 So, as I mentioned in the previous lecture, the analysis that we are interested in will use a macroscopic or a black box approach that ignores internal details. So, what do we mean by this? So, when we say that a certain amount of heat is transferred to or from a device, we do not look at the exact process by which this heat is transferred. For instance, we do not worry about whether it is due to conduction heat transfer or convection heat transfer or radiation, we simply say that a certain amount of heat is transferred to or from the device, we do not worry about the details. Similarly, when we say that a certain amount of work or power is supplied to a device, let us say to a compressor, we do not really bother about whether the compressor is a centrifugal compressor or an axial flow compressor or even for instance a reciprocating compressor for that matter. We simply say that a certain amount of power is transferred, work is transferred to the device and that is used for compressing a certain amount of air, either a certain quantity of air or air is compressed with a certain mass flow rate, that is the only difference. But we do not worry about the internal details of how the work that we are transferring to the compressor is actually realized in the form of an increased pressure. So these details are usually covered when mechanical engineering students later on in their IR semesters do a course on heat transfer or fluid mechanics or turbo machines. In the same manner, we do not really worry about how a turbine converts the enthalpy of the incoming fluid into power. So, it could be an axial flow turbine, it could be a radial turbine, the exact details are immaterial. We treat the turbine itself as a black box and say that fluid enters with a certain enthalpy, leaves with a certain enthalpy and we determine the amount of power that the turbine generates. So that is one important thing that we understand or we should understand when we say that we follow the macroscopic approach. The second thing that is important in the context of macroscopic approach is that molecular level details are also ignored. We assume the working substance to be a single entity with a unique value for the properties such as pressure, density, temperature and so on. What we mean by this is this. So let us say I have a vessel like this which is filled with air and if I measure the pressure of the air at this location or at this location or at this location or any location for instance, not only pressure, any property of the air, they must all have the same value. So the density measured here, here, here or any other location should all be the same. Temperature measured at all these locations should all be the same. So the working substance, that is what we mean when we say the working substance is assumed to be a single entity. Now for this thing to hold, certain conditions have to be met which we will discuss next. And we must also keep in mind that mixing and stirring processors seem to be microscopic in nature. For instance, you know when two gases are allowed to mix and we want to do a thermodynamic analysis, we do not use concepts like Graham's law of diffusion and so on. We assume that you know they are mixed well and it is a homogeneous mixture. So we can go ahead with the analysis. We do not look at molecular effects in mixing and stirring also. Now when we say that the pressure, density and temperature must be the same. So when we say that pressure, density and temperature must be the same everywhere, what we actually mean by that is that the continuum must prevail. So what do we mean by continuum? Before we define continuum, let us just complete this thought. So when continuum prevails, properties such as pressure, density and temperature of the system under consideration, they are known without any ambiguity anywhere. Otherwise, when I measure temperature or pressure at one location, I get some value. I measure it at another location, I get some other value. Then we do not have an unambiguous value for that particular property of the system. So what is required is that we must know the property values without any ambiguity. So when we say that the pressure of air in a vessel is let us say 200 kilopascal, that means that it is 200 kilopascal everywhere and continuum prevails. So what do we mean by continuum? Let us do a thought experiment. So we assume that we have a cubical vessel of dimension L and let us say that it contains a certain amount of gas. Now we have on one of the walls of the vessel, let us say we have a viewport made out of glass which allows us to make observations of the contents within a fixed observation volume. So the observation volume is known, we can make observations of the contents of the observation volume within that volume. Now let us say that we now propose to measure the density of the gas using the following methodology. We basically count the number of the molecules at any instant in time, we count the number of molecules within this observation volume. We know the mass of the molecule and so we know the observation volume. So we can then calculate the density as the mass of the total mass of the molecules divided by the observation volume. So that is how we propose to evaluate the density of the gas at an instant. So at an instant we look at the observation volume, count the number of molecules, count the molecule mass or mass of all the molecules together divided by the observation volume. Now let us say that we have 100 molecules in the vessel. So if we start with 100 molecules inside the vessel then the measured density values will actually fluctuate quite a bit, sometimes even going down to 0, we may not have any molecule at all inside our observation volume at some instance when we look and at some other