Envlron. Sci. Technol. 1093, 27, 1690-1692
COMMUNICATIONS Mechanistic Model for Colllslonal Desorption Steven J. Severtson and Sujlt Banerjee'
Institute of Paper Science and Technology, 500 Tenth Street, Northwest, Atlanta, Georgia 30318
Introduction Solute partitioning between solids and water decreases with increasing solids concentration at high particle levels, e.g., at levels exceeding 1mg/L (1-10). DiToro and others (1-4)have proposed that particle-particle collisions lead to increased desorption and have developed an empirical equation that accommodates the data quite well. Others have invoked a "third phase", which is assumed to separate from the solids and associate with solute (7-9). Since this material is included in the aqueous phase, it lowers the apparent distribution coefficient. There is ample evidence that third-phase arguments apply to many situations where sorption is lower than expected (11). However, in an elegant series of experiments, DiToro factored out third-phase effects and demonstrated that they cannot completely account for decreased sorption in all cases (2). He also demonstrated the phenomenon with glass beads where third-phase effects are unlikely (12). In this paper, we provide a mechanism and thermodynamic support for the collisional model and validate DiToro's empirical equation.
Model Development Environmental transport of suspended particles in water occurs principally through Brownian diffusion (13).When two particles collide, all or part of their kinetic energy is deposited at the particle surface. Einstein has shown that the particles involved have surprisingly high mean velocities through Brownian motion; for instance, the mean velocity of a 2.5-fg platinum particle in water at 19 "C is 8.6 cm/s (14).The mean displacement,however, is smaller by 4 orders of magnitude, since the motion of the particle is random. On the basis of simple collision theory (151, the disparity between displacement and velocity requires that the two particles collide extensively through translational Brownian motion before they fully separate. Rotational Brownian motion further increases the collision frequency if the particles are nonspherical. The energy transferred to the surface will be dissipated at the surface (16) and is essentially equivalent to heat. Collision of the two platinum particles of the type described above will lead to a surface temperature rise of a fraction of 1K, depending on the depth of the surface impacted. Since the sorption coefficient is inversely proportional to temperature (IT), collision will lead to a spontaneous desorption of a small fraction of solute. If the collisions are infrequent, the displaced solute will simply resorb, and conventional partitioning behavior will result. However, if the collisional frequency is high enough, the 1690 Environ. Scl. Technol., Vol. 27, No. 8, 1993
conventional equilibrium concentrations will not be restored. In other words, a balance will be struck between the frequency of collision and the rate of resorption. Consider the equilibrium
kl S+P=SP
k, where S, P, and SP are solute, particle, and reversibly sorbed solute, respectively. It follows that d[SP]/dt = kl[Sl[P]- k,[SP] Equation 1 integrates to ISPI = [SP,le-k't + (kl/k')[Pl[Stl(l -
(1)
(2)
where the total number of adsorbable sites is considered to be in excess over occupied sites, i.e., [PI [Pol.Poand SP, are the number of occupied and unoccupied sites, respectively, at t = 0
[S,I = [SI + CSPI
(3)
and
k' = k,[Pl
+ k,
(4)
At the moment of collision, the particle receives an energy pulse that desorbs a small fraction of solute. If the fraction remaining on the particle is designated as f , then multiplying the [SP] term in eq 2 by f gives [SPI = f([SP,le-k't
+ (kl/k')[Pl[Stl(l - e?)
