J. Phys. Chem. 1988, 92, 1151-1155 that the kinetics of this reaction have been the subject of so much controversy. Free energy profiles for a typical industrial stripping reaction are shown in Figure 17, for 5 and 24%stripping. In the stripping direction equilibrium is reached at about 24% so the changing concentrations make less difference to the free energy profiles. The rate-limiting free energy difference throughout is the conversion of CuL+ to aqueous Cu2+ and organic HL, followed by the adsorption of a fresh CuL, molecule through transition state 3. We have discussed previously the evolution ofthe ideal enzyme catalyst.29 A feature of such a catalyst is that it sould work reversibly-that is, close to thermodynamic equilibrium. Secondly, the transition states should have similar free energies to each other and the same applies to the intermediates. In this waq the system is not trapped in a deep hole as a particularly stable intermediate nor does it have to climb over a particularly high transition state. Although enzymes will evolve to be perfect catalysts by natural selection, it is a tribute to IC1 that, through the pressures of commercial selection, Acorga P50 is almost a perfect catalyst for the solvent extraction of copper.
Acknowledgment. We thank IC1 and the SERC f’or a Cooperative award, and it is a pleasure to acknowledge helpful con(29) Albery,
W. J.; Knowles, J. R. Biochemistry
1976, 15, 5631.
1151
versations with Dr. John Middleton of IC1 New Science Group and Dr. Ray Dalton of IC1 Organics. Appendix In this Appendix we give the deriution of eq 13. We assume that each species obeys a Langmuir isotherm for its adsorption on the surface. We write the fraction of sites occupied by a species, X, as xx; VS symbolizes a vacant site. Then we have for the flux j o through each transition state the equations jo =
~ ( ~ I ~ X- H k-ihxcu~) L
(35)
j o = n ( k , k l L - k-2hXCuLJ
(36
1 0 = 4 k 3 X C ” L 2 - k-3cxvs)
(37
and
The adsorption of H L is given by = Kolxvs
(38
+ X C ~ L+ X C U L ~ + xvs = 1
(39)
XnL
Finally XHL
Elimination of the four x? by using eq 36-39 yields eq 13. Registry No. Acorga P 50, 50849-47-3;Cu, 7440-50-8.
Transport and Kinetics In Reactions Involving Two Liquid Phases W. John Albery* and Riaz A. Choudheryt Department of Chemistry, ImperiGl College, London SW7 2AY, England (Received: April 17, 1987)
There can be four possible rate-limiting processes for a reaction taking place in a system of two liquid phases. The rate may be limited, first, by the kinetics of a bulk homogeneous reaction, second, by the transport of the reactant to the bulk phase, third, by the kinetics of a homogeneous reaction in a thin reaction layer, and, finally, by the kinetics of a reaction taking place on the liquid/liquid interlace. The kinetic/transport equations are solved for the linear case, for diffusion away from a drop, and for diffusion into a drop. A common treatment is derived for these three cases, and it is shown that the behavior can be described in terms of four characteristic lengths of the system. The conditions for each case are derived in terms of inequalities of these lengths. Case diagrams are constructed to show the interrelation of the different cases. Only two different patterns are found. It is shown that modest stabilization (-20 kJ mol-’) of the interfacial transition state by the two solvents will result in the interfacial mechanism being dominant.
Introduction In the last paper we explored in some detail the kinetics and mechanism of the solvent extraction of copper.’ Therl: has been much controversy as to whether this reaction is an interfacial reaction or whether it takes place in the aqueous phase .2-10 The results from the rotating diffusion cell experiments show that under rotating diffusion cell conditions the reaction is an interfacial one. However, it may be dangerous to make the facile assurr ption that a mechanism, established by a technique where the interfacial area is small and the bulk volume relatively large, will still be the operative mechanism in a stirred reactor, where the interfacial area compared to the homogeneous volume is many orders of magnitude larger. The case of mass transport with r1:action in a macroscopic reactor is a familiar problem in chemical engineering.I1-l6 In this paper we present a unified treatment which describes both the macroscopic case and the case where the same system is dispersed into droplets. We derive the criteria that decide where a two-phase reaction takes place-on the interface, in a thin reaction layer, or in the bulk of solution. We will present ‘Present address: IC1 Paints Division, Wexham Road, Slough, Berkshire,
England.
