J. Phys. Chem. 1984, 88, 5385-5391
P PO
R S
T t tM
tnl tR
U U
UO
number of moles of solute in the gas phase at equilibrium (es 3) numer of moles of solute in the stationary phase at equilibrium (eq 1) local pressure at ascissa z and at time t (eq 1) vapor pressure of solute ideal gas constant (eq 1) detector response factor (eq 34) column absolute temperature (eq 1) time; time origin at the injection of solute (eq 1) retention time; time of the peak maximum (eq 41) retention time of a nonretained compound (eq 4a) limit retention time, obtained for a zero sample size (eq 41) auxiliary variable: uo/(l + K l / c ) (eq 24b) local velocity of the mobile phase (at abscissa z and time t ) (eq 1) outlet carrier-gas velocity, under steady-state conditions (eq 12)
ub V
approximation for the gas velocity (eq 21) propagation speed of a slice of constant mole fraction (eq 14)
5385
column volume available to the gas phase (eq 1) auxiliary variable: e - (eq 29b) mole fraction of the solute in the gas phase (eq 1) detector signal (eq 34) fictive signal pulse height during injection (eq 3 6 ) abscissa along the column (eq 1) V o / Adefined in eq 19 auxiliary variable: z - Ut (eq 24a) auxiliary integration variable (eq 28) auxiliary integration variable (eq 28) auxiliary variable: 2[(K2cv- K,)/(CV(KI + e))] (eq 26b) auxiliary variable: (XU/2D9L0C0(eq 29a) standard deviation of the Gaussian profile obtained for a very small sample size; length unit (eq 4b) sample pulse duration (eq 36)
VO W
X Y Yo z €
b e x
7
P UL 70
Supplementary Material Available: Appendices I and I1 containing derivations of the relationship between tM and tR (7 pages). Ordering information is given on any current masthead page.
Study of Peak Profiles in Nonlinear Gas Chromatography. 2. Determination of the Curvature of Isotherms at Zero Surface Coverage on Graphitized Carbon Black A. Jaulmes, C. Vidal-Madjar, M. Gaspar, and G. Guiochon* Laboratoire de Chimie Analytique Physique, Ecole Polytechnique, 91 12% Palaiseau Cedex, France (Received: January 3, 1984; In Final Form: June 1 , 1984)
The elution profiles observed for benzene and n-hexane on graphitized carbon black are well described by the theoretical model of nonlinear chromatography proposed in our previous paper (part 1). The model is a function of four parameters: the sample size (peak area), the apparent axial dispersion (band broadening), the slope (retention time), and the curvature of the isotherm at the origin (peak leaning). The physical reliability of the theoretical model is demonstrated by the fact that the parameters of the adsorption isotherm determined by fitting experimental data on the elution profile predicted by the model are in good agreement with the results obtained by independent techniques of determination of isotherms. The coefficient of apparent axial dispersion is found to be independent of the sample size within the range studied. The method of determination of the isotherm described here permits a rapid investigation of the effect of temperature.
Introduction In a previous paper,’ we have shown that as soon as the sample size is not zero, the profile of a chromatographic zone is modified by the curvature of the isotherm and that this profile is a function of four parameters: (i) the sample size itself, which can be represented by the peak area if a linear detector is used, (ii) the apparent diffusion coefficient which accounts for band broadening due to axial diffusion and resistance to mass transfer and can be related to the height equivalent to a theoretical plate, (iii) the slope, and (iv) the curvature of the isotherm at zero concentration. As a consequence, the time of the peak maximum, i.e. the classical chromatographic retention time, is not constant but varies with increasing sample size except in the coincidental case when the isotherm and sorption effects compensate for each other. Otherwise, it either increases or decreases, depending on which effect predominates. The aim of this paper is to apply these theoretical results in the case of the band profiles obtained by elution of n-hexane and benzene on graphitized carbon black and to compare the parameters of the adsorption isotherms derived from that model to those obtained by independent methods. Experimental Section The chromatographic equipment specially designed for this work incorporates a fluidic sample switch injector and a flame ionization detector (FID) (Varian). The whole system is placed in an air-
stirred oven (Servomex), the temperature of which is maintained within 0.1 OC with a proportional and integral controller. The “fluidic logic” injector design of Gaspar et al.2,3was used in this study in order to obtain symmetrical, reproducible injection profiles whose width does not change when the amount of sample injected into the column is increased. The general injection system, which includes a bistable fluidic logic gate (Model 191454, Corning glass), is shown Figure 1. The fluidic device has been enclosed inside a pressurized vessel similar to the one previously described. The column is connected to gate 0 1 . The command pulse is actuated by a solenoid valve (Clippard EVO-3) for a time which can be chosen by program between 10 and 100 ms. The command port C2 is connected to C1 to obtain a rectangular injection profile of constant time width, which throughout this work has been set at 50 ms. In order to measure reliable values of k‘, a mixture of methane and solute vapors has been injected into the column. Methane is used to determine the gas holdup time t , since it is the least adsorbed gas which gives a signal with the flame ionization detector. To determine this time, methane has been added to the nitrogen sample gas. The sample is taken from a stream of hydrocarbon vapors diluted in nitrogen. The input sample pressure of nitrogen, P,, and methane, P,, and the command pressure, Pm, are controlled with Negretti and Zambra pressure regulators. The (2) Gaspar, G.;Arpino, P.; Guiochon, G. J . Chromatogr. Sci. 1977, 15,
256. (1) Jaulmes, A,; Ladurelli, A.; Vidal-Madjar, C.; Guiochon, G.,preceding paper in this issue.
