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a Department of Chemical Engineering, Environmental Engineering Program, Yale University, 9 Hillhouse Avenue, P.O. Box 208286, New Haven, CT 06520-8286
b Department of Soil and Water, Connecticut Agricultural Experiment Station, 123 Huntington Street, P.O. Box 1006, New Haven, CT 06504
* Corresponding author (joseph.pignatello{at}po.state.ct.us)
Received for publication August 5, 2004.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONDERIVATION OF THE THERMODYNAMIC...RESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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Abbreviations: AS, Amherst soil • BZL, Beulah-Zap lignite • DCB, 1,4-dichlorobenzene • PP, Pahokee peat • TII, Thermodynamic Index of Irreversibility
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONDERIVATION OF THE THERMODYNAMIC...RESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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Possible explanations for true hysteresis include (i) the formation of metastable states of adsorbate in fixed mesopores (i.e., capillary condensation hysteresis) (Sing et al., 1985; Burgess et al., 1989; Liu et al., 1993; Rouquerol et al., 1999; Neimark et al., 2000; Aharoni, 2002), and (ii) irreversible deformation of the sorbent by the sorbate (sometimes called "low-pressure hysteresis") (Bailey et al., 1971). In both cases, sorption is "irreversible" (i.e., the microscopic pathways for uptake and release are different). The pore deformation mechanism has precedence in the sorption of gases to glassy synthetic polymers (Kamiya et al., 1989; Bourbon et al., 1990; Stamatialis et al., 1997; Kamiya et al., 1998). In brief, the unrelaxed free volume of the sorbent is postulated to increase during the sorption step as a result of pore ("hole") creation and dilation of existing holes. The free volume is the volume not occupied by macromolecules. The term "unrelaxed" means the excess free volume resulting from structural metastability of the solid compared with the volume of the solid at true equilibrium. Upon desorption, the sorbent does not relax freely due to the structural rigidity of the macromolecules (i.e., relaxation is kinetically hindered). Since the free volume of the solid is greater during desorption than sorption, the solid exhibits increased affinity for the solute during desorption. Recent studies (Tvardovski et al., 1997; Xia and Pignatello, 2001; Lu and Pignatello, 2002, 2004a, 2004b) support irreversible deformation as the cause of hysteresis in flexible solids, such as natural organic matter and organoclays.
Pore deformation during the sorption–desorption cycle may also result in a fraction of sorbate transferred to sites in the solid where free exchange with molecules in the bulk fluid phase is no longer possible (Weber et al., 2002; Braida et al., 2003; Kan et al., 1997). The premise of the entrapment mechanism is that sorption at a relatively high concentration leads to a swollen (pore-opened) physical state of the solid that then collapses around some of the sorbate molecules when the external concentration is abruptly lowered. The entrapped fraction of sorbate does not reequilibrate with the decreased solute concentrations achieved during subsequent desorption steps, resulting in the phenomenon of hysteresis.
Several empirical indices for quantifying hysteresis in soils exist. These indices can be subdivided into groups based on one of the following (see Table 1 for details and citations): (i) sorbed concentration q (mol kg–1); (ii) the exponent of the Freundlich equation, N; (iii) the distribution coefficient (i.e., ratio of the solid-phase to the liquid-phase concentration), Kd (L kg–1); (iv) the area in-between the sorption and desorption branches of the isotherm; or (v) the slope of the desorption isotherm in relation to the slope of the sorption isotherm. However, these indices are unsatisfactory for one or more of the following reasons: they (i) rely on a specific isotherm model (most commonly the Freundlich); (ii) depend arbitrarily on the dilution ratio used in constructing the desorption branch; (iii) are applicable only to single step, but not multistep desorption isotherms; (iv) make invalid assumptions, such as linearity between a sorption point and its corresponding desorption point; or (v) are thermodynamically flawed because they are based on "desorption isotherms" constructed by connecting single-step dilution points originating from different points along the sorption isotherm.
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This paper comprises three primary objectives. The first objective is to derive a measure of hysteresis based on fundamental thermodynamics (the Thermodynamic Index of Irreversibility, TII). This index is applicable to sorption from the solution or gaseous state and only requires data on concentration changes occurring in the fluid phase. The second objective is to demonstrate practical aspects of obtaining TII values. For this objective, (i) an equilibrium expression for hysteretic desorption isotherms is derived on the basis of TII; (ii) TII is applied to computationally generated data sets that were selected to represent characteristic sorption–desorption systems; and (iii) a sensitivity analysis on TII with respect to measurement errors is performed. Based on the latter, general recommendations on how to best perform aqueous sorption–desorption experiments are given. Finally, the third objective is to apply the TII to sorption of 1,4-dichlorobenzene (DCB) to three natural sorbents.
