Experimental Determination of Diffusion Coefficients

8 Other Voltammetric Techniques Differential Pulse Voltammetry 

As introduced earlier, voltammetry so far has been concerned with applying a potential step where the response is a pulse of current which decays with time as the electroactive species near the vicinity of the electrode surface are consumed. This Faradaic process (Ip) is superimposed with a capacitative contribution (7 ) due to double layer charging which dies away much more quickly, typically within microseconds (see Fig. 2.31). 

The current (for a reversible system) is in the form of the Cottrell Equation where I and charge, 2, is 2 When a step in pulse is applied the 

The base potential is implemented in a staircase and the pulse is a factor of 10 or more shorter than the pulse of the staircase waveform. The difference between the two sampled currents is plotted against the staircase potential leading to a peak shaped waveform as shown in Fig. 2.33. 

For a reversible system the peak occurs at a potential Ep = Exj-i — bEjl where A is the pulse amplitude. The current is given by  

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M Southard, L Dias, K Himmelstein, V Stella. Experimental determinations of diffusion coefficients in dilute aqueous solution using the method of hydrodynamic stability. Pharm Res 8 1489-1491, 1991. 

For experimental determination of diffusion coefficients, a large database is already available. Nonetheless, data for specific applications are often difficult to find because the data may not cover the right temperature range, mineral compositions, or fluid conditions. In geospeedometry applications, data often must be extrapolated to much lower temperatures and the accuracy of such extrapolation is difficult to assess. Because the timescale of geological processes is often in the order of Myr, and that of experiments is at most years, instrumental methods to measure very short profile are the key for the determination of diffusion coefficients that are applicable to geologic problems. 

Experimental determination of diffusion coefficients using HPLC E. Grushka and S. Levin, in Quantitative Analysis Using Chromatographic Techniques, E. Katz (ed.), John Wiley Sons, Inc., New York, 1987, pp. 360-374. 

Experimental determination of diffusion coefficients. A number of different experimental methods have been used to determine the molecular diffusivity for binary gas mixtures. Several of the important methods are as follows. One method is to evaporate a pure liquid in a narrow tube with a gas passed over the top as shown in Fig. 6.2-2a. The fall in liquid level is measured with time and the diffusivity calculated from Eq. (6.2-26). 

This section describes the experimental methods and focuses on the estimation of diffusivity after the experiment. The analytical methods are not described here. Estimation of diffusivity from homogeneous reaction kinetics (e.g., Ganguly and Tazzoli, 1994) is discussed in Chapter 2 and will not be covered here. Determination of diffusion coefficients is one kind of inverse problems in diffusion. This kind of inverse problem is relatively straightforward on the basis of solutions to forward diffusion problems. The second kind of inverse problem, inferring thermal history in thermochronology and geospeedometry, is discussed in Chapter 5. 

The Nernst-Einstein reiation can be tested by using the experimentally determined tracer-diffusion coefficients D,. to calcuiate the equivalent conductivity A and then comparing this theoreticai vaiue with the experimentally observed A. It is found that the vaiues of A caicuiated by Eq. (5.61) are distinctly greater (by 10 to 50%) than the measured values (see Table 5.27 and Fig. 5.33). Thus there are deviations from the Nernst-Einstein equation and this is strange because its deduction is phenomenological. ... 

Empirical parameters governing atmospheric dispersal pervade the literature on this subject. Like most cases of turbulent transport, elimination of a disposable coefficient in one place leads to a reappearance of one somewhere else. The present work uses an experimentally determined turbulent diffusion coefficient, D, in Equation 19. Near the ground and near the inversion base we must assign a height (z) dependence to the diffusion coefficient. 

Experimental determination of the diffusion coefficients D is enough complicated and labor intensive process, whereas the experimental determination of the coefficients of viscosity doesn t strike the great eomplieations. Established long ago the empirical Walden s rule... 

Concentrations of COi and Oiin the analyser chamber as a function of time are presented in Figure 8.4. The fitting of experimental data to Equation 8.2 allows the determination of diffusion coefficients. The values obtained for diffusion coefficients of LDPE to O2 and C02were 1.68 x 10 cmVs and 2.77 x 10" cmVs, respectively. 

I. Experimental determination of diffusivities. Several different methods are used to determine diffusion coefficients experimentally in liquids. In one method unsteady-state diffusion in a long capillary tube is carried out and the diffusivity determined from the concentration profile. If the solute A is diffusing in B, the diffusion coefficient determined is D g. Also, the value of diffusivity is often very dependent upon the concentration of the diffusing solute A. Unlike gases, the diffusivity does not equal Dg for liquids. 

The techniques described in this chapter (see Chapter 11 for detailed description of techniques) for the determination of diffusion coefficients are smnmarized in Table 19.1. Each of these techniques has its own strengths and weaknesses that should be evaluated before selecting a method. One important consideration is the dependence of D on the experimentally measured quantities (e.g., current or electrode dimensions). Under conditions of... 

Matyukhin, V. J., and 0. N. Prokofyev. 1966. Experimental determination of the coefficient of vertical turbulent diffusion in water for settling particles. Soviet Hydrol. (Am. Geophys.Union), No 3. 

