Ind. Eng. Chem. Res. 1989,28, 1025-1030
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Alumina Desiccant Isosteres: Thermodynamics of Desiccant Equilibria Stephen C. Carniglia* and Wendy L. Pingt LaRoche Chemicals, Znc., P.O. Box 877, Pleasanton, California 94566
T h e equilibrium p ( H 2 0 )(in atmospheres) over HzO sorbed in any stable solid desiccant a t loading X must obey log p , = 6.231 0.2186AS *, - (2.303 0.2186AH*X)(1000/T),where AS*, (cal mol-' K-l) and AH*, AH,,, (kcal mol-l). Sets of log p x versus 1/T plots a t fixed X (i.e., isosteres) disclose fundamental differences between the types of desiccants, shown, e.g., for activated aluminas and zeolites. For a representative commercial activated alumina, AH*,and A S *, have been experimentally and analytically determined over the entire range 0 < X < 41 g/100 g. The AH*x curve agrees with selected prior calorimetric data. At X < 9 (monolayer sorption), AH*x = 7.12 - 2 In A, while A S = 7.0. Isosteres are presented for the range 0.02 < X < 5.0, based on these functions. These curves and the thermodynamic quantities are useful for design and analysis of cyclic systems employing aluminas in drying, refrigeration, and heat-pump applications.
+
Activated alumina desiccants, long used in drying industrial liquids and gases, are also suitable for refrigeration and similar heat-exchange uses. The typical fixed-bed operating cycle consists of a water sorption step at or near ambient temperature, followed by a desorption or desiccant regeneration step using a heated fluid, usually the same as that being dried. In such cyclic applications, the reversible phase change may be expressed as H,O(,) = H20(,). Here H20(,) means water sorbed in the desiccant a t some loading X in grams of HzO/lOO g or pounds of H20/100 lb of dry desiccant. In dynamic processes, this phase change is not necessarily conducted at equilibrium. Nevertheless, at every location in the desiccant bed, both thermal and phase equilibria are nature's aim a t each instant. The discipline of thermodynamics connects both the thermal and the mass aspects of equilibria, and it is important to know what these are for purposes of system and cycle design and performance analysis. Commercial desiccant-grade activated aluminas are exemplified by a material bearing the trade designation A-201. We have determined a new set of A-201/Hz0 equilibrium "isosteres", drawn in Figure 1 on linear log p , versus 1000/T (K-l) coordinates with the latter transformed to "C and OF. Points A-D on the figure schematically indicate one possible fixed-bed operating cycle. An exposition of the thermodynamic imperatives governing such curves is given here first; then we shall describe how Figure 1 was developed.
Thermodynamics of Desiccant Equilibria For the phase change written above, equilibrium is described by a fixed desiccant water loading A, a fixed temperature T, and the corresponding equilibrium partial pressure of water vapor p x . It is common to use for p x the value for H,Oe) acting as an ideal gas. Familiar graphical representations of the relationships are sets of (a) curves of p , versus T a t selected X values, called isosteres; (b) curves of p hversus X at selected T values, called isotherms; and (c) curves of X versus T a t selected p Avalues, called isobars. Isosteres are especially convenient in relation to conventional drying cycles. But all curves for all desiccants obey the same basic thermodynamic equation, as will be shown. A limiting case, "desiccant absent", is the dewpoint curve. *Present address: 115 Wilshire Court, Danville, CA 94526. Present address: Greenleaf Corp., P.O. Box 2420, Livermore, CA 94550. 0888-5885/89/2628-1025$01.50/0
+
Dew-Point Curve: Pure Water Isostere. For the pure-water equilibrium, H20(,) = H20(,), the following thermodynamic equation governs: ASO1-g lnp=----
AH"1-g
R
R
1
T
(1)
where H is the molal enthalpy, S is the molal entropy, p is the equilibrium gas partial pressure, T is the absolute temperature, R is the gas constant, and the superscript O denotes standard states (viz., pure liquid and ideal gas at 1 atm, respectively). A similar equation governs the equilibrium H20(,).= H20(,),i.e., below 0 "C. Equation 1 and its mate for ice relate p and T by the two joined dew-point curves of Figure 2. Desiccant equilibrium vapor pressures are often restated as their corresponding dew points (e.g., scale included in Figure 1);hence, Figure 2 is needed to interconvert p , and Tdp. Equation 1and Figure 2 also provide a ready background for the following. Universal Equilibrium Equation for Desiccants. For desiccant equilibria, described as HZO(,) = H,Oe), the corresponding thermodynamic statement is
The only change from eq 1 is the specification of a new initial state, H20(,),which is a variable state with variable (partial molal) thermodynamic properties. These are sometimes designated by an asterisk, e.g., H *, and S *,. Equations 1 and 2 are linear in In p versus 1/T only if AH is invariant with temperature. In fact, AH is never quite T-independent. For any TI and Tz,