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DERIVATION OF THE THERMODYNAMIC INDEX OF IRREVERSIBILITY |
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TOPABSTRACTINTRODUCTIONDERIVATION OF THE THERMODYNAMIC...RESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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 | [1] |
aq is the solute activity coefficient, Caq (mol L–1) is the solute concentration, and 
aq (L mol–1) is the molar volume of water.
If gas–solid sorption is examined, the appropriate equation is:
 | [2] |
Equations [1] and [2] require no assumptions about the properties of the sorbent or the "activity coefficient" of sorbate in the sorbent. Equations [1] and [2] also hold true for metastable states of the sorbent that are persistent over the time frame of the experiment. However, they require sorbate to be in free exchange with chemical in the fluid state, such that the fluid-phase chemical potential accurately reflects the sorbed state chemical potential. In the following, we focus for the sake of brevity on aqueous–solid systems, where aqueous and sorbed-phase concentrations are symbolized by C and q, respectively.
Consider a single solute in an aqueous bath of infinite volume at a concentration of CS (mol L–1). This bath is equilibrated with three identical sorbent particles reaching a final sorbed concentration of qS (mol kg–1) (Fig. 1)
. Desorption is initiated by separately taking the particles out of the sorption bath and placing them into three different infinite desorption baths. The four systems are chosen to represent different states of the sorption–desorption cycle: (i) State S (C = CS, q = qS) corresponds to the experimental sorption point at which desorption is initiated; (ii) State D (C = CD, q = qD) corresponds to the experimental desorption point, which is displaced from its expected position on the sorption branch (State 
) due to the irreversibility of sorption (irreversibility is symbolized by the distorted shape of the particle in Fig. 1); (iii) State 
(C = C, q = q = qD) is the hypothetical reversible desorption state corresponding to the same sorbed concentration as in State D; and (iv) State (C = C = CD, q = q) is the hypothetical reversible desorption state corresponding to the same partial molar free energy as State D (µ = µD).
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and D is:
 | [3] |
The upper limit loss of molar free energy due to irreversible effects during the sorption–desorption cycle is found by letting State approach State S (i.e., qD 
qS). The difference in partial molar free energy between States S and is then:
 | [4] |
Equation [4] is an upper limit because if State reaches State S, the equilibrium assumptions of Eq. [1] and [2] are violated; that is, if no molecules desorbed, the affinity of sorbate for sorbent is infinite. Physically, this would correspond to sorbate entrapment, and the solution phase chemical potential would no longer represent the sorbed phase chemical potential.
The TII is defined as the ratio of the observed to the upper limit loss of free energy due to irreversibility, which is given by:
 | [5] |
 | [6] |
The index is 0 for completely reversible systems and approaches 1 as the process tends toward complete irreversibility.
Equations [5] and [6] show that the degree of hysteresis is logarithmically rather than linearly dependent on concentration and pressure. Calculation of TII is straightforward: CS and CD are experimentally determined values and C is easily computed given a model fit to data on the sorption branch near qD = q. Figure 1 shows that C is obtained by projecting qD onto the sorption branch; that is, C = Fct–1sorb
, where Fct–1sorb represents the inverse function describing the sorption branch of the isotherm.
In real systems it is possible that some sorbate molecules become entrapped while the remaining undergo irreversible sorption still in thermodynamic contact with the aqueous phase. In that case, the entrapped sorbed concentration (qentr) would have to be independently determined. If this is accomplished, then C or P in the above equations are replaced by C' or P', which are equal to Fct–1sorb(qD – qentr).
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONDERIVATION OF THE THERMODYNAMIC...RESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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, a desorption isotherm, qDdesorb = Fctdesorb
, that allows for irreversibility by incorporation of TII, can be derived by substituting C = Fct–1sorb
into Eq. [5] and subsequently rearranging for qDdesorb:
 | [7] |
Application of Eq. [7] to the Freundlich and Langmuir models gives, respectively:
 | [8] |
 | [9] |
Equations [8] and [9] are general expressions, as no assumptions are made on the dependency of TII on the desorption concentration CD. Note that for a constant TII along the desorption branch, Eq. [8] is of the true Freundlich form, whereas the iso-TII desorption branch for a Langmuir forward isotherm is of Langmuir–Freundlich form. Although Eq. [8] or [9] are not necessarily the ones that will be observed, they provide a basis for incorporating hysteresis into equilibrium expressions in contaminant fate models.
