Lecture 06 : Understanding of thermodynamic equilibrium - IV

[MUSIC] We're going to start lecture six today. What we've done so far is we've defined some thermodynamic variables, then slowly we've tried to understand thermodynamic equilibrium, and then we've seen in the last class that thermodynamic equilibrium is defined by the combined fast and second laws. Which is D u equal to t del s minus p div but the difficulty is that where with this equation if we try to understand the system means the state of the system if we try but it is very difficult because of the two independent variables one is entropy One is volume and entropy and volume measurement it is not easy means it is a very difficult thing but volume can be done but entropy is very difficult so we have introduced an auxiliary function which is free energy and we have introduced pure given energy. We didn't bring the work function, there is another auxiliary function called work function, that work function can be described as A equal to u minus ts ok. There thermodynamic equilibrium will be easy to understand on the basis of easily measurable independent variables and whenever we come to G we see that there are two independent variables which are pressure and one is temperature but for bulk system measurement is very easy and we I saw an equation that is del g equal to vdp minus sdt in differential mode so we start with that equation today we repeated the equation that is del g equal to vdp sdt did we condition closed system fixed composition and since the closed system and Fixed composition so we mean mass is also fixed means size also means system size but fixed here by system size we mean how many number of moles there are and fixed composition also means that no reaction is happening in the system ok in that case we have seen it and it is seen there We are trying to define the system with two variables pressure and temperature. Here remember system definition means defining the state of the system and here state means liquid solid liquid or gas here state of system means thermodynamic state that means when the system is in equilibrium When we get to that we understand the system better with the help of these two independent variables and here too but since we've seen it before when we follow this equation we follow this equation there we see that del u equal to zero constant entropy constant volume equal to Zero at the same time del s constant internal energy and volume equal to zero these two mean the equilibrium condition of equilibrium and there in this case it looks like my internal energy is minimization and here it looks like entropy maximization is right we have but this time this is equilibrium because we have this time Equilibrium has to be understood with the help of G and to understand that equilibrium we have to take the help of some equations, we have to do some analysis, let's do that analysis today. Go U plus PV if I assume state one to state two I am not saying anything now weather this is reversibly right or irreversibly process followed but I see state one to state two and here GH and S are three but The state function is always thermodynamics one thing we should always understand is that we always subtract the initial from the final, we go from state two that means the final minus the initial or I can also write state two minus state one or I can write it like this if State Two is G One. If State One is State One, I write the others as H Two minus H One minus T One S One T Two S Two because I always drop the initial from the final T One S One and put it in brackets and since my I have this equation in my hand and I can write it like this: H to minus H one equals to u to minus u one plus p to v to minus p one v one so I'm going to substitute this equation in here so I get u two minus u one Plus p to v to minus p one v one minus t to s to minus t one s one putting it in brackets we're getting to this equation again and I know that u to minus u one is from the fast law q minus w so I put it here as well. If I replace here, if I replace it, it becomes Q minus W plus P two V two minus P one V one minus T two S two minus T one S one I have this equation and here instead of G two I can write G two minus G one Instead, I can write Del G. I mean the equation becomes Del G equals to Q minus W plus P to V minus P one V one minus T to S to minus T one S one From this equation we need to go through some analysis. And what is our purpose first if I see the subject today's class subject is again thermodynamic system we have to understand and its equilibrium we have to understand it and our equilibrium because we tried to understand it with particle fast law we faced difficulty but we have this time Let's try to understand it with G. Now let's try to understand it with G. If I take W to be the total work of the system, I can write it as W prime plus PV to minus V one. That means this is mechanical work and this is other forms. OF WORK Other forms of work mean electrical work magnetic work or chemical work all the other works we are considering in terms of w prime then I am imposing some more conditions here those conditions one condition is constant pressure another condition is constant temperature Fine, that means I go back to this equation, let's see, first I can write w prime would be equal to w minus p del v ok and if that's the case I see from this equation here I get delta