Mass action expressions linearization

Kurbatov Plots. Kurbatov plots (6) are made by applying the linearization of a general overall mass action expression (e.g. Equation 4)... 

The brute force method is the classical approach where mass action expressions are substituted directly into the mass balance conditions and solved for total concentrations which are then compared to the analytical values. In the continued fraction method, the non-linear equations are rearranged to solve for free ion concentrations which are initially assumed to be equal to the total concentrations, as detailed by Wigley (42). These two methods are best illustrated by a simple example. Assume a solution which contains free Ca2+ ions, free CQ ions, and only one ion pair CaCO. The mass balance s conditions are given by... 

We now have 1 charge balance equation (19.22), a total of W — M — 1 mass balances (19.23), and M mass action expressions (19.24) for a grand total of N equations in N unknowns. Because of the mass action equations, this is a non-linear system. It could be solved by any of a number of numerical techniques. We will use the Newton-Raphson method in this example. 

Most analyses of kinetic data have the object of identifying the constants of a rate equation based on the law of mass action and possibly some mass transfer relation.. The law of mass action Is expressed In terms of concentrations of the participants, so ultimately the chemical composition must be known as a function of time. In the laboratory the chemical composition Is determined by some instrument that is suitably calibrated to provide the needed information. Titration, refractive index, density, chromatography, spectrometry, polarimetry, conductimetry, absorbance, magnetic resonance — all of these are used at one time or another to measure chemical composition. In some cases, the calibration to chemical composition is linear with the reading. 

As C increases, the thermal transition range narrows, and Tg increases in a roughly linear fashion. Assuming that the effect of Na" " ions on the transition could be expressed in the mass-action form... 

Composition The law of mass action is expressed as a rate in terms of chemical compositions of the participants, so nltimately the variation of composition with time mnst be fonnd. The composition is determined in terms of a property that is measnred by some instm-ment and calibrated in terms of composition. Among the measnres that have been used are titration, pressure, refractive index, density, chromatography, spectrometry, polarimetry conductimetry absorbance, and magnetic resonance. In some cases the composition may vary linearly with the observed property, but in every case a cahbration is needed. Before kinetic analysis is undertaken, the data are converted to composition as a function of time (C, t), or to composition and temperature as functions of time (C, T, t). In a steady CSTR the rate is observed as a function of residence time. 

T3 uptake values are customarily expressed as matrix-bound counts divided by the total counts added (typical euthyroid reference iateiwal 25% to 37%). This calculation, however, is theoretically unsound and has led to problems of interpretation, particularly when serum TBG concentrations are very high or very low. Instead the uptake value should be expressed tis the matrix-bound counts divided by the counts that remain bound to serum proteins (typical euthyroid reference interval 33% to 50%). This result corresponds more closely to the mass action law and is more linearly related to the free hormone fraction over a wide range of values. 

Summary. The classical mass action law of chemical kinetics was proved, in fact, in the linear fluid mixture as the general constitutive equations for the reaction rates which were reproduced in this section as (4.470). This law generally states that the rates depend only on temperature and composition expressed by densities, molar concentrations or activities or, alternatively, even by (molar) chemical potentials. The equilibrium constant of independent reactions was defined by (4.474). Then we have shown on several reaction examples how the general function reaction rate-concentrations (or reaction rate-activities) can be approximated by a suitable... 

Another model has ofl en been applied to RBa2Cu306+ , based on the S-shaped isotherms which become nearly linear when the composition is expressed as log[w/(l - w)]. A mass-action term of the w/(l - w) type appears when the number of the structure sites involved is considered constant (Maier and TuUer 1993). This means that for every defect on the product side of the formal defect reaction, there occurs one participating stmctural site, appearing at the left hand side of the equation. The slope of the log/ro, vs. log[w/(l - w)] dependences then refers to the number of such defects corresponding to... 

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