Gaines-Thomas model

No value for dmq/dmp is needed to evaluate the and Freundlich models. For the Langmuir model and the ion exchange model under the Gaines-Thomas and Gapon conventions,... 

Adsorption of a dissolved ionic species is always part of an exchange reaction that involves a competing ionic species. The desorbing species creates the vacant site to be occupied by the adsorbing one. Soil scientists use a variety of ion exchange models (e.g., the Gaines-Thomas, Gapon, Vanselow, and Rothmund-Kornfeld models) in which different conventions are used to write the concentrations of dissolved and adsorbed species. These models are adequately described and compared elsewhere (cf. Bolt 1979 Sposito 1981, 1989 Appelo and Postma 1993). 

Model constraints equilibrium between pure water and calcite (Ca control), dolomite (Mg control, dissolution only), fluorite (F control) and chalcedony (Si control). Electroneutrality is ensured by charge-balancing using Cl x (M +/Na) = 3.2 (where A pp is the cation selectivity constant and M is a divalent metal cation in milli-equivalents Gaines and Thomas model). K is not considered because of too high uncertainty on its A pp. Porosity calculated from leachable [Cl ] (around 17 ppm by weight of rock) and computed [Cl ] concentration in the water. 

The models applying the law of mass actions (Gaines and Thomas 1953 Howery and Thomas 1965 Argersinger et al. 1950 Ekedahl et al. 1950 Hogfeldt 1950) introduce the term of surface activity coefficient. 

In addition, there is interest in further extending the discussion to a variety of situations, that have recently gained much attention in the nonrelativistic case, as time-dependent systems [49], excited states [45] or finite temperature ensembles [110]. As an example of work along these lines we mention the gradient expansion of the noninteracting, relativistic free energy [110], leading to a temperature-dependent relativistic extended Thomas-Fermi model. 

Bayesian statistics is based upon manipulation of the probability p(0 j) that the model has a parameter vector 9, given a set of measured response data y. While the Bayesian approach dates to the work of Thomas Bayes in the mid- 1700s, it was slow to gain acceptance because, by treating 0 as a random vector, it violates the philosophical principle of Laplacian determinism, stating that nature is deterministic and predictable. Such criticisms were muted by the interpretation of probabilities as statements of belief, leading to a resurgence in the Bayesian approach. 

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