Anal. Chem. 1988, 60, 2358-2364
2358
Effects of Temperature and Solvent Composition on Conductometric Titrations in Nonaqueous Mixed Solvents Giancarlo Franchini, Carlo Preti, Lorenzo Tassi, and Giuseppe Tosi* Department of Chemistry, University of Modena, Via G . Campi, 183, 41100 Modena, Italy
The effects of temperature and composition of solvent mixtures on conductometric tltratbns have been lnvestlgated for the solvent system 2methoxyethand/ethane-l,2dloi. Good analytical results were obtained for a series of monocarboxylic aiiphatk and ammatk aclds and phenols by uslng 0.10 M N,N'-dlphenylguanMlne as standard titrant. Conductance titration curves, performed at -10, 25, 50, and 75 O C in the abovementioned pure solvents and in their four binary mixtures, are plesented and discussed. The peculiar behavlor of pick ackl in the lnvestlgatedsolvent system is emphasized in the discusslon.
The titrimetric determination of weak acids or bases in nonaqueous solvents of the type
HA
+ B + A-.BH+ + A- + BH+
(1)
in which a ion-pair-formation step is followed by a dissociation equilibrium, is one of the most powerful tools in analytical chemistry because it can provide information unavailable by other techniques, such as about the formation of ion pairing or other ion aggregation phenomena at the various points of the titration ( I , 2). Ion pairiig can influence the reaction rates and the equilibria for many solute-solvent systems at low concentrations, and consequently, it is an important factor in selecting solvents and reagents for analytical purposes (3). Conductance measurements are particularly effective in the determination of ion-pair formation constants and ion mobilities and remain a very useful technique especially for acquiring information about nonaqueous solutions. Furthermore, if used for titrations, a careful selection of the solvent system makes possible a differentiation of single functions present either in different molecular units or in the same molecule. Alcohols have received good attention as solvents for acid-base determinations (1,2) but not many conductometric studies appeared in the past about alcoholic solvent mixtures, only in the last years are they becoming more common (3). In our previous works (4-7) studies were reported on the feasibility of conductometric titrations of monoprotic and diprotic acids, phenols, and aromatic nitroderivatives in 2methoxyethanol (hereafter abbreviated Gliem) and ethane1,2-diol (Gliet) as solvents, using N,"-diphenylguanidine (DPG) as base titrant dissolved in the above media. The choice of the two solvents was suggested by considering that they are good solvents for the titration of acidic compounds, that the most common organic acids are soluble enough to do excellent titrimetric work, and that they are completely miscible. DPG was chosen because it can be conveniently recrystallized, it is a stable standard material, quite soluble in nonaqueous media, and it has a high equivalent weight; furthermore, DPG shows a low conductance in the solvents used, as revealed by the curvature in a X vs c1I2 plot. The results indicated also the possibility of performing differential titrations of acid mixtures in cases where such determinations are very difficult, if not impossible, in aqueous solutions.
The subsequent developments of this research should aid in the investigation of the influence on the shape of the titration curves and on the accuracy of the analysis of the solvent properties (i.e., for example, the solvating power and the dielectric constant), of the temperature, of the molecular size of the base, and of the distance between the acidic groups of the polyprotic acids. The present work deals with the study of the effects on the conductometric titrations of the solvent properties and of the temperature, i.e. of the variable quantities that are the most important in determining the best titration conditions. An important section of this paper regards also the results obtained by working with a series of binary mixtures of Gliem and Gliet and using, at various temperatures, the picric acid as titrand; in this way a continuous variation of the dielectric constant of the solvent between 13 and 49 is possible.
