Activity II For solids or liquid solutions ai Xi i For gases ai Pi i fi Xi mole fraction of component i Pi partial pressure of component i mi molal concentration of component i For aqueous solutions ai mi i Activity Coefficients Debye Huckel approximation 2 Az I log 1 2 I aBI 1 2 Where A and B are constants depending on T see table 10 3 in your book and a is a measure of the effective diameter of the ion table 10 4 I mi z i 2 i Equilibrium Constant GR G0R RT ln Q AT equilibrium GR 0 therefore G0R RT ln Keq where Keq is the equilibrium constant Equilibrium constants G0R RT ln K Rearrange ln K G0R RT K e G R0 RT Find K from thermodynamic data for any reaction Q is also found from the activities of the specific minerals gases and species involved in a reaction in turn affected by the solution they are in products Q reactants n i n i 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 G0R RT ln Ksp Higher values more soluble CaCO3 calcite Ca2 CO32Fe3 PO4 2 8H2O 3 Fe2 2 PO43 8 H2O Ion Activity Product For reaction aA bB cC dD C D IAP a b A B c d Q GR RT ln K eq For simple mineral dissolution this is only the product of the products activity of a solid phase is equal to one CaCO3 Ca2 CO32IAP Ca2 CO32 c d C D IAP 1 Saturation Index When GR 0 then ln Q Keq 0 therefore Q Keq For minerals dissolving in water Saturation Index SI log Q K or IAP Keq When SI 0 mineral is at equilibrium when SI 0 i e negative mineral is undersaturated Q G RT ln K eq o R Q G 2 303RT log K eq 0 R Calculating Keq GR0 ni Gi0 products ni Gi0 reactants G0R RT ln Keq i i Look up G0i for each component in data tables such as Appendix F3 F5 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 Using Keq to define equilibrium concentrations G0R RT ln Keq K eq n products i n reactants i Keq sets the amount of ions present relative to one another for any equilibrium condition AT Equilibrium K eq Q n products i n reactants i This is the law of mass action If the system is at equilibrium then the ratio is a constant Example CaCO3 calcite Ca2 CO32 pK 8 4 T 25 Ca2 CO32 what s the concentration of Ca2 What if there is already some CO 32 there Ca2 CO32 Summation of reactionthermodynamic properties Can sum a set of reactions cancelling out equivalent terms on opposite sides to form a new reaction and derive that reaction s K eq from it s constituents Consider rxn CaCO3 calcite CO2 g H2O Ca2 2 HCO3 SUM of reactions CaCO3 calcite Ca2 CO32CO2 g H2O H2CO30 H2CO30 H HCO3H CO32 HCO3 log Keq CaCO3 calcite Ca2 CO32 8 48 CO2 g H2O H2CO30 1 47 H2CO30 H HCO3 6 35 H CO32 HCO3 10 33 CaCO3 calcite CO2 g H2O Ca2 2 HCO3 5 97 Another way to do this is to simply combine the Keq data algebraically K eq K calcite K CO2 K1 K2 Still another way is recompute the G0R for the reaction of interest and calculate Keq Mineral Chemistry How could we describe the complete chemistry of a mineral or rock What processes affect how this complete chemistry might change Pauling s Rules for ionic structures 1 Radius Ratio Principle cation anion distance can be calculated from their effective ionic radii cation coordination depends on relative radii between cations and surrounding anions Geometrical calculations reveal ideal Rc Ra ratios for selected coordination numbers Larger cation anion ratio yields higher C N as C N increases space between anions increases and larger cations can fit Stretching a polyhedra to fit a larger cation is possible Pauling s Rules for ionic structures 2 Electrostatic Valency Principle Bond strength ion valence C N Sum of bonds to an ion charge on that ion Relative bond strengths in a mineral containing 2 different ions Isodesmic all bonds have same relative strength Anisodesmic strength of one bond much stronger than others simplify much stronger part to be an anionic entity SO42 NO3 CO32 Mesodesmic cation anion bond strength charge meaning identical bond strength available for further bonding to cation or other anion Pauling s Rules for ionic structures 3 Sharing of edges or faces by coordinating polyhedra is inherently unstable This puts cations closer together and they will repel each other Pauling s Rules for ionic structures 4 Cations of high charge do not share anions easily with other cations due to high degree of repulsion 5 Principle of Parsimony Atomic structures tend to be composed of only a few distinct components they are simple with only a few types of ions and bonds Isoelectronic series Ions of different elements with equal numbers of electrons are said to be isoelectronic Example Al3 Si4 P5 S6 Cl7 each have how many e Size patterns can be related to the concept of shielding vs electrostatic interaction For any element how would charge and size be related Goldschmidt s rules of Substitution 1 The ions of one element can extensively replace those of another in ionic crystals if their radii differ by less than about 15 2 Ions whose charges differ by one may substitute readily if electrical neutrality is maintained if charge differs by more than one substitution is minimal Goldschmidt s rules of Substitution 3 When 2 ions can occupy a particular position in a lattice the ion with the higher ionic potential forms a stronger bond with the anions surrounding the site 4 Substitution may be limited when the electronegativities of competing ions are different forming bonds of different ionic character Goldschmidt s rules updated After Wood 2003 Equilibrium partitioning depends primarily on 2 energies of substitution into the crystal a the energy of elastic strain generated by inserting an ion which is either too large or too small for the site b the electrostatic work done in inserting an ion which is either more or less highly charged than the major ion normally occupying the site The theory requires modifications to Goldschmidt s rules 2 and 3 Rule 2 should now be The site has a preferred radius of ion r O which enters most easily For ions of the same charge those which are closest in radius to rO enter most easily Ions which are smaller or larger are discriminated against Rule 3 The site has a preferred charge ZO For ions of similar size but different charge the one whose charge is closest to Z O enters most easily What ions would substitute nicely into pyrite S radius 219 …