(5)
Let T be the period between collisions. Just before the next collision, [SP] will be given by substituting [SP,] in eq 2 with the quantity in eq 5. Hence
[SPI = (k,/k')[PI[S,I(l- e-"') + fe-k"([SP,le-k" + (k,/k')[Pl[Stl(l = (~,/~')[PI[s,I(I- e-'')
fe-'"(k,/k')[Pl
(6)
+
[Stl(l - e-'")
+ f[SP,le-2k"
(7)
Just before the subsequent collision
[SPI = ( k , / k ' ) [ ~[SJ l (1- e-'') + fe-'"(k,/k') [PI[S,I(~ + fe-2k'7(kl/k')[~~ [s,I(~ - e-'") + ~[sP,I~-"") (8)
Equation 8 was obtained by multiplying eq 7 by f and substituting the result for the [SP,] term in eq 2 as before. 0013-936X/93/0927-1690$04.00/0
0 1993 Amerlcan Chemical Society
Discussion
After n collisions
DiToro's model for collisional desorption is based on the scheme pl[Sp,]e-(n+l)k'T (9) For large n, the summation C(fe-k'7)nconverges to 1/(1fe-k'7),and the last term in eq 9 approaches zero. Hence
[SPI = (k,/k')[~l[st] (1- e-k'7)/(1- fe-9
The (reversible) partition coefficient K can be written as
The conventional (reversible) distribution coefficient (i.e., without the collision effect) K, is kllk2. Equation 12 can be rearranged to
K,(I 1- fe-'" + K,[Pl (1- fie-'"
(13)
Equation 13 is the general equation governing both particle-induced and conventional sorption. At low solids, the exponential term vanishes since T is large and K = Kc. Thus, the difference in magnitude between conventional and collisional desorption is governed by 7 , the interval between collisions.
A potential difficulty with the above scheme is that Brownian collision of water molecules with the particle surface occurs much more often and with energies not much different than those involved in interparticle collision (14). However, energy transfer in collisions between species of different size is much less efficient than those between similar sized particles. For example, the fraction of kinetic energy transferred (ftrm8) from one particle to the other in an elastic head-on collision is given by
ftrm8 = (4m1/m2)/(l+ m1/m2)2
ka
k2
from which
(10)
Iff = 1,eq 10reverts to the conventional sorption equation. As an order of magnitude illustration, if kl = 1 X min-l (18) and T = 0.1 min, then [SPI decreases by half if f decreases from 1 (conventional) to 0,999. Thus, collision-induced desorption of 0.1 % is sufficient to cause a sizable decrease in sorption under these conditions if, on the average, a collision occurs every 6 s. The collision frequency will depend upon the shape of the particles and diffusion-related parameters. The f term will be governed by the heat of sorption, surface effects related to vicinal water, the depth of the surface affected, particle size distribution, and other factors.
K=
kl
s + P = SP + P- s + 2P
(14) where ml and m2 are the masses of the two particles involved, with particle 2 being considered to be initially at rest on a relative basis (19).For two particles of equal mass, ftrm8 = 1,i.e., all the kinetic energy is transferred. Hence, the energy transferred during particle-particle collision is several orders of magnitude greater than that involved in water-particle collision.
Lande (5) has objected to the scheme on the grounds that an equilibrium cannot include an irreversible step. Thus, the basis of eq 15 is uncertain, but it applies quite well to the experimental data at an empirical level. Equations 13 and 15 predict similar dependencies of K on particle concentration. Equation 13 can be rearranged to K=
1- fe-k"l 1- e-k'r
Kc K,[P] (1- fie-'" + 1- e-'7
(16)
Sincekl> k2,thenathigh [PI,kl[Pl >k2andk'~=kl[P17 (from eq 4). Since T is inversely proportional to [PI,the exponential term will be independent of [PI. Also, since f approaches 1,the term (1-fe-k'7)/(l - e-9 will be only slightly greater than 1,and eqs 15 and 16 will be similar in form. DiToro (1)notes that the coefficient (kalkl) in eq 15 tends to approximate one. The equivalent term in eq 16 is (1- fieJ"'7/(l Setting this term to one gives f
= 2 - ek'7
