J. Phys. Chem. 1986, 90, 1885-1891 Acknowledgment. The simulations reported herein were performed on the IBM 370/3081 at Clemson University with time made available through Clemson’s College of Engineering. Figure 8 and 13 were created on an Apple Macintosh computer using a modified version of a program originally written by E. J. Kirklat~d.~’During the period in which this work was done,
1885
M.C.W. and J.M.H. were partially supported by a grant from the Chemical and Process Engineering Division of the National Science Foundation. Additional support was provided by a National Science Foundation Presidential Young Investigator Award (1984) to J.M.H. J.P.0C. is grateful to the University of Florida Consortium for Enhanced Oil Recovery for financial assistance.
STATISTICAL MECHANICS AND THERMODYNAMICS The Characteristic Function Method in the Stochastic Theory of Chromatography Francesco Dondi* and Maurizio Remelli Analytical Chemistry Laboratory, Department of Chemistry, University of Ferrara, I-441 00 Ferrara, Italy (Received: June 27, 1985)
The stochastic theory of chromatography is reconsidered through the fundamental characteristic function method of the probability theory. The means of obtaining expressions for plate height H , specific asymmetry Z,and specific flatness F is described, and six different chromatographic models of progressively greater complexity are analyzed. All the models assume a constant mobile-phasevelocity and equal sorption sites and do not take into account flow pattern and mobile-phase diffusion effects. The role of the random nature of both the stationary-phase entries and the sorption processes is considered from a general point of view. The rate of convergencetoward the Gaussian law is described by using the general Berry-Essden theorem of the probability theory, and the approximation properties of the Edgeworth-Cramdr series are described for a general chromatographic model. Examples of mixed retention on a single site by a bifunctional solute molecule are given, and influence on the peak shape is described.
Introduction The stochastic theory of chromatography, founded by Giddings and Eyring,’ is the most complete and the most exhaustive, among the various descriptions of the chromatographic process, presented to date. In fact the chromatographic process is described from a microscopic point of view as a probabilistic dynamic process, and a complete solution of the peak shape function is given. However, the development of the theory has been hindered by mathematical difficulties, and the advancement of chromatography has been aided by other theoretical approaches, such as the mass balance approach,2 nonequilibrium a p p r ~ a c hand , ~ the simplified stochastic approach of the random ~ a l k . ~ ~ ~ In this paper the stochastic theory is reconsidered through the use of the so-called characteristic function method, which has played a central role in the development of the modern probability theory.68 The aim here is to present a method which is able to overcome the mathematical complexities, preserving the completeness of the theory. Certainly many unsolved or incompletely understood practical and theoretical problems in chromatography would be enlightened by such an advancement. (1) Giddings, J. C.; Eyring, H. J. Phys. Chem. 1955, 59, 416. (2) Grubner, 0. Adu. Chromatogr. 1968,6, 173. (3) Giddings, J. C. “Dynamics of Chromatography”; M. Dekker: New York, 1965; Chapters 2-4. (4) Giddings, J. C. J . Chem. Educ. 1958, 35, 5 8 8 . ( 5 ) Weber, S. Anal. Chem. 1984, 56, 2104. (6) Cramtr, H. “Mathematical Methods of Statistics”, 7th ed.; Princeton University Press: Princeton, NJ, 1956. (7) CramEr, H. “Random Variables and Probability Distributions”; Cambridge University Press: Cambridge, U.K., 1961. ( 8 ) Feller, W. “An Introduction to Probability Theory and its Applications”; Wiley: New York, 1966; Vol. 11.
