 So, welcome to the 20th lecture of cryogenic engineering under the NPTEL program. In the earlier lecture on cryogenic engineering we are talking about gas separation and we are going to continue talking on gas separation. A mixture composition can be represented by either by volume, mass or molar basis molar fractions this is what we did this is what we understood in the last two lectures. Then we found out the expression for work of separation and this work of separation is represented by W im by N n that is work of separation of mixture ideal separation ideal work of separation of mixture per mole of mixture. Similarly, the ideal work of separation of mixture per mole of gas A or similarly per mole of gas B when a mixture is made up of gas A and gas B additionally it is also represented by work of separation of A only from a mixture of A and B per mole of A or work of separation of B only per mole of B or work of separation of C only per mole of C for a mixture of gas A gas B and gas C. So, there are two possibilities three possibilities three component mixture possibilities there could be N number of components in a mixture. This N number of components can all be separated at one time or only one component can be separated at out of this N components and the work of separation of this one component for a mixture from mixture is different than a work of separation of all the components of the mixture at the same time is different. This is what we have seen last time we also solve some tutorial problem to understand the values associated with this. The ideal work of separation per mole of mixture with N constituents is given by this formula. So, we had studied the derivation of this formula at the same time we solved some problems to understand what is the relevance or what is the significance of this formula in order to get ideal work of separation per mole of mixture per mole of gas A per mole of gas B or per mole of gas C. So, it is RTM is equal to sigma j to 1 y j log 1 upon y j this is the ideal work of separation per mole of mixture where y j is a mole fraction of the j th component. This is what we studied last time what is important here we are going to talk about now the mixture composition of different moles different components and therefore, all these mixtures are governed by some rules and therefore, it is very important to understand these rules there are some laws associated with this and now we will spend some time in understanding what are those basic rules basic laws which govern this mixture from this point of view the outline of today's lecture is basically we want to try to understand what is this Gibbs phase rule what is the phase equilibrium curve what is the temperature composition diagram because the information which we get from these rules and curves and diagrams is very important in order to understand the separation of different gases and from that point of view this lecture is dedicated to understand Gibbs phase rule phase equilibrium curve and temperature composition diagram. So, a mixture can have various components and can exist in various phases in thermal equilibrium as you know we can have different components and therefore, different phases which are in thermal equilibrium. For example, a mixture of ice and water is a one component and a two phase mixture obvious you got a solid and liquid phase stick in thermal equilibrium with each other and it is only a one component. If number of components and number of phases in thermal equilibrium are denoted by C and P respectively C is number of components and P is number of phases then for above mixture C is equal to 1 and P is equal to 2 similarly we got other possibilities of having mixture wherein we got a boiling liquid air boiling liquid air will have locks that means liquid oxygen liquid nitrogen nitrogen gas and oxygen gas. So, we got a two components who are present in two phases liquid and gas liquid and gas. So, the values of C and P here will be two two component that is nitrogen and oxygen and two phases that is liquid and gas every mixture now can be uniquely defined by state of properties called thermostatic properties. So, when we got a two components one phase two phase whatever number of phases we will have to define this mixture by some unique properties which are called as thermostatic properties. So, all these things which we are doing is basically to understand if I want to define a particular mixture how many property should there be in order to define that mixture. These properties can either be intensive properties or extensive properties what is intensive means they are independent of mass extensive means they depend on mass. What are these properties for example, pressure temperature density are examples of intensive property they do not depend on how much mass is there. These are independent of mass properties and there are standard characteristics of a given gas or basically any constituent we are talking about while volume enthalpy entropy are full example of extensive properties because they will depend on what is the mass of the system. So, these parameters definitely depend on the mass and therefore, they are called extensive properties. For any mixture there are certain minimum number of intensive properties normally I would look for intensive properties in order to get all the details of a any component or any mixture. So, for any mixture we need to have certain minimum number of intensive properties to define the complete state alright. In other words these properties are required to look at a unique point on temperature entropy diagram or the pressure temperature diagram or pressure enthalpy diagram etcetera corresponding to the unique state of mixture. So, as soon as I know that a mixture should have 2 or 3 properties to be known. So, that I can locate the properties of these mixtures they are called a important properties or they are called a intensive properties of this particular mixture. The Gibbs phase rule is used to determine these properties or we can call them as degrees of freedom also because the degrees of freedom will define a particular system alright. So, similarly the number of properties or degrees of freedom will define a particular mixture or a component also for a given mixture. So, how many components are there, how many phases are there which will decide how many properties should be known to us, how many degrees of freedom should be known to us to define this particular mixture. So, the Gibbs phase rule was formulated by Josiah Wilyard Gibbs an American physicist in 1878. So, this