Diffusion equilibrium times

This diffusion takes time. If cooling is slow, time is available and equilibrium is maintained. But if cooling is rapid, there is insufficient time for diffusion, and, although the new primary (Pb), on the outside of the solid, has the proper composition, the inside (which solidified first) does not. The inside is purer than the outside there is a composition gradient in each (Pb) grain, from the middle to the outside. This gradient is called segregation, and is found in almost all alloys (see Fig. A1.36). 

The test, theoretical relationship between the non-dimensional relative concentration (cRC), and the root time factor (r) may be seen in Fig. 5. Mohamed and Yong [142] analyzed the results obtained from the diffusion experiment shown in Fig. 5 a, b, using the information from solution of the equation above. The theoretical correlation in Fig. 5 c shows a linear relationship up to a relative concentration of 0.2 (80% equilibrium). At a relative concentration of 0.1 (90% equilibrium), the abscissa is used to determine the point on the experimental curve corresponding to a relative concentration of 0.1 (i.e., 90% of the steady state equilibrium time). 

CE is a technique with a very high power of resolution. This is attributed to low diffusion and high plate numbers obtained from the absence of band-broadening factors (e.g., eddy diffusion, equilibrium dynamics, etc.) other than diffusion, which is also minimized by short analysis time. 

Unsteady State Diffusion. The apparatus, experimental procedures, and the computational procedures used to calculate the diffusion parameter D /r (where D is the diffusion coefficient and r is the diffusion path length) have been described in detail previously (6, 8). A differential experimental system was used to avoid errors caused by small temperature fluctuations. In principle, the procedure consisted of charging the sample under consideration with argon to an absolute pressure of 1204 12 torr (an equilibrium time of about 24 hours was allowed) and then measuring the unsteady state release of the gas after suddenly reducing the pressure outside the particles back to atmospheric. 

This standard method of density gradient centrifugation, however, has a serious disadvantage. Because of the sedimentation and diffusion equilibrium formation of the density profile the time needed for orie experiment amounts to more than 15 hours. 

Subsequently a well-defined area at the surface is depleted from the adsorbate layer by a focused laser pulse. Since thermal equilibrium at the surface is rapidly recovered, the bare spot can be refilled only by surface diffusion of adsorbates from the surrounding areas [31]. A second laser impulse is applied to desorb the transported adsorbates after a time interval t from the first pulse. The corresponding amount of material can be quantified by mass spectrometry. For the idealized case of a circular depletion region, with a step-like coverage gradient and a concentration-independent diffusivity, the time-dependent refilling from Fick s first law is [32,33] ... 

The equilibration time for the adsorption in some microporous materials, like CMS and carbonized chars, may be extremely long that may be a source of error for the evaluation of microporosity. For example, this occurs for N2 at 77 K in samples with narrow microporosity (size below 0.7 nm), where the size of the adsorbate molecule is similar to the size of the pore entrance. In this case, contrary to the exothermic nature of the adsorption process, an increase in the temperature of adsorption leads to an increase in the amount adsorbed. In this so-called activated diffusion process, the molecules will have insufficient kinetic energy, and the number of molecules entering the pores during the adsorption equilibrium time will increase with temperature [9,23],... 

Figure 3.5.13 (A) Equilibrium times for diffusion on macroscopic (1 mm) and nanoscopic (10 nm) length scales. (B) Illustration of ionic and electronic wiring, with hierarchical porosity as Li+ distribution network and a carbon second-phase e- distribution network. Reprinted from [58] with permission, copyright 2007 John Wiley Sons.

From all these factors the time necessary fo a diffusion equilibrium may be a few years, as shown by Kiister (Fig. 33b). Carbon dioxide which is often used as a blowing agent has a lower heat conductivity (0.014 kcal/mxh°C) than air (0.023 kcal/m x h °C at 20 °C). Thus, COj would seem tobe of advantage in the manufacture of high-quality heat insulating fomns. However, most polymers are much more permeable to COj than to air, thus the former tends to be rapidly displaced by the latter. Hence, despite its attractively low thermoconductivity. 