0022-3654/86/2090-1885$01.50/0
The migration of an individual molecule through the chromatographic column is a highly erratic process. Its most general description consists of a complex chain of random processes of different kinds: ordinary molecular diffusion, flow pattern effects called eddy diffusion, and sorption-desorption kinetics. For the purposes of this first treatment and for the sake of simplicity, only the last one, the sorption-desorption kinetics, is considered here. A single molecule, moving through the chromatographic medium is described as constantly changing between the state of being sorbed and that of being in the mobile phase (desorbed state). To make this point clear, three different paths of the molecule through the column (or three different “solute histories”) are described by means of their trajectories on a coordinate plane (Figure 1). The x and y axes are respectively the elapsed time t from the injection and the column axis covered distance I. The chromatographic elution peak is simply the probability density function of trajectory cross sections with a horizontal axis located at 1 = L, where L is the column length. A horizontal At, segment on these trajectories represents the solute molecule remaining in the stationary phase (s). A slanting segment of the slope u, represents the solute molecule moving forward in the column at the mobile phase (m) velocity v,,,. The covered distance AI between two successive sorption steps is u,,,At,, where At, is the time elapsed between these steps. Both At,,, and At, are random variables. Thus, even if the total time spent in the mobile phase t, is, in this model, a constant quantity (Llu,,,), the total number of entries n in the stationary phase will be a random integer quantity. The total time spent in the stationary phase, t,, proves to be a sum of random quantities (horizontal segments in Figure 1). However the number of terms in this sum is not constant but it is the above random integer quantity n. 0 1986 American Chemical Society
1886
Dondi and Remelli
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 Iff(x) exists and g(x) is defined as
thejth moment about the origin,
XI,
pJ’, is
Jx’f(x)
p,f = p I f is
dx
(5)
the mean m: p,‘
=m
(6)
If g(x) is (x - m y , the j t h central moment p,
= J(x
-
p,
is defined as
m)’f(x) dx
(7)
- m)2f(x) dx
(8)
Variance u2 is p 2 : g2
= p2 =
J(
Y
The characteristic function (cf) 4 ( z ) is defined as the expected value of the function g(x) = exp(izx) (ref 6), where z is an auxiliary variable and i is the imaginary unit: $ ( z ) = E[exp(izq)] = Jexp(izx)
dF(x)
(9)
If the frequency functionf(x) exists or if 7 is a discontinuous random variable, one obtains
4 ( z ) = Jenp(ixz)f(x)
dx
(10)
4(z) = Cexp(ixkz)fk
(1 1)
k
The characteristic function completely defines a distribution function6 Moreover a relationship exists between the jth derivative value of characteristic function at z = 0 and the jth moment about the origin: @J)(O) = [d’4(z)/d~’],=~= Pp,‘
(12)
The Taylor expansion of @(z)can be written in terms of the moments p,f (ref 6):
(b)
m
Figure 1. (a) F tamples of random trajectories ( I , t ) of solute molecules moving forwara in chromatographic medium. 1, 2: unsorbed, sorbed solute. (b) Molecule status during migration inside the column, referred to the full line case in (a).
In brief, the chromatographic problem of finding the elution peak shape consists of finding the distribution of a sum of a random number of independent random quantities as a function of the distribution of both these random quantities. The characteristic function method allows one to solve this problem. Theory Integer or continuous random quantities q, which build up the previous general stochastic model of the chromatographic process, are completely described by their cumulative distribution function F(x), defined as F(x) = Pr(q5x)
(1)
where Pr(q5x) means the probability that the random quantity q will be less or equal to x. If g(q) is a function of q , the expectation or the mean value of g is denoted and defined by E[g(v)l = J d x ) dF(x)
4(z) =
1
+ C(pJ’/j!)(iz)’ 1
For many applications the logarithm of characteristic function (called second characteristic function, IIcf)6 is taken: ic(z) = In 4(z)
$U)(O)
= [dhJ(z)/dz’],=,
dx
(3)
If F(x) is a step function with steps of heightfk at the points x k , eq 2 is taken to be E[g(q)l = zdXk)fk k
(4)
Equations 3 and 4 deal, respectively, with continuous and discontinuous random quantities.
= i’KJ
(15)
The Taylor development of $(z) is9
$ ( z ) = In 4 ( z ) = E ( ~ ~ / j ! ) ( f z ) J
(16)
1
The expression of +(z),p,’, pJ, and K, for fundamental distributions such as exponential, rectangular, normal, Poisson, r, compound Poisson, and others are reported in many textbooks.8.’0 The relationships between the first four moments about the origin, central moments and cumulants are:I0 PI = P2
0
(17)
= Pi -
(18)
p 3 = p3’ - 3 & ’ k f
E[g(q)l = Jg(x)f(x)
(14)
The j t h derivative at z = 0 defines the cumulant of order j , K,:
(2)
The integral here is taken over all values of x. Equation 2 should be interpreted as Stieltjes integral and deals with the most general case.6 If F(x) has a derivativef(x) (frequency function), eq 2 can be rewritten as
(13)
/A4
=
p4f
+2
+
~
~
- 4p11/.L3’ 6 p 1 ’ 2 ~ 2 ’ 3/.4.1f4
’
~
(19)
(20)
PI’ =
KI
(21)
P2 =
K2
(22)
=
K3
P3
fi~q = K 4
+
3K22
(23)
(24)
(9) Gnedenko, B. V.; Kolmogorov, A. N. “Limit Distributions for Sums of Independent Random Variables”; Addison Wesley: Cambridge, MA, 1968. (10) Abramowitz, M.; Segun, I. “Handbook or Mathematical Functions with Formula, Graphs and Mathematical Tables”; Dover: New York 1965; Chapter 26.