0022-3654/88/2092-1151$01.50/0
“case diagrams” which show the interrelation of the different cases to each other. These diagrams are useful for seeing the effects (1) Albery, W. J.; Choudhery, R. A. J . Phys. Chem., preceding paper in this issue. (2) Flett, D. S. Acc. Chem. Res. 1977, 10, 99. (3) Whewell. R. J.: Huahes. M. A.: Hanson. C. International Solvent Exiruction Conjerence’1977; Canadian Institute of Mining and Metallurgy: Montreal, 1979; p 185. (4) Danesi, P. R.; Chiarizia, R.; Vandegrift, G. F. J. Phys. Chem. 1980, 84, 3455. (5) Albery, W. J.; Fisk, P. R. Hydrometallurgy 81; Society of Chemical Industry: London, 1981; p F5/1. (6) Freiser, H.; Akiba, K. Anal. Chim. Acta 1982, 136, 329. (7) Freeman, R. W.; Tavlarides, L. L. Chem. Eng. Sci. 1982, 37, 1547. (8) Albery, W. J.; Choudhery, R. A,; Fisk, P. R. Faraday Discuss. 1984, 77,53. (9) Freiser, H. Acc. Chem. Res. 1984, J 7 , 126. (10) DjugumoviE, S.; Kreevoy, M. M.; Skerlak, T. J . Phys. Chem. 1985, 89, 3151. (1 1) Astarita, G. Mass Transfer with Chemical Reaction; Elsevier: Amsterdam, 1967; p 49. (12) Danckwerts, P. V. Gas-Liquid Reactions; McGraw-Hill: New York, 1970; p 107. (13) Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975; p 313.
0 1988 American Chemical Society
Albery and Choudhery
1152 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
of changing parameters such as reactant concentration or drop size and for seeing how valid conclusions drawn under one set of conditions are likely to be when the reactant conditions are varied. The System and the Four Characteristic Lengths We assume that a species A present in phase a is reacting with a species B present in phase /3. We make the assumption that the concentration of B is in sufficient excess that the reactions of A either at the interface or in the bulk of solution can be described by first-order kinetics with the rate constants k’ and k, respectively: k’ = k;[B] and k = k*[B] The units of k‘and k are cm s-l and s-l, respectively; k’is similar to a normal electrochemical rate constant. We assume that an interface of area A is producing reactant for a volume V ,of the /3 phase and that at the interface there is a Nernst diffusion layer of thickness ZD. In the special case of the rotating diffusion ce1116 this thickness is given by the Levich equation.” For diffusion from a droplet of a phase, ZDwill be given by the radius of the droplet r,. Another crucial length is the thickness of the reaction layer, Z,. This is also the average distance that A diffuses before reacting with B and is given byIB Zk
F
(D/k)1/2
(2)
For the reaction layer model the flux j (in mol cm-2 s-l) is given by j = kaozk (3) where a, is the concentration of A in phase /3 at the interface. We assume throughout that is no interfacial resistance to the partition of A between the two phases. Now our treatment will be carried out by comparing the sizes of four characteristic lengths. We write for each of the possible mechanisms j = kaoZ?
