J. Phys. Ckem. 1984,88, 5315-5319 of added salt to remove the broad peaks corresponds approximately to the same condition:
-
At Cs N the scattered intensity is high at q loT3k'and decreases as a function of q typically like the form factor of a single rod (Z(q) = (ql)-I though the electrostatic length decreases strongly. Another study on dynamics of NaPSS28shows that the amplitudes of both slow and fast relaxation modes depend on added salt. For (Cm/2Cs)(a/Q)> 1 the amplitude of fast modes is very low and we observe generally the slow relaxation mode. An important transition appears; when (C,/2Cs)(a/Q) 1, the interactive peak disappears, the amplitude of slow relaxation time becomes very low, and we observe a large increasing of scattered intensity in the presence of very fast relaxation mode. If (Cm/ 2Cs(a/Q) < 1, only the fast relaxation time is present and corresponds to the cooperative diffusion. The correlation between the presence or the disappearance of the peak and the modification on the dynamics of solution is new information on the properties of polyelectrolyte solutions.
Conclusion In this paper we have investigated the properties of polyelectrolytes by light scattering. In particular, we have focused our attention on the conformation of linear polyelectrolytes in the dilute/semidilute regime. The scattered intensity is characterized by a small intensity at q < .&-'and by a single broad peak whose position is shifted toward higher momentum transfer q as the polyion concentration increases (qM (28)Drifford, M.; Dalbiez, J. P., to be submitted for publication.
5375
= C'I2). The same behavior (shift and shape of peak) is observed in both neutron and light scattering-where the concentration range is very different. The measured diffusion coefficient Deff is q dependent and shows a broad minimum at q = qM. This suggests, as the neutron results do,I8 that the origin of the peak is due to repulsive interchain interactions giving rise to some short-range order between the chains. The equality of the value of 2?r/qM and the spacing between aligned macromolecules, which is molecular weight independent, confirm that the conformation of polyelectrolytes at low concentrations can be described as a model of aligned stiff chains. A local short-range order of a few interparticle spacings due to the Coulombic interaction is probable. The degree of stiffness of single-chain conformation is difficult to determine precisely. For the very low polyelectrolyte concentration, the "rod limit" condition is obtained and the chains are very stiff. For intermediate concentration the macromolecules are more or less stretched. An acceptable description of the polyelectrolyte solution without added salt at the dilute/semidilute transition can be based on the presence of stable and anisotropic domains of stiff chains. These domains can move isotropically in the medium. The fundamental parameter I' (Debye distance) gives a strong stability of the domains. Only which varies as C1/z the addition of salt screens the electrostatic interaction and destroys the structure of domains. A large effect is observed with both the disappearance of the peak and the modification of the dynamics of macromolecules. Acknowledgment. We are indebted to M. Nierlich for providing the NaPSS(380000) sample. We thank T. Odijk, M. Nierlich, G. Jannink, P. G. De Gennes, F. Brochard, and L. Belloni for advice and helpful comments. We also thank P. Tivant for critical reading of the manuscript. Registry No. Sodium poly(styrenesulfonate), 9080-79-9.
Pulse Radiolysis Measurements of the Solvation Rate of Benzophenone Anion in Liquid Alcohol: Effect of Temperature J. L. Marigniert and B. Hickel" Centre d'Etudes Nucleaires de Saclay, DZpartement de Physico-chimie, (LA331), 91 191 Gif-sur- Yvette Cedex, France (Received: November 22, 1983)
The pulse radiolysis of benzophenonein liquid 1-propanolwas investigated within a large temperature range. At temperatures lower than 180 K the absorption spectrum of benzophenone anion is time dependent. The blue shift observed after the pulse is attributed to the reorientation of alcohol molecules around the aromatic anion which was formed in a presolvated state. The kinetics of this spectral shift was studied as a function of temperature. Comparison with the dielectric relaxation times in 1-propanol shows that the rate-determining step is the breaking of solvent hydrogen bonds. The possibility that part of the spectral shift is due to the formation of a hydrogen bond between the solvent and the anion is discussed.
