Lecture 15 : Solution thermodynamics - Part IV

[MUSIC] [MUSIC] Today we will start lecture 15 what we have seen in the previous classes is that we have considered gas mixtures and for gas mixtures we have calculated the free energy of mixing we have calculated the free energy enthalpy of mixing and we have calculated the entropy of mixing and in each case But we have taken the ideal gas mixture and when the gases are mixed together they form a homogeneous gas mixture and we have seen that they are described by partial pressure and we have seen those expressions not only partial pressure but also the mole fraction of different gases means the mole of each gas. We can describe it with fractions. This time we will slowly enter into solid solution or liquid solution. We have taken gas mixture before. Now we will take mixture of different elements or components in solid phase or liquid phase or when it means solution form will be solid or liquid. 15 Topics to be considered we will continue with solution thermodynamics and after knowing the fundamentals of solution thermodynamics we will finally get to phase equilibria and that phase equilibria for multicomponent systems but we will understand and also draw phase diagrams which we will do today. That's Delta G Mix, which we did in the case of the gas mixture, now we're going to do it in the case of the condensate system, then we're going to take the liquid or solid form, and we're going to do another thing when we do this, and that's Raoultial. Solution We will discuss a little bit and try to understand that this applies to ideal systems. Ok these two things we will consider in lecture 15. What we saw in lecture four. X pl n x p described like this. Let's look at what we did at the end of last class. This is what we write here. Fine and delta h mix. This is zero and delta s mix. This is what we wrote minus r x l n x a plus x b. LNXBR here ideal gas mixture is considered for two components one A gas and ND gas it is considered OK so now if I go into condensate system when we consider condensate system also there but how to use my concept of ideal gas mixture OK Let's put that to work and then we'll understand that one thing we understood before is the saturated vapor pressure. Let's take that saturated vapor pressure as ideal and we'll see that we get a very nice equation that gives us the free energy for a mixture, which is that it's a gas. A mixture can be a liquid solution or a solid solution. I'll get it. For the condensation system, we'll take the saturated vapor pressure, the pressure, and where it's coming from. That temperature specific will reach an equilibrium and when equilibrium is reached then the pressure that will extract the vapor molecules of that solid and liquid in equilibrium with solid and liquid surface that pressure is called saturated vapor pressure and I take it as PI OK and if pure If it's a pure solid and liquid, we're going to put Knot and this is the saturated vapor pressure for a pure substance. G mix we also took as RTLNP by P mix plus and I can sum it up and if in AB's system then RTLNP by P mix plus RTLNP by P mix it is capital p. will be not small p means system pressure ok and know this for ideal case p is equal to x a pp mix PB equal to x pp mix that means i will get the same equation this equation is nothing but this equation now it will be seen that we are ideal In the case of gas mixture, the calculation that I brought about can be done in condensation system in almost similar way, but the treatment is a little different. How can we do the different treatment? Final M is our Delta G mix. I is in a condensed state i.e. let's assume liquid is in liquid state or solid state and at a particle temperature t partial pressure equilibrium partial pressure which will be pi not this time pure component is liquid pure component i while liquid its vapors We are taking the pressure that will be created PI NOT Now if IT is in a solution and that too is in liquid form means it is in condensate form then the pressure will be PI NOT because whenever I put IT in the solution but The concentration of I can be captured by X i means the mole fraction of I in a solution where multiple components are present. Equal to XIPI not and this I am taking for ideal system in case of ideal system means when the gas will act as ideal and the whole system will behave as ideal this whenever I enter ideal system ok let's discuss a little bit about that ideal system ok I am taking If there is an ideal gas, assuming ideal gas mixture, if I assume ideal gas mixture and take very high temperature high temperature and take very low pressure or N value which is the number of moles it is taken very small by these three cases let us see what is meant physically. When the system satisfies these three conditions but can behave as an ideal, the reason is that whenever the temperature is high, the pressure is low and the number of moles in that system is very small, it can be seen that the volume will be Very large very large And in this case if I keep the pressure temperature fixed and the volume is a certain initial working with a finite volume if I reduce the number of moles then I will see that the king of my molecules are atoms or molecules of two. But the interaction tendency between gases will decrease because there is a certain amount of moles. If we increase the volume too much, that interaction probability means the probability of one molecule hitting another molecule. But if the number of moles is reduced too much, that means I have very few atoms or few molecules and if there is a very small amount of them in that particle container, then it will be seen that the probability of interaction between these two molecules is very small. That will also decrease and what do you mean by ideal system and by ideal system I mean that there is no interaction between my sphere of molecules negligible interaction there is no