0022-3654/84/2088-5385$01.50/0
(3) Gaspar, G.; Olivo, J. P.; Guiochon, G. Chromatographia 1978,lI. 321. (4) Gaspar, G.; Annino, R.; Vidal-Madjar, C.; Guiochon, G. Anal. Chem. 1978, 50, 1512.
0 1984 American Chemical Society
5386
The Journal of Physical Chemistry, Vol. 88. No. 22, 1984
Jaulmes et al.
11-71 inlet pressure
P-
d
regulator
Figure 1. Scheme of the gas chromatographic injection system: F, fine metering valve; C, column; I, injector; M, manometer; P, pressure controller; SV, solenoid valve; Fluidic gate connections: Ps, solute inlet; C1, C2, command inlets; 0 1 , 0 2 , corresponding outlets.
sample flow of about 1 L m i d is adjusted with a metering valve (Nupro). The solute liquid sample is injected manually and vaporized in the injector I. The vapors are carried to the fluidic gate by the nitrogen flow. The solenoid valve is actuated after a given delay time set between 0.5 and 5 s. The amount of sample injected into the column is a complicated function of this delay and depends on the vaporization time of the liquid and the design of the tube between the injector I and the fluidic logic gate. For a given time delay, the amounts of the compounds studied (n-hexane or benzene) injected into the column are reproducible within a few percent. We checked that the sample size has no influence on the solute bandwidth at injection by measuring the peak width at half-height for methane. Within experimental errors it was found to be independent of the solute amount used, even for large variations (from 0.04 to 30 Mg). A 10-s delay is enough to completely remove the vapors from the fluidic logic lines. The amounts injected into the column are known from the results of previous calibration of the detector with direct on-column injection of dilute solutions of the hydrocarbons in n-heptane. Helium is used as the carrier gas. The inlet pressure is controlled by a Texas Instruments pressure regulator working with reference to the outlet pressure, with fluctuations smaller than 0.015 mbar.5 The retention time of methane is taken as the retention time of an unretained compound, without correction for the small retention of methane on graphitized carbon black. The gaseous volume of the column has been calculated by the classical formula
Vo = tJFo (1) where Fois the flow rate at column outlet, t, is the residence time of the unretained solute, and j = ~(PZ 1 ) / [ 2 ( ~ 3- 1)1
(2)
where P is the inlet to outlet pressure ratio. For the independent determination involving the "step and pulse" method: a Varian thermal conductivity detector was used with helium as carrier gas. Mixtures of this gas and the vapor of the compound studied are generated by a diffusion cell.' The concentration of the carrier gas mixtures are determined by quantitative analysis using an FID gas chromatograph (Varian 3700).
Results 1. Fitting of the Theoretical Model Curve on the Elution Peak Profile. The theoretical model of the elution peak is given by the following equation: 2 D' 112 exp(-f2/4D't) C(t,{) = XU( at) coth ( , ~ / 2 )+ erf ({/2(D't)''2) (3) ( 5 ) Goedert, M.; Guiochon, G. Anal. Chem. 1973, 45, 1180, 1188. (6) Valentin, P.; Guiochon, G. J . Chromarogr. Sci. 1976, 14, 56, 132. (7) McKelvey, J. M.; Hoescher, H. E. Anal. Chem. 1957, 29, 123.
110
120
130
140
tIm. (.)
Figure 2. Elution peaks of hexane on graphitized carbon black showing experimental points (computer data acquisition, 100 Hz) and best fit of the theoretical profile (eq 3) (temperature 100 'C; carrier gas, helium; column pressure drop 600 mbar; flow rate 32 mL min-I; peak area normalized to 100). Mass injected: (1) 0.7, (2) 4.8, (3) 16, (4) 31 pg.
L
70
80
90 tiua
(E)
Figure 3. Elution peaks of benzene on graphitized carbon black (same conditions as in Figure 2). Mass injected: (1) 1.6 (2) 5.5, (3) 11, (4) 23 Pg.