g = qw - w - pdv plus minus TS to minus S one OK what makes this equation is that I'm importing this condition here at the same time I can see that when constant pressure becomes P DELVI means this part stands for This is nothing but this and this The part is going to be mine This part is OK That means I can write I've checked all ready This means Combine everything Combine this Combine this Combine this Combine this What I get is delta G equals to Q minus W prime minus TS to minus S one I try to write it another way delta g equal to q minus w prime minus t del srt del s part ta ki nothing but q reversible ok that means i see again that means delta g equal to q minus w prime minus q Reversible This is very important equation and remember here W prime is other work other gives mechanical work and here mechanical work is getting canceled out because it is canceling out but not happening remember mechanical work but we are not canceling because the mechanical work is through this Getting involved OK I know Q is reversible and Q Q is reversible is nothing but reversible heat absorbed by the system and this is irreversible heat absorbed by the system and if we subtract this we know this is nothing but loss of work or for isothermal system I know this is nothing but w reversible minus w irreversible and whenever I get this it means t del s irreversible so this part stands for delta g equal to minus w prime minus q reversible minus q now if I put the w prime on the left Let's take the rest to the other side plus Q reversible minus Q minus W prime it would be equal to delta G plus T del SIr so that means I'll have a component which is loss of work plus free energy change would be equal to minus W prime here I am. If I see that this particular term this term is always positive so that means w prime would be equal to delta g when it is reversible and when it becomes reversible I know delta s i r t would be equal to q reversible minus q that too. Reversible means going to zero means then the work I will get from the system but nothing but decrease in free energy or change in at this moment we don't understand it is decreasing not increasing atlist change in free energy the work we are getting From this equation we understand that the free energy change in G is nothing but work and when it is possible it is reversible case OK this work this work we get from the system is the system doing work and the system doing work we have already taken the convention that is the positive term this is the positive term fine and next to the positive term there is a minus sign so what does that mean that the free energy is decreasing so that means here I can introduce another term that is decrease in free energy ok this we will introduce later decrease in Free energy during reversible process is equal to the work done by the system is ok and whenever is reversible what do we mean that work done by the system is maximum ok but if irreversible comes we are already from our old our earlier consideration we I know that in irreversible process but we lose work and if we lose work in irreversible process that means I always have minus term because the system is working and it is always positive but since we are losing work then for reversibility we have total amount of work that will be available to us. But will decrease so energy has to decrease to get some work from the system but if it is irreversible but the amount of work which was supposed to be obtained will be less than in reversible process because entropy has increased and it destroyed the work so if it is irreversible process But then this equation would be minus w prime less than delta g here the reason why it becomes less than is because it's an irreversible process work loss due to irreversible tropic change ok that means I can combine these two and combine and get an equation that is w prime Less Than Equal To Delta G We always say in general that free energy change is the work done by the system and here work done by the system is the effective work how much will I get this term effective comes only when the system goes to natural process because natural process It is irreversible there will be entropy increase and if entropy increase then work will be loss and my total amount of work that sud we get effective work this effective word came now but it will decrease ok but it is always seen that to get work to get net work must be given B DECREASE IN GIVES FREE ENERGY OK THIS IS A VERY IMPORTANT STATEMENT WE HAVE NOT ENTERED EQUILIBRIUM YET WE DON'T UNDERSTAND BUT THIS IS UNDERSTANDING THAT THE FREE ENERGY MUST DECREASE TO GET SOME WORK BY THE SYSTEM THAT WORK MAY BE MECHANICAL OR OTHER FORMS OF WORK But there must be change in free energy towards negative side that means if we look at state one and state two the free energy of state two must be less than that of state one if it is a natural process ok it is understood now let us understand where equilibrium is coming from If I start again from this equation from this equation minus w prime equals to delta g plus t δ s i r if it's reversible reversible now let's pose the condition one is reversible second is no other work the other gives mechanical work the other gives mechanical work The second condition becomes what happens if I look at these two cases because it is reversible that means t del sis sud be zero or del si r equal to zero t can not be zero else give