 actually I should now write 7. What is thermometry? Thermometry is a method providing and remember one thing, this is what distinguishes temperature from other properties of thermal dynamics. For example, when you talk of pressure, we talk of pressure of A system. When you talk of energy, we talk of energy of a system, but temperature is something which goes across systems. When you talk of temperature, is temperature has a system A has some temperature, but there is corresponding value of temperature for some other system. So water at a given temperature say 30 Celsius and air at 30 Celsius have something in common, but water with an internal energy with respect to some reference state of so many kilo joules and air with the same value of internal energy with respect to some reference state have absolutely nothing to do each other. They may not, there may be nothing common between those two states, but water at 50 Celsius and air at 50 Celsius have something in common. If you take one system containing water at 50 Celsius, another system containing air at 50 Celsius, bring them together across a diathermic wall. They will say that look, we know we can interact by heat interaction, but since we belong to corresponding isothermal states, we will not interact with each other and that is the reason why we have been labored with the same temperature 50 Celsius. Now, thermometer provides us a simple way to label isotherms or provide values of temperature and get rid of this arbitrariness. Although the arbitrariness does not totally go away, some sort of arbitrariness remains and hence we have different temperature scales. For example, traditionally we know of the centigrade scale now called the Celsius scale. We also have the Fahrenheit scale to some extent we still use it for medicinal purposes. When you go to a doctor, you never talk that my temperature is 38 degree Celsius or 304 Kelvin. We will not understand. We say that I am running fever, my temperature is 100 degree Fahrenheit. So, Fahrenheit scale is used there and for more detailed experiment, we use the Kelvin scale in thermodynamics, in radiation using the Kelvin scale is very convenient. So, we have a number of scales and we must have some relation between them, but before that we must have some method by which we are able to assign a temperature to a given state of a system. How to do that? That is the job of thermometry. A typical thermometric procedure is like this. We define a reference system. At this stage, we will soon see that defining or creating a rudimentary system is convenient because such a system will have only one property of significance and hence that property can be directly used for labeling temperatures. Then define one or more reference states of that system. These then define temperatures for those reference states. These reference states and temperatures are known as reference points or fixed points and then we define an interpolation schemes for scheme, not schemes for continuous labeling of temperature scale. We define thermometry as a method of providing proper and convenient labels to isotherms. For this, a more or less not standardized, but a typical procedure involves for defining and implementing a temperature scale is define a reference system. This reference system is known as the thermometer. Define one or more reference states of that system. Then we define temperatures for those reference states. These are the so called fixed points, labels to reference states and then these are only a few points, sometimes as little as 1, sometimes 2, usually 2, but quite often 1 and then in between or from that reference states we have to label other temperatures. So, we have a relation which is known as the interpolation scheme. With just one reference point interpolation is not the correct word, but let us continue using it. There is with one point there is no range. So, interpolation or extrapolation are not the proper words to use there. Let us look at the way the temperatures were defined. Let us look at the classical scales. So, the Celsius scale earlier called centigrade. Here look at first reference system, reference states, values assigned to values of temperature assigned to those reference states and interpolation scheme. Now, here the reference system is mercury in glass capillary. So, what you have is a bulb of mercury in a long glass capillary, enclosed of course in a glass shell, a very crude way of showing things. So, this is our so called thermometer and because it is solid inflexible, non expandable, untwistable, the only state which you can change, you can observe change is as you play with it, as you expose it to different environments, the length of the mercury in the capillary goes up and down. The capillary contains mercury and vacuum after that. So, there is no sort of pressure acting on the mercury, except perhaps its own vapor pressure which is pretty low. So, the with some reference point the location of the mercury thread represented by L is the only variable of state or it is only the property. Remember that this is a rudimentary system. Hence, only one property will define