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Anal. Chem. 1983, 5 5 , 78-81
Recovery Factor for Extraction from a Solid, Extractant-Retaining Matrix David Emlyn Hughes Analyfical Chemistry Division, Norwich Eaton Pharmaceuticals, Inc., Box 19 1, Norwich, New York 138 15
The theory and calculation of the recovery factor for an extraction from a solid extractant-retaining sample are presented. The recovery factor Is seen to consist of two multiplicatlve factors, one factor for extraction efficlency and one for extract retention by the sample. Several examples are presented.
Although liquid-liquid extraction has attracted a great deal of theoretical modeling, solvent extraction from a solid medium has drawn less attention. Liquid-solid extraction has been used for centuries for the recovery of metals, sugars, salts, and medicinals (1). Recently, much of the interest in the field has been in empirical considerations in solvent/metal-containing compounds and refinements of Soxhlet extraction. Although liquid-liquid extraction is well understood, liquidsolid extraction theory is absent from the chemical literature. The theoretical considerations presented here do not concern themselves with why a particular extract-matrix partitioning has occurred. The theoretical problem is deriving an analytical expression for the recovery factor for a liquid extraction from a solid, solvent-retaining sample. The results differ substantially from liquid-liquid systems, since liquid-solid extraction systems frequently retain extract. When the volume of extractant is comparable to the volume of the solid matrix or when a "reversen extraction is performed on another liquid (extraction of an organic layer with an aqueous solution), some of the extract is often physically trapped in the medium. Both solid and liquid media are therefore able to retain extract. Multiple extraction often removes sufficient analyte such that the extracts may be combined and successfully assayed. A somewhat less defined situation exists when extraction yields insufficient extract to produce an acceptable recovery. If extract is associated with the medium in a fixed proportion, the association is reasonably constant from sample to sample. The concentration of the analyte may then be calculated. In what at f i s t appears to be the most analytically complex situation, some indeterminate amount of the analyte is physically trapped in the matrix. The precise volume of extract in the matrix prior to assay is dependent on the prior sample handling and may vary significantly from determination to determination. A familiar and consistent physical model is aqueous extraction from vegetable oil. Upon extraction, apparently intact droplets of extract (or extractant) appear within the vegetable oil layer. If extract is retained within vegetable oil, some unspecified amount of the analyte is still within the vegetable oil phase and is not available for assay. A mathematical model will now be presented for an extraction from a solid which retains a variable amount of extract. After the initial extraction, the extract trapped in the sample: (1)may or may not have the same concentration as the extractant; (2) may or may not be able to be extracted further. The following discussion is consistent with IUPAC recommended nomenclature and terminology (2).
THEORY If the sample contains ni mol of analyte in a volume V,,, and a volume V,, of a thermodynamical ideal extractant is added, the available volume of the extract is (V, + V , - V), where V is the volume occupied by the extract within the sample. If the solution is given sufficient time to equilibrate, the concentration OnilV may be such that
where Mext is the molarity of the solution external to the matrix. The molarity of the extract concentration within the sample may therefore be less than, greater than, or equal to that of the surrounding extract. In eq 3, the extract concentration is the same internally and externally to the sample. The internal and external extracts in this case have the same concentration and if the system is assumed to be isothermal, the expression (3) for the molal distribution constant (KD)A of the analyte A is
where yint= molal activity coefficient within the sample extract, Text = molal activity coefficient of the surrounding extract, 4 O e X t = partial molal free energy of the extract, and 40ht = partial molal free energy of the retained extract. Therefore
and
where mint= molality of the matrix extract and inext = molality of the extract. Henceforth, the less cumbersome KD will be used in place of (ITD)*since only one substance is being considered. The conclusion is that the physically obvious KD = 1 (since mext= mint and KD LZ mext/mint)is not an approximation but is rather thermodynamically exact (4, 5). KD = 1 allows the derivation of analytical expressions for the recovery of the analyte using two extraction models. It is clear that the KD = 1 assumption is simply a mathematical convenience and not a requirement of the present model. If an apparent (nonunity) value of KD is assumed, then the derivation is unchanged and the model may be extended to cases (1)and (2) above. The mathematical model is then completed by calculation of the recovery factor for the procedure. In the first extraction model procedure, sufficient extractant is supplied to perform a single, simple extraction such that the liquid recovered has a volume Vf. The molarity of the resulting solution is
0003-2700/83/0355-0078$01.50/00 1982 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 55, NO. 1, JANUARY 1983 79
(7)
l
.
