Linear dependence concentration profiles

The exact solution for the time-dependence of the current at a planar electrode embedded in an infinitely large planar insulator, the so-called semi-infinite linear diffusion condition, is obtained. Solving the diffusion equation under the proper set of boundary and initial conditions yields the time-dependent concentration profile. 

The preceding graph shows the time-dependent concentrations of each component. The profile for B drops nearly linearly with time and that of product rises the same way The concentration of A is very small until most of B is used up and then it rises sharply with time. 

It should be stressed that in the case of linear isotherm, the peak broadening effect results from eddy diffusion and from resistance of the mass transfer only, and it does not depend on Henry s constant. In practice, such concentration profiles are observed for these analyte concentrations, which are low enough for the equilibrium isotherm to be regarded as linear. 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

In addition, a linear dependence was found between the concentration of DNB and its fluorescence response profile. All these characteristics demonstrate that this sensor array is suitable for use in detecting explosive vapors. 

If the concentration profile can be determined the moduli can be evaluated. In principle there is no reason why this should be a non-linear measurement, it depends upon the magnitude of the gravitational Peclet number. Buscall35 suggested that a low speed centrifuge could be used to apply different acceleration gradients to the dispersion. If the angular velocity of the rotor is cor and if X is the distance from the centre of the rotor to the top of the sediment then the pressure balance equation becomes... 

Note that the sensitivity of the net flux between the soil and water to the worms activities depends on the relation between the rate R and the solute concentration. For the calculations in Figures 2.13 and 2.14, R varies linearly with concentration as specified in Equation (2.40), and the flux is sensitive to worm activity. But where the rate is independent of concentration, as for NH4+ formation in Equation (2.39), the net flux, which in this case is roughly Ro/a + LRi, is necessarily independent of worm activity, though the distribution of the flux between burrows and the sediment surface and the concentration profile are not. In practice the rate will always depend to some extent on concentration. But the predictions here for the idealized steady state indicate the expected sensitivities. 

Next we turn to the inference of cooling history. The length of the concentration profile in each phase is a rough indication of (jDdf) = (Dot), where Do is calculated using Tq estimated from the thermometry calculation. If can be estimated, then x, Xc and cooling rate q may be estimated. However, because the interface concentration varies with time (due to the dependence of the equilibrium constants between the two phases, and a, on temperature), the concentration profile in each phase is not a simple error function, and often may not have an analytical solution. Suppose the surface concentration is a linear function of time, the diffusion profile would be an integrated error function i erfc[x/(4/Ddf) ] (Appendix A3.2.3b). Then the mid-concentration distance would occur at... 

At the RDE, various approximate analytical treatments have been presented by dropping the highest order convective term [237], neglecting convection completely [238], and by assuming a linear concentration profile within a time-dependent mass transfer boundary layer [239]. The last of these gives... 

It is important to recognize the unique relationship that exists between the responses to an impulse and step change in concentration. The derivative of the step response (Eq. 2.14) is identical to the impulse response (Eq. 2.4), and the integral of the impulse response is identical to the step response. This reciprocity is an important property of linear systems in general. The reader should now appreciate that under linear conditions, the time dependence of any concentration profile can be treated by adding the response functions for its component impulses. 

The result of this approach was a 100-fold increase in the hydrolytic activity of the imprinted polymer compared with the background at pH = 7.6. As a control, another polymer was made using a complex between amidine and benzoate, showing a surprisingly 20-fold increase in the hydrolysis of the substrate. The authors also reported a kinetic investigation of the TSA-imprinted and the benzoate-imprinted polymers, in addition to the free catalyst in solution. Although the ratio substrate/catalyst is not specified, and therefore the steady-state conditions could not be verified, the authors claimed for the two polymers a Michaelis-Menten kinetic behaviour, with a higher profile for the TSA-imprinted polymer. On the other hand, the free catalyst in solution showed, as expected, a linear dependence of the rate from the substrate concentration. The TSA also showed a moderate selectivity towards its own substrate. 