which assumes a Dirac function profile for the injection signal. U, D', X, and M are defined by
u = tug/(€ + K , )
+ K,) + K,)
(3a)
D' = cD/(c X = 2(KZC" - Kl)/C"(t
= XULoCo/2D'
(3b) { = z - ut
(3c) (34
A least-squares fitting of the elution profiles on this curve has been made. To characterize the elution peak profile, only four parameters are needed: the area of the elution peak, a, the retention time at infinite dilution, tR, the axial dispersion coefficient of the zone D', and A, a parameter which characterizes peak leaning. tR is directly related to the slope of the isotherm at zero concentration while X is a function of the curvature of the isotherm at the 0rigin.l An example of the modification of peak shape with increasing amount injected is given in Figure 2 for n-hexane eluted on graphitized carbon black at 100 O C . For easier comparison of peak shapes, peak areas are normalized. In this example the adsorption isotherm is concave, with X > 0. The least-squares fit adjustment is very good for the whole set of data, thus confirming the prediction of the model that only four parameters are necessary to define the elution peak in this range of concentrations and that eq 3 is at least a good representation of peak profiles. With a convex adsorption isotherm (X < 0), similar changes in peak profile are observed as illustrated in Figure 3 which shows the elution of benzene on the same adsorbent at 100 O C . The adjustment is somewhat less satisfactory than for hexane as the lowest part exhibits an additional tailing which most probably has
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5387
Peak Profiles in Nonlinear Gas Chromatography
TABLE I: Influence of Sample Size on the Parameters of Peak Profile" sample fitted peak area, mass, r g fimol s cm-3 tM, s fitted t R , s fitted LY,cm2 s-l 0.0119 f 2 X 0.04 8.6 X 116.7 117.4 f 0.2 0.0121 f 6 X 0.18 0.0039 116.6 116.9 f 0.7 0.0124 A 2 X 0.7 0.015 117.0 117.0 f 0.2 0.0124 4 X lo-' 1.9 0.040 117.4 117.1 0.2 0.0124 4 X lo-' 118.3 116.8 f 0.2 4.8 0.104 0.0121 f 9 X lo-' 16 0.35 121.8 116.2 0.4 122.6 115.3 f 0.2 0.0118 f 2 X 22 0.46 0.0115 f 31 0.67 124 115.1 0.2
* *
* * *
fitted A, cm3 pmo1-I -61 f 30 -3.5 6 -0.69 f 0.8 0.35 0.09 1.3 f 0.04 1.48 0.05 1.35 f 0.04 1.2 0.15
* * * *
K, I
cm3 m-z 3.09 3.08 3.08 3.08 3.08 3.07 3.04 3.03
K2, cm6 m-2 fimoi-I
2.14 f 0.06 2.41 0.08 2.19 0.06
* *
"Solute, hexane; temperature 100 O C ; pressure drop 597 mbar. a kinetic origin caused by some active sites on graphite surface, a phenomenon which is not accounted for in eq 3. Now that we have established that eq 3 is a good representation of peak profiles, it is necessary to compare the values of the parameters of this equation to those derived from independent values of the physicochemical data involved. 2. Methods of Determination of the Adsorption Isotherm at Low Surface Coverage. As demonstrated in a previous paper,' the characteristics of the adsorption isotherm at low surface coverage can be derived either from the variation with sample size of the retention time of the peak maximum or by curve fitting of the elution peak to the model equation. These constants are K , , the Henry's adsorption coefficient, and K2,a parameter which is related to the curvature of the adsorption isotherm:
K2 = (eVo/2)(s2ns/snG2)C=o
(4)
( a ) Plot of the Retention Time of the Peak Maximum with Increasing Sample Size. If increasing sample sizes are injected into the column and the elution bands are recorded, it is easy to plot the retention time of the peak maximum vs. the concentration at peak maximum, derived from the detector calibration. From this plot, it is possible to derive K , and K2 from the following set of equations: M t = t R ( 1 Xc,) (5)
C.(nmol. cm-3
+
K2 = X(KI
+ t)/2 + K , R T / P
(7)
where tR is the solute retention time at infinite dilution, t , the retention time of an unretained component, p the mean column pressure, R the ideal gas constant, T the absolute temperature, M t the retention time of the peak maximum, and CMthe corresponding solute concentration. In parts a and b of Figure 4, examples of the determination of these parameters are given for hexane and benzene, respectively. The set of experimental points is located around a straight line for low values of C M . A slight scattering of experimental values of the retention times (ca. 0.5%) is observed, much larger than the reproducibility expected from this equipment under classical condition^.^ It arises from the lack of stability of the flow at the column inlet caused by the turbulent conditions necessary for operating the fluidic device. In the same figure are also reported the theoretical plots of the maximum concentration vs. the first-order moment A, obtained by numerical integration of eq 3 for the same values of the parameters. Obviously, the slopes of the two theoretical curves (C, with the time of maximum or first moment) are different, but they will yield the same time value at infinite dilution, t R . One can determine the limit retention time tR by measuring the first-order moment and extrapolating to zero concentration, but the determination of the isotherm curvature will be more difficult. The approximation of the theoretical model, t M = f ( c M ) (eq 5 ) , by a straight line is correct within only and the error is negligible. It is represented by the solid line in the Figure 4. Therefore, the deviations from a straight line observed at the largest values of the concentration are mainly caused by the shape of the sorption isotherm which cannot be approximated by a parabola at high concentrations anymore. A three-term expansion
Figure 4. Variation of the retention time of peak maximum with the maximum concentration for (a) hexane (tR = 116.7 s, D' = 0.012 cm2 s-l, X = 1.3 cm3fimol-I) and (b) benzene (tR = 75.1 s, D'= 0.020 cm2 S-I, X = -1.1 cm3 fimol-I): (0)experimental results; (-) theoretical curve for CM vs. t,, (- - -) theoretical curve for CM vs. first-order moment (same conditions as in Figure 2).
