Peaks, shape equation

Thede, R. Haberland, D. Fischer, C. Below, E. Langer, S.H. Parametric studies on the determination of enantiomerization rate constants from liquid chromatographic data by empirical peak shape equations for multi-step consecutive reactions. J. Liq. Chromatogr. 1998, 21, 2089-2102. 

Lange, J. Haberland, D. Thede, R. Separate determination of rate constants from reversible reactions in chromatographic column and eluent using empirical peak shape equations. J. Liquid Chromatogr. Relat. Techol. 2003, 26, 285-296. 

The Beer-Lambert Law of Equation (2) is a simpliftcation of the analysis of the second-band shape characteristic, the integrated peak intensity. If a band arises from a particular vibrational mode, then to the first order the integrated intensity is proportional to the concentration of absorbing bonds. When one assumes that the area is proportional to the peak intensity. Equation (2) applies. 

Comparison of the measured peak shape with simulations based on Equations (2-5) and (2-6) reveals that a nucleation and growth model best describes the reduction... 

The cyclic voltammetric response is no longer peak-shaped but rather, plateau-shaped, the plateau current being given by equation (4.19), independent of the scan rate. 

It has been found that the equations describe correctly the peak shape determined experimentally, and can be applied for the prediction of peak distortion [118]. 

As Beer s law in absorption spectroscopy has a path length dependence, the observe volume, Vobs, or active volume of an NMR probe is an important determinant of the sensitivity of NMR measurements. The observe volume is the fraction of the total sample volume, Vtot. that returns a signal when a sample is inserted in an NMR tube or is injected into a flow system. The relationship between chromatographic peak shape, peak volume and flow rate, and sensitivity in hyphenated NMR measurements is complex and is discussed in greater detail in Section 7.2. For the purpose of this discussion, the sample is assumed to be present at a uniform concentration in a sample volume, Vtot. The probe observe factor, /o, is calculated as shown in the following equation ... 

The normalized peak-shape function PS introduced by equation (1) must be determined in order to figure out the dependence of PS on several crystallite parameters, such as average size of crystallites, misorientation of crystallites in the sample etc. These parameters lead to a broadening of reflections, which must be taken into account. 

The second method is to integrate the intensities under the whole arcshaped reflections. The separation of overlapping peaks can be done automatically, once the peak shape has been parameterized from a few well-separated peaks. Occasionally there is a possibility of full overlapping when two or more reflections have the same or almost (within given precision) the same geometrical properties, like dm and Dm values (see equations (8) and (11)). Only then the overlapping reflections cannot be separated. 

The presence of the central spot (the primary beam) and diffuse rings Idiff from the film support brings significant errors into estimated intensities. The shape of the primary beam feam can be approximated by one of several peak-shape functions such as pseudo-Voigt, Gaussian or Lorentzian [16], The diffuse background can be described by a polynomial function of order 12. Then equation (1) becomes... 

The ideal model and the equilibrium-dispersive model are the two important subclasses of the equilibrium model. The ideal model completely ignores the contribution of kinetics and mobile phase processes to the band broadening. It assumes that thermodynamics is the only factor that influences the evolution of the peak shape. We obtain the mass balance equation of the ideal model if we write > =0 in Equation 10.8, i.e., we assume that the number of theoretical plates is infinity. The ideal model has the advantage of supplying the thermodynamical limit of minimum band broadening under overloaded conditions. 

A chromatogram without noise and drift is composed of a number of approximately bell-shaped peaks, resolved and unresolved. It is obvious that the most realistic model of a single peak shape or even the complete chromatogram could be obtained by the solution of mass transport models, based on conservation laws. However, the often used plug flow with constant flow velocity and axial diffusion, resulting in real Gaussian peak shape, is hardly realistic. Even a slightly more complicated transport equation... 

The alternative is the use of a descriptive mathematical model without any relation with the solution of the transport equation. On the analog of the characterization of statistical probability density functions a peak shape f(t) can be characterized by moments, defined by ... 

Analysis of the digitized peak shapes is critical for the calculation of M2. The least-squares method was used to determine the base line for the moment calculations. Detailed description of the data analysis may be found in Ref. 1. The integrations of the integrals in Equations 1 and 4 were done by Bode s rule (Newton-Cotes four-point formula). 