o
where ni = moles of solute originally in the matrix, 0 = mole (decimal) fraction of ?itrapped in matrix after one extraction assuming mint = mesrt,and Vf = final volume of extract, in milliliters. In the second extraction procedure, Vf mL of extractant is added to the matrix. The molalities of the internal (sample) and external extracts are again equal by assumption mint = mext = 1o3ni/Vfp
(8)
where p = density of the extractant (g/mL). Converting to molarity
Mz
=
103~solmext -
lo3 + Mwmext
0.7
and substituting in me,, from (8)
1 2 Number of E x t r o c t l o n s , n
3
Figure 1. Extraction from an extractant-retaining sample with K, = 4: curve 1, On,lV < M& curves 2-5, 0 n , / V 2 Me,, and limiting R, Y
Note that although thie total volume of the sample and extract system is indeterminate, the volume of extract is Vp If an aliquot of each resulting solution is now assayed by a hypothetically perfect method (since we are interested in errors inherent in the extraction, not in the subsequent assay) the recovery factor
0.80,0.90, 0.95, and 1.0.
ion, and Setchenow (11, 12)), and generalized distribution coefficients (13-15).
EXAMPLES Examination of eq 16 reveals that the recovery factor consists of two multiplicative terms
RA = RretRext
(17)
where where M = 103ni/Vf,that is
and for a single extraction step in the first extraction procedure. If the retained extract is available for further extraction (7) = ((KDvf/n)
+v
>"
(13)
providing the volumle of extractant is divided by n and n extractions are performed. The recovery factor becomes vfPsol[l - [V/((Vf/n)KD Vfp + Mwn.i
RA = -
+ v)]"]
(14)
A slightly more general form would allow n extractions with volumes V1, V,, V,, .,.,such that n
c
vi
= Vf
i=l
and
Although the recovery equation is somewhat involved, further mathematical complications result if the system is not assumed to be isothermal. I t is clear that although .the physical model presented here lent direction to the mathematical model derivation, eq 16 could have been derived solely from conservation equations. The derivation could easily be extended to allow asslociation and dissociation reactions (8, 9), mutual dissolution, of the matrix and extracting solvent, side reactions (IO), salt effects (salting-in, salting-out, common
Rret is the recovery factor due to the retention of extract by the sample and Rextis the recovery factor for the extraction process. Example 1. In Figure 1,curve 1, the sample extract is more dilute than the surrounding extract, as in eq 1. Further extraction allows more analyte to be retained by the sample and the recovery factor drops asymptotically until Oni/ V = Mext Equations 2 and 3 have similar shapes and are represented by curves 3, 4, and 5, with respective limiting Rretvalues of 0.90, 0.95, and 1.0. Curve 2 represents that special case for eq 3 only when ViKD >> V and the extract is fully retained after the first extraction with an Rret= R E 0.80. The samples in Figure 1would not show significantly higher recovery factor values with more elaborate (and numerous) extraction schemes. Example 2. When the analyte is retained in a solid sample, at least two mechanisms may be involved. The intact extract may be trapped in the sample and the analyte may be irreversibly bound to the solid sample. The extent of these mechanisms may be determined by performing the following analyses: (a) single extraction on identical samples using NV, mL of extractant for N = 1, 2, 3, and 6; (b) a multiple extraction (n = 3) using Vl mL for each extraction; (c) assay of the seven samples. For the analysis of the single extraction data, recall eq 17 and notice that Rex. may be rearranged to
l/Rext = 1 + ( V / K d ( l / V f )
(20)
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 1, JANUARY 1983
Table I. Parameters for Extractant-Retaining Systems 1 / R (at V, )” slope/Vi, intercept 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
1.5.
( b ) I R R E V . BOUND EXTRACTANT RETAINING
(c,d,e)
a Calculated 0.0 5
1.00 0.996 0.982 0.960 0.934 0,900 0.862 0.818 0.772 0.720 0.660
from eq 22 for (Vf -- VI )/V, = 1, 2, and 3.
0.1
7Vf Flgure 2. Dependence of for a single extraction.
