Page 91 - Geothermal Energy Renewable Energy and The Environment

76 Geothermal Energy: Renewable Energy and the Environment

a solution so that the mass action law for a given reaction is satisfied. For the halite reaction this
means the mass action expression takes the form

[(γ Na + × m Na +) ⋅ (γ – × m –)] = K,
Cl
Cl
where γ is the activity coefficient for species i. For dilute solutions, γ is close to 1 and, for most
i
i
purposes, the measured concentrations can be used in calculations. Dilute solutions are those that
have total dissolved solute loads equal to or less than that of sea water, which has a dissolved load
of about 35 parts per thousand. For more concentrated solutions, the effects of nonideality can
be significant and need to be taken into account. The available data for activity coefficients that
are experimentally determined are relatively limited, especially for highly concentrated solutions.
Compilations of activity coefficients have been published or discussed by Harvey and Prausnitz
(1989), and papers in Palmer et al. (2004a).

affiniTy
Establishing whether a chemical reaction such as the halite dissolution reaction will actually take
place is accomplished by comparing the composition of the solution to the value of the equilibrium
constant using the following expression:

A = R × T × ln (Q/K), (5.6)
where A is called the affinity (J/mole), R is the universal gas constant (8.314 J/mole-K), T is tem-
perature (Kelvins), Q is the activity quotient for the relevant species in the applicable reaction, and
K is the equilibrium constant for that same reaction. If the measured solution composition results
in an activity product that is equal to K, the affinity will be zero, indicating that the solution is in
equilibrium with the solids, and no net dissolution or precipitation will occur. If the affinity were
greater than zero, the concentration of the solutes exceed the equilibrium value and the solution
would begin to precipitate solid. This situation is one in which the solution is supersaturated in the
solid involved in the reaction. For affinities less than zero, the reactant (in this case, our halite solid
phase) would continue to dissolve until it is either completely dissolved, or the activity product of
the solutes becomes equal to K, and equilibrium would be achieved.
This situation with respect to the values of affinity suggests that, mathematically, A must be
equivalent to the Gibbs energy of a reaction

A = G products – G reactants = (H products – H reactants ) – T × (S products – S reactants ). (3.16)

This relationship leads to the fundamental expression

∆G = –R × T × ln (K), (5.7)

which is a statement of the relationship between the activities of the species involved in a reaction
and the conditions under which they will coexist in thermodynamic equilibrium.
At 100°C, the value for log K for the halite dissolution reaction is about 1.6 (Figure 5.2). On
a molar basis, and assuming ideal behavior of the ions in solution, this means that a little more
+
than 6 moles, each, of Na and Cl must be dissolved in solution to achieve saturation. Given that
–
the molecular weight of sodium is about 22.99 gm/mole and that of Cl is about 35.39 gm/mole,
the respective masses of Na and Cl in a saturated solution will be a little more than 145 gms and
220 gms. Clearly, our original 10 grams of salt added to the pan would completely dissolve, the
solution would remain strongly undersaturated, and the resulting affinity and Gibbs energy would
be much less than zero.