EXPERIMENTAL SECTION Conductance titrations in 2-methoxyethanol,ethane-l,a-diol, and their binary mixtures were performed by using Amel Model 123 and 134 conductivity bridges and platinized platinum electrodes (cell constanta equal to 0.98 and 1.04 cm) operating at -10 (or 0), 25,50, and 75 "C in the 0.1 rS to 0.3 S (scale-end) range with a sensitivity of f1.0%. The temperature control was provided by a Lauda K2R thermostatic bath maintained to i0.02 "C. The solvents employed were supplied by Carlo Erba in high purity grade and containing less than 0.10% (ethane-1,2-diol)and 0.05 % (2-methoxyethanol)water, checked by Karl-Fischer titrations; the absence of conductivity impurities was tested for in many blank titrations before the study. The water utilized in the mixtures was bidistilled water having a specific conductance less than 2.36 X 10" S. The solvent mixtures were prepared by weight, with the exception of the ethane-1,2-diol/watermixtures (v/v). N,"-Diphenylguanidine (DPG), supplied by Fluka (purum 98%), was twice purified by recrystallization from hot toluene (mp 150 "C; lit. 150 "C (8)). All the acids and phenols, commercially available or prepared and purified according to literature methods, were reagent grade. The accurately weighed acid samples were dissolved in 50 mL of the solvent or solvent mixture and then were titrated with 0.10 M DPG in the same medium through an automatic digital buret, Amel Model 233. The titrant was added in 0.5- or 1.0-mL portions under magnetic stirring and then, with the stirrer off, the equilibrium reading of the bridge wm taken after each addition; equilibrium was reached within 5 min and the conductance readings were stable with time. We have never observed under the experimental conditions the formation of any precipitate. Volume and solvent conductance corrections were applied to all the conductance data. RESULTS AND DISCUSSION The present work can be divided into three parts: study of the titrations at different temperatures of the monoprotic acids and phenols (i) in pure Gliem and (ii) in pure Gliet; (iii) study of the titrations at different temperatures of picric acid in pure Gliet and Gliem and in a series of their binary mixtures. The investigated temperatures are -10,25,50, and 75 "C and the base titrant solution is N,"-diphenylguanidine 0.10 M in the appropriate solvent or mixture of solvents. All
0003-2700/88/0380-2358$01.50/00 1988 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988
Table 11. Conductometric Titrations of Picric Acid in Ethane-1,2-diol/2-MethoxyethanolSolvent System at 25 "C"
Table I. Conductometric Titrations in 2-Methoxyethanol (Gliem) and Ethane-1,Z-diol(Gliet) of Monocarboxylic Aliphatic or Aromatic Acids and of Phenols at 25 'C compound
mmol taken
recovery, % '
Titrations in GLIEM 3-nitropropanoic acid 0.6667 0.8310 2-nitrobenzoic acid 0.6511 0.8875 3-nitrobenzoic acid 0.6205 0.7632 4-nitrobenzoic acid 0.7131 0.8985 4-nitrophenol 0.8070 0.8386
100.4, 99.8, 99.1 100.5 102.2 103.0 99.6 100.7 99.7, 100.2 100.0, 100.2
Titrations in GLIET 4-nitrobenzoic acid 0.6594 0.7655 4-nitrophenol 0.7144 0.9632
100.8 99.9 100.5, 99.5 100.3, 100.2
2359
solvent system
mmol taken
recovery, %
A (pure Gliet)
0.6422 0.8053 0.6445 0.8330 0.6489 0.8411 0.6542 0.7930 0.6476 0.8107 0.7544 0.8179
99.6, 100.8 99.5, 100.4 100.4, 99.1 99.9, 99.5 99.3, 99.1 99.5, 100.4 98.6, 99.6 99.0, 99.4 99.8 100.6 99.1 100.7
B C D E
F (pure Gliem)
Italic values refer to the semineutralization break Doints.
