Tncta rrxn rS1

As already discussed, yncl is a net probability normalized to the number of gas-surface collisions and is the parameter actually measured in experiments (and hence also often referred to as ymcas). In Eq. (QQ), each conductance represents one of the processes involved; i.e., Tg involves the conductance for gas-phase diffusion, rran that for reaction in the aqueous phase, and rsol that for solubility and diffusion into the bulk. Each of the terms has been normalized and made unitless by dividing by the rate of gas-surface collisions, Eq. (PP), except for a, which by definition is already normalized to this parameter.

Let us now examine each of these terms individually.

Diffusion of the gas to the surface (T ). As described by Fuchs and Sutugin (1970, 1971) in their comprehensive treatment of highly dispersed aerosols, the rate of transfer of mass to the surface of a spherical particle by diffusion of a gas is described by dN

dx where N is the gas concentration a distance x from the center of the particle and the constant depends on the boundary conditions at the surface of the particle. Take the case where the Knudsen number, Kn, defined as the ratio of the mean free path of the gas to the radius of the particle (a), approaches 0 (i.e., the mean free path is small compared to the particle size); all molecules colliding with the surface are taken up at the surface and the boundary condition is (N)x=a = 0. Fuchs and Sutugin show that Eq. (RR) can be integrated to obtain the gas concentration as a function of x, N = Nj,i - a/x), where A^ is the gas concentration at x = which can be taken as N for a constant gas-phase concentration. They also show that for small Knudsen numbers where Kn -» 0, the rate of diffusion of the gas to the surface of the particle of radius a is given by

Number of molecules per second diffusing to the surface = AirDgaN%. (SS)

The surface area of a spherical particle of radius a is 4-7rfl2. Thus, the rate of diffusion to the surface

^interface

FIGURE 5.16 Schematic of resistance model for diffusion, uptake, and reaction of gases with liquids. Tg represents the transport of gases to the surface of the particle, a the mass accommodation coefficient for transfer across the interface, rso] the solubilization and diffusion in the liquid phase, rKn the bulk liquid-phase reaction, and rjnlcrlacc the reaction of the gas at the interface.

^interface

FIGURE 5.16 Schematic of resistance model for diffusion, uptake, and reaction of gases with liquids. Tg represents the transport of gases to the surface of the particle, a the mass accommodation coefficient for transfer across the interface, rso] the solubilization and diffusion in the liquid phase, rKn the bulk liquid-phase reaction, and rjnlcrlacc the reaction of the gas at the interface.

per unit surface area is given by

Rate of diffusion to surface per cm2 per second

where d = 2a is the particle diameter. Normalizing to the rate of collisions, T becomes re = 8 D/Umd.

As discussed by Fuchs and Sutugin (1970, 1971) and Motz and Wise (1960), in this continuous regime, distortion of the Boltzmann velocity distribution in the region close to the surface occurs if there is rapid uptake. In effect, the normal thermal velocity distribution is distorted so that the effective speed toward the surface is higher. In the case of a surface where the uptake occurs on every collision, the net speed toward the surface is effectively doubled. This adds an additional term to the rate of transfer of the gas to the surface, which when normalized using Eq. (PP), gives an additional "resistance" term of — f /2. The overall normalized conductance is therefore given by f

"a yd

8 Dg

This applies to diffusion to a planar surface in the continuous regime, when the Knudsen number is small, and is the expression for gas diffusion most often encountered in the atmospheric chemistry literature.

For the other extreme of the free molecular regime where Kn —> the particle radius is small compared to the mean free path. In this case, the thermal velocity distribution of the gas is not distorted by uptake at the surface. In effect, the gas molecules do not "see" the small particles. For this case, Fuchs and Sutugin (1970, 1971) show that for diffusion to a spherical particle of radius a

Number of molecules diffusing to surface per second

where aK is the probability of uptake at the surface and Mav is the mean thermal velocity.

For intermediate regimes of Kn, which are common both in the atmosphere and in many laboratory studies, exact calculations are not readily carried out. Fuchs and Sutugin (1970, 1971) suggest the form

Number of molecules diffusing to surface per second = 4irDgoArg/(l + A K„). (XX)

As Kn 0, Eq. (XX) approaches Eq. (SS), as expected. Values of A are provided in the literature (Fuchs and Sutugin, 1970, 1971); as an approximation,

Interface

Interface

which reduces to x + dx

FIGURE 5.17 Schematic diagram for treatment of diffusion of a species in one dimension in a liquid.

The net flux in given by

8c 82c

8x 8x is the combination of the net flux in and out, i.e.,

8c 8x

+ D, which reduces to

8c 82c

8x 8x

The rate of transfer (Rt, in units of molecules or moles per cm2 per second) of the species across a plane at x = 0, is given by

FIGURE 5.17 Schematic diagram for treatment of diffusion of a species in one dimension in a liquid.

Uptake across the interface into solution (a). By definition, this is described by the mass accommodation coefficient, a, and l/a is the "interfacial resistance."

Solubility and diffusion in the liquid phase (rso,). Consider the diffusion of a dissolved species in one dimension into the bulk solution from the interface region. The concentration of the species in the liquid will depend on time as well as on the distance from the interface. It is assumed first that no reaction is taking place in the aqueous phase.

Figure 5.17 shows a general case of diffusion in the x direction across a plane of unit area. The concentration of the diffusing species is taken as c.

8c 8x

The concentration gradient (8c/8x)x=0 at the surface depends on time because as uptake occurs, reevaporation back to the gas phase becomes increasingly important and the magnitude of the concentration gradient decreases.

Equation (CCC) can be solved to obtain the rate, Rt, under certain boundary conditions. Take the case where the concentration in the bulk liquid is given by clibulk at time t = 0 as well as at x = for times t > 0. It is assumed that there is a thin layer at the surface that contains the dissolved species in equilibrium with the gas immediately adjacent to the surface. This interface concentration is denoted as ^i.inicriacc Under these conditions, Eq. (CCC) can be solved (see Danckwerts, 1970, pp. 31-33) to obtain the rate of transfer per unit surface area after exposure time t, as

where D, is the diffusion coefficient in the liquid phase (units of cm2 s~ ') and (<5c/<5x) is the concentration gradient at x. The concentration gradient (i.e., slope of the concentration versus distance) is also changing with distance. At a distance dx from the position x, this concentration gradient is given by [(8c/8x) + dx(82c/8x2)] and the net flux across the plane at dx, i.e., out of the volume bounded by x and (x + dx), is given by

As intuitively expected, the rate of transfer across the interface depends on the difference in the liquid-phase concentrations at the interface and in the bulk and on the diffusion coefficient in the liquid. In addition, it depends inversely on the time of exposure of the liquid to the gas because of the increasing importance of reevaporation back to the gas phase at longer times. When c, bu|k = 0, Eq. (EEE) becomes

The net flux in the region bounded by x and (x + dx)

The dissolved species at the interface is considered to occur in a thin layer of thickness (D^t)^2

0 0

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