Phases, Components,
Species & Solutions
Lecture 5
Chapter 3
Thermodynamics of multi-component systems
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.
Some Definitions
• Phase
o 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.
o NaCl dissolved in seawater is not a phase, but seawater with all its
dissolved components (but not the particulates) is.
• Species
o 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
o Components are more carefully defined. But:
• We are free to define the components of our system
• Components need not be real chemical entities.
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
o 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?
The CO2–H2O System
•
•
•
•
•
•
Carbonate Solution Recipe:
o
o
Distill water
Place in a beaker and let stand exposed to the atmosphere
o
o
o
o
H+, OH–, H2O, CO2, H2CO3, HCO3–, CO32–
What is the minimum number of components?
Our rule was c = n – r
Just 3 components, e.g., CO2, H2O, H+
In the distilled water, some of the water molecules will dissociate
to form hydrogen and hydroxyl ions:
1. H2O ⇋ H+ + OH–
Some atmospheric CO2 will dissolve in the water and reacts to
form carbonic acid:
2. CO2 + H2O ⇋ H2CO3
Some of that carbonic acid dissociates to form H+ ions plus
bicarbonate ion:
3. H2CO3 ⇋ H+ + HCO3–
Some of the bicarbonate will dissociate to form carbonate ions:
4. HCO3– ⇋ H+ + CO32–
How many species and how many components?
Graphical Approach
• 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
Phase diagram
for the system
Al2O3–H2O–SiO2
The lines are called joins because they
join phases. In addition to the endmembers, 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, αquartz, and β-quartz).
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.
Gibbs Phase Rule
• The phase rule is
ƒ= c - ϕ + 2
o where ƒ is the degrees of freedom, c is the number of components, and f
is the number of phases.
o 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.
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?
• How many to
uniquely fix the
system?
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:
d∆ Gr = ∆ Vr dP -∆ Sr dT = 0
• And
dP ∆ Sr
=
dT ∆ Vr
Solutions
Solutions
• Solutions are defined as homogenous phases
produced by dissolving one or more substances in
another substance.
• Mixtures are not solutions
o Salad dressing (oil and vinegar) is not a solution, no matter how much you
shake it.
o The mineral alkali feldspar (K,Na)AlSi3O8 is a solution (at high temperature).
o 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).
Molar Quantities
• Formally, a molar quantity is simply the quantity per
mole.
• For example, the molar volume is
V
=V
N
• 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
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:
Pi = Xi Pi o
• Where Pi is the partial pressure exerted by gas i:
Pi = Xi Ptotal
• 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.
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.
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
o where hi is Henry’s Law ‘constant’. It can be (usually is) a function of T and
P and the nature of the solution, but is independent of the concentration
of i.
Vapor Pressures in a
Water-Dioxane Solution