Table 111. Conductometric Titrations of Picric Acid in Ethane-1,2-diol/Water Solvent System with DPG (0.1 N) or NaOH (0.1 N) at 25 'C
75'C
solvent system
mmol taken
recovery, %
50'C
Gliet/water 90:lO (v/v) with DPG Gliet/water 8020 (v/v) with NaOH Gliet/water 6040 (v/v) with NaOH Gliet/water 4060 (v/v) with NaOH Gliet/water 2080 (v/v) with NaOH
0.6475 0.6853 0.6488 0.6183 0.6476 0.7441 0.6340 0.7393 0.6355 0.8810
101.2 99.5 100.3 99.7 99.3 99.2 100.3 99.6 99.5 99.6
25.C
with the exception of a few particular cases, which will be pointed out. In Tables 1-111 we have reported as examples only the data relative to the titrations performed at 25 "C, but the recoveries at -10,50, and 75 "C are satisfactory. The titration curves have been plotted in units of specific conductance vs moles of base per mole of acid and volume corrections have been applied to all the data. The least-squares method was used to calculate the straight lines that intersect at the theoretical base-acid integral ratio (see the figures); in order to present clearly the results, a vertical shift of the conductance curves has been made. Monoprotic Acids and Phenols in Gliem. The analytical results at 25 "C of the species investigated are reported in Table I. Figure 1 shows the titrimetric behavior of 3-nitropropanoic acid; one can see that a decrease of the temperature (-10 and 25 "C)produces the appearance of a second break at the 2:l molar ratio with a recovery always very poor. The
-10.C
I
I
I
1
I
0
1
2
3
Flgure 1. Conductometric titration curves at different temperatures of 3-nitropropanoic acid in pure 2-methoxyethanol.
the analytical data are referred at least to duplicate titrations. The recoveries are in general quite satisfactory (error ca. *l%)
5D.C
25.C
-1O.C
0
I
2 -1.
or b . .
3 I -1.
or .cld
Flgure 2. Conductometric titratlon curves at different temperatures of 2-, 3-, and Cnitrobenzoic acid (from left to right) in pure 2-methoxyethanol.
2360
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988 750c
SO'C
25O C
-1ooc
the titration at 25 "C for the 3-derivative (the same difficulty, even if less evident, is observed for the other isomers). On the contrary the 4-nitrobenzoic acid exhibits smooth curves beyond the fiist break point working at 50 "C and higher. Also here the curves have been fitted as straight lines with acceptable correlation Coefficients. In this case a hypothesized ternary adduct involving the nitro group should have properties of temperature stability completely opposite to that of 3-nitropropanoic acid. 4-Nitrophenol in Gliem (Figure 3) shows the evidence of two breaks at base/acid molar ratio 1:l and 1:2 overall at 25 and 50 "C, while any difficulty may be present at the other temperatures (smooth curves as above). Other authors (9,lO) have observed for similar phenols a behavior that is analogous for the presence of two equivalent points in the conductometric titration curves, but substantially different: in fact, the break points were at base-acid molar ratios 1:2 and 1:l and the first was attributed to the formation of a complex anion, phenol-phenolate, stabilized by hydrogen bonding
Figure 3. Conductometric titration curves at different temperatures of 4-nitrophenoi in pure 2-methoxyethanol.
In alcoholic solutions and at low concentrations, which are our experimental conditions, the formation of the above complex is very unlikely. In our titrations the 2:l break should arise from a donor-acceptor adduct in which the nitro group can be involved in the delocalization of the negative charge of the anion
titration curves beyond the first equivalent point may appear as smooth curves, that we have approximated as two straight lines with quite satisfactory correlation coefficients (ranging between 0.986 and 0.999). A tentative explanation of this behavior could be the presence of a ternary temperatureunstable adduct formed by 1 mol of acid and 2 mol of base, in which the nitro group should be involved in donor-acceptor interactions with the second base molecule. Figure 2 reports the neutralization curves of three isomeric nitrobenzoic acids, to discover information about possible correlations between molecular structure and shape of the titration curves. For the 2- and 3-nitrobenzoic acids one equivalent point (molar ratio 1:l) is observed at all temperatures, the recoveries being acceptable with the exception of
The presence of this second break at 75 "C also suggests that the adduct is quite temperature stable. Monoprotic Acids and Phenols in Gliet. In this solvent we have restricted our interest to a comparison of two nitro-containing compounds, the 4-nitrobenzoic acid and the 4-nitrophenol, Table I. 4-Nitrobenzoic acid has supplied at all the temperature conditions very sharp curves with only one equivalent point (base/acid = l/l);therefore, in this solvent the 2:l adduct disappears (and consequently the second break observed in
I
o
l
3
2
1
moles of ba,e
1 mole
of a c l d
75'C
/ I
\
P
\
4
/
\
5O'C
b\
h
25'
0
c
I
I
1
1
2
0
m o l e s of base
1 mole of acid
0.5
1
1.5 m o l e s of base
2
/ mole of acid
Figure 4. Conductometric titration curves at different temperatures of picric acid in pure 2-methoxyethanoi (left)and in pure ethane-l,2diol (right).
