2410
Anal. Chem. 1982, 5 4 , 2410-2421
Adsorption Isotherm at the Liquid-Solid Interface and the Interpretation of Chromatographic Data Ngoc
Le Ha, J h o s
UngvBrai, and Ervin
sz. KovBts*
Laboratoire de Chimie Technique de I’Ecole Pol’echnique
Fidirale de Lausanne, 10 15 Lausanne, Switzerland
A method Is presented for the chromatographlc determlnatlon of adsorption Isotherms at the llquld-solid Interface. It Is based on the lnterpretatlon of the retentlon data of labeled compounds and of concentratlon perturbations of the eluent. The examples glven are binary mixtures In contact wlth lowenergy surfaces, typical In reversed-phase chromatography. The results fully support the theory of adsorptlon chromatography descrlbed In a recent paper. The notion of the holdup volume Is clarlfled and Its experlmental determlnatlon Is dlscussed. Models are presented for the holdup volume and for a “surface phase” accommodating adsorbed materlal at the Ilquld-solid Interface.
Contrary to the gas-solid adsorption experiment, adsorption at liquid-solid interfaces can only be deduced from composition changes in a liquid mixture before and after contact with the solid. If such a change is not observed as in the case of a pure liquid, there is no way of determining the surface concentration of that part of the liquid which is adsorbed. Nevertheless, it is known from calorimetric measurements (heat of immersion) that adsorption did take place. In liquid mixtures the effect of adsorption can be described following Gibbs’ procedure (1-4). A hypothetical dividing plane is placed parallel to the interface and it is supposed that the liquid has homogeneous composition up to this plane. Zero volume but finite material content is attributed to this strictly bidimensional “surface phase”. The position of the Gibbs dividing plane is determined by a convention. Usually an internal convention is applied meaning a convention which refers to the adsorption process itself. The position of the Gibbs dividing plane determines the volume and material content of the hypothetically homogeneous liquid (4). The adsorbed material is now given as the difference between the total material content in the system and that of the homogeneous liquid. Gibbs’ description of the adsorption equilibrium is abstract and has nothing to do with the physical reality in the vicinity of the liquidaolid interface; however, it summarizes in precise language all information about the adsorption of components relative to each other. Obviously, if a model is proposed for a chromatographic process involving adsorption there can be no contradiction between model and Gibbs’ description of the adsorption. In this paper, chromatographic determination and the method of presentation of adsorption isotherms will be described in binary liquid systems important in reversed-phase chromatography. The method is based on the evaluation of the retention data of labeled components of the eluent (4-7). The necessary equations and the underlying mathematics are described in a recent paper where exact relationships were derived for binary liquid mixtures (4). It will be demonstrated that the theory presented for the chromatographic process in this reference is in accordance with experiment. The exact meaning of “holdup” volume will also be clarified (6-13). It will be shown, fially, by one example, that models of a surface 0003-2700/82/0354-2410$01.25/0
phase can be easily developed from the abstract isotherm. In this way there will be no contradiction between model and the underlying thermodynamics.
THEORY The adsorption isotherm at the liquid-solid interface was derived by injecting labeled, deuterated components A* and B* into a mixture of nonlabeled compounds A and B used as eluent in a liquid-solid chromatographic experiment. The binary mixture was composed of compounds often used in reversed-phase chromatography: acetonitrile (AN), tetrahydrofuran (THF), and water (H,O). The necessary information about partial volumes in these systems was published recently (14). Two stationary phases were applied. The f i s t was a silicon
m DMB
TD
dioxide covered by a dense layer of (3,3-dimethylbutyl)dimethylsiloxy groups. This substituent doubly protects the surface. The maximum coverage of l? = 3.9 pmol m-2 is determined by the van der Waals space requirement of the ”methylen”-dimethylsiloxy-part, the base of the substituent. The size of the “second stage“, the tert-butyl group is such that it forms at this coverage a slightly less dense second protective layer. This “biplane” shielding covers completely the underlying unreacted silanols a t the silicon dioxide as suggested on the basis of wetting experiments (15). The second surface examined was a silicon dioxide preparation modified by a dense layer of tetradecyldimethylsiloxy groups with r = 4.1 pmol m-2 as an example of a long chain covered surface. The choice of the C14-substituentinstead of the widely used CI8-coveredsilica is justified by calorimetric measurements, as yet unpublished. The C18-covered surface shows a phase transition at about 15 f 10 “C; thus the surface is not in a well-defined state at the temperatures applied in our experiment (20 and 40 “C) (15,16). The necessary expressions for calculating points on the adsorption isotherms are taken from ref 4. The surface concentration of A with reference of the “B-is-Not-Adsorbed” (BNA) convention is given by where rA/BNA is the surface concentration of A applying the BNA convention expressed in number of moles per unit surface area and x , , ~ ”is the molar fraction of component A 0 1982 American Chemlcal Society