(4)
For the interfacial reaction, since k’describes the reaction of A in a with B in 6, we have j = k’K-’ao
(5)
where K is the partition coefficient for transferring A from a to /3. Comparison of eq 4 and 5 allows us to define the characteristic length Z,: 21
= k’/(Kk) = k,’/(Kk,)
(6)
ZI compares the free energies of the homogeneous and heterogeneous transition states. For reaction taking place in the bulk volume of the solution, when the whole volume is saturated with A j = kaoV,/A
(7)
Comparison of eq 4 and 7 allows us to define the characteristic length ZB where ZB = V,/A
For a stirred reactor we consider two cases. First we take the case where the volume of the a phase, V,, is much smaller than that of the /3 phase, V,. Under these conditions we assume that the a phase is dispersed in droplets of radius r,. The number of droplets N will then be given by N = 3V,/(4rra3)
The volume of phase /3 that each droplet has to service with its area of 4rrm2is given by V,/N. Hence from eq 8 and 9 we find ZB = r,VB/3V,
(10)
If we make the opposite assumption that the volume of the CY phase is much larger than that of the /3 phase, then the /3 phase will be distributed as droplets of radius r,. Now each volume of (4/3)ar83 is serviced by an area of 4 r r 2 and hence ZB = r a / 3
-
-
(1 1)
It is satisfactory that when V, V,, r, r,, and there is a good join between eq 10 and 11. We shall show that under some circumstances the flux is limited by the rate of transport of A across the diffusion layer; A is then consumed in the bulk of the solution. For this case the flux is given by j = D a o / Z D = kao(zk2/zD)
(12)
The characteristic length, z?,for this case is equal to zk2/zD. We have now defined the four characteristic lengths of the system, Zk, ZI, ZB and ZD, which are important in the reaction layer, interface, bulk, and mass-transport cases, respectively. The Relative Sizes of ZDand ZB Next we note that ZBcan be larger or approximately equal to ZD,but that it can never be smaller than ZD.This arises because the total volume V, is equal to the area A times ZB. The volume of the diffusion layer must be smaller than the total volume, and hence ZDmust be smaller than ZB. For the particular case of a rotating diffusion cell ZBcan be as large as 200 cm, while ZD is typically cm. Under these conditions the large bulk volume of the solution has to be fed through a diffusion layer of a relatively small area. In the case of a stirred reactor where V, is much larger than V,, the same conclusion holds. The volume of the diffusion layer around each small droplet of phase a is much smaller than the volume of phase /3 that each droplet has to service. When V, is much larger than V,, we then find that ZDand ZBare of similar size, being determined by the radius of the droplet of /3 phase. Next we shall consider the interrelation of the three homogeneous cases-reaction layer, mass transport, and bulk reaction. We shall do this for three systems. First, we consider a macroscopic apparatus like the rotating diffusion cell. Second, we consider the stirred reactor with V , >> V, and, third, the stirred reactor with V, >> V,. Macroscopic Case For this case we assume that outside the diffusion layer the concentration of A is uniform and equal to a,. Then in the steady state and in the diffusion layer at a distance z from the interface we have to solve
D d2a/dz2 = ka
(8)
For an apparatus like a rotating diffusion cell V, and A will be defined by the geometry of the apparatus.
(9)
(13)
with the following boundary conditions. At the interface a = a,,
(14)
(da/dz)o = - j / D
(1 5 )
and (14) Astarita, G.; Savage, D. W.; Bisio, A. Gas Treating with Chemical Soluenrs; Wiley: New York, 1983; p 98. (15) Treybel, R. E. Mass-Tranfer Operations; McGraw-Hill: New York, 1985; p 45. (16) Albery, W. J.; Burke, J. F.; Leffler, E. B.; Hadgraft, J. H. J . Chem. SOC..Faraday Trans. 1 1976, 72, 1618. (17) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962; p 60. (18) Koutecky, J.; Levich, V. G. Zh. Fiz. Khim. 1956, 32, 1565.