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21,NOVEMBER 1, 1988
2361
U
c" 4
0 3 U 0
-E
0
1.5
1
0.5
niolcr of base
2
0
1 mole
0.5
1.5
1
of acid
2
niolcs of biisc
ill
1 niok oC d
i d
I,)
U
/
E
4s
Q
1; 0
0.5
I
1.5 2 molcs of bare / mole of acid C)
I
I
1
0
0.5
1
1
I
1.5 2 molcs of b * c
1 inole
d)
Flgure 5. Isothermal titration curves of picric acid in the solvent system ethane-1,2-diol/2-methoxyethanol: (a) -10 O C ; (b) 25 (d) 75 O C .
Gliem) and the curve is similar to those already observed in Figure 2 for 2- and 3-nitrobenzoic acid; this behavior could be probably ascribed to a lower interaction between the nitro group and the second DPG molecule as a consequence of stronger solute-solvent interactions. On the contrary 4-nitrophenol exhibits in Gliet the same behavior already observed in Gliem; the same explanation above suggested for Gliem can be considered in this case. There is difficulty in detecting the second break and poor
or a z d
OC;
(c) 50 O C ;
recovery working at -10 and 75 OC. Picric Acid in Gliem, Gliet, and Their Binary Mixtures. The titrating behavior of 2,4,6-trinitrophenol (picric acid) in pure Gliem and Gliet is quite interesting, as can be seen from Figure 4. In the solvent Gliem one equivalent point at baselacid stoichiometric ratio 1/1 is present and this fact differs from that observed for 4-nitrophenol, probably because of delocalization of the negative ionic charge on more than one nitro group.
2362
ANALYTICAL CHEMISTRY, VOL. 60,NO. 21, NOVEMBER 1, 1988
On the contrary, in pure Gliet two break points are always detected at base/acid molar ratios 1:2 and 1:l; beyond the 1:l ratio and up to 3:l the conductance values have a perfectly linear trend without any change of slope. It is evident from the above considerations that the solvent plays a fundamental role in the formation of the 1:2 adduct, which should be, however, present as a nonconducting ion pair, as suggested by the continuous decrease of conductance and in accordance with the reaction 2HPi
+ B + (PiHPi)-.BH+
(2)
A similar adduct was observed by other authors (11)in other solvents by means of spectrophotometric measures. In order to clarify the role of the solvent, we have performed a series of titrations of picric acid at various temperatures in some binary Gliet/Gliem solvent mixtures. The mixtures will be identified by the letters A, B, ... F and refer to the following mole fraction values of component Gliet, XGliet: A = 1.0000 (pure Gliet), B = 0.8499, C = 0.6799, D = 0.4856, E = 0.2614, F = 0.0000 (pure Gliem). As base titrant, 0.10 M solutions of DPG in the same mixtures have been used. The titration curves obtained are summarized in Figure 5. A comparison between the results obtained for the mixtures E and D is quite interesting; while for E the conductometric trend is that observed in pure Gliem (system F), the titrations performed in D exhibit two equivalent points at stoichiometric 1:2 and 1:l base:acid ratios as in pure Gliet (system A) with the exception of the titration at 75 “C in which only the first break is present. In the C and B systems the conductometric behavior is equal to that of pure Gliet with two clear break points. The increase of the temperature induces graph deformations corresponding to changes in the slope sign of the curve relative to the reaction BH+.(PiHPi)- + B + 2(BH+.Pi-) + 2BH+
\
+ 2Pi-
(3)
Some considerations can be put forth after analyzing in detail this set of titration curves. In system A (pure Gliet) there is a continuous decrease of conductivity, different for the two titration steps until the 1:l break point; after this point the conductivity increases. This evidence, observed at any temperature, might find a possible explanation in the fact that picric acid, giving a conjugation with this base in this solvent, exhibits a behavior typical of difunctional compounds. Therefore, the acidic adduct that takes place at the first titration step (equilibrium 2) is probably ionized and present as an ion pair scarcely dissociated in solution. The formation of this adduct is reflected in the big decrease of conductivity until the equivalent point at the stoichiometric 2:l ratio. On the contrary, the neutral adduct (BH+.Pi-) of equilibrium 