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1887
Stochastic Theory of Chromatography Other quantities are used in peak shape coefficients Tk defined as Yk-2
= Kk/(K2)k’2
the cumulant (25)
The first two cumulant coefficients y l and y2 are respectively the skewness S and the excess E, Yl
=s
(26)
Y2
=E
(27)
Cumulant coefficients are the standardized random variable distribution cumulants. The standardized random variable is c = (x - m ) / u (28) The total retention time a molecule spends in the stationary phase when performing two entries in this phase is again R random variable. Its frequency function, hot(x), is the convolution of frequency functionfi(x) andf,(x) of the time spent in each entry
perform a variable number of entries in the stationary phase during its trip in the mobile phase from the inlet to the end of the column. If the only hypotheses put forward are that sorption time distribution on a single site is continuous with finite moments of all orders and that the column sorption sites are equal, we define a general chromatographic model denoted here as model I. Let pn be the probability of performing exactly n sorption-desorption steps. The frequency function would be a “mixture” of different “histories” each characterized by different n values taken on by the number of sorption-desorption steps as well as by proper weighing factor pn. Thus eq 40, or eq 41 describing the mixture effect, is to be combined with eq 36 or 37 which take into account the “random sum” effect of n entries in the stationary phase:
hot
=
CP,f,“’
or 4tOdZ) = CP,4,”(4 n
which is symbolically written as8
hot
= fi*f2
When n entries are performed hot
= fi*fi*f3*...*f,
(31)
The use of characteristic function enables one to replace the convolution with the more easily performed product:6
= ?4k(Z)
k
CKkj k
k
=P
(34)
(35)
(36)
4tot(z) = 4S(Z)”
(37)
$tot(z) = n$s(z)
(38)
=
(39)
KtOtJ
nKsJ
Here and in the following the subscript s specifies a stationary phase site property and the subscript tot a property of the total resulting sorption time. When a molecule can perform alternatively a rest on a site 1 with probability pI or a rest on a site 2 with probability p 2 ( = I - pl) we have
+ Pd2(X)
(40)
4tot(z) = Pld,(Z) + P242(Z)
(41)
hodx) = PLfib)
CP, exp(izn) n
(44)
x‘ = exp(r In (x))
Equations 3 1-34 can be interpreted as giving respectively the frequency function, the cf, the IIcf, and the cumulants of the total time spent by n molecule in the stationary phase, performing n sorption steps in n different sites. When the sites are equal, instead of eq 31-34 we have hot
+ill@) =
(33)
valid for cumulants of all orders. Recalling eq 6, 8, 21, and 22, we obtain the well-known variance addition theorem from eq 34: utot2= Cak2
The sums in eq 42 and 43 are to be extended to all possible values of n entries in the stationary phase. 4sis the characteristic function of the sorption time on a single sorption site. Equation 42 describes the frequency function of the total time spent by a molecule in the stationary phase. The characteristic function of the number of adsorption-desorption steps is defined by making use of eq 11:
The subscript m, employed here and in the following, underlines the fact that this is a property of the mobile phase. Recalling the fundamental identity
Thus, applying the Taylor expansion (eq 16) to both members of eq 33 and equating the coefficients, the following relationship is obtained Ktotj =
(43)
(32)
When the I1 cf is employed one obtains
= C$k(z)
(42)
and6.73
we can rewrite eq 43 as follows: 4todZ) =
CPil exp(in(ln 4s(z))/4 n
(46)
or, by combining eq 44 and eq 46, as 4tot(z) = drn[In +Az)/il
(47)
Equation 47 is the solution of the general stochastic model of the chromatographic elution (model I), written in terms of characteristic functions. Equation 47 can also be written in terms of I1 c f +tot(z) = $rn[$s(z)/iI
(48)
Equation 48 can be inverted to give the chromatographic peak, that is, the distribution or the frequency function of the residence time of the molecule in the stationary phase. The chromatographic elution peak will be equal to the previous frequency function plus the constant term t,, which is the time spent in the mobile phase. This route has been followed, in the past, by McQuarrie12 but it involves complex integration problems and only leads to a simple solution in a limited number of cases. One can, more easily, make use of eq 12 and 15 to compute moments and cumulants and, from these, conventional chromatographic quantities will be derived. The problem of the peak shape will be dealt with in the last section. The first four cumulants computed for the general model (model I) are reported in Table I. The three fundamental chromatographic quantities employed in column efficiency studies are the plate height ( H ) , the specific asymmetry ( Z ) ,and the specific excess (Q2
Chromatographic Models Model I . The general chromatographic model previously described takes into account the possibility that a molecule will (11) Dondi, F.; Betti, A.; Blo, G . ; Bighi, C. Anal. Chem. 1981, 53,496.
