Anal. Chem. 1999, 71, 4472-4479
Stochastic-Dispersive Theory of Chromatography Attila Felinger,*,† Alberto Cavazzini,‡ Maurizio Remelli,‡ and Francesco Dondi‡
Department of Analytical Chemistry, University of Veszpre´ m, Egyetem utca 10, H-8200 Veszpre´ m, Hungary, and Department of Chemistry, University of Ferrara, Via Luigi Borsari, 46, I-44100 Ferrara, Italy
The stochastic model of chromatography has been combined with mobile-phase dispersion. With the combined model, both the effect of slow mass transfer or adsorptiondesorption kinetics and dispersion on the band profile can be characterized. The stochastic model of chromatography is addressed with the characteristic function method. The moments of the peaks are calculated analytically for homogeneous and heterogeneous surfaces. It is shown that even in cases when the characteristic function cannot be calculated in closed form, the moments of the peak, and therefore the retention time, the number of theoretical plates, the peak asymmetry, can be calculated with simple expressions. Therefore, a full description of the chromatographic peak is available for homogeneous and any heterogeneous surfaces provided that the distribution of the sorption energies is known. The stochastic model of chromatography was developed by Giddings and Eyring in the 1950s.1 This model depicts the separation at a molecular level. A random migration of a molecule along the column is studied. The chromatogram is determined as the probability density function of the residence time of the molecules in the column. Usually, the stochastic model neglects the dispersion occurring in the chromatographic column; only the effect of slow mass transfer or adsorption-desorption kinetics is taken into account. Giddings and Eyring assumed that, while migrating along the column, a molecule performs a random number of adsorption and desorption steps.1 This random quantity can be described by a Poisson distribution being n the average number of sorption steps. Furthermore, when a molecule is adsorbed on a site in the stationary phase, the sojourn time of the molecule on the site is another random variable. This latter random variable follows an exponential distribution with a time constant τ which is the average sojourn time. These assumptions led to a complex peak profile containing the Bessel function of the first order and first kind. A significant effort has been devoted to the extension of the stochastic model to heterogeneous surfaces, but the handling of the problem in time domain resulted in very complex expressions, inappropriate for the analysis of the peak profiles.1-3 * Corresponding author: (e-mail)
[email protected]; (fax) + 36 88 421869. † University of Veszpre ´ m. ‡ University of Ferrara. (1) Giddings, J. C.; Eyring, H. J. Phys. Chem. 1955, 59, 416-421. (2) Giddings, J. C. Anal. Chem. 1963, 35, 1999-2002.
4472 Analytical Chemistry, Vol. 71, No. 20, October 15, 1999
The method of characteristic functions (CF) was first used by McQuarrie to enhance the stochastic model of chromatography.4 A thorough discussion of the applicability of the CF to exploit the stochastic model was given by Dondi and Remelli.5 They showed that many models of chromatography can identically be derived by means of the method of characteristic functions, including, for instance, the Martin-Synge plate model or the molecular dynamic model of Giddings and Eyring.6 A recent publication evidenced that the stochastic theory can be extended to two-site heterogeneous surfaces with simple mathematics.7 By means of the CF method, closed form expressions are obtained in Fourier or Laplace domain for the band profiles; thus, the moments can directly be calculated even if the transformation into time domain is possible only numerically.8 Very recently, we developed the stochastic theory of any type for multiple-site heterogeneous chromatography.9 