3 should exhibit a larger dissociation than the acidic adduct and this fact corresponds to a less dramatic negative variation of conductivity. In the first step of the titration curves relative to systems B and C, the same characteristic found in the A system and attributable to the same mechanism of acidic adduct formation is observed; in particular, for the B system one notes a gradual and progressive deformation of the second titration step, corresponding to the formation of a neutral adduct, which leads to the sharp charge in the slope sign at high temperatures. The conductivity increase corresponding to the neutral adduct formation probably provides an appreciable change of the peculiar solvent system characteristics with respect to those of pure Gliet, although for values of xGliet still high; as a consequence, this increase is due to the increased mobility of the ions and/or to the consistent increase of the ionic species concentration in solution and this arises from a dissociation
75’C 50’C 25‘C 0.C
I 0
2
1 m o l e s of base
1 mole
of acid
Flgure 6. Conductometric titration curves at different temperatures of picric acid in the mixed solvent system ethane-l,2doi/water90/10 (v/v).
degree of the species higher than that of the acidic adduct. The analytical data at 25 “C for the six investigated solvent systems are reported in Table I1 and are quite accurate for both break points; also at -10,50, and 75 O C the accuracy is acceptable, with the exception of the data relative to the 1:2 ratio at -10 “C for the solvent system D. The isothermal graphs of Figure 5 show clearly that the characteristic feature is the possibility of distinguishing two groups of binary mixtures: the former has curves with only one equivalent point (1:l) and corresponds to the mixtures containing more Gliem, and the other having both the breaks. This feature is evident at all investigated temperatures. On the basis of our experimental data we could hypothesize the existence of a particular mixture that is the “limit” between chair- and boat-shaped curves and no slope change should be expected. In our previous papers (12, 13) regarding the conductometric determination of the dissociation constant of picric acid in this Gliet/Gliem solvent system at different temperatures, the existence of this so-called “limiting mixture” was suggested on the basis of a sharp change in the trend of the dependence of the K values on the temperature. A value of xGllet = 0.2677 for the “limiting mixture” was calculated by considering bo vs l / e graphs (13). The observations of the present work confirm the above mentioned results; in fact, the shape of the titration curves changes between D (x = 0.4856) and E (x = 0.2614) mixtures. Another interesting question is that of the appearance of the break point at the base:acid 1:2 molar ratio. We have made an attempt to try to clarify the role of the solvent and of the titrant in the formation of the (PiHPi)-.BH+ adduct; we have chosen the binary Gliet/water solvent system as comparing system, even if some problems arise from the fact that DPG is sufficiently soluble only up to Gliet/water 90/10 volume to volume ratio and, obviously, at temperatures not lower than 0 O C . Figure 6 contains the titration curves performed in the 90/10 (v/v) mixture with DPG 0.10 M titrant solution; only one break point at the 1:l stoichiometric ratio is evident, excluding the possibility of the formation of the semineutralization adduct. For mixtures containing more water (Gliet/water 80/20,60/40,40/60, and 20/80 (v/v)) NaOH 0.10 M titrant solutions have been employed; the curves show only the 1:l equivalent point (Figure 7) and the analytical results
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988
75'C
2363
75'C
V
-
U
4
J
50'C
a
2
s
50'C
25'C
25'C
O'C
1
I
I
0
1
2 inolcs of bnrc
0.C
0
1 mole of acid
1
2
molcr of h s e
a)
1 mole of a c i d
b)
75'C u
u
g
C
Y
L
U
a
-0
s
0
U
50'C
25'C
O'C
1;
VI
Ii 1
I
1
0
1
2
molcs or bnrc
1 mole of acid
C)
2
1
molcs of I)~lsc1 mole of a i i d
d)
Figure 7. Conductometric titration curves at different temperatures of picric acid in the mixed solvent system ethane-l,2diol/water: (v/v); (b) 60/40 (v/v); (c) 40/60 (v/v); (d) 20/80 (v/v).
