Error function table

According to the error function Table 13.1 in Appendix 13A, we have... 

At low temperatures, using the original function/(T ) could lead to greater error. In Tables 4.11 and 4.12, the results obtained by the Soave method are compared with fitted curves published by the DIPPR for hexane and hexadecane. Note that the differences are less than 5% between the normal boiling point and the critical point but that they are greater at low temperature. The original form of the Soave equation should be used with caution when the vapor pressure of the components is less than 0.1 bar. In these conditions, it leads to underestimating the values for equilibrium coefficients for these components. 

Relative uncertainties for absorption spectrophotometry as a function of absorbance for the three categories of indeterminate instrumental errors (see Table 10.8 for equations). 

TABLE 7.12 Practical Modifications of the Gauss Error-Function Velocity ... 

Application of the Gauss error-function equation for velocity profile in the form proposed by Shepelev (Table 7.12) in Eq. (7.39) results in the following formula for the centerline velocity in Zone 3 of the compact jet ... 

The equation for the centerline temperature differential in Zone 3 of the compact jet derived" from Eq. (7.61) using the Gauss error-function temperature profile (Table 7.14) is... 

Using Tables of error functions (Appendix A3, Tabic 12) ... 

The error function cannot be evaluated analytically, although it is readily available in the form of tables and evaluated in many computer programs. 

The error function is tabulated in most handbooks of mathematical tables. It is useful to note that erf( —x) = — erf(x) and that... 

Table 7.2 gives tabulated values of the error function and related functions in the solution of other semi-infinite conduction problems. For example, the more general boundary condition analogous to that of Equation (7.27), including a surface heat loss,... 

Note that s is a dummy variable the value of the integral depends only on the value of the upper limit. Tables of the error function are available and values can be calculated from power series [Dwight (1961), Kreyszig (1988)]. The error function has the properties erf(0) = 0 and erf(°°) = 1. Equation 10.31 can be written in terms of the error function as... 

For selected values of C/Co, the value of E may be found from the tables of error functions given in the Appendix in Volume 1. From E a ratio r /X may be calculated and hence a corresponding time. These calculations are summarised as follows ... 

A diagram of erf(z) vs. z is shown in Figure A2-1. Values of error functions are given in Table A2-1. 

Table A2-1 Values of error function and related functions...

Dipole moments from 6-31G and 6-311+G calculations are nearly identical both for the amines (Table 10-3) and for the full set of molecules found in Appendix AlO. Comparison of Figures 10-3 and 10-4 further drives home the point. It is difficult to justify use of the larger basis set for this purpose. On the other hand, dipole moments resulting from all density functional models (including local density models) and from MP2 models, display some sensitivity to basis set. In most (but not all) cases, agreement with experiment improves, as generally reflected by the mean absolute errors in Table 10-4, and by... 

The error function is also useful for solving chemical diffusion problems (Chapter 13) and thermal conduction problems and the details of the error function and the table of the error fiinction are given in Chapter 13. 

The solution to this partial differential equation depends upon geometry, which imposes certain boundary conditions. Look np the solution to this equation for a semi-infinite solid in which the surface concentration is held constant, and the diffusion coefficient is assumed to be constant. The solution should contain the error function. Report the following the bonndary conditions, the resulting equation, and a table of the error function. 

Tables are also available (e.g., see Beyer 1987) for the area under the normal curve between 7 (7 = 0) and one value of tt (i.e., they apply to one tail only). This area is known as the error function and is often symbolized as erf (t) ...

The error function erf(x) is defined in Section A.2. Table A. 1 gives specific numerical values of the normal distribution and its integral 7,(x). 

Apart from general updating and correction, the main alterations in the second edition of Volume 3 are additions to Chapter 1 on Reactor Design and the inclusion of a Table of Error Functions in the Appendix. 

The stochastic description of barrierless relaxations by Bagchi, Fleming, and Oxtoby (Ref. 195 and Section IV.I) was first applied by these authors to TPM dyes to explain the observed nonexponential fluorescence decay and ground-state repopulation kinetics. The experimental evidence of an activation energy obs < Ev is also in accordance with a barrierless relaxation model. The data presented in Table IV are indicative of nonexponential decay, too. They were obtained by fitting the experiment to a biexponential model, but it can be shown50 that a fit of similar quality can be obtained with the error-function model of barrierless relaxations. Thus, r, and t2 are related to r° and t", but, at present, we can only... 

The parameters in Eq. (2.59) are usually determined from the condition that some function mean-square deviations between the experimental and calculated curves (the error function). The search for the minimum of the function Nelder-Mead algorithm.103 As an example, Table 2.2 contains results of the calculation of the constants in a self-accelerating kinetic equation used to describe experimental data from anionic-activated e-caprolactam polymerization for different catalyst concentrations. There is good correlation between the results obtained by different methods,as can be seen from Table 2.2. In order to increase the value of the experimental results, measurements have been made at different non-isothermal regimes, in which both the initial temperature and the temperature changes with time were varied. 

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