Examples of models that exhibit nonunique solutions

This page follows Chapter 12 of Bethke (2007).

Boehmite equilibrium

Assume:

  • temperature is C;

  • the mineral Boehmite is used in place of H in the component basis;

  • there is 1cm of free Boehmite;

  • supersaturation of all minerals is ignored;

  • the concentrations of the other components are given in Table 1.

Table 1: Bulk concentration in the Boehmite example

SpeciesConcentration (mmol/kg(solvent water))
K100
Cl100
SiO(aq)3
HCO0.06

Case 1

Assume the bulk concentration of Al is mol/kg(solvent water).

Case 2

Use Al(OH) instead of Al in the component basis, with a concentration of mol/kg(solvent water).

Results

Bethke (2007) predicts the composition at equilibrium to be as shown in Table 2. MOOSE produces the result ????

Table 2: Equilibrium composition in the Boehmite example

QuantityCase 1Case 2
pH3.16.3
total Al (molal)1010
free AlOH (molal)4.3
free Al(OH) (molal)3.0
free Al(OH) (molal)

Pyrite example

Assume:

  • temperature is C;

  • the mineral pyrite is used in place of O(aq) in the component basis;

  • there is 1cm of free pyrite;

  • the pH is 4

  • supersaturation of all minerals is ignored;

  • the concentrations of the other components are given in Table 3

Table 3: Bulk concentration in the Pyrite example

SpeciesConcentration (mol/kg(solvent water))
Na1
Cl1
Fe

Case 1

Assume the bulk concentration of SO is 10mmolal

Case 2

Use HS(aq) instead of SO in the component basis and its bulk concentration is 10mmolal.

Results

Bethke (2007) predicts the composition at equilibrium to be as shown in Table 4. MOOSE produces the result ????

Table 4: Equilibrium composition in the Pyrite example

QuantityCase 1Case 2
pH44
-50-67
total Fe (molal)0.010.01
total S (molal)0.010.01
SO (molal)0.010
HS(aq) (molal)0.010

References

  1. Craig M. Bethke. Geochemical and Biogeochemical Reaction Modeling. Cambridge University Press, 2 edition, 2007. doi:10.1017/CBO9780511619670.[BibTeX]