is a very old rule, but with a very important rule. If f is the degrees of freedom or the minimum number of intensive properties required to define a thermodynamic state of a system this is what we talked about then the Gibbs phase rule is given as f is equal to c minus p plus 2. Now, many of you possibly will know what is this Gibbs phase rule, but this Gibbs phase rule is play a very important rule as far as cryogenic mixtures are considered and that is why I am repeating it from basics. So, f is equal to f is nothing, but the degrees of freedom or minimum number of intensive properties c is number of components p is number of phases and therefore, f is equal to c minus p plus 2. Now, let us see how do we define different mixtures for example, for gaseous nitrogen we have c is equal to 1 that is nitrogen only p is equal to 1 that is gas only and therefore, f is equal to 2 that means, we require two minimum properties to be known to us in order to define a state of nitrogen at any given time. This can be pressure temperatures or pressure specific volume etcetera. Let us see for example, on this diagram which is a temperature entropy diagram and most of you are familiar with this temperature entropy diagram. Now, let us see that known points are p and t this is a pressure and temperature here and I need to locate a point in the gaseous this is the gaseous reason as you know this is a dome inside this dome what we have is a two phase region, but we are talking about a single component and single phase and therefore, we are in gaseous region and therefore, I want to locate a point in the gaseous region what do I need to know here in order that I locate this point a I note to know pressure and temperature. So, if I know these two points the intersection of this point is a unique point and this defines this point a. So, let us say that for a given or known p and t lines in the gaseous region the intersect at point a as shown and there are pressure and temperature. It is clear all other properties having known this point a we can find all other properties of this gas component at point a and this is unique point now and therefore, all the properties like specific enthalpy specific entropy specific volume etcetera all this point can be located from the same diagram. So, basically what is important here to know the degrees of freedom are two intensive properties or the intensive properties required to define this component is only two. Now, I got a second example similarly for a two phase mixture now instead of having one phase we got two phases now. So, for a two phase mixture of let us say helium this diagram is all of you know this diagram this is a phase diagram of basically liquid helium both the phases are in thermal equilibrium what we have therefore, is p is equal to 2 c is equal to 1 only component only one which is helium, but the phases are 2 c is equal to 1. So, by Gibbs phase rule f is equal to c minus p plus 2 which will give me 1 which means that if I have got two phase of helium I will need to know only one property and once I know this property I can look at this point how do I know this thing this property can either be saturation temperature or saturation pressure. So, how will I do I got this pressure temperature diagram of helium as shown this is the line which gives vapor and liquid in thermal equilibrium. So, if I want to locate any point any two phase point which is in thermal equilibrium both the phases are in thermal equilibrium as soon as know I know pressure I can locate this point on this curve. So, or even if I know temperature I can draw a vertical and I will locate this point on this vapor liquid phase diagram here I got a vapor I got a liquid on this side and this is nothing but a vapor pressure curve. So, if I know the vapor pressure curve because this is a known characteristic of helium. So, as soon as I know one property may be pressure or may be temperature I can locate this point A. So, let us say the pressure p is known it intersect the vapor line at A the corresponding temperature can be known immediately and that means I can calculate all the values associated this point A which means that I should know only one point or one degree of freedom or one intensive property may be pressure or temperature in order to locate this A point in this phase diagram why because there are two phases which are in thermal equilibrium with each other and only one component. So, this gives me the second part now let us come to the phase equilibrium curve and I got a mixture of A and B and again for a mixture of gas A and gas B now there are two components till now we are talking about about only one component but if I got two components gas A and gas B I got C is equal to 2 now I got p is equal to 1 in this case. So, they are both of them are in gaseous state therefore, p is equal to 1 C is equal to 2 by using Gibbs phase rule I now have a phase equal to 3. That means I should know three properties intensive properties or degrees of freedom is 3. So, what are these properties these properties are pressure temperature this is what we had for single phase in addition to that we are going to have a mole fraction of one of the components. So, when you have got a two components mixture both of them in the same phase which is gas phase I should have three properties to be known to me in order to locate and their pressure temperature and mole fraction of one of the components this will give me all the properties associated with this gas mixture and this is very important. In general phase of one component system is governed by pressure and temperature but for a two component mixture at a given pressure the mole fractions of the components in vapor and liquid phases change with the temperature and that is why we need to have a molar fraction the third property. The two component mixture at a given pressure the molar fractions of component in vapor and liquid phases change with temperature and therefore, we need to know this molar fraction. So, variation of this mole fraction why with temperature T is what is given by temperature composition diagram or a phase equilibrium curve this is what we basically are going to know this we just found that for a two component mixture we should know temperature pressure and molar fraction and this is what is given by a temperature composition diagram or T versus mole fraction for a given pressure and this is exactly what we want to study now for mixtures which is temperature composition diagram or phase equilibrium curve there are three typical curves of which I will explain to you