(45)
(12) McQuarrie, D. A, J. Chem. Phys. 1963, 38, 437
1888
Dondi and Remelli
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986
where tR = t,
TABLE I: Cumulant Expressions for Model I
+ t,
(52)
K K
and t, = m
(53)
m is the first moment about the origin of the time spent in the stationary phase; p2, p3, and p4 are the central moments of the elution peak. They are equal to the corresponding central moments computed on the distribution of the time spent in the stationary phase, since, for all the models here considered, the two distributions are only shifted by the constant quantity t,. By introducing the retention ratio R3 R = t,/tR
= mmms = ~mS2am2 ~ , + ~a?m, = ~mS3~,,3 ~ , + ~3m,a,2am2+ K~ ,m, ~ = ~m,4K,,4 ~ , ~3u,4am2 6 r n , 1 ~ , ~ ~+, ,4~m , ~ , , j u , ~+ ~ , , 4 m ~
K101.1
K
~ ~
+
+
(1.1) (1.2) (1.3) (1.4)
TABLE 11: Chromatographic Quantities for Models I-VI (11.1) (11.2) (11.3)
(11.4)
(54)
and noting that in the present model we have (55)
(11.5) (11.6)
making use of eq 21-24 and 52-55, eq 49-51 can be rewritten as follows:
(11.7) (11.8)
urn = L / t m
(56)
H = R(1 - R)(Kmt,2/m)Um = R2(1 - R)(Ktot,3/m)Um2
(57)
F = R3(1 - R)((KtOt,4+
(11.9) (11.10) (11.11) (11.12)
(58)
Expressions for H , Z , and F of the general model I are obtained by combining equations of Table I and eq 56-58 (see Table 11). Different particular chromatographic models can be described if some hypotheses about the nature of both random processes, in the mobile phase and in the stationary phase, are put forward. In order to appreciate the power of the characteristic function method and to make comparisons with previous results obtained by the stochastic theory of chromatography, two basic hypotheses about the character of fs and f, will be considered. The sorption time on a single sorption site is considered to be an exponentially distributed random variable: ” a t )= k, exp(-tk,)
(11.13) (11.14) (11.15) (11.16) (11.17)
(11.18) (11.19) (11.20)
(59)
The average sorption time r, is 1 /k,.” This quantity is not easy to measure e~perimenta1ly.l~Its theoretical evaluation can be obtained from the Frenkel equationT4 7s
=
70 exp(Q/RT)
(60)
s, Q is the sorption energy, R is the where ro is about 1.7 X gas constant, and T is the absolute temperature. In physical adsorption processes Q ranges from 1 to 20 kcal/mol (ref 13) and r,, computed by eq 60, varies from 1 X s up to 100 s. However, in a real physical sorption process, a molecule can remain in single sorption site in many different configurations, each one characterized by a weighing factor piand a time constant r,,, or a sorption constant k , , (ref 3). A more realistic distribution would be a mixture of exponentials fs(t) = P1kS.l exp(-tk,,,) + P2ks.2 exp(-tk,,,) +
”.
(61)
with the condition cpk = 1 k
(62)
The cumulants of all orders j , K , ~ ,of a homogeneous and exponentially distributed sorption time arelo = (j- I)!/k,
K , ~
(63)
The first four moments about the origin, of the same case, are computed by using eq 63 and general expressions between cumulants and moments 17-24: psi=
j!/kj
j = I, 4
(64)
(1 3) Steele. W. A. “The Interaction of Gases with Solid Surfaces”; Pergamon Press: Oxford, U.K., 1974, Chapter 6. (14) de Boer, J. H. “The Dynamical Character of Adsorption’, 2nd ed.; Clarendon Press: Oxford, U.K., 1968; Chapter 111.