zero kelvin ok and that is the condition of equilibrium condition of equilibrium And since I no other work else gives mechanical work and mechanical work is already involved but here is involved in this equation so what I stands for is also going to zero means delta G is equal to zero and when is I constant pressure P constant end t constant fine that means pt what it stands for this is condition of equilibrium that means whenever the system changes from state one to state two free energy change at constant temperature and pressure if it is zero then we say it has reached equilibrium and if it is zero On one hand we have seen entropy maximization and already seen that whenever state changes from state one to state to free energy but it is decreasing and if I write it in differential mode delta g pt equal to zero what does it mean g sud go to minimization fine what is it Now this is statement of equilibrium is very important statement we said it means that if we understand a system it means if we understand the state of system if we achieve this condition we will reach equilibrium and how we reached what we learned from here are two very Important thing I learned is one is free energy decrease is nothing but effective work right by the system another is when free energy decrease becomes minimum but free energy change will be zero at constant temperature pressure and system will reach equilibrium It is understood that first slowly we reach system equilibrium What I did to understand First I saw that entropy change zero at constant internal energy volume Internal energy change zero constant entropy and constant volume From there slowly I found a variable that will help me understand the system with the help of easily measurable parameters pressure and temperature there From we saw that free energy change gives free energy change it gives us effective work and if it is reversible then it gives maximum work and from there when we proceed slowly analysis we see that when free energy change is zero at constant temperature pressure then the system reaches equilibrium will go and the free energy will be minimum so this time whenever we know that we need to understand what phase equilibrium is till now we were looking at system equilibrium and there are many phases in the system so there can be single phase there can be multiple phase and in the system There can be multiple components so by multiple components I mean species like one two three four five species and our course is binary single component and we will analyze in binary system means we will treat as single component or binary component but we have a Generic equation but we need to develop to develop that generic equation we need to capture not single component but multicomponent in Ever's system and there is another thing to capture that is the size effect so far in all the derivations we have done but we have not considered the size and always system and we have to come up with a generic equation that's valid for open as well as closed systems and there's one more thing that we have to take into account is the composition that we're going to have to bring into some consideration another variable we're going to have to bring in OK I think Let's assume that I have a system. Pressure, temperature, volume, and composition in the system. If I have variables in this chart, but we always need three variables to define the system. If it is a single component, a single gas. If there are two variables then pressure temperature or temperature or volume temperature volume or PV pressure volume and the reason why we are not taking one is because there would be one equation of state if ideal gas taking PV equal to NRT from there but we see if PR vita I know temperature is automatically known because I have an equation of state which refers to Equilibrium. Also here we have to bring an equation of state. It can be seen that there are three variables and those three variables are not needed because already the other variable is fixed because I have three variables. If we know that the other will be fixed then let's get that thing now. To get that thing we have to think about the system as a function of pressure temperature and size or composition. Okay, so whenever we do this, the free energy as we wrote earlier is Del G. Equal to vdp minus sdt we wrote this free energy when we wrote that our composition fix was closed system and now what condition do we impose that the system can have multiple components or species there could be phases multiple phases and whenever we capture the phase this phase Let's take a homogeneous system fine and size. We'll solve this equation for a particulate homogeneous phase. If I take it as a particulate homogeneous phase, that means the next state is going to be multiple. And there's something else off-course that's pressure, temperature, and volume. Fine, OK. That is multiple component I put the second put single phase already assumed size put and multiple component and size we can correlate pressure put and put temperature ok because if we do pressure temperature and multiple component and size then I know the volume then we have this multiple component We will handle this with the number of moles. Another thing can be reaction. Here is another thing. In case of my multiple components, what will happen if it is a reactive system? Here I associate the reactive system and in all