Anal. Chem. 1984, 56,1561-1566 (15) Weinberger, R.; Yarmchuk, P.; Cline Love, L. J. In “Surfactants in Solution”; Lindman, Bjorn, Mittal, K. L., Eds.; Plenum Press: New York, 1984;pp 1139-1158. (16) Yarmchuk, P.; Weinberger, R.;Hirsch, R. F.; Cline Love, L. J. J . Chromatogr. 1984, 283, 47-60. (17) Pramauro, E.; Pelizzetti, E. Anal. Chim. Acta 1983, 154,153-158. (18) Yarmchuk, P.; Weinberger, R.; Hirsch, R. F.; Cllne Love, L. J. Anal. Chem. 1982, 54,2233-2238. (19) Weinberger, R.; Yarmchuk, P.; Cline Love, L. J. Anal. Chem. 1982, 54, 1552-1558.
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(20) Mukerjee, P. J . Phys. Chem. 1962, 66, 1733-1735.
RECEIVED for review September 27, 1983. Accepted March, 19,1984. This work was presented in part at the March 1984 Pittsburgh Conference, Abstract No. 47. This work was supported, in part, by the National Science Foundation Grant NO. CHE-8216878.
Model for Chromatographic Separations Based on Renewal Theory Dan M. Scott Statistics Department, Iowa State University, Ames, Iowa 50011 James S. Fritz*
Ames Laboratory, Iowa State University, Ames, Iowa 50011
A simple but reasonably reallstlc model is formulated for describing the behavior of chromatographic peaks. Our approach Is based on statlstlcal concepts and completely avolds the physically nonexistent “theoretical plates” of classical theory. Thls work complements the “rate theory” of chromatography In that we provide a more detailed look at the “reslstance to mass-transfer process” (what we call the “Interphase process”). The model Is a stochastlc one; because molecular level processes are random In nature, we feel that thls is a natural approach. Although a varlety of stochastlc models have been proposed prevlously, they have been damaged by the necesslty of assumlng a particular mechanism. The present theory is largely immune from thls crltlclsm. The paper makes use of results from the theory of renewal processes, but the resun8 should be comprehensible to anyone wlth only a modest acquaintance wlth statlsticai notions.
One of the major conceptual defects of the plate theory of chromatography is the division of the column into so-called “theoretical plates”. As this completely arbitrary division of the chromatographic column bears no relation to any physical realities, the plate theory, for all its descriptive successes, remains unsatisfying. (Giddings (1)gives an excellent discussion of the plate theory and its limitations.) Rate theory has been very useful in explaining the contribution of various dynamic processes to chromatographic peak broadening, but the idea of theoretical plates is still present in that various broadening effects are almost always given in terms of plate height and therefore in terms of theoretical plates. The present work is an attempt to provide a useful theory that does not depend on the use of theoretical plates. There continues to be an active interest in chromatographic theory as evidenced by the selection of recent publications (2-9). The present authors have reviewed some of the past work on chromatographic plate theory and have proposed a simple statistical approach (10). Some of the drawbacks of existing plate theory were discussed and a plate number was introduced that is independent of the capacity factors ( k ) of the various chromatographic peaks.
A statistical approach is really a simple and understandable way to look at chromatography. Each molecule of a sample component alternately enters the stationary phase and returns to the mobile phase many times during its passage through a column. Since all molecules of the chemical do not behave identically, there is a distribution of exit times and the recorded peak is essentially Gaussian in nature. The statistical plate model published recently (10) gives viable results, but the method is not a realistic representation of the physical process that is in progress. The same limitations apply to another statistical approach, the random walk model (11,12). In this model a molecule is assumed to take a step of random length down the column at fiied intervals of time. Or, in what amounts to almost the same thing, it is assumed to take a step of fiied length in a random direction at fixed intervals of time. In either of the above cases the idea of a fixed length of time between steps seems artificial and unrealistic. In this paper we present a model which gives the chromatographic practitioner a nonplate way of thinking of chromatography that can be readily understood. Its basis is a statistical model which is called the “renewal model”. In a sense this is a generalization of the random walk model in that our molecule is assumed to take steps of random length with a random time between these steps. In the theory of stochastic processes, a stochastic process which “renews”or “regenerates” itself is called a renewal or regenerative process. As applied to chromatography, this means that each time a molecule enters the stationary phase (or alternatively each time it enters the mobile phase) the process is “renewed”because from this point the future motion of the molecule is the same, probabilistically speaking, as it was (and will be) at any other such point of revewal. Technically what we are investigating is called an alternating renewal process. It is known that several factors contribute to the broadening of chromatographic peaks; these include extracolumn broadening in the connecting lines and detector, injection broadening resulting from injecting the sample over a finite time period, broadening resulting from axial diffusion in the column, and multipath effects in packed columns. These effects are now well understood and are, in effect, artifacts of the actual chromatographic separation process. Since it is generally assumed that the variances of various contributions to
0003-2700/84/0356-1561$01.50/0 .. . 0 1984 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 9, AUGUST 1984
peak broadening are additive and independent of one another, we will subtract these variances from the observed peak variance and focus major attention on the difference, which will be termed “interphase broadening”. This is clearly the most important aspect of chromatographic behavior because it is the action of the various sample molecules repeatedly entering the stationary phase and later returning to the mobile phase that is the basis of chromatographic separations.