Assuming constant TII in Eq. [8], and given that qdesorb(CS) = qsorb(CS) and qdesorb(CD) = qsorb(C), Eq. [8] can be simplified to:
 | [10] |
Interestingly, Eq. [10] is related to Eq. [15] and [16] in Table 1, which are based on the Freundlich exponent N (i.e., TII = 1 – HI and TII = 1 – HI/100 for Eq. [15] and [16], respectively, where HI represents previously proposed hysteresis indices). Therefore, these empirical indices happen to have thermodynamic justification. However, unlike them, the TII does not rely on a specific sorption–desorption model and simplifying assumptions such as constant degree of hysteresis during the desorption.
Figure 2 shows the results of applying TII to data sets representing linear desorption isotherms (Fig. 2a) and single desorption points originating from different CS (Fig. 2b). Two important points can be made. First (Fig. 2a), the TII for points along a linear desorption branch increases with the degree of dilution, converging to 1 as dilution approaches infinity (CD approaches zero). This is true whether the sorption branch is linear or not. Intersection of the desorption branch with the ordinate signifies the formation of an entrapped fraction. Second (Fig. 2b), the TII depends on the sorption point at which desorption was initiated. Therefore, quantification of irreversibility requires information of the preexposure history. This is consistent with Everett (1954) who shows that systems that reach the same point on the sorbed vs. solution concentration graph through hysteresis, but by different pathways, are not necessarily in the same state and may behave differently on further changes in the experimental conditions.
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When desorption is initiated by replacing a fraction of the total solution with solute-free solution, the mass-balance expression using the Freundlich equation is given by:
 | [11] |
Each single sensitivity analysis calculation was performed in the following manner. First, input parameter combinations were chosen to represent different model scenarios. Second, the corresponding degree of irreversibility TII was determined by solving Eq. [11] using a generalized reduced gradient nonlinear optimization algorithm (Solver in MS Excel). Third, preset values of CD were varied in the range –5 to +5% to reflect experimental uncertainties in the solute concentrations. Finally, a new index of irreversibility TII' was calculated and plotted against percent measurement error in CD. Underestimating CD overestimates qD and causes TII' to be greater than TII, whereas overestimating CD causes the opposite. Calculated values of TII' can therefore be larger than unity or smaller than zero in some scenarios.
Assuming constant V and M and accurately determined CS for State S, calculated TII' values are dependent on (i) the true TII, (ii) the error of CD, (iii) the fraction of solution replaced, v/V, and (iv) the fraction of total mass sorbed at State S, fSsorb. We did not separately study the effect of errors in CS on TII, because this concentration is obtained by performing a mass balance using a model fit of the isotherm around State S. Therefore, it is the quality of the isotherm fit rather than the measurement error in CS that is important.
The first scenario (Fig. 3a) represents a model system with Nsorb = 0.8 and v/V = 0.9. For a constant M, solution volumes V were varied such that fSsorb was 0.1, 0.5, or 0.9.
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Figure 3a further illustrates the effect of varying fSsorb: for TII > 0.75, the effect of fSsorb is only minor but becomes more pronounced as TII declines. The difference between TII and TII' at a given error in CD decreases in the order fSsorb = 0.9 > fSsorb = 0.1 > fSsorb = 0.5. This trend can be explained as follows (Fig. 3b). At high fSsorb, only a small fraction of total mass is removed when the solution is replaced to initiate desorption, so CD ends up being close in magnitude to both CS and C, especially as TII decreases. Thus, errors in CD lead to large errors in TII, as can be seen by inspecting Eq. [5]. At low fSsorb, the fraction of molecules sorbed at the desorption point will be correspondingly low. Since qD is calculated on the basis of CD, uncertainty in CD introduces a large uncertainty in qD, and, hence, in C. Figure 3c plots the absolute error in TII over fSsorb for a measurement error in CD of 5%. Clearly, the sensitivity of TII to a given error in CD is lowest in the range 0.3 < fSsorb < 0.6. This range was found to be independent of Nsorb (not shown).