AHz - AH1 = s T z A C , d T Tl
(3)
But the heat capacity difference AC, is so small relative to AH in these equilibria that Figures 1 and 2 cannot depart significantly from linearity. Figure 2, which is faithful to reference data, exhibits barely discernible curvature. This near-constancy of AH as f ( T ) has been found experimentally in many gassolid sorption systems, as it should. A similar constraint on A S as f ( T ) also follows. Hence, these quantities in eq 2 can be treated as functions of X alone over reasonable intervals of T. In eq 2, evidently AHo,- = AHA-, + AHo.+ and similarly for ASo,-. Shorten AHx+ to read this is commonly called the "net heat of adsorption". Likewise, AS,,, = A S * , is the net entropy change, i.e., that from 0 1989 American Chemical Society
1026 Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 0
25
50
TemperaturC,OC 75 100 I50
20 ?50300350
0.
1.0
0.1
0.01
0.001
E 4
--
10-P
B =
10”
10b l@@UlT
dF ’
h.’
Figure 2. Dew-point curves for water, from reference data.
Temperature,O F Figure 1. A-201 alumina desiccant isosteres, this work.
H,O,,, to H20a,. The best reference values for and AS”,, at 294 K are 10.533 kcal/mol and 28.50 cal/ (mol-If), respectively; and R is 1.9865 cal/(mol.K). Substituting the above in eq 2, converting In p,, to log p x ,and consolidating produces log ph = 6.231 + 0.2186AS*h (2.303 + 0.2186AH*,)(lOOO/T) (4) Equation 4 expresses the universal water-vapor equilibrium equation (a thermodynamic imperative) for all stable, reversible solid desiccants. The sole differentiation among desiccants is the identification of A S and AH with X for each. Equation 4 defines all isosteres, isotherms, and isobars for any given desiccant, once those identifications are made. Its numerical terms correspond to p x in atm (1 atm = 101.33 kPa), AS*, in cal/(moEK) (1 cal = 4.186 J), AH** in kcal/mol, and T in K. Calories are used here in deference to trade practice. On log p i versus 1000/T coordinates, from eq 4 and the preceding analysis, there are three thermodynamic imperatives on all desiccant isosteres as individual curves: (a) each curve must be sensibly linear over the range 273-573 K; (b) whatever its X label, its slope is -(2.303 + 0.2186AH*J; and (c) whatever its X label, its extrapolated intercept at 1000/T = 0 is 6.231 + 0.2186AS*h. Isostere families describing each single stable desiccant embody continuous variations of AH and A S *,, with 1. Ordinarily in an interval of constant chemical nature of H201,,,,either AH*,, must increase with decreasing X or A S must decrease, or both. These three characteristic types of isostere sets are shown in parts A-C of Figure 3, whose outer frames encompass extrapolations to 1000/ T
-
= 0 and whose inner frames are the common limits of isostere presentation, viz., from about 273 to 573 K. increases with deType A, Figure 3A. Here AH*,, creasing X while A S is constant. This “fan” pattern will be approached when sorption sites differ in energy so that water molecules must condense on them in order, and there is little multiplicity of sites of equal energy. Activated alumina appears to be of this type in the X range from 0 up to a filled H20monolayer on all pore surfaces (vide infra). Type B, Figure 3B. Here AH*,, is constant while AS decreases with decreasing X. This “parallel” pattern will be approached when a very large number of sorption sites are of identical energy and hence S *,, increases with decreasing occupancy of sites. Zeolites or molecular sieves, whose sorption sites are identical voids in the crystal interior, should show parallel isosteres at the lowest X values. Isosteres for the type 5-A zeolite published by Barry (1960) do show this: his set is represented in Figure 4, extrapolated so the parallelism of the lowest four curves can be seen and AH*,, determined. Type C, Figure 3C. Here AH*h increases and A S decreases simultaneously with decreasing A. Barry’s last three isosteres at higher X values, Figure 4, exemplify the pattern of Figure 3C: a “fanlike” pattern but not emanating from a common point. The thermodynamic statement of eq 2 has of course been made use of previously in the gas-sorption literature, though often in alternative or even obscured forms and in other notations (e.g., Sircar and Gupta (1981) and Hacskaylo and LeVan (1985)). Its simple restatement here as eq 4 and the properties of the isosteres illustrated in Figure 3A-C facilitate the accurate description of desiccant/ water-vapor equilibria using only recognized thermodynamic quantities, as well as the testing of experimental relationships or data against readily visible and interpretable features of log p versus 1 / T plots. These qualities
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 1027
n
8.