attraction there is no repulsion there is no kind of nothing ok that is ideal mixture and then this thing works for gas mixture Now let's see why this happens. If I write the state of equation that is RT equal to PV equal to NRT. Now I keep N fixed and increase T and decrease pressure. Then volume is NRT by P OK and gas constant. And is gas constant it is constant n also assumed constant that means if p or t increase or decrease p then volume has to go up that means i have certain amount of molecules the volume has increased so definitely their interaction ability will decrease a lot ok it can be considered Let's give a small example like when I see during the day time means when it's office hours there's a lot of traffic on the road and what does that mean the traffic is moving so the bus trumps everything and people are moving just right and then it will be seen that one person with another People can sometimes be seen walking on the street touching. OK, if I consider this touching as an interaction. OK, too many molecules in office hours. It's ok this time I realized that night started to fall around 12:00 pm on any street my people are very less. Maybe one or two are ok and they can be seen. People are decreasing means the number of molecules is decreasing that is an ideality comes no interaction again it is found that in a particular station which I assume fixed volume then this thing is coming means the number of molecules is decreasing the number of people is decreasing If I see the same number of people, let's take a playground. Okay, now 10 people are playing in the playground. 10 people can touch each other. If I increase the playground, it's okay. Going means that the tendency of the two to touch is decreasing. OK, it means that for an ideal solution, the volume must be large or the number of molecules must be small. What happens if I assume that in my solid liquid the molecules are more closely spaced if I assume an alloy system I assume a metallic alloy system then I assume the atoms have a bond between the atoms fine ok that means the atoms are more closely spaced like a gas Not that widely spaced means here widely spaced and here means molecules molecules and widely spaced and here atoms if I take metallic alloy and if I take homogeneous means which is nothing but solution I can call it solid solution or liquid solution I can catch then atoms And the closely spaced and bonded to the bonded to is well bonded to what we call a metallic bond and another thing that can be done when making a metallic bond is that we assume that the coordination number is well the coordination number is that there is a particular atom around There are many more atoms and the nearest neighbor is with them but that atom will form a bond. If I see such a system, the atoms are distributed like this, then my bonds will be like this. We indicate it as Z, fine. Now here, what is the coordination number in this case, four, if I take BC, in the case of body centered cubic iron, which is alpha or delta iron, its structure is crystal structure, body Q center cubic, and the coordination number is Z, which is eight. As you can see we can see like I can draw a crystal cubic crystal structure body centered cubic means there is one atom in the body center and one atom in each corner. bond this is a bond this is a bond this is a bond four bonds above four bonds below will be four bonds let's see that's three another one this is another point of the cubic that but that but behind this plane here's an atom and this is it's become a bond ok total eight The bond is formed and the nearest neighbor is eight, there comes eight. In the case of FC, face centered cubic, for example, gamma iron or aluminum or nickel, they have a face- centered cubic structure, where z is 12. I will see that each atom that has a nearest neighbor is the nearest neighbor number. 12 And each atom is going to bond with its closest 12 neighbors, so these bonds are not there in the case of gases, so there are almost no widely spaced bonds, but the bonds that are here are keeping the atoms closely spaced, so I mean, if that's the case, but these atoms dislodge from the solid or liquid surface, it's not easy for the vapor, which is not easy for the solid. It takes an energy to leave the liquid or solid atom surface and become part of the wave phase. In the case that I was thinking about it easily but in this case but I am not so easy but I have to think of some more parameters which can break that bond energy ok for that what parameters I decide that tendency of liquid atom or solid atom or solid atom which Condensation system to get out of there to get detached from solid or liquid surface ok that will be decided by few factors atomic size electronegativity or electron atom ratio will function on these that means function of this kind these many things ok and this tendency is that Detach or tendency is also deciding that these are again function of bond energy ok and this bond energy what is bond energy means an interaction ok and that interaction nature as well as magnitude of this introduction will finally decide vapor phase formation OK or we know there's a condensate system from there if we take a container let's take a container let's take a fixed container let's put a liquid in there let's put a pure liquid this is at a particular temperature and before we put the liquid in this chamber I evacuate the process is the fast step is evacuation and then the second step is keeping pure liquid pure liquid a ok when I do this I know why the system will come to saturated vapor pressure saturated vapor pressure because I know that the system will have to rich equilibrium that means I will come to PIP not this will be reached which is function of temperature again this also but as a function of not only temperature other factors if I take it with bond energy then I mean nature as well as magnitude of interaction or atomic interaction