Diffusion coefficient ratio

A number of approaches have been suggested for the determination of the molecular diffusion coefficient, D, of a component in water (Othmer and Thakar, 1953 Scheibel, 1954 Wilke and Chang, 1955 Hayduk andLaudie, 1974 Thibodeaux, 1996). Based on these five references, the diffusion coefficient ratio />/Jl2s / Dlq2 was found to vary within the interval 0.78-0.86 with an arithmetic mean value equal to 0.84. This value can be inserted in Equation (4.22) as a first estimate to determine Km. Equation (4.22) and the empirical expressions for KLC>2 outlined in Table 4.7 are the basis for the determination of the mass transfer coefficient for H2S, KL i S, and thereby, the emission of H2S from the wastewater into the sewer atmosphere. Further details relevant in this respect are dealt with in Section 4.4. 

In this equation, is the total interaction energy between the two colliding particles defined in the previous section. The stability ratio, W, for the system gives the ratio of rapid coagulation, Jp, to slow coagulation, J[= J W], DQi) is the position-dependent diflusion equation. This diffusion coefficient ratio is a factor that decreases the collision rate because of the difficulty in draining the liquid between the two solid surfaces. This diffiision coefficient ratio is given by [60,61]... 

Thus the diffusion coefficient ratio of the oxidized and reduced forms ( y) can be found by measuring the ratio of the feedback and collection currents at the same d (35). No prior knowledge of a or d is required for this determination. One should also notice that the transient current flowing at the macroscopic substrate electrode vanishes at long times. The steady-state current is confined to the microscopic area of the substrate surface facing the tip. This largely eliminates the IR-drop problem. 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation Dx lkT)a = C holds, which forms the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the t5q)e of boundary conditions used in deriving Stokes law. It follows that the diffusion coefficient ratio is given by ... 

Equation 19 determines the mean value of the reaction-influenced to nonreaction diffusion coefficient ratio between the inlet and maximum membrane concentrations. Derivation and discussion of Equations 19 through 23 are detailed elsewhere (39, 0). 

There is thus a critical value of the diffusion coefficient ratio, = 8 > 1, above which H k ) > 0 over a range of wavenumbers < k < /fema. Perturbations with wavenumbers within this range will grow because the associated temporal eigenvalues are positive. Perturbations with wavenumbers outside this range will decay exponentially to the homogeneous steady state. Figure 11 shows the variation oiH(k ) with k for the cases of 8 > 8, and 8 <... 

In 1940, Simha derived dependencies for [ j] of the freely rotating monodispersed ellipsoids. The derivation considered the viscosity increase due to the disorienting influence of the thermal motion. At the limit of the shear rate to the rotational diffusion coefficient ratio, y/Dr 0, [ j] of the prolate and oblate ellipsoid suspension with high aspect ratio, p 1, was derived as, respectively [Simha, 1940 ... 

Figure 18 Illustration of the double-oil self-diffusion experiment with a cyclohexane-hexadecane mixture. K = D /D2, where D and Dj are the cyclohexane and hexadecane diffusion coefficients, respectively. In the pure oil mixture the ratio of the two diffusion coefficients is K = Kq— 1.69. For a water-in-oil droplet structure the two oil molecules have the same diffusion coefficient, that of the micelle, and the ratio A equals unity. In a bicontinuous structure, on the other hand, a molecular diffusion mechanism is dominating and the ratio K equals that of the pure oil mixture, Kq. By monitoring the diffusion coefficient ratio, the droplet-to-bicontinuous transition could be studied.

Figure 19 Double-oil diffusion experiment with nonionic surfactant, (a) Self-diffusion coefficients and (b) diffusion coefficient ratio A" as a function of temperature in a water-rich microemulsion with nonionic surfactant. A transition from oil-in-water droplets to a bicontinuous microstructure occurs with increasing temperature (decreasing spontaneous curvature of the C12E5 surfactant film). The maximum in K indicates that an attractive interaction between the micelles is operating prior to the formation of a bicontinuous structure. Kq = 1.69 is the diffusion coefficient ratio in the pure oil mixture and is indicated as a broken line in (b). Note that the initial decrease of the self-diffusion coefficients shows that the droplets grow in size before the bicontinuous transition. The phase boundary at 25.7 C is indicated as a vertical broken line. (Data from Ref 43.)...