Mackay and Powers (4) calculated that the interval between Brownian collisions corresponding to a particle density of 1.3 X 10l1 particles/L is approximately 1 h. However, this value applies only to the initial collision, and T will be very much lower since multiple collisions will occur before the particles fully separate. Finally, given DiToro's success in applying eq 15 to a large body of sorption data (11,we recommend that the equation be retained with its constants redefined. If (1 - f t ~ ~ " ) / -( l = 1 in eq 16, then eq 16 simplifies to
where Cfis designated the collision factor and is defined as
C, = (1- fi/(ek" - 1) Cfis expected to be constant for a given solute/sorbent combination. Equation 17 is similar in form to eq 15, but has a defensible thermodynamic basis. In summary, our model assumes that (a) Brownian collision between particles leads to energy transfer that desorbs a small fraction of solute, (b) two particles will collide extensively through rohtional or translational Brownian motion before they fully separate, and (c) the collision frequency is high enough to prevent conventional equilibrium from being reached. The degree of desorption in item (a) depends upon the magnitude of the volume element of the particle through which the energy is dissipated. In amorphous media, energy penetration is typically limited to a few Angstroms (16). The collision Environ. Sci. Technol., Vol. 27, No. 8 , 1093 1691
frequencyin assumption (b)will depend on particle shape; e.g., rotational Brownian collisions will be absent for spherical particles. These assumptions affect the magnitude o f f and 7. Finally, we note that the model, as formulated, requires sorption to be reversible. It is well-known that sorption may contain both reversible and resistant components (18). The latter is usually associated with material diffused into the body of the sorbent. Our model only applies to the reversible component since the solute diffused into the sorbent should not be affected by local events at the particle surface. Nomenclature S solute P particle reversibly sorbed solute SP sorption rate constant kl desorption rate constant k2 time t interval between collisions 7 f fraction of solute remaining on the particle after collision conventional distribution coefficient (reversible) KC distribution coefficient covering both conventional K and collision-induced processes (reversible) collisional factor Cf
Literature Cited (1)DiToro, D. M. Chemosphere 1985,14,1503-1538.
1692 Environ. Scl. Technol., Vol. 27, No. 8, 1993
(2) DiToro, D. M.; Mahony, J. D.; Kirchgraber, P. L.; O’Bpne, A. L.; Pasquale, L. R.; Piccirilli,D. C. Environ. Sci. Technol. 1986,20,55-61. (3) O’Connor,D. J.;Connolly,J. P. Water Res. 1980,14,15171523. (4) Mackay, D.; Powers, P. Chemosphere 1987, 16, 745-757. (5) Lande, S.S.Chemosphere 1988,17,1085-1088. (6) Schrap, S.M.; Opperhuizen,A. Chemosphere 1992,24,12591282. (7) Gshwend, P. M.; Wu, S. Environ. Sci. Technol. 1985,19, 90-96. (8)Voice, T. C.; Rice, C. P.; Weber,W. J. Enuiron. Sci. Technol. 1983, 7, 513-518. (9) Voice, T. C.; Weber, W. J. Enuiron. Sci. Technol. 1985,19, 789-796. (10) Van Hoof, P.L.; Anders, A. W. In Organic Substances and Sediments in Water. Volume 2,Processes and Analytical; CRC Press: BocaRaton, FL, 1991;Chapter 18,pp 149-167. (11) Killey, R. W. D.; McHugh, J. 0.;Champ, D. R.; Cooper, E. L.; Young, J. L. Environ. Sci. Technol. 1984,18,148-157. (12)DiToro, D. M. Unpublished results. (13) O’Melia, C. R. Environ. Sci. Technol. 1980,14,1052-1060. (14) Einstein, A. Investigations on the Theory of the Brownian Mouement; Dover: New York, 1956. (15)Frost, A. A.; Pearson, R. G. Kinetics and Mechanism; Wiley: New York, 1961. (16) Kittel, C . Introduction to Solid State Physics; Wiley: New York, 1986. (17)Szecsody, J. E.;Bales, R. C. Chemosphere 1991,23,11411151. (18)Karickhoff, S. W.; Morris, K. R. Enuiron. Toricol. Chem. 1985,19,469-479. (19) Weidner, R. T. Physics; Allyn and Bacon: 1985.
Received for review September 9, 1992.Revised manuscript received April 12, 1993.Accepted April 14, 1993.
