1. Phases, Components, Species & Solutions
Lecture 5

2. Thermodynamics of multi-component systems
Chapter 3
Thermodynamics of multi-component systems

3. The real world is complicated
Our attempt to estimate the plagioclase-spinel phase boundary failed because we assumed the phases involved had fixed composition. In reality they do not, they are solutions of several components or species.
We need to add a few tools to our thermodynamic tool box to deal with these complexities.

4. Some Definitions Phase Species Components
Phases are real substances that are homogeneous, physically distinct, and (in principle) mechanically separable. For example, the phases in a rock are the minerals present. Amorphous substances are also phases.
NaCl dissolved in seawater is not a phase, but seawater with all its dissolved components (but not the particulates) is.
Species
A species is a chemical entity, generally an element or compound (which may or may not be ionized). The term is most useful in the context of gases and liquids. A single liquid phase, such as an aqueous solution, may contain a number of species. Na+ in seawater is a species.
Components
Components are more specifically defined. But:
We are free to define the components of our system
Components need not be real chemical entities.

5. Minimum Number of Components
The minimum number of components of a system is rigidly defined as the minimum number of independently variable entities necessary to describe the composition of each and every phase of a system.
The rule is:
c = n – r
where n is the number of species, and r is the number of independent chemical reactions possible between these species.
How many components do we need to describe a system composed of CO2 dissolved in H2O?
Memorize!

6. The CO2–H2O System Carbonate Solution Recipe:
Distill water
Place in a beaker and let stand exposed to the atmosphere
In the distilled water, some of the water molecules will dissociate to form hydrogen and hydroxyl ions:
H2O ⇋ H+ + OH–
Some atmospheric CO2 will dissolve in the water and react to form carbonic acid:
CO2 + H2O ⇋ H2CO3
Some of that carbonic acid dissociates to form H+ ions plus bicarbonate ion:
H2CO3 ⇋ H+ + HCO3–
Some of the bicarbonate will dissociate to form carbonate ions:
HCO3– ⇋ H+ + CO32–
How many species and how many components?
7 species: H+, OH–, H2O, CO2, H2CO3, HCO3–, CO32–
What is the minimum number of components?
Our rule was c = n – r = 7 - 4
Just 3 components, e.g., CO2, H2O, H+

7. Graphical Approach The system Al2O3–H2O
If it can be graphed in 1 dimension, it is a two component system, in 2 dimensions, a 3 component system, etc.
Consider the hydration of Al2O3 (corundum) to form boehmite (AlO(OH)) or gibbsite Al(OH)3. Such a system would contain four phases (corundum, boehmite, gibbsite, water).
How many components?
The system Al2O3–H2O

8. Phase diagram for the system Al2O3–H2O–SiO2
The lines are called joins because they join phases. In addition to the end-members, or components, phases represented are g: gibbsite, by: bayerite, n: norstrandite (all polymorphs of Al(OH)3), d: diaspore, bo: boehmite (polymorphs of AlO(OH)), a: andalusite, k: kyanite, s: sillimanite (all polymorphs of Al2SiO5), ka: kaolinite, ha: halloysite, di: dickite, na: nacrite (all polymorphs of Al2Si2O5(OH)4), and p: pyrophyllite (Al2Si4O10(OH)2). There are also six polymorphs of quartz, q (coesite, stishovite, tridymite, cristobalite, α-quartz, and β-quartz).

9. Degrees of Freedom of a System
The number of degrees of freedom in a system is equal to the sum of the number of independent intensive variables (generally T and P) and independent concentrations of components in phases that must be fixed to uniquely define the state of the system.
A system that has no degrees of freedom is said to be invariant, one that has one degree of freedom is univariant, and so on.
Thus in a univariant system, for example, we need specify the value of only one variable, T for example, and the value of pressure and all other concentrations are then fixed and can be calculated at equilibrium.

10. Gibbs Phase Rule The phase rule is ƒ= c - ϕ + 2
where ƒ is the degrees of freedom, c is the number of components, and f is the number of phases.
The mathematical analogy is that the degrees of freedom are equal to the number of variables minus the number of equations relating those variables.
For example, in a system consisting of just H2O, if two phases coexist, for example, water and steam, then the system is univariant. Three phases coexist at the triple point of water, so the system is said to be invariant, and T and P are uniquely fixed.
Memorize!

