ActivityCoefficients;EquilibriumConstantsActivityCoefficients;EquilibriumConstants
Lecture 8
How do deal with individualions in aqueous solution?How do deal with individualions in aqueous solution?
•We can’t simply add Na+ to a solution (positive ionswould repel each other).
•We can add NaCl. How do we partitionthermodynamic parameters between Na+ and Cl–?
•For a salt AB, the molarity is:
•mA = νAmAB and mB = νBmAB
oas usual ν is the stoichiometric coefficient
•For some thermodynamic parameter Ψ (e.g., µ)
•ΨAB = νAΨA + νBΨB
•So for example for MgCl2:
Practical Approach toElectrolyte ActivityCoefficientsPractical Approach toElectrolyte ActivityCoefficients
Debye-Hückel and Davies
Debeye-Hückel TheoryDebeye-Hückel Theory
•In an electrolyte solution, each ion exerts an electrostatic force on every other ion.These forces will decrease with the square of distance between ions.
•The forces between ions will be reduced by the presence of water molecules, dueto its dielectric nature.
•As total solute concentration increases, the mean distance between ions willdecrease. Thus activity should depend on the total ionic concentration.
•The extent of electrostatic interaction will also depend on the charge of the ionsinvolved: the force between Ca2+ and Mg2+ ions will be greater at the samedistance than between Na+ and K+ ions.
•In the Debye-Hückel Theory, a given ion is considered to be surrounded by anatmosphere or cloud of oppositely charged ions (this atmosphere is distinct from,and unrelated to, the solvation shell). If it were not for the thermal motion of theions, the structure would be analogous to that of a crystal lattice, though looser.Thermal motion tends to destroy this structure.
•The density of charge in this ion atmosphere increases with the square root of theionic concentrations, but increases with the square of the charges on those ions.The dielectric effect of intervening water molecules will tend to reduce theinteraction between ions.
Debye-Hückel ExtendedLawDebye-Hückel ExtendedLaw
•Assumptions
oComplete dissociation
oIons are point charges
oSolvent is structureless
oThermal energy exceedselectrostatic interaction energy
•Debye-Hückel ExtendedLaw
•Where A and B areconstants, z is ioniccharge, å is effectiveionic radius and I is ionicstrength:
Debye-Hückel Limiting &Davies LawsDebye-Hückel Limiting &Davies Laws
•Limiting Law (for lowionic strength)
•Davies Law:
oWhere b is a constant (≈0.3).
Assumption of completedissociation one of main limitingfactors of these approaches:ions more likely to associate andform ion pairs at higherconcentrations.
Activities in SolidSolutionsActivities in SolidSolutions
•Many minerals are alsosolutions.
•How do we computeactivites ofcomponents in thesolutions?
•For example, if wesubstitute Fe2+ for Mg2+in forsterite, what arethe activities of Fe2+and Mg2+?
Olivine Structure:
Fe and Mg ions are located betweenunlinked silica tetrahedra
Mixing-On-Site ModelMixing-On-Site Model
•Many crystalline solids, for example olivine, can be treated as idealsolutions. A simple ideal solution model is the mixing on site model, whichconsiders the substitution of species in any site individually. In this model,the activity of an individual species is calculated as:
ai,ideal = (Xi)ν
•where X is the mole fraction of the ith atom and ν is the number of sitesper formula unit on which mixing takes place (the stoichiometriccoefficient). For example, ν=2 in the Fe-Mg exchange in olivine,(Mg,Fe)2SiO4. Olivine has two sites that can be occupied by Mg and Fe.We could treat them separately, then the activity of Mg would be, ineffect, the sum of its activity in the two sites:
oFor example, if Mg were distributed randomly between the two sites, the total mole fraction were 0.5, theactiity would be 0.52 + 0.52 = 0.5.
oWhile olivine has two sites that can be occupied by Mg and Fe (M1 and M2), they are effectivelyequivalent. So we could simplify things by choosing (Mg,Fe)Si1/2O2 as the formula unit (we must then chooseall other thermodynamic parameters to be 1/2 those of (Mg,Fe)2SiO4). In this case, the activity of Mg2+ issimply its mole fraction.
