Error function curves

Thus, according to this theory, polarity of transport is produced by an asymmetry in cellular permeability to the auxin anions (see also Sect. 3.2.3.3), but polar transport is basically an asymmetric diffusion (Goldsmith 1977, p 457). Therefore, the theory requires that profiles of the distribution of radioactivity in plant parts which had been supplied with a constant concentration of labeled auxin should fit to a solution of Pick s second law of diffusion, which relates the concentration to both the time of transport and the distance from the source, namely to error function curves. Such curves are constructed from... 

The profile of an eluted peak closely resembles the Gaussian or error function curve where the independent variable is volume of mobile phase and the dependent variable is solute concentration. If the flow rate through the column is kept constant, then the independent variable can be replaced by time. Thus, a linear detector will provide an accurate representation of the Gaussian... 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

A velocity profile and a temperature difference profile have shapes that can be approximated by an error-function type curve. 

Fig. 12. Interface width a as a function of annealing time x during initial stages of interdiffusion of PS(D)/PS(H) [95]. Data points are obtained by a fit with error function profiles of neutron reflectivity curves as shown in Fig. 11. Different symbols correspond to different samples. The interface width a0 prior to annealing is also indicated (T) and is subtracted quadratically from the data (a = [

Ccs is the constant concentration of the diffusing species at the surface, c0 is the uniform concentration of the diffusing species already present in the solid before the experiment and cx is the concentration of the diffusing species at a position x from the surface after time t has elapsed and D is the (constant) diffusion coefficient of the diffusing species. The function erf [x/2(Dr)1/2] is called the error function. The error function is closely related to the area under the normal distribution curve and differs from it only by scaling. It can be expressed as an integral or by the infinite series erf(x) — 2/y/ [x — yry + ]. The comp-... 

Indirect evidence of isotopic fractionation among different complexes was obtained by Marechal et al. (1999) and Marechal and Albarede (2002) who observed different elution rates of Cu and Cu on anion-exchange columns (Fig. 11). These experiments were confirmed by Zhu et al. (2002) and Rouxel (2002) with similar results on fractionation coefficients. Figure 11 shows that, in HCl medium, the heavier isotope 65 is less well retained on the column than the lighter isotope 63. Marechal and Albarede (2002) used an error function approximation to the elution curve to derive the ratio of fractionation coefficients for the 63 and 65 isotopes between the resin and the eluent. From the relationship between the elution volume (position... 

Figure 11. Left elution curve of Cu from a chalcopyrite sample in HCI 7 M on anion exchange resin AG-MP1 (Marechal and Albarede 2002). The points are fitted by an error function. The small misfit of the curve from the curve for small and large fractions eluted reflects that this function is only an approximation. Right evolution of the 5 Cu values in each fraction. The curve represents the isotopic values derived from the error function model.

Figure 3-16 H2O diffusion profile in (a) a dehydration experiment (Zhang et al., 1991a) and (h) a hydration experiment with data read from a figure in Roberts and Roberts (1964). Two outlier points are excluded in (a). The solid curves are fits to the data assuming D is proportional to C, and the dashed curves are fits assuming constant D (error function).

Figure 3-28 H2O diffusion profile for a diffusion-couple experiment. Points are data, and the solid curve is fit of data by (a) error function (i.e., constant D) with 167 /irn ls, which does not fit the data well and (b) assuming D = Do(C/Cmax) with Do = 409 /im ls, which fits the data well, meaning that D ranges from 1 /rm /s at minimum H2O content (0.015 wt%) to 409 firn ls at maximum H2O content (6.2 wt%). Interface position has been adjusted to optimize the fit. Data are adapted from Behrens et al. (2004), sample DacDC3.

Figure 3-29 A half-space diffusion profile of Ar. Ca, = 0.00147 wt% is obtained by averaging 45 points at 346 to 766 /an. Points are data, and the solid curve is a fit of (a) all data by the error function with D = 0.207 /im /s and Co = 0.272 wt%, and (b) data at v < 230 fim (solid dots) by the inverse error function. In (b), for larger x, evaluation of erfc [(C —Cot)/(Co —Coa)] becomes increasingly unreliable and even impossible as (C - Cot)/(Cq — Coo) becomes negative. Data are adapted from Behrens and Zhang (2001), sample AbDArl.

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 interdiffusivity, D, which measures the interdiffusion between Cu and Zn in the laboratory frame, is a strong function of the concentration of Zn. The curve describing D(czn) is concave upward and roughly parabolic in shape, and D(czn) increases by a factor of about 20 when the Zn content increases from 0 to 30 at. % [8]. Describe how the shape of the diffusion-penetration curve for a diffusion couple made of Cu/Cu-30 at. % Zn is expected to deviate from the symmetric form of the constant diffusivity error-function solution. 

Figure 3. Gel permeation data for linear randomly coiled polypeptides on various agarose resins, plotted according to the method of Ackers (9). M0 555 is plotted vs. the inverse error function complement of Kd (erfc 1 Kd). Lines drawn through the data points represent best fits obtained from linear least-squares analysis of the data. Numerical designation of each curve represents the percent agarose composition for the resin used. Filled triangles on the curve for the 6% resin, and the filled squares on the curve for the 10% resin are points determined using fluorescent proteins. Data for the labeled polypeptides were not included in the least-squares analysis.

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. 

The parameter a adjusts the width of the curve, but does not change the value of the error function. This definition makes erf(0) = 0 and erf(oo) = 1, but in some ways it is not particularly convenient usually we would rather find the area between limits expressed as multiples of the standard deviation a. The normalized area between 0 and za is... 

To gain quantitative information on the profile characteristics, the profile shape must be evaluated mathematically. The parameter Dt (D, diffusion constant t, exposure time) that describes the depth of the diffusion front that penetrated into the sample was determined by fitting the data with an error function (erf). The resulting curve describes the result of an undisturbed diffusion process. If the exposure time t is known, e.g. by radiocarbon dating, the diffusion constant D, a material constant, can be derived from this data. 

An attempt was made to correlate the slope of the sensor response curve to the initial diffusible hydrogen concentration in the sample. The steady state portion of the curve could be assumed to be proportional to the flux of hydrogen from the weld metal. To investigate this possibility theoretical curves were generated using an equation derived from the error function erf(x). 

In order to compare results of studies that are expressed in different quantities, dimensionless representations are always preferred. Examples of dimensionless quantities are the relative concentration c/c0 already mentioned above and the parameter appearing in the error function z = x / 2 (D t)1/2 in Fig. 7-6. Systems described with help from the same model but differing from one another with respect to material constants, e.g. D values, can have the same z and c/co values at different times. As a result, whole series of curves can be represented by a single, easy to read curve. 

Because the original curve is already represented as the sum of two error functions, the complete system is represented by a series of error functions ... 

The dependence of 50 on E for molecules in different vibrational and rotational states [19, 20] shown in Fig. 2 are characterised by an S-shape curve, where 50 saturates at high energy and falls exponentially as the energy decreases before flattening out at low energy. The sticking function is represented by a sigmoid curve based on an error function form ... 

The upper curve shows a current transient produced by a voltage step applied to an unilluminated sample. The applied field was 107Vm-1 for a sample 6.5 pm. thick measured at 298 K. The lower curve shows a TOF transient recorded under the same conditions, and the points represent a theoretical fit using an analytical form based on an error function. This function is a reasonable approximation at high temperatures, when the photo-generated carriers rapidly achieve a dynamic equilibrium in the Gaussian DOS of the DEH molecules. 

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