Introduction In a previous publication' we reported that the spectrum of the benzophenone anion (C,H,),CO- formed by pulse radiolysis in liquid propanol near the melting point is time dependent. This behavior observed before only in the glassy state2J has been attributed to the reorientation of solvent molecules around the aromatic At low temperature in liquid 1-propanol the electron solvation is a comparatively slow process, easily obervable in the microsecond Some aromatic compounds like benzophenone are able to scavenge electrons before solvation: giving the corresponding anion. It is then possible to inject Present address: Laboratoire de Physico-chimie des rayonnements (LA 75). Bit. 350,91405 Orsay Cedex, France.
0022-3654/84/2088-5375$01.50/0
aromatic anions in the liquid phase with the same shell of solvent molecules as the neutral parent compound if the electron capture rate is faster than the dielectric relaxation time of the medium. Owing to the ion dipole interaction, reorientation of polar solvent molecules would take place around the anion, and this can be (1) J. L. Marignier and B. Hickel, Chem. Phys. Lett., 86, 95 (1982). (2) M. Hoshino, S.Arai, and M. Imamura, J . Phys. Chem., 78, 1437 ( 1974). (3) R. K. Huddleston and J. R. Miller, Radiat. Phys. Chem., 17, 383 (1981). (4) J. H.Baxendale and P. Wardman, J . Chem. Soc., Faraday Trans. 1 , 69,524 (1973). ( 5 ) L. Gilles, M. R. Bono, and M. Schmidt, Can. J . Chem., 55, 2003 (1977). ( 6 ) K.Y.Lam and J. W. Hunt, Int. J . Radiat. Phys. Chem., 7, 317 (1975).
0 1984 American Chemical Society
5376
Marignier and Hickel
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 15t
O _L
O'
LbO
500
660 Ahmi
700
800
Figure 1. Transient absorption spectra following pulse radiolysis of 0.4 M benzophenone in 1-propanol at room temperature after (a) 0.05, (b) 0.1, (c) 0.2, (d) 1, and (e) 10 ps.
Results Experiments at Room Temperature. ( a ) Neutral Solutions. When a solution of 0.4 M benzophenone in 1-propanol is irradiated with a 20-11s electron pulse, a transient spectrum is observed with maxima at 555 and 620 nm (Figure 1). The band at 555 nm has been attributed to the ketyl radical (C6H5)zCOH.'2-'5 The other band, which is much more solvent dependent, is due to the benzophenone anion (C6H5),CO-.'2-13~15 As a function of time between 50 ns and 10 p s , the band at 620 nm decreased almost completely whereas the band at 555 nm remained approximately unchanged. The decay at 620 nm follows second-order kinetics (7) D. W. Davidson and R. H. Cole, J . Chem. Phys., 19, 1484 (1951). (8) R. H. Cole and D. W. Davidson, J . Chem. Phys., 20, 1389 (1952). (9) H. Fellner-Feldegg, J . Phys. Chem., 73, 616 (1969). (10) B. Lesigne and R. Sauneuf, Rev. Sci. Instrum., 47, 1063 (1976). (11) B. Hickel, J . Phys. Chem., 82, 1005 (1978). (12) G . Porter and F. Wilkinson, Trans. Faraday SOC.,57, 1686 (1961). (13) E. J. Land, Proc. R. SOC.London, Ser. A, 305, 457 (1968). (14) E. Hayon, I. Ibata, N. N. Lichtin, and M. Simic, J . Phys. Chem., 76, 2072 (1972). (15) G. E. Adams and R. L. Wilson, J. Chem. SOC.,Faraday Trans. 1,69, 719 (1973).