ANALYTICAL CHEMISTRY, VOL. 54, NO. 14, DECEMBER 1982
in the eluent. The superscript refers to eluent concentration at the beginning of the chromatographic process (t = 0). The symbol VR is the retention volume, S is the surface area of contact, and upo is the mean molar volume of the eluent
upo -=
+ uBxpqB0
(2)
VAX~,AO
a t the composition x,bAo,The asterisk refers to labeled components of the element, A* or B*.The necessary information about partial molar volumes, uI (mL mol-*),and partial specific volumes, OI (mL g-l), are found in ref 14, Isotherms in terms of the m-p-II language (mass, mass fraction, and mass adsorbed per unit surface area) and that of the v-4-q language (volume, volume fraction, and volume adsorbed per unit surface area) are simple to calculate from the surface concentration given in the n-x-r language if this convention is applied. By definition r B = 0, consequently also IIB = 9~= 0 and so nA/BNA
= M A T A J B N A and
*A/BNA
=
vArA/BNA
=
(VIt,A*
-
VR,B*)Pp,AoPp,Bo / s o p o
(4a)
=
=
(\'R,A*
- VR,B*)$p,Aodp,Bo/S
(4c)
where surface concentrutions and conventions are coherent, Le., expressed in the same units. T h e holdup volume referring to the BNA convention is given by j'p/BNA
=
VR,B*
(5)
For holdup volumes referring to NA conventions (nothingis-adsorbed), the necessary relationships are given in eq 6a-c. Vp/mNA
= I)p,Ao VR,A* + PpJ3O VR,B*
(64
Vp/nNA
= zp,Ao VR,A* + xp,Elo VR,B*
0%)
*B/nNA
=
(9)
rB/nNAvB
= -rB/nNA
rA/nNA
(10)
Combination of eq 8, 9, and 10 gives *A/vNA
=
rA/nNA(@p,BoUA+
@p,AoUB)
(11)
By definition $p,i
= Vixp,i/Vp
(12)
Substitution of eq 12 in eq 11and numerous rearrangements of the result gives the desired relationship *A/vNA
=
(13)
rA/nNA(vAUB/vp?
A second example is calculated by using the definition 4&,i = fiiP,,i/fip
(14)
Analogous derivation gives the relationship
= nAJmNA(oAcB/a>)
*A/vNA
(15)
Equating eq 13 and 14 gives after rearrangement a third example nA/mNA
*A/vNA
and
rA/nNAUA
An incoherent surface concentration does not refer to a NA convention formulated in the same units as the surface concentration. By use of coherent concentrations the total amount adsorbed is zero by convention and so
(3)
where M is the molecujlar mass. The adsorption isotherm of A with reference to the "nothing-is-adsorbed"N A convention is different if it is stated in the m-p-II, n-x-r, or ithe v-4-9 language. Therefore, surface concentrations referring to the "A, nNA, or the vNA convention will be different. They can be calculated from chromatographic data as follows: nA/mNA
*A/nNA
2411
=
rA/nNA(MAMB/Mp")
(16)
where M is for the molecular mass. Relationships between holdup volumes referring to different languages and conventions can also be derived. As a first example, combination of eq 6a and 6c with eq 12 gives after a few rearrangements Vp/nNA
=
Vp/vNA
+ s ( O B - EA) [(VR,A* - VR,B*)Pp,AoPp,Bo / s f i p O 1 (17)
Comparison with eq 4a shows that the expression in brackets is the absorption isotherm ~ A / ~ NsoAthat Vp/mNA
=
Vp/vNA
+ s(fiB
- DA)nA/mNA
(l8)
Analogous derivation gives by using eq 6
Relationships betwetm surface concentrations in different units referring to different conventions permit their interconversion; obviously they are not independent of each other, in every set of isothermri complete and equivalent information is summarized about the adsorption equilibrium. Actually, the necessary link between isotherms is given by partial molar volumes, vi, or by partial specific volumes, Bi. A first relationship is derived by dividing eq 1by 4b. After rearrangement, eq 7 gives the relationship between the surface conrA/nMA = Xp,BorA/BNA (7) centration referring to tlhe nNA and that referring to the BNA convention. In order to find relationships between concentrations referring to different NA conventions let us calculate as an example the isotherm \kAlvNAfrom I ' A / ~ A . As a general rule, the surface concentration \k*lvNAcan be calculated by using eq 8 if the surface concentration is known, in any *A/vNA
=
@jr,Bo*A/CX
- $p,Ao*B/CX
(8)
well-defined convention, CX. For the first example let us choose the nNA convention as the well-defined "convention X", CX. The calculation of the incoherent surface concentration, \kA/vNA, referring to the convention vNA is straightforward
Vp/nNA
=
Vp/vNA
+ s(VB
- UA)rA/nNA
(19)
where the partial volumes are to be taken at the initial composition pp,Ao (or at the corresponding value, x,,A0). The net retention volume of a substance i, VN,i,can now be calculated by using the general expression vN,i/CX
=
vR,i
- VpjCX
(20)
where CX can be any of the conventions enumerated. If the surface area of contact, S, is known, the surface specific retention volume is given by vS,i/CX
=
vN,i/CX/s
(21)
The last equation necessary for the evaluation of chromatographic data is Vs,i/cx
=
VpXi/CX
(22)
where the conversion factor, vp, is a property of the eluent and xilcx is the peak propagation resistivity. Its interpretation in terms of adsorption isotherms is given in Table I for the retention volume of a labeled component of the eluent, or a solute, su, and for the concentration perturbation of the eluent, cc.