6.
I
I
r--l
I
38 -
4.
2.
--.
0
.
E
5 e4 - 2 . 0 i
W
2
-4.
-6.
-8.
-10. 4.
1000/T,
K-'
Figure 3. Allowable types of desiccant isostere sets: 3A, type A; 3B, type B; 3C, type C. 5.
J.
I
I
I
I
1
I
c
3.
-
2.
-
1.
-
3
I
I
I
1
I
I
I
x -
15.7
18.
I
9.5
4.0 3.2 2.2 1.6
X
t I I
-
0 A
0.
7.
I
I
6.
5.
4.
3.
2.
1.
0.
I
-1.
1
-2.
I
1
-3.
-4.
I
-5.
-6.
L O G l O P A (ATM)
Figure 4. Type 5-A zeolite isosteres, adapted from Barry (1960).
are preferred here over the more complicated expressions and exclusive use of isotherms presented by the abovecited authors and others, valuable as those approaches may be in addressing much wider ranges of gas-solid systems, p , and T. One case has been omitted from Figure 3 which does occur in gas sorption, although only over narrow intervals of X: namely, dAH*JdX > 0 (cf. Sircar and Gupta (1981)). In Figures 1, 3, and 4, we show AH*x as monotonically decreasing with increasing X or as constant with A. In zeolites up to the filling of all intracrystalline voids and in activated alumina up to a filled surface monolayer, the state or structure of sorbed H 2 0 argues against the occurrence of a slope reversal. In particular, for activated
alumina, Rhone-Poulenc (1982) showed no indication of a slope reversal in AH*x over the entire range 1 < X < 12. Our experimental work with this material, to be described next, agrees.
Thermodynamics of H20,;) in A-201 Alumina: Experimental and Analytical No satisfactory description of both AH*x and A S of eq 4 as simultaneous functions of X has been found for activated alumina, and no thermodynamically confirmed set of equilibrium water-vapor isosteres has been published for this material. The work described below was intended to fill these voids, using the typical commercial product designated A-201.
1028 Ind. Eng. Chem. Res., Vol. 28, No. 7 , 1989
“t” ”
!I
.
10.
20.
30.
40.
50.
60.
R E L A T I V E HUMIDITY, R . H . ,
70.
80.
90.
m”
100
%
Figure 5. Equilibrium isotherm a t 21 “C: A-201 activated alumina.