-
LOO
-
L
- L 500
600
700
800
Ah
m) Figure 2. Transient absorption spectra following pulse radiolysis of 1O-* M benzophenone in 1-propanol at room temperature after (a) 0.2, (b) 0.4, (c) 0.8, and (d) 1.6 ps.
observed as a spectral shift as a function of time. If this interpretation is correct, a study of time-dependent spectra of anions formed by electron capture would give information on the response of solvent molecules to the strong field due to the negative charge of the ion. It would then be interesting to compare the rate of anion solvation with the macroscopic relaxation time of 1-propanol which is measured at low In this article we report the effect of temperature on the spectral relaxation time of benzophenone anion in liquid 1-propanol. This solvent is particularly suitable for such measurements because dielectric relaxation times are available for a large temperature ra~~ge.~-~
Experimental Section The pulse radiolysis experiments were carried out with a modified Febetron 707 delivering single pulses of electrons in the energy range 1.6-1.8 MeV.l0 The total duration of the electron pulse is less than 20 ns. Dose variations from pulse to pulse were monitored with a charge integrating circuit which gives a voltage proportional to the dose. The suprasil irradiation cell has an optical path length of 2.5 cm. The temperature inside the cell can be varied between -160 and +50 O C S 5 The temperature was measured by calibrated thermocouples. Absorption spectra were recorded point by point with a spectrophotometric detection system. The analyzing light from a 450-W Osram xenon arc was intensified for 1 ms to increase the signal to noise ratio. The light at the entrance of the irradiation cell was collimated by a I-mm slit to minimize dose and temperature gradients. Details on the pulse radiolysis setup are given in previous publications.'J1 The solvent, 1-propanol (Merck pro analysis), was purified by distillation. The other reagents were of analytical grade and used without purification. The solutions before irradiation were deaerated by bubbling with high-purity argon.
L
oL--------l----
L c 600
500
LOO
>\ 700
. L
800
mi
+
Figure 3. Pulse radiolysis of 0.4 M benzophenone 8.3 X lo4 M HC104 in 1-propanol at room temperature after (a) 0.05, (b) 0.1, (c) 0.2, and (d) 1.6 ps.
*
with a slope k / ( d ) = (1.4 0.2) X lo6 s-' and k/r = (3.5 f 0.5) X lo6 cm s-l. A different behavior is observed when the concentration of benzophenone is lowered to M. The relative intensities of the two peaks are inverted, and after a short time the ketyl radical band at 555 nm appears only as a shoulder on the main absorption at 620 nm (Figure 2). About 800 ns after the pulse the two bands are clearly distinct. The benzophenone anion band decreases as a function of time, and again only a small change is observed at 555 nm during the same period. When the benzophenone concentration is lower than M, the absorption spectrum of solvated electrons is observed immediately after the pulse. The decay of the solvated electron at 720 nm, where the absorption of (C6H5)&O- is negligible, follows pseudo-first-order kinetics with a rate proportional to the concentration of benzophenone. The rate constant for the reaction of solvated electrons and benzophenone in I-propanol is k(e-, (C6H5)@) = (6 f 0.5) X lo9 M-' S - ' . (b) Acid Solutions. When the 0.4 M solution of benzophenone is acidified with HClO,, the decay of the band at 620 nm is faster than in neutral solution (Figure 3). The benzophenone concentration is much higher than the acid concentration so the solvated electrons react only with benzophenone molecules. For [HC104] = 8.3 X lo4 M the decay of benzophenone anion follows pseudo-first-order kinetics with k = (6 f 0.6) X lo6 s-l. If we assume that HCIOl is completely dissociated, then k((C6H5),COH+) = (7.2 f 0.8) X lo9 M-l s-I. In 1-propanol at 22 O C the dielectric constant is 20.7. Extrapolation to zero ionic strength gives ko((C6H5)2CO- H+) = (1.2 f 0.2) X 1O'O M-' s-' for the reaction
+
+
+
( c 6 H ~ ) z c O - H+
-+
(C6H5)2COH
After 2 p s all the benzophenone anions are converted into ketyl radicals which decay in the millisecond range by second-order kinetics with 2k/e = (1.2 0.2) X lo4 cm s-*. 2(C6H5)2COH ((C6H5)2C0H)2
*
+
If we take for the value 2800 M-' cm-' found in water,I5 then 2k = (3.4 f 0.6) X lo7 M-' s-l. This value is comparable with those obtained by flash photolysis of benzophenone in 2-propanol so1utions.l6
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5317
Solvation Rate of Benzophenone Anion in Liquid Alcohol
0.4 -
.OD
0.3-
1
? m
0.2-
v
n 0 I
I
I
1
500
600
700
800
0.1 -
A(nm) Figure 4. Pulse radiolysis of 0.4 M benzophenone lo-' M KOH in 1-propanol at room temperature after (a) 1, (b) 2, (c) 5, and (d) 15 ps.