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ANALYTICAL CHEMISTRY, VOL. 54, NO. 14, DECEMBER 1982
Table I. The Meaning of the Peak Propagation Resistivity, xi/cx, in Equation 22 in Terms of the Adsorption Isotherms ~ I A[g m-'1, r A [mol m-'1, and \ k [mL ~ m-'1 and the Composition of the Eluent Expressed asp^ (Mass Fraction), X A (Molar Fraction), and @ A (Volume Fraction) of A in the Eluent' solvent property
convention
cx
"P
BNA
UPO
mNA nNA vNA
1
interpretation of xilcx solute, su
labeled, A* A/BNAo/xp,Ao
( a su/BNA laxb ,su)ti ,Ao
A/mNi/Pp,Aa
(a
concn perturbation, cc (arA/BNA/x/.L,A)tiP,Ao
a Pfi,su)pP,Ao
(anAImNA/aPb,A
) p p , ~ O
rA/nNi/xp,A"
(a r ~ / l l N A / a X / . b , ~ ) a P , A a
(a r A / n N A / a X p , A ) t i P , ~ o
qA/vNAo/@p,An
( a @' s u / v N A l a @ p , s u)GP,A0
(a\kA/vNA/a@p,A)@P,Ao6
a Analytical, isochratic, isothermal conditions. Also listed are the conversion factors, v P . A/B mixtures. For the small correction in real mixtures see ref 4.
Strictly valid only for ideal
Table 11. Properties of the R-dimethyl(dimethy1amino)silanesUsed for the Preparation of Surface Modified Silicon Dioxidesa
R
nDa0
d 2 0 9gcm
purity by TITR, bp, "C (torr)
by GC, %
%
3,3-dimethylbutyl 1.4282 0.787 156-1 59 (720) 99.4 99.2 tetradecyl 1.4476 0.815 115-118 99.3 99.3 'Symbols: n D a 0 is for the refraction index at 20 "C at the sodium D line; d , , for the density at 20 "C; bp is the boiling point, "C. The purity can be concluded from the relative quantity of dimethylamine determined after hydrolysis with water: TITR and the purity of the thoxy derivative (gas chromatography) formed in the reaction with ethanol, GC (surface percent on the chromatogram; flame ionization detector).
EXPERIMENTAL SECTION General Procedures. The structure of compounds synthetized in our laboratory was confirmed by UV, IR, and 13Cand lH NMR spectra as well as by elemental analysis. The instruments used were a UV spectrophotometer from Varian-Techtron (Model 635), an IR spectrometer from Perkin-Elmer (Model 521), an NMR instrument from Bruker (Model WP-80), and the elemental analyzer from Perkin-Elmer (Model 240B). Materials. The solvents tetrahydrofuran, acetonitrile, methanol, ethanol, 2-propanol, and isopentane (research grade, Merck, Darmstadt, FRG) were distilled in a Vigreux column discarding a fore-run and a residue of 10% each. The distilled solvents were stored in amber bottles and used within the next 3 days. Deionized water was boiled with KMn04 for 2 h and then distilled in a Pyrex glass column. The solutes CD&N and D20 (research grade, Fluka, Buchs, Switzerland) and hetrahydrofuran-d8 (THF-d,; research grade Merck) were used without further purification as well as a gift of partially deuterated tetrahydrofurans (THF-dt3 and THF-d5.0, with deuterium statistically distributed; from the physical chemistry section of our department). Dibutyl phthalate and dimethyl phthalate were from Fluka. The silylating agents (3,3-dimethylbutyl)dimethyl(dimethylamino)silane and tetradecyldimethyl(dimethy1amino)silane were synthetized in our laboratory (for synthesis see ref 17 and the forthcoming paper (18)).The purity of the dimethylaminosilanes was determined by first dissolving an aliquot in 0.1 N aqueous HC1 and determining the excess acid by titration. In a second experiment a small quantity of silane was dissolved in very little absolute ethanol and the purity of the resulting ethoxy derivative was determined by gas chromatography by injecting this solution on Apolane-87 as stationary phase (19). For purity and physical properties of the silylating agents, see Table 11. As silicon dioxide reaction partner, Lichrosorb-SI-100 with particle diameter of 10 pm and nominal pore size of 100 A (Merck) was used after drying at 110 0C/10-3torr overnight. The specific surface area was calculated by the "BET" method (20) from the Nz isotherm determined with an apparatus from Carlo-Erba (Milano, Italy; Model Sorptomatic). Experimental points used 0.05 < prel< 0.23, surface occupied by a Nz molecule, 16.2 A2. The sample used in this work gave a specific surface area of 270 3 m2 8-l (average of five determinations). Following information from the manufacturer the surface of the Lichrosorb is fully hydrated by the method of preparation (21).
