EQUILIBRIUM CONSTANTS, CHEMICAL AFFINITY, REACTION PROGRESS AND THE PHASE RULE

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TOPIC 5

• EQUILIBRIUM CONSTANTS, CHEMICAL AFFINITY,
  REACTION PROGRESS AND THE PHASE RULE

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I. THE EQUILIBRIUM CONSTANT
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DERIVATION OF THE EQUILIBRIUM CONSTANT

• Consider the reaction aA bB ? cC
• For each constituent we can write
• ?A ?A? RT ln aA
• ?B ?B? RT ln aB
• ?C ?C? RT ln aC
• But we now write
• ?r? c?C - a?A - b?B
• c(?C? RT ln aC) - a(?A? RT ln aA) - b(?B?
  RT ln aB)
• (c?C - a?A - b?B) cRT ln aC - aRT ln aA -
  bRT ln aB
• (c?C - a?A - b?B) RT ln aCc - RT ln aAa -
  RT ln aBb

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• In general notation, this is
• we normally then write
• so
• At equilibrium we have

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• We call K the equilibrium constant. Because the
  standard states usually refer to pure substances,
  in which ? G, we often write this equation
• The equilibrium constant is not a function of the
  composition of the system. It depends on
  temperature, and depending on the standard state
  chosen, pressure.
• This is a remarkable equation, because it relates
  the free energy difference of the reactants and
  products in their standard states, with the
  activities of the constituents in their actual
  state!

The ai here are the activities at equilibrium!
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CHANGE OF K WITH TEMPERATURE

• Start with dG VdP - SdT
• But ?rG ?rH - T?rS 0 at equilibrium, so
• ?rS ?rH/T

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DOING THE INTEGRATION

• CONSTANT ENTHALPY (?rCP 0)
• This is the equation of a straight line with a
  slope of - ?rH/R on a plot of log K vs. 1/T.
• CONSTANT HEAT CAPACITY

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• The previous equation can be rewritten in the
  form
• log K a b/T c ln T
• where

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• HEAT CAPACITY KNOWN (Maier-Kelley Expression)

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How log KTr, ?rH and ?rCP contribute to the
variation of log K with temperature.
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CHANGE OF K WITH PRESSURE

• STANDARD STATE AT FIXED PRESSURE
• If the standard state has a fixed pressure, then
• and the equilibrium constant is independent of P.

and
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• VARIABLE PRESSURE STANDARD STATE
• Integration is best accomplished by dividing up
  the free energy terms into separate terms for
  condensed phases, gases and solutes.
• In general

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• For condensed phases, a constant volume
  approximation may be appropriate
• For gases and supercritical fluids, the standard
  state is normally chosen as the ideal gas at the
  system temperature and one bar, so
• However, if we have a variable pressure standard
  state (this is hardly ever used for gases), then

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• where Q is given by
• For solutes, no general expression exists to
  substitute for the volume integral, so we write
• So the general expression for the change in K
  with pressure is

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II. HETEROGENEOUS AND OPEN SYSTEMS
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THE CHEMICAL AFFINITY

• For systems that are not in equilibrium, it would
  be convenient to have some quantitative measure
  of the degree to which the system is out of
  equilibrium. This measure is given by the
  affinity (A)
• where ? is called the reaction progress variable.
  This variable takes on a value of 0 at the start
  of some spontaneous process and ends at some
  value, perhaps 1, in the equilibrium state.

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• The differential d? is therefore an increment of
  progress of any spontaneous reaction. For an
  irreversible process, we can therefore write
• If we do not include the affinity term, then all
  we can write for an irreversible process is

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• Another way to understand the meaning of the
  affinity is to consider open systems. We have
  previously seen that at equilibrium in an open
  system, the chemical potentials of each component
  must be balanced in each and every phase. Let us
  consider a metastable state in which the chemical
  potentials are under some constraint and are not
  balanced. When we release this constraint, there
  will be a spontaneous, irreversible
  redistribution of potentials until equilibrium is
  obtained. Now let us imagine that this
  irreversible process takes place in very small
  increments, i.e., d?, in moles.