With the stochastic model, any unimodal or multimodal distribution of sorption energies can be taken into account. It has been revealed that the stochastic model and the lumped kinetic model of chromatography are identical theories to model the separation process. The real benefit of the stochastic model is that it can include with unforeseen ease the effect of any surface heterogeneity; most often the only numerical step is the inverse transformation of the peak profile into time domain. It has also been shown by these authors that kinetic tailing is probably more often observed than previously presumed: a minor dispersion of the sorption energies on an otherwise homogeneous surface will cause peak tailing, and this tailing depends directly on the value of the standard deviation of the energy dispersion.9 In this study, our aim is to further generalize the stochastic model embracing the effect of axial dispersion and to give more generic expressions for the fundamental quantities characterizing the efficiency of the separation and the peak shape parameters. THEORY We assume that molecules move at random in the mobile phase. This random walk model is the simplest molecular model (3) Giddings, J. C. Dynamics of Chromatography; Marcel Dekker: New York, 1965. (4) McQuarrie, D. A. J. Chem. Phys. 1963, 38, 437-445. (5) Dondi, F.; Remelli, M. J. Phys. Chem. 1986, 90, 1885- -1891. (6) Dondi, F.; Cavazzini, A.; Remelli, M. Adv. Chromatogr. 1998, 38, 51-74. (7) Cavazzini, A.; Remelli, M.; Dondi, F. J. Microcolumn Sep. 1997, 9, 295302. (8) Felinger, A. Data Analysis and Signal Processing in Chromatography; Elsevier: Amsterdam, 1998. (9) Cavazzini, A.; Remelli, M.; Dondi, F.; Felinger, A. Anal. Chem. 1999, 71, 3453-3462. 10.1021/ac990412u CCC: $18.00
© 1999 American Chemical Society Published on Web 09/18/1999
of chromatography.10 The one-dimensional random walk model assumes that the migrating particle moves in the positive or negative direction depending on the outcome of a random experiment. In a symmetrical case the movement in both directions has a probability of p ) 1/2. The probability that the particle moves exactly k times in one direction is given by the binomial distribution
P(k))
()
n k p (1 - p)n-k k
(1)
After n steps (k steps to the right and n - k to the left), each of δ length, the expected location of the particle is11
E(ln) ) E{[k - (n - k)]δ} ) [2E(k) - n]δ
(2)
As the first moment of the binomial distribution is E(k) ) np, the most probable location of the particle remains at the origin: E(ln) ) 0.10,11 The variance of the spread is
σ2l
)
E(l2n)
equation8,13
f(t) )
u
x4πDt0
[
exp -
2
2
]
(7)
where u is the linear velocity of the mobile phase. The discussion of this assumption will be given later. D is the mobile-phase diffusion coefficient. In gas chromatography, the diffusion in mobile phase is about 104 times faster than in the stationary phase; therefore, the effect of the latter on D can be ignored. In liquid chromatography, however, the diffusion in the two phases is similar, and D is composed of the diffusion in the mobile and stationary phases. The characteristic function of the random variable X is the mathematical expectation E{eiωX}, where ω is an auxiliary variable and i is the imaginary unit.14 For a continuous random variable, the CF is calculated by the following equation
Φ(ω) ) E{eiωX} )
) E{[(2k - n)δ] } ) [4E(k ) - 4E(k)n + n ] δ (3) 2
u2 (t - t0)2 4Dt0
2
∫
∞
-∞
eiωtf(t) dt
(8)
where Φ(ω) is the CF. The CF of a discrete random variable is As the second moment of the binomial distribution is E(k2) ) n2p2 + np(1 - p), we shall get
e
-l2/4Dt
x4πDt
(9)
pj
The probability density function and the CF of a random variable form a Fourier transform pair. When the molecule travels along the column, the number of adsorption-desorption steps is governed by Poisson statistics. The CF for various mobile-phase velocity assumptions is discussed by Dondi et al.13 For the CF of the residence time in the column, they obtained the following generic expression:
ΦR(ω) ) Φm
(
)
ln Φr(ω) i
(10)
(5)
The variance of the above distribution is
σ2l ) 2Dt
iωxj
(4)
When molecules diffuse randomly, the number of steps per second is enormous, so the binomial distribution given in eq 1 can be replaced by a Gaussian distribution with a mean and variance identical to those of the binomial distribution. Using the substitutions l ) (2k - n)δ, t ) nτδ, and D ) δ2/2τδswhere τδ is the time needed for one step and D is the diffusion coefficientswe get the distribution derived by Einstein for a randomly migrating particle:12
P(l) )
∑e j)1
σ2l ) nδ2
1
n
Φ(ω) ) E{eiωX} )
(6)
where Φm is the CF of the time spent in the mobile phase and Φr is the CF of the residence time in the column on a time scale normalized with respect to t0. Assuming that the number of sorption steps follows a Poisson distribution, Dondi et al. obtained the following CF for the normalized residence time:
Therefore, a connection is made between the molecular and macroscopic model of random walk, and diffusion is described as a result of independent random movements. In chromatography, the diffusion described above is combined with a constant-velocity mobile-phase convection. Furthermore, since the distribution given by this model deviates from zero just in the vicinity of the hold-up time, the residence time of a molecule in the mobile phase is, therefore, modeled here with the following
The CF of the residence time in the mobile phase (eq 7) is
(10) Giddings, J. C. J. Chem. Educ. 1958, 35, 588- -591. (11) Berg, H. C. Random Walks in Biology; Princeton University Press: Princeton, NJ, 1993. (12) Einstein, A. Investigations on the Theory of the Brownian Movement; Dover Publications: New York, 1956.
(13) Dondi, F.; Blo, G.; Remelli, M.; Reschiglian, P. In Theoretical Advancement in Chromatography and Related Separation Techniques; Dondi, F., Guiochon, G., Eds.; Kluwer: Dordrecht, 1992; pp 173-210. (14) Crame´r, H. Mathematical Methods of Statistics; Princeton University Press: Princeton, NJ, 1957.
Φr (ω) ) exp[(n/t0)[Φs(ω) - 1] + iω]
[
Φm(ω) ) exp iωt0 -
] [
ω2Dt30 L
2
(11)
]
L ω2DL ) exp iω u u3
Analytical Chemistry, Vol. 71, No. 20, October 15, 1999
(12)
4473
The CF of the distribution of the residence times in the column is obtained when we write eqs 11 and 12 into eq 10:
[
ΦR(ω) ) exp n[Φs(ω) - 1] + iωt0 +
µ1 ) nτ +
D (n[Φs(ω) - 1] + Lu
k′ + 1 L ) nτ u k′
µ′2 ) 2nτ2 +
(18)
2DL(k′ + 1)2
(19)
u3
]
iωt0)2 (13)
where Φs(ω) is the CF of the sojourn time distribution of a single adsorption. It is natural to assume that the sojourn times of a single adsorption step follow an exponential distribution. Hence, when the mean sojourn time is τ, and the surface is homogeneous, we get the following CF:
Φs(ω) ) 1/(1 - iωτ)
(14)
The mean sojourn time is the reciprocal of the desorption rate constant (τ ) 1/kd), employed in the conventional kinetic model of chromatography. For heterogeneous adsorption-desorption processes, we have the following expression for the CF:
Φs(ω) )
∑p /(1 - iωτ ) j
As we expected, the dispersion has no influence on the retention time, and the variance is the sum of the variance of kinetics and that of dispersion. When the mass balance equation of linear chromatography is solved by applying the so-called open-open boundary condition, one obtains slightly different expressions for the moments. The first moment, in the latter instance, contains an extra 2D(k′ + 1)/u2 term.15-17 In our study, the contribution of the mobile phase to the chromatographic band profile is modeled by a Gaussian model (see eq 7) rather than the band profile obtained for an unretained component with the massbalance equation. The difference is minuscule when Ndisp ) Lu/ 2D is at least 500 and completely negligible at Ndisp ) 1000.8,15 From the above moments we can express the reciprocal of the number of theoretical plates as