*
Table 111. Properties of the Column Packings: Surface Modified Lichrosorb-SI-100 Covered by R-dimethylsiloxy Groupsa rsnox >
symbol
R
,mol m-2
Pc,%
m g-'
DMB TD
3,3-dimethylbutyl tetradecyl
3.9 4.1
9.06 17.00
233 209
",.r
3
a Surface concentration of the siloxy groups, rsnoxy, calculated from the percentage carbon, P c , and the specific surface area of the original untreated Lichrosorb, ssiO2= 270 f 3 ma g-I (eq 23).
The Stationary phases were prepared by reacting vacuum dried Lichrosorb-SI-100with R-dimethyl(dimethy1amino)silane. In a typical experiment a quantity of 20 g of Lichrosorb was wetted with a solution of 16.4 g of tetradecyldimethyl(dimethy1amino)silane in 52 mL of isopentane in an ampule (5.5 mmol of (dimethy1amino)silanecorresponding to about 10 pmol/m2). The isopentane was evaporated at -30 "C at 0.1 torr and then the ampule was sealed and placed in an oven at 140 "C. After 200 h the ampule was opened, the product washed with peroxide-free diethyl ether, and dried at 90 0C/10-3 torr. The (3,3-dimethylbuty1)dimethylsiloxycovered product was prepared similarly but without dissolving the silylating agent in isopentane. Surface concentrations were calculated following ref 22 from elemental analysis. For the properties of the column packings see Table 111. Following unpublished results, the surface area of the product is about the same as that of the original silicon dioxide (23). Therefore, the specific surface area of the surface modified product, spr:was calculated from the specific surface area of the silicon dioxide, ssio2,as follows spr= Ssio,(l - Psi,yl/lOO) = ssio,[l
- (MBilYl - l)Pc/1201b] (23)
where Psilyl is the weight percent of the surface layer, PCis that of the carbon in the product (the result of element analysis), and MBilyl is the molecular mass of the silyl substituent having b carbon atoms in its elemental formula. The silyl group is that which substituted the proton of the surface silanol of the silicon dioxide (a small correction for water desorbed from the surface of the
ANALYTICAL CHEMISTRY, VOL. 54, NO. 14, DECEMBER 1982
silicon dioxide during preparation is neglected). Eluents of the appropriate composition were prepared by weighing. The mixture was filtered through Teflon filters with 0.5-pm pores from Millipore before use. The Preparation of the Columns. The columns were precision stainless steel tubeci of 4.00 mm i.d. and 25.00 cm long closed at both ends with sintered metal plates with 2-pm pores (Brechbuhler AG, Urdorf, Switzerland). The void volume of the columns was 3.141 f 0.002 mL. For the preparation of the column an apparatus from Haskel Engineering Co. was used equipped with a pneumatically operated pump (type MCP). An exactly weighed quantity (-2 g) of surface modified Lichrosorb wag3 suspended in filtered 2-propanol (polyamide filters with 0.5 prn pores from Millipore) and the resulting slurry was exposed to ultrasonic vibrations (Model Marin from Elgasonic, Biel, Switzerland) for 5 min. The suspension was filtered at 500 atm onto the sintered plate closing the outlet of the column. The filling was compreeaed by pasaing at this pressure successively 2-propanol (800 mL), distilled water (500 mL), and 2-propanol again (300 mL), The column was disconnected and aJl material which did not enter the column was carefully collected by washing with methanol and then isolated by filtering dried at 95 oC/lO-a torr and weighed. The mass of the column filling was calculated as the difference of total and waste. Reproducibility of column fillings was as ffollows: 1.40 A 0.01 g for DMB and 1.61 A 0.01 g for TD columns. Completed columns were tested with a mixture of dimethyl phthalate and dibutyl phthalate by using HzO/methanol (30/70 (w/w)) as eluent. They had typically theoretical plate heights of 70 and 100 pm with TD and DMB fillings, respectively. Chromatograph. A Hewlett-Packard liquid chromatograph (Model 1084A) was usedl with the following modifications. Detector. A differential refractometer from Waters (Model 401) was used. Cell volume was 10 pL. The degassing of the eluent was made following the idea of ref 24 by means of a homemade stopcock system. Under continuous stirring (magnetic stirrer) the bottle containing the eluent was rapidly evacuated at room temperature until the solvent began to boil and then the atmlospheric pressure was reestablished by bubbling helium through the liquid. The operation was repeated. This degassing was Sufficient and did not change the composition of the eluent by more t h m 0.05 wt %. Column Temperature. The chromatographic columns were placed in a copper cylinder onto which a copper spiral was soldered. The cylinder was thermostated by circulating low viscosity oil in the spiral (thermostat from Haake, Berlin, FRG Model F3); its temperature was measured with a chromel-alumel thermocouple and an instiiunent from Doric (San