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• If we compare the two expressions
• we see that
• or
• To see what this means, we need a relationship
  between dn and d?.
• Because reactants and products in a reaction are
  related by fixed stoichiometries, changes in
  their number of moles are related by the
  following

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• The only natural cause of changes in the number
  of moles is a spontaneous or irreversible
  reaction such as we are now considering. Thus, it
  seems natural to write
• It thus follows that

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• For example, if we have the reaction
• A ? 5B
• we would write
• This relation allows us to make a connection
  between the reaction progress and the chemical
  potentials. Starting with
• we now sum over all constituents to get

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• But we have and
• so it follows that
• Now, the criterion of equilibrium is
• So it is clear that A 0 is another way of
  expressing the criterion of equilibrium. That is,
  at equilibrium, the chemical affinity will be
  zero there is no drive towards equilibrium,
  because the system is already there!

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• For example, for the reaction O2 2H2 ? 2H2O
• A ?O2 2?H2 - 2?H2O
• In a metastable mixture of O2 and H2, A is the
  amount by which the chemical potentials could be
  lowered by spontaneous reaction upon release of
  the metastable constraint. The chemical affinity
  is the driving force of the reaction. The
  driving force becomes zero when ?O2 2?H2
  2?H2O.
• Another relation can be derived starting with
• ?i ?i RT ln ai

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• where
• but
• so
• From what we have done, we can expect that we
  have the following relations

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THE PHASE RULE

• Let us consider a closed system that has C
  components and P phases. In this system, the
  compositions of each of the phases are potential
  variables, for a total of P(C-1) compositional
  variables (there are only C-1 compositional
  variables for each phase, because the sum has to
  total 100). In addition, temperature and
  pressure are variables. So the total number of
  variables are P(C-1) 2.
• At the same time, at equilibrium, we have for
  each component, P - 1 constraints of the type
• ?i1 ?i2 ?i3 ?i4 ?iP
• This gives us a total of C(P-1) constraints.

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• If we define the number of degrees of freedom (F)
  as
• F of variables - of constraints
• we can then write
• F P(C-1) 2 - C(P - 1)
• F PC - P 2 - CP C
• F C 2 - P
• This expression is known as the Gibbs phase rule.
  It is an amazingly powerful relationship, but
  care must be taken in its application. Recall
  that the components are defined as the minimum
  number of mathematical entities required to
  express the compositions of all phases in the
  system. The choice of components is arbitrary,
  and they need not correspond to actual chemical
  entities.

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CHOICE OF COMPONENTS

• Example 1 Consider the system containing
  forsterite (Mg2SiO4), enstatite (Mg2Si2O6) and
  quartz (SiO2).
• In this case, only two components are required to
  fully describe the compositions of all the
  phases.
• Valid Component Choices
• 1) Mg2SiO4, SiO2 2) Mg2SiO4, Mg2Si2O6 3) MgO,
  SiO2 4) Si2O4, MgO 5) Mg2Si2O6, -SiO2 etc.
• Invalid Component Choices
• 1) Mg, Si 2) Mg, Si, O 3) Mg2SiO4
• F C 2 - P 2 2 - 3 1
• So the system is univariant.

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• Example 2 Consider a system composed of the
  phases dolomite (CaMg(CO3)2), quartz (SiO2),
  diopside (CaMgSi2O6) and carbon dioxide vapor
  (CO2).
• Only three components are required to describe
  this system. One valid choice is CaMgO2, CO2 and
  SiO2.
• Note that Ca and Mg do not give rise to separate
  components because they occur in the same ratio
  wherever they occur.
• F C 2 - P 3 2 - 4 1
• so this system is also univariant.

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• Example 3 Consider a system containing calcite,
  dolomite, quartz, and a single fluid containing
  H2O and CO2.
• In this system, five components are required to
  describe the system. One valid choice is CaO,
  MgO, SiO2, H2O, and CO2.
• F C 2 - P 5 2 - 4 3
• This system is a trivariant one. If the vapor
  phase formed two immisicible fluid phases, then
  the system would have one more phase, and one
  less degree of freedom and would be divariant.
• F C 2 - P 5 2 - 5 2

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• Example 4 Consider a system containing garnet
  ((Fe,Mg)3Al2(SiO4)3), cordierite
  ((Fe,Mg)2Al4Si5O18), spinel ((Fe,Mg)Al2O4),
  olivine ((Fe,Mg)2SiO4), quartz (SiO2) and
  orthopyroxene ((Fe,Mg)2Si2O6).
• This one is a bit tricky. If Fe and Mg substitute
  for one another ideally in each of these phases,
  i.e., there is no differential partitioning of Fe
  and Mg, then they behave as a single component,
  so we have three components (Fe,Mg)O, Al2O3, and
  SiO2. This system has F 3 2 - 6 -1 and
  cannot be in equilibrium. If Fe or Mg prefer one
  phase to the others, then they are separate
  components and C 4. In this case F 4 2 - 6
  0 and the system is invariant.