RENEWAL THEORY A statistical approach will be taken. We consider the passage of a single molecule through the column. As the molecule moves through the column it alternates between being “free” in the mobile phase and being “stuck” in the stationary phase. We assume, for the sake of convenience, that our molecule begins a t time zero stuck in the stationary phase. It spends an amount of time W , stuck, and then it breaks free where it spends an amount of time F1. During this free time it travels a distance X1. Then the molecule sticks again for a time W z and then breaks free for a time F2-traveling a distance X 2 during this free time. Then the molecule sticks for a time W , .... Eventually the molecule leaves the column. The Wi (i = 1,2, ...), the Pi(i = 1,2, ...), and the X, (i = 1,2, ...) are called the stuck times (or waiting times), the free times, and the free distances, respectively. We make the assumption that, while in the mobile phase, the molecule moves at the constant velocity, u, down the column. Thus the free times and free distances are intimately related. This relationship is described by eq 1.
xi = UFi
(i = 1, 2, ...)
(1) The stuck times, free times, and free distances are all random. From physical considerations we see that they can take on any positive value, but, because they are random, just which values they will take on cannot be known in advance. In the jargon of probability and statistics we say that W,,F;, and Xi(i = 1,2, ...) are continuous positive random vyiables. We make the following reasonable assumptions about these random variables: (1) Random variables of the same type (the three types are stuck times, free times, and free distances) share a common distribution. (2) All of ow random variables are assumed to be independent of each other except Fi and Xi (i = 1, 2, ...). These are clearly dependent from eq 1. For convenience we shall denote by W, F, and X random variables which share the common distribution of their respective types. The following, rather common notation will be adopted. If Y is any random variable, then either E ( Y ) or p y can be used to denote the expected value of Y , while either Var (Y) or ayz can be used to denote the variance of Y. Note that the strong connection between the free times and free distances (eq 1) implies the following:
E ( X ) = uE(F)
Var ( X ) = u2 Var (F)
(2)
=
uWF
ax2 =
u2CF2
F5 w5
F4
aJ
F3
E
c
w3
F2 w2
FI W,
Distance
Figure 1. Diagram illustrating random motion of a sample molecule
through a column. set PW, PF, uw2, circumstances.
or the set pw,p x , awz, q2, depending on
FUNDAMENTAL STOCHASTIC PROCESSES Basically this paper will investigate the behavior of two stochastic processes, T(x)and S(t). Their definitions are as follows: T(x)= time when the molecule’s position along the column first exceeds x (4a)
S(t) distance along the column that the molecule has travelled by time t (4b) Figure 1is designed to illustrate a portion of the passage of a particular molecule through the column and in doing so to clarify some of the foregoing definitions. To determine S(t) from this diagram, simply locate t on the time axis and move horizontally to the graph. Then move down to the distance axis to read off S(t). To determine T(x) locate x on the distance axis and move up from there until you first reach the graph, then move as much higher as you can and still stay on the graph. Then move horizontally to the time axis to read off T(x). Of course, the graph should extend on the distance axis over to I , the length of the column. But, since a typical molecule can be expected to change phase many times, this is not practical to show in our diagram. T(x)and S(t)are both said to be “stochastic processes”. We must define one additional stochastic process: n*(x) is the minimum value of n such that
k X k>x
These equations can also be written as PX
t
(3)
This completes the basic statement of the model and associated notations. The next step is to determine just which consequences of the model we wish to investigate and then to do so. It is clear that nothing quantitative can be accomplished without some kind of knowledge about the distributions of the stuck times, free times, and free distances. Quite significant results can be obtained from surprisingly little information. We shall assume that we know the six quantities defined above: bw, pF, px, uw2, aF2, and ax2. Because of the relationships in eq 3, there are really only four independent parameters here and we will focus our attention on either the
k=l
Note that n*(x) is the number of the free period which is in progress when the molecule first exceeds the distance x along the column (i.e., at time T(x)).Because there is one free period for each stuck period, n*(x)is also the number of stuck periods which have elapsed when the molecule first exceeds the distance x . We shall defer the analysis of S ( t ) until after we have analyzed T(x),discussed the significance of our results, and explored some examples.