the state and the obvious property for us to define the state is the length L. Now, if we define its temperature T or its energy E, all those things will have to be at least the thermal internal energy E or U will have to be defined with respect to this L. Forget the other part, our main interest here is to define a temperature and hence let us start mapping by proper thermometric means this length all to temperature which leads to the definition of the Celsius scale of temperature. So, this is the first step. Second step is we have to define some fixed points that means situations and systems which we can and systems and states which we can recreate without much difficulty. So, that fixed states and fixed points which we are considered where the so called ice point which is approximately the normal mating point of water which water naturally available water the appropriate natural appendance of H2O and D2O the so called ordinary water substance and all other impurities removed from it. So, the purest water that you can get pressure of one standard atmosphere that is 101325 Pascal 1.01325 bar that is the same and state of equilibrium between solid which is the ice in liquid which is water exposed to atmosphere. So, if you look at the detail it is solid ice in equilibrium with air saturated water. This is one reference point and the second reference point is known as the steam point which is the normal boiling point of water that is state of equilibrium between pure water and it is vapor at a system pressure of one standard atmosphere 1.01325 bar. So, these are the reference points reasonably easy to create in our laboratories and I am sure many of your institutes will have a ice point used as a reference for thermocouples and for other instruments and the steam point is the ipsometer for calibration of thermometers and thermocouples. We also have to provide numerical values of temperature. Why numerical values? Because unlike color it is difficult to interpolate between colors. It is difficult to interpolate between names of famous great getters. So, numbers is something which can vary continuously and you can have to between two numbers any subdivision to any precision which you want. So on the Celsius scale the ice point is labeled 0, the steam point is labeled 100 and because we are defining the Celsius scale we call it degree Celsius. The degree is unfortunate because that brings us to the old idea of temperature being a representation of the degree of hotness. Remember the first law does not tell us which is a higher temperature which is a lower temperature. Actually this 0 C and 100 C 100 degrees C is not the first cut Celsius wanted ice to be at 100 degrees C and steam to be at 0 degrees C. But other people surrounding him told him that look hotter stuff preferably should have higher numbers that is what we feel like psychologically. Higher temperature means hotter more dangerous thing and lower temperature means cooler. Well at that time perhaps liquid nitrogen and such temperature dangerously low temperatures were not easily createable. So low temperatures were not considered as dangerous as ice temperatures and high sorry steam temperatures and high temperatures. So this where the so called reference states and reference points. So three of the four things we define the reference system that is the mercury in glass thermometer. We define two reference states and we define temperatures at those two so called fixed points reference points or fixed points. And finally the interpolation thing what you do is we measure the length of the capillary at 0 degrees C let it be L0 length of the location of the capillary at the other fixed point L100. And now if you want to measure the temperature of say your body or temperature of this water in this glass in this bottle then what will we do? We will bring this thermometer in thermal equilibrium with this. That means bring it in contact and do not do anything which would prevent heat transfer. So essentially you are providing a diathermic wall separating the two. Wait till thermal equilibrium is reached and measure the location L. And then when this position L exists then we say temperature when the state is L is defined by this interpolation law L minus L0 divided by L100 minus L0 multiplied by 100 the unit is provided degree Celsius. So these are the four steps 1, 2, 3 and 4. By this means using the basic Celsius scale idea of basic Celsius scale we could easily determine temperatures between 0 and 100 Celsius. This was not the only scale defined using the mercury in glass thermometer. For example another scientist Fahrenheit another in a cranky way he said that look 0 is not cold enough. So I think he experimented with a sort of eukaryotic mixture of water and I think ammonium chloride and he reached temperature lower than that of the ice point and he called it 0 degree Fahrenheit in his own name 0 degree F. At the upper end maybe he was sort of a cooler person than Celsius. I do not want to go as high as the steam point. He decided that his own body temperature be the other arbitrary. Why his body temperature? We know our body