A-201 activated alumina consists of mixed x,p , and II (from X-ray-amorphous to defect-cryptocrystalline) transition-alumina phases. The mean skeletal density is about 2 g/cm3. The typical total pore volume is about 0.47 cm3/g, and the typical BET surface area is about 325 m2/g as-delivered. The LO1 is about 6 wt %, of which an estimated 4% is residual chemically combined or skeletal H,O, the remainder being mostly combined CO,. Numerous 21 “C (294 K) A-201/H20 equilibrium isotherms have been reliably measured in our laboratory, of which Figure 5 is representative. The X scale in the figure is divided into three zones: zone “l”, 14 d X d 41, H,O(,, is liquid, filling the pore volume; zone “m”, 0 6 X 6 9, H,O,,, is a monolayer on pore surfaces; and zone “t”,9 < X < 14, transition zone between zones “m” and “1”. Zone 1,14 g d X d 41 g/100 g. In this zone, bulk liquid water gradually fills the pores of the solid (Kaiser Chemicals, 19851, ending at the “active capacity” where the capillary effect of pore radius becomes imperceptible. At X = 41 g/100 g, all of the active pore volume is filled. The lower X limit of this zone is that a t which the residual surface-sorbed water (approximating a bilayer) transmits very little of the alumina surface activity to the overlying liquid (Fubini et al., 1978). The state of sorbed water in zone 1 is H,O(,! = HzO(l, throughout; hence, A S *, = 0 throughout. Reading log p x and the corresponding X at frequent points along the curve of Figure 5 and entering these into eq 4, we obtained an experimental curve of AH*,versus A, which is AH*,= 0.347 - 0.01OOX + 2.700/X kcal/mol ( 5 ) Equation 5 , obtained by inspection, is an excellent fit to Figure 5, as will be shown presently. As it falls within our chosen framework of using only simple thermodynamic quantities, it is preferred here to the use of the modified Antoine equation proposed by Hacskaylo and LeVan (1985). Their equation could surely be as well fitted to Figure 5 but uses one additional parameter and loses an immediate and simple connection to eq 4. Zone m,0 < X 9 g/ 100 g. In this zone, water is sorbed as “solid”, as a fractional monolayer on the active alumina pore surfaces. The monolayer is full a t the upper limit of the zone, X = A, = 9 g/100 g (Rhone-Poulenc, 1982). This rough value of A, agrees well with the typical surface area (Kaiser Chemicals, 1985), with the assumption of the condensation process as nominally H,O,,, + >A1=0
-
>Al(OH), (Pearson and Rigge, 1974) and with the estimated average surface density of A1 atoms. Thermodynamic properties of HzO(,, in this zone must be obtained by equilibrium (X,T,p,) measurements, since the above structural description of H,O(,) is realized only very slowly at low temperatures (Borello et al., 1974). Della Gatta et al. (1975) implied this may require at least 1 2 h, even at 150 “C. Microcalorimetry (loc. cit.) is incapable of accurately measuring AH at equilibrium under these circumstances. Prior to the present work, we had no equilibrium data at X < 5 g/100 g and no pertinent entropy data at all. We first estimated A S *, from sorbed-water structural considerations, arriving very closely at 7.0 cal/(mol.K). We also made observations that suggest near-constancy of A S *, with X for activated aluminas in zone m, from examination of prior analogous work on crystallized transition aluminas (GBry et al., 1962; Borello et al., 1974; Della Gatta et al., 1975; Fubini et al., 1978). This fixed value of AS’*, with eq 4 and with the 294 K equilibrium data of Figure 5 in the range 5 g d X 6 9 g/100 g gave very acceptable AH*,values when compared with those given by Rhone-Poulenc (1982). This lent initial support to our entropy estimate. Next we measured a new matrix of equilibria comprising portions of isotherms at 294,329, and 363 K and designed to extend the gravimetric data to as low as we could accurately measure, namely X = 0.1 g/100 g. Recomputing log ph at 294 K from the data obtained at the higher temperatures and plotting this versus log X using all our data at X 6 9, we obtained an assuredly linear regression curve which is a Freundlich isotherm (G6ry et al., 1962): log p,,294 = 3.426 log X - 5.367
(6)
From this, the following was obtained by combining with eq 4:
AH*, = 7.12 - 2.001 In X kcal/mol
(7)
Equation 7 gave final and independent confirmation of the near-constancy of A S in this zone. Its coefficient, 2, which is the number of surface OH groups formed per HzO sorbed, could not have resulted from the data under any other assumption. This equation leads by algebraic manipulation to In (X/h,) = (H*, - H*,)/2 = -(e, - e), (8)
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 1029 I
1
FROM 294
ISOTHERM
FROM SMOOTHED HIGH-TEMPERATURE ISOTHERMS
A FROM COMPETITIVE ALUMINA “A“
A
I Am 1
0.6 w
I
z
15.
Figure 6. Full-range plot of A H
20.
t
10.
5.
WATER LOADING, A , G / 1 0 0 G I
I
I
25.
30.
35.
versus A: A-201 activated alumina.
