 Hello everyone, welcome to the module where we are discussing intermolecular forces and its impact on the properties of systems. We are going to continue our discussion on the impact of intermolecular forces on real gases and in this lecture we shall look at what are called as van der Waals equation and what are the different properties and what are the different caveats of it. Before we get into it let us just refresh our memories on what we discussed in the last lecture. In the last lecture first we had looked at what is the influence of intermolecular interactions on the properties of real gases. We had seen that there are two aspects to the interaction that is one is the attractive force that is when you bring in two molecules from far distance close to one another there is invariably an attraction but if you push the molecules even further then the repulsive interaction would kick in and that would lead to some sort of an excluded volume effect and this has its important consequences in the way the actually the gases behave that is how they liquefy and their behaviour with respect to temperature, pressure and volume. So in this context we had seen a term called as a compression factor which is basically nothing but a ratio of the molar volume of a real gas to that of an ideal gas and we had seen that this factor or the compression factor Z also depends heavily on the intermolecular interactions that is it is not a flat line as it is for ideal gas it has its own curvatures which again capture the attractive and the repulsive part of the interaction between molecules. And taking this discussion further we had looked at what is called as an Virial Equation of State where the idea was to start with an ideal gas equation and modify it in such a way that one could take into account the real part or the interaction between the molecules into account and this is what we ended up with the Virial Equation and we are told that the Virial Coefficient that is B and C are two dependent temperature however we are mostly interested in the second Virial coefficient or the second Virial till the second term because third and higher order terms are generally negligible in their contribution to the pressure. So with this we shall now try and look at another formalism to understand real gases which is called as Van der Waals equation of state and let us begin by looking at this Van der Waals equation of state. So to in order to understand this let us start again from the ideal gas equation and see how do we get to this Van der Waals equation from an ideal gas equation. So again we start with PV is equal to NRT right and this is for an ideal gas and we know that for a real gas it has two components one is attractive interactions, attractive forces and the other is the repulsive part. So now having understood or having known these two let us see if we can now try to understand or try to modify the ideal gas equation so as to capture both the attractive as well as the repulsive parts. So I am going to write the in terms of P that is P is equal to NRT divided by V alright and on this side I am going to draw a box which has gas molecules which are shown by these beads okay and let us call the volume of this container as V and now you have different gas molecules around it. So now if I were to understand the pressure of this particular let us say real system that is a gas let us say of methane or ethane contained in a particular in a container. So then let us see what do I look at. So now I have the ideal gas equation however we do know that the volume or the let us say molecules of methane or ethane do have a finite volume. So they are not the volume is not negligible or 0 they do have a finite volume. So that means the this equation the ideal gas equation from that I should subtract some part of the volume which is occupied by the molecules of ethane, methane or any of the molecule to get towards a real situation. So for that I will write minus some factor okay so that would correct my volume part. Now coming to the pressure so if I have to actually capture the real pressure of a real gas molecule such as methane or ethane I should now start taking into account the attractive or the repulsive parts. So now let us see what happens with these attractive or repulsive parts. So with this there are two things which happen one is that if I actually have if the molecules actually come together then they would attract with one other if they let us say if these molecules are now coming together or if they are forming an attract if they are having undergoing an association then the force and the frequency with which they impinge on the walls that is this would decrease right compared to them being free. So that means the attractive interaction between the two molecules actually reduces the force as well as the frequency with which the gas molecules impinge on its walls. So again repeat it the attractive part of the interaction means that I will no longer be able to hit the walls of the gas molecule with the same frequency as well as a force if it were a ideal gas because now the some of the molecules are actually held together and they formed a small complex or a cluster. So that means the pressure which will exert on the walls of this container would be lesser than it were for a ideal gas. So that means I need to subtract this by some term let us leave this blank for the time being but I need to subtract. I hope this logic is at least clear to everyone that the volume would be the actual volume which is available for the molecules to move around would be less than the ideal volume because the gas molecules do have a finite volume that is methane, ethane or other molecules and if you try to actually push one into another you would end up into the repulsive interactions and this repulsive interaction means the actual volume is actually less than what it were for ideal gas that is point number 1. And coming to the pressure term what we see is that the pressure exerted by a real gas on the walls of a container is also lower than it if it were a ideal gas and this is because of two reasons that is one is the force and the second one is the frequency. Frequency of impingement of gas molecules on the walls of a container. So these both these factors that is force in the frequency get influenced if the molecules associate with one another and form a small clusters or small kind of a complexes. So having understood that the repulsive part would lead to a reduction in the actual volume available for the gas molecule and the attractive part let us just write it here that this comes because of the attractive interaction and this comes because of the repulsive part or the repulsive forces which forces the volume cannot be 0 you will have to have finite volume for the gas molecules right. So having understood this by general arguments now let us try to go ahead and put some numbers or put some actual values for this okay. So what people typically write is that P is equal to nRT divided by V minus n times B, B is a factor which actually captures the repulsion between the molecules or it is actually nB is together is what can be regarded as the volume occupied by the gas molecules and minus A and coming to this part the attractive part let us look at these two terms that is the force and the frequency. So what do