Heat of reaction DH0R

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Heat of Reaction

• Heat of reaction DH0R
• DH0R is positive ? exothermic
• DH0R is negative ? endothermic
• Example 2A 3B ? A2B3
• DH0R H0f(A2B3)-2H0f(A) 3H0f(B)

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Entropy of reaction

• Just as was done with enthalpies
• Entropy of reaction S0R
• When DS0R is positive ? entropy increases as a
  result of a change in state
• When DS0R is negative ? entropy decreases as a
  result of a change in state

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J. Willard Gibbs

• Gibbs realized that for a reaction, a certain
  amount of energy goes to an increase in entropy
  of a system.
• G H TS or DG0R DH0R TDS0R
• Gibbs Free Energy (G) is a state variable,
  measured in KJ/mol or Cal/mol
• Tabulated values of DG0R are in Appendix F3-F5

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G is a measure of driving force

• DG0R DH0R TDS0R
• When DG0R is negative ? forward reaction has
  excess energy and will occur spontaneously
• When DG0R is positive ? there is not enough
  energy in the forward direction, and the BACKWARD
  reaction will occur
• When DG0R is ZERO ? reaction is AT equilibrium

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Increasing energy with temp?

• The added energy in a substance that occurs as
  temperature increases is stored in modes of
  motion in the substance
• For any molecule modes are vibration,
  translation, and rotation
• Solid ? bond vibrations
• Gases ? translation
• Liquid water complex function

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Heat Capacity

• When heat is added to a phase its temperature
  increases (No, really)
• Not all materials behave the same though!
• dqCVdT ? where CV is a constant (heat capacity
  for a particular material)
• Or at constant P dqCpdT
• Recall that dqpdH then dHCpdT
• Relationship between CV and Cp

Where a and b are coefficients of isobaric
thermal expansion and isothermal compression,
respectively
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Enthalpy at different temps

• HOWEVER ? C isnt really constant.
• C also varies with temperature, so to really
  describe enthalpy of formation at any
  temperature, we need to define C as a function of
  temperature
• Another empirical determination
• Cpa(bx10-3)T(cx10-6)T2
• Where this is a fit to experimental data and a,
  b, and c are from the fit line (non-linear)

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Does water behave like this?

• Water exists as liquid, solids, gas, and
  supercritical fluid (boundary between gas and
  liquid disappears where this happens is the
  critical point)
• Cp is a complex function of
• T and P (H-bond affinities),
• does not ascribe to Maier-
• Kelley forms

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Volume Changes (Equation of State)
For Minerals
Volume is related to energy changes
Mineral volume changes as a function of T a,
coefficient of thermal expansion Mineral
volume changes as a function of P b, coefficient
of isothermal expansion
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Volume Changes (Equation of State)

• Gases and liquids undergo significant volume
  changes with T and P changes
• Number of empirically based EOS solns..
• For metamorphic environments
• Redlich and Kwong equation
• V-bar denotes a molar quantity, aRw and bRK are
  constants
• EOS then equates energy (Helmholtz fee energy) to
  P

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• Now, how does free energy change with T and P?
• From DGDH-TDS

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Phase Relations

• Rule At equilibrium, reactants and products have
  the same Gibbs Energy
• For 2 things at equilibrium, can investigate the
  P-T relationships ? different minerals change
  with T-P differently
• For DGR DSRdT DVRdP, at equilibrium, DG0,
  rearranging

Clausius-Clapeyron equation
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V Vº(1-bDP)
DSR change with T or P?

• DV for solids stays nearly constant as P, T
  change, DV for liquids and gases DOES NOT
• Solid-solid reactions linear ? S and V nearly
  constant, DS/DV constant ? slope in diagram
• For metamorphic reactions involving liquids or
  gases, volume changes are significant, DV terms
  large and a function of T and P (and often
  complex functions) slope is not linear and can
  change sign (change slope to )

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Example Diamond-graphite

• To get C from graphite to diamond at 25ºC
  requires 1600 MPa of pressure, lets calculate
  what P it requires at 1000ºC

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Phase diagram

• Need to represent how mineral reactions at
  equilibrium vary with P and T

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Gibbs Phase Rule

• The number of variables which are required to
  describe the state of a system
• pfc2 fc-p2
• Where p of phases, c of components,
• f degrees of freedom
• The degrees of freedom correspond to the number
  of intensive variables that can be changed
  without changing the number of phases in the
  system

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Variance and f

• fc-p2
• Consider a one component (unary) diagram
• If considering presence of 1 phase (the liquid,
  solid, OR gas) it is divariant
• 2 phases univariant
• 3 phases invariant

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Heat of Reaction

• Heat of reaction DH0R
• DH0R is positive ? exothermic
• DH0R is negative ? endothermic
• Example 2A 3B ? A2B3
• DH0R H0f(A2B3)-2H0f(A) 3H0f(B)

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Entropy of reaction

• Just as was done with enthalpies
• Entropy of reaction S0R
• When DS0R is positive ? entropy increases as a
  result of a change in state
• When DS0R is negative ? entropy decreases as a
  result of a change in state

21
J. Willard Gibbs

• Gibbs realized that for a reaction, a certain
  amount of energy goes to an increase in entropy
  of a system.
• G H TS or DG0R DH0R TDS0R
• Gibbs Free Energy (G) is a state variable,
  measured in KJ/mol or Cal/mol
• Tabulated values of DG0R are in Appendix F3-F5