Diego, CA, type K). The long range temperature stability of the column was better than *0.05 K. Flow Rate of the Eluent. The detector outlet was connected to a glass outlet tube on which a graduated 5-mL vertical tube was soldered. The tube was equipped with a mantle and was thermostated at 20 A 0.06 "C by circulating water (thermostat from Colora, Wurtemberg, FRG; type NB). By manipulation of a three-way cock, the eluent raised in the vertical tube and the time between two marks was measured with a chronometer. The graduated tube was calibrated with mercury. Flow rate measurements were reproducible to 10.01 mL mi&; the long range flow rate stability was around 10.5%. Experimental Parameters. Mean Column Pressure. By calibration of the chromatograph the same flow resistance was found for the injection system and the outlet tubing (both giving about 2-7 atm pressure drop at normal operating conditions; eluent flow of 0.8-1.5 mL imin-l, 20-40 "C). The flow resistance of the sintered plates in the columns was negligible. Thus the mean column pressure was given with sufficient precision (15%) by P, = Pi/2 (24) where Pi is the inlet pressure measured before the injection system (typical values: 40-250 d m ) . Mean eluent flow rate in the column, v, was calculated by correcting the flow rate measured by the flometer at 1 atm and 20 "C, Vf, to the mean clolumn pressure, P,, and column temperature, T,. The coefficient of isothermal compressibility, K',,
2413
Table IV. Coefficient of Isothermal Compressibility, K , and Coefficient of Thermal Expansion, a, Used in This Work for Water, H,O, Acetonitrile, AN, and Tetrahydrofuran, THF a
H,O AN THF 1.00 (27) 0.45 ( 2 5 ) 1.60 (26) K X lo4 atm-' 3.03 ( 2 5 ) 13.7 (28) 11.4 (28) a X l o 4 K-' a Values at 30 "C. Data from references indicated. and the coefficient of the thermal expansion of the mixture, a,, were approximated by K,
=
$,,AKA
+ &BKB
a, = ~,,A"A + ~,,B"B
(25) (26)
where d, is the volume fraction of component A or B (data necessary for the calculation see Table IV). The flow rate is then given Corrections were of the order of 0-2%: For example, eluert = 0.3 and 4,m = 0.7, V f = 1.90 mL m i d , P, composition 4,, = 100 atm, and !fc= 40.1 OC. Calculation gives K, = 1.26 x 104 atm-l and a, = 10.5 X lo4 K-' resulting in V = 1.92 mL min-'. Retention times, tR,were short (on the order of 1-10 min) and were calculated from retention distances measured on the chart paper run at a speed of 10 cm mi&. Reproducibility was *Oh%. Retention volumes were calculated as
v, = tRv
(28) Eluent composition as mass fraction was given by the method of preparation. The corresponding molar fraction and volume fraction were calculated by using the data of ref 14. A Typical Experiment. In a typical experiment the following samples were injected into the system eluted by eluent of composition dp,A0: pure components of the eluent, A and B; pure deuterated components, A* and B*; a mixture of labeled A and nonlabeled B with the same composition as that of the eluent ( 4 ~ * = 4,,,A0), A*B; an analogous sample, AB*; a mixture of labeled A and labeled B of the same composition as the eluent = 4r,Ao),A*B*. The resulting elution patterns permitted the evaluation and sound identification of the retention volume of labeled A, VR,A* (sample A*B), that of labeled B, V R , p (sample AB*), and that of the concentration perturbation, V,,,, (samples A and B). For an example see Figure 1. The experimental results are summarized in Table V. Calculations. The common factor in eq 4a-c is the function
R = (VR,A*- V R , B * ) / ~
(29)
Experimental points, Ri, determined at given composition of the eluent plotted as a function of prAo,xpb0, or show S-shaped curves difficult to describe in either variable by low degree polynomials. Therefore, first the following expression was fitted on the experimental points where the indexes were dropped for simplicity: p P,,~". This expression was found by trial and error, there is no theoretical justification for its actual form. Equation 30 gives a nonlinear regression problem. The six coefficients Ao-A5 were determined by iterative approximation; the sum of squares of the deviations between the function and the experimental points was minimized with the aid of a computer (PDP-11). The coefficients are listed in Table VI. In Figure 2 on one example the regression curve is compared with the experimental points. Typical deviations are of the order of the experimental error of 10.5% determined by the stability of the eluent flow. In a second step Margules' method was applied to represent the isotherms referring to a NA convention (29) where E is II, T', or and P i s a polynomial in w p = p,,, x,, or 4,, respectively. In order to calculate coefficients of the polynomials
2414
ANALYTICAL CHEMISTRY, VOL. 54, NO. 14, DECEMBER 1982 st
0.0
- 0.1
II I
- 0.2 AYE'
;
,
"
"
J
-
!