1 2 k′ 2 2D ) + N n k′ + 1 Lu
(
(15)
j
j
where τj is the mean sojourn time on sites of type j and pj is the relative amount of those sites. When the mean sojourn times can be approximated by a continuous distribution, Φs(ω) becomes
Φs(ω) )
f(τ)
∫1 - iωτ dτ
(16)
The CF of the peak profile is obtained when the proper expression for Φs(ω) is written into eq 13. RESULTS AND DISCUSSION Homogeneous Surfaces. On a single-site, homogeneous surface, the CF of the sojourn time distribution is given by eq 14. Therefore, for the CF of the peak profile we get
{ (1 -1iωτ - 1) + iωt + D 1 - 1) + iωt ] } (17) n Lu[ (1 - iωτ
ΦR(ω) ) exp n
0
2
0
From this expression it is obvious that the peak profile is not a simple convolution of the respective mobile- and stationary-phase terms. Due to the random diffusion, some molecules leave the column sooner and thus perform fewer adsorption steps, while others stay in the column for a longer time undergoing more adsorption steps. The difference is important only when the mobile-phase velocity is extremely slow or the axial dispersion is rather high, though. The moments of the peak profile can be determined from the derivatives of the CF.5 The first moment about the origin (retention time) and the second central moment (variance) are, respectively 4474
Analytical Chemistry, Vol. 71, No. 20, October 15, 1999
)
(20)
In a previous study, we showed that the number of masstransfer units is identical to the average number of sorption steps.9 Therefore, the above expression is identical to the one obtained with the lumped kinetic model of chromatography,15 demonstrating again the equivalence of the two approaches. When expressing the plate height, we may utilize the fact that the number of sorption steps can be expressed as n ) t0/τm, where τm is the average sojourn time of the molecule in the mobile phase waiting for an adsorption. τm is the reciprocal of the adsorption rate constant: τm ) 1/ka. With these considerations, we shall get
H ) 2τm
2D u+ (k′ k′+ 1) u + 2Du ) (k′2k′τ u + 1) 2
2
(21)
The above expression accounts for the B and the C terms of the van Deemter equation and has already been obtained by Dondi et al. by means of the stochastic model.13 It is absolutely identical to the plate height equation obtained with the lumped kinetic model of chromatography.15 The higher-order central moments of the peak profile can also be determined from the CF.
µ′3 ) 6nτ3 +
12DLk′(k′ - 1)τ
µ′4 ) 24nτ4 + 3(µ′2)2 +
u3 24DLk′τ2 (3k′ + 2) u3
(22)
(23)
From the moments, the most important peak shape parameters, (15) Guiochon, G.; Shirazi, S. G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, MA, 1994. (16) Kucˇera, E. J. Chromatogr. 1965, 19, 237-348. (17) Grushka, E. J. Phys. Chem. 1972, 76, 2586-2593.
Figure 1. Band profiles calculated for a homogeneous surface. L ) 25 cm, u ) 0.5 cm/s, k′ ) 2, and n ) 1000. The mobile-phase diffusion coefficient is varied between D ) 10-4 and 10-1 cm2/s.
Figure 3. Band profiles calculated for a homogeneous surface. L ) 25 cm, u ) 0.2 cm/s, k′ ) 4, and n ) 4. The mobile phase diffusion coefficient is varied between D ) 10-3 and 10-2 cm2/s.
than four.9 This phenomenon is illustrated in Figure 3. Due to the very same reasons as in the cases of Figures 1 and 2, the height and the width of the peak emerging at the holdup time is significantly affected by the mobile-phase diffusion coefficient, while the broad, elongated band of the retained fraction is only very slightly altered. Multiple-Site Model (Either Discrete or Continuous). For a generic multiple-site surface, when there are M types of sites, each with a relative abundance of pj, the ith moment about the origin of the distribution of the average sojourn times is M
mi )
Figure 2. Same as Figure 1, except for n ) 10.