Fine whenever I come to this situation this situation comes immediately whenever I am putting pure liquid after evacuation the liquid molecules will detach from the liquid surface and become part of the gas molecules that means when this process is going on liquid to vapor we call that process. Evaporation will take place in this process, there will be a rate that is R e of a face and at the same time when my atoms become part of the gas they are moving in different directions and while moving those atoms again here one atom will want that atom again. To hit the liquid surface and they also want to be part of the liquid ok that means there will also be an opposing process that is vapor to liquid ok that means process one process two and from vapor I will call it condensation that will also have a rate R C of A I mean condensation and E means evaporation and it is at a particular temperature that means these two rates will reach equilibrium at some point means equal and EA is equal to RC then I will say the system has reached equilibrium has reached ok and when it is But then we can also express it with a partial pressure, whenever it reaches equilibrium, that means it's going to P a knot. OK, this time, it's a function of interaction, and interaction means that the atoms are interacting. I know there's a bond. Bonds are formed. Bond energy is one. What do you do for that bond energy? Let's say it's a liquid atom. This bond. These atoms will always be in a potential well. The surface atoms and other atoms are in the potential well. They are in a low energy situation in this potential well. If I want to dislodge this atom to come out, it's going to go through my potential barrier. It's liquid. It's vapor. Okay. This potential barrier is fine. So this potential energy is going to be a star. So it follows the Arrhenius equation and I can write that as exponential of minus E star by RT here because the atom is being captured so I'm going to take KT so there's a minus sign over there so that means REA is proportional to exponential minus E star by KT OK and this is this E star value is the activation energy for breaking off I of liquid atom from liquid surface fine OK that means this is a relation I'm getting on the other side if I'm taking with evaporation point off I'm taking with point of view if I'm taking condensation point From view it means that the condensation rate is RCA but it depends on the frequency of heating gas molecules on the liquid surface per unit time and another determine that is the number of molecules or gas molecules present in the gas phase. Okay so that means I can say this because these two things are mine but as a function of partial pressure and this is going to be proportional taking proportional proportionality there's going to be another constant we're taking proportionality because we don't know that constant but it's got to have a direct relationship with it right and having a direct relationship So I can write rca equal to proportional to p a not because I know that at a particular temperature the equilibrium partial pressure that I will develop is p a not and if I call that the reaction equilibrium reaction constant or the reaction equilibrium constant If I take the interaction that is from gas phase to liquid phase I can write RCA equal to KPA not and I know what it is in case of Equilibrium I can also do this that means I do n't follow this process I follow partial pressure process The reason I'm doing this is to determine the partial pressure, but it's relatively easy and we're taking an easy route. Okay, this time if I know that REA is equal to the partial pressure of a knot, it's the same for pure if I take a B and take a B liquid. I can do the treatment if it is pure and EB would be equal to PB not now I am mixing ARB in condensate system if I assume liquid is mixing and while mixing I instead of considering it means liquid phase If we don't consider if I take it with the gas face, but I'm thinking of a how can it be mixed, what can be the rate of mixing to mix, but we can think of it with gas because I've started to think of condensation with evaporation, okay? We're only thinking of condensation because if I have condensation then I'll get solid solution or liquid solution but we're taking it in reverse by evaporation route and we're taking partial pressure. It will be X B and X A it is the mole fraction if we add these two it will be one because I know that let's say here there is NA number of moles and here there is NB number of moles in that mixture if I give it the mole fraction Well then I can write na by na plus NB it becomes x and NB by na plus NB it becomes xp ok that means it will be 1 when it is because my pure condition will have a partial pressure when it is not pure When the solution is done it means that my mole fraction decreases from one to fraction one so after solution it will be seen that the number of molecules or atoms of B or A which will be available to convert to gas phase means that there is such a surface such molecules There are and the size of each molecule is assumed to be the same, that means or the size of the atom is equal to the size of the atom. If I am in a situation like this, if I want to take B into the gas phase, well, if I want to take B into the gas phase, then the partial pressure of B will be that would be a fraction of PB. will be PB x p into PB not let's see what it's doing but one thing is clear whenever i first what was first situation was all b atom now some atom b that means my number of atom which will be gas part which will be the rate That rate will definitely be different if I assume RBR evaporation would be proportional to PB but it is not and this PB would be decided by number of B atoms that are available which means what is available on the surface should be captured. We can also write like this and equal to k prime PB k prime is assumed to be the equilibrium constant