(11.21)
For mixed and exponentially distributed sorption time, the moments about the origin are P,,’
= Cp,o’!)/k,.i
j = 1, 4
(65)
I
This equation is obtained by applying the moment defining eq 5 to eq 61 and by using eq 64. The Poisson lawlo f,(n=r) = exp(-m,)m,‘/r!
r = 0, 1, ..., m
(66)
is a reasonable model for the distribution of the number of entries in the stationary phase; m, is the constant of the Poisson distribution. The waiting time tw for a successive sorption step is, according to this distribution, an exponentially distributed random variable8 f(tw) = km exp(-twkm)
(67)
where k, is the constant of the process. The mean waiting time T , is l/k, and the mean number of entries in the stationary phase during a time t , is tm/r,. This quantity has proved to be equal to m, (ref 8). Thus we can also write m, = Lk,/cm
(68)
recalling the relationship 55 between t , and the mobile phase velocity D,. The cumulants of all ordersj for Poisson distribution are equal:I0 K,J
= m,
(69)
Thus the stationary phase entry process described through this distribution is a highly erratic process, with variance equal to the
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1889
Stochastic Theory of Chromatography TABLE 111: Models Description hypotheses on entries distribution in hypotheses on the stationary single site sorption phase, f,, time distribution,f, model I none none model I1 Poisson none (eq 6 6 ) homogeneous expon model 111 none (eq-59) model IV Poisson homogeneous expon (eq 66) (eq 59) model V constant homogeneous expon (eq 59) constant homogeneous expon model VI (eq 7 8 )
equivalence
Giddings-Erying (ref 1) Martin-Synge (ref 16)
mean value of the process. Even in this case a more realistic process should be a mixed Poisson distribution, but this point has not been considered here. The intrinsic random character of the stationary-phase entry process will be evaluated with respect to a hypothetical highly ordered entry process, with zero variance and zero higher cumulants, that is, a process with constant number of entries in the stationary phase. By using the previously described hypotheses regarding f, and fm, five other more specific models have been derived. Their features are specified in Table 111. It can be noted that model IV is the stochastic model of chromatography described by Giddings and Eyring’ and analyzed by McQuarrie.’* Model V is mathematically equivalent to the classical Martin-Synge plate theory.l5~l6 These models are mathematically developed in the following. Their physical properties will be more extensively considered in the Discussion of the Models section. Model ZZ. The characteristic function of the Poisson distribution of the number of entries in the stationary phase is7 $m(z) = exp{m,[exp(iz) - 111
-- ~-
Figure 2. Three-dimensional plots of repeated n-convolutions; (a) convoluted exponential distribution, k, = 1 (eq 59); (b, c) convoluted mixed exponential distribution (eq 7 8 ) . (b): p = 0.5, k , , = 1, kr,2= 0.5; (c): p = 0.9, kS,’= 1, k3,2= 0.1.
Expressions for peak skewness S and excess E depend respectively on l/(L)I/* and on 1/L:
(70)
S=
By introducing eq 70 into the $tot expression of the general model I (eq 47) cf and I1 cf expressions are obtained: 4todz) = exPhn[&(z) - 111
(71)
Thus +.,,,(z,L) is linearly dependent on the continuous parameter L, the column length, and the inverse of the mobile-phase velocity l/vm. This is the most relevant property of the model I1 which allows one to place it within the important class of stochastic processes with stationary and independent increments.’ The only qualifying condition is thatf, is continuous with finite moments, which is normal in chromatography. This property will allow one to use approximating functions (Edgeworth-Crambr series) for the peak shape and to define the rate of convergence toward the normal law (see below). By applying the Taylor development (eq 13 and 16) to both fibt(z) and $,(z), and equating the coefficients of (iz)equal powers, the following general relationship between cumulants of the stationary-phase residence time distribution K~~~~ and the moments about the origin of the sorption time distribution on a single site is obtained K~~~~
= mmpsj
all j
(74)
or, by using eq 68 = L(km/um)psj’
~totj
all j
(75)
Pertinent expressions for the first cumulants and for H , F, and Z are obtained by using eq 74, 17-24, and 56-58. ( 1 5 ) Dondi, F. Anal. Chem. 1982, 54, 473. (16) Martin, A. P. J.; Synge, L. R.Biochem. J . 1941, 35, 1358.