Figure 2.1b shows the time dependence of the concentration profiles. It can also be observed that the perturbed region of the solution adjacent to the electrode surface grows with time and the relative difference between the linear diffusion layer and the accurate diffusion layer (determined as the value x for which cQ reaches the 99 % of its bulk value) is greater for shorter times [12]. 

When compared with the linear concentration profiles of Fig. 2.1a, it can be observed that, in agreement with Eq. (2.146) for spherical electrodes, the Nemst diffusion layer is, under these conditions, independent of the potential in all the cases. As for the time dependence of the profiles shown in Fig. 2.14b, it can be seen that the Nemst diffusion layer becomes more similar to the electrode size at larger times. Analogous behavior can be observed when the electrode radius decreases. 

Okamoto et al. investigated the skin-moisturizing effect of glycerol depending on the absorbed amount in SC and the concentration profile. The skin-moisturizing effect increased linearly with the amount of absorbed humectant in the SC and was dependent on the hygroscopicity of the humectants. A repeated application twice daily for 10 days leads to an accumulation of glycerol in SC.16... 

Equation (96) is known as the linear diffusion equation since the lowest-order field dependence is linear. Thus we have a microscopic derivation of the Einstein relation, eqn. (98). This relation is normally derived from quite different considerations based on setting the current equal to zero in the linear diffusion equation and comparing the concentration profile C (x) with that predicted by equilibrium thermodynamics. 

Here it is not very correct to assume that the concentration gradients vary linearly through the junction, especially because the concentration profiles depend on the technique of junction formation. Assuming that activities are equal to concentrations and that there is, in fact, a linear transition, we obtain the Henderson equation... 

When dealing with currents in ionic solutes, one must take into account the finite diffusion of ions within the electrolyte. As mentioned in Section 6.21, Fick s83 second law of diffusion states that the time-dependence of the concentration profile in a one-dimensional planar system Co(x,t) depends linearly on the derivative of the concentration gradient ... 

At the initial stage the dependence of fractional conversion F on (Fo) is linear. The exchange rate is dependent on the value of ag/Cg and independent of Dy. This is in accordance with the analytical solution (11) for the rectangular isotherm of the B counterion. The greatest spread of concentration profiles appears in situations where both factors act together to spread the concentration distribution of the B ion i.e., at Df/ Dg < 1 and Kg /Kgg < 1 (Fig. 3, Vg, curve Il.e). In this instance the diffusion of the A ion is slower than the diffusion of the B ion (D, < Dg) and this results in accumulation of the A ion (Fig. 3, curve Il.e). The accumulation is partially attenuated due to the effect of the selectivity factor when Kg /Kgs < 1 continues to prevail (compare concentration profiles in variants II and Il.e). Co-ion Y also enters the bead (Fig. 3, Cy, curve Il.e) although not as vigorously as in the case with variant II. 

Let us consider, for example, the simple nernstian reduction reaction in Eq. (221) and a solution containing initially only the reactant R. Before any electrochemical perturbation the electrode rest potential Ej is made largely positive to E . At time zero the potential is stepped to a value E2, sufficiently negative to E , so that the concentration of R is close to zero at the electrode surface. After a time 6, the electrode potential is stepped back to El, so that the concentration of P at the electrode surface becomes zero. When this potentiostatic perturbation, represented in Fig. 21a, is applied in a steady-state method, the R and P concentration profiles are linear and depend only on the electrode potential but not on time, as shown in Fig. 20a (for k 0). Yet when the same perturbation is applied in transient methods, the concentration profiles are curved and time dependent, as evidenced in Fig. 21b. Thus it is seen from this figure that a step duration at Ei, much longer than the step duration 0 at E2, is needed for the initial concentration profiles to be restored. This hysterisis corresponds to the propagation of the diffusion perturbation within the solution, which then keeps a memory of the past perturbation. This information is stored via the structuring of the concentrations in the space near the electrode as a function of the elapsed time. 

Fig. 7.21. Dependence of mobility-gel concentration profiles of DNA on conformation. Samples are supercoiled SV40 DNA (sc), nicked circular SV40 DNA (NC) and HeLa cell linear double-stranded DNA. A single-stranded species, 28 S HeLa cell ribosomal RNA is also shown (Harley et al. 1973).

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