Differential measured curve

Fig. 2a,b. Principle of integral and differential measurement of reaction kinetics a concentration-time course b rate-time course. Symbols indicate initial and final concentrations in parallel integral measurements. Corresponding curves use same dashes... 

In order to minimize external (bed) diffusion resistance and maximize the heat transfer rate it is desirable to use a very small adsorbent sample with the crystals spread as thinly as possible over the balance pan or within the containing vessel. To minimize the effect of non-linearities, such as the strong concentration dependence of the diffusivity, measurements should be made differentially over small concentration changes. Variation of the step size and comparison of adsorption and desorption curves provide simple tests for linearity of the system. The large differences between adsorption and desorption diffusivities, reported in some of the earlier work, have been shown to be due to the concentration dependence of the diffusivity(8) and in differential measurements under similar conditions no such anomaly was observed. 

In a flow system, the rate parameters can therefore be determined in two essentially independent ways. (1) From the pressure drop as a function of time, as indicated on the previous page. (2) By measuring the amount of gas desorbed after different adsorption intervals At. Differentiating the curve of n vs At obtained under reproducible flow conditions then yields the net rate of adsorption. This can again be separated into contributions from adsorption and desorption rates by determining the pressure dependence. 

Each electroanalytical technique has certain characteristic potentials, which can be derived from the measured curves. These are the half-wave potential in direct current polarography (DCP), the peak potentials in cyclic voltammetry (CV), the mid-peak potential in cyclic voltammetry, and the peak potential in differential pulse voltammetry (DPV) and square-wave voltammetry. In the case of electrochemical reversibility (see Chap. 1.3) all these characteristic potentials are interrelated and it is important to know their relationship to the standard and formal potential of the redox system. Here follows a brief summary of the most important characteristic potentials. 

It should be noticed that the Galvani potential difference, A(p2, cannot be measured directly but its value can be derived from the Gouy-Chapman theory of the diffuse double layer. If the excess charge in the metal, q , is determined experimentally, that is, from the integration of differential capacity curves, then for a 1 1 electrolyte of concentration c, then A(p2 can be calculated ... 

Relevant physical properties of the adsorbents are summarized in Table 6.1, Integral uptake curves measured over large concentration steps show the expected large difference between adsorption and desorption rates as may be seen from Figure 6.7. The experimental curves arc well approximated by theoretical curves calculated from the solution of Eq. (6.19) and the time constants obtained in this way are consistent with the values derived from differential measurements at low concentrations, as may be seen from Table 6.3. 

Value of Dq for propane derived from differential measurements at low sorbate concentration —2 X 10 cm D = apparent diffusivity from sorption curve, Dq = limiting value of D as ->0,... 

Measurements of the differential reactivity as a function of water height for each lattice yielded the common straight lines of (differential reactivity) as a function of water height. Numerical integration of the differential reactivity curve yields the excess reactivity as a function of core radius, shown in Figures 2 and 3. 

Since log c depends linearly on (E ,ax)" nd on log c, the line connecting the tops of the adsorption peaks on the differential capacitance curves, measured at different concentrations of the organic substance, must be of parabolic shape [3], This is valid only for equilibrium differential capacitance corresponding to zero frequency (see Section 2.1.3.3). 

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