in the next slides. So, we basically have understood now I need to know three components and these three components I can understand from temperature composition diagram. So, this is the typical temperature composition diagram which we will come across. So, there are two components consider a two component system A and B. So, here I got A is equal to 0 that means B is equal to 1 on the x axis I am plotting basically mole fraction on the y axis I have got temperature and this curve is given for a constant pressure P. We will go in detail of this diagram, but what you can understand here this is A is equal to 0 and A is equal to 1 that means in this direction A increases and in opposite direction B decreases. So, consider a two component system which is A and B at some pressure P for which this diagram has been drawn and temperature T. The critical pressure P c of both these components are more than the pressure P. When you got a such a diagram and we will go in the details of this diagram later what I want to say is when this diagram goes from A is equal to 0 to A is equal to 1 it means that the critical pressures of both the components are more than the pressure basically. So, basically in this case the P c of these components are more than the pressure for which this diagram has been drawn. It means that both the gases now are below the critical pressure. Both the pressure for which this diagram has been drawn at this pressure both the gases have this pressure is less than the critical pressure for both the gases basically. The plot shows the variation of mole fraction of the components with temperature. So, this is one of the types of the diagram. For example, consider a mixture of nitrogen and oxygen at one atmosphere ok. So, this is the case become oxygen equal to 0, oxygen equal to 1 and oxygen equal to 1 will have its boiling point as 90 Kelvin at this point and nitrogen equal to 1 will have 77 Kelvin at its boiling point because the mixture is at one atmosphere. It is clear that above 90 Kelvin the mixture is in gaseous phase and below 77 Kelvin it is going to be liquid phase. So, this is 90 Kelvin this is 77 Kelvin below 77 Kelvin the mixture is going to be completely liquid phase above the 90 Kelvin is going to be gaseous phase. The area formed by these two curves indicated two phase region. So, this area is going to be a two phase curve here which is going to be liquid plus vapor. So, below is liquid top is gas and what you have is a vapor liquid and liquid plus vapor in this case. Now, in this case this is the second time of a diagram which is a phase equilibrium curve. You have got a two component system again A and B at some pressure P and temperature T, but you see this diagram is not going from this end to that end. That means, the critical pressure is playing a role here. The plot shows in which one of the components has a critical pressure which is less than the pressure P. So, this diagram has been drawn for a some pressure P, but one of the components here has a critical pressure less than this P value and that is why what you see here is only gas what you see here not two phase over region. This was not true in the earlier case in the earlier case the critical pressure was more than this P for both the gases, but here the critical pressure for one of the gases is less than pressure for which this diagram has been drawn. And therefore, above a particular mole fraction of let us say A what you see is not this region at all there are only gas in this case. And this is a very typical case normally we will not come across such a case unless we want to draw specific for some cases. There is no liquid phase after a certain temperature and mole fraction and that is what you can see after particular temperature and mole fraction there is no liquid phase. And this is a very typical diagram of a two component mixture. So, you can see beyond this value there is no beyond this mole fraction there is no liquid. There is one more typical diagram which we hardly will come across, but I just want to complete it for the sake of giving you all the three types of this phase equilibrium curves. And what you see here is a typical behavior on this side. So, at this point you can see that both the vapor and liquid are actually at one point that means you cannot possibly differentiate between vapor and liquid at this point. So, few substances when mixed in certain proportions. So, this happens typically at one molar fractions when A is equal to this much and B is equal to so much. At this particular molar fraction only you find that both the phases are coming together when mixed in certain proportion they physically behave as one substance. They are actually behaving like one substance you cannot differentiate between these two vapor phases or liquid phases at this particular molar fraction. That means they are actually showing a single component behavior over here. And therefore, they behave as one substance in this case. For example, mixture of acetone and chloroform at a particular proportion a mixture of acetone and chloroform will show such kind of behavior. In the figure at T and fraction of Y. So, a particular temperature T and vertical fraction Y the mixture behaves at as one substance. Such mixtures are called as azeotropic mixtures or constant boiling liquids. What are azeotropic mixtures? The liquid is constantly boiling you know you cannot differentiate between vapor and liquid that is why it is called as constant boiling liquids over here. It is undesirable to separate such mixtures by rectifications. Now, normally such mixtures cannot be separated by what we call as a distillation or rectification because vapor and liquid cannot be differentiated. They are being azeotropic in nature. The mixtures cannot be separated past this composition. Once we heat this composition you see on the left side of this your vapor and liquid can be differentiated and therefore, this mixture can be separated. So, below a particular mole fraction if you got a mixture then one can separate because the vapor region and the liquid region you can see that you got a vapor and liquid separated out over here. But on the right side of this molar fraction the material behaves as if this is only one component and therefore, it cannot be separated on the right side of it. Cryogens rarely exhibit such mixtures. We will not come across this situation in cryogens and therefore, they will not be taken. So, having understood the importance of temperature composition diagram for a mixture, let us see the temperature composition diagram for a binary