[Big-Daddy]

Is there any way of mathematically defining or finding the critical or triple point of a substance in a phase diagram from the Gibbs' energies or standard Gibbs' energies, at any required temperature/pressure value, of each phase? Let's say the phase diagram corresponds to component A in a mixture of A, B and C (so the phase diagram is single-component but represents the phases of component A alone, whilst accommodating the fact that A is part of a mixture and the system is not pure A). I suppose then the Gibbs' energies will have to accommodate this fact as well?

So two main questions: 1) can we mathematically define critical and triple point in terms of the G and G° values for a given phase at each temperature, pressure, 2) can this be extended to phase diagrams of components in mixtures?

[Wystale]

Hi,
1) The triple point is a characteristic of the substance, so there is only 1 value for temperature and 1 value for pressure.
The 3 phases coexist in thermodynamic equilibrium (->the Gibbs free energy of each phase is the same)

2) If you have a mixture you must consider the interactions between the substances and therefore you have another diagram.

[Big-Daddy]

Hi,
1) The triple point is a characteristic of the substance, so there is only 1 value for temperature and 1 value for pressure.
The 3 phases coexist in thermodynamic equilibrium (->the Gibbs free energy of each phase is the same)

Even so there can be various points of equilibrium between 3 phases. After all many (most) components have more than 3 phases. So, what I'm really asking is, can I mathematically define the triple point or critical point in terms of the temperature and pressure required and G or G° of the phases?

2) If you have a mixture you must consider the interactions between the substances and therefore you have another diagram.

Ok, but you can still get a diagram representing each component (within that mixture) showing under what conditions it will assume what phase? Then, I'm looking for a mathematical definition of triple or critical point that extends to such a phase diagram and curve as well.

[curiouscat]

Even so there can be various points of equilibrium between 3 phases.

Nope.

[Big-Daddy]

Nope.

Then why does Wikipedia say

"In addition to the triple point between solid, liquid, and gas, there can be triple points involving more than one solid phase, for substances with multiple polymorphs. Helium-4 is a special case that presents a triple point involving two different fluid phases (see lambda point). In general, for a system with p possible phases, there are [tex]{p\choose 3} = \tfrac16p(p-1)(p-2)[/tex] triple points" http://en.wikipedia.org/wiki/Triple_point

:p In any case, what is the mathematical reason for there not to be multiple triple points for a given set of 3 phases?

All this would come naturally if you could help me with my main question - how to find triple point and critical point mathematically.

[MrTeo]

Nope.

Then why does Wikipedia say

"In addition to the triple point between solid, liquid, and gas, there can be triple points involving more than one solid phase, for substances with multiple polymorphs. Helium-4 is a special case that presents a triple point involving two different fluid phases (see lambda point). In general, for a system with p possible phases, there are [tex]{p\choose 3} = \tfrac16p(p-1)(p-2)[/tex] triple points" http://en.wikipedia.org/wiki/Triple_point

:p In any case, what is the mathematical reason for there not to be multiple triple points for a given set of 3 phases?

All this would come naturally if you could help me with my main question - how to find triple point and critical point mathematically.

True, other substances with more than one triple points are sulfur (orthorhombic and monoclinic) and tin (α and β forms).
I think that you look at the high part of the phase diagram of water you will also find quite a lot of additional triple points due to all the equilibria between the different types of ice phases.

The reason why there is no "quadruple point" or why there is only one triple point between three phases is the so-called "phase rule" which you can derive quite easily thinking of how many equations you need to describe such an equilibrium and how many parameters you need to fix the state of your system. Turns out that if you want to work on the simultaneous equilibrium of three phases (and one substance) you have 0 "variance" (the number of variables you can change "freely" around your equilibrium point without losing the equilibrium).

http://en.wikipedia.org/wiki/Phase_rule

[Big-Daddy]

The reason why there is no "quadruple point" or why there is only one triple point between three phases is the so-called "phase rule" which you can derive quite easily thinking of how many equations you need to describe such an equilibrium and how many parameters you need to fix the state of your system. Turns out that if you want to work on the simultaneous equilibrium of three phases (and one substance) you have 0 "variance" (the number of variables you can change "freely" around your equilibrium point without losing the equilibrium).

http://en.wikipedia.org/wiki/Phase_rule

Thanks for the info. So number of variables F that can be changed around the equilibrium point, is related to number of components C and number of phases P by the equation F = C - P + 2. A couple of questions then: 1) for a mixture, what is P? The total number of phases across all components? 2) You mentioned "parameters you need to fix the state of your system". What are all of these parameters? There is temperature and pressure obviously but what else?

[MrTeo]

[...] is related to number of components C and number of phases P by the equation F = C - P + 2.

"Indipendent components" actually, which sometimes is a bit tricky because you have to subtract from the number of components the number of equations that link them (eg. take water autoprotolysis: you have three components (H₃O⁺, OH^- and H₂O) but C=1 because they are linked by the equilibrium constant K_w and the electroneutrality condition (which in this case is equal to the mass conservation: [H₃O⁺]+[OH^-]=[H₂O])).