+
01 500
I
I
600
700
I
A h
OL-
800 m)
900
Figure 6. Transient absorption spectra of benzophenone anion in liquid 1-propanol at 159 K after (a) 0.05, (b) 0.1, (c) 0.2, (d) 0.4, and (e) 1.6 ps. The inset shows a kinetic plot for the decay at X = 825 nm.
- 03-
T ('C)
-122
3
-140
rb v
n 0
02-
01-
1
1
I
1
I
\ ' i t
\
i
Figure 7. Effect of temperature on the kinetics of the spectral shift at 826 nm: (+) fast process, ( 0 )slow process.
absorption at 555 nm is complete within the duration of the pulse (less than 20 ns) and remains stable for hours at 144 K. After the first pulse, therefore, no absorption measurement can be performed at X < 600 nm. Figures 5 and 6 show two examples of transient spectra at 137 and 159 K over the wavelength range 600-1000 nm. Immediately after the pulse the maximum lies in the near-infrared region, then with time the band shifts to shorter wavelengths. This spectral shift, which i s solvent and temperature dependent, has already been attributed to the solvation process of benzophenone anion in liquid' or in g l a s ~ e s . ~ ~ ~ The kinetics of the spectral shift was studied at 825 nm where the absorption of the anion, when fully solvated, is negligible. Some experiments were performed at longer wavelengths and gave identical results. At short times there is a deviation from a simple first-order plot (insets of Figures 5 and 6). This deviation can be accounted for by another parallel first-order process (full circles in the insets) which is faster by about 1 order of magnitude.' During the spectral shift there is no loss of (C,H,),CO-, and the decay of the solvated anion takes place at a later stage. The fast process, responsible for most of the spectral shift, has an activation energy of 5.9 0.7 kcal/mol. The slow process has an activation energy of 7.4 0.8 kcal/mol (Figure 7).
* *
5318
Marignier and Hickel
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984
Discussion Formation of the Ketyl Radical. W e have seen that ketyl radical is completely formed during the electron pulse in concentrated benzophenone solution. At room temperature and particularly at low temperature a reaction between benzophenone and alcohol radicals is too slow to explain the very rapid formation of the ketyl radical. The more likely mechanism is a fast protonation of the presolvated benzophenone anion taking place inside the spurs. The same mechanism has been proposed to account for the rapid formation of ketyl radicals in acidic matrices of alcohol^;'^ ROH
--
ROH'
+ e-
+ (C6H5)2CO (C,L,H~)~COROH+ + R O H ROH2+ + R'OH (C6H5)2CO-+ ROH2+ (C,&)@H 4- R O H e-
+
-
-+
In this mechanism e- is an electron before solvation (this can be a subexcitation electron) and (C6H5)&O- a presolvated benzophenone anion since at low temperature the appearance of the ketyl radical is faster than the solvation of the electron or the benzophenone anion. At room temperature it is possible that at least part of the reactions takes place between solvated species, but most of the electrons will be scavenged by benzophenone before solvation.6 The relative value of the extinction coefficients of (C6HJ2CO- and (C6H5)2COHcan be estimated from experiments in acid and alkaline solutions (Figures 3 and 4). At the (C6H5)2CO- absorption maximum the contribution of (C6H5)2COH is negligible (Figure 3, curve d). The reverse is not true, and at 545 nm the protonation or deprotonation reaction occurs with only a small change in optical density which means that the extinction coefficients of both species are approximately the same at this wavelength. Comparison of the relative intensities of the two peaks in neutral solution (Figure 1) shows that about half the benzophenone anions are protonated inside the spurs: (C6H5)2CO-4H+ (C&)2COH. The comparatively slow decay a t 620 nm