Experimental methods for determining diffusion coefficients are described in the following section. The diffusion coefficients of the individual ions at infinite dilution can be calculated from the ionic conductivities by using Eqs (2.3.22), (2.4.2) and (2.4.3). The individual diffusion coefficients of the ions in the presence of an excess of indifferent electrolyte are usually found by electrochemical methods such as polarography or chronopotentiometry (see Section 5.4). Examples of diffusion coefficients determined in this way are listed in Table 2.4. Table 2.5 gives examples of the diffusion coefficients of various salts in aqueous solutions in dependence on the concentration. 

In a basalt-rhyolite interdiffusion experiment (Alibert and Carron, 1980), potassium concentrations CK were measured in a basalt at a given arbitrary distance y in pm between rhyolitic and basaltic liquids experimentally heated for 5000 seconds (Table 5.5 and Figure 5.4). In order to determine the diffusion coefficients, a fit of the experimental points with a polynomial is requested. Use the reduced concentration u, (the fractional deviation of the concentration at a, from the concentrations in the original liquids) given by... 

Geochemical kinetics is stiU in its infancy, and much research is necessary. One task is the accumulation of kinetic data, such as experimental determination of reaction rate laws and rate coefficients for homogeneous reactions, diffusion coefficients of various components in various phases under various conditions (temperature, pressure, fluid compositions, and phase compositions), interface reaction rates as a function of supersaturation, crystal growth and dissolution rates, and bubble growth and dissolution rates. These data are critical to geological applications of kinetics. Data collection requires increasingly more sophisticated experimental apparatus and analytical instruments, and often new progresses arise from new instrumentation or methods. 

Experimental determination of the diffusion coefficient matrix is time-consuming and labor-intensive. Nonetheless, diffusion studies have advanced significantly in recent years. Hence, with persistence and concerted effort, it is possible that reliable and reproducible diffusivity matrices for major components in some natural melts will become available in the near future. 

Loomis T.P., Ganguly J., and Elphick S.C. (1985) Experimental determination of cation diffusivities in aluminosilicate garnets, II multicomponent simulation and tracer diffusion coefficients. Contrib. Mineral. Petrol. 90, 45-51. 

Equation (26) is a differential equation with a solution that describes the concentration of a system as a function of time and position. The solution depends on the boundary conditions of the problem as well as on the parameter D. This is the basis for the experimental determination of the diffusion coefficient. Equation (26) is solved for the boundary conditions that apply to a particular experimental arrangement. Then, the concentration of the diffusing substance is measured as a function of time and location in the apparatus. Fitting the experimental data to the theoretical concentration function permits the evaluation of the diffusion coefficient for the system under consideration. 

Diffusion-Controlled Reactions. The specific rates of many of the reactions of elq exceed 10 Af-1 sec.-1, and it has been shown that many of these rates are diffusion controlled (92, 113). The parameters used in these calculations, which were carried out according to Debye s theory (41), were a diffusion coefficient of 10-4 sec.-1 (78, 113) and an effective radius of 2.5-3.0 A. (77). The energies of activation observed in e aq reactions are also of the order encountered in diffusion-controlled processes (121). A very recent experimental determination of the diffusion coefficient of e aq by electrical conductivity yielded the value 4.7 0.7 X 10 -5 cm.2 sec.-1 (65). This new value would imply a larger effective cross-section for e aq and would increase the number of diffusion-controlled reactions. A quantitative examination of the rate data for diffusion-controlled processes (47) compared with that of eaq reactions reveals however that most of the latter reactions with specific rates of < 1010 Af-1 sec.-1 are not diffusion controlled. 

The rate of diffusion is proportional to the concentration gradient, and the proportionality constant is defined as the diffusion coefficient (D) in Fick s first law of diffusion. Experimental determination of D is commonly performed ex vivo due to the difficulty of measuring concentration gradients in the interstitium. In vivo measurement can be performed in specific tissues, using transparent chamber preparations in combination with the FRAP technique (Berk et al., 1997 Jain et al., 1997 Pluen et al, 2001). However, the in vivo approach is limited only to fluorescent molecules or solutes whose D is not affected by labeling with fluorescent markers. 

In Section IB we presented experimental evidence that diffusion coefficients correlate with PVC main-chain polymer motions. This relationship has also been justified theoretically (12). In the previous section we demonstrated that the presence of CO2 effects the cooperative main-chain motions of the polymer. The increase in with increasing gas concentration means that the real diffusion coefficient [D in eq. (11)] must also increase with concentration. The nmr results reflect the real diffusion coefficients, since the gas concentration is uniform throughout the polymer sample under the static gas pressures and equilibrium conditions of the nmr measurements. Unfortunately, the real diffusion coefficient, the diffusion coefficient in the absence of a concentration gradient, cannot be determined from classical sorption and transport data without the aid of a transport model. Without prejustice to any particular model, we can only use the relative change in the real diffusion coefficient to indicate the relative change in the apparent diffusion coefficient. 

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