~
0
2
'
1l1 4
v,
6mi
I
1 1
-5
5
0
4,"M
10ylm-2
wn20*
0 1
Flgure 1. Elution patterns in a typical series of experiments as recorded by a differential refractometer detector. Component A of the eluent is water, P,,,~' = 0.747 (X_,,,A" = 0.871; $ p , A o = 0.705), B is acetonitrile (AN); T , = 20 OC; P , = 82 atm; V = 1.99 mL min-'; samples of 1 pL. Column filling: 1.40 g of DMB (Llchrosorb-SI-100 surface modified with 3.9 pmol m-2 (3,3~imethylbutyl)dimethylslloxy substituents) with S = 326 m2, Labeled compounds: A* = HDO B* = AN-d,. Labeling ignored, mixed samples were always of the same composition as the eluent; for example, in the sample A"B, 4A. = 4 Symbols: st is for start; cc is for concentratlon perturbation of the eluent; VR is retention volume; VSIVNA is for surface speclflc retention volume referring to the vNA convention (for the determlnatlon of VSIVNA see later).
**
' wCHrCN
0
Flgure 2. An example of the description of the experimental points, (\/,,A* - VR,B.)/S,by the regression equation, Rdp) (eq 30), and the traces of the polynomial representations of the eq 33a-c: Pdp) (1l t h degree), P ( x ) (21th degree), and P ( 4 ) (1 I t h degree). Experimental conditions: T , = 20.0 OC; eluent, water-acetonitrile; column filling, For experimental points TD. The symbol w p is for p,,', x C o ,or see Table V; for regression coefflcients and coefficients of polynomials see Tables V I and V I I .
ml
T I
P(W,,AO)f i i t the followingthree sets of 201 values were calculated, each equidistant in p,,, x,, and $*, respectively, with the aid of the function R ( p ) Pibrr,A(i)')
= R i ( p p , A ( i ) O )/OF;
= Pp,A(i)' -k (1/200)
Pr,A(i+l)'
,
(32a) Pi(xp,A(i)O)
Ri(Pfi,A(i)' ( x p , A ( i ) " ) )/ u p o ;
xp,A(i+l)O
x,,,.qi)' Pi(4p,A(i)o)
= Ri@rb(i)'($p,A(i)O));
$r&i+l)
=
=
+ (1/200) $p,A(i)
(32b)
+ (1/200) (32~)
i = 0-200 On the 201 equidistant points polynomials of degree N were fitted in each language by the Gaussian method of the least sum of squares J=O
(Pp,.4°)JBJ
(33a)
(XF,A')~CJ
(33b)
($N,A")~DJ
(33c)
N
P(xpbo)=
J-0 N
p($,,Ao)
=
J-0
In order to decide the degree N necessary to describe the function, the first derivative was taken as a test. If the first derivative gave an oscillating function, a higher degree polynomial was made and tested. The most difficult was to describe the function in the n-x-r language; two functions, P ( x ) ,had to be calculated, each valid in separate domains. The regression coefficients Bj, CJ, and D j are listed in Table VII. In Figure 2 traces of three polynomials
I
0
. I
8
Flgure 3. Examples of retention volumes, V,* = V , - k,,of deuterated tetrahydrofuran samples as a function of the degree of deuteration, L. The choice of the constant, k,,is purely arbitrary and serves to bring experimental polnts to scale. Experimental conditions: for (1) = column filling, TD; eluent, H20-THF; T,, 20.0 OC; pp,H200 0.775 and (2) = 0.288. The retention volume of the nondeuterated sample is found by extrapolation. The difference between the retention volume of perdeuterated THF and that of the nondeuterated compound (found by extrapolation, AV,) for (1) is 0.03 mL (-0.1 pL m-' and for (2) is 0.06 mL (-0.2 pL m-*).
