Additive and Subtractive Methods

According to the Lambert-Beer law the absorbance of two different substances is, 

If it is assumed that both substances are first mixed and the derivatives are computed afterwards, one comes to identical conclusions. For a 1 1 mixture 

The sum of the derivative spectra of two (or more) individual substances should be equal to the derivatives of the mixture of all components. Upon examination, this statement could, in fact, be confirmed if the substances have a mutual influence and the absorbances are additive. Under these circumstances it is possible, for example, to investigate weak intermolecular forces or the influences of the solvent on the spectra of the components, in that case, the two methods of computation would not give identical results. 

Similar to the addition of two (or more) derivatives (A method) it is also possible to subtract one or more derivative(s) from a sum of derivatives S method). If 

The S method of evaluation makes it possible, for instance, to separate an unknown derivative spectrum from the sum of this unknown derivative and a well-known derivative. 

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In addition techniques, the test substance concentration is determined from the difference in the ISE potentials obtained before and after a change in the sample solution concentration. The main advantage Ues in the fact that the whole measurement is carried out in the presence of the sample matrix, so that results with satisfactory accuracy and precision can be obtained even if a substantial portion of the test substance is complexed. Several addition techniques can be used, namely, single, double or multiple known addition methods, in which the sample concentration is increased by additions of a test substance standard solution single, double or multiple known subtraction methods, in which the sample concentration is decreased by additions of a standard solution of a substance that reacts stoichiometrically with the determinand and analyte addition and subtraction methods, in which the sample is added to a test substance solution or to a reagent solution. 

H. Li, Improvement of Gran s Plot Method in Standard Addition and Subtraction Methods by a New Plot Method, Anal. Lett., 24 (1991) 473. 

The Third Dimension Additive and Subtractive Construction Methods / 151... 

THE THIRD DIMENSION ADDITIVE AND SUBTRACTIVE CONSTRUCTION METHODS... 

Define additive and subtractive clay construction methods... 

You are femihar with the addition and subtraction of algebraic equations. This method of combining thermochemical equations is analogous. 

Hansch and Leo [14] do not give the uncertainties for each of their f and F values, but a typical value of 0.03 log K w unit can be assumed for common fragments and factors and 0.05 for less common ones. The total uncertainty in any estimate derived via the method outlined in Eq. 1-4 can then be calculated by the method outlined in Appendix C of this handbook. Since simple addition and subtraction of terms is involved here, the total method error is the square root of the sum of the squares of the individual uncertainties. Using the example given previously in Eq. 1-5, and assuming that the uncertainties for fci and fBr are both 0.03 log K w unit (Cl and Br are common fragments), the total uncertainty is (0.03 + 0.03 ) = 0.04 log Kow unit. Note that this does not consider any uncertainty in the measured value of log Kow for the base chemical. Accordingly, method errors of 0.04 to 0.1 log K w unit should be expected when this method is used. 

Since calculation of MO s from first principles is difficult, the usual approximate approach is the linear combination of atomic orbitals (LCAO) method. It seems reasonable that MO s of a molecule should resemble atomic orbitals (AO s) of the atoms of which the molecule is composed. From known shapes of AO s, one can approximate shapes of MO s. The linear combinations (additions and subtractions) of two atomic s orbitals to give two molecular orbitals are pictured in Figure 2.16. One MO results fi om addition of the parts of AO s that overlap, the other Irom their subtraction. 

Having made these few remarks, we shall turn to some specific examples. A method especially attractive for its simplicity is due to Verlet. If jc is the x component of a particular molecule s center of mass at time r , and is the X component of the molecule s translational velocity, the Verlet algorithm is obtained by addition and subtraction of the Taylor series for Xn+i = x(t +M) andx -i = x(t -AO ... 

In the incremental or decremental technique, another designation for the standard addition (or subtraction) technique, one adds increments of standard solution to the sample, or vice versa. (In the decremental technique the standard precipitates or complexes the ion under test.) When the sample itself is incrementally added to the standard, the latter may have received a previous addition of ISA and/or pH adjuster, but in the reverse method this addition may be made to the sample. However, for the specific example of a univalent anion we shall show how the normal incremental method works38 and that in fact the addition of ISA is not necessary. 

The simplest addition methods are the known addition method (KAM) and known subtraction method (KSM). Two potential values are obtained in the sample solution, before the addition (Ei) and after the addition (E2 ) of volume 1 of a standard solution to sample volume Vq. The sample concentration is Cx, the standard solution concentration is c. In KAM, the standard solution contains... 

The known addition method has an advantage in greater rapidity and somewhat improved precision compared with calibration with two standards. It is claimed that it requires about half the time necessary for two-standard calibration [17] and that the relative error is about 2% [22]. A disadvantage is the necessity of determining the value for most systems. The method should be used only for concentrations corresponding to the linear part of the calibration curve. To eliminate the effect of a change in the ionic strength, it has been recommended [19] that a mixture of a standard with the test solution be added to the sample. The known subtraction method is much less used in practice. 

The analyte addition method (AAM) involves adding the sample solution to a standard solution of the determinand, whereas in the analyte subtraction method (ASM) the sample is added to a standard solution of an ion that reacts stoichiometrically with the test substance and is sensed by an ISE. These methods are advantageous for determinations on small samples for which microelectrodes would otherwise have to be used. pH adjustment and masking of interferents in the sample is unnecessary because all these operations can be done beforehand on the standard solution. Furthermore, the analyte subtraction method widens... 

Potentiometric titration is actually a form of the multiple known subtraction method. The main advantage of titration procedures, similar to multiple addition techniques in general, is the improved precision, especially at high determinand concentrations. ISEs are suitable for end-point indication in all combination titrations (acid-base, precipitation, complexometric), provided that either the titrand or the titrant is sensed by an ISE. If both the titrant and the titrand are electro-inactive, an electrometric indicator must be added (for example Fe ion can be titrated with EDTA using the fluoride ISE when a small amount of fluoride is added to the sample solution [126]). 

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