0 0.104 0.218 0.340 0.466 0.600 0.738 0.882 1.03 1.18 1.22
the recovery factor on the extraction volume
For eq 18, if the decimal fraction of analyte (1 - RirJ is irreversibly bound and if the concentration of trapped extractant is linearly dependent on VI, then
or
which approximates a straight line (f0.4%,1/R I 1.7) if l/Rmt is plotted vs. 1/ Vf and ( Vf - Vl)/ Vl 5 3. Hence both Red and Rret are linear if 1/R is plotted vs. 1/Vf for experimental extraction data Vf 5 3V,. In Figure 2, 1/R data (R = assay value/accepted value) vs. l/Vf data are plotted for Vl = 10 mL. By simply relabeling the axis, any VI, 2V1, and 3V1 scheme may be used. If the multiple extraction scheme improved the recovery factor significantly, the data for the single extraction scheme will fall on lines similar to (a) and (b) on Figure 1. If the points fall on line a, then the solid is being extracted as if it were a classical liquid. If the 1/R intercept is statistically larger than zero, some (1 - R at l/Vf = 0) is being irreversibly retained. For line b, 1/R = 1.2 (R = 0.833) or about 17% of the analyte is irreversibly retained. If an assay with a higher accuracy is required, the extractant system is to be abandoned. If the multiple extraction scheme is ineffective in improving the recovery factor, the experimental points for the single extraction will lie on a line similar to c in Figure 2. If extractant is retained in accordance with eq 3 and eq 22, then the slope and intercept of the line may be obtained from Table I and Ri, = assay value (N = G)/accepted value within + Me, (eq 2) may have an apparently anomalously high 1/R at l/Vf = lI3Vl due to curvature of the defining equation. Increasing VI for lines (c-e) efficiently increases the recovery factor when compared to classical line a. When an extractant-retaining system is encountered, failure of a multiple extraction scheme implies that a single (large VI) extraction may significantly improve the recovery. Although lines c, d, and e in Figure 2 have intercepts corresponding to recovery factors in excess of 1, the actual function defined in eq 22 is nonlinear and converges to R = 1 after N = 4. In summary, the procedure outlined here need not be done in detail; an examination of the effect of multiple extraction and single extraction in general terms would be sufficient for I
simple systems. Single extraction recovery data could be plotted as in Figure 2 and evidence for a significant intercept evaluated. If the multiple extraction appeared promising, plotting the data would indicate the presence or absence of a positive intercept and therefore irreversibly bound solute. If the multiple extraction does not greatly improve the recovery, plotting the data might reveal a statistically significant negative intercept and hence extractant retention. It may become apparent from the above examples that solid-liquid extraction systems become increasingly similar to classical liquid-liquid systems as the extractant to matrix ratio increases. For these near-classical systems, an unsatisfactorily low recovery factor may be improved by multiple extraction, providing an extractant with a suitable KD is chosen. For a solid-liquid extraction, multiple extraction may only slightly increase (or actually decrease) the recovery factor. These nonclassical cases are a strong indication that the solid sample is retaining some of analyte and that improved recovery may be a function of the initial extraction variables, principally, the mass of the analyte and the volume of the initial extraction. If adjustment of the initial extraction variables does not improve recovery, the analyte may be irreversibly bound to the matrix (either before or upon extraction) and a different separatory system must be employed.
GLOSSARY number of grams of analyte g molal activity coefficient of extract retained within Tint the sample molal activity coefficient of extract external to the Text sample WD)Aor molal distribution constant for species A KD molality of the extract external to the sample mext molality of the extract retained within the sample mint the molarity of an R = 1 extract, Le., 103ni/Vf M molarity of the extract in model 1 MI molarity of the extract in model 2 M2 general term for molarity of the extract external Mext to the sample molecular weight of the analyte Mw the number of extractions n the number of moles of analyte ni multiple of V1 for single extraction N decimal fraction of moles of analyte retained within 0 the sample standard partial molal free energy of the extract 4 O ext external to the sample standard partial molal free energy of the extract 4Oint retained within the sample density of the extractant P density of the extract Pa01 recovery factor for species A RA extraction recovery factor Rext extract retention recovery factor Rret irreversibly bound analyte recovery factor Rirr
Anal. Chem. 1983, 55, 81-88
R T V Vf Vi vo
gas constant, 8.314 J/(deg mol) Kelvin temperature volume of extract within the sample final volume of extract volume of extractant of the ith extraction total volume of extractant
LITERATURE CITED (1) Berg, Eugene W. “Physical and Chemical Methods of Separation”; McGraw-Hill: New York, 1963; p 73. (2) Irving, H. M. N. H. PuruAppl. Chem. 1970, 21(1), 111-113. (3) Morrison, George H.; Frelser, Henry “Solvent Extraction In Analytical Chemistry”; Why: New York, 1963; p 9. (4) Berg, Eugene W. “Physical and Chemical Methods of Separation”; McGraw-Hill: New York, 1963; pp 50-52. (5) Castellan, Gilbert W. “Physlcal Chemistry“: Addison-Wesley: Readlng, MA, 1964; pp 287-280.