1-------1
(vm/Lkm)1’2~s,3’/~s,23’2
E = (Um/Lkm)ps,4’/Ys,22
(76) (77)
Equations 76 and 77 are obtained from eq 75 and eq 26 and 27. Models ZZZ-VI. The expressions in Table I1 pertinent to model 111 and to model IV have been derived respectively from those referred to model I and I1 and making use of eq 63 and eq 56-58. Model V can be derived from model I11 by inserting K ~ , ’= n and by equating to zero all the cumulants K , ~ for j > 1. The sorption time distribution on a single site for model VI has been taken as f,(t)
= P ~ ,exp(-tks,i) I
+ (1 - ~ ) k , ex~(-tk,2) 2
(78)
that is a bifunctional molecule on a sorption site has been con~ i d e r e d .The ~ moments about the origin are (see eq 65) P,,,’ = j!b/k,,,i + (1 - ~ ) / k , i l
(79)
Only the H expression is reported in Table I1 for the sake of simplicity.
Computational Procedure The same numerical procedure previously d e s ~ r i b e dwas ’ ~ used in computing EC series, in obtaining convoluted peaks of Figure 2 and in determining LEVYdistances reported in Figure 3. The computation of L6vy distances (Figure 3) was run on Digital Minc 11 under RT-11, as previously d e ~ c r i b e d . ’ ~ Three-dimensional plots of Figure 2 and plots of Figure 1 were run on CDC Cyber 76 computer (CINECA, Casalecchio, Bologna, Italy), and Calcomp software was employed. Discussion of the Models It must first be noted that the present characteristic function method is superior to the previous methods followed in the stochastic theory of chromatography. Not only the expressions for H but also those for Z and F can be obtained. In addition three
1890 The Journal of Physical Chemistry, Vol. 90, No. 9, 1986
Dondi and Remelli mobile-phase contribution,
K,,~/K,,,
and the stationary site effect
K,,2/K,,I2.
‘gl!
- 1.(
- 1,5
- 2.(
Expressions for Z and F a r e explained by following this same method. The only difference, with respect to the H case, is that mobile-phase and sorption site contributions can no longer be completely separated. The general scaling factors are respectively um2R2( 1 - R ) K , , and , ~ um3R3(1 - R ) K , , ~ ~ . Having given this brief overview regarding the H , 2,and F quantities, their relationships with physical parameters such as the diffusion coefficients in the mobile, stationary, and stagnant phase, stationary-phase thickness, and other parameters could be established. This might be obtained by making use of the usual relationships between the various cumulants and the above physical parameters. For example, the relationship among the mean sorption time in the stationary liquid-phase K s , l , the stationary liquid-phase thickness d, and the diffusion coefficient in this phase D,is known to be3
Putting eq 80 in eq 11.6 (Table 11) and by defining the following configuration factor -2.E
Figure 3. Lbvy distance L vs. n, number of sorption-desorption processes, in the cases a, b, and c of Figure 2. k = 0, 1, 2: Gaussian approximation, first and second EC series expansions Data for case a are taken from ref 15.
more general models (I, 11, and 111 in Table 111) are completely described (the case of different sites has not been considered here). Two main general results of Table I deserve particular comment. First, the retention time is the product of two mean quantities, m , and m,, which are respectively the mean number of sorption-desorption steps and the mean time spent in each sorption site (eq 1.1). Second, the peak variance Ktot,2, which characterizes peak broadening, depends on the combined dispersion of the number of sorption steps, urn2,and of the time spent on a single site (see eq 1.2 of Table I). These cooperative effects also act on the higher cumulants ~ ~ and~ Ktot,d. ~ , 3 Discussion on chromatographic quantities H , F, and Z must be started by noting that the H expression for model IV is the usual C term of the van Deemter equation, repeatedly derived in different ways.3 The factor 2 of this expression doubles the term u,R( 1 - R)/k,, which is the H term in model V. These two models differ in the dispersion of the mobile-phase entry process into the stationary phase, model V being a completely ordered entry process. Thus the Poisson random contribution in the mobile phase determines half of this total value. But this Poisson quantity is only a particular value of a more general quantity appearing in the square brackets of the general exponential model 111 (eq 11.10, Table 11). In fact for the Poisson distribution K , , ~ and K,,] are equal, and the square bracket term in eq 11.10 is 2. If, instead, the entry process is more disordered, in eq 11.10 a factor greater than 2 is to be expected. This can be the case of diffusion-controlled processes, nonhomogeneous distributions of the sorbing phase on a column cross section, or voids in some parts of the column. All these physical circumstances can cause an additional broadening in the distribution of the number of sorption-desorption