At z = Z D a = a,
and AD da/dz = -Vka,
(17)
Reactions Involving Two Liquid Phases
The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 1153
This last boundary condition matches the flux of A leaving the diffusion layer with the A consumed in the bulk of the solution. Solution of eq 13 with the boundary conditions in eq 14 and 15 gives a
= a. cosh (z/zk) - (Z,J/D) sinh ( z / Z , )
(18)
K~
= kr2/D = ZD2/2k2
(27)
Substitution in eq 22-24 gives d2U/dp2 = K’U
(28)
with the boundary conditions that a t p = 1
Elimination between eq 16, 17, and 18 then gives
u = l
(29)
du/dp = 1 - g
(30)
g = jra/Dao
(31)
and
Most existing treatments of mass transfer with reaction consider that the volume of phase @ is very large. Under these conditions zB/zk >> 1 and eq 19 reduces to Hatta’s result for a macroscopic This equation describes two cases and the transition between them, namely, complete reaction of A within the diffusion layer and rate-limiting transfer of A across the diffusion layer. The inclusion of ZBdescribing a finite volume of phase @ means that eq 19 now also includes a third case when the kinetics are so slow that phase @ becomes saturated with A. We now consider how eq 19 describes the three cases, and we find the conditions for the existence of the different cases. First, if zD> 3Zk, then tanh (zD/zk) = 1, the square bracket is unity, and eq 19 reduces to eq 3, the equation for the reaction layer model. Under these conditions A is destroyed before it crosses the diffusion layer. If on the other hand z k > 3ZD, then tanh (zD/zk) = Z D / Z k . Substitution in eq 19, and remembering that for this case ZD> (ZBZD)’”
(21)
then eq 20 reduces to eq 4 with ZBand we have the case where the bulk volume is saturated. On the other hand, when (Z&ZD)’/* >> zk, then eq 20 reduces to eq 12. The thick diffusion layer supplying a large volume through a small area means that the mass transport of A is rate limiting. Hence we have demonstrated that eq 19 describes the three homogeneous cases for the macroscopic case. Stirred Reactor with V , >> V , Next we apply the same type of analysis to the case where Vs is much greater than V, and A is diffusing from small dispersed droplets of a phase into a sea of p phase. Instead of eq 13 we have the slightly more complicated equation for spherical diffusion: D d2a/dG
+ (2/r)(da/dr)
= ka
(22)
The boundary conditions for this equation at the interface, where r = r,, are a = a.
(23)
d a / d r = -j/D
(24)
and
where Integration of eq 28 with the boundary conditions in eq 29 and 30 gives p I)] u = cosh [ ~ ( -
+ (1 - g)K-’
da/dr = 0
(25)
We now define the dimensionless radial variable p, where p = r/r,, and we write u = pa/ao
(26)
and
u = p,(du/dp)
(33)
From eq 32 and 33 we then obtain g= 1
+
KbP, tanh [K(p,
- I)] - 1)
-
Kpm - tanh [ ~ ( p , I)]
(34)
+
When [ ~ ( p ,- I)] > 3, eq 34 simplifies to g = 1 K . When K > 1, we then have the reaction layer case; zk is smaller than ZD = r, and A does not penetrate far across the spherical diffusion layer. In the intermediate case where KP, > 1 > p , then g = 1. From eq 31 we than find that the rate is controlled by the mass transport of A across the spherical diffusion layer; it is then consumed in the bulk. When [ ~ ( p ,- l)] < 0.3, then, approximating tanh x as x - x3/3 and remembering that p, >> 1, we find that g = K2p3/3
Substitution from eq 27 and 31 then gives 4 ~ 2 =j (4/3)rrm3kao
(35)
A is now so stable that the whole sphere of action is saturated and we have a bulk reaction. It is satisfactory that the rigorous treatment gives the same answer for ZBas that given in eq 10. It is also satisfactory that the same three homogeneous regimes are found for the spherical case as for the linear case. Furthermore the criteria in terms of the lengths, zk, ZB, and ZD, for the boundaries between the different cases are the same. Stirred Reactor with V , >> V , We now turn to the case where we assume that V, >> V,. Small droplets of phase @ are now engulfed in a sea of phase a. We follow the same transformations of the spherical diffusion equation. Instead of eq 24 we have that, at p = 1, da/dr = j / D leading to
+g
(36)
Hence the expression for u is u = cosh [ ~ ( p I)]
+ (1 + g)K-’
sinh [ ~ ( p l)]
(37)
The boundary condition of zero concentration gradient in eq 33 must now be applied at p = 0 rather than p,, and so we obtain g = K coth K - 1 (38) When K >> 1, we find that g = K and we have a thin reaction layer on the outside of the droplet. When K < 0.3, we can use the approximation that coth x = l / x x/3 to find g = ~’13. substitution from eq 27 and 31 then gives
+
4?rrB2j= (4/3)rrB3kao (19) Hatta, S . Technol. Rep. Tohoku Imp. Univ. 1932 10, 119. ( 2 0 ) Albery, W. J.; Bartlett, P. N. J. Electroanal. Chem. Interfacial Electrochem. 1982, 131, 1 3 1 .
