Equilibrium concentration gradient

Whenever die rich and the lean phases are not in equilibrium, an interphase concentration gradient and a mass-transfer driving force develop leading to a net transfer of the solute from the rich phase to the lean phase. A common method of describing the rates of interphase mass transfer involves the use of overall mass-transfer coefficients which are based on the difference between the bulk concentration of the solute in one phase and its equilibrium concentration in the other phase. Suppose that the bulk concentradons of a pollutant in the rich and the lean phases are yi and Xj, respectively. For die case of linear equilibrium, the pollutant concnetration in the lean phase which is in equilibrium with y is given by... 

Figure 5-2 shows schematically the dependence of the relative concentration of the diazo equilibrium forms on the pH (for the diazoanhydride mentioned in this figure see Sec. 5.2). The relative concentrations of the two major equilibrium forms, the diazonium ion and the diazoate ion, decrease on the right and left sides, respectively, of the pH value corresponding to equal concentrations of these two forms ([ArNj] = [ArN20-]). The gradients correspond to a factor of 100 per pH unit, compared with only 10 per pH unit in the case of dibasic Bronsted acids. The equilibrium concentrations of the diazohydroxide and the diazoanhydride (except for very reactive diazonium ions such as the benzene-1,4-bisdiazonium dication mentioned above) are very small at all pH values, with a maximum at pH = pKm. 

First, we must consider a gas-liquid system separated by an interface. When the thermodynamic equilibrium concentration is not reached for a transferable solute A in the gas phase, a concentration gradient is established between the two phases, and this will create a mass transfer flow of A from the gas phase to the liquid phase. This is described by the two-film model proposed by W. G. Whitman, where interphase mass transfer is ensured by diffusion of the solute through two stagnant layers of thickness <5G and <5L on both sides of the interface (Fig. 45.1) [1—4]. 

A portion of volatile LNAPL in the subsurface vaporizes into air-filled pore spaces until the vapor-liquid equilibrium concentrations are established. If the soil zone is naturally permeable from the fluid surface to the soil surface, a concentration gradient is established. Eventually, most of the volatile mass can be transferred from the LNAPL to the atmosphere, minus that portion retained by sorption on the soil, or biologically degraded. 

A unique solution for the equilibrium concentrations of each ion is obtained by fixing the temperature and chloride concentration. The resulting atmospheric level of CO2 can also be calculated. An example of the numerical solution to this multicomponent equilibrium concentration calculation is shown in Table 21.10. The predicted major ion concentrations are close to the observed values. Nevertheless, this model is not widely accepted as realistic because little evidence has been found for the establishment of equilibria between seawater and the solid phases. In feet, concentration gradients in the bottom and pore waters suggest that equilibrium is not being attained (Figure 21.2). This model is also not able to predict chloride concentrations because the major sedimentary component (halite) is nowhere near saturation with respect to average seawater. 

Particle-Diffusion Control. Here the activation barriers are most pronounced in the condensate, and so is the concentration gradient. Equilibrium pertains at the surface, but the mass-transfer coefficient varies with time. Pressure in the vapor phase is constant. 

In the sedimentation-equilibrium method a lower centrifugal field is applied and the processes of sedimentation and diffusion are brought to equilibrium [13]. In this case the governing equation contains sedimentation equilibrium concentrations of species at different positions from the axis of rotation, but one does not need to know D. It should be pointed out that sedimentation and diffusion are more complicated when the species are electrically charged. This is because the smaller counterions sediment at a slower rate than do the colloidal-sized species. This creates an electric potential gradient that tends to speed up the counter-ions and to drag the colloidal species. The reverse effect occurs for diffusion. 

In a sedimentation equilibrium experiment the cell is rotated at a relatively low speed (typically 5000-10000 rpm) until an equilibrium is attained whereby the centrifugal force just balances the tendency of the molecules to diffuse back against the concentration gradient developed. Measurements are made of the equilibrium concentration profiles for a series of solutions with different initial polymer concentrations so that the results can be extrapolated to c = 0. A rigorous thermodynamic treatment is possible and enables absolute values ot Mwand Mz, to be determined. The principal restriction to the use of sedimentation equilibrium measurements is the long time required to reach equilibrium, since this is at least a few hours and more usually is a few days. 

When the chemical reaction is fast (with large Damkohler numbers or with very low diffusivities) the reactant and product reach their equilibrium concentration throughout most of the film. The concentration gradients are very steep at the nonequilibrium region. The set of parameters Das = 1.0, DaP = 0.5, y = 0.0 represent slow reaction and nonequilibrium film. 

Up to now, our equations have been continuum-level descriptions of mass flow. As with the other transport properties discussed in this chapter, however, the primary objective here is to examine the microscopic, or atomistic, descriptions, a topic that is now taken up. The transport of matter through a solid is a good example of a phenomenon mediated by point defects. Diffusion is the result of a concentration gradient of solute atoms, vacancies (unoccupied lattice, or solvent atom, sites), or interstitials (atoms residing between lattice sites). An equilibrium concentration of vacancies and interstitials are introduced into a lattice by thermal vibrations, for it is known from the theory of specific heat, atoms in a crystal oscillate around their equilibrium positions. Nonequilibrium concentrations can be introduced by materials processing (e.g. rapid quenching or irradiation treatment). 

The operation pressure can influence the second part of the desorption mechanism. If the partial vapor pressure inside the product is equal to the value of the desorption isotherm at that temperature, no desorption can take place. Therefore, the pressure in the chamber has to be small compared to this equilibrium pressure. The necessary pressure drop from the place of desorption to the pressure in the chamber is difficult to evaluate theoretically because water molecules will be transported not only by diffusion in the capillaries but by migration on the solid structure. To transport the possible volume of water vapor would require a high chamber pressure. On the other hand, a large gradient in concentration for the diffusion of molecules would require a very low pressure in the chamber. In most practical cases the pressure limit given by the condenser temperature (T ) will be used, e.g., = -55°C limits the chamber pressure to 0.02 mbar. This may not be... 

Note that this equilibrium involves only the membrane potential, and is independent of the pH gradient. Estimation of membrane potential from this Nemst equation involves the determination of the equilibrium concentration gradient of the indicator ion across the membrane, either by the use of isotopes, or by using an electrode in the medium responsive to the decrease in external concentration as the ion is accumulated by the organelle [13,24]. 

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