A A A A A A A A A A A A A A A A A A A A A A B B B B B B B B B B B B B A A B B B B THE GIBBS FREE ENERGY BEHIND THE PHASE DIAGRAM
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1 THE GIS FREE ENERGY EHIND THE PHSE DIGRM OF INRY, ISOMORPHOUS SYSTEM temperature, T sol + liq G = H - TS (1) H = enthalpy T = temperature S = entropy Define molar quantities: = G/mole h = H/mole s = S/mole composition, X X i = mole fraction of i = N i / ΣN i Consider the ibbs free enery of one mole of atoms (molar ibbs free enery), some of which are atoms and some of which are atoms For the moment, let a partition sit between the two types of atoms Consider (X ), that is, as we chane the relative amounts of and 0% (100% ) 33% 67% 100% If the (molar) ibbs free enery of pure is, and that of pure is, then the (molar) ibbs free enery for the combination of pure components is (pure, combined) = X + X (2)
2 MS15a, Gibbs Free Enery and Phase Diarams Note: the molar ibbs free enery of a pure element is often ive the symbol µ o, as it is equivalent to the chemical potential of the pure element s a function of composition, the ibbs free enery for the combination of pure and pure is a straiht line connectin and, as shown below: (pure, combined) X Now, let us remove the imainary partition and let the and atoms mix There should be some chane in due to this mixin, and it is iven by mix = h mix - T s mix (3) The enthalpy term, h mix, represents the nature of the chemical bondin, or, put in different words, the extent to which prefers, or prefers as a neihbor (No social commentary implied) The entropy term, s mix, sinifies the increase in disorder in the system as we let the and atoms mix It is independent of the nature of the chemical bondin We can approximate our isomorphous, binary system as an ideal solution That is, couldn t care less whether it is sittin next to another to a atom This is much like our Ni-Cu system Under these circumstances, h mix = 0 Let us qualitatively examine the entropy of mixin If we have a system of pure, and let the atoms mix with one another, there is no increase in entropy because they were mixed to bein with The same holds true for a system of pure If we have just one atom of in a mole of, removin the invisible partition hardly chanes the amount of disorder ut, as we et to a 50:50 composition of :, the increase in entropy as we allow the system to mix is enormous Therefore, s mix has the followin dependence on composition: 2
3 MS15a, Gibbs Free Enery and Phase Diarams s mix 0 X From that we can easily et the chane in ibbs free enery due to mixin: 0 mix = -T s mix X Now we sum (pure, combined) and mix to et the total ibbs free enery of the solution as a function of composition: 3
4 MS15a, Gibbs Free Enery and Phase Diarams (pure, combined) mix X t this point we have the eneral shape of the (X ) curves for phases in a binary system This overall shape holds for both the and phases, so lon as both make ideal (or close to ideal) solutions We are now ready to compare sol (X ) with liq (X ) for various temperatures, and examine how these correlate to the phase diaram (1) Hih temperature: t temperatures above the meltin points of both pure and pure, the is the stable phase for all compositions Therefore, sol (X ) > liq (X ) and the ibbs free enery curves look like: molar ibbs free enery, composition, X 4
5 MS15a, Gibbs Free Enery and Phase Diarams (2) Low temperature: t temperatures below the meltin points of both pure and pure, the is the stable phase for all compositions Therefore, sol (X ) < liq (X ) and the ibbs free enery curves look like: molar ibbs free enery, composition, X (3) Intermediate temperature: s the temperature is brouht down from hih to low, the sol (X ) starts to move below that of liq (X ) The minima in the two curves, in eneral, do not occur at the same point, so for some compositions, sol (X ) > liq (X ), while for others sol (X ) < liq (X ) Therefore, at temperatures between the meltin points of pure and pure, the and curves look like: molar ibbs free enery, composition, X There are two reasons why the lines don t pass exactly over one another: (1) the slope of the line (pure, combined) will be different for the and, because () () and () () 5
6 MS15a, Gibbs Free Enery and Phase Diarams (2) no solution is really ideal, so the enthalpy of mixin needs to be taken into account What are these last pair of free enery curves tellin us? They contain the information we need to show that we must pass throuh a two-phase, + reion as we cool from only to only To facilitate the discussion, let s redraw the last fiure with a few points labeled, and compare it to the phase diaram, also redrawn with important points labeled temperature, T T o X S X O X L composition, X molar ibbs free enery, (1) (4) s (2) (3) (5) L X S X O X L composition, X Let s say we start out with a of composition X O and cool it to T o The ibbs free enery of the would be iven by point (1) on the (X ) diaram The system realizes it could lower its ibbs free enery by transformin to a The ibbs free enery of that would be iven by point (2) on the (X ) diaram ut, how low can you o? 6
7 MS15a, Gibbs Free Enery and Phase Diarams The system can, in fact, lower its free enery even further by splittin up into a of composition X S and a of composition X L (shown on both diarams) The ibbs free enery of the is iven by point (4) on the (X ) diaram and that of the by point (5) on the same diaram Now let s prove that the phase separation (into + ) does indeed lower The ibbs enery of a system composed of a and a is (total) = liq F liq + sol F sol (4) where F liq and F sol are the mole fractions of and of, respectively (these can be determined by the lever rule) This looks a lot like equation (2), if we simply notice that in the case of the pure components, the mole fraction of is X and of is X Notice also that the line connectin points (4) and (5) acts like the line in the plot of the ibbs free enery of the pure, combined components as a function of composition For example, let s say we had a system of overall composition X S (at T o ) Then, we only have phase present, and the ibbs free enery is iven by point (4) (in areement with equation 4) If we has a system of overall composition X L, then we only have phase present and the total ibbs free enery is iven by point (5), also in areement with equation 4 If we had a system of a composition exactly in the middle of X S and X L, then half the moles of our system would be in the phase and the other half in the phase The ibbs free enery would be iven by a point half-way between points (4) and (5), sittin precisely on the line that connects them Similarly, any system with an overall composition that is between X S and X L and that underoes the phase separation, will have a ibbs free enery on the line connectin points (4) and (5) So, for the composition X O, splittin into a and lowers the ibbs free enery to point (3) Et viola: a two phase reion exists on the phase diaram because its the lowest ibbs free enery state! 7
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