11. Back to Al2O3–H2O-SiO2
What does our phase rule (ƒ=c - ϕ + 2) tell us about how many phases can coexist in this system over a range of T and P?
Φ = c- ƒ + 2 =
How many to uniquely fix the system?

12. Clapeyron Equation
Consider two phases –graphite & diamond– of one component, C.
Under what conditions does one change into the other?
It occurs when ∆G for the reaction between the two is 0.
Therefore:
And
How many degrees of freedom in this system?
Another important one, but easily derived!

13. Solutions

14. Solutions
Solutions are defined as homogenous phases produced by dissolving one or more substances in another substance.
Mixtures are not solutions
Salad dressing (oil and vinegar) is not a solution, no matter how much you shake it.
The mineral alkali feldspar (K,Na)AlSi3O8 is a solution (at high temperature).
A mixture of orthoclase (KAlSi3O8) and albite (NaAlSi3O8) will never be a solution no matter how much you grind and shake it. (Of course, if you were to heat that mixture sufficiently, the two minerals would eventually react to form alkali feldspar).
Alloys such as steel are generally solutions.

15. Molar Quantities
Formally, a molar quantity is simply the quantity per mole.
For example, the molar volume is
Generally, we will implicitly use molar quantities and not necessarily use the overbar to indicate such.
Another important parameter is the mole fraction:
Xi = Ni/ΣN

16. Raoult’s Law
Raoult noticed that the vapor pressures of a ethylene bromide and propylene bromide solution were proportional to the mole fractions of those components:
Where Pi is the partial pressure exerted by gas i:
and P˚ is the vapor pressure of pure i
Raoult’s Law states that the partial pressure of an ideal component in a solution is equal to the mole fraction times the partial pressure exerted by the pure substance.

17. Ideal Solutions
Turns out this does not hold in the exact and is only approximately true for a limited number of solutions.
Such solutions are termed ideal solutions. Raoult’s Law expresses ideal behavior in solutions.
In an ideal solution, interactions between different species are the same as the interactions between molecules or atoms of the same species.

18. Henry’s Law
As we’ll see, most substances approach ideal behavior as their mole fraction approaches 1.
On the other end of the spectrum, most substances exhibit Henry’s Law behavior as their mole fractions approach 0 (Xi ⟶ 0).
Henry’s Law is:
Pi = hiXi
where hi is Henry’s Law ‘constant’. It can be (generally is) a function of T and P and the nature of the solution, but is independent of the concentration of i.

19. Vapor Pressures in a Water-Dioxane Solution
Ideal
Henry’s Law

20. Partial Molar Quantities and the Chemical Potential

21. Partial Molar Quantities
Now that we have introduced the mole fraction, X, and variable composition, we want to know how the variables of our system, e.g., V, change as we change composition.
These are partial molar quantities, usually indicated by the lower case letter.
For example:
Such that
This is the partial molar volume of component i. For example, the partial molar volume of O2 dissolved in seawater.
This tell us how the volume of water changes for an addition of dissolved O2 holding T, P, and the amounts of everything else constant.

22. Partial Molar Volumes of Ethanol and Water
If you add a shot (3 oz) of rum to 12 oz of Coca Cola, what will be the volume of your ‘rum ‘n coke’?
Less than 15 oz! Blame chemistry, not the bartender.

23. Other Partial Molar Quantities
We can also define partial molar quantities of other thermodynamic variables, such as entropy, and enthalpy.
One partial molar quantity is particularly useful, that of the Gibbs Free Energy.

24. Chemical Potential
The chemical potential is defined as partial molar Gibbs Free Energy:
such that
or, dividing each side by the total number of moles:
The chemical potential tells us how the Gibbs Free Energy will vary with the number of moles, ni, of component i holding temperature, pressure, and the number of moles of all other components constant.
For a pure substance, the chemical potential is equal to its molar Gibbs Free Energy (also the molar Helmholz Free Energy):