Single point equations

Extrapolation to infinite dilution requites viscosity measurements at usually four or five concentrations. Eor relative (rel) measurements of rapid determination, a single-point equation may often be used. A useful expression is the following (eq. 9) (27) ... 

Single-point equations suppose that kn, kK and kss are constants and that kn + kK = 0.5, as is indicated by the combination of equations Huggins and Kraemer. They all include the values for relative viscosity, increment of viscosity and concentration. For example, Solomon-Ciuta (1962) proposes ... 

Tisp, and relative viscosity, riiei, at only one low concentration. Chuali et al. [37] examined die application of several single-point equations for PTT. They found tiiat when tire solution concenttation is <0.005 g/dL, Psp can be approximated to [p] witiiin 3 %. The single-point equation used in tiiis author s laboratoiy is from Solomon and Ciuta [38], as follows ... 

The measurement of intrinsic viscosity using capillary viscometers can be a labour intensive and time-consuming exercise. However, polymer chemists undertaking characterisation studies in this way have been spared a significant amount practical work as a result of the development of so-called single point equations . These provide a method by which intrinsic viscosity can be determined when the flow time for the polymer solution is determined at only one concentration and compared to the flow time for that of the solvent alone. Solomon and Ciuta [23] proposed the following equation for use ... 

Th c eigen value of ih is Sch riidiri gcr equation. the electron ie energy deperi ds parametrically, as sh own, on the coord in ales of th e nuclei (assumed to he fixed for the purposes of calcti lali ri g each Heie.lK), bin variable in general), I h e electronic energy, combined with y, (K,K) is the total energy of Single Point semi-em pirical calculation s. 

In a single-point internal standardization, a single standard is prepared, and K is determined by solving equation 5.10... 

Quantitative Analysis for a Single Analyte The concentration of a single analyte is determined by measuring the absorbance of the sample and applying Beer s law (equation 10.5) using any of the standardization methods described in Chapter 5. The most common methods are the normal calibration curve and the method of standard additions. Single-point standardizations also can be used, provided that the validity of Beer s law has been demonstrated. 

In the modem scanning mass spectrometer, it is more convenient that ions arrive at a single point for monitoring (collection), so r (or r ) is kept constant. Therefore, B or V must be varied to bring all ions to the same focus viz., one of the relationships in Equation 24.5 must apply ... 

For a given set of nuclear coordinates, this corresponds to the total energy predicted hy a single point energy calculation, although such calculations, of course, do not solve this equation exactly. The approximation methods used to solve it will be discussed in subsequent sections of this appendix. 

Several points are worth noting about these formulae. Firstly, the concentrations follow an Arrhenius law except for the constitutional def t, however in no case is the activation energy a single point defect formation energy. Secondly, in a quantitative calculation the activation energy should include a temperature dependence of the formation energies and their formation entropies. The latter will appear as a preexponential factor, for example, the first equation becomes... 

Frequently occurs that extrapolations do not have a common value at their origin ordinates. These deviations may be caused by inadequate lineal extrapolations. The above mentioned is the routine method used for [q] determination. The procedure is laborious and consumes a considerable amount of time and reactive because of this, several equations were developed which estimate intrinsic viscosity at one single concentration and do not require a graphic. They are known as "single-point" methods. 

Chee (1987) and Rao Yaseen (1986) have examined the applicability of the single-point method and have found that some equations are inadequate or applicable only to some specific macromolecule-solvent systems. 

Curvale, R.A., Cesco, J.C. 2009. Intrinsic viscosity determination by "single-point" and "double-point" equations. Applied Rheology 19, 5, 53347. 

A concentrated heat capacity. We now consider the boundary-value problem for the heat conduction equation with some unusual condition placing the concentrated heat capacity Co on the boundary, say at a single point X = 0. The traditional way of covering this is to impose at the point a = 0 an unusual boundary condition such as... 

Thus taking the single rate equation dy/dt = f (y, t) and knowing the solution at any point Pnlyn tn) the value of the function at the next point can be... 

These assumptions are partially different from those introduced in our previous model.10 In that work, in fact, in order to simplify the kinetic description, we assumed that all the steps involved in the formation of both the chain growth monomer CH2 and water (i.e., Equations 16.3 and 16.4a to 16.4e) were a series of irreversible and consecutive steps. Under this assumption, it was possible to describe the rate of the overall CO conversion process by means of a single rate equation. Nevertheless, from a physical point of view, this hypothesis implies that the surface concentration of the molecular adsorbed CO is nil, with the rate of formation of this species equal to the rate of consumption. However, recent in situ Fourier transform infrared (FT-IR) studies carried out on the same catalyst adopted in this work, at the typical reaction temperature and in an atmosphere composed by H2 and CO, revealed the presence of a significant amount of molecular CO adsorbed on the catalysts surface.17 For these reasons, in the present work, the hypothesis of the irreversible molecular CO adsorption has been removed. 

Anfalt and Jagner [57] measured total fluoride ion concentration by means of a single-crystal fluoride selective electrode (Orion, model 94-09). Samples of seawater were adjusted to pH 6.6 with hydrochloric acid and were titrated with 0.01 M sodium fluoride with use of the semi-automatic titrator described by Jagner [28]. Equations for the graphical or computer treatment of the results are given. Calibration of the electrode for single-point potentiometric measurements at different seawater salinities is discussed. 

In previous chapters, we deal with simple systems in which the stoichiometry and kinetics can each be represented by a single equation. In this chapter we deal with complex systems, which require more than one equation, and this introduces the additional features of product distribution and reaction network. Product distribution is not uniquely determined by a single stoichiometric equation, but depends on the reactor type, as well as on the relative rates of two or more simultaneous processes, which form a reaction network. From the point of view of kinetics, we must follow the course of reaction with respect to more than one species in order to determine values of more than one rate constant. We continue to consider only systems in which reaction occurs in a single phase. This includes some catalytic reactions, which, for our purpose in this chapter, may be treated as pseudohomogeneous. Some development is done with those famous fictitious species A, B, C, etc. to illustrate some features as simply as possible, but real systems are introduced to explore details of product distribution and reaction networks involving more than one reaction step. 

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