which follows second-order kinetics may then be due to protonation in the bulk of the solution where the concentrations of H+ and (C6H5)2CO-are equal. At 620 nm k / e = (3.5 f 0.5) X lo6 cm s-l; if we take for k the value found in acid solution, then €620 = 3400 f 800 M-l cm-'. This value, lower than those found in water at 600 nm (e = 4800 M-' cm-'),I5 is however a lower limit because we have assumed for the determination of k that HC104is completely dissociated in 1-propanol. Solvation of the Benzophenone Anion. In a previous publication' we compared the rates of solvation of the benzophenone anion with the dielectric relaxation time of l - p r ~ p a n o l .It~ was ~~ found that at 146 K,just above the freezing point, the slow process of solvation is correlated with the first dielectric relaxation time, attributed to the breaking of the hydrogen bond. The effect of temperature on the rate of benzophenone anion solvation strengthens this interpretation. In the same temperature range the mean activation energies for the dielectric rate constant of solvation are 7.6 f 0.9 and 7.4 f 0.8 kcal/mol, respectively. The fact that the slow process is only responsible for a small spectral shift may be an indication that the reorientation of the second or more distant shells of solvent molecules is involved. The effect of the charge on their reorientation rate would be screened by the first solvent shell arid should not be very different from the first dielectric relaxation time as observed experimentally (Table I). The fast process may be then attributed to the reorientation of solvent molecules close to the negative charge, which for the benzophenone anion is mainly located on the carbonyl group.18 The fast process would involve also, as a rate-determining step, the breaking of hydrogen bonds prior to the reorientation of solvent molecules. Although an exact theory is still lacking, it has often
-
(16) A. Beckett and G. Porter, Trans. Faraday SOC.,59, 2028 (1963). (17) M. Hoshino, S. Arai, M. Imamura, K. Ikehara, and Y. Hama, J. Phys. Chem., 84, 2576 (1980). (18) T. Shida and W. H. Hamill, J . Am. Chem. SOC.,88, 2683 (1966).
TABLE I: Dielectric Properties and Rate Constants of Solvation in Liquid 1-Propanol at 146 K E, kcal/mol
k, s-I electrons (C6H5)&O- (fast process) (C6H&CO- (slow process) dielectric relaxation ( l / r ) dielectric relaxation at constant charge ( ( l / r ) c ~ / c . d
1.5 x 1.5 X 1.1 x 4.7 x 4.8 X
107 lo6 105 104 lo5
ref
5.9 5.9 7.4
4
7.6
8 8
7.3
been s u g g e ~ t e d 'that ~ , ~ in ~ the presence of a charge the dielectric relaxation would be speeded up by a factor close to eo/€-, where eo and t, are respectively the dielectric constant at low and high frequency on the Cole and Davidson p l ~ t . For ~ , ~1-propanol this ratio is slightly temperature dependent and close to 10 at the melting point. The activation energy for 7' is 7.3 f 0.9 kcal/mol. It is interesting to note that for I-propanol a good correlation exists between the dielectric relaxation time and the viscosity which follows the Tamman and Hesse e q ~ a t i o n .If~ the fast process is due to reorientation of the first shell of solvent molecules around the anion, the rate must depend also on the local viscosity, which is not expected to be exactly the same as in the bulk solvent. It would be tempting to state that the activation energy difference between 7' and the fast process reflects this situation, but this difference is too small to be really significant. Influence of Hydrogen Bonding. Although the time-dependent spectral shift is generally attributed to stabilization of the ground state by solvation, the possibility remains that part of this shift is due to the formation of a hydrogen bond between the solvent and the anion. This process was invoked to explain the remarkable - ' ~ in ethanolblue shift observed in hydroxylic ~ o l v e n t s ' ~and acetonitrile mixture.20 The magnitude AV of the spectral shift caused by hydrogen (H) bonding has been discussed by PimenteL21 A y = ya - yo = Wo - Wl ~1