(32)
The boundary condition in eq 25 gives us that at p,
du/dp = 1
Each droplet has a sphere of actionZoof radius r, where the droplets are separated on average from each other by 2r,. At this boundary
sinh [ ~ ( p- I ) ]
(39)
Since K is small, A is sufficiently stable to saturate the drop and we have a bulk reaction. Again it is satisfactory that the rigorous
Albery and Choudhery
1154 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 Zk =
zc
TRANSPORT LAYER
z,
ZI
/ /
MASS
REACTION
log[?]
INTERFACIAL
log[;]
zo
Zk = REACTION
1
TRANSPORT LAYER
z,
:
ZI
= Zk
2,
Z? = Zk Zk
,/
ZI
Z , / Z D ,a homogeneous mechanism will be found. When Z I / Z D> Z , / Z D ,the interfacial mechanism will be found.
treatment yields the same expression for ZBas that deduced in eq 11. In this case there is no separate mass-transport case because both ZDand ZB are -r8. Homogeneous Case Diagram
We can now construct the homogeneous case diagram, shown in Figure 1. This shows the conditions for the three different cases in terms of the ratios Zk/ZDand ZB/ZD As discussed above, ZB/ZD3 1. The length to be used for Z? in eq 4 is also given for each case, and it is satisfactory that good joins are found on each of the boundaries. The case of the stirred reactor with V, >> V ,is found along the bottom edge of the diagram, and it can be seen that there is no mass-transport regime. Increasing ZB means that the mass-transport regime becomes more significant. Increasing the rate of the reaction, for instance, by increasing [B], decreases Zkand means that one moves from right to left on the diagram. One could pass through the sequence of cases: bulk, mass transport, and reaction layer. The order of the kinetics with respect to B is first for bulk, zero for mass transport, and half
Figure 4. Case diagram including the interfacial reaction for ZI > ZD. The conditions for the existence of each region are displayed.
for reaction layer. The case diagram holds for macroscopic and microscopic systems. Interfacial Reaction We now allow the interfacial mechanism to enter the competition. If ZI is greater than the appropriate Z? from the homogeneous case diagram, then the interfacial mechanism will be observed. Conversely if ZI is smaller than the appropriate Z , , a homogeneous mechanism will be observed. In order to illustrate the competition we can construct the contour diagram shown in Figure 2. In this diagram we have plotted contours for the appropriate (Z?/ZD)for the homogeneous mechanisms. There is a mountain toward the top right-hand corner of the diagram. Now somewhere the mountain will be slicg by the Zr/ZDcontour. Areas of the homogeneous case diagram that are higher than this contour will exhibit homogeneous mechanisms; areas that are lower will yield to the interfacial mechanism. Inspection then shows that there are two and only two different patterns of behavior that may be found. First, if ZI C Z,, then the ZI/ZDcontour will slice the homogeneous mountain in the reaction layer region. The competition will then give the pattern depicted in Figure 3, where the conditions required for the different regions are also displayed. All the homogeneous mechanisms may
The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 1155
Reactions Involving Two Liquid Phases
log IZk/cm)
log rate
I
/
/
INTERFACIAL
I
REACTION LAYER
-6
Zk =
zs =
ZD
TRANSPORT 0
2
I,
6
U
log lk/s-’l
log t B 1
Figure 5. Effect on the reaction rate of varying [B]. The line in the top left-hand corner (Z, 7 2, is a slice across the lower half of Figure 4. The next line ( Z B 7ZI> Z,) is a slice across the top half of Figure 4. The line ZB> ZD 7 Z I is a slice across the top half of Figure 3, and the line ZB ZD> ZIis a slice along the x axis of Figure 3.
