Thermodynamics: (pure) phase equilibrium

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Thermodynamics(pure) phase equilibrium
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Phases
The idea of phase has an experimental basis and
is not strictly required by theory. In pure
chemical thermodynamics, a phase is defined as a
region of space where the properties are uniform
(they do not depend on position in space).
When other (surface, electric, magnetic,
gravity,) effects are considered, the definition
must be modified some properties actually are no
more uniform and are allowed to change
continuously within a phase, but there is a
discontinuous change at the phase boundaries.
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Phases - examples
Liquid water water vapor gt two phases. Liquid
water water vapor air gt two phases. (Only)
solid iron gt one phase. Solid iron air (which
possibly includes a fairly small amount of iron
atoms) gt two phases. A solution of sugar in
water gt one phase. A solution of sugar in water
sugar particles gt two phases.
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Phases
The region of space corresponding to a phase does
NOT need to be connected in space.
For instance, in the case of a solution of sugar
in water solid sugar particles, we have two
phases independently of the fact that solid sugar
is in a single crystal or in many solid
particles. In most cases, solid samples are not
single crystals, but are polycrystalline (several
grains separated by grain boundaries).
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Phases
A phase can be made of a single substance (pure
phase). When the whole system is made of more
than a substance, each phase ideally contains
(small or large) amounts of each substance.
There are cases, however, where these amounts are
so small that can be neglected. This depends from
chemical nature of the substances and nature of
the phase
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Phases
There is always only one gas phase (complete
miscibility in the gas phase is the rule).
Complete miscibility of two or more substances in
a liquid phase is frequently achieved, but this
is not the rule miscibility gaps are possible in
liquids (for instance water a hydrocarbon).
Miscibility gaps are much more frequent in solids.
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Solid (crystalline) Phases
As a rule, different crystal structures define
different phases. Things are not so simple,
however for two reasons. One is because two
different crystal structures can be symmetry
related. Then, it may happen that the properties
of a structure switch continuously into those of
the other by moving along a state variable (e.g.
T or composition). The other reason is related to
miscibility gaps (in liquid state or in the same
crystal structure), and is important in solids
because miscibility gaps are so frequent. More on
these points later.
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Free energy in heterogeneous systems
When discussing phase equilibria, the general
rule that (at given T, P conditions) the free
energy (G) is at its minimum has a very nice
graphical representation. To show this, it is
necessary to look at the graphical representation
of an heterogeneous system, that is a system
where two (or more) phases are in equilibrum with
each other. The discussion is presented for a
two-components systems, but is trivially
generalized to many components.
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g of phase a with comp. xa
Let us consider an heterogeneous system (hs) with
overall composition xc and made of appropriate
amounts of Phase a with composition xa lt
xc Phase b with composition xb gt xc
g of phase b with comp. xb
Overall composition xc
The amounts of each component in each phase (n1a,
n1b, n2a, n2b) are constrained by mass
conservation
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g of phase b
By considering these constraints, one (easily)
founds that
g of phase a
Overall composition
The heterogeneous system is fully represented by
c, that is by the point where the ab segment
crosses the vertical line corresponding to the
overall composition.
The ordinate of point c is the g of the hs.
The amounts of phases a and b are in the ratio
(LEVER RULE)
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Achieving minimum G or g for a two phase system
Let us consider a system with specified overall
composition and (possibly) divided into two
phases.
Overall composition
It can be single phase (point a only phase a or
point b only phase b), it can be divided into
various amounts of both phases (point
c). Thermodynamic stability corresponds to point
d.
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Achieving minimum G or g for a two phase system
Thermodynamic stability corresponds to a common
tangent line to the gs of both phases.
This also means that (at thermodynamic
equilibrium) the chemical potential of component
1 must the same in the two phases (the same holds
for component 2).
Composition of phase a
Composition of phase b
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Phase equilibrium
Going into general results, in a system with n
(independent) components, thermodynamic
equilibrium is achieved when (for each
independent component) the chemical potential
takes the same value in all phases.

• Discussing phase equilibria using g or using the
  chemical potentials is mathematically equivalent
  
• Using g usually provides better insights and
  makes simpler to discuss many applications in
  Materials Science.
• Using the chemical potentials makes easier to
  generalize.

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The (Gibbs) Phase rule

• Let us say that we have
• n (independent) components,
• f phases
• Let us ask how many are (v) the degrees of
  freedom of the system (how many variables we must
  declare to specify the state of the system).
• The answer is
• v n f 2

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Gibbs Phase rule
This is easily obtained. We have n independent
components, so we do not have to care about
chemical reactions. The total number of chemical
concentrations or other intensive chemical
variables is (n-1) for each phase, that is
f(n-1). Moreover, we must state 2 physical
variables (let us say T and P) nf f 2. For
each component, there are f-1 equalities between
chemical potentials. Then nf n
equalities. Then v nf f 2 nf n n f
2.