+
where yo is the transition energy in a hydrocarbon and ya is the transition energy in an H-bonding solvent, Woand Wl are the H-bond energies in the ground and excited states, respectively, and w1 is the extra transition energy imposed by the FranckCondon principle. The absorption spectrum of the benzophenone anion in the visible and near-IR region has been attributed to an electron charge transfer from the carbonyl group to the benzene rings.I9 The H bond in the ground state is stronger than in the excited state, and a blue shift is expected. In a dilute solution of alcohols in a nonpolar solvent benzophenone itself is relatively weakly H bonded (AH 2-4 k ~ a l / m o l ) . ~In ~ -pure ~ ~ alcohols, however, the H-bonded benzophenone molecule fraction is unknown. If the benzophenone were already H bonded with the 1-propanol, the formation of a negative ion would only strengthen and shorten the bond. This process is probably too fast to be observed. On the other hand, if benzophenone were not H bonded, the appearance of a negative charge on the carbonyl group would favor the establishment of a strong H bond. This process involves the rotation of an alcohol molecule and the concomitant breaking of a H bond. Clearly, the temperature effect cannot distinguish between solvation and the formation of H bonds. Work is in progress to study the solvation of the biphenyl anion in solvents where H bonding is absent. Comparison with Electron Solvation. In the same solvent and over the same temperature range electron solvation is at least 1 order of magnitude faster than benzophenone anion solvation (Table I). Although the activation energies for the fast process and for electron solvation are equal, this does not prove that the solvation mechanisms are the same. Electrons can jump from a shallow trap to a deeper trap, whereas big organic anions cannot. N
(19) T. Shida, S.Iwata, and M. Imamura, J. Phys. Chem., 78, 741 (1974). (20) J. D. Simon and K. S. Peters, J . Phys. Chem., 87, 4855 (1983). (21) G . C. Pimentel, J . Am. Chem. SOC.,79, 3323 (1957). (22) G. J. Bealey and M. Kasha, J . Am. Chem. SOC.,77, 4462 (1955). (23) E. D. Becker, Spectrochim. Acta, 17, 436 (1961). (24) R. G. Lewis and J. J. Freeman, J . Mol. Spectrosc., 32, 24 (1969).
J. Phys. Chem. 1984,88, 5379-5385
5379
again the latter is at least 2 orders of magnitude faster than the former. Under these conditions the equality of the activation energies would simply mean that establishment of solvation shells around the anions, hydrogen-bond formation between anions and alcohol molecules, or creation of deep traps to which electrons can jump involves in each case the breaking of solvent hydrogen bonds as a primary step.
The possibility of anion migration, by hopping, to a more favorable solvation site has been ruled out in glassy ethan01.~ In liquid phase too, when some solvent reorientation has occurred around the anion, the probability of finding a neutral molecule with a “better“ solvation shell would be low. Electron transfer between benzophenone anion partly solvated and benzophenone is unfavorable and must be comparatively slow. Recent work in n-propanolalkane mixtures at low temperature shows good correlation between dielectric relaxation rate and electron solvation rate,25but
Acknowledgment. The authors thank J. Potier for technical assistance.
~
Registry NO. (CsHS)2CO, 119-61-9; (C,Hs),COH, 4971-41-9; (C6H,),CO-, 16592-08-8; 1-propanol, 7 1-23-8.
( 2 5 ) J. Mayer, J. L. Gebicki, E. Szajdzinska, and J. Kroh, Radiat. Phys. Chem., 19, 491 (1982).