In this model system (Fig. 3d), Nsorb and fSsorb are kept constant at 0.8 and 0.5, respectively, and v/V is varied: 0.9 (dash), 0.5 (solid), and 0.3 (dash dot dot). Figure 3d clearly illustrates decreasing sensitivity of TII with v/V. This trend, again, results from decreasing absolute errors in CD with v/V. For systems with v/V < 0.9 and fSsorb < 0.3 or fSsorb > 0.6, the TII becomes highly sensitive to errors in CD even for highly irreversible systems (i.e., TII 
0.75) (simulations not shown).
The following recommendations can therefore be made. First, before setting up a sorption–desorption experiment, the overall measurement error in C including sampling, extraction, and analytical errors should be independently determined in sorbent-free systems and minimized. Second, for any point where desorption is initiated, the percentage of total molecules sorbed should preferably be between 30 and 60%. If the isotherm is nonlinear attaining this range may require progressively adjusting the sorbent-to-solution ratio. Third, the dilution ratio, v/V, should be as high as possible.
Application of the Thermodynamic Index of Irreversibility to Experimental Data
Sorption–desorption experiments were conducted for the model compound 1,4-dichlorobenzene (DCB) on three natural sorbents: Amherst soil (AS) (Yuan and Xing, 2001), Pahokee peat (PP) (reference soil of the International Humic Substance Society), and Beulah-Zap lignite (BZL) (reference low-rank coal from the Argonne Coal Sample Program, Argonne National Laboratory). The latter was chosen to represent "hard" natural organic matter. The DCB and [U-14C] DCB (7.36 x 1011 Bq mol–1, 99%+ radiolabel purity) were purchased from Sigma-Aldrich (St. Louis, MO). For all experiments, the background solution was 0.01 M CaCl2 containing 200 mg L–1 NaN3 to inhibit microbial activity. Sorption–desorption experiments were performed in 64-mL Teflon-lined septum screw-cap vials placed on a rotary shaker at 20 ± 1°C for sorption and desorption equilibration periods of 35 d (PP and BZL) or 14 d (AS). Desorption was initiated by centrifuging at 1800 rpm for 20 min and then replacing more than 80% of the clear supernatant with fresh background solution. The "colloids effect" was tested for by constructing separate sorption isotherms on all solids at different particle concentrations Cp (kgsorbent L–1): 0.03, 0.01, 0.0041, and 0.00015 for PP (all 14-d equilibration); 0.003, 0.001, and 0.0003 for BZL (all 21-d equilibration); and 0.03, 0.01, and 0.003 for AS. The liquid phase was extracted with hexanes (4:1 volume ratio) containing 1,3-dibromopropane as internal standard and analyzed by gas chromatography with electron-capture detection or liquid scintillation counting (Tricarb 2900; Packard Bioscience, Meriden, CT). (The analytical technique used is indicated in the relevant figure legends). For PP and BZL, solid-to-solution ratios were adjusted to attain 40 to 60% uptake during the sorption step. Sorbed concentrations were calculated by mass balance. Sorption and desorption kinetics of DCB on PP and BZL were independently studied using 160-mL glass vials capped with Mininert valves (Pierce Biotechnology, Rockford, IL). Additionally, vials containing only background solution were run as controls to quantify solute loss during the sorption and desorption period as well as the dilution step itself.
Sorption–desorption isotherms of DCB in PP, BZL, and AS are shown in Fig. 4a, 4b, and 4c , respectively. The corresponding Freundlich parameters for the sorption branch are given in the legend. The TII was calculated separately for each replicate sorption–desorption data pair and the average value assigned to CD (Fig. 4d). (Assigning TII to CD is arbitrary and does not imply that each CD has a unique value of TII.)
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Past studies of sorption hysteresis have often been plagued by artifacts. The most common artifacts are: (i) nonattainment of diffusive equilibrium; (ii) mass loss due to volatilization or reaction with vessel components; (iii) analyte degradation; and (iv) adsorption to an "invisible" third phase such as colloids (Gschwend and Wu, 1985; Schrap et al., 1995; Huang et al., 1998) whose concentration changes in the dilution step.