To obtain the diffusion constant, D, we consider two alternative equilibrium time correlation function approaches. First, D can be obtained from the long time limit of the slope of the time-dependent mean square displacement of the electron from its starting position. The quantum expression for this estimator is... 

Equation (3) applies strictly to the diffusive equilibrium with Gint = G jk at the interface. However, for large ratios of thermal to solutal diffusivity, it has been shown to hold also in the transient stage , if G c/k is replaced by the time-dependent interfacial concentration Cint t)- Hence... 

The characteristic time for the relaxation process described here is referred to as Tl (the spin-lattice relaxation time), as the spin system returns to equilibrium with the external surroundings. The idealized temperature dependence of Ti is shown in Figure 4.11 this emphasizes the importance of a minimum in Ti, since at this temperature the diffusion correlation time tc is approximately equal to the Larmor frequency. 

When the adsorption/desorption kinetics are slow compared to the rate of diffusional mass transfer through the tip/substrate gap, the system responds sluggishly to depletion of the solution component of the adsorbate close to the interface and the current-time characteristics tend towards those predicted for an inert substrate. As the kinetics increase, the response to the perturbation in the interfacial equilibrium is more rapid and, at short to moderate times, the additional source of protons from the induced-desorption process increases the current compared to that for an inert surface. This occurs up to a limit where the interfacial kinetics are sufficiently fast that the adsorption/desorption process is essentially always at equilibrium on the time scale of SECM measurements. For the case shown in Figure 3 this is effectively reached when Ka = Kd= 1000. In the absence of surface diffusion, at times sufficiently long for the system to attain a true steady state, the UME currents for all kinetic cases approach the value for an inert substrate. In this situation, the adsorption/desorption process reaches a new equilibrium (governed by the local solution concentration of the target species adjacent to the substrate/solution interface) and the tip current depends only on the rate of (hindered) diffusion through solution. 

The degree of separation obtainable in thermal diffusion (the difference in composition between hot and cold walls) is much less than in other diffusion processes, so that use of a column to multiply the composition difference is practically essential. The stage type of thermal diffusion has been used only to measure the thermal diffusion coefficient and is never used for practical separations. In some thermal diffusion columns, htu s are as low as 1.5 cm, and as many as 800 stages of separation have been obtained from a sin e column. Even with such a great increase in separation, it is often necessary to use a tapered cascade of thermal diffusion columns for isotopic mixtures, to minimize hold-up of partially enriched isotopes and to reduce equilibrium time. 

Recently, Ferri and Stebe [62] proposed a scaling low in order to directly compare the adsorption dynamics of different surfactants. By plotting dynamic surface tensions in a dimensional format n( t/ToVrio, where no=y(t)-yo is the equilibrium surface pressure and the diffusion relaxation time tq is defined by the following relationship... 

There is one point important to note here, the experimental data plotted as y( - 1) must cross the ordinate at a value identical to the surface tension of the surfactant-free system, i.e. the surface tension of water for a water/air interface. This is often not the case, in particular for drop volume or maximum bubble pressure experiments where due to the peculiarities of the measurement an initial surfactant load of the interface exists. It has been demonstrated in the book by Joos [16] that even in these cases, assumed it is the initial period of the adsorption time, the slope of the plot y( /t) yields the diffusion relaxation time defined by Eq. (4.26) and hence information about the diffusion coefficient. For small deviation from equilibrium we have the relationship... 

The number of collisions of particles with radii f i and f 2 in a unit time when they approach each other via the diffusion mechanism is equal to the flux of particles of radius f 2 towards the particle of radius l i. If we assume that diffusion equilibrium is established much faster than concentration equilibrium, the problem reduces to solving the stationary diffusion equation in a force field [27]... 

With the gradient IPN s ", the composition is varied within the sample at the macroscopic level. This is conveniently carried out by soaking a sheet of network I in monomer II for a limited period of time, and then polymerizing II rapidly, before diffusion equilibrium can occur. 

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