Here is the mean molal excitation energy of active OHsorption sites produced by thermal regeneration of the desiccant, and t, is the value of e, at A,. The form of eq 8 has been seen previously in crystalline transition aluminas, both quantitatively by GBry et al. (1962) and qualitatively in nonequilibrium AH measurements (Borello et al., 1974;Della Gatta et al., 1975; Fubini et al., 1978). If the energy distribution of these active surface sites in A-201 alumina is governed by the Boltzmann law (with t in kcal/mol), In (X/X,) = -lOOO(eA - e,)/RT, (9) then the computed regeneration temperature T , is 503 K or 230 “C. The actual regeneration temperature used in preparing all of our specimens for equilibrium measurements was 250 “C. The small discrepancy can be explained by our laboratory conditions of postregeneration cooling prior to specimen weighing at room temperature. Thus, eq 7 for AH*,and a near-fixed AS*, = 7.0 were confirmed by every available inference. They should serve for all of zone m, including X values well below our lowest experimental measurement. Zone t, 9 g < X < 14 g/ 100 g. We postulated a statistical function for A S *, that would effect the transition from solid H 2 0 monolayer sorption (zone m) to liquid H 2 0capillary sorption (zone 1) in A-201 alumina. Our curvefitting equation for A S in this zone is
from which AH*,is readily obtained versus X from eq 4 and the data of Figure 5 in zone t. This joins AH*A smoothly to the other two zones. Overall Display of AH*,. Figure 6 is the AH*,plot thus obtained for A-201 alumina over the entire range from X = 0 to X = 40 g/100 g. Along the solid curve, points are included representing (a) our own 294 K isotherm data, (b) our own (smoothed) higher temperature isotherm data, and (c) the calorimetric data given by Rhone-Poulenc
(1982) (R-P) for a like alumina “A”. The solid curve comes solely from our own measurements; its general agreement with the R-P data is the first known independent confirmation of the latter. A-201 Alumina Desiccant Isosteres The isosteres of Figure 1 fall entirely in zone m. These were obtained from eq 4, with A S *,, = 7.0 and using eq 7 for AH*,. Any other isosteres, isotherms, or isobars can now be drawn for A-201 alumina, using eq 4 with the AH*, and A S *, values of zones m, t, and 1 as appropriate. Small departures from the X labels of Figure 1 are implied depending on the regeneration temperature (Borello et al., 1974),but 250 “C is representative of field practice. Figure 1 extrapolated to T-’ = 0 is the generic type of Figure 3A. It remains only to decide at what maximum AH*,to cut off the isostere set. As-manufactured activated aluminas contain roughly 4% combined H 2 0 (as OH groups) in the interior of the solid, and this water is in principle exchangeable with the surface (Pearson and Rigge, 1974). We estimated from the bulk thermodynamic properties (Carniglia, 1983) that this exchange is energetically favored for surface sites whose AH*Ais above about 15 kcal/mol. By eq 7, this corresponds to X r 0.02. That is where we have terminated the isostere set in Figure 1. Integrating AH*,dX from X = 0 to 1, using eq 7 alone, gives an integrated “net heat of adsorption” of 9.12 kcal/mol over this range. This same integration with a cutoff AH*Aof 15 gives 9.10 kcal/mol, sensible the same. Both are in excellent agreement with a n experimental estimate of roughly 9.5 kcal/mol given by PapBe et al. (1967). Summary and Conclusions Reliable thermodynamic descriptions of the equilibrium sorption of water vapor by solid desiccants are needed for engineered design and performance evaluation of regenerative systems using these materials, e.g., in drying and in refrigeration and heat-pump applications. A single thermodynamic equation describes the equilibrium between sorbed H 2 0 and H,O(,, for all stable solid
Ind. Eng. Chem. Res. 1989,28, 1030-1036
1030
desiccants, as a function of T and of the loading, X (g of HzO/lOO g), of the desiccant. Isosteres, plots of log pHlO versus 1 / T at selected X values, are revealing representations of this equation. The common thermodynamic characteristics of these equilibrium isosteres and their families are described, exemplified by activated aluminas and zeolites, respectively. A t low A, it appears that adjacent isosteres for the former are separated by differences in partial molal enthalpy of sorbed HzO, whereas partial molal entropy separates isosteres for the latter. The pertinent H 2 0 equilibrium has been exhaustively described for a representative commercial activated alumina, trade designation A-201. Experimental (X,p,T) measurements were made at selected T from 294 to 363 K and X from 0.1 to 41, coupled with appropriate analytical and confirming methods and with conservative extrapolation, deriving AH and A S of H20 between the sorbed and liquid states. In the sorbed-monolayer region, 0 < X < 9, A S appeared sensibly constant, while AH = a - b In X, giving Freundlich isotherms. Expressions for A S and AH were also developed for 9 < X < 41, giving a continuous thermodynamic description up to the full HzO capacity of the desiccant and relating this to the state of sorbed H20 in each of three specified ranges of A. The AH versus X curve agreed well with a prior set of calorimetric AH data for a very similar alumina. A family of equilibrium isosteres for A-201 alumina is presented, numerically representing the thermodynamic functions over the loading range 0.02 < X I 5.0. A rationale is described for cutting off the set at a minimum A = 0.02, where the net heat of adsorption is approximately 15 kcal/mol. These isosteres and their underlying thermodynamic quantities should serve the sorption equilibrium data needs of engineered systems using activated alumina
desiccants, in the absence of other competitively sorbed species. Registry No. H20, 7732-18-5; alumina, 1344-28-1.