would become necssary, but the mathematics are too complex for the derivation of an analytical solution. ( b ) Profile of the Elution Peak. As explained above, the peak profiles can be characterized by the four parameters involved in eq 3. Numerical values of these parameters can be derived from any experimental profile by conventional procedures pf curve fitting. Two of these parameters, X and tR, are used to calculate the constants, K , and K2 of the absorption isotherm at low surface coverage (cf. eq 5-7). An example of the determination of these parameters by this method of peak profile fitting is given in Tables I and I1 for series of peaks of hexane and benzene, respectively,obtained by injecting increasing sample sizes of these hydrocarbons in the range 0.05-30 fig, at 100 O C , on graphitized carbon black. These data correspond to the same experiments as those used to study the variation of
5388 The Journal of Physical Chemistry, Vol. 88, No. 22, 1984
Jaulmes et al.
TABLE 11: Iafluence of Sample Size on the Parameters of Peak Profile'
sample mass, rg 0.058 0.14 0.66 1.6 1.9 2.2 5.5 10.6 23
fitted peak area, pmol s cm-3 1.36 x 10-3 3.35 x 10-3 0.0157 0.039 0.046 0.052 0.130 0.251 0.544
fitted A, cm3 *mol-' -20 f 25 -9.4 f 5 -3.2 f 2
75.1 75.1 75.1
fitted t ~ s , 75.7 f 0.6 75.6 f 0.2 75.7 f 0.1
fitted D', cm2 s-I 0.020 f 0.001 0.020 f 0.0004 0.020 f 0.0004
74.8
75.9 f 0.1
0.020 f 0.001
-2.2
74.0 73.0 71.2
76.2 f 0.5 76.2 f 0.2 76.7 f 0.2
0.020 f 0.001 0.020 f 0.002 0.021 f 0.002
-1.6 f 0.2 -1.24 f 0.07 -1.02 f 0.04
tM, s
f 0.3
K'9
cm3 m-z 1.96 1.95 1.96
K2 3
cm6 m-2 pmol-'
1.96 1.97 1.97 1.99
-1.24 f 0.08 -1.02 f 0.04
"Solute, benzene; temperature 100 O C ; pressure drop 597 mbar. TABLE III: Influence of Temperature on the Slope and Curvature of the Isotherm at Origin
benzene method model fitting
peak apex locus
0,
oc
70 80 90 100 110 70 80 90 100 110
K , , cm3 m-2
K2, cm6 m-' pmol-l
5.14 3.65 2.59 1.94 1.28 5.14 3.65 2.59 1.94 1.28
-5.3 1.1 -2.6 f 0.2 -1.4 f 0.1 -1.1 f 0.2 -0.5 f 0.1 -4.0 f 0.4 -2.5 f 0.2 -1.2 f 0.1 -0.74 f 0.08 -0.29 f 0.04
*
hexane Kl/Cva 0.11 0.08 0.06 0.04 0.03 0.11 0.08 0.06 0.04 0.03
K,, cm3 m-*
K2, cm6 m-2 mol-'
K , ICv'
8.59 5.91 4.15 3.07 1.93 8.60 5.91 4.14 3.07 1.93
28 f 5 11 f 1 4.0 f 0.8 2.3 f 0.3 0.5 f 0.2 31 f 4 13 f 2 4.4 f 0.3 2.2 f 0.3 0.8 f 0.1
0.18 0.13 0.09 0.07
0.05 0.18 0.13 0.09 0.07 0.05
'CV is the average concentration (in pmol cm-3) of the carrier gas in the column (concentration at average pressure). the retention time of the peak maximum with increasing sample size (cf. Figure 4). The theoretical model predicts that the values of three parameters (tR, X, and 0') are independent of the amount injected. The relative confidence interval for the determination of t R and a value calculated from D'is small and ranges around 5 X the standard deviation, multiplied by the Student coefficient corresponding to six repeated injections for a confidence interval of 95%. The correspondingcoefficient interval for X is reported in Tables I and 11. It is a few percent of the X values obtained at large solute concentrations but becomes large at small concentrations, as the peak shape is distorted, and a significant tailing occurs which may arise from kinetic phenomena caused by residual heterogeneity of the support surface or from extracolumn disturbances. Within the experimental errors the dispersion coefficient of the solute zone D'is constant when the sample size is varied. This is in excellent agreement with all the chromatographic theories based on the mass-balance equations. The value of fitted tR in Tables I and I1 should remain constant and equal to the limit value of t M = t o M . For the lower solute amounts injected, the value of t R remains close to toM. For the larger amounts a significant deviation is observed (a few percent) which may be explained by the increasing discrepancy between the adsorption isotherm and the assumed parabola. In order to derive meaningful values of K1 and K2 from the values of the fitted parameters tR and A, it is necessary that the theoretical model of the elution peak be valid, i.e. that the adsorption isotherm be a parabola in the range of concentrations covered by the peak i.e. from 0 to the concentration corresponding to the peak maxima.' In this case we have seen that the plots of maximum peak height (or corresponding concentration) vs. its retention time are a straight line (cf. Figure 4a,b). Therefore, the results given in Tables I and I1 can be used to determine the adsorption isotherm slope and curvature at origin, only when the maximum concentration is 0.35 nmol cm-3 for n-hexane and 50 nmol cm-3 for benzene (cf. Figure 4a,b). On the other hand, if the solute concentration is too low, X values can be affected by large uncertainties. Therefore, a correct determination of the parameters at the origin can only be made using a rather narrow sample size range, to avoid both the extratailing influences at low concentrations and the deviation of the adsorption isotherm from
the two-term expansion used to approximate it at large concentrations. This range is, however, easy and rapid to determine. ( c ) Comparison of Methods. Both methods discussed above have been applied to calculate Kl and K2 at various temperatures. The results are given in Table 111. The values obtained for K1by the two methods are in excellent at the agreement, with a relative experimental error of 5 X confidence level of 95%. For the same confidence rate, the relative interval of K2 derived by the method of the peak apex locus is 10-15%, while it is somewhat larger with the peak-fitting method because of the presence of residual tailing of the experimental elution profiles which is not accounted for by the theoretical model. Within these limits, however, the agreement between the two sets of results is good.