mixtures that means two component mixtures. In gas separation the first type of diagram which is what we I just showed you three types of diagrams. The first type of diagram is of great importance and this is this diagram. So, from here you can see that we got temperature on the y axis and the mole fractions on the x axis. This diagram is a very characteristic diagram of A and B components for a given mixture. This is a constant pressure P temperature and molar fractions are the three degrees of freedom required for a two component mixture for two phases. So, here is a diagram which gives you all the details and now we will understand this diagram and we will be using this diagram hence forth for all the understanding later. So, let us come to the process of rectification is best understood with these curves and therefore, as I said we will stick to these diagrams now. Hence, it is important to study the temperature composition diagrams to estimate the composition of vapor and liquid phases. So, basically from here from this diagram I can get the relationship between temperature and composition of A and B or a mixture and then I can understand what is the state of that particular mixture how much vapor how much liquid how much molar fraction of A how much molar fraction of B will be there and this is very important when I go for gas separation technique or rectification technique. So, this is the typical diagram we just talked about it can be put for let us say nitrogen and oxygen I have got a nitrogen equal to 1 from this point and nitrogen equal to 0 over here oxygen equal to 0 at this point oxygen is equal to 1 at this point. So, consider a mixture of oxygen and nitrogen at a given constant pressure and this is a typical diagram. Now, this diagram can be put in this form and here what you get is oxygen equal to 1 and therefore, this is nothing, but the boiling point of oxygen at 1 atmosphere which is 90 Kelvin and this point is the boiling point of nitrogen at 1 atmosphere which is 77 Kelvin in between lies both the components here O 2 is equal to 0 and N 2 is equal to 1. That means, single component at this point there is a single component at this point which is oxygen and therefore, what they have is a boiling point of oxygen and nitrogen only. Now, this diagram can also be put in this way also in some certain books in reference books you can have instead of having oxygen equal to 0 you can have oxygen equal to 1 that means, the higher boiling component and the lower boiling component could be on any side. So, once you see this point this is a higher boiling component because the boiling point of this is higher than the boiling point of this. So, by looking at this diagram one should be able to and immediately understand that in this case the higher boiling component is on the right side and therefore, oxygen equal to 1 in this case. While here the left side of the diagram shows higher boiling point because this temperature is more than this temperature and therefore, one knows that here oxygen equal to 1 in this case. So, you may come across different diagrams from different reference books you do not have to get worried about this is only that is O 2 is equal to 1 and N 2 is equal to 0 that means, low boiling component is 0 in this case while low boiling component N 2 was equal to 100 percent or 1 in this case. So, if the components on the x axis are interchanged the diagram would look like this everything remains the same the whole theory remains the same what is different is only the axis are changed. So, here what we have 90 Kelvin at this point and 77 Kelvin at this point instead of having it at this point in this case it is important to note these two plots are one and the same as I said either of these plots are commonly used in literature. So, as I said you can find both the types given in different reference books. Now, similarly these diagrams could be at different pressures. So, moment I say one atmosphere I know 90 77 Kelvin are the respective boiling points moment I know the two atmosphere I got 97 and something like 87 or 88 Kelvin as the boiling point of nitrogen at two atmosphere. Similarly, at three atmosphere I got a boiling point of oxygen and boiling point of nitrogen. So, temperature composition diagram for nitrogen oxygen mixture are shown at different pressures these plots are obtained experimentally. These values these are very important plots and we will need these plots normally they are obtained experimentally and they are a very strong function of intermolecular forces. So, experimental values will take care of all these things one can compute however theoretically also the theoretical plots can be drawn, but based on some assumptions. So, one one will be able to draw these diagrams on a theoretical basis on ideal basis also and these are the very important diagrams as I said temperature composition diagram temperature concentration diagram whatever they are called as for a different pressures. Now, look at this typical diagram which will come across all the time. So, consider a temperature composition diagram for a mixture of oxygen and nitrogen at a pressure of one atmosphere when I say one atmosphere the boiling points are fixed 77 Kelvin 90 Kelvin 77 Kelvin nitrogen equal to 1 90 Kelvin oxygen is equal to 1. So, this is what I just said let the initial state of the mixture be at 0.1. I want to show you what happens when the mixture any mixture of oxygen nitrogen is cooled from very high point and it comes through this down. So, what happens to different phases basically. So, let us look at a point at 0.1 which is in gaseous phase it is a mixture at the molar composition at of this mixture is going to be this point let us say 0.5 0.5. So, oxygen is increasing in this direction when nitrogen is decreasing in this direction. So, at this point oxygen is 50 percent nitrogen is 50 percent the molar fraction of oxygen is 0.5 the molar fraction of nitrogen is 0.5. Since the temperature at 0.1 is above 90 Kelvin that means a higher boiling component also the mixture exists in completely gaseous state right. So, at this point the phase is equal to 1 actually the upper curve is called the now this is a curve which we are going to talk about which is a red curve and you got a violet curve. So, red curve is called dew line and this bottom one is called as bubble line and why do we call by thisness we will understand this. So, there are basically dew point or dew line and bubble lines. The area formed by these two curves indicated two phase region as I return here you got a vapor over here you got a l