x
2D(k′ + 1) + τu2 u3 2L [D(k′ + 1)2 + k′τu2]3/2
(24)
(26)
where τj is the average sojourn time on site of type j. When the distribution of the average sojourn times is continuous, we shall have
mi )
2
6k′τ u D(3k′ + 2) + τu Ex ) L [D(k′ + 1)2 + k′τu2]2 2 3
i j j
j)1
the skew and the excess, are
S ) 3k′τ
∑p τ
∫τ f(τ) dτ i
(27)
(25)
The combined effect of the rate of sorption kinetics and axial dispersion is illustrated in Figures 1 and 2. In Figure 1, the kinetics is fast, the average number of adsorptionsdesorption steps is n ) 1000. In this case, the chromatogram is fairly symmetrical (the skew changes between S ) 0.067 and 0.02 as D increases, and the axial dispersion significantly broadens the peak. When the same retention time is reached with 100-fold slower kinetics, the average number of adsorption-desorption steps reduces to n ) 10; accordingly we observe tailing peaks whose width is much larger due to the fact that peak width is proportional to the average sojourn time. Due to the increased peak widths, the axial dispersion has now much smaller effect on the peak shape and peak width. In this instance, the skew changes between S ) 0.67 and 0.65 as D increases. When the kinetics is extremely slow, a fraction of the sample elutes at the holdup time, and the split peak phenomenon is observed. On a homogeneous surface, split peaks emerge due to slow kinetics when the number of mass transfer units is not higher
In a previous study, we demonstrated that the moments of the peak profiles can be calculated in closed form even if we cannot do that for the CF.9 Although it was not discussed in detail in that study, we can calculate the mean, the variance, the skew, and the excess of the peak profile in the case of a generic homogeneous or multiple-site heterogeneous surface summarizing the expressions reported in Table 1 of ref 9:
µ1 ) nm1
(28)
µ′2 ) 2nm2
(29)
S)
3 m3 x2n m3/2 2
Ex )
6 m4 n m2
(30)
(31)
2
Note that the above equations describe the contribution of the slow kinetics to the band profile. Both the mobile-phase convection Analytical Chemistry, Vol. 71, No. 20, October 15, 1999
4475
and the axial dispersion are neglected. Therefore, for instance, µ1 gives the corrected retention time. With an extension of our previous results, from the derivatives of the CF, we can determine the moments and the peak shape parameters for any type of surface heterogeneity taking into account the axial dispersion too, provided that we know the distribution of the average sojourn times. We obtain the CF for multiple-site heterogeneous adsorption when substituting eq 15 or eq 16 into eq 13. For the first moment and the second central moment of the peak shape we shall obtain
µ1 ) nm1 +
k′ + 1 L ) nm1 u k′
µ′2 ) 2nm2 +
2DL(k′ + 1)2 u3
(
1
)
2k′m2 2D 2D k′ 2 m2 H ) 2τm u+ u+ ) 2 2 k′ + 1 m u u (k′ + 1) m 1 1
(
)
2m2τm (m1 + τm)2
u+
2D u
(33) S ) 3τm
Ex )
x
(39)
τ2mu3
24DLm22 τ2mu3
+
48DL(m1 + τm)m3 τ2mu3
2 u3 2Dm2(m1 + τm) + m3u τm 2L [D(m + τ )2 + m u2τ ]3/2 1
m
2
2
(41)
m
6u3τ2m D[2m3(m1 + τm) + m22] + m4u2τm L [D(m + τ )2 + m u2τ ]2 m
(40)
(42)
m
(34)
(35)
(36)
(37)
Hence, the minimum plate height is 4476
(38)
12DL(m1 + τm)m2
1
By differentiating the plate height equation, we can determine the optimum linear velocity as
uopt ) (m1 + τm)xD/(m2τm)
µ′3 ) 6nm3 +
µ′4 ) 24nm4 + 3(µ′2)2 +