Discussion The chromatographic theory of band profile previously described' and applied here to the determination of adsorption isotherms at low surface coverage takes into account, in addition to the nonlinearity of the isotherm, the dispersive term (conventional band broadening) and the sorption effect with three basic assumptions: ideality of the gas phase, a Dirac pulse function for the injection signal, and a uniform carrier-gas pressure along the column. The sorption effect as derived from the mass-balance equations has been correctly accounted for by using the method of characteristics described by Valentin, Jacob, and G u i o ~ h o n . ~It, ~is completely and quantitatively described by the term K ; R T / p or K l / c v in eq 3c. It appears only in the expression of X, which characterizes peak leaning, and is related to the slope at the origin of the apex locus function (the plot of the maximum solute concentration vs. the retention time of peak maximum). The value of this term is reported in Tale IV. Although its value remains lower than the confidence interval on K2, its contribution in the cases studied here is not always negligible as it may reach 5-10% of the value of K2. The correction for the contribution of the gas-phase nonideal behavior to the retention volume measurements is B , , p / R T , as (8) Guiochon, G.; Jacob, L.; Valentin, P. J. Chim. Phys. Phys.-Chim.B i d . 1969, 66, 1097. (9) Guiochon, G.; Jacob, L. Chromatogr. Reu. 1971, 14, 77.
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5389
Peak Profiles in Nonlinear Gas Chromatography
TABLE I V Comparison between the Characteristics of Adsorption Isotherms Derived by Different Methods (0 = 100 "C)
method model fitting peak apex locus
AD, mbar 400 600 800
benzene K,, cm3 m-2 K l r cm6 m-2 fimo1-l 2.02 -1.0 f 0.1 1.94 -1.1 f 0.2
400 600 800
step and pulse
K,, cm3 m-2 3.08 3.07 3.00
2.1 f 0.2 2.3 f 0.1 2.0 0.1
*
0.08 0.07 0.06
3.06 3.07 3.00
2.3 f 0.08 2.2 f 0.08 2.1 f 0.1
0.08 0.07 0.06
3.08
2.4 f 0.3
1.92
-0.9 f 0.3
0.05 0.04 0.04
2.01 1.94 1.91
-1.2 f 0.02 -0.74 f 0.08 -0.79 f 0.1
0.05 0.04 0.04
1.89
-0.71
* 0.15
hexane K2, cm6 m-2 pmol-'
K,/Cva
K,/Cva
" Cv is the average concentration (in pmol ~ m - of~ )the carrier gas in the column (concentration at average pressure) TABLE V Comparison between the Results Obtained for K , and K 2by Gas Chromatograpbic and Static Methods (0 = 100 "C)
benzene ,409
method peak fitting" peak apex locus" step and pulse" step and pulseb infinite dilutionC infinite dilutiond infinite dilutione
KI
9
m2 g-'
cm3 m-2
10.0 10.0
1.95 1.95 1.90 1.65 2.35
10.0 8.7 7.6 7.6 8.7 12.7 12.5
hexane K1
Qd,
K2r
cm6 m-2 pmol-' kJ mol-' cm3 m-2 Chromatographic Method 31 37
-1.0 -0.8 -0.7 -0.3
1.55 1.65 1.70
9
cm6 m-2 pmol-I
3.1 3.1 3.1
2.1
2.1 2.4
Qd,
kJ mol-' 40 40
concn range, pmol cm-3 10-4-5 X lo-' 10-4-5 X lo2 0-0.4
0-1.5
39 39 37 38 38 39
1.90
K2
I
3.6
42
0- 10-4 0-10-4 0- 10-4
Static Method adsorption isotherm
7.6
3g
-2g
408
3g
(509
43g
0.02-1.0
"This work. bReference 13. CReference17. dReference 18. Reference 11. /Reference 16. EExtrapolated for zero surface coverage. demonstrated by Everett,Iowhere B I 2is the second virial coefficient characterizing the interaction between the molecules of the carrier gas and the sample vapor. In previous work," where we used experimental conditions near the present ones (helium carrier gas), we have shown that this corrective term is about 0.2% of the value of K , and can thus be neglected. The shape of the injection signal is roughly triangular, with a width at half-height of 50 ms. This width has been maintained constant throughout the experiments whatever the amount injected. It is one of the advantages of the fluidic logic injection device to make this easily possible. As the standard deviation of the elution peak studied was between 2 and 6 s, depending on the experiment, the contribution of the injection signal to the chromatographic band broadening which is given by the rule of variance addition is at most 0.1%.3,'2 Moreover, the injection function is symmetrical and in the worst case would affect mainly the determination of the coefficient D,the apparent diffusion term. The model does not account for pressure-drop effects. Therefore, a short (50 cm long) chromatographic column was chosen to minimize this effect. The experimental validity of this