1) for a mixture, what is P? The total number of phases across all components?

Define "mixture": if you're thinking about something heterogeneous or microheterogeneous I think you're right, you probably have to take into account the whole picture (all the phases and all the components) to describe such a mess. Maybe when you work with colloids or similar systems you can actually make some approximations but I'm pretty sure you'll have to consider surface effects of which I don't know anything so maybe it's better if you refer to a monograph in that field if you want additional information. If you stick to something that looks much more like an everyday chemical system, well, the homogenous solutions will be a unique phase with additional variables such as the composition (you only need n-1 molar fractions as usual to fully determine the composition of such a phase with n components). Obviously I'm talking about solid, liquid and gas solutions (nay, mixtures in the case of gases). A common application of the phase rule is in distillation diagrams where you want to figure out what's the vapor composition varying the liquid components ratio or the pressure of your system: you have one liquid phase (considering fully miscible liquids) and one gas phase. Another example of a more complex system could be an eutectic: there you have a liquid phase and a solid phase (or two of them, one pure crystalline solid which is either the solvent or the solute and the eutectic, a solid mixture with a fixed composition). As you can see things can become quite complex (and quite easily: we're talking about the freezing of a salt solution).

2) You mentioned "parameters you need to fix the state of your system". What are all of these parameters? There is temperature and pressure obviously but what else?

p and T are the usual parameters of phase diagrams, for a number of good reasons, but no one stops you from working with p and V (in fact you do that when you study Andrews' isotherms and vapor pressure, you can apply phase rule to that graph too). When you have solutions you will have to use molar fractions as I told you in the previous answer. To sum up when it comes to variables I think that it's all a matter of what you're taking into account: for example studying surface effects you may need to add a surface tension term and that would bring an additional variable too, the interfacial area (dE=γdS).

[Big-Daddy]

"Indipendent components" actually, which sometimes is a bit tricky because you have to subtract from the number of components the number of equations that link them (eg. take water autoprotolysis: you have three components (H₃O⁺, OH^- and H₂O) but C=1 because they are linked by the equilibrium constant K_w and the electroneutrality condition (which in this case is equal to the mass conservation: [H₃O⁺]+[OH^-]=[H₂O])).

Ok, thanks. And the number of equations which link your component(s) is 0 if you have only one component and its phases are involved in phase transition reactions, but nothing else?

p and T are the usual parameters of phase diagrams, for a number of good reasons, but no one stops you from working with p and V (in fact you do that when you study Andrews' isotherms and vapor pressure, you can apply phase rule to that graph too). When you have solutions you will have to use molar fractions as I told you in the previous answer. To sum up when it comes to variables I think that it's all a matter of what you're taking into account: for example studying surface effects you may need to add a surface tension term and that would bring an additional variable too, the interfacial area (dE=γdS).

Thanks very much, this helps a lot.

You don't happen to know the answer to my original question, do you? (The mathematical calculation of triple point or critical point using G or G° values, or if it's even possible?)

[MrTeo]

"Indipendent components" actually, which sometimes is a bit tricky because you have to subtract from the number of components the number of equations that link them (eg. take water autoprotolysis: you have three components (H₃O⁺, OH^- and H₂O) but C=1 because they are linked by the equilibrium constant K_w and the electroneutrality condition (which in this case is equal to the mass conservation: [H₃O⁺]+[OH^-]=[H₂O])).

Ok, thanks. And the number of equations which link your component(s) is 0 if you have only one component and its phases are involved in phase transition reactions, but nothing else?

Exactly.

You don't happen to know the answer to my original question, do you? (The mathematical calculation of triple point or critical point using G or G° values, or if it's even possible?)

Talking of the triple point I find it quite straightforward: since it's an equilibrium between the three phases the condition you need to apply is (just an example with water triple point, μ is the chemical potential, defined as the derivative of G over the number of moles of the i-th species, which is some sort of "molar" free Gibbs energy):

[tex] \mu_{solid}=\mu_{liquid}=\mu_{gas} [/tex]

Which will have some sort of form like:

[tex] \mu_{solid,\,liquid,\,gas} = \mu_{0;\,solid,\,liquid,\,gas} + RT\ln{a_{solid,\,liquid,\,gas}} [/tex]

that can become awfully complicated (a is the activity of your species, or fugacity for gases) if our substance doesn't really behave as we would like it to, but in the most common case I think you can put the activity equal to 1 for solids and liquids and equal to the pressure for gases. I think it's also necessary to refer all the energies to the same standard state, which I'm not sure if it's the same one (and in general I'm pretty sure that it is not, because there are quite a lot of "standard chemical potentials" depending on the system you're working with) for all the species involved. Whith different triple point you will have to work with different chemical potentials, depending of what equilibrium you're talking about, but I think that is the only condition you need, though I'm not absolutely sure (I'll check if I don't completely forget about this, which is quite possible).