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G is a measure of driving force

• DG0R DH0R TDS0R
• When DG0R is negative ? forward reaction has
  excess energy and will occur spontaneously
• When DG0R is positive ? there is not enough
  energy in the forward direction, and the BACKWARD
  reaction will occur
• When DG0R is ZERO ? reaction is AT equilibrium

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Free Energy Examples

• DG0R DH0R TDS0R
• H2O(l)-63.32 kcal/mol (NIST value
  http//webbook.nist.gov/chemistry/)
• Fe2 ¼ O2 H ? Fe3 ½ H2O
• -4120(-633200.5)--21870(39540.25)
• -67440--19893-47547 cal/mol

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Chemical Potential

• Enthalpy (H), entropy (S), and Gibbs Free Energy
  (G) are molal (moles/kg) quantities
• Chemical potential, m, is the Gibbs free energy
  per molal unit
• In other words, the "chemical potential mi" is a
  measure of how much the free energy of a system
  changes (by dGi) if you add or remove a number
  dni particles of the particle species i while
  keeping the number of the other particles (and
  the temperature T and the pressure P) constant

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Mixing

• Putting two components into the same system
  they mix and potentially interact
• Mechanical mixture no chemical interaction
  where X is mole fraction of A, B
• ms XAmA XBmB
• Random mixture particles spontaneously (so m
  must go down) orient randomly
• Dmmixms mmechanical mixing
• Mixing ideal IF interaction of A-A A-B B-B ?
  if that is true then DHmix0, so DSmix must be gt0
  (because mmixlt0 (spontaneous mixing)
• DSid mix -RSXilnXi

Rmolar gas constant Xmole fraction component i
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Mixing, ideal systems
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Mixing, real systems

• When components interact with each other
  chemically and change the overall solution energy
• Dmreg ?XAXB
• Particularly this formulation is important in
  geochemistry for solid solutions of minerals,
  such as olivine (ex Fo50Fa50)

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Law of Mass Action

• Getting out of the standard state
• Accounting for free energy of ions ? 1
• mm0 RT ln P
• Bear in mind the difference between the standard
  state G0 and m0 vs. the molar property G and m
  (not at standard state ? 25 C, 1 bar, a mole)

GP G0 RT(ln P ln P0)
GP G0 RT ln P
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Equilibrium Constant

• For a reaction of ideal gases, P becomes
• for aA bB ? cC dD
• Restate the equation as
• DGR DG0R RT ln Q
• AT equilibrium, DGR0, therefore
• DG0R -RT ln Keq
• where Keq is the equilibrium constant

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Assessing equilibrium

• If DGR DG0R RT ln Q, and at equilibrium DG0R
  0, then QK
• Q ? reaction quotient, aka Ion Activity Product
  (IAP) is the product of all products over product
  of all reactants at any condition
• K ? aka Keq, same calculation, but AT equilibrium

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Saturation Index

• When DGR0, then ln Q/Keq0, therefore QKeq.
• For minerals dissolving in water
• Saturation Index (SI) log Q/K or IAP/Keq
• When SI0, mineral is at equilibrium, when SIlt0
  (i.e. negative), mineral is undersaturated

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Calculating Keq

• DG0R -RT ln Keq
• Look up G0i for each component in data tables
  (such as Appendix B in your book)
• Examples
• CaCO3(calcite) 2 H ? Ca2 H2CO3(aq)
• CaCO3(aragonite) 2 H ? Ca2 H2CO3(aq)
• H2CO3(aq) ? H2O CO2(aq)
• NaAlSiO4(nepheline) SiO2(quartz) ?
  NaAlSi3O8(albite)

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Application to ions in solution

• Ions in solutions are obviously nonideal states!
  
• Use activities (ai) to apply thermodynamics and
  law of mass action
• ai gimi
• The activity coefficient, gi, is found via some
  empirical foundations

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Ion Activity Product

• For reaction aA bB ? cC dD
• For simple mineral dissolution, this is only the
  product of the products ? activity of a solid
  phase is equal to one
• CaCO3 ? Ca2 CO32-
• IAP Ca2CO32-

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Solubility Product Constant

• For mineral dissolution, the K is Ksp, the
  solubility product constant
• Use it for a quick look at how soluble a mineral
  is, often presented as pK (table 10.1)
• DG0R RT ln Ksp
• Higher values ? more soluble
• CaCO3(calcite) ? Ca2 CO32-
• Fe3(PO4)28H2O ? 3 Fe2 2 PO43- 8 H2O

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Activity

• Activity, a, is the term which relates Gibbs Free
  Energy to chemical potential
• mi-G0i RT ln ai
• Why is there now a correction term you might ask
• Has to do with how things mix together
• Relates an ideal solution to a non-ideal solution

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Activity II

• For solids or liquid solutions
• aiXigi
• For gases
• aiPigi fi
• For aqueous solutions
• aimigi

Ximole fraction of component i Pi partial
pressure of component i mi molal concentration
of component i
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Activity Coefficients

• Where do they come from??
• We think of ideal as the standard state, but
  for dissolved ions, that is actually an
  infinitely dilute solution
• Gases, minerals, and bulk liquids (H2O) are
  usually pretty close to 1 in waters
• Dissolved molecules/ ions are have activity
  coefficients that change with concentration (ions
  are curved, molecules usually more linear
  relation)