property. So combination of property is also a property and that combination we write as R and we end up with mRT the equation of state of an ideal gas where R is defined as pV by T at the reference state and this is known as the gas constant for that gas. You take a different ideal gas with a different identity you will get a different gas constant. For example, we know the ideal gas is an idealization we can only think about it it is an approximation. However it is a very good approximation for real gases which we have in practice provided they are at low densities that means low pressures and reasonably high temperatures. I say reasonably high temperatures because if we go to very high temperatures real gases will start dissociating and the moment they start dissociating that system does not remain a simple system. Its composition changes it can be charged and discharged so all complications arise. So for many gases low pressures that is near ambient below ambient may be few bars few tens of bars. Temperatures from may be about do not go very near the liquefaction point but say from a few hundred degrees Kelvin to may be a 1000, 1500 degree or 1500 Kelvin that is good enough. This is a good range of approximation in which many real gases behave like an ideal gas. Lower the pressure lower the density better is the approximation. So you can make say nitrogen behave like an ideal gas. You can make oxygen behave like an ideal gas. You can make say argon behave like an ideal gas helium behave like an ideal gas. But each gas will have its own identity and hence each gas will have its own different gas constant. That is where thermodynamics ends. But the story of ideal gas is not complete yet there are two further points to be noted. So let us one equation of state of an ideal gas. We have soon we have just seen that it is p v equals m r t where r is a constant specific to that gas. It is known as that particular gas constant or the specific gas constant. I will write this as 1a. Now we take help from physical chemistry or the molecular simple molecular theory of gases and then we realize that there is something called a universal gas constant which relates to individual gas constants by their molecular weight. So we have the gas constant for any gas related to the universal gas constant divided by molecular weight. So this is the 8.314 kilo joule per k mole kelvin. I think I have the right number. And this is the molecular weight which is in kilogram per. This is the characteristic of a gas for hydrogen it is approximately 2 for oxygen it is 32 nitrogen 28 and so on. So if you know the molecular weight of any gas molecule you know the universal gas constant you can determine the specific gas constant for that gas. The second thing it is something which we have not yet realized but an important thing which will help us later when setting up the kelvin scale on absolute thermodynamic foundations is the ideal gas obeys two laws. This equation of state comes out of Boyle's law plus definition of kelvin scale of temperature. But there is another law of ideal gas experiments done by joule in standard physics textbooks on thermodynamics. A good explanation will be found I think both in Sears as well as Ziemanski definitely in Sears and Salinger is the so called porous plug experiments or the so called throttling experiments by joule. The generalization of that is joule's law. In fact joule's law and joule's experiments in some other way have resulted in the first law of thermodynamics but joule's experiments on gases approximating ideal gases have been generalized by joule and what we call joule's law today. And this says that the thermal energy of an ideal gas of a system containing fixed mass or let us say specific thermal internal energy of a given ideal gas depends only on temperature this is important because ideal gas is a gas it is a simple compressible system or a system containing an ideal gas is a simple compressible system. So in principle any property must be function of two variables we need two properties to define a state say for example pressure and volume or temperature and volume or temperature and pressure thermodynamics is two properties it does not say which two properties but what joule's law says that for an ideal gas if you measure the thermal energy as a function of temperature and pressure or as a function of temperature and volume you will realize that it is not a function of pressure it is not a function of volume when temperature is fixed it is only a function of temperature this is something which we will note just now use in our exercises but we will revisit this and see how important it is when we come to the second law of thermodynamics. So now we are at a stage where we have understood what is meant by temperature we have understood what is meant by isothermal states thermal equilibrium scales of temperature and we understood the ideal gas Kelvin scale of temperature and the ideal gas equation of state again let me summarize and extend any ideal gas will be characterized by one its molecular mass which gives us its gas constant this is one characteristic we will find that there are other characteristics which are needed from here we come to its equation of state which is pv equals mrt or pv equals rt where this small lower case v is the specific volume since r can be written down in terms of the ideal gas constant and specific sorry and the molecular