are quite satisfactory at all the investigated temperatures; the data obtained at 25 "C are reported in Table 111. From the above considerations it is evident that the appearance of the 1:2 equivalent point, i.e. the formation of the (PiHPi)-.BH+ adduct, is strictly connected with the specific solvent-solvent and solvent-solute interactions. CONCLUSIONS The study of the shape of the acid-base titration curves obtained at different temperatures and for different binary solvent systems probably represents the most characterizing aspect of this work, in particular for that regarding picric acid. In fact, the trend of a titration curve, expressed as a variation of slopes in the various steps, might be correlated "step by step" to the formation, ionization, and dissociation constants of the adducts and salts formed during the titration.
(a) 80120
All these constant values, which are at the moment unknown, probably are susceptible of correlation with the dissociation constant values of picric acid, experimentally determined at the same temperature conditions and in the same solvent systems. Now, with a three-dimensional model, K = K(T,x),available for this acid in these binary solvent systems (14), it is possible to choose the most appropriate conditions, i.e. the best values of T and x, to obtain the best results or clearest experimental evidence. It should be very useful to have similar models for as many solvent-solute systems as possible in order to obtain any generalization of the problem. ACKNOWLEDGMENT The authors wish to thank Dr. Paola Rossi for her experimental work and the Centro Interdipartimentale di Calcolo
Anal. Chem. 1880, 60, 2364-2374
2364
Automatic0 ed Informatica Applicata (C.I.C.A.I.A.) of Modena University for the computing facilities. Registry No. 3-Nitropropanoicacid, 504-88-1;2-nitrobenzoic acid, 552-16-9; 3-nitrobenzoic acid, 121-92-6;4-nitrobenzoic acid, 62-23-7; 4-nitrophenol, 100-02-7; picric acid, 88-89-1; 2-methoxvethanol. 109-86-4:ethane-1.2-diol, 107-21-1:. N,”-diphenvl. - guanidine, 102-06-7.. LITERATURE CITED (1) Chantooni, M. K., Jr.; Kdthoff. I. M. J. Phys. Chem. 1978. 82, 994-1000. and references therein. (2) . . Kolthoff. I. M.: Chantooni. M. K.. Jr. Anal. Chem. 1978, 5 0 , 1440-1446, and references therein. (3) Kratochvll, 8. Anal. Chem. 1982, 5 4 , 105R-121R. (4) Preti, C.: Tosl, 0. Anal. Chem. 1981, 5 3 , 40-51. (5) Pretl, C.; Tessl, L.; Tosi, 0.Anal. Chem. 1982, 5 4 , 796-799. (6) Neviani Qillberti, E.: Preti, C.; Tassi, L.; Tosi, G. Ann. Chlm. (Rome) 198% 7 3 , 527-532.
(7) Preti, C.; Tassi, L.; Tosi, 0.Ann. CMm. (Rome)1985, 7 5 , 201-206. (8) Handbock of Chemlshy and Physlcs, 66th ed.; Weast, R. C.. Ed.; The Chemical Rubber Co.: Cleveland, OH, 1985 p C-293. (9) Van Meurs, M.; Dahmen, E. A. M. F. Anal. Chim. Acta 1958, 79, 64-73. (lo) Van M. Anal, chim, Acte 19gg, 2 , , 443-455. (11) Kolthoff, I. M.; Bruckenstein, S.; Chantooni, M. K., Jr. J. Am. Chem. S0C.1961.83.3927-3935. (12) Franchini, G. C:: Ori, E.; Preti, C.; Tassi, L.; Tosi, G. Can. J. Chem. 1987, 6 5 , 722-726. (13) Franchini, G. C.; Tassi, L.; Tosi. G. J . Chem. SOC.,Farady Trans. 7 1987, 83, 3129-3138. (14) Franchini, G.C.; Preti, C.; Tassi, L.; Tosi, G., submitted for publication in Can. J . Chem
.