Activities of PhaseComponentsActivities of PhaseComponents
•Now suppose we chose of component to be not anion, but a pure phase, such as pyrope, Mg3Al2Si3O12,in garnet, whose general formula is X3Y2Si3O12.
•Suppose the garnet of interest has the composition:
•The chemical potential of pyrope in garnet containsmixing contributions from both Mg in the cubic siteand Al in the octahedral site. The activity of pyropeis thus given by:
•For example:
Free Energy of Solution inIdeal Solid SolutionsFree Energy of Solution inIdeal Solid Solutions
∆G = ∆H – T∆S
oBut ∆H = 0 in the ideal case so
∆Gideal mixing = -T∆Sideal mixing
owhere ∆Sideal mixing is simply the configurational entropy:
•In the pyrope example, the chemical potential ofpyrope in garnet would be:
The Equilibrium ConstantThe Equilibrium Constant
•Consider the reaction
aA + bB = cC + dD
•The Free Energy change of reaction is:
∆G = cµc + dµd – aµa – bµb
•At equilibrium:
•Expanding the right side
•or
•We define the right term as the equilibrium constant:
Free Energy and theEquilibrium ConstantFree Energy and theEquilibrium Constant
•Since:
•then
•and
•Note of caution: our thermodynamic parameters areadditive, but because of the exponential relationbetween the equilibrium constant and free energy,equilibrium constants are multiplicative.
Manipulating EquilibriumConstantsManipulating EquilibriumConstants
•Suppose we want to know the equilibrium constant for areaction that can be written as the sum of two reactions,
oe.g., we can sum
oto yield
oThe equilibrium constant of the net reaction would be the product of theequilibrium constants of the individual reactions.
•For this reason and because equilibrium constants canbe very large or very small numbers, it is oftenconvenient to work with logs of equilibrium constants:
pK = - log K
o(we can then sum the pK’s).
Apparent Equilibrium Constantsand Distribution CoefficientApparent Equilibrium Constantsand Distribution Coefficient
•In practice, other kinds of equilibrium constants areused based on concentrations rather thanactivities.
•Distribution Coefficient
•Apparent Equilibrium Constant
Other FormsOther Forms
•A ‘solubility constant’ is an equilibrium constant. Forexample:
oSince the activity of NaCl in halite = 1, then
•Henry’s Law constants for describing solubility ofgases in solution (e.g., CO2 in water).
oSince Pi = hiXi
Law of Mass ActionLaw of Mass Action
•Important to remember our equation
describes the equilibrium condition. At non-equilibriumconditions it is called the reaction quotient, Q.
•Written for the reaction H2CO3 = HCO3- + H+
•We can see that addition of H+ will drive the reaction tothe left.
•“Changing the concentration of one species in areaction in a system at equilibrium will cause a reactionin a direction that minimizes that change”.
Le Chatelier’s PrincipleLe Chatelier’s Principle
•We can generalize this to pressure andtemperature:
dG = VdP - SdT
•An increase in pressure will drive a reaction in adirection such as to decrease volume
•An increase in temperature will drive a reaction in adirection such as to increase entropy.
•“When perturbed, a system reacts to minimize theeffects of perturbation.”
Temperature and PressureDependenceTemperature and PressureDependence
•Since ∆G˚ = ∆H˚ - T∆S˚ and ∆G˚ = -RT ln K then
•Temperature and pressure dependencies found bytaking derivatives of this equation with respect to Tand P.
Oxidation and ReductionOxidation and Reduction
Oxidation refers to processes in which atoms gain or losselectrons, e.g., Fe2+ Fe3+
Valence and RedoxValence and Redox
•We define valence as the charge an atom acquireswhen it is dissolved in solution.
•Conventions
oValence of all elements in pure form is 0.
oSum of valences much equal actual charge of species
oValence of hydrogen is +1 except in metal hydrides when it is -1
oValence of O is -2 except in peroxides when it is -1.
•Elements generally function as either electron donors oracceptors.
oMetals in 0 valence state are electron donors (become positively charged)
oOxygen is the most common electron acceptor (hence the term oxidation)
•Redox
oA reduced state can be thought of as one is which the availability of electronsis high
oAn oxidized state is one in which the availability of electrons is low.