Diffusion Calculated versus experiment

Either Transwell inserts or side-by-side diffusion chambers can be used for transport studies. Bode et al. have provided an excellent review on this subject [60], Briefly, cells are incubated for 30-60 min with a buffer solution. To initiate the transport study, a transport buffer containing the drug under investigation is added to either the apical or the basal chamber depending on the transport direction of interest. At predetermined time points, the respective receiver chamber is sampled and the withdrawn volume is replaced with the same volume of fresh buffer. The permeability coefficient (Papp) is calculated and the ratio of /apP in the basolateral-to-apical direction versus that in the apical-to-basolateral direction gives the efflux ratio. These sort of transport experiments are well suited to determine if drugs/xenobiotics are substrates of the placental efflux proteins. 

If D only depends on temperature (and thus not on concentration or time), the diffusion process is called Fickian. Simple gases show Fickian diffusion and so do many dilute solutions (even in polymers).The diffusivity can be determined directly either from sorption or from permeation experiments. In the first case the reduced sorption, c(t) / (cfX, cG), is plotted versus the square root of the sorption time and D is calculated from the equation ... 

Figure 10 Relative intensity of the cross-peaks versus MAS rate for the P NMR spin-diffusion experiments on hydrated VPI-5 with a 3-s mixing time [77]. The relative intensity is defined as a fraction of the total intensity of the 2D spectrum. Computer-fitted polynomial curves of the fourth order provide a guide for the eye and serve for the calculation of the inflection points (see text).

Figure 11.13 illustrates the behavior of A [, = C /Cq versus t. This family of plots is very useful for those who would like to use a batch adsorber to determine the effective diffusivity of solute in an adsorbent particle. With a few modifications of symbols, it could also be used to determine the thermal conductivity of solid. In an experiment, the bulk fluid concentration is monitored as function of time, t. Since the partition coefficient K is available from equilibrium experiments, that is, the slope of the linear isotherm is known, the parameter B can be calculated. Knowing B, the curve in Fig. 11.13 corresponding to this value of B will be the characteristic curve for the given system. From... 

A polymer sample has been applied in a closed, constant volume. The volume has been evacuated for a certain period to remove present interfering molecules and then a gas is applied at a cenain pressure. Due to sorption of the gas in the polymer the pressure decreases in time until equilibrium has been reached and the amount of penetrant inside polymer can now be calculated. From the sorption experiments an effective diffusion coefficient can be determined as well. By plotting the ratio of mass uptake at time t (Mt) over the mass uptake at infinite time (MJ versus the square toot of time, the diffusion... 

Long-Time Diffusion Coefficient In various experiments and computer simulations, the mean square displacement ((r - r ) ) is often measured or calculated as a function of time t. If the double logarithmic plot of ((r - r ) ) versus t has a slope of 1, we can say that the relevant motion is diffusional. It often happens that the proportionality is reached only after a sufficiently long time. It is therefore customary to define the diffusion coefficient in the long-time limit ... 

The most important use of residence time theory is its application to equipment that is already bnilt and operating. It is usually possible to find a tracer together with injection and detection methods that will be acceptable to a plant manager. The RTD is measnred and then analyzed to understand system performance. In this section we focns on such uses. The washout function is assumed to have an experimental basis. Calculations using it will be numerical in nature or will be analytical procednres applied to a model that reproduces the data accurately. Data fitting is best done by nonlinear least squares using untransformed experimental measurements of W(t), F(t), or f(t) versus time, t. Eddy diffusion in a turbulent system justifies exponential extrapolation of the integrals that define the moments in Table 1-2. For laminar flow systems, washout experiments should be continued until at least five times the estimated valne for t. The dimensionless variance has limited usefnlness in laminar flow systems. 

Solvent-free films about 30-90 pm thick were prepared (see ref. 8). Their gas permeation properties were measured with pure gases using a self-built vacuum time-lag apparatus. Permeate pressure increase with time was recorded at 30°C by two MKS baratron pressure sensors (10 mbar maximum (permeate), 5 bar maximum (feed)) which were connected to a computer. Feed pressure was varied from 0.2 to 1 bar. Permeate pressure was <10 mbar at the beginning of the experiment and was recorded up to 0.5-9 mbar, depending on the feed gas. Permeability was calculated from the slope of the permeate pressure versus time data in the steady state region. Apparent diffusion coefficients. Da, were estimated from the time-lag 0 by Da= f/60 (I being the film thickness). Apparent solubility coefficients, Sa, were calculated from Sa P/Da. 

FIGURE 11.12 Calculated polarization curve for the ORR over Pt(l 11). The insert shows the Tafel plot of log current versus potential. The black curve includes diffusion Umitations present in the experiment, while the sohd curve is the kinetic current density in the absence of such limitations. Adapted from Hansen et al. (2014). 

The long range of double-layer forces in nonpolar media is demonstrated by force measurements. The first direct force experiments were carried out with the SFA [479]. More recently, forces in nonpolar liquids have been measured with optical techniques [464, 478]. Optical techniques are more suitable for such long-range interaction potentials. One example is shown in Figure 4.12. Sainis et al. [464] measured the force between two PMM A particles by first bringing both particles into close proximity by optical tweezers. Then, the particles are released and they start to diffuse apart, driven by the electrostatic force. Their movement is hindered by Stokes friction. Their position is tracked by optical microscopy. From the trajectories of many events, the force versus distance was calculated. 

They argued that the layer forms by reaction of Cu at the CusSi-Si interface. They also maintained that this reaction is controlled by diffusion of Cu through CusSi. (a) Show that the variation of thickness versus time is consistent with a diffusion mechanism, (b) Discuss what reaction stoichiometry would be required to produce this same variation. (c) Calculate the diffusion coefficient and compare it with other values for diffusion in solids. In this experiment, the driving force of Cu3Si is believed to be 1.10 mol% Cu. Answer 2 10 cm /sec. 

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