Study of Peak Profiles in Nonlinear Gas Chromatography. 1. Derivation of a Theoretical Model A. Jaulmes, C. Vidal-Madjar, A. Ladurelli, and G. Guiochon* Laboratoire de Chimie Analytique Physique, Ecole Polytechnique, 91 128 Palaiseau Cedex, France (Received: January 3, 1984; In Final Form: June I , 1984)
A theoretical model is proposed for the profile of elution peaks in gas chromatography. This model accounts both for the influence of the isotherm curvature at zero concentration and for the perturbation of the flow rate due to solute exchange between the mobile and stationary phases. The model relies on a pulse injection function and needs only four parameters: peak area, axial dispersion coefficient, limit retention time, and leaning coefficient. The last two parameters are directly related to the slope and curvature of the adsorption isotherm at zero partial pressure of the solute in the gas phase. The theoretical method ascertains the convenience of determining these two isotherm parameters from the variation of the peak maximum retention time with solute amount
Introduction Detailed studies of the profiles of elution peaks obtained in gas chromatography show that they are rarely Gaussian curves or even symmetrical. The problem of properly accounting for these experimental profiles has remained frustratingly unsolved in spite of numerous investigations. This problem is complex because whereas on the surface chromatography looks deceptively simple, elution profiles result from the interaction between various physical chemical phenomena, none of them being simple. The first series of phenomena which may be considered as responsible for band broadening’ are axial diffusion, the resistance to mass transfer in the mobile phase, in the stagnant gas phase (inside particles), and in the stationary phase, at the various interfaces, the effects of irregularities of the packing, and the resulting unevenness of flow pattern. Although they will always be present, however, these phenomena cannot explain the often unsymmetrical peak profiles observed. Peak tailing is often blamed on another phenomenon, the kiIf the latter is slow and the netics of ad~orption-desorption.~~~ surface is heterogeneous, molecules adsorbed on high-energy sites can be markedly delayed while most of the sample does not interact with these sites. This certainly explains a number of results, although no quantitative study of experimental data has even been published. But it does not account for most of the unsymmetrical peaks with tailing and certainly cannot explain leading peaks. In an attempt to study this previous effect in detail, we observed that significant tailing was often a result of the injection method.4 Samples eluted under identical conditions on the same column would give unsymmetrical peaks when injected with a valve or (1) Giddings, J. C. “Dynamics of Chromatography”, Part I; Marcel Dekker: New York, 1965. (2) Giddings, J. C. Anal. Chem. 1963, 35, 1999. (3) Villermaux, J. J . Chromatogr. Sci. 1974, J2, 822. (4) Jaulmes, A.; Vidal-Madjar, C.; Guiochon, G.,unpublished results.
(5) Gaspar, G.; Arpino, P.; Guiochon, G. J. Chromatogr. Sri. 1977, 15, 256. (6) Houghton, G. J . Phys. Chem. 1963, 67, 84.
0022-3654/84/2088-5379$01.50/0 , I
,
a syringe and nearly Gaussian-shape peaks when injected with the fluidic logic gate developed in this laboratory and used in this work.5 This instrumental effect seems to be often prevalent in experimental data and should be either eliminated by careful design or corrected for. A more fundamental source of band broadening and an essential phenomenon in determining band profiles are to be found, however, in the thermodynamics of chromatography. All theoretical models should and did rely on the resolution of the mass-balance equations for the solute and mobile phases, implying a partition phenomenon between the mobile and stationary phases. This system of partial differential equations is not simple, however, and its solution is difficult.22 At the low concentrations used in analytical chromatography, the simplest models assume the partition coefficient to be independent of solute concentration. This assumption, which would be strictly valid only at infinite dilution, results in an easy solution and a Gaussian band profile. It is seldom corroborated, however, by experimental results, even at very low solute concentrations. It is therefore interesting (both from a theoretical and an experimental point of view) to study the influence of a nonlinear equilibrium isotherm on the shape of the elution profile. Houghtod was the first, as far as we know, to give an analytical solution of the second-order differential equation accounting for a nonlinear isotherm. In this model, however, the carrier-gas mass balance is omitted; hence, the sorption effect is neglected. This effect, which results from the presence of the solute in the mobile phase, is proportional to the difference between the partial molar volumes of the solute in the mobile and stationary phases and is a function of the solute concentration. It is important in gas chromatography at temperatures where the solute vapor pressure
~
0 1984 American Chemical Society