assumption has been proven by checking the influence of the pressure drop on the parameters of the model. The results obtained for K , and K2, at 100 OC, are given in Table IV, as determined from profiles recorded with different pressure drops (and flow velocities). The influence of pressure on the values of K1 and K2 is low, in the range of the experimental dispersion. The range of pressures investigated is not very large: for AP varying from 0.4 to 0.8 atm, the average absolute pressure increases from 1.21 to 1.44 atm. The fact that the effect on the determination of the parameters of the isotherm is negligible is interesting but not conclusive. Determinations in a wider pressure range should be made to check this effect. As expected, on the other hand, D' is largly influenced by the carrier-gas velocity and thus by the pressure drop, for it is a measure of the global band broadening during elution: D' is related to the classical column HETP. (10) Everett, D. H. Trans. Faraday Soc. 1965, 61, 1631. (1 1) Vidal-Madjar, C.;Gonnord, M. F.; Goedert, M.; Guiochon, G. J . Phys. Chem. 1975, 79,132. (12) Sternberg, J. C. Adu. Chromatogr. ( N . Y . ) 1966, 2, 205.
It is interesting at this stage to compare our results with those of other independent determinations. We have compared them with results obtained on the same column, using a completely different chromatographic method of isotherm determination, the "step and pulse" procedure introduced by Valentin,6 which takes into account the pressure-drop effects; we have also compared our results to other ones obtained by different authors on the same systems. The principle of the "step and pulse" method is to replace the carrier gas by a gas stream containing a constant adjustable concentration of the vapor of the solute studied in the same inert gas (He) and to measure the retention time of a small perturbation pulse of this solute as a function of the vapor concentration. From these results, it is possible to derive the parameters of the isotherm equation.6 We have assumed a virial model for the adsorption isotherm e q ~ a t i o n ' ~ . ' ~
+ C1 + C 2 ( n S / A+) C , ( ~ Z ~ /+A...) ~
In p' = In ( n S / A )
(8)
wherep'is the solute partial pressure and rrS is the number of moles of adsorbed solute in equilibrium with a partial pressure p'in the gas phase. The coefficients C1, C2,C, ... were determined by using a nonlinear least-squares fit program (MINUIT) according to the procedure described by Dondi et al.I3 K 1 and K2 are related to the first two parameters of eq 8 through the relationships
K l = R T exp(-C1)
K2 = -KI2C2
(9)
Excellent agreement is observed between the results derived from the peak profiles or the apex locus method and those given by the step and pulse procedure (Table IV). The satisfactory concordance between results supplied by these independent experiments proves the validity of the procedure developed in this work to determine the sorption isotherm at low surface coverage. Quantitites which are important in adsorption thermodynamics can be derived from K , and K 2 . K , is the adsorbate-adsorbent (13) Dondi, F.; Gonnord, M. F.; Guiochon, G. J . Colloid Interface
Sci.
1977, 62, 303, 316. (14) Ross, S.; Olivier, J. P. "On Physical Adsorption" Wiley: New York, 1964.
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Jaulmes et al. isotherm presents at 100 O C a positive curvature in a narrow concentration range near the origin, the value of K2 measured by a static method and reported in Table V is meaningless, since the static method operates by extrapolation to zero surface coverage of data obtained in a concentration range much larger than the one accessible by chromatographic experiments.' The step and pulse chromatographic method has been used previously by Dondi et al.13 to determine the adsorption isotherm of benzene on graphitized carbon black at 100 OC. From their data we have calculated the constants K1 and K2 and obtained results which are significantly lower than those measured in this work. However, the aim of this last work13 was quite different from ours. It was to modelize the isotherm in a much larger concentration range than studied here. Therefore, even small deviations of their model isotherm equation from experiment near the origin strongly affect the K2 values, which are related to the curvature at the origin.