instance we may have 5 molecules, 6, 10 and so on and so forth. So the density value that we calculate will vary wildly starting from 0 to some value which keeps changing with time. So we do not know the value of density unambiguously. Now let us say that if you have another observation port and we measure the density there, the value that we get there at an instant will be different from the value that we measure at another observation port. So the values are not only varying with time at a given observation volume, they also vary across observation volumes if you have only 100 molecules. Let us say that we now increase the number of molecules progressively to say 10,000, 100,000 and so on. So we increase it to 1000, 10,000 and 100,000 then we will notice that as we increase the number of molecules the fluctuations that we are seeing in the density values begin to diminish. They do not seem to even go down to 0, we always seem to have a finite number of molecules in our observation volume and the fluctuation within the values themselves do not seem to be very high. In fact beyond a point, we notice that the fluctuations die out completely, it does not seem to matter whether we increase the number of molecules beyond the threshold or how much we increase the number of molecules beyond the threshold, we get a single value for the density not only at a given observation volume but at any observation volume in the vessel that is intuitively clear to us. Now let us do the same experiment but this time measuring the pressure instead of the density. So let us say that we have kept sensors at different locations on the walls of the vessel and as you know from your high school physics pressure is nothing but force exerted by the molecules on the wall per unit area. So we have pressure sensors which measure the force that is exerted and then we convert that depending on the area of the sensor we convert that into a pressure. So again when we do this experiment we notice that just like density the values that we measure for pressure also show similar trends or similar fluctuations which die out as the number of molecules is increased and once we increase the number of molecules beyond a certain threshold the values remain constant and the fluctuations die out completely. So we can summarize based on these two thought experiments that when the number of molecules is less the molecules travel freely for a considerable distance of time the number of molecules inside the vessel is less so the molecules can travel freely from one place to another without encountering another molecule or encountering the wall. But as we increase the number of molecules the distance that the molecules can travel freely diminishes. So the distance that they can travel before encountering another molecule or the wall diminishes and this distance is usually termed as the mean free path. The distance between collisions is on an average is termed as the mean free path and as the mean free path decreases intuitively we know that the collision frequency also increases because the number of collisions increases so the mean free path also decreases. Once the mean free path decreases below a certain limiting value then the measured property values do not change anymore. So once the mean free path falls below a limiting value which is what happens when we increase the number of molecules beyond a certain value as we said earlier when we increase it beyond a certain value we saw that the fluctuations died down altogether. And we can say now is that as we increase the number of molecules the mean free path decreases and once the mean free path decreases below a certain value and the property values do not exhibit any fluctuations and they have a constant value not only at one location but at any location in the vessel. So the gas is then said to behave as a continuum that is a very important concept that we just talked about. So for the macroscopic approach to hold continuum must prevail. So what do we mean by continuum must prevail this is what we mean that the mean free path should be very very small. Now how do we quantify very very small? Very very small can mean many things in different contexts. So how small is small is defined relative to the physical dimensions of the vessel. If the vessel itself is very very small then you know mean free path being small actually is not quite meaningful. So what we need to do is relate the mean free path to the physical dimensions of the vessel and that is what we do next. So we define a parameter known as the Knudsen number which is defined as the ratio of the mean free path lambda to the characteristic dimension L okay you may recall that we started this thought experiment by saying that the vessel is cubical with dimension L. So we take that to be the characteristic dimension okay. Typically Knudsen number when the Knudsen number is very very small continuum is said to prevail okay. Again general guideline is that once the Knudsen number falls below 10 rise to minus 2 like its 10 rise to minus 3 or even smaller then we can safely say that continuum prevails. Once it increases to 10 rise to minus 2 or above then we cannot safely assume that continuum prevails and the effect of rarefaction meaning less number of molecules in the gas will become more pronounced as the Knudsen number becomes higher and higher. But general guideline is that it should be below 10 rise to minus 2 and smaller the better for continuum to prevail. So this is what we assume when we do microscopic thermodynamics that the property values at any instant are known without any ambiguity and the property values are the same everywhere in the system regardless of where we measure the property value in the system okay. So this brings us to a close of the first module.