plus v over here and therefore, it is a two phase region these two lines and enclosed region actually indicate the liquid plus vapor region and below this what you have is a liquid region. It means that it has both liquid and vapor phases this is what I just talked about. Now, let us see what happens with this point let the mixture which is at 90 plus temperature at this point in time is being cooled at constant pressure and therefore, I am cooling it down keeping the same molar concentration. So, I am coming now I am cooling it from 90 plus temperature to somewhere here which is at this point and it hits this dew line as I said. What is happening at this point? Suddenly this single phase mixture gets converted to two phase region. So, you got a l plus v now and therefore, the first dew will appear and that is why this is called as dew line. When the mixture is cooled it gets condensed at this point or some liquid and some vapor or a two phase region would appear over here and that is why this is called as dew line. When the temperature of the mixture reaches to 0.2 g which is this the mixture starts condensing. So, the 0.2 g now lies on the dew line and the first drop of the dew appears in the mixture this what I just said and therefore, this is called as dew line. Therefore, the mixture at 0.2 g is a two phase mixture with liquid vapor in equilibrium. The moment you decrease the temperature for 1 to 2 g what you have is a two phase mixture. At two phase now you have got a liquid and vapor both of them are in equilibrium both these phases are in equilibrium. So, I say the mixture is having a molar concentration which is same as this which is at 0.2 g, but it has got vapor phase and liquid phase. What are the concentrations of oxygen and nitrogen in vapor and liquid that is what we want to understand now. The condensate liquid has mole fractions of both high boiling component and low boiling liquid. So, we can say high boiling and low boiling sometime they are referred as high boiling component low boiling components high volatile and low volatile we will talk about that later also. The liquid content how do I get liquid this is a dew line which normally is a kind of a saturated vapor line and this is a liquid line and below this what you have got a liquid line. As I said the vapor and the liquid are in thermal equilibrium that means, the temperatures of both the phases is same. So, the temperature remaining same that means, I will have to draw a horizontal line this horizontal line will heat dew line at one point will heat the bubble line at one point and therefore, these two points will give me the vapor and liquid points respectively. Extending a constant temperature line about 0.2 g it intersects the bubble line at 0.2 f that means, because I have just said that the vapor and the liquid line they will be in thermal equilibrium. Therefore, it will heat this point at 2 g and retreat at 2 f respectively alright. So, this is intersecting the bubble line at 0.2 f. So, 2 g and 2 f denote the mole fractions of high boiling liquid in gaseous and liquid phases high boiling component in gaseous and liquid phases respectively. So, you have got a 0.2 g which has got oxygen is equal to 0.5 which is a high boiling component while you have got a 0.2 f which has got oxygen in very high proportion 0.8 or 0.9 proportion. So, this is liquid and this is basically the vapor point. The compositions are y vapor I will refer to this as a y vapor. So, in a vapor region the y vapor will have a 2 g concentration. So, around 0.5 is a oxygen vapor and 0.5 is a nitrogen vapor, but in liquid which is at 2 f I got a y liquid at 2 f that means, almost 80 percent of this liquid will be having oxygen while 20 percent will be nitrogen. And this is very important that this liquid will have more of high boiling component at this particular temperature while the molar fraction in a vapor will have 0.5 and 0.5 each. This we have just cooled up to 2 g now. The mixture is now cooled to a 0.3. So, if I come down further I will come to a 0.3 as shown in this figure and therefore, I am cooling this liquid now cooling this mixture from 2 g to 0.3 that means, it has come straight in the heart of this two phase diagram. Are you understanding now? Basically, I was at some point over here and now I have come down to something like 85 Kelvin or something like that or 83 or 84 Kelvin like that where I am in the heart of this L plus V region. So, if I now want to see what is my molar fractions of A and B or oxygen and nitrogen in vapor or in liquid fractions. So, I want to see the molar composition of vapor and molar composition of liquid. What will I do in that case? I will again draw a horizontal line because the vapor and liquid are going to be in thermal equilibrium with each other. The horizontal line where it hits the dew point or dew line I will get a vapor composition at this point and the line which touches the bubble line at one point I will get a liquid composition at that point. So, again extending a constant temperature line to the left and the right of this point 3 we have the following. The line extended to the left side intersects 3 g. So, from 3 to I go to 3 g when I am extending this line on the left side and where it intersects the dew line. Similarly, if I extend this line to the right side it hits the bubble line at point 3 f and the line extended to the right side intersects the bubble line at point 3 f. All right. So, I have got now two more points which is 3 g and 3 f and 3 g basically will give me the molar fraction of the vapor in which oxygen will be so much percentage and nitrogen will be so much percentage while the point 3 f will give me the molar composition of the liquid. How much oxygen is there and how much nitrogen is there in the liquid which is in thermal equilibrium with this 3 g point all right. The compositions of the high boiling component at point 3 are given as at point 3 g are given as y vapor is equal to 3 g. So, oxygen now which is the high boiling component has got less number of molar fraction as compared to the previous point. At the previous point it was 0.5, 0.5, but as the mixture is cooled further down the high boiling component oxygen has become less in this while the y liquid this is the liquid point you see the oxygen percentage in this case also has gone has decreased as compared to what it was at point 2 f and this is a straight result of cooling of this mixture from 2 g to 3 point. So, in short what has happened is when you cooled from 2 g to 3 in a in a vapor region the high boiling component decreased also in