 Welcome back. I hope you are able to solve the exercises from the exercise sheet B. Exercise sheet B consists of a few exercises in which the working fluid is an ideal gas. In real life all the gases that we encounter do not exactly follow the equation of state as modelled by an ideal gas. There are two issues here. For real gases the equation of state is an approximation. If it were not an approximation, if it were to follow this relationship exactly we would call that gas an ideal gas. Now what happens is for a real gases if you study kinetic theory which we will not, but we will appreciate that the kinetic theory says that a gas which for kinetic theory is a collection of small molecule, a large number of them has two primary effects which take us away from the ideal gas. The one is molecules have a volume. Molecules themselves have a volume. Say they occupy some space and number two is molecules have a force between molecules. This force acts at small distances, at larger distances it becomes very, very weak. But because of this volume which the molecules themselves occupy, the volume available for a molecule to move reduces. And the second one is because two molecules attract each other, the pressure existed by the molecules on the walls of the system, the enclosing chamber that also reduces. Van der Waals was a scientist who took these two effects into account and provided a better approximation to real gases than the ideal gas approximation. The ideal gas approximation is simply PV equals RT. The Van der Waals model modifies this P plus A by V square V minus P is RT. You should notice that this equation of state is known as Van der Waals equation of state. And this term is the term modeling intermolecular forces A by V square. And this term is the term which models the volume occupied by the molecules themselves. So there are two effects which are attempted to be modelled. The second thing you should notice is that now there are three parameters. Our familiar r still exists but we also have two parameters A and B. This continues to be called the gas constant whereas A and B are known as Van der Waals parameters for that particular gas. Notice that as volume increases that means low densities, V will become much larger than B. Similarly, P will turn out to be much larger than A by V square. And under these conditions the equation of state will approximate an ideal gas equation of state. It is now time for us to look at the state space of a Van der Waals gas. And let us look at the PV diagram on which a few isotherms are plotted. Here we have the Van der Waals gas equation of state plotted as a set of isotherms, pressure on the y-axis, volume on the x-axis. So what we have is essentially the PV diagram for a Van der Waals gas. Now notice that what we have said here is the reduced pressure. And what we have said here is the reduced volume. Now what does it mean? Before we see what it is, let us look at the isotherms. I have not put values on the isotherms but just the way we have a reduced pressure, reduced volume, we also have a reduced temperature. It is important to look at this point. You will notice that the isotherms above this are of one kind. As you go far above this the isotherms essentially become rectangular hyperbolas. So in this zone pressure is large but the volume is also very large, temperature is also large. In this zone the Van der Waals gas behaves somewhat like an ideal gas and hence the isotherms look like the ideal gas isotherms. The other zone is this zone. Compared to the earlier zone this zone is at lower pressures. And in this zone the isotherms have a funny shape. There is a minimum here, there is a maximum here and then beyond that the pressure keeps on reducing as the volume increases. But in this particular zone you will notice that let us take this isotherm as volume increases, pressure decreases. But then it goes through a minimum, starts increasing, goes through a maximum and then starts decreasing again. This behavior is rather unrealistic. Let us come back to this point. This point is on a given isotherm, a particular isotherm where here we neither have a minimum nor have a maximum. But this is a situation which is an inflection point. That is if you take a tangent to the isotherm the tangent is horizontal and the tangent crosses this line. This point is known as the critical point. The importance of the critical point we will appreciate very well when in a few days from now you will start looking at the properties of a very important real fluid steam, steam, water. That is what we are going to study maybe next week. And at that point you will appreciate the utility of the critical point. You will notice that below the critical point there is a zone of confusion because of this minimum and maximum. But in real life what happens is the pressure does not go through a minimum and maximum. In this zone there is a change of phase with liquid on one side and vapor on the other side. And we know that the liquid vapor evaporation process takes place at a constant pressure if the temperature is given. So at a given temperature you will have a variation like this. What we will do is we will look at the next slide where the variation is plotted. So what you notice is that at the critical point the tangent is horizontal. But in this zone where the pressure reduces goes through a minimum again increases goes through a maximum and then starts reducing again. The actual process is this process at this temperature and this process at a still lower temperature. So this is the process of evaporation. Now let us come to the terms reduced pressure, reduced volume and reduced temperature. This point which is known as the critical point let me say it is Cp. It will have a specific temperature, specific pressure and specific volume associated with it for a given fluid. And if we divide the temperature, the real temperature by the temperature of the critical point then we get the reduced pressure. Then we get the reduced temperature. If we divide the volume by the volume at the critical point we get the reduced volume. And if we divide the pressure by the pressure at the critical point we get the reduced pressure. Using reduced pressure, reduced volume and reduced temperature we can plot the isotherms of all gases which follow the van der Waals equation of state on one single plot. Now if you have appreciated this, this isotherm pertains to a reduced temperature of 1. Whereas this pertains to 1.1, this pertains to 1.2, this pertains to 1.3. Similarly these two isotherms below that they pertain to 0.95 and 0.90 respectively. Now another exercise sheet follows which puts you through a set of exercises on the van der Waals equation of state. I hope you enjoy doing those exercises. Thank you.