End-member model

In conclusion, the inclined U-Pa and U-Th arrays appear to have some time significance because their interpretation as simply the result of recent mixing with a fluid containing Th and Pa as well as U requires fluid partition coefficients for Th and Pa well in excess of those observed experimentally. The corollary is that that there must be a decoupling between Ra-Th and Th-U disequilibria. A further possibility is a combination of the two end-member models discussed above into one in which some Th and Pa addition by fluids is followed by some in-growth due to ageing. In this case (discussed further below) the age inferred from the U-Th and U-Pa arrays is necessarily less straight forward to interpret. 

The amount of reduced sulfur in freshwater lacustrine sediment and in most marine sediment is a function of the availability of the limiting reactant during sulfate reduction—whether sulfate or organic matter. This simple two end-member model must frequently be modified for saline lacustrine sediment and for some marine sediment in order to reflect the capacity of the... 

For the calculation of solid-solution mineral behavior, two conceptual models may be used the end-member model (arbitrary mixing of two or more phases) and the site-mixing model (substituting elements can replace certain elements only at certain sites within the crystal structure). 

Although there is no a priori reason to assume that the core contains only one light element, a binary iron-light-element alloy core serves as a useful end-member model until suflhcient experimental data become available to allow consideration of more complicated and realistic alternatives. 

Abstract Voluminous anorthosite intrusions are common in mid-Proterozoie crust. Historically, two end-member models have been proposed for the origin of these anorthosites. 

The two-dimensional, single-cylinder model can be considered the far superior of the two end-member models for the bioturbated zone presented in this article. Although idealized, it is basically realistic and allows direct input of measurable physical and biological parameters with little mysticism. It will now be shown that this simple but relatively successful description of transport conditions within the bioturbated zone can provide insight into thermodynamic equilibrium controls on pore-water distributions. 

In the first instance, a number of workers (Varekamp et al. 1992 Sano and Marty 1995 Sano and Williams 1996) approximated the C-output at arcs using a three end-member model consisting of MORE mantle (M), and slab-derived marine carbonate/limestone (L) and (organic) sedimentary components (S). Sano and Marty... 

The formation and growth of the continental crust, and so the rate at which noble gas parent elements are extracted from the mantle, clearly affects the rate of daughter noble gas production in the mantle. Also, continental extraction is likely to also impact the rate of mantle degassing. Two end member models exist rapid and early continental growth, with subsequent additions balanced by subduction (Armstrong 1968, 1991), and continuous or episodic continental growth (see Taylor and McLennan 1985). Although recent publications tend to favor the latter (e.g., McCulloch and Bennett 1998), the issue remains open, and so noble gas models must assume some rate of parent element extraction from the mantle. 

The perfect mixing model frequently provides a poor fit to the distribution of petroleum properties in vertically stacked reservoirs and fails to account for the compositional grading within the individual reservoirs of the stack. Conversely, the second end-member model often fits the observed data surprisingly well, considering that it implies no mixing at all. This supports the contention that the petroleum in many, if not all, fields is poorly mixed. We show two examples here. 

Chemical potentials for the constituents of minerals are defined in a similar manner. All minerals contain substitutional impurities that affect their chemical properties. Impurities range from trace substitutions, as might be found in quartz, to widely varying fractions of the end-members of solid solutions series. Solid solutions of geologic significance include clay minerals, zeolites, and plagioclase feldspars, which are important components in most geochemical models. 

Non-stoichiometry in solid solutions may also be handled by the compound energy model see for example a recent review by Hillert [16]. In this approach the end-member corresponding to vacancies is an empty sub-lattice and it may be argued that the model loses its physical significance. Nevertheless, this model represents a mathematically efficient description that is often incorporated in thermodynamic representations of phase diagrams. 

Constraints are relations of equality or inequality, which must be exactly obeyed by the unknowns of a model. A familiar example is the mineral abundances in a rock or the end-member proportions in a mixture, which must sum up to unity whatever the errors on the data. 

The results of two sets of computations are shown in Figure 12. Open circles represent calculations for the binary adsorbent systems which used the Kurbatov coefficients for the end-member systems. This is the approach used by Davies-Colley et al., (9) in their examination of metal ion adsorption in mixtures of model... 

Figure 12. Comparison of the observed Cd(II) fractional removal f = [Cd(II). g/Cd(II)T] with model results incorporating different characteristics of x- Open circles Kurbatov coefficients for end-members (Tltania Xr = 0.99, pKj, = 1.77 Amorphous Iron ...

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