the liquid region high boiling component decreased. It means that the low boiling component increased both in vapor and liquid region and this is what we want to do when we are cooling a mixture from a high temperature to low temperature that means as you go on go down cooling this mixture the percentage of low boiling component will start increasing both in vapor region and in liquid region. So, let us see what happens if we cool this mixture further down on further cooling of the mixture the temperature reaches a point 4 f. So, if I am now going to first cool down from 1 to 2 g from 2 g I got 2 g and 2 f then I cooled it further from 2 g to 3 point where I got 3 g and 3 f in which we just saw that the high boiling component in vapor and in liquid both decreased it means that the low boiling component in the mixture in the liquid and both in the liquid and vapor from increased. Now, if I cool it further down to 4 f at this point it will hit my bubble line. So, now I have cooled the mixture from dew line to bubble line. So, I have passed through complete two phase region and now I have below this bubble line what we have only liquid. So, if I want to see now what is the composition of this mixture when I am reaching this temperature which is going to be it is around 80 Kelvin right at 77 Kelvin what you have is a nitrogen boiling point. So, for any mixture which is between 77 and 90 Kelvin I got a bubble line at this point let us see what are the compositions of this mixture. So, at this point most of the high boiling component of the vapor is condensed alright as you are coming down the high boiling component is getting condensed and also low boiling component is showing up now basically. So, as you cool down the temperature the high boiling component gets condensed first and that is what we saw here at point 2 f and as you come down further the low boiling component also start showing up because the low boiling now starts getting condensed slowly and steadily. So, extending a constant temperature line. So, if I want to see now what is the molar composition of the vapor at this point and what is the molar composition of the liquid at this point the liquid is going to be given by the same point which is 4 f while if I draw a horizontal line I will get a 4 g point which will give me the vapor composition. So, if I draw horizontal line which is the constant temperature line I will hit 4 g. So, this is my point 4 g as mentioned earlier this curve is called bubble line because when the liquid mixture is heated the first bubble or vapor will appear. So, here we have come from dew line and we we understood that when this gas mixture is cooled the first dew or condensate would appear at this point and therefore, it is called as dew line. Similarly, if I am at this point and if I increase the temperature of this liquid slowly at this point this is going to the first point where the bubble or vapor will appear and therefore, this line is called as bubble line. So, bubble line will show up for the first time bubble when we start hitting this liquid while the dew line will show when we start lowering the temperature of a mixture and when it hits this particular temperature the first dew will appear or the first liquid or the condensate would appear and therefore, it is called as dew line I hope you have understood this. The composition of the high boiling component now let us come back to this 4 f and 4 g and now if I am coming at point 4 g what is going to be the composition the composition at this point is the vapor composition is going to be 4 g that means, you can see now the high boiling component is only this much when we are cooling from here to here and we are coming close to the low boiling component boiling temperature the vapor also has got very less quantity of oxygen that means the high boiling component has actually becoming less and less while the percentage of vapor in of nitrogen is much more all right. So, oxygen is only going to be around 10 percent while 90 percent is going to be nitrogen vapor all right and similarly at for liquid what we have is a 4 f. So, if I take a liquid at this point this is a liquid line it is 0.5 0.5. So, 50 percent oxygen and 50 percent nitrogen well in the first case 2 f it was let us say 90 and 10 90 percent oxygen and 10 percent nitrogen at 0.3 f we had around 80 percent oxygen and 20 percent nitrogen, but here at 0.4 f what you have is a 50 percent nitrogen and 50 percent oxygen which means that as you go on lowering the temperature of the mixture the low boiling components in the vapor and liquid form have increased, but if you are at this point the high boiling components are more when you are at a dew point it basically depends on molar fraction you start with because we started at 0.1 instead of this if I started at 0.1 being here then I would have come very close to this point because my high boiling component were very less in this case. So, it all depends on what is your molar composition of mixture and at what point you enter this curve down if I start from the right side at this point that means my high boiling components or oxygen percentage is going to be very high initially only. So, initial molar composition is what will decide what is going to be your relative molar compositions at different vapor and liquid lines. So, this is a very important thing to understand one should be very clear about this dew line and bubble line the vapor compositions and a liquid composition. So, there is the respective 4 g and 4 f points which will give you vapor composition and the liquid composition of the mixture at this point. So, what do we understand by cooling the mixture which is what I have been saying all through by cooling the mixture the percentage of low boiling components in liquid have increased. When I am cooling it from 1 to 2 g 2 g 2 3 and 3 to 4 f what is happening is the percentage of low boiling component has increased while the percentage of high boiling component is decreasing slowly the percentage of high boiling component has decreased in vapor and liquid phases. If you see this vapor the percentage of high boiling component oxygen has decreased if you see this liquid the percentage of oxygen which is high boiling component has decreased. What is happening to mixture? The mixture molar composition is remaining the same. If I am coming down vertically and my initial molar composition of mixture is 1 that is going to be remaining same I am not going to do anything with 0.5 0.5. But if I say respective vapor and liquid composition that is going to be different all