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Therefore, it is possible to compare the data obtained in this work with those published in the literature (cf. Table V and Figure 5). The values obtained for the equilibrium adsorbate-adsorbent constant, K 1 ,do not always agree with one another (cf. Table V), the difference not exceeding 20%, however. This dispersion may be due in part to differences in the graphitized carbon black samples used. From the variation of KI with temperature the differential molar heat of adsorption Qdis calculated. Within a few percent (for benzene the average is 38.2 kJ mol-', with standard deviation 1.1 kJ mol-'; for hexane the average is 41.2 kJ mol-', with standard deviation 1.5 kJ mol-'), the values of Qd measured in this work are in agreement with those published in the literature, which is most satisfactory. The variations of BzDvs. temperature are compared in Figure 5. The plots obtained are very similar to those predicted by theory or those determined experimentally for rare gases adsorbed on graphite.l5 The values of B ~ are D positive for benzene and negative for hexane in the range of temperatures studied, a fact explained by weaker adsorbateadsorbate interactions for benzene than for hexane." The BzD values for benzene measured in this work are in good agreement with the values derived from the adsorption isotherms obtained by static rnethods.l6-'* For hexane, as the adsorption (15) Steele, W. A. "The Interaction of Gases with Solid Surfaces"; Pergamon Press: London, 1974. (16) Avgul, N. N.; Kiselev, A. V. "Chemistry and Physics of Carbon"; Walker, P. L., Ed.; Marcel Dekker: New York, 1970; Vol. 6, p 1. (17) Kalashnikova, E. V.; Kiselev, A. V.; Petrova, R. S.; Shcherbakova, K. D. Chromatographia 1971, 4 , 495.
Conclusion The results obtained from the analysis of peak profiles convincingly prove that significant deviations from symmetry originates from a nonlinear isotherm, at low concentration. Adsorption isotherms are known to deviate markedly from linearity and exhibit important curvature at relatively low concentrations, so it is not surprising to observe marked effects, including large variations in the retention time of the peak maximum with increasing sample size whose direction depends essentially on the sign of the curvature of the isotherm. Smaller effects, noticeable only at simple sizes larger by orders of magnitude, can be anticipated in the case of gas-liquid chromatography, as solubility isotherms usually have a much smaller curvature around the origin. In such a case the sorption effect may become of major importance in determining the direction in which the peak leans. Better results would be obtained in gas-solid chromatography if a three-term expansion of the isotherm could be used, while the effect of the variation of carrier-gas pressure along the column does not seem significant. The reverse may be true in gas-liquid chromatography. These problems are currently under investigation. The method described here offers several important and useful features. First, it provides a theoretically sound and experimentally convenient four-parameter equation to account for peak profiles. Thus, the problems related to peak shape in chromatography can be investigated again with a solid theoretical background. The different sources of band broadening can be studied separately. Second, it is possible to determine the curvature of the isotherm at zero concentration very easily, from a peak height vs. retention time of peak maximum plot and a calibration curve of the detector response, as easily as its slope. This provides a tool for qualitative analysis in gas chromatography, as the curvature of the equilibrium isotherm of a solute on a given stationary phase is certainly very dependent on the nature of the solute. It also provides us with a convenient and rapid way of determination of isotherms in some concentration range which may be pretty large for gas-liquid and liquid-liquid isotherms. Also, the marked effect of concentration on the actual retention time of peak maximum may explain some errors and inconsistencies observed in the past in thermodynamic studies. This method can be used to study concentration ranges in adsorption studies which are not easily or even not at all reached by conventional static methods. Finally, the principles of this work are easy to extend to the case of liquid chromatography (LC) and could be used to account for the effects observed in preparative LC at moderate sample sizes. We did not take into account in this work any of the kinetic phenomena, although we encountered them in our experiments on the adsorption of benzene and hexane on graphitized carbon black. Some discrepancies between results obtained through the two methods using peak profile data, either the entire band profile or just the part around peak maxima, can most probably be (18) Kalashnikova. E. V.; Kiselev, A. V.; Petrova, R. S.; Shcherbakova, K. D.;Poshkus, K. D.;Chromatographia 1979, 12, 799.
J . Phys. Chem. 1984,88, 5391-5397 explained by these spurious adsorption kinetic effects. So far all models which include kinetic effects assume linear isotherms, an assumption we have proven to be rather unrealistic. The convolution of all these phenomena makes the problem extremely difficult to solve, however. Work is in progress in these different areas and results will be reported later.