N
p(P,,Ao)
I
P(w,,,AO)
are plotted for an example.
RESULTS AND DISCUSSION The method of calculation of adsorption isotherms and holdup volumes is based on the existence of solutes which are in every respect identical with components of the eluent with the exception of one property permitting their detection. These solutes were approximated by deuterated compounds and it is certainly not true that they are in every respect identical with nondeuterated compounds. Therefore, in preliminary experiments the retention volumes of a series of partially deuterated tetrahydrofurans were determined in order to explore the effect of the degree of labeling. Figure 3 illustrates results under typical experimental conditions. It
ANALYTICAL CHEMISTRY, VOL. 54, NO. 14, DECEMBER 1982
2415
Table V. Experimental Retention Volumes of Labeled Components of the Eluent Referred to Unit Surface Area: Deuterated Water, HDO, Acetonitrile, AN-d, , and Tetrahydrofuran, THF-d, as well as That of the Concentration Perturbation, cc, at Different Eluent Compositionsa TD
Col.filling Tc
40.0
20.0
40.0
20.0
LOCI
Eluent
AI8
[-I
.i-A*
:-I
cc
B*
i=A*
B*
[-I
cc ~~
0.020
10.1
8.72
8.M
( A = W20)
MITHIF
(A = M )
7.691
7.69
7.51
6.J5
8.23
0.263
7.48
8.01
6.77
0.126
7.92
7.83
7.27
7.79
0.357
7.09
8.04
6.38
0.351
7.39
8.04
6.74
7.42
0.3'53
8.32
9.00
7.61
0.447
7.06 8.37
6.53
0.561
7.00
8.43
7.00
9.06
7.33
0 . M
8.13
9.28
7.61
0.565
6.88
8.64
6.82
0.751
7.21
9.05
8.01
9.34
7.27
0.563
7.98
9.56
7.92
0.625
6.94
8.87
7.24
0.021
7.51
9.85
9.17
7.55
0.661
8.w 9.87
8.35
0.700
6.88
9.11
7.86
0.959
7.57
10.4
9.50
0.803
8.23
9.34
0.757
7.06
9.35
8.46
0.984
11.1
10.1
9.61
8.84
1.000
7.57 7.35
?
-
8.10
7.80
-
8.26
8.94
0.342
8.13
0.447
7.86
9.53 10.0
8.75
10.4
10.8
9.87
0.810
7.27
8.643
11.8
10.8
0.905
7.54
10.0
9.26
8.69
12.6
11.4
0.950
7.57
10.4
9.47
8.75
13.3
-
0.980
7.69
11.3
10.1
10.8
0.990
7.60
11.3
10.6
12.1
11.6
1 .ooo
7.36
11.2
-
12.7
8.64
7.80
-
0.000
0.870
8.41
9.68
0.960
9.99
0.980
11.4
10.4
1 .OOo
8.57
11.8
0.973
8.57
1 .ooo
8.44
0.751
8.04
i0.4
9.43
0.794
8.17
10.5
0.847
8.29
10.1
0.921
8.51
0.950
o.m*
0. 000*
H20/THF
7.74
8.63 8.85
0.301
8.72
8.37
9.33
8.85
O . m
8.69 8.69 8.72
{y;
8.72
-
0.000
8.57
0.M7
9.28
8.82
8.48
0.028
?
7.72
7.42
0.025
?
7.69
7.51
8.3'5
0.W7
?
1'
8.10
0.054
7.69
7.77
7.27
0.054
7.57
7.69
7.24
0.23,2
8.13
6.88
7.64
0.114
7.45
7.74
6.91
0.114
7.45
7.80
7.00
0.221
7.00
7.86
6.53
7.95
!
0.027
8.m 0.123 7 . ~ Q.2f40
8.85
0.113
7.83
-
0.064
8.48
9.34
7.80
7.76
7.21
8.97
0.056
8.46
{::
7.86
0.240
?
0.010
D.OOO*
7.80
0.127
0.025
8.66
!