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• Example 5 Consider the system containing the
  phases calcite (CaCO3), fluorite (CaF2),
  fluocerite (LaF3), bastnäsite (LaCO3F), parisite
  (CaLa2(CO3)3F2) and synchysite (CaLa(CO3)2F). How
  many components are required to describe this
  system?
• This one is also tricky. It would seem that F-
  and CO32- should be separate components. However,
  there are charge-balance requirements that tie
  these together as a single component. It takes 3
  components to describe this system. One choice
  is CaCO3, LaCO3F and F2(CO3)-1. The latter is an
  example of an exchange operator component, in
  which 2 F- are exchanged for one CO32-.

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APPLICATION OF THE PHASE RULE TO PHASE
DIAGRAMS THE Al2SiO5 SYSTEM
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PHASE RULE FOR OPEN SYSTEMS

• The activities of mobile components in an open
  system are not controlled by the phases in the
  system they are controlled by some external
  factor. Therefore, there is an additional
  constraint on each of the mobile components, and
  the degrees of freedom should be reduced
  accordingly.
• F C 2 - P - M
• where C number of total components and M the
  number of mobile components.
• This version of the phase rule explains why
  increased metasomatism tends to reduce the number
  of minerals in a rock.

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TYPES OF COMPONENTS(AFTER KORZHINSKII, 1959)

• Trace components Usually chosen as elements and
  occur at very low concentrations in the minerals
  of a system (e.g., V, Cr, Co in a granite). They
  are not an essential part of any phase. Their
  concentrations can very over a range without
  affecting the number of phases.
• Accessory components Components that occur
  almost entirely in a single phase. For example
  TiO2 in rutile, titanite or ilmenite P2O5 in
  apatite ZrO2 in zircon. In each case, the
  presence of the component causes a phase to
  exist, but no net increase in the degrees of
  freedom occurs.

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• Isomorphous components These are components that
  substitute freely for one another on various
  lattice sites. If the substitution its truly
  ideal, in other words, the mineral properties are
  unaffected by substitution, then the two
  components behave as one and can be considered a
  single component.
• Excess components Those that occur in the pure
  form, and also in some or all of the other phases
  of the system. For example SiO2 occurs in all
  silicates and also in the form of quartz. Because
  the activity of SiO2 is fixed by the presence of
  quartz, it is often possible to neglect SiO2 as a
  component in the counting of components.

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• Determinative components All those that do not
  fit into any of the above categories. They are
  therefore the set of independent components that
  control which phases will occur and how much.
  These are the components used to plot diagrams
  showing phase relations.

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MINERALOGICAL PHASE RULE

• Natural systems would normally have at least two
  degrees of freedom, e.g., temperature and
  pressure, in other words F ? 2. This allows us to
  write the mineralogical phase rule as
• P ? C
• this simply states that the number of minerals
  one would find in a rock in equilibrium is
  limited by the number of components.

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BUFFERED SYSTEMS

• Buffered system - a system in which the presence
  of a certain number of phases fixes all the
  properties of the system at a given temperature
  and pressure. In other words, the activities of
  all the components are fixed if temperature and
  pressure are specified.
• The phase rule can help us to identify buffered
  systems.
• For example, consider a rock containing both
  gypsum (CaSO42H2O) and anhydrite (CaSO4). This
  is a two-component, two-phase system, so F C
  2 - P 2 2 - 2 2.

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• This is a divariant system, and if we fix
  temperature and pressure, and gypsum and
  anhydrite remain in equilibrium, the the activity
  of H2O is fixed according to
• CaSO42H2O ? CaSO4 2H2O
• The equilibrium constant for this reaction can be
  written
• Because the solids are pure, their activities are
  unity, so
• and the activity of water is fixed!

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• Of course, any component can be buffered in
  theory. Consider a rock containing only
  forsterite and enstatite. This is a two-component
  system which will fix silica activity when P and
  T are specified.
• Mg2SiO4 SiO2 ? Mg2Si2O6
• Because enstatite and forsterite are pure phases,
  their activities are unity. However, this is not
  the case for the activity of silica.