cases of multiple components and reactive systems, I am one. number of moles of different species let's say let's take an example closed system and reactive system and here also there is multicomponent multicomponent or species let's say there is co2 to co2 H to this system only then I can reach the equilibrium that means I have state of To know the thermodynamic state of the system, to know the thermodynamic state of the system, what I need to know is pressure, temperature, and it can only be a state that I can understand when a unique composition is reached, and since a reactive system can reach a unique composition only when the chemical The reaction will reach equilibrium and that means co+ H to equal to C o to plus H to this reaction considering at the temperature of the system I reach an equilibrium ok what does that mean temperature pressure sorry pressure temperature unique composition and that unique composition can Only be aware that chemical equilibrium is rich when there are multiple components or species reacting to each other. Since what happens in a closed system can not be exchange mass but in an open system which species can I take in from outside or take out from inside all two possibilities can exist those possibilities must exist and my free energy will also change. Why free energy will change because free energy is extensive quantity ok and if extensive quantity it means mass of the system that will have influence on free energy of system ok it means in that case also number of moles will also be effective in DECIDING SYSTEM EQUILIBRIUM That means I can see combining all I can see G can be function of pressure temperature and number of moles of different species Here I is species I N one can be N one N I could be N one N two N three like This ni or i end it can go to nk so this mini species can exist one two three four five six and each species number of moles is n one n two n three n four n five like that let's their rk number of species so that the whole number of moles will be in the system and then we can write it in differential mode as delta g equal to delta g del g tao g del p constant temperature ni dp plus del g del t do t pressure ni so like That way we can extend this like this we can write this now it can be written as VDP minus S DT we know that if it is a closed system and fixed composition means if I compare it with this is one equation this is another equation If I compare these two equations, what do I get Delta G Dow Dow P equal to equal V minus S again this whole part I can write this way summation I equal to one I equal to K OK like that way I can express it means the equation is standing del t equation is ok so here basically i component one to now this equation whenever i developed it here comes an important component this component this is very important and remember already we defined here that it applies to particulate homogeneous phase that Means this equation for a this equation is for a homogeneous system and that can be also told s for a homogeneous face that means I'm bringing two things here one is that I'm no longer in single component can go to multi component and it's unique because it's Multicomponent also applies to single component, it can be two, it can be three, it can be four, and at the same time this equation is the particular equation, but we are coming for a particular homogeneous phase, that homogeneous phase can be alpha, beta, gamma, if I call it Take alpha means I will write for alpha phase if it is beta then for beta phase I will write like that there could be equations for all the phases and every time those phases but homogeneous phase will be this quantity but very important quantity what do we consider that quantity IDNI stands for Mu I. This part is Mu I. This part is Mu Y and Mu Y is also called chemical potential. So there are two quantities here one is mu and one is i that means is and another thing we got already is for the homogeneous phase that means this is i but that component is in that homogeneous phase so that means I can write alpha here if I alpha phase So mu i alpha means that the chemical potential of rectangle component in phase alpha is correct, the chemical potential of rectangle and all the components between that alpha phase but this is the statement that I gave here we said it is a very important statement. The statement will help us to draw the phase diagram and when we can draw the phase diagram we will see if the equilibrium is reached and there we will hold the equilibrium that two phases can be in equilibrium when the chemical potential of component in that phase is equal to the chemical potential. Of that same in other face considering temperature and pressure and constant ok then but we can get that and here if the equation is this chemical potential we will discuss more in next class ok but for now let's see delta g if constant pressure and temperature Either that means this will go to zero that will also go to zero that means then mu i pt constant temperature pressure this equation is one of the equations which is part of the gives two equation there are three equations the givesm equation but will help us understand phase equilibrium ok we End here today we will talk about chemical potentials and phase equilibria in the next class and we will wait until we get into the single component slowly and draw phase diagrams thanks [MUSIC] [MUSIC] [MUSIC]