For any heterogeneous adsorption-desorption process, the effect of sorption kinetics and that of axial dispersion on the variance of the peak, as well as on the plate number or plate height, is clearly separated. Only the first and the second moments of the distribution of the average sojourn times (or desorption rate constants) are needed to describe the effect of surface heterogeneity on N or H. The equation written for the plate number and plate height should be further modified because the retention factor depends on the mean value of the average sojourn times. The average sojourn time of a molecule in the mobile phase is τm ) t0/n as the molecule stays in the mobile phase for a period of t0 and performs n adsorption steps in the meantime. The retention factor, on the other hand is the ratio of the average sojourn times in the stationary and the mobile phases k′ ) τ/τm, or for a heterogeneous surface k′ ) m1/τm. Thus, the plate height equation can be rewritten as
H)
2xDm2τm m 1 + τm
Note that both uopt and Hmin are proportional to the square root of D. The higher-order moments and the peak shape parameters in the instance of a generic heterogeneous surface are
(32)
where the retention factor is now k′ ) nm1/t0. Also the reciprocal of the plate number and the plate height can be calculated with ease for any type of heterogeneous adsorption:
1 2 m2 k′ 2 2D ) + N n m2 k′ + 1 Lu
Hmin )
Analytical Chemistry, Vol. 71, No. 20, October 15, 1999
Equations 32-42 are general equations relating surface heterogeneity and dispersion to the most fundamental chromatographic peak characteristics. The homogeneous surface and or the simplest heterogeneous case, the two-site model, are all special cases of the general heterogeneous model. To calculate the peak characteristics with eqs 32-42, all we need are the moments of the average sojourn time distribution. Although those quantities are derived via the CF method of the stochastic model, the numerical or analytical calculation of the CF is necessary only when we need the peak profile itself. All the important moments and peak shape parameters can be deduced directly from eqs 3242. In the following, we show this procedure with two very common types of surface heterogeneity: for the two-site surface and for the unimodal Gaussian adsorption energy distribution. Two-Site Surface. We show the effect of surface heterogeneity by means of the plate height calculated for a stationary phase containing two types of sites. When we calculate the optimum linear velocity of the mobile phase (eq 37) and the minimum plate height (eq 38), a behavior as plotted in Figure 4 can be observed. In the main figure, the proportion of the quick sites is zero on the left-hand side of the figure. As the proportion of the quick sites increases, the minimum plate height also increases and passes through a maximum. At the right-hand side of the figure, where the proportion of the slow sites is zero, the minimum plate height is only slightly smaller than when the stationary phase contains only slow sites. Remember that for each stationary phase composition a different optimum mobile-phase velocity gives the minimum plate height (see eqs 37 and 38). Figure 5 shows the optimum mobile-phase velocity for the homogeneous surfaces that compose the heterogeneous stationary phases studied in this instance. When we write the moments of the two-site model (mi ) pτi1 + [1 - p]τi2) into eq 38 or 36, after differentiation, we can
Figure 4. Minimum achievable plate height on a two-site heterogeneous surface. τm ) 0.005 s and τ1 ) 0.01 s. Note that in the main figure the proportion of the fast sites increases from left to right; in the inset, a semilogarithmic plot of the main figure is shown, but the proportion of the slow sites increases from left to right now.
Figure 6. Plot of the skew against the proportion of slow sites as calculated with eq 41. τm ) 0.005 s, τ1 ) 0.01 s, and (a) L ) 25 cm, D ) 0.001 cm2/s, u ) 0.5 cm/s, (b) same as in (a) except D ) 0.01 cm2/s, (c) same as in (a) except u ) 1 cm/s, and (d) same as in (a) except L ) 12.5 cm.
