 So, having understood the phase diagram, the phase change processes, the PT diagram and the PV diagram, let us now try to understand importance of Gibbs phase rule because of involvement of various phases as far as water is considered. Let us also see a specific application region that we are going to concentrate from engineering perspective on water and then I will introduce you to steam tables which are required for specific properties of water for given applications. So, Gibbs phase rule, this is the picture of Emma scientist Gibbs from Yale University USA 5 period was 1839 to 1903. He is known for his very important theoretical contributions to physics, chemistry and mathematics. Also Gibbs phase rule is a very famous rule that is normally used to define properties for different phases together or independent phases. Gibbs phase rule is very famously given by this formulation F is equal to C minus P plus 2 where F is nothing but degrees of freedom or number of intensive properties. So, when I say degrees of freedom, it means that how many properties are there which I can vary but I can still be in that same phase, I can still remain in the same phase. So, one can define it as how many number of intensive properties that are required to define a particular thermodynamic state of a thermodynamic system. C is number of components as far as water is considered, we have got only one component be it in solid phase, liquid phase, vapor phase be it solid plus liquid, liquid plus vapor or solid plus vapor as we know that it is a pure fluid and therefore our component as far as water is concerned it is always equal to 1. And P is number of phases where we can have single phase, two phase or three phases together depending on what region of the phase diagram we are focusing on. This is a very famous Gibbs phase rule F is equal to C minus P plus 2. Let us try to apply this phase rule now using the phase diagram using the PT diagram for example. So this is our famous PT diagram now and I am sure now most of you had become conversant with this SLV region given in this Y, crooked Y on the PT diagram. Also this line LV line as you know goes from TP to CP triple point to critical point. Let us try to apply this Gibbs phase rule for different region for single phase region for two phase region and for the three phase point. The Gibbs phase rule is F is equal to C minus P plus 2 mathematically as we give this as let us concentrate on this single phase region which is SL or V region. We can apply it to any of these regions and all these three regions are nothing but single phase region this is what is given by this blue dots over here. For states in blue dot which actually depict single phase in this PT diagram. Let us apply this Gibbs phase rule as I have already said in this case C is equal to 1 because there is water everywhere whether it is solid region liquid phase or vaporized region in the phase diagram and because I am dealing with single phase region 100% solid 100% liquid or 100% vapor I am having P is equal to 1 in all these three cases. Let us now put this C and P values in this Gibbs phase rule and see how we complete the value of F. If I say F is equal to 1 minus 1 plus 2 I will get F is equal to 2 it means that for the single phase region the degrees of freedom is equal to 2 it means that for example let us be in this region if I vary this T here in this region I can still be in single phase region or I can vary pressure little bit and I could still be remaining in the single phase region. So I can vary in P direction or in P direction I can still be in the same phase that is why my degrees of freedom in this case is 2 I can still remain in the same phase. Also I can say that in order to define any position any thermodynamic state in this thermodynamic state space I need to define two properties minimum two properties pressure and temperature and that is why I say the degrees of freedom in this case are 2 for this particular single phase region this is true for L region this is also true for S region. So we say that for single phase region the degrees of freedom is equal to 2. Let us now concentrate on this green dots the system that lies on this saturation lines SL saturation line, LV saturation line and SV saturation line. So our thermodynamic system now is a two phase system where the two phases coexist they are in equilibrium and they lie on this region. See if I apply the Gibbs phase rule for these for these green systems which are two phases systems. So again I will say C is equal to 1 but P is equal to 2 because there are two phases coexisting and if I apply the Gibbs phase rule for these which would give me degrees of freedom I can compute the values of F for these systems now and F is equal to 1 minus 2 plus 2 in this case and I say F is equal to 1 that means the degree of freedom in this case is equal to 1 what does it mean it means that when the system lies on these I can change pressure and I can still be in the same two phase region. The moment I change pressure you can see from here the temperature also automatically gets changed. So pressure and temperature are not independent parameters they are dependent on each other. I can change the temperature if I change temperature the pressure also gets changed which was not the scenario when I was in single phase region I could independently change pressure and temperature and still be in the same phase region but here I cannot change pressure and temperature independently as soon as I change the pressure the temperature also gets fixed accordingly or I can change temperature and the pressure gets fixed accordingly and that is why my degree of freedom in this case is equal to 1 alright. I can also say that I need to define only one condition here either pressure or temperature the moment I say pressure you can see because of this line my temperature is fixed or I can define the temperature for the two phase region system and my pressure is also fixed accordingly and therefore my degree of freedom in this case 1 I need to define only one parameter in this case to define this system. Let us now try to apply this formulation for three phases coexisting together which is the triple point. So for state which is given as red point over here which is nothing but lying on the triple point or three phases coexisting together in this case therefore I will have p is equal to 3, c is equal to 1 and p is equal to 3 and in this case my f will come out to be equal to 0 that means the degrees of freedom for a tp point the system which is lying on a triple phase t phase region triple point at this point the degree of freedom is equal to 0 in this case what does it mean it means that I can neither change pressure or temperature I can still remain in a three phase region this is the point basically so because it is a point I have got its coordinate fixed I cannot change pressure I cannot take temperature and therefore my degrees of freedom in this case is equal to 0 because the triple point is already defined for water the triple point has already been defined by given pressure and temperature and therefore you cannot change those things it is a given parameter it is a given specific property of water and therefore f is equal to 0 in this case this is the way I need to define degrees of freedom or properties for a given system which may lie in a single phase region two phase region or three phase region point or it on the triple point this is the application of Gibbs phase rule for water thank you very much.