0.135
8.m
7.83
8.75
-
8.57
8.78
0.663
9.93
7.83
cc
9.6% 6.75
9.46
7.73
0.0%
{;;
B*
~
8.13
0.060
0.527
0. OOOf
O.WJ3f
0. goo*
i=A*
0.222
8.17
8.82
7.45
0.433
7.61
9.12
7.24
0.222
7.12
7.86
6.50
0.326
7.92
8.94
7.39
0.64
7.67
10.0
8.38
0.328
6.88
7.98
6.41
0.327
6.74
7.98
6.20
0.433
7.76
9.12
7.61
0.82Q
8.10
11.3
9.68
0.431
6.74
8.16
6.50
0.432
6.62
8.25
6.23
0.531
7.70
9.37
8.01
0.893
8.23
12.5
9.93
0.532
6.74
8.49
6.97
0.534
6.53
8.52
6.59
0.627
7.70
9.68
8.44
0.940
8.35
15.2
10.7
0.627
6.79
8.84
7.39
0.628
6.65
8.96
7.33
0.743
7.89
10.3
9.06
0.985
8.57
20.5
12.9
0.726
6.91
9.29
8.01
0.731
6.85
9.55
8.13
0.774
8.10
10.5
9.28
1.000
8.75
?
-
0.820
7.15
9.94
8.58
0.771
6.97
9.82
8.43
0.m
8.29
12.3
9.53
0.914
7.33
11.3
9.14
0.914
7.33
11.8
9.41
0.950
8.29
14.4
10.2
0.956
7.39
13.3
9.61
0.955
7.45
13.8
10.2
0.983
8.48
19.8
12.7
0.980
0.981
7.54
18.3
12.0
8.44
?
-
1.000
17.3 ?
11.3
1 .m
7.45 7.36
-
1.000
7.35
?
-
O.MI0
6 - 9 4 8.82
-
O.Oo0
7.66
7.80
-
0.000
7.61
7.80
-
0.187
8.82 8.78 8.57 8.60 8.76 8.63
0.179
7.43
7.73
7.43
0.182
7.50
7.74
7.50
0.390
0.370
7.4b 7 85
7.52
0.373
7.46
7.84
7.53
D.W
8.63
8.82
8.7'5
0.57n
0.W
7.46
7.96
7.74
0.770
8.66 9.126 8.m 9.25
8.9il
0.7"
7 . w 7.645 7.5% 8.251 1.95
0.780
7.60
8.18
7.94
n .m
7.83 8.643
1.OOo
7.76
8.51
l.m
-
7.39
-
!
!
-
a Experiments at 20.0 and 40.0 "C. Column filling: 1.40 g of (3,3-dimethylbutyl)dimethylsiloxy-, DMB, and 1.61 g of tetradecyldimethylsiloxgr-, TD, covered Lichrosorb-100 with nominal pore diameter of 100 A . Particle diameter of 10 pm. Surface: 326 m 2 for DlMB and 337 m2 for TD. Retention volumes in p L m - 2 . At conditions marked by an asterisk
double peaks were observed for which we cannot account for. Exclamation mark means that data were uncertain. Question mark means that for some reason (in general too long retention times) data are not available. is seen that the retention volume of a labeled but nondeuterated compound woulld be some 0.2 pL m-2 different from that of the perdeuterated compound. On comparison of this
result with the maximum value of the function (VR,A*VR,B*)/Sof about 4 p L m-2 it is seen that a systematic error is certainly introduced by the use of perdeuterated compounds,
2416
ANALYTICAL CHEMISTRY, VOL. 54, NO. 14, DECEMBER 1982
Table VI. Regression Coefficients in Equation 30 for Binary Mixtures Formed by Water, H,O, Acetonitrile, AN, and Tetrahydrofuran, THFa stationary phase DMB
regression coefficients
eluent A/B H,O/AN
Tc, OC A0 AI A, A, A4 A, 20 -176.15 -46.49 177.45 0.216 -2.750E-9 20.5 40 -87.13 19.30 88.46 -0.300 -4.23E-9 20.1 H,O/THF 20 -0.20 -2.85 1.44 -8.121 -4.95E-8 18.9 40 -0.18 -3.31 1.12 -6.76 -1.12E-7 18.1 AN/THF 20 & 40 0.05 -0.56 TD H,O/AN 20 0.65 -5.02 0.047 3.89 -5.773-12 24.3 40 -101.22 60.80 101.28 -0.065 -7.783-11 23.9 H,O/THF 20 -0.05 -3.22 0.69 -16.5 -9.57E-11 25.2 40 -0.17 -3.43 0.46 -7.87 -8.963-10 22.9 AN/THF 20 & 40 -0.17 -0.60 a Experiments at T, = 20.0 and 40.0 "C with (3,3-dimethylbutyl)dimethylsiloxy, DMB, and tetradecyldimethylsiloxy, TD, covered silicon dioxides.
I
0
1
s.
&"
0
.
0l 1
S
LZ#,O. b @
+
L
~oxg"
4 ., C - * U j
- I
0
P