The fact that actually the reaction product was identified as Cu(II) and not Cu(I) as indicated in Equation 12.5 also confirms that Equation 12.4 corresponds to the first oxidation wave. At potentials where Cu(I) is formed (second oxidation wave), it is immediately further oxidised to Cu(II), according to Equation 12.4. The peak shape of this wave can be explained by the limited supply of metallic copper, and the current drops to zero once the copper layer is stripped from the electrode surface. 

From Eq. (5.62) or (5.65) it is clear that when the electrode radius decreases the second term in the right-hand side of both equations becomes dominant and the current becomes stationary (see below). Thus, the typical peak-shaped signal of macroelectrodes evolves toward a sigmoidal or quasi-sigmoidal shape, indicative of stationary or quasi-stationary behavior, and therefore, under these conditions, the peak is no longer an important feature of the signal. 

The differential curves are peak shaped in all cases even under the stationary state, with a peak potential equal to the formal potential if the current is plotted versus 1Ildex (given by equations (7.3) or (7.7)), and the peak current is given by... 

In order to apply these equations to a femtosecond pump-probe experiment, an additional assumption has to be made regarding the shape of the time resolved signal. We wish to account for the finite relaxation time of the transient polarisation and so the signal must be described by a double convolution of an exponential decay function with the pump and probe intensity envelope functions. We will assume a Gaussian peak shape so that the convolution may be calculated analytically. As we will see, the experimental results require two such contributions, and hence, the following function will be used to fit the experimental data... 

Eqn.(3.19) describes the ideal case in which the adsorption isotherm of the solute is linear and the carrier gas does not adsorb onto the stationary phase. This simple situation is not always encountered, but analytical equations can be derived for many other cases [308]. In fact, the practical conditions in GSC are more often non-ideal than is the case in GLC. The adsorption isotherm can only be approximated as linear at very low concentrations. In other words, solute capacities are usually lower in GSC. Surface heterogeneities play a role, especially on inorganic adsorbents such as silica and alumina. These stationary phases are also sensitive to contaminations. Consequently, the observed peak shapes and retention times may be affected by the history of the column ( conditioning ) and by the water content of the carrier gas. 

Equation 3.42 is a partial differential equation that will, upon solution, yield concentration as a function of time and distance for any sample pulse undergoing uniform translation and diffusion. In theory, we need only specify the initial conditions (i.e., the mathematical shape of the starting peak) along with any applicable boundary conditions and apply standard methods for solving partial differential equations to obtain our solutions. These solutions tend to be unwieldy if the initial peak shape is complicated. Fortunately, a majority of practical cases are described by a relatively simple special case, which we now describe. 

The most common performance indicator of a column is a dimensionless, theoretical plate count number, N. This number is also referred to as an efficiency value for the column. There is a tendency to equate the column efficiency value with the quality of a column. However, it is important to remember that the column efficiency is only part of the quality of a column. The calculation of theoretical plates is commonly based on a Gaussian model for peak shape because the chromatographic peak is assumed to result from the spreading of a population of sample molecules resulting in a Gaussian distribution of sample concentrations in the mobile and stationary phases. The general formula for calculating column efficiency is... 

In utilizing the Scherrer equation, care must be exercised to properly account for instrumental factors which contribute to the measured peak width at half maximum. This "intrinsic" width must be subtracted from the measured width to yield a value representative of the sample broadening. When the experimental conditions have been properly accounted for, reasonably accurate values for the average crystallite size can be determined. Peak shapes and widths, however, can also provide other information about the catalyst materials being studied. For example, combinations of broad and sharp peaks or asymmetric peak shapes in a pattern can be manifestations of structural disorder, morphology, compositional variations, or impurities. 

A reversible one-electron transfer process (19) is initially examined. For all forms of hydrodynamic electrode, material reaches the electrode via diffusion and convection. In the cases of the RDE and ChE under steady-state conditions, solutions to the mass transport equations are combined with the Nernst equation to obtain the reversible response shown in Fig. 26. A sigmoidal-shaped voltammogram is obtained, in contrast to the peak-shaped voltammetric response obtained in cyclic voltammetry. 

The majority of chromatographic separations as well as the theory assume that each component elutes out of the column as a narrow band or a Gaussian peak. Using the position of the maximum of the peak as a measure of retention time, the peak shape conforms closely to the equation C = Cjjjg, exp[-(t -1] ) The modelling of this process, by traditional descriptive models, has been extensively reported in the literature. 

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