-
be observed. The interfacial reaction is favored by fast kinetics (large [B]),but changing ZBdoes not affect the interfacial/homogeneous boundary. Second, if ZI > ZD, then the ZI/ZDcontour will slice through the mass-transport and bulk regions, giving the pattern depicted in Figure 4. Now the reaction layer case can never be observed. Fast kinetics and low values of ZBboth favor the interfacial reaction. In this case a homogeneous mechanism may be observed in a macroscopic experiment at large ZB 10’ cm,and when ZBis reduced by 6 orders of magnitude in the stirred reactor to lo4 cm, the mechanism may change to an interfacial one. On the other hand, if an interfacial mechanism is observed in a macroscopic experiment, then reduction of ZBwill not change the mechanism. Hence the conclusions in the previous paper about the solvent extraction of copper will hold in the industrial reactor. From Figures 3 and 4 we can now summarize the conditions under which each mechanism will be observed:
-
bulk mass transport
zk > (ZDZB)’” > (ZDZI)’” (ZDZB)’/’ > z k > (ZDZI)’/’ + ZD
reaction layer
(41)
> Zk > zl ZI > ZB
(42)
> ZI > (Z[zD)”’ > zk
(44)
> ZI > zk
(45)
ZD
interfacial
(40)
(43)
or ZB
or ZD
In Figure 5 we illustrate the effect of changing the concentration of B and thereby going from right to left on either Figure 3 or 4. Any particular system is unlikely to show all of these changes, but the changes in order with respect to B can help identify the different cases. Heterogeneous vs Homogeneous Kinetics If we assume that V, and Vpare comparable, so that we can simplify by assuming that ZD ZB,and further if we take a plausible value for D of cmz s-I, then we can construct the diagram in Figure 6. This diagram shows which case is found as a function along the x axis of the first-order homogeneous rate constant k and along they axis of Z , which measures the difference
-
Figure 6. Diagram showing how existence of the different cases depends on the homogeneous rate constant, k, and ZI. A value of cm2 s-I has been assumed for D, and we have also assumed that Z B = ZD.The solid and broken lines are for ZB= IO-’ and cm, respectively. A scale showing the difference in the free energies of the homogeneous and heterogeneous transition states, AAG*, is also given. The location of the solvent extraction of copper is also shown.
in free energy of the homogeneous and heterogeneous transition states. We assume that this difference is zero when Z1 = cm. The boundaries for the bulk region have been drawn for ZB = cm. The broken lines show the pattern for ZB= lo-’ cm. The diagram is bounded on the right-hand side by the fact that k < cm the distinction between a reaction layer and when z an interfacial reaction becomes blurred.” In fact k2 has to be close to the diffusion-controlled limit for k to be as large as lo9 s-l. Using our data from the previous paper,’ we have also plotted the region for the solvent extraction of copper. Finally, it can be seen that the interfacial mechanism only requires quite modest stabilization of its transition state by the two solvents for that mechanism to the dominant route of reaction, and therefore one should not be too surprised to find systems that react by that mechanism. Conclusions Finally we summarize the conclusions of this work. 1. The location of a reaction in a biphasic liquid/liquid system may be at the liquid interface, in a thin reaction layer close to the interface, or in the bulk phase. If the location is in the bulk phase, the rate-limiting process may be either the kinetics of the reaction or the mass transport of a reactant across a diffusion layer next to the interface. Hence we have four possible cases: interfacial, reaction layer, mass transport, and bulk. 2. Which case is found is determined by the relative sizes of k (eq 2), ZI (eq 6), and ZB the four characteristic lengths, ZD,z (eq 8, 10, and 11). 3. The conditions for the existence of each case in terms of the lengths are summarized in eq 40-45. These rules can be used for both macroscopic and microscopic systems. 4.Two and only two different patterns may be found as depicted in the case diagrams in Figures 3 and 4. 5. For the ZI < ZDpattern (Figure 3) reduction of ZB (macroscopic to microscopic) will not change the case. For the ZI > ZDpattern (Figure 4) reduction of ZBmay lead to the sequence: mass transport, bulk, interfacial. 6. Increasing [B] can give the four different sequences shown in Figure 5 . The same sequences will be found by increasing k Z . 7. When V, 2 Vp,Figure 6 shows which case will be found in terms of the homogeneous rate constant, k , and the length Z I . 8. A small amount of stabilization (-20 kJ mol-’) of the interfacial transition state by the second solvent will lead to the interfacial reaction being the preferred route. (21) Albery, W . .I. Faraday Discuss. 1984, 77, 144