weight this can also be written down as pv equals n universal gas constant t where n is the number of moles the extent of mass can be measured directly in terms of mass or can be measured in terms of moles. So, let it be n the number of moles or p molar specific volume is universal gas constant into t as mechanical engineers we will generally be using this form but we should not hesitate to use this form if it is so convenient to us the third one is the specific internal energy of any ideal gas is a function of temperature. Now, there are certain subsidiary properties which we define for any gas for example, for any fluid a very useful quantity which we will come across is the enthalpy which is defined as the thermal energy plus the pv product we will work with the specific enthalpy which is specific thermal energy plus the pv product where v is the specific volume. Now, notice that for any general fluid u will be a function of two variables. So, h will only also be a function of two variables say pressure and volume temperature and pressure temperature and volume but let us say for an ideal gas h is u which is a function only of temperature plus what is pv for an ideal gas r t. So, this implies that for an ideal gas h is a function only of temperature next for any fluid cases or otherwise we are interested in knowing how does the internal energy vary with temperature and some other convenient variable. Remember that only for an ideal gas is a u function only of temperature for other gases and fluids u will be a function of temperature and one other variable and we are also interested in knowing how does the enthalpy vary with temperature and one other variable. Later on we will realize that when we work with the thermal energy the most convenient second variable is the specific volume and when we work with enthalpy a very convenient second variable is pressure. This is not dictated by thermodynamics this is this comes out of our experience and when we develop it further you will notice that for internal energy or thermal energy temperature and specific volume or temperature and volume are the most convenient independent variables whereas, for enthalpy temperature and pressure will be the most convenient variables and not only that we are interested in knowing how does the thermal energy vary with temperature and volume and how does the enthalpy vary with temperature and pressure. It turns out that in a large part of the state space of any fluid although u depends on t and v the variation with t is significant the variation with u is not that significant and the same thing is true for enthalpy. Enthalpy is significantly affected by t not that significantly affected by p and hence we define two ratios one C v which is defined as variation of thermal energy specific thermal energy with respect to temperature at constant volume and another ratio C p which is defined as variation of h with respect to temperature at constant pressure. We will specifically write volume as a subscript here to indicate that we consider u as a function of temperature and volume and for this differentiation volume is being maintained constant and for C p we will write p to indicate that h is considered as a function of temperature and pressure and for this differentiation pressure is being maintained constant. Unfortunately I am not writing it but you know and all of us know that we still continue with the traditional names for C v and C p. The traditional name for C v is specific heat at constant volume specific heat at constant pressure but the definition and the realization should be there that C p and C v although they are known as specific heats have absolutely nothing to do with heat. Heat is a interaction it is not a property whereas u, t, v are all properties so C v is a property h, t, p are all properties so C p also is a property. So unfortunately they are called specific heats at constant volume and constant pressure respectively but they have absolutely nothing to do with heat that is only a historical wastage we continue with it. Now remember this is the last thing and I hope I complete in this slide for an ideal gas. What happens for an ideal gas? u is a function only of temperature so v does not come in. So C v turns out to be equal to d u by d t again C p turns out to be d h by d t mind you this is only for an ideal gas and since h is u plus p v you should be able to show that C p minus C v is remember that u is a function of t only for an ideal gas it does not say what type of function. So since u is a function only of t h is a function only of t C v will also be a function only of t C p will also be a function only of t. So C p is a function of t C v also is a function of t but r is not a function of t remember this. Now for simplicity quite often we will notice as we solve problems that although we have said that u can be an arbitrary function of t h can be an arbitrary function of t and hence C p and C v will also be functions of t. Generally you will find that over a reasonable range of temperature u is almost a linear function of t h is almost a linear function of t to a very good approximation and hence for solving simple problems we can assume C p and C v to be independent of temperature and we end up with ideal gas with constant specific heats and when the specific heats are constant we define another parameter the ratio of specific heats which also we will come across.