ANALYSIS OF T(X) The time elapsed when the molecule moves past position x (Le., T(x))can Le broken up into two parts, the time spent
ANALYTICAL CHEMISTRY, VOL. 56, NO. 9, AUGUST 1984
in the mobile phase and the time spent in the stationary phase. Since the molecule moves at the constant speed u while in the mobile phase, and since it moves a distance x , it follows that the time spent in the mobile phase is x / v . The amount of time spent in the stationary phase is the sum of the stuck times which have occurred, that is n*(d k=l
w k
Thus, since T(x)is the s u m of the free time and the stuck time, we have
T(x) = ( x / v ) +
the mean and the variance of the distribution. Since these are given by eq l l a and l l b , we see that the probability distribution of T(x)is completely determined. Of course, most of our interest is concentrated on T(1), where 1 is the length of the column. That is, we are interested in the exit time from the column. This is merely a special case of eq 11,but because of its importance we shall describe it below. Note: henceforth, for the sake of brevitry and convenience, we shall use T to describe the exit time from the column, rather than T(1):The exit time, T, is approximately normally distributed; and
n*(d k=l
w k
1563
E(T) =
(7)
;[
1+
-1
Pt /U
We can use the theory of random sums to obtain the following results: (1)T(x)is asymptotically normally distributed. (2) The expected value of T ( x )
E[T(x)l = ( x / u ) + E[n*(x)lE(W)
(W
(3) The variance of T ( x )
Var [T(x)]= E[n*(x)]Var (W)
+ [E(W)I2Var [n*(x)] (8b)
Notice that the variation in T(x)comes from two sources. The first term comes from the variation in the stuck times; the second term comes from the variation in the number of stuck times. Although the above relationships are very interesting, in order to accomplish ow desired goal the expected value and variance of T(x)must be expressed in terms of “known“ parameters. (That is, the expected values and variances of W , F, and X.) To do this we must express E[n*(x)]and Var [n*(x)]in terms of the known parameters. Fortunately, this is a problem which is solved in statistical renewal theory (13). The results are as follows:
-
E[n*(x)]
X
X
-= II,
-
Note: the used in eq 9 and 10 is used here to show that the two sides are asymptotically equal as x becomes large. It is accurate enough, for our purpose, to assume that these quantities are actually equal, and we shall do this in the remainder of the paper. More precise results than eq 9 and 10 can be obtained, but use of the longer equations does not improve the results significantly. If we substitute for E[n*(x)]and Var [n*(x)]in eq 8 by using eq 9 and 10, the following system is obtained: T(x)is asymptotically normally distributed; and
It is instructive to compare eq 12a with the classical formula for retention time. Observe the following facts: (a) l / v is the dead time for the column, to. (b) pw is the average length of a stuck time. (c) p x / u is the average length of a free time. Since the molecule has the same number of free periods as stuck periods it follows that PW/(P,/U)
=k
(13)
That is, the ratio of eq 13 is the long run ratio of time spent in the stationary phase to time spent in the mobile phase, which is referred to in the literature as k,the capacity factor. Thus, eq 12a is seen to be equivalent to
E(T) = t,[l
+ k]
(14)
And this, of course, is the classical formula for the retention time. Note that we needed to make no assumption that equilibrium is attained, as is usually done. Furthermore, the mean and variance of the exit time, T, can be rewritten by using the free time parameters, pF and U F ~rather , than the free distance parameters, px and. :a To do this simply substitute for px and:u in eq 12 by using eq 3. The results are
E(T)=
i[ U 1+
E]
In this form it is even easier to see that eq 14 will result. An alternate way of writing Var (7‘)may be of some interest. We make use of
k = -P W
to =
PF
-1
E[n*(l)]= 1 W F
U
(16)
and eq 15b to show that
+
Var (2‘) = E[n*(l)][aw2 k2aF2] These equations summarize everything we need to know about T(x). According to the first statement, the random variable T ( x ) has a probability distribution which is approximately normal (i.e., Gaussian). Or, at least, it says this is so if x is large enough. We should translate this as meaning that if the number of stuck times, n*(x),is large enough (say 20 or 301, then T(x)will have approximately a normal distribution. Since this is the case during a typical chromatographic separation, we can take the distribution to be normal. A normal distribution is completely described by two things,
(17)
RENEWAL THEORY: DISCUSSION Here we discuss how the renewal theory supplements and improves the rate theory of chromatography. The renewal theory concentrates on what is known in the rate theory as the resistance to mass-transfer broadening. (The authors prefer the term “interphase broadening”, since this broadening is the result of transfers between two phases.) The renewal theory supplements the rate theory in several ways. First, it makes it clear precisely what the cause of interphase broadening is. This is by no means obvious from rate theory alone.