right. The liquid actually will remain the same basically because we are going to be touching the same molar composition. So, we can say that the molar composition of mixture at all the points is going to be remaining same because we are travelling vertical down. But the vapor and the liquid will have different molar compositions of respective high boiling component and low boiling component right. This is what I am going to say at the end the mole fractions of mixture is unchanged. But the mole fractions of vapor and in vapor and liquid phases have changed and this is what we understand from cooling of this mixture. So, having understood this temperature composition diagram and having understood the what happens to the molar composition when we start cooling this mixture. Let us see what we have understood from this the temperature phase diagram for a one component system is normally like this. So, if I start cooling a one component I am coming down from this vapor region and during this phase transition I got a two phase region and therefore, I got a l plus v over here and as you know for a single component during the phase transition the temperature remains constant. So, when I got a l plus v shown over here this is a completely horizontal line if I plot temperature versus phase. So, here the vapor composition is coming down this is a l plus v and then what you see is l further down and then it becomes solidification. So, what is important to understand is for a given pressure during this phase transition vapor to liquid you got a l plus v region which is a two phase region and during this transition temperature remains constant. So, it is clear that during phase change the temperature remains constant and this is a very known fact when ice becomes water or when water becomes vapor during this phase transitions you got a temperature remaining constant as 0 degree centigrade and at 100 degree centigrade and it is a known fact. Now, let us see what happens in a not a single component but what we have got a mixture of two components and let us see what happens to that and what happens to the temperature during the phase transition. So, in a single component what we say is a phase change is a isothermal process is an isothermal process because the temperature remains constant. Now, this is a diagram which is what we have just seen where we plot temperatures for a given mole fraction and if I go on reducing the temperature how you see the phase change now. Similarly, the temperature composition diagram for a two component mixture is as shown. What you see here is this vapor will touch due line you get a condensed set at this point and then you got a l plus v region. Now, when I am hitting this l plus v region my temperature is going down and still I am having a phase change which is possible over here and here I meet liquid region now. So, the vapor will get completely converted to liquid when I am cooling down this vapor pass through l plus v region and then go to liquid. What does that mean? It means that the phase transition is happening in this region but during this phase transition as you can see over here the temperature is not remaining constant the temperature is reducing and this is a major change when we go from a single component system to a two component system. There is a change in the temperature when the mixture condenses or boils. So, it is not an isothermal process. The phase change in a two component mixture when it is subjected to cooling or boiling or heating whatever whenever there is a phase change that occurs one needs to increase or decrease temperature in order to have complete phase change phase transformation. In order to go from vapor to liquid you pass through a two phase transition, but in order to achieve that completely you have to decrease the temperature all through. This means that the phase transformation in a two component system or a in a mixture of two or three different components is not an isothermal process the temperature does not remain constant during the phase change. Therefore, the phase change is is a non isothermal process for mixture while it was isothermal for a single component system and this is a major change when you go from single component to multi component mixtures. Having understood this now let us come to a very important aspect why are we doing all these things we are trying to understand the temperature composition diagram. So, as to understand the mixture separation. So, what is this and I just want to give a small glimpse because we will go in the details of this mixture separation further, but I just want to give a some basics as to why I am doing all these things and this is what these three figures will show you why are we doing all these things how does this help me in separating mixtures oxygen and nitrogen for example, in this case. So, mixtures are separated by rectification and this is explained using this adjacent diagram. Consider a mixture of nitrogen and oxygen at one atmosphere let us say the figure has three diagrams which is A B and C they are all the same actually they are all having 77 and 90 Kelvin that means they are all drawn at one atmosphere only I am just showing this repeat of one A B and C are essentially absolutely same diagrams, but because I want to show different processes it may not be possible to show all the processes one diagram I am showing it in three different diagrams they are all the same diagrams, but are placed one over the other for ease of understanding alright do not get confused by A B C basically essentially they are the same I am just showing some processes in B some processes in C and some processes in A. So, let us initial condition of the mixture B at 0.3 let us say the molar fraction of the mixture to be separated have 0.5, 0.5 oxygen and nitrogen let us say. So, when I come in this middle diagram I start with B first and I am let us say at 0.3. Now, let us find out corresponding vapor and liquid molar fractions as we have done earlier if I draw a horizontal at this line the point where it intersect the bubble line will give me the liquid fraction the point where it intersects the due line it will give me a vapor fraction. So, the composition of the high boiling component at 0.3 are given by 3 F and 3 G. So, why vapor 3 G give me vapor condition 3 F give me liquid conditions the vapor composition at 3 G of oxygen high boiling component oxygen is going to be given by 3 G. So, this 3 G is basically nothing, but vapor composition or vapor fraction of high boiling component while the liquid composition of the high boiling