Literature Cited Barry, H. M. Fixed-Bed Adsorption. Chem. Eng. 1960,67, 105-20. Borello, E.; Della Gatta, G.; Fubini, B.; Morterra, C.; Venturello, G. Surface Rehydration of Variously Dehydrated Eta-Alumina. J . Catal. 1974, 35, 1-10. Carniglia, S. C. Thermochemistry of the Aluminas and Aluminum 1983, 66, 495-500. Trihalides. J . Am. Ceram. SOC. Della Gatta, G.; Fubini, B.; Venturello, G. Heats of Different Surface Rehydration Processes for Various Aluminas. Conf. Znt. Thermodynam. Chim., 4th 1975, 7, 72-79. Fubini, B.; Della Gatta, G.; Venturello, G. Energetics of Adsorption in the Alumina-Water System: Influence of Temperature. J . Colloid Znterf. Sci. 1978, 64, 782-91. GBry, M.; Lenoir, J.; Eyraud, C. Chemisorption of Water on Alumina. Compt. Rend. 1962, 254, 128-9. Hacskaylo, J. J.; LeVan, M. D. Correlation of Adsorption Equilibrium Data Using a Modified Antoine Equation: A New Approach for Pore-Filling Models. Langmuir 1985, 1, 97-100. Kaiser Chemicals Fixed-Bed Dehydration with Activated Aluminas. Technical Service Bulletin 1985; LaRoche Chemicals Inc., Baton Rouge, LA. PapBe, D.; Maniere, S.; Bellier, A.; Jouanneault, F. Optimize Alumina Gas Drying Systems. Hydrocarbon Process. 1967, 46, 142-6. Pearson, R. M.; Rigge, R. J. Unpublished work, Kaiser Alum. & Chem. Corp., Pleasanton, CA, 1974. Rhone-Poulenc, Activated Alumina. Technical Document SC-MinF-82-02-03, 1982. Rhone-Poulenc, Paris. Sircar, S.; Gupta, R. A Semi-Empirical Adsorption Equation for Single Component Gas-Solid Equilibria. AZChE J . 1981, 27, 806-12. Received for review August 22, 1988 Revised manuscript received February 27, 1989 Accepted April 10, 1989
GENERAL RESEARCH Batch Extraction with Reaction: Phenolic Antioxidant Migration from Polyolefins to Water. 1. Theory T h o m a s P. G a n d e k , T. A l a n Hatton,* a n d R o b e r t
C.Reid
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Before food-packaging materials are approved for use, batch extraction tests are performed to determine whether harmful amounts of packaging components are extracted by the contents. In interpreting the test results, it is important to have a n adequate level of understanding of the parameters that affect migration rates. Two possible factors which are often ignored are the possibility of migrant degradation in the extracting phase and back-diffusion of degradation products into the packaging materials. This paper presents a theoretical analysis of the migration process including treatment of migrant diffusion from the packaging material t o a poorly stirred liquid extractant with a first-order degradation reaction in the liquid. T h e reaction products, which have different solubilities in each phase, are allowed t o diffuse back into the solid phase. T h e following paper in this issue uses these tools to analyze laboratory migration data which exhibit all of these features. Some components of plastic food-packaging wraps may migrate into the food. These "indirect food additives"
* Author t o whom correspondence should be addressed. 0888-5885/89/2628-1030$01.50/0
often consist of residual monomers and oligomers, plasticizers, antioxidants, UV absorbers, and antistatic and slip additives (Arthur D. Little, Inc., 198313; Crompton, 1979). Before approval of these wraps by regulatory agencies, 0 1989 American Chemical Society