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ANALYTICAL CHEMISTRY, VOL. 56,
NO. 9, AUGUST 1984
Second, renewal theory raises the possibility that the influence of various factors such as eluant concentration, thickness of stationary phase, particle packing size, etc. can be predicted by determining either theoretically or experimentally their effect on the stuck times and free distances (or free times). For example, a smaller particle size could be expected to reduce free distances, while having no significant effect on stuck times. (This excludes any effects of the stagnant mobile phase.) On the other hand, a thicker stationary phase could be expected to lead to longer stuck times while having little or no effect on the free distances. (Molecules which sink into the stationary phase would take longer to break free.) This is perhaps a good place to make some remarks about the form of the stuck time and free distance and free time distributions. A molecule which has just broken free from the stationary phase has a higher probability of sticking again during the next brief time interval At than does a molecule which has been free for some time. This is because we can guarantee that the molecule which has just broken free is, at least, still near the stationary phase and hence is vulnerable to readsorbtion. No such guarantee can be made for a molecule which has been free for some time. Similar arguments can be made for the free distance and stuck time distributions. Statisticians call distributions of this type “decreasingfailure rate” or DFR distributions. (The odd name comes from studies of reliability theory.) The most important distribution of this general type is the exponential distribution which has a “constant failure rate”. If the thickness of the stationary phase is not too great, then it becomes reasonable to assume that the stuck times have an exponential distribution. Things are not so straightforward for the free distances or free times, but if we make the assumption that the free distances (and hence free times) are also exponentially distributed, then we have the special case considered by Giddings (14) and by Beynon et al. (15). We shall also consider this case as an example of how to apply eq 12. Since this process has been called the Bessel process in statistical literature, we shall use that name here.
0
I
2
3
4
Capacity F a c t o r
~
5
6
7
k
Figure 2. Number of stuck time periods as a function of capacity factor.
peak. To do this we solve eq 20a and 20b for X and P in terms of E(T)and Var (T).
We shall now investigate some connections between the Bessel process and the classical plate theory (10). We shall let r and k denote the true number of theoretical plates and the capacity factor, respectively. (r = N [ k / ( l + k ) ] = (NNeff)1/2, where N is the classical theoretical plate number and Neffis the effective theoretical plate number.) From plate theory we have the following:
BESSEL PROCESS For this special case we assume that the stuck times, free distances, and hence free times are all exponentially distributed. (The density function for an exponential distribution with parameter A is f ( t ) = Ae-xt ( t 1 O).) That is
W
-
exp(p)
X
-
F
exp(X)
N
exphu)
(18)
(The third distribution is a consequence of the second one.) The approach taken by other authors is to find the exact distribution of the exit time T and then to observe that this exact distribution is approximately normally distributed with a certain mean and variance. In this approach, while the exact exit time distribution has been derived, it is the normal approximation that is of primary importance. We can obtain this same normal distribution, but with vastly greater ease. From the general properties of the exponential distribution it follows that
,
I
I
Using the above equations and eq 21 we can determine the relationship between the parameters A, p and r, k. The results are given below: A = 2kr/[l(l P
= 2r/[to(l
+ k)]
(234
+ k)l
(2%)
k=X V / ~
(244
r = t0(p + Au)/2
(24)~)
These relationships enable us to investigate a question that is not considered in plate theory. How many times can we expect a molecule to become stuck as it moves through the column? In other words, what is E[n*(l)]?Equations 9 and 19 tell us that
E[n*(l)]=
l/px
= XI
(25)
But, utilizing eq 24a in the above we obtain If we take these equations and use them in eq 12, we obtain the following: T is approximately normally distributed; and
Var (T) = 2A1/p2
(20b)
It is worth noting that the parameters A and I.L can be computed from the mean and variance of the observed exit
E[n*(l)]= 2kr/(l
+ k)
(26)
Now, it is an experimental fact that the true number of theoretical plates, r, is approximately constant as k is changed. Thus, we can graph the expected number of adsorptions against the value of k, as we have done in Figure 2. A typical example might be r = 6000, k = 2. In that case the expected number of adsorptions is 2(2)(6000)/3 = 8000. This sort of computation lends strong support to the assumption that there
ANALYTICAL CHEMISTRY, VOL. 56, NO. 9, AUGUST 1984
are enough stuck and free periods to result in a Gaussian exit time distribution.