these two terms depend on that is the force and the frequency. So I hope you agree or at least see that it makes sense to say that these two quantities that is the force and the frequency are directly proportional to the molar volume to the molar concentration or the number of molecules per unit volume. In other words it is n divided by V because if you have more number of molecules then the frequency of collision would also be more and the force would also be higher as a result these two quantities that is force and the frequency are directly proportional to the molar concentration of the gas molecules right. So now what we can do is since these are two terms which are interacting so the net effect will be the square of it. So I will put a proportional to the constant A times n square by V square. So this entire thing can be written in a slightly different form that is P is equal to RT divided by Vm minus B minus A divided by Vm square. So I have just divided both numerator and denominator by n and that would give me this particular expression that is P pressure of a real gas is equal to RT divided by the molar volume minus term called as minus a factor called as B which is a constant for a given gas and this whole thing minus A which is again a constant for a given gas divided by the molar volume square. So this is what is called as the Van der Waals equation of state and A and B are the Van der Waals constants Van der Waals coefficients or Van der Waals constant for a given particular gas and what these indicate is B indicates the repulsion the extent of repulsion and A indicates the kind of a measure for the attractiveness for a particular gas and these are actually constants for a given gas and they do not invariably depend on temperature. So a point which is important to be noted here is that this equation of this kind of an argument was given by Johannes Van der Waals in about 1875 and he did not give a rigorous proof of this expression. He took the ideal gas expression or he took the ideal gas arguments and then he said let us say that the molecules have an attractive interaction because if you bring them close together they have the induced dipole-induced dipole interaction and if you bring them even further then they would have a repulsive interaction. So they have a volume which you cannot which you cannot occupy that is the molar volume of a given gas or the volume of a given gas molecule right. So with these two simple arguments he came up with this kind of equation of state and this equation of state has actually stood the test of time and it is a very nice way to phenomenologically explain the behavior of real gases. So I hope you appreciate that you do not have to mug up or you do not have to remember anything as long as you remember that how gas molecules interact with one another. Going back to our previous analogy of gas molecules being similar to human beings if they are all separated well separated then there would be no interaction among them or minimal interaction among them but the moment you try to bring molecules of people together they would interact with one another and that is what would result in this or would result in the deviation from the ideality and Van der Waals has captured this by using phenomenological arguments rather than a rigorous mathematical proof which is very important to note. So having seen the equation of state or the Van der Waals equation of state now let us go ahead and look at how does the occur or what is the nature of this curve. So for that what is done here is a plot or three dimensional plot of a pressure, volume and temperature. So you could for the timing you could just look at one of the temperature curves and neglect the others and what you see is that if you start looking at this temperature curve it goes up and then sort of comes down and then finally you do not see it here because it is cut off and then it finally again comes up and the same thing you see as you go for the next curves as well right. So if you now take a two dimensional cross section of these isotherms then one would get a graph which would look something like this right and if you remember from a previous lecture we had said that initially you would the molecules would behave similar to ideal gas until a certain reduction in the volume but beyond a certain reduction they would actually sort of undergo a liquefaction because of the attractive forces between them and that would result in a flat line correct and finally once you have all the liquid and if you try to again further reduce the volume that would lead to a repulsion because you cannot compress the liquid beyond a certain point. So that is the picture what we had in the last lecture if you now look at this particular let us say the P versus V diagram which you get out of Van der Waals equation it looks slightly different so you again start at a higher volume you keep decreasing the volume and then the pressure goes up which is normal and then at a certain juncture it actually as you reduce the volume the pressure actually goes down and then it comes up which is a bit of an anomaly so I will just try and point that out to you so that to make it more clear okay so if we if you try and take a look at this particular graph we have come here the volume is been reduced starting from the here to here right so you still keep producing the volume the pressure goes up but from here and beyond what is do is as you reduce the volume you also see a drop in the pressure which is actually counterintuitive or you do not expect that right so this is where actually the Van der Waals equation of the state does not capture the real behavior completely well that it has a deviation from the what one would expect for a real gas because if it were a real gas you would expect it to be remain constant something like this however here the Van der Waals equation predicts a drop in the pressure with a decrease in the volume which is counterintuitive to rectify this people have come up with what are called as Maxwell's lines or they have just drawn some horizontal lines such that let's say if I take this particular curve here I will use a different ink now say if I take this particular curve so what people have done is they draw horizontal line here so that an equal portion is above the curve and equal portion is below so as to sort of nullify the whole effect which the Van der Waals equation predicts so this is a bit of unrealistic or unphysical or a situation which does not take place in the real word with real gases that is a drop in the pressure with decreasing volume so to account for that you draw these kind of lines which are called as Maxwell's lines and this is used to do a correction to the Van der Waals equation alright so now let's go ahead and try to look at what are the main characteristic features of this Van der Waals equation so I will again rewrite the Van der Waals equation so that we all remember what we are talking about that is the P is equal to RT divided by Vm minus B minus 8 divided by Vm square okay so I hope everyone remembers this and once we have this particular equation so then the following features become pretty obvious that is the first one is at high temperature and at a very large molar volume then the equation should or equation 10 towards an ideal gas that is let's say if I