steps. The same arguments can be applied in discussing the coefficient 6 in the Z term (eq 11.15) and the coefficient 12 i n the F term (eq 11.16). The contribution to the specific asymmetry Z arising from the Poisson dispersion in the gas phase is */, of the total, whereas the contribution to the specific flatness F is 3/4. In the more general model TI the relevance of sorption site structural properties is given by K s , 2 / K s , 1 2 , which becomes 1 for the specific case of an exponentially distributed sorption time (compare eq 11.2 with eq 11.10). Thus the omnipresent factor u,R(1 - R)K,I is to be considered the scaling factor of the band broadening process. The other general term, which is contained in the square brackets of eq 11.2, is built u p by two dispersion effect terms: the
the classical C term for liquid-phase partition is ~ b t a i n e d .The ~ factor q depends on K,,2/K,,12, which in turn is most likely dependent on liquid-phase distribution on solid support. These last points have not been further developed here because they require more specific and detailed treatment. However it must be stressed that the results of the present theoretical analysis are useful not only for column performance studies but also to characterize the sorption-desorption process d y n a r n i ~ s . ” ~ In ’~ fact the average desorption time on a single site K,,] and its variance K , , ~ can be determined from band broadening measurements, according to general eq 11.6, provided that the statistical factor of the entry process Km,2/K,,1 is determined. Conversely the band broadening process can be simulated, e.g., by the Monte Carlo method, by determining all the quantities appearing in eq 11.6.
Approximation of the Chromatographic Peak and Rate of Convergence Toward the Gaussian Form Chromatographic elution peaks are not generally symmetrical, but nevertheless they recall the normal or Gaussian curve, whose frequency function Z(c) is Z ( c ) = 1 / ( 2 r ) I l 2 exp(-c2/2)
(82)
and whose cumulative distribution function P(c) is
P(c) = j c Z ( c )dc -m
(83)
The central role of the normal curve in chromatography has been extensively discussed3 and it appears as the limiting solution for the “long column approximation” of various chromatographic theories. The most appropriate and exhaustive explanation is given in the stochastic theory of chromatography through the fundamental central limit theorem of the probability theory. It is, in fact, a general phenomenon that the sum of a great number n of random independent variables or, its equivalent, the n-fold repeated convolution of distribution functions with finite variances, builds up the Gaussian curve when n is great enough.8 The part of the stochastic theory of chromatography that has never been discussed from a general point of view is the rate at which the Gaussian form is attained. When the rate of convergence to this form is low this point also calls for better approximation to the chromatographic peak. The first question is answered in probability theory by the fundamental theorem of Berry-EssCen8 which states that for sums of n random and identically distributed variables with finite variance (as is the case ( 1 7 ) Vidal-Madjar, C.; Guiochon, G. J . Phys. Chem. 1967, 7 1 , 4031. ( 1 8 ) Grushka, E. J . Phys Chrrn. 1972, 76, 2586.
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1891
Stochastic Theory of Chromatography of the present chromatographic models V and VI), the following relationship holds true
(Fn(c)- P(c)l ITpS/nl'' where ps is the reduced third absolute moment of the single random variable:
Constant T is equal to 3; Fn(c)is the resulting cumulative distribution in the standardized variable c. A similar theorem also holds for stochastic processes with independent and stationary increments7 and, thus, for the present chromatographic models I1 and IV, which have proved to belong to the above class. In this case n must be replaced by m , or by its equivalent Lk,/V, (see eq 68). However the constant T i s in this case undefined. Thus for models I1 and IV the rate of convergence toward the Gaussian law depends on 1/L'I2 and on a "deiicate property"8 of the sorption time distribution on a single site, Le., ps. If ps is abnormally high, as has been found for bifunctional sorbed solutes (see below), the Gaussian shape cannot be attained with the column lengths employed in chromatographic practice and peaks exhibit long tailing. This point is illustrated in Figure 2, where constant numbers n of repeated convolutions of different distributions are reported ( n = 1, 12, 20). The Gaussian shape is almost attained for the simple exponential case a, whereas for the two other cases which represent a bifunctional molecule performing successive sorption entry processes (b, c), tailing still persists even a t n = 20. The Edgeworth-Cramir series (EC series)&*is superior to the Gaussian curve in approximating the convolution process. This series is formally written as k
