Proportional equations, method

Fla. 2.8 Flow diagram of the program for the resolution of ammonia-hydrazine mixtures by the iterative proportional-equation method. 

There are several types of differential kinetic methods and the most frequently used of which are the logarithmic-extrapolation and the proportional-equation methods, both of which are based on the application of eqn [12],... 

Application of the proportional-equation method to a binary mixture entails measuring C, at two reaction times and formulating two equations similar to [12], the resolution of which provides the initial concentration of the two components as a function of their respective rate constants, which should be known in advance. Not only time, but any other experimental variable such as the temperature or one of the physicochemical properties of the reactants are usable for this pmpose. The two equations in question will be of the form... 

Determination of a singie species in a mixture Simuitaneous kinetic-based determinations Classical differential kinetic methods Logarithmic-extrapolation method Proportional-equation method Multipoint methods Curve-fitting methods Kalman filter algorithm Artificial neural networks Multivariate calibration methods... 

Corticosteroids Blue Tetrazolium Use of the Kalman filter algorithm to resolve mixtures of cortisone and hydrocortisone with a pseudo-first order rate constant ratio as low as 1.8 Comparison with logarithmic-extrapolation and proportional-equations methods... 

Significant distinction in rate constants of MDASA and TPASA oxidation reactions by periodate ions at the presence of individual catalysts allow to use them for differential determination of platinum metals in complex mixtures. The range of concentration rations iridium (IV) rhodium (III) is determined where sinergetic effect of concentration of one catalyst on the rate of oxidation MDASA and TPASA by periodate ions at the presence of another is not observed. Optimal conditions of iridium (IV) and rhodium (III) determination are established at theirs simultaneous presence. Indicative oxidation reactions of MDASA and TPASA are applied to differential determination of iridium (IV) and rhodium (III) in artificial mixtures and a complex industrial sample by the method of the proportional equations. 

An alternative approach (the method of proportional equations) is to use Equation (21-21) and measure C, at two different reaction times to obtain two simultaneous equations that can be solved for [A]q and [B]o-... 

Three of the more commonly used first-order differential kinetic methods are discussed in detail in this section. These are the logarithmic-extrapolation method, the method of proportional equations, and the method of Roberts and Regan. 

Species to be Measured. The method of Roberts and Regan can only be used if the concentration of some species (reagent or product—or a parameter proportional to one of those concentrations) can be measured. The logarithmic-extrapolation method or the method of proportional equations can only be used if the total concentration of some species can be measured. These are relatively unimportant considerations, but in some instances could eUminate one or two of the potential methods at the outset. 

M-hTCSP-M-TCSP Zn-Hg Using the method of proportional equations in an unsegmented continuous-flow system at two different pH values... 

Ephedrine/phenylephedrine IO4- Use of stopped-fiow mixing. Absorbance resulting from formazan formation was monitored at 620 nm. Method of proportional equations... 

In the Monte Carlo methods, the evolution of (discrete) number densities is followed in time by randomly selecting a sequence of processes. The probability of selecting a process is proportional to its rate, which is determined in a similar manner to the rate equation method. Because Monte Carlo methods use random numbers and probabilities instead of analytical expressions, coupling between the two methods (rate equations for the gas and a Monte Carlo procedure for the grain) is harder to achieve. Different (kinetic) Monte Carlo implementations are used and usually a distinction between macroscopic and microscopic Monte Carlo is made. In the macroscopic simulations, only the number density is followed in time in the microscopic simulations the exact positions of the species are also considered. Recently, macroscopic Monte Carlo simulations of both the gas phase and grain surface chemistry have been carried out for a proto-planetary disk [52]. 

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. 

The method of standard additions can be used to check the validity of an external standardization when matrix matching is not feasible. To do this, a normal calibration curve of Sjtand versus Cs is constructed, and the value of k is determined from its slope. A standard additions calibration curve is then constructed using equation 5.6, plotting the data as shown in Figure 5.7(b). The slope of this standard additions calibration curve gives an independent determination of k. If the two values of k are identical, then any difference between the sample s matrix and that of the external standards can be ignored. When the values of k are different, a proportional determinate error is introduced if the normal calibration curve is used. 

There are two ways in which the sensitivity can be increased. The first, and most obvious, is to decrease the concentration of the titrant, since it is inversely proportional to the sensitivity, k. The second method, which only applies if the analyte is multiprotic, is to titrate to a later equivalence point. When H2SO3 is titrated to the second equivalence point, for instance, equation 9.10 becomes... 

Standardizing the Method Equations 10.32 and 10.33 show that the intensity of fluorescent or phosphorescent emission is proportional to the concentration of the photoluminescent species, provided that the absorbance of radiation from the excitation source (A = ebC) is less than approximately 0.01. Quantitative methods are usually standardized using a set of external standards. Calibration curves are linear over as much as four to six orders of magnitude for fluorescence and two to four orders of magnitude for phosphorescence. Calibration curves become nonlinear for high concentrations of the photoluminescent species at which the intensity of emission is given by equation 10.31. Nonlinearity also may be observed at low concentrations due to the presence of fluorescent or phosphorescent contaminants. As discussed earlier, the quantum efficiency for emission is sensitive to temperature and sample matrix, both of which must be controlled if external standards are to be used. In addition, emission intensity depends on the molar absorptivity of the photoluminescent species, which is sensitive to the sample matrix. 

Standardizing the Method Equation 10.34 shows that emission intensity is proportional to the population of the excited state, N, from which the emission line originates. If the emission source is in thermal equilibrium, then the excited state population is proportional to the total population of analyte atoms, N, through the Boltzmann distribution (equation 10.35). 

The copolymer composition equation relates the r s to either the ratio [Eq. (7.15)] or the mole fraction [Eq. (7.18)] of the monomers in the feedstock and repeat units in the copolymer. To use this equation to evaluate rj and V2, the composition of a copolymer resulting from a feedstock of known composition must be measured. The composition of the feedstock itself must be known also, but we assume this poses no problems. The copolymer specimen must be obtained by proper sampling procedures, and purified of extraneous materials. Remember that monomers, initiators, and possibly solvents are involved in these reactions also, even though we have been focusing attention on the copolymer alone. The proportions of the two kinds of repeat unit in the copolymer is then determined by either chemical or physical methods. Elemental analysis has been the chemical method most widely used, although analysis for functional groups is also employed. 

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