Diffusive nonequilibrium was addressed as follows. In Fig. 5a , isotherms of DCB in PP and BZL constructed after 35 d of equilibration (replotted from Fig. 4) superimpose on the corresponding isotherm constructed after a shorter time, 14 d for PP and 21 d for BZL. This indicates that 35 d was sufficiently long for DCB to diffuse to all sorption sites in PP and BZL during the sorption step. Figure 5b shows representative desorption rate profiles for DCB in PP and BZL. The ordinate is the ratio of measured Ct to the hypothetical value, C*, if all DCB molecules ended up in solution after the desorption step. For both solids sorbate release appears to be complete in a few days, certainly after 35 d. (Note that the release is plotted over the square root of time.) Similar results were obtained at a higher concentration for PP and at a lower concentration for BZL (data not shown). Thus, desorption periods of 35 d for PP and BZL were sufficiently long to enable us to rule out diffusive nonequilibrium during the desorption step as an artificial cause of the observed hysteresis.
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Vessel losses during sample equilibration were determined in control vials containing no solid that were equilibrated for 35 d (6 replicates) or 70 d (6 replicates). Mass recoveries were 98.5 ± 1.9% and 98.1 ± 3.0%, respectively. These recoveries were used to correct total DCB mass spiked to the bottle.
For all three solids, Freundlich model parameters obtained by fitting the separate isotherms constructed at different Cp were statistically indistinguishable (data not shown). Furthermore, the isotherms at all Cp on both PP and BZL were found to overlap the respective isotherms obtained after 35 d of equilibration (results for the lowest Cp for PP and BZL are in Fig. 5b); consequently, a "colloids effect" is ruled out.
We conclude that the observed hysteresis of DCB to PP and BZL is true; that is, it results from thermodynamically nonreversible processes during the sorption–desorption cycle. Reversible sorption of DCB to AS further supports the finding that hysteresis on PP and BZL was not artificial, as the same experimental technique was employed for all three solids.
As mentioned in the introductory paragraphs, a possible explanation for true hysteresis in deformable solids is irreversible pore deformation during the sorption–desorption cycle, which may or may not lead to the formation of an entrapped fraction of sorbate. An isotope exchange technique is currently employed in our lab to further investigate the mechanistic causes of DCB sorption irreversibility on PP and BZL (results to be published separately).
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONDERIVATION OF THE THERMODYNAMIC...RESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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APPENDIX |
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TOPABSTRACTINTRODUCTIONDERIVATION OF THE THERMODYNAMIC...RESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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C, equilibrium aqueous concentration of sorbate (g L–1)
Cp, particle concentration (kgsorbent L–1)
D, experimental desorption point with C = CD and q = qD
desorb, subscript indicative of desorption branch of the isotherm
Fct, functional relationship between q and C
, hypothetical reversible desorption state with C = C = CD and q = q
g, subscript indicative of the gas phase
, activity coefficient of solute or sorbate
, hypothetical reversible desorption state with C = C and q = q = qD
fSsorb, fraction of total sorbate sorbed at State S
Kd, distribution coefficient (L kg–1)
KF, Freundlich affinity coefficient [g(1–N) kg–1 LN]
kL, Langmuir affinity coefficient (L g–1)
µ, partial molar free energy of the sorbate in a given phase (J mol–1)
µ0, pure organic liquid reference potential (J mol–1)
M, sorbent mass (kg)
N, Freundlich exponent
P, partial pressure of the compound at equilibrium
Q, Langmuir sorption capacity of the sorbent (g kg–1)
q, equilibrium sorbed concentration (g kg–1)
R, universal gas constant (J mol–1 K–1)
S, sorption state from which desorption is initiated with C = CS and q = qS
solid, subscript indicative of the sorbent phase
sorb, subscript indicative of sorption branch of the isotherm
T, absolute temperature (K)
v, volume of solution replaced to initiate desorption (L)
V, total volume of solution in the system (L)
aq, molar volume of water (L mol–1)
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ACKNOWLEDGMENTS |
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REFERENCES |
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TOPABSTRACTINTRODUCTIONDERIVATION OF THE THERMODYNAMIC...RESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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, squares 
, and circles 
. (c) Absolute error in the TII, |TII – TII'|, over fSsorb for Nsorb = 0.8 and TII of 0.00 (solid), 0.25 (long dash), 0.50 (short dash), and 0.75 (dash dot dot). The fraction of solution displaced to initiate desorption is v/V = 0.9. (d) Calculated values of TII' for v/V ratios of 0.9 (dash), 0.5 (solid), and 0.3 (dash dot dot). The terms Nsorb and fSsorb are set to 0.8 and 0.5, respectively. The term v (L) is the volume removed to initiate desorption and V (L) is the total volume of solution.