component will be 3 F. So, why liquid is 3 F? So, when I have mixture it at 0.3 the liquid have higher molar fraction of high boiling component while the vapor will have 3 G mole fractions of the high boiling component. Now, what I am going to do is I am taking this liquid from here. So, this is my first process in rectification I am going to take this liquid and I am going to have this molar fractions of liquid plus gas entering the same temperature composition curve again. So, I will get some vapor fraction now added to this. So, I got a 2 phase now again and this will enter the same this liquid plus vapor will enter the same vapor temperature composition curve at this point now. Now, consider the rectification of a mixture with composition at 0.3 F. So, what is that going to be 3 F? I got oxygen so much and nitrogen so much and now I will have a 2 phase flow entering the same region at point which is the composition given by this point. So, what will happen? I will bring this point in a 2 phase region which I have got oxygen and nitrogen given as same composition. All right what will happen? This point will further have 3 F F as a liquid composition and the corresponding vapor composition will be 3 F G, but now I am interested in only liquid composition and what you can see now when the molar composition of the mixture is 3 F when it enters this vapor composition curve the percentage of high boiling component in the liquid has increased and this is what I am doing repetitively now. What I will do now further? I will further have this composition and the liquid plus vapor at this composition will enter this diagram again. So, that I am going towards right side and my right most point is nothing, but oxygen is equal to 100 percent that is 100 percent high boiling component. The objective of doing this, objective of repeating this process and entering again the same curve is basically to reach the right most point which is 100 percent oxygen. That means, my liquid is getting richer and richer in high boiling component all right. So, again extending the constant temperature line about 0.3 F we have 3 F F and 3 F F G respectively the liquid composition of the high boiling component 3 F will be given as Y liquid which is 3 F at this point which is richer than what it was at this point. So, what I am going to do is I will repeat this again and again so that I will reach to the right most quantity. So, the liquid is going to be condensed again and again and reached right. Now, let us come back to what will I do with the vapor again with this composition now I will enter this region and now I will use A in order to explain that. So, consider the rectification of mixture with composition at 0.3 G and now I will go into this diagram again and let us see I have entered with this composition in a two phase region over here and let us see what is happening to the liquid and gas composition. Extending the constant temperature line about 0.3 G we have 3 G F which is the liquid part of it and 3 G G which is the vapor part of it. The vapor composition of the lower boiling component at this 3 G G is given as Y vapor 3 G G. So, what is happening is at 3 G now I have got a vapor at this point in which the high boiling component has decreased the high boiling component is less than what it was at this point while the lower boiling component has increased. So, what is basically happening is the liquid which I get at this first point I will go on condensing it till I hit the right most point and I will get 100 percent oxygen liquid form. My condensate will be 100 percent richer in the high boiling component while my vapor is going to be richer and richer because if I go on condensing this vapor of this left composition again and again I will ultimately would reach the 100 percent low boiling component which is nitrogen and at ultimately my vapor is going to be 100 percent rich in low boiling component while my liquid is going to be 100 percent rich in high boiling component and this is what basic principle of rectification is. Thus the rectification of mixture at 0.3 the vapor is enriched in low boiling component. Similarly, the liquid is getting enriched in high boiling component. So, you can see that liquid is getting richer in high boiling component and the vapor is getting richer in low boiling component and this is what we call as separation. Ultimately this vapor is going to be condensed at the last point and I will get liquid nitrogen at one point and liquid oxygen at other point. So, depending on what is my entry point ultimately what I am going to do is reach to the right most point and reach to the left most point. So, right most point is 100 percent oxygen left most point is 100 percent nitrogen and this what I do is basically go on the repetitively going on condensing my mixture and reaching to the right most and the left most point and this is what we call as rectification. So, this is what is basic principle of gas separation at in cryogenic condition. This process forms the fundamental step of the rectification column exactly this is going to be achieved by a device called as rectification column which we will see in detail in the coming discussion. So, in order to summarize what we have done till now, first we talked about Gibbs phase rule. If number of components number of phases and degrees of freedom for mixture in thermal equilibrium are denoted by C, P and F respectively the Gibbs phase rule give you F is equal to C minus P plus 2 and we have seen several examples for different components and different phases. The variations of mole fractions Y with temperature at constant pressure P is given by temperature composition diagram or a phase equilibrium curve and we understood that how important it is in order to understand that characteristic of any mixture. Condensation or boiling of a mixture is a non isothermal process which is a very important thing while it is an isothermal process for a single component. Repeated rectification of a mixture enriches the liquid and vapor phase with high and low boiling component. So, the liquid is getting richer in high boiling component the vapor is getting richer in low boiling component and this form the basic principle of rectification. We got a self assessment test at the end please go through those slides kindly assess yourself because that will give us a kind of feedback feedback to you as to what you have understood from this lecture. So, just small little questions and at the end of this we got answers given here. So, please go through this self assessment. Thank you very much.