Fn(c) = P ( c ) + CQj(-P) + R n , k ( C ) j= I
(86)
where the first two Qj(-P) terms are
Ql(-P) = - ( S / 3 ! ) f i 3 ) ( c ) Q2(-P) = (E/4!)P'4'(~)+ ( 1 0 S ' / 6 ! ) ~ ( ~ ) ( ~ ) (88)
and R n , k (is~the ) remainder term, which has the following property
in which Mk is an undefined constant, depending only on k but not on n and c; P'")(c) is the nth derivative of P(c); k is the expansion order of the EC series. When k is equal to 0 the Gaussian approximation is obtained. According to eq 89, the EC series is an asymptotic approximation with respect to the quantity n; this means that a great enough n can be found for which the remaining term is of the same magnitude as the last neglected term, and limited EC expansions will approximate the cumulative frequency function. These properties also hold true for the frequency f ~ n c t i o n . ~ , ~ If these conditions are met, the laborious procedure of characteristic function inversion can be avoided and the peak shape is easily built up once S, E , and other cumulant coefficients are given. A second point of fundamental interest for an extended stochastic theory of chromatography is that similar approximation properties also hold true for the present general model 11, thanks to its being a stochastic process with stationary and independent
increments. This topic is dealt with in Chapter VI11 of Cramir's mathematical tract7 which is a basic reference for stochastic theory of chromatography. For model I1 the column length L, or the inverse mobile phase velocity l/u,, play the same role as n in eq 89. The only limitation of the EC series approximation theory is that the effective approximation degree is to be determined for each particular case. Some work has already been performed and the EC series approximation properties toward the present models IV and V have been proved.15 The same procedure has been repeated here for model VI cases reported in Figure 2. The aim has been to take on the peak tailing phenomenon arising from the presence of bifunctional sorbing sites. The remainder term &(c) is numerically evaluated by the Evy distar~ce~.'~ between the true cumulative distribution function F,(c) and its limited development ( k = 0, 1, 2) as a function of n. The LEVYdistance between the function G and its approximation F is the minimum h value satisfying the following relationship9 F(c - h) - h IG(c) IF(c h ) h (90) for all c values. The results are reported in Figure 3. The limiting slopes for rather large n have the values expected from eq 89, that is ll2,1, and 3/2, respectively for k = 0, 1, and 2. A certain intercrossing is observed at low n values for the most skewed case c, but this simply reflects the previously mentioned asymptotic character of the EC series. In Figure 3 the Gaussian approximation lines are shifted apart by a quantity of the same magnitude as the difference in log ps. In fact, for the three cases a, b, and c the ps values are respectively 2.4, 2.8, and 4.2, and the A log ps values for the cases b and c with respect to case a are respectively 0.24 and 0.07; the corresponding shifts in Figure 3 are 0.9 and 0.1. This behavior shows the content of the Berry-Eden theorem previously mentioned. Thus the following simple rule characterizes the tailing phenomenon: the greater the sorption time dispersion on a single site (measured by its ps value), the longer the column or the slower the mobile-phase velocity must be. The EC series, which was in the past suggested for the peak shape fitting or a p p r ~ x i m a t i o n , " , ~must ~ ~ 'be ~ ~considered ~ a faster general approximation of the chromatographic elution peak. We have not tried to analyze the Poisson dispersion effect on EC series approximation properties. To do so the same computations on model I1 cases, similar to the model VI cases a, b, and c considered above, would have to be made, but the problem's numerical complexity is prohibitive. Results of such an analysis are most likely the same as those given in Figure 3, as was observed when model IV and V analyses were ~ 0 m p a r e d . IIn ~ fact the differences between models I1 and VI are the same as those between models IV and V. Thus this study on sorption-desorption kinetics must be considered sufficiently enlightening. This topic will be more complete once the combined mobile-phase diffusion and flow pattern effects and site heterogeneity are considered. Even these points deserve specific treatment.
+ +
Acknowledgment. Financial support from MPI is gratefully acknowledged. (19) Kaminskii, V. A.; Timashev, S . F.; Tunitskii, N . N. Russ. J . Phys. Chem. 1965, 39, 1354. (20) Kelly, P. C.; Harris, W. E. Anal. Chem. 1971, 43, 1170. (21) Kelly, P. C.; Harris, W. E. Anal. Chem. 1971, 43, 1184. (22) Dondi, F.; Pulidori, F. J . Chromatogr. 1984, 284, 293. (23) Dondi, F.; Remelli, M. J . Chromatogr. 1984, 315, 6 7 .