Figure 20 Double-water experiment, the aqueous analogy of the double-oil experiment, performed on an AOT microemulsion as a function of temperature. The polar solvent is a 5% A-methyl formamide (NMF) solution in heavy water (D2O). The ratio of the water (here measured as trace impurities of HDO) and NMF diffusion coefficients is monitored as a function of temperature (c). Also shown as (a) the individual self-diffusion coefficients of water (O). NMF ( ), and AOT ( ) and (b) the relative diffusion coefficient of water. Kq = 1.73 is the diffusion coefficient ratio in the pure water-NMF mixture and is indicated as a broken line in (c). The phase boundary at 75"C is indicated as a vertical broken line. The behavior with increasing temperature is completely analogous to that of the nonionic system (Fig. 19) and illustrates a transition from reverse micelles to a bicontinuous structure via growing droplets that become attractive. (Data from Ref 49.)... 

Figure 7 Diffusion coefficient ratio, K, as a function of temperature in the same system as that in Fig. 6. The maximum in K indicates that an attractive interaction between the micelles is operating prior to the formation of a bicontinuous stracture. (Data from Ref 20.)...

Flux was defined in (17) and we must here take note of the fact that the two species diffusion coefficients might not be the same. For this reason, we define diffusion coefficient ratios, all referred to that of the main species, whose bulk concentration is also the reference, c. Thus, for species B, we have... 

Figure 4.40 - Concentration profile during an electrolysis experiment at steady state for a membrane cell with a diffusion coefficients ratio, Di/Di, equal to 100...

J = - Dll Vci - >12 Vc2 where Vci and Vc2 represent the concentration gradient of elements 1 and 2. [1964Bro] measured experimentally the diffusion coefficient ratio >12 / Du over the temperature range 800-1050°C and showed that, within a good approximation, this ratio may be deduced from thermodynamic considerations. The diffusion coefficient of carbon in an (aFe,Ni) alloy has been measured by [1964Hel] which proposes the following relationship ... 

Bro] Carbon diffusion in a (yFe,Ni) framework 800-1050°C, diffusion coefficient ratio... 

Table 2. Calculated and Computer Values of Diffusion Coefficient Ratio DfD ...

Martin RD, Unwin PR (1998) Theory and experiment for the substrate generation tip collection mode of the scanning electrochemical microscope application as an approach for measuring the diffusion coefficient ratio of a redox couple. Anal Chem 70 (2) 276-284. doi 10.102l/ac97068Ip... 

The diffusion coefficient ratio of a redox couple can also be measured with SECM using an approach developed by Unwin (13,14). With this technique, the ratio of the steady-state collector current for the substrate generation/tip collection (SG/TC) mode to the steady-state collector current measured in the feedback mode (at the same tip-substrate separation) gives directly the ratio of diffusion coefficients for the redox couple. The advantage of this approach is that no knowledge of the tip-substrate separation, the electrode sizes, or the mediator concentration is required. 

With nonionic surfactants, it is possible to generate an oil-in-water droplet to bicontinuous structural transition at constant composition by increasing the temperature. This transition can be studied in more detail by using self-diffusion experiments with mixed solvents. In a system where the oil was an equal weight mixture of cyclohexane (1) and hexadecane (2) the transition from normal oil-swollen micelles to a bicontinuous microstructure was studied by monitoring the variation of the self-diffusion coefficient ratio, K = as a... 

Figure 17.24. The variation of the self-diffusion coefficient ratio, K = DxjDi, with temperature D and D2 are the self-diffusion coefficients of cyclohexane and hexadecane, respectively (data taken from ref. (14))...

As discussed in Sections 7.2.1 and 7.2.2, the analysis of SECM chemical kinetic data assumes that the electroactive precnrsor (A in the terminology of our example case) and the electrogenerated species (B) have the same diffusion coefficient. A simple approach for confirming this is to measure the tip currents at a fixed (close) tip/substrate separation, under positive feedback and then SG/TC control. For a chemically stable redox couple, the ratio of the tip feedback and collector currents under steady-state conditions reveals the diffusion coefficient ratio, p, directly ... 

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