which I assumed for a k k assumed for b is found to be k prime ok it is k prime I took k prime same equilibrium constant if I take same element it is ok found again at the same time I know that and EB I take it prime because it is prime because it is when the solution is that means in case of solution again know when it is pure that is prime PB not I these two OK that we also saw the equation When pure is fine then it should be fraction of B atoms present in the liquid into R EB when pure it means in pure case I had all B atoms while in pure case B atoms will decrease while in solution B atoms will decrease By that fraction my evaporation rate will also change, so if I write this, that means I can write k bk prime pb equal to xp into k prime pb not so i get pb equal to xp pb not so i got this and the other way around is If I take in this system it is for B atom in solution and if I take atom in solution then we will stand as p is equal to x pp a not I got these two equations earlier but I said one thing that is atom size rb Atom Size Same Let's do a little bit more here and that is I see that whenever I can write this in terms of B in terms of what I can write in terms of R E prime A equal to X A R A that means this is for pure A R It is A in solution and it is mole fraction of A in solution. Some conditions are needed to write these two conditions. Let's see what are the conditions. This is that a mole fraction multiplied by the evaporation rate of A or B in pure condition is only when It can be written that when the A atom pulls out from the liquid surface or the B atom pulls out of the liquid surface the other atom or its neighboring atoms have no influence. And when the B atom was pure it was in the same potential well as it was in the potential well when it was mixed but the character of that potential well didn't change that means the activation barrier did n't change either. The activation barrier will act to pull A out and when B is in pure condition the activation barrier will act to pull B atoms out to B part of the vapor phase with similar activation energy when it goes into solution but will maintain the same So that means this can be written as this and this can be written as OK so what does that mean that E activation of A in pure is equal to E activation of A in solution similarly E activation of B is equal to E activation of B Solution And for this to happen, of course atom size B atom size must be same at the same time bond energy as many bond energies as possible must be same what bond energy can be A bond can be BB bond or can be AB bond and If I take these bond energies as E, if I take these AB and AB, they're all going to be equal. When A atom or B and B atoms are brought closer, there is no different interaction, the interaction remains the same, that means when before solution making and after solution making, the interaction parameter remains the same, but the condition of an ideal system is correct. The interaction that was there is not going to change at all so that means here also but it appears that the interaction is not changing that means it is acting as an ideal system acting as an ideal system and then I would call it an ideal solution not a mixture because the solid and the liquid Ideal solution and ideal solution is then another thing since this condition is maintained AB and A or BB it will be a random distribution it will be a random distribution and it can also be said that no preference no preference I suppose both A and A like What does that mean, they will be on one side, BB will be on one side, and if it is found that A and B are trying to attract each other equally, it means that they want it, it means that A does n't want to be together, B doesn't want to be together, rather they want to be together. If AB wants to be with AB then AB will form a form A or BB will not form. Well, we are not going to any such situation. Rather, we are saying that both A and B will be homogeneously distributed and there will be no preference between them. But the character of ideal solution is fine that means we get that in case of ideal solution the bond energies should be same atomic size should be same and there will be no preference between them then we see the evaporation rate which is independent of means not depending on the number of takers. Rather it depends only on the concentration of that particular element in the solution on the surface and we can say that p is equal to x apa not and we can also say PB is equal to x p PB not when this condition maintains then it is called this ideal solution is called the Raoultian solution and its ideal solution is applied to the ideal solution by Fine and this rule is called Rawls's law where do we start let's see where we started we said that we said that when the pure component is We assume that the pure component will have I PI knots and whenever we make the solution and make the ideal solution we will see that the partial pressure of the right component of that vapor and that vapor is ideal then the vapor in the ideal solution that is made of that particular component is is going to relate to mole fraction and that relationship is this and this pi is going to decrease because mole fraction is half of it sorry mole fraction could be half one by third could be one third could be one fourth could be x i never From where x i will vary from zero to one is fine and in every case the partial pressure is decreasing fine this is what we came up with this Raoultian concept to explain fine this Raoultian concept we had to come up with this Raoultian concept and this is our basis of this but our whole solution Let's continue with the thermodynamics ok later we'll get into real solutions then we'll discuss Henry's lot but for a bit ok we'll stop here for now we'll take that to the next class on how to calculate the value of the Delgy mix from this Raoultian concept thanks [MUSIC] [ music] [music]