APPLICATION TO MULTISITE MODELS As an application of the renewal model we shall show how it might be used to analyze a multisite problem. In a multisite problem there are assumed to be several different types of adsorption sites, different in the sense that the stuck time distributions are different for the different sites. The information we must have in order to perform our analysis is the expected value and the variance of the stuck times for each of the different types of sites, the expected value and variance of the free distance, and the probabilities of adsorption into each type of site. The type of site that the molecule is adsorbed onto a t the end of any particular free period is a random variable which we shall denote by M. Now the stuck time is a random variable which depends on M. Thus, we will denote the stuck time by W(M). We must discover the mean and variance of W(M). From probability theory we have the following:
Var
W v M ) l = ~bwwoIW1 (274 [W(M)I = Var VW(M)lW1+ W a r [W(M)IMl) (27b)
The vertical lines in the above are used to indicate “conditional” expectations and variances, i.e., the expected value of W(M),given M. The above mean and variance can now be substituted into eq 12 along with the (presumed) known mean and variance of the free distance to obtain the mean and variance of the exit time. We shall illustrate the above by considering the following special case which has been considered by other authors (16, 17). The free distance is assumed exponential with parameter A. We have n different types of sites and the sticking time at a type i site is assumed to be exponential with parameter pi. The probability that adsorbtion has occurred onto a site of type i is ai,where we naturally have n
cai = 1 i=l
If we use this information in eq 27a and 27b, we obtain the following:
Of course the free distance has expected value and variance equal to 1 / X and 1/X2, respectively, as we saw before in our analysis of the Bessel process. So these results can simply be substituted into eq 12 to give us the mean and variance of the exit time.
ANALYSIS OF S ( T ) For column chromatography T ( x )is the stochastic process of primary interest, but in certain cases, notably thin-layer chromatography, we are interested in the position of the molecules at time t. That is, we are interested in S ( t ) . This problem is somewhat more resistant to statistical analysis than the exit time problem. However, in his study of “sojourn times”, Takac (18)has worked out results which are readily
1565
applicable to our problem. His results are summarized as follows: S ( t ) is asymptotically normally distributed; and r
’ I
In thin-layer chromatography the time t is generally the same for all sample components. The eluant velocity u is also known. Therefore, these equations should be directly applicable to TLC provided some reasonable measures or estimates of pw, px, uw2,and ax2are available for each of the sample components.
CONNECTION BETWEEN S (T)AND T(X) Considering the strong relationship between S(t ) and T(x) which is apparent from either their definitions or Figure 1, it is not surprising that we can discover relationships between their means and variances. These relationships are given below:
E [ T ( x ) ] that value of t such that E [ S ( t ) ]= x
(31a)
E[S(t)] that value of x such that E[T(x)]= t
(314
Var [S(t)]= Var (T[E(S(t))])V,2 = Var [T(Vct)]V,2 (3W
V, is the velocity of the component under investigation (see eq 30a):
Equations 31a and 31b are particularly useful for programmed chromatography where it is often simpler to investigate the process S(t). Then, under the assumption that the component velocity remains relatively constant during the time while the component is leaving the column, we can use those two equations to compute the mean and variance of the exit time.
RENEWAL THEORY WITH RANDOMIZED COLUMN LENGTH In our previous analysis it was assumed that if the molecule moved a longitudinal distance, d, then it did so in a time diu. But this is not necessarily so. This is because the velocity may not be constant, and even if the velocity is always u the total distance traveled may be larger than d, since there may also be a transverse component of motion. We shall assume here that u is constant but that the actual distance traveled is not necessarily d. We can model this situation by assuming that the length of the column is a random variable L . Thus the exit time, T , is given by n*(4
T = T ( 6 )= ( 6 / u )
+ kC= l W ,
(33)
We shall assume that L is independent of the other random variables we have defined. Then we have the following results: E(T) = E ( E [ T ( L ) J L l l
Var ( T ) = Var ( E [ T ( L ) I L ]+JE{Var [ T ( L ) l L l )
(34) (35)
(This notation refers to conditional expectations and variances.) These formulas yield
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 9, AUGUST 1984
E(?') =
"'[
U
1
+
$1
(36a)
Var (79 =
. (36b)
This model may be appropriate as a model of multipaths. (Though that situation may be more complicated than this. Note the similarity of eq 36 to eq 12.)
RENEWAL THEORY WITH RANDOMIZED DEAD TIME A development similar to the above can be made by assuming that