have the temperature term which is pretty high that means this term the first term would be very high and if the molar volume Vm is also very very high or we are looking at a very large volume then the second term actually becomes vanishingly small compared to the first so then this equation would tend towards P is equal to RT divided by Vm with the B factor being very small in comparison to Vm that is if Vm is much much bigger than B right that is what we are talking about that Vm is very very large so in other words the Van der Waals equation of state can actually be also be used to describe in the limiting cases the ideal gas equation that is when the temperature is very high and the molar volume is also very large and the second important feature of the Van der Waals equation is it shows the existence of the liquids and the gases in what are called as Van der Waals loops or if you just to refresh your memories I hope you remember in this I will just show a small curve here volume versus P we had these kind of curves and we had called them as Maxwell's lines this is a Maxwell line Maxwell line and these are what are called as the these are what are called as Van der Waals loops so these loops actually predict or at least tell us about the coexistence of gas and liquid so this is very similar to what we saw even for the even in the last lecture where going from the going till the if you remember the the letters we are going from C to E there was a complete liquefaction which was taking place from E onwards the liquid was incompressible so you had a huge rise in the temperature you had a huge rise in the pressure correct so this is very similar to that and the third and the most important the point is that the Van der Waals equation of state can also be used to derive what are called as critical constants in terms of the Van der Waals coefficients so I will just try and explain that to you in a minute so if you again take this particular kind of an expression or let me try and draw it here a small one so if I if you have a point where actually the if you have along the PV curve a point where actually you do not have the or the liquid and the gaseous coexistence line actually coincide or they become in they merge into single point and then it goes up then this one would call it as a critical point or the TC or the or the volume as we see in the pressure as Pc the corresponding component that is the temperature pressure and the volume are called as critical temperature critical volume and critical pressure so at this let us say at this line if you now take the derivative and the double derivative they should both equate to 0 and if you use just use that the this mathematical statement that is at this critical temperature and pressure and volume the Van der Waals equation of state or the first and the second derivative of Van der Waals equation can be equated to 0 then just by doing a bit of algebra you can end up on these terms that is the VC and the Pc and the TC in terms of the Van der Waals coefficients that is A, B and the universal gas constant R so this is a very useful way to actually calculate the critical constant for a given gas and also to actually check back or to go back and look at the Van der Waals coefficient which you obtain for a given system are they actually correct based on the experimental measurement of the critical constant that is temperature, pressure and volume alright so now we have looked at what are the different features of the Van der Waals equation of state and we also talked a bit about the critical temperatures in the critical volumes so now let us go ahead and look at this in a little more detail what are these critical temperatures pressure and volume mean so to understand that again let us go back to the pressure volume graphs so I hope you remember that we started a very high volume and then we as we start compressing then the at a certain point you at around point G you have a liquefaction which starts taking place and that continues till F and from there on it is the liquid becomes incompressible right and in between the G and F you always have a coexistence of the gas and the liquid and this is what I tried to draw in the previous graph where you have a what Van der Waals equation predicts are these dotted lines and the shaded areas are called as Van der Waals loops so if you now keep on changing the temperature that is if you keep on recording that at various temperatures because these are isotherms so this is at let us say T1 and this is at T2 and this is at say T3 and you see a point where these two the G and the F actually come back and merge to this K right and this is the point which is called as a TC where the where both the liquefaction is complete at the single point and this is what is called as a critical temperature and the corresponding pressure and the volume are called as the critical pressure in the critical volume all right and this can be represented in a slightly different way in what are called as PT phase diagrams or a pressure and temperature phase diagrams where you have a temperature on the x axis and let us just try and look at this part that is you have a vapor and a liquid phase because that is what we are trying to do you go from a gas to the vapor phase to the liquid phase and what you see is that beyond a certain point you have what is called as a critical point and here and beyond beyond this pressure and temperature here what is one calls it as a supercritical fluid or a state in which the properties of the system are in between those of a liquid and a gas and so you must be wondering what is this sort of an enigmatic term is calling it as a supercritical fluid. So the main idea of the importance is that the properties of this particular state which is called a supercritical fluid are somewhere in between those of a pure liquid and a pure gas so what that means is that you can actually have you can make use of both the properties and you use it in some sort of an application to give you a feel for the applications if you think of carbon dioxide and if you take a look at the critical temperature and pressures of carbon dioxide the critical temperature is around 31 degrees C and the critical pressure is about 72 atmosphere so by now varying the pressure you can go to this critical supercritical fluid phase and in this supercritical fluid phase you have a entirely different properties and the properties are they have very different solubilities they have very different let us say densities and this can be made use to extract different materials in a mixture. For example if you have a coffee beads and if you are trying to extract caffeine out of it actually using a supercritical carbon dioxide is a very very benign and a sustainable way to actually extract caffeine out of the coffee beans and also by just playing around with the pressure one can selectively extract different components so that is the reason why a lot of technological importance and relevance is given to supercritical fluids and I hope this convinces you or at least gives you a flavor of why one should study the supercritical phenomena or the supercritical fluids so with this we shall stop our discussion here and in the next class we shall look at potential energy diagrams and what do they tell us about bonding in molecular systems thank you