81
(6) Shoemaker, Davld P.; Garland, Carl W.; Steinfeld, Jeffrey I. “Experiments in Physlcal Chemistry”; McGraw-Hill: New York, 1974; p 173. (7) Morrlson, George H.; Freiser. Henry “Solvent Extraction in Analytical Chemistry”; Wiley: ,,New York, 1963; pp 58-60. (8) Berg, Eugene W. Physical and Chemical Methods of Separation”; McGraw-HIII: New York, 1983; pp 7-15. (9) Morrison, George H. Anal. Chem. 1950, 22, 1388-1393. (10) Masterson, W. L.; Lee, T. P. J . Phys. Chem. 1970, 7 4 , 1776. (11) Ringbom, A. “Complexation in Analytical Chemistry”; Interscience: New York, 1963; pp 10-50. (12) Groenewald, Theo. Anal. Chem. 1971, 4 3 , 1678-1683. (13) Likussar, Werner; Bolt.?, D. F. Anal. Chem. 1971, 43, 1265-1272. (14) Likussar, Werner; Bolt.?, D. F. Anal. Chem. 1971, 43, 1273-1277. (15) Atklnson, G. F. Anal. Chem. 1972, 4 4 , 1098.
RECEIVED for review March 9, 1982. Accepted September 29, 1982.
Interpretation of Sets of Pyrolysis Mass Spectra by Discriminant Analysis and Graphical Rotation Wlllem Wlndlg’ Centraalbureau voor Schlnimelcultures, Oosterstraat 1, Baarn, The Netherlands, and FOM-Institute for Atomic and Molecular Physics, Krulslaan 407, 1098 SJ Amsterdam, The Netherlands
Johan Haverkamp and Piet
G. Klstemaker”
FOM-Institute for Atomic and Molecular Physics, Krulslaan 407, 1098 SJ Amsterdam, The Netherlands
Pyrolysis mass spectra of complex blologlcal samples can be interpreted for partlal chemical composttion, even when exact reference spectra of the components Involved are not avallable. The method is biased on a factor analysis method: prhclpal component analysls followed by dlscrlmlnant analysls and graphical rotation. Appllcations of the procedure are discussed for various sets of samples, viz., fungi (taxonomy), bacteria (detectlon of capsular polysaccharlde), and human bile (differentiationof choiellthlasls and cholecystltis patients and normals).
Pyrolysis mass spectrometry (Py-MS) is a fast method foh fingerprinting complex biological materials like macromolecules, cells, and microorganisms (I). However, chemical interpretation of pyrolysis mass spectra of complex samples is hampered as: the spectra seldom exhibit single masses specific for a particular compouiid (so-called “pure masses” (2)),accurate reference spectra of pure components are usually not available or even not obtainable, intermolecular reactions during pyrolysis can occur so that the pyrolysis spectra are not additive. Consequently automated mixture analysh techniques like library search procedures are not easily applicable to pyrolysis mass spectra. In Py-MS studies the main analytical goal is often to determine the nature and extent of differences within a set of‘ related samples. Factor analysis of spectral data offers a powerful method to describe the differences efficiently. It has been shown in various applications on spectral data sets that by an adequate transformation of the factors, mixture comI P r e s e n t address: Biornaterials P r o f i l i n g Centre, 391 S o u t h Chipeta Way, Research Park, Salt L a k e City, UT 84108. 0003-2700/63/0355-0081$01 S O / O
ponents can be identified and quantified (2-4). In particular target rotation methods, as developed by Malinowski, have been applied successfully ( 4 ) . However such an evaluation relies on a number of assumptions with regard to the structure of the dataset, i.e., additivity of spectra, availability of accurate reference spectra, and sometimes the presence of “pure masses”. As outlined before these conditions are not fullfilled in Py-MS data sets and, consequently, other-less elegant-factor transformation procedures have to be applied. Recently we described a combination of factor analysis and graphical rotation procedures, for chemical evaluation of small sets of Py-MS spectra (5-7). The graphical rotation method consists of a systematic-stepwise-rotationof principal factors. Visual examination of a series of transformed factors can lead to recognition of mixture components. It will be clear that this method does not provide the speed and rigidness of automated procedures and that the results depend on the experience of the operator. Nevertheless, in practice the graphical rotation procedure has been proven to work out satisfactory with respect to both the time afforded and quality of the results. One advantage of this procedure over other methods is that during examination of a series of transformed factors the experienced operator can be stimulated to make reasonable guesses about the nature of a mixture component; this by the associative ability of the human mind. In this way components can be identified of which no accurate reference spectra were available at that time. Although the factor analysis and graphical rotation method were successful for interpretation of small data sets, the technique did not work out satisfactory for larger data sets including multiple replicated spectra. The problems were %fold: the large number of factors obtained complicated the rotation procedure and the single factors did not discriminate well between the groups of replicated spectra. This was due to the fact that factor analysis does not describe optimally 0 1982 American Chemical Society