Acknowledgment. The technical assistance of Guy Preau is greatly appreciated. We thank Francesco Dondi (Ferrara) for his help in performing the calculations for the step and pulse method. Glossary U peak area (eq 35 of preceding article) A total area of adsorbent surface in the column Two-dimensional second virial coefficient (eq 10) B2D C concentration (pmol L-I) of solute at z and t (eq 3) average concentration of carrier gas (eq 16 of preceding CV article) solute concentration at peak maximum (eq 43 of preceding CM article) height of the sample injection pulse (eq 29a of preceding CO article) c,,c2, c, coefficients of the virial model of the adsorption isotherm (eq 8)
D
global or apparent diffusion coefficient in the gas phase (eq 1 of preceding article) auxiliary variable: De/(K1 + e) (eq 3b) outlet carrier-gas flow rate (eq 1) average plate height (eq 46 of preceding article)
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R
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James and Martin factor (eq 2) first coefficient of the two-term expansion of the isotherm (es 6) second coefficient of the two-term expansion of the isotherm (eq 4) Limit capacity factor for zero sample size (eq 6 ) column length width of the sample pulse (eq 29a); sample size, CoLo number of moles of solute in the gas phase at equilibrium (eq 4) number of moles of solute in the stationary phase at equilibrium (eq 4) average column pressure solute partial pressure inlet to outlet pressure ratio molar heat of adsorption (kJ mol-') ideal gas constant (eq 7 ) column absolute temperature (eq 7 ) time; time origin at the injection of solute (eq 3) retention time, time of the peak maximum (eq 5 ) retention time of a nonretained compound (eq 6) limit retention time, obtained for a zero sample size (eq 5) auxiliary variable: uo/(l + K , / E )(eq 3a) outlet carrier gas velocity, under steady-state conditions (eq 34 column volume available to the gas phase (eq 1) abscissa along the column (eq 1 of preceding article) V o / Adefined in eq 19 of preceding article auxiliary variable: z - Ut (eq 3c) auxiliary variable: 2[(K2cv - Kl)/(Cv(e - K,))] (eq 3c) auxiliary variable: (XU/D?LoCo (eq 3d)
Mixed Micelles of Nonionic and Ionic Surfactants. A Nuclear Magnetic Resonance Self-Diffusion and Proton Relaxation Study Per-Gunnar Nilsson and Bjorn Lindman* Physical Chemistry I , Chemical Center, Lund University, S-220 07 Lund, Sweden (Received: February 6, 1984; In Final Form: April 25, 1984) The surfactant self-diffusion coefficient of mixed micellar solutions of ionic and nonionic surfactants has been measured under various conditions by the NMR pulsed field gradient technique. In addition, the line widths of the proton NMR signals have been monitored. The systems investigated are C12H25(OCH2CH2)50H (C12E5)/C12H25S04-Nat(SDS)/D,O, C,zE5/ClzH25N(CH3)3+C1(DTAC)/D20, and C,2H25(OCH2CH2)80H (C12E8)/SDS/D20. In the experimental series, the molar ratio D 2 0 to surfactant (ionic + nonionic) was kept constant while the surfactant mixing ratio was varied. For the C12E5 systems, the surfactant self-diffusion coefficient goes through a minimum when the surfactant mixing ratio is varied between 0 and 100% ionic surfactant. The observed decrease in self-diffusion coefficient as one starts to replace the nonionic surfactant by ionic surfactant is interpreted to mainly be due to an increased micelle-micelle repulsion. Then a diffusion mechanism in which monomers can be exchanged between different aggregates is partly inhibited. Such a mechanism is important for pure nonionic micellar solutions at temperatures close to the cloudpoint temperature, because then attractive interactions between the aggregates are present. The increase in self-diffusion coefficient occurring at higher fractions of ionic surfactant is shown to be due to a decrease in micelle size. For the C12E8system, the effect of the surfactant mixing ratio is much weaker which can be understood by considering the molecular geometry (large headgroup area) and the fact that the experimental temperature is far below the cloud-point temperature. Therefore repulsive interactions between the micelles are present also in the absence of ionic surfactant. The proton NMR line widths correlate well with the self-diffusion coefficients and broadening of the alkyl chain methylene signals is found when the self-diffusion coefficient is low. The broadening is interpreted to mainly be due to a partial inhibition of motions existing in the pure nonionic micellar solution which average the proton-proton dipolar couplings. The effect of various parameters, such as temperature and total surfactant concentration, on the self-diffusion coefficients and proton NMR line widths has also been investigated and interpreted according to the model described above. The dependence of the self-diffusion coefficient upon critical fluctuations is also briefly discussed. Small additions of ionic surfactant dramatically increase the cloud-point temperature but negligibly affect the observed self-diffusion coefficients. It is suggested that the micellar growth observed with increasing temperature for certain nonionic surfactant systems is determined by the absolute temperature rather than the distance from the phase separation limit.
Introduction Nonionic surfactants of the poly(ethy1ene oxide) variety are known to form mixed micelles with ionic surfactants in aqueous solution.'-6 Depending on the particular surfactants chosen as (1)
Ruriyama, K.; Inoue, H.; Nakagawa, T. Kolloid Z.Z.Polym.
1962,
183, 68.
0022-3654/84/2088-5391$01 S O / O
well as their mixing ratio, the total surfactant concentration, temperature, etc. one expects the micelle size and shape as well (2) Corkill, J. M.; Goodman, J. F.; Tate, J. R. Trans. Faraday SOC.1964, 60,986. ( 3 ) Schick, M. J.; Manning, D. F. J . Am. Oil Chem. SOC.1965, 43, 133. (4) Tokiwa, F.; Moriyama, N. J . Colloid Interface Sci. 1969, 30, 338.
0 1984 American Chemical Society