Figure 5. Plot of plate height against linear mobile-phase velocity for homogeneous surfaces. τm ) 0.005 s and D ) 0.001 cm2/s.
determine the worst composition of the two-site stationary phase. For this we get
pworst )
τ2 τm 1 1 ) (43) τ1 + τ2 τ2 - τ1 k′1/k′2 + 1 k′2 - k′1
The least efficient separation is achieved when the proportion of the slower sites is pworst even if the linear velocity of the mobile phase is optimized. Note that the value of pworst depends on two factors: (i) the ratio of the mean sojourn times on the two sites, i.e., the ratio of the desorption rate constants (τ1/τ2 ) kd,2/kd,1, which ratio is also identical to k′1/k′2); (ii) the difference of the retention factors corresponding to each site (k′1 and k′2). The location of the maximums in Figure 4 is at pworst. As the difference between the kinetics of the two types of sites increases, the maximum value also increases, and the worst efficiency is achieved when the proportion of the slower sites is very small. With the current values, we get that the least efficient is the separation when the proportion of the slow sites is 14.65, 1.5, and 0.15%, respectively, when there is a 10-, 100-, and 1000-fold difference in the rate of desorption kinetics on the two sites. We can calculate the effect of slow sites on the peak asymmetry, too. In Figure 6, a series of curves are plotted for different experimental setups. The fundamental difference between Figures
Figure 7. Chromatograms calculated for two-site heterogeneous kinetics. τm ) 0.005 s, τ1 ) 0.02 s, τ2/τ1 ) 1000, L ) 25 cm, D ) 0.001 cm2/s, and n ) 104. The proportion of the slow sites is varied between 10-6 and 10-3.
4 and 6 is that in Figure 6 the mobile-phase velocity is unchanged when p is varied. The maximum tailingsdepending on the experimental conditions, axial dispersion, and heterogeneous kineticssis found at a smaller proportion of slow sites than the least efficiency. When, for instance, the ratio of the desorption rate constants is 10-, 100-, or 1000-fold, the worst tailing is found when the proportion of slow sites in the stationary phase is 4, 0.054, and 0.0006%, respectively. The effect of the small proportion of slow sites is illustrated in Figure 7. When the proportion of slow sites is 10-6 (solid line in Figure 7), the peak is fairly symmetrical. In this instance the amount of slow sites is negligible, due to their extremely small amount. When the relative amount of slow sites is slightly increased, the effect of slow sites emerges and an elongated tailing is observed. When the relative amount of the slow sites is 10-5 (thick dotted line in Figure 7), the peak shape still seems symmetrical, but with a closer look we can discover that the tailing part extremely slowly returns to the baseline. From the drop of the peak height, we can judge the amount of sample that elutes in the tailing part. The average number of sorption steps is n ) 10 000, a number that produces a symmetrical peak on a homogeneous surface. But as the proportion of the slow sites is Analytical Chemistry, Vol. 71, No. 20, October 15, 1999
4477
1 - p ) 10-5 now, out of the 10 000 steps, on the average only 0.1 step is performed on the slow sites. When we further increase the proportion of slow sites, they will overcome the quick sites. When average number of sorption steps on the low sites is larger than one, the peak at the retention time due to the quick sites gradually disappears and the peak starts to shift toward the retention time determined by the slow sites. Normal Distribution of Sorption Energies. The Frenkel equation connects the adsorption energy and the mean sojourn time18
τ ) τ0eE/RT
(44)
where τ0 is a constant, typically around 10-14 s.18 When the distribution of adsorption energies follow a normal distribution with an average of E/ and variance of σ2E, the average sojourn times in adsorbed state can be described with the following lognormal distribution:
f(τ) )
[
]
(ln τ - ln τ*)2
RT exp 2(σE/RT)2 x2πτσE
(46)
where for the sake of simplicity, we used q ) exp[(σE/RT)2]. In the following equations, k′/ stands for the retention factor obtained on a homogeneous surface with τ/ average sojourn time: k′/ ) τ//τm. With these notations, the first moment and the variance of the peak are
µ1 ) nτ*xq +
k′*xq + 1 L ) nτ*xq u k′*
(47)
and