RECEIVED for review January 21, 1988. Accepted June 10, 1988* The deuaPubblicaIsh’~ione(MpI) of is gratefully acknowledged for the financial support.
Analytical Solution for the Ideal Model of Chromatography in the Case of a Langmuir Isotherm Sadroddin Golshan-Shirazi and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
An exact solutlon of the Ideal model of chromatography for one compound Is derived in the case of a Langmulr isotherm (C,= aC,,,/(l bC,), where C, Is the amount of solute sorbed per unit mass of packing material, In equllibrium wlth a concentration C , In the mobile phase). A set of reduced coordinates can be chosen that pennits the representation of the elution profiles as a set of curves that are a function of only the reduced sample she. The reduced time Is ( t to)/(tR,o to),whkh provides a t h scale bnlependent of the first isotherm parameter, a. The reduced ordinate Is bC,. The reduced sample size Is the loading factor, Le., the ratlo of the sample size to the amount of sample that would completely saturate the cdumn. WHh the parameters of the Isotherm known, It Is easy to predlct the elution band profile for any sample size. With the sample slze known, it Is easy to calculate the parameters of the “best” Langmulr Isotherm. From these the prolile ot the elutlon band of any sample slze can be predicted. The bsnds eluted from real cokmne having several thousands of theoretical plates are so sharp that the effects of axial diffusion and of the radlal resistance to mass transfer Introduce only very smaii devlatlons that can be corrected eadly by uslng a dlmenslonless plot. This method is IJgorous. It Is mole general than the mplrkai approaches recently wggested. It Is easier to use and provides more 8CCW8te W e d k t h S . It IS bound t0 f8l4 hOWeVBI, WheIWVW the Langmulr model does not account correctly for the adsorption behavior of the compound under study In the concentratlon range Investigated. I n this way, It provldes 8 sensltivo test of the SuItabUIty of a Langmulr isotherm to represent adsorption In a glven system.
+
-
-
There is a considerable interest in the prediction of band profiles in chromatography. Although this is a very old
* Author to whom correspondence should be sent. 0003-2700/88/0360-2364$01.50/0
theoretical problem, originating with the work of Wilson (1) and De Vault (2),it has not been solved rigorously yet, in spite of many efforts, notably by Glueckauf ( 3 , 4 ) ,Guiochon and Jacob (5, 6) and Rhee et al. (7, 8). Recently, a numerical solution has been published, which permits the accurate prediction of the elution band profile of a pure compound when its equilibrium isotherm in the chromatographic system used is known (9). An experimental demonstration of the validity of this numerical solution has been published (10). The difficulty in applying this solution is that the exact equilibrium isotherm should be known. The elution profile depends very strongly on this isotherm, and it is very sensitive to small changes of the numerical values of its constants. Moreover, the determination of equilibrium isotherms, even with the simple, rather straightforward,and fast ECP method, is time-consuming (10). Besides the theoretical interest in solving well-formulated problems, such as the prediction of the elution band profile of a pure compound, there is an important practical interest in being able to optimize experimental conditions in preparative liquid chromatography. This method is largely used in the pharmaceutical and biotechnological industries for the extraction and purification of valuable intermediates or of finished products, to be used as drugs. The elimination of trace impurities is required. Rapid optimization procedures are sorely needed. Recently an empirical method has been suggested (11,12). It seems to be valid essentially in the case of a Langmuir-type isotherm. This is certainly an important restriction from a theoretical viewpoint. In practice, however, it would be potentially very useful, because the equilibrium isotherms of pure compounds in most adsorption systems can be fairly well approximated by a Langmuir-type equation. Although the experimental and the predicted results seem to agree reasonably well regarding the retention times of the band maxima, the agreement is merely fair for band widths or column efficiencies (12). Furthermore, as all empirical results, the method does not give much of a clue concerning the physical phenomena involved and cannot be extended to other related problems. Especially taxing is the optimization 0 1988 American Chemical Society
