Mixing in water• Solutions dominated by water (1 L=55.51 moles

H2O)

• aA=kHXA where KH is Henry’s Law coefficient – where is this valid? Low concentration of A

Mol fraction AH2O A

Act

ivity

0.0

1.0

0.0 1.0

Ideal mixing

aA

aH2ORaoult’s Law – higher concentration ranges (higher XA):

A=A0+RTlnAXA

where A is Rauolt’s law activity coefficient

Activity

• Activity, a, is the term which relates Gibbs Free Energy to chemical potential:

i-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

Activity II

• For solids or liquid solutions:ai=Xii

• For gases:

ai=Pii = fi

• For aqueous solutions:

ai=mii

Xi=mole fraction of component iPi = partial pressure of component imi = molal concentration of component i

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 lines relating concentration and activity coefficients, molecules usually more linear relation)

Application to ions in solution

• Ions in solutions are obviously nonideal mixtures!

• Use activities (ai) to apply thermodynamics and law of mass action

ai = imi

• The activity coefficient, i, is found via some empirical foundations

Dissolved species i

• First must define the ionic strength (I) of the solution the ion is in:

Where mi is the molar concentration of species i and zi is the charge of species I

2

izmI

ii

Activity Coefficients

• Debye-Huckel approximation (valid for I:

• 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)

2

1

2

12

log

aBII

IAz

Different ways to calculate i

• Limiting law

• Debye-Huckel

• Davies

• TJ, SIT models

• Pitzer, HKW models

Neutral species

• Setchnow equation:

• Logn=ksI

For activity coefficient (see table 4-2 for selected coefficients)

Law of Mass Action

• Getting ‘out’ of the standard state:

• Accounting for free energy of ions ≠ 1:=0 + RT ln P

• Bear in mind the difference between the standard state G0 and 0 vs. the molar property G and (not at standard state 25 C, 1 bar, a mole)

P

P

P

PP

dPRTdG

00

GP – G0 = RT(ln P – ln P0) GP – G0 = RT ln P

Equilibrium Constant

• For a reaction of ideal gases, P becomes:

for aA + bB cC + dD

• Restate the equation as:

GR – G0R = RT ln Q

• AT equilibrium, GR=0, therefore:

G0R = -RT ln Keq

where Keq is the equilibrium constant

QRTPP

PPRT

bB

aA

dD

cC lnln

Assessing equilibrium

IfGR – G0R = RT ln Q, and at equilibrium G0

R = 0, then: Q=K

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

i

ni

n

reactants

productsK

][

][

i

ni

n

reactants

productsQ

][

][

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+ + CO32-

Fe3(PO4)2*8H2O 3 Fe2+ + 2 PO43- + 8 H2O

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 = [Ca2+][CO32-]

1

dc DCIAP

ba

dc

BA

DCIAP

eqR K

QRTG ln

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

eq

oR K

QRTG ln

eqR K

QRTG log303.20

Calculating Keq

G0R = -RT ln Keq

• 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)

)reactants()( 000i

iii

iiR GnproductsGnG