µ′2 ) 2nτ2*q2 +
2DL(k′*xq + 1)2 u3
(48)
respectively. When we express the reciprocal of the plate number, the following expression is obtained:
(
k′*q 1 2 ) N n k′ xq + 1 *
)
2
+
2D Lu
x
S ) 3k′*q2τ*
5/2 2 u3 2D(k′*xq + 1) + τ*q u (50) 2L [D(k′ xq + 1)2 + k′ q2u2τ ]3/2 *
*
*
and
Ex )
6k′*q4u3τ2/ D(k′* + 2xq + 2k′*q) + q4u2τ* L [D(k′ xq + 1)2 + k′ q2u2τ ]2 *
*
(51)
*
respectively. In the homogeneous limiting case, when σE ) 0, q ) 1 should be substituted into eqs 47- 51 and expressions identical to eqs 18-25 are obtained. The comparison of eqs 4751 with eqs 18-25 indicates in what manner the spread of sorption energies alters the peak characteristics. The skew of the peaks are plotted in Figure 8, for Gaussian sorption energy distribution. Since the standard deviation of sorption energies affects the peak shape through an exponential expression, minor deviation from a homogeneous surface increases only slightly the skew. However, as σE increases, the skew, i.e., the peak asymmetry, increases to rather high values. This can be explained by the fact that a Gaussian sorption energy distribution results in a log-normal distribution of the mean sojourn times. When σE increases, the tailing effect of the log-normal distribution amplifies. Accordingly, a rather small fraction of very slow sites will be present on the stationary phase. This situation will introduce an enormous peak broadening and tailing effect similarly to the one observed for a two-site surface when the proportion of the slow sites is very small. The effect of axial dispersion can also be seen in Figure 8. As we expect it, axial dispersion broadens the peak, thus reduces the asymmetry.
(49)
The skew and the excess for normal distribution of adsorption (18) de Boer, J. H. The Dynamical Character of Adsorption; Clarendon Press: Oxford, U.K., 1968. (19) Keeping, E. S. Introduction to Statistical Inference; Dover: New York, 1995.
4478
energies are
(45)
where τ/ ) τ0eE*/RT. The moments about the origin of the lognormal distribution are needed to calculate the moments of the peak profile and the peak shape parameters. For the log-normal distribution, the r th moment about the origin is19
mr ) τr/q(r2/2)
Figure 8. Plot of the skew against the linear velocity of the mobile phase for Gaussian sorption energy distribution. τm ) 0.005 s, τ* ) 0.02 s and L ) 25 cm. For each pair of curves, the upper line is for D ) 0.001 cm2/s and the lower one is for D ) 0.01 cm2/s.
Analytical Chemistry, Vol. 71, No. 20, October 15, 1999
CONCLUSIONS The stochastic model of chromatography has been extended by means of the characteristic function method to include the effects of axial dispersion and mobile-phase convection. When modeling homogeneous mass-transfer kinetics, we showed, by means of the plate number and the plate height equation, that the stochastic theory and the lumped kinetic model of chroma-
tography give identical results. One of the essential benefits of the stochastic model is that the only numerical step is usually the calculation of the inverse Fourier transform of the CF to obtain the peak profile. Anothersobviously more importantsbenefit of the CF method is that it provides a rather intelligible procedure to model the effect of heterogeneous adsorptionsdesorption kinetics. Our calculations confirmed that all the important characteristics of the peak profilesretention time, variance, skew, and excessscan be determined with simple expressions without the need for the CF. The distribution of the sorption energies or, what is equivalent, that of the mean sojourn times is all the information needed to fully describe the effect of heterogeneity. The theory here developed is promising for the investigation of surface heterogeneity in gas or liquid chromatography by fitting
the different models of heterogeneous kinetics to measured chromatograms. ACKNOWLEDGMENT This work was supported by research grants T 025458 from the Hungarian National Science Foundation (OTKA) and FKFP 0609/1997 from the Hungarian Ministry of Education, by NATO Linkage Grant OUTR.LG971480, as well as by the Italian Ministry of University and Scientific Research (MURST, 60%) and by the University of Ferrara, Italy.
Received for review April 20, 1999. Accepted August 2, 1999. AC990412U
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