Isotopes, mathematical calculation

Parts of a conceptual model, or all of it, may in many cases be quantified, using mass or energy balances, chemical equilibria and degree of saturation calculations, flow velocities calculated by hydraulic parameters, or velocities calculated via isotopic age indicators. In presenting the mathematical calculations, knowns and unknowns have to be clearly defined, and basic assumptions have to be discussed. Each calculation should be accompanied by an evaluation of its degree of confidence. 

ThAJ age is calculated. Mathematically, however, the choice of ratios is unimportant, provided that there is a common-denominator isotope to all three axes, and the errors and error correlations for the data points are correctly calculated. Note that, unlike paired x-y isochrons, not only can the errors of the detrital-corrected Th/ U and be... 

A double spike technique is essential for TIMS analyses of Se and Cr, and may also be useful in MC-ICP-MS analysis. Briefly, two spike isotopes with a known ratio are added to each sample, and the measured ratio of the spike isotopes is used to determine and correct for instrumental bias. Examples of Se and Cr double spikes currently in use are given in Table 1. The fact that small amounts of the spike isotopes are present in the samples and small amormts of nominally unspiked isotopes are found in the spikes is not a problem, as the measurements allow highly precise mathematical separation of spike from samples. Algorithms for such calculations are described by Albarede and Beard (2004) and, specifically for Se, by Johnson etal. (1999). 

If the process of APIO is properly described by Equation (19), which infers the presence of a soluble Fe(III) intermediate species, it will be difficult to analyze this species directly, given the low levels that are expected. We must therefore develop mathematical approaches to estimating the isotopic composition of this component, as was done for DIR. The equations used in the previous chapter (Chapter lOA Beard and Johnson 2004) to describe abiotic Fe(II) oxidation are useful for illustrating possible isotopic fractionations that may occur during APIO. We will assume that the overall oxidation process occurs through a series of first-order rate equations, where relatively slow oxidation of FefTI) to a soluble Fe(III) component occurs, which we will denote as Fe(III)jq for simplicity. The oxidation step is followed by precipitation of Fe(III)jq to ferrihydrite at a much faster rate, which maintains a relatively low level of Fe(III)jq relative to Fe(II)jq. The assumption of first-order kinetics is not strictly valid for the experiments reported in Croal et al. (2004), where decreasing FefTI) contents with time do not closely follow either zeroth-, first-, or second-order rate laws. However, use of a first-order rate law allows us to directly compare calculations here with those that are appropriate for abiologic Fe(II) oxidation, where experimental data are well fit to a first-order rate law (Chapter lOA Beard and Johnson 2004). 

Like all radioactive isotopes, C-14 decays at a predictable rate. Its half-life of 5,730 years means that one-half the amount of C-14 normally present in a living organism is present in an organism that has been dead for 5,730 years. By suitable manipulation of the mathematics involved in half-life calculations, the approximate age of the remains of plants and animals can be determined. 

Mathematical methods for the calculation of theoretical relative abundances within the isotopic duster, for comparison with experiment, usually rely on expansion of the polynomial expression based on an extension of the binomial probability distribution [12, 13]. Indeed, for an element with x isotopes and relative abundances /1,/2, ., /, and for... 

In arsenic assays, one possible way to eliminate interference resulting from the isobaric ions referred to above is application of correction equations. The contribution of argon chloride to the value of the signal measured for the As ion can be calculated because chlorine occurs as two isotopes, Cl and Cl. The latter forms the " Ar CE ion in plasma (signal at w/z 77). The intensity of the signal makes it possible to assess the necessary correction of ion signal intensity at w/z 75 by applying an appropriate mathematical equation [24, 62]. The approach is only possible if the ratio between the As and Cl content in the sample does not exceed... 

Table 3 shows that the measured and calculated bulk-rock 6 0 values generally exhibit a good correlation (r = 0.82). Minor discrepancies between the two data sets are probably due to small errors in the estimation of the volume of individual cement generations in some samples. The bulk-rock 6 C composition of each sample is also controlled by the relative proportion of each generation of siderite cement, and the same mathematical formula as shown above (substituting 6 C for 6 0) can be applied. However, the correlation coefficient for calculated 6 C values (r = 0.64) is lower than the one for calculated 6 0 compositions. We are uncertain about the reason(s) for this phenomenon, but believe it may be due to the fact that 6 in a cement, at any given pore-water isotope composition, is a strongly temperature-dependent variable. 

This study emphasises the special behaviour of ceria-zirconia catalysts and related compounds exhibiting very high oxygen diffusivity. The use of refined mathematical models and computer-assisted calculations are required for processing the data of isotopic exchange with the highest efficiency and accuracy. 

Fortunately all modem mass spectrometer data systems include algorithms that calculate isotope patterns for ions with any combination of elements, at both unit mass resolution (as above) and at high resolution a brief review (Meija 2006) lists the mathematical approaches that have been applied to devising efficient algorithms for this purpose. A simple example of application of these principles (determination of the cross-over value ) was given in Section 1.5. 

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