The critical point can be defined as the inflection point of Andrews' isothermal so you only need to put the first and second  derivatives of the pressure with respect to volume (obviously you need a state equation for that) equal to zero and you've got it. You can try with van der Waals' equation if you want, after some boring algebra you'll get all the values you need (you can find T_c from the state equation when you're done with the other two conditions).

[Big-Daddy]

Talking of the triple point I find it quite straightforward: since it's an equilibrium between the three phases the condition you need to apply is (just an example with water triple point, μ is the chemical potential, defined as the derivative of G over the number of moles of the i-th species, which is some sort of "molar" free Gibbs energy):

[tex] \mu_{solid}=\mu_{liquid}=\mu_{gas} [/tex]

Thanks. So quite simply, the chemical potentials of the phases are equivalent. Now I think about it, this is a criterion for equilibrium so it should have been obvious! Complications then ensue as one tries to work out the chemical potentials as functions of activity/fugacity, G°, T and P.

The critical point can be defined as the inflection point of Andrews' isothermal so you only need to put the first and second  derivatives of the pressure with respect to volume (obviously you need a state equation for that) equal to zero and you've got it. You can try with van der Waals' equation if you want, after some boring algebra you'll get all the values you need (you can find T_c from the state equation when you're done with the other two conditions).

Thanks! So I need to express p(V) and then find the solution(s) when both d(p(V))/dV=0 and d²(p(V))/dV²=0. I'm guessing multiple solutions, if they ever came, would mean multiple critical points? (I don't think I've ever heard of this actually happening though.)

What happens to p(V) at a point of equilibrium between phases? Because then there are multiple different phases of the component present, and each may have a slightly different pressure (e.g. if I've got liquid and gas in a beaker, the pressure in the gas will be the same, and dependent on the V of the gas, but the pressure in the liquid will vary with the height from the surface, and not with the volume of the liquid).

Finally, what happens if my phase diagram corresponds to the component for which the phase diagram is drawn, being in a mixture? "Mixture" meaning there are other components present, which may themselves adopt various different phases depending on P,T, as well as the one you're interested in. Then can you say p is the partial pressure of your component, V is the volume of your component's phase, and then if you can express p(V) you will be able to find the critical point in the same way?

[MrTeo]

Thanks! So I need to express p(V) and then find the solution(s) when both d(p(V))/dV=0 and d²(p(V))/dV²=0. I'm guessing multiple solutions, if they ever came, would mean multiple critical points? (I don't think I've ever heard of this actually happening though.)

Hm. I think the only way to get that is to have quite a strange system, so maybe there things are not actually that easy and you can't find your second "critical point" with a single state equation just like with gases. But I'm not really sure as I've never heard of a second "critical point".

Because then there are multiple different phases of the component present, and each may have a slightly different pressure [...]

Wrong, they will all have the same vapour pressure: with some approximations, at least for the everyday phase changes you get that condition for phase transitions. This is obviously true for the boiling water, but also for the freezing water (which is maybe not as clear).

[...], but the pressure in the liquid will vary with the height from the surface, and not with the volume of the liquid).

This is kind of different: you're taking into account the effects of an external field on you system: you need to rewrite the first principle of thermodynamics with the gravitational potential energy on the left side (you would have to do the same if you had water sprouting out of a hole and you wanted to consider the kinetic energy term too, but usually these effects are negligible). I'm not sure but I suppose that in that case you won't have the whole liquid boiling (you won't actually see bubbles forming at the bottom *and* at the top, I'm saying this as I don't really know right now where they're going to form first) but a continuous phase change from the lower part to the upper part which gives slightly different vapor pressures as you move through the liquid. This is insane, if you want my opinion. I think that you can really forget about it.

Finally, what happens if my phase diagram corresponds to the component for which the phase diagram is drawn, being in a mixture? "Mixture" meaning there are other components present, which may themselves adopt various different phases depending on P,T, as well as the one you're interested in. Then can you say p is the partial pressure of your component, V is the volume of your component's phase, and then if you can express p(V) you will be able to find the critical point in the same way?

Wild thinking. In plain words is quite a mess: consider that the phase diagram of your phase will depend, in most of the real cases, on all the other components phases so you will get this huge network of equilibria which you probably can't even describe (so no, finding the critical point won't be that easy I suppose). Just to make an example I suggest you to take a quick look at this: this approximation (Lewis-Randall rule) is actually applicable in only a quite small range of conditions (we're talking of gas mixtures), even though it seems legit, and we're only thinking about really well-behaving gases!

http://books.google.ch/books?id=GjlO9MA9edUC&pg=PA415&lpg=PA415&dq=lewis+randall+rule&source=bl&ots=IbtJBSb9Pg&sig=84XMOGg2Dn3xot4VNCyBa0Bj1Vc&hl=en&sa=X&ei=lp4SUrivCKmJ4ASI5IHwAw&ved=0CCkQ6AEwADgK#v=onepage&q=lewis%20randall%20rule&f=false

I think the only way to work on such a system is to use empyrical parameters and series expansions, but that looks much more like chemical engineering and I don't actually know anything about all that.