temperature also goes up and down and our body temperatures are different from person to person from time to time under condition to condition. So that is not a good point but he defined some temperature as 96 degree Fahrenheit. Why 96? Perhaps he thought 96 is a better number than 100 and then similarly he decided on an interpolation. Now although this gives us a basic idea of how the Celsius scale and later on how the Fahrenheit scale was decided or defined. Soon we realize that there are disadvantages in this scale. People started using boilers, people started creating high temperatures, people started melting metals including gold and they found that 100 degree Celsius is perhaps too lower temperature. They wanted to go to really, they wanted to measure really higher and higher temperature. But before we talk of higher and lower, remember that this higher and lower is not in a proper thermodynamic sense. This higher and lower is only as defined by the arbitrary Celsius scale or some other arbitrary scale. Celsius scale is arbitrary depends on mercury, depends on glass and their properties and hence there is nothing thermodynamic about a higher value of temperature on the Celsius scale and lower value of temperature on the Celsius scale. So when people started extending it by lengthening it, providing larger and larger lengths, they realized that perhaps the absolute limit for mercury in glass thermometer was something like minus 35, 36 degree C by extrapolating this on the lower end because mercury freezes around that time. Similarly as you go to temperatures higher than 300 Celsius, the thermometers tend to crack and mercury comes out because the normal boiling point of mercury is between 350 and 360 degree C. So the vapor pressure increases and the vacuum does not remain vacuum anymore. The behavior becomes complex. So maybe the mercury in glass thermometer is good enough from something like minus 20, 25 to maybe something like 250 or 300 Celsius. It is very convenient between 0 and 100, not so convenient beyond that. So scientists and engineers started looking at other systems by which we could define temperatures and physical chemists and engineers realized soon that air and many components of air like oxygen, nitrogen and gases created in the laboratory like hydrogen and helium etc. Those gases remain in the same gaseous form over a very wide range of temperatures. For example helium remains in a proper gas form till really very very low temperatures. Today we know that temperature is almost minus 269 Celsius but using the Celsius scheme there is no way we can really measure that temperature. Similarly helium, hydrogen, oxygen they remain in the gaseous form without much change in their characteristic right up to the melting point of lead, melting point of silver, melting point of gold. So scientists thought of using the gases as thermometric fluids or a system containing a gas as a thermometer and Robert Boyle he did lot of experiments and he tried to plot the isotherms of a gas. He plotted on a PV diagram various isotherms. He plotted a black isotherm, he plotted a blue isotherm, he plotted a light blue isotherm, there may be a red isotherm, may be a green isotherm and he noticed the following. He noticed that isotherm of a gas are approximately rectangular hyperbole. That means a set of isothermal states if you try to fit an equation to any one of these turns out to be PV is approximately constant not exactly constant but approximately constant. And then he noticed that if he repeats these experiments with lower and lower density of the gas that means in the same chamber put lower amount of the gas. So the same volume less mass, so less density. So he found that this approximation becomes better and better at low density and he proposed the so called Boyle's law. The Boyle's law says that at low density any isotherm of a gas is representable by PV equals constant and this constant different for different isotherms. Now physicists being perfection and mathematically oriented immediately define what is known as an ideal gas. An ideal gas is a fluid or a gas which obeys Boyle's law all over its state space. That means we do not have to worry about the density. Now ideal gas means that wherever you go PV equals constant represents an isotherm, a different constant for a different isotherm. So that means isotherms will be represented by PV equal to constant exactly. And this gives us the idea as for defining ideal gas scales remember plural of temperature. All that the idea uses is it uses the PV product of an ideal gas as the measure of temperature and if you go through the history of thermometry you will find that initially the ideal gas temperatures were based on the Celsius temperature use the same two fixed points the ice point and the steam point. And we had the two fixed point definition of the ideal gas scale of temperature. But later on it was realized that we do not really need two fixed points for defining the ideal gas scale of temperature. You need only one fixed point and it was from a physics point of view that was a better way and a more consistent way of doing things. And based on that the ideal gas scales of temperatures were redefined using one single fixed point. And one could have selected either the ice point or the steam point as the single fixed point. But then there were some disadvantages the ice point required not only pure water but it required one atmospheric pressure so some other arbitrary number does not depend on the property of water. Similarly it was difficult to create in a laboratory to a very high precision because that was an equilibrium between solid of pure water but liquid which is air saturated water. Similarly with steam point although it is only equilibrium with water and its vapor again that one atmosphere a rather arbitrary pressure comes into picture. And tomorrow if you go on to some other planet where there will be a different atmospheric pressure or maybe no atmosphere at all the atmospheric pressure as a standard does not remain does not retain its sanctity. So the ideal gas scale of scales of temperature will redefine using just one single fixed point. And since water was agreed by many as a abundant reasonably pure material which can be purified to any extent desired in a lab. So the triple point of water was used as a fixed point. The advantage here is all that we need is pure water and triple point because of the phase rule is a point in the state space of water which has a fixed pressure and a fixed temperature. We are not much worried about the pressure here we leave it to whatever it turns out to be but we define that temperature to be our fixed temperature. So the ideal gas scales of temperature you can use this as a fixed point you can use many other fixed points. But the basic idea is for any scale of the ideal gas temperature define a reference state and its temperature that is reference temperature the system would be a system closed system containing an ideal gas whose pressure and volume we can measure. And then the standard way of defining this is major PV at the state whose temperature is to be assigned major PV ref at the reference state and this is defined to be T by T ref. In this PV ref and T ref are PV ref and PV are measured T ref is assigned. So T is what is computed out. So remember that in this T ref and the reference state are defined and PV ref and PV are measured. For a given system PV ref will be measured just once and PV will need to be measured for every measurement of temperature that you wish to undertake. But again here you will notice that the reference state can be defined by different people in a different way. The T reference can also be defined by different people in different ways and there are ideal gas scales of temperature. For example, there is a Rankine scale of temperature, there is a Kelvin scale of temperature and I am sure there could be other scales of temperature. So again there is a certain amount of arbitrariness but the standard scale today is the ideal gas Kelvin scale which allows us to assign the temperature on the Kelvin scale to a state of any system. So what we do is we have our reference state which is water at its triple point and we have our system whose temperature is to be measured. All that we do is take an ideal gas of fixed mass, bring it in thermal contact with our reference state, see to it that we discover a state in thermodynamic equilibrium with this. Major pressure, major volume obtain its product we call it PV ref. This is our ideal gas thermometer that means a system containing an ideal gas. Then we bring that system in contact with our system and maybe we will have to change the pressure, we will have to change the volume so that it again attains a state of equilibrium with our system thermal equilibrium with our system T. And now we measure the pressure P volume V and major PV and using the earlier relation I will write it in a slightly different way. We now define the temperature T of this system on the ideal gas Kelvin scale as T ref into PV divided by PV ref. PV ref is measured, PV is measured, all that remains to be defined is T ref and for the ideal gas Kelvin scale T ref is defined to be 273.16 K. Now a few things again there is some idealization in this although we can create a system containing water at triple point exactly that is not an issue at all. An ideal gas is an idealization or expected perfection from a real gas. However in a laboratory you can create an almost ideal gas by selecting the gas appropriately. It is known that hydrogen and in particular helium behave like an ideal gas at reasonably low pressures for over a wide range of temperatures and pressures. So helium is a good candidate if you really want wide enough application of this. So a real gas at low pressures particularly a gas like helium is a very good candidate for ideal gas temperature measurement that is one. The second thing is why this T ref of 273.16 K? The reason is the Kelvin scale was defined much after the Celsius scale and when you define a new scale much of the data much of the information will be based on temperatures measured on the earlier scale say Celsius scale. And if you have a new standard like the Kelvin scale then naturally you want that the old data be easily transferred to the scale to the new scale and hence we want a very simple transfer method for temperatures on the Celsius scale to temperatures on the Kelvin scale. All of you are even from our school days we are conversant with the conversion between the Celsius scale temperature and the Fahrenheit scale temperature. We have all the formula which is known to us is temperatures take the temperature in Fahrenheit scale subtract 32 multiply by 5 or divide by 9 multiply by 5 and you get the Celsius scale or you take the Celsius temperature multiply by 5 by 9 and then add 32 you temperature on the Fahrenheit scale. So remember that one multiplication, one division and one addition or subtraction is involved in this. We why not have just a scale change which requires just one multiplication or better still just one addition or subtraction which is much simpler than multiplication. And again we have this idea that on the Celsius scale the ice point and the steam point are separated by 100 units the so called 100 degrees of the Celsius scale that 100 gave the old name centigrade on it that means something which is divided into 100 different graduations or 100 different parts then we started calling it Celsius. So maybe on the Kelvin scale also we would like to have ice point and steam point separated by 100 units so that maybe we have a very simple conversion formula between Celsius and 100 and that is the reason why we have selected this 273.16 Kelvin. What happens is temperature on the Kelvin scale I will write three points one is ice point one is triple point of water and the third is the steam point. On the original Celsius scale the ice point was 0 degrees C exactly whatever is doubly underlined here means exact. The steam point was defined to be 100 degrees C. So by definition these two fixed points were exact. The triple point of water turned out to be 0.01 degrees C not exactly major to a good precision when we have the ideal gas Kelvin scale we define this temperature to be 273.16 Kelvin exactly that is the definition you cannot measure this temperature on the Kelvin scale it is defined to be 273.16. If you measure on the Kelvin scale the triple point of water temperature to be 274 Kelvin then people will say something wrong in your measurements the measurement has to be 273.16 because it is defined that. Now it turns out that because of this particular selection of temperature the ice point turns out to be almost exactly 273.15 Kelvin and the steam point turns out to be 373.15 Kelvin and because of that the difference between these two temperatures ice point and steam point is 100 degrees C also turns out to be 100 Kelvin that means the reason for selecting this 273.16 and because of this the conversion between Celsius and Kelvin turned out to be at 273.15 to Celsius and you get the Kelvin scale or subtract 273.15 from Kelvin and you get the Celsius scale. Now for some time it was the situation and two temperatures were supposed to be related by this conversion but then what we have done is we have got rid of this original Celsius scale and we have today's Celsius scale. Today's Celsius scale is defined by the relation temperature of any system in degree Celsius is nothing but temperature of the same system in Kelvin minus 273.15 this is today's Celsius scale because of this the old Celsius scale has been forgotten and we have on today's Celsius scale the triple point of water to be 0.01 degrees C exactly because just subtract 273.15 from the corresponding Kelvin temperature the ice point of water now is 0.0 degrees C not exactly it is excellent approximation but it is not exactly defined to be or is measured to be 0.0 degrees C but it is good up to a very small fraction may be 1000 or even lower than that. Similarly the steam point turns out to be 100 degrees C but not exactly the exact temperature on the Celsius scale is today just 1 and that is 0.01 degrees Celsius at the triple point of water. Now this brings us to the definition of the ideal gas Kelvin scale the today's definition of the Celsius scale and before we go further we should remember that the degree symbol in degree C is just historic there is nothing about degree you can call it Celsius you can just leave C there but we continue to use the degree formula number one for historic reasons and number two in the SI system of units the unit of charge coulomb also has the symbol C and if you use degree C for temperature it reduces the confusion about the unit of charge and the Celsius unit of temperature the standard temperature scale in standard temperature unit in the SI system international system of units is the Kelvin and only Kelvin the Celsius scale or degree C temperature is measured and is noted and used only for general convenience because historically we are used to Celsius all meteorological data etc is announced in Celsius so we continue using Celsius but Kelvin scale is the standard and we never use degree that is wrong temperature is 350 Kelvin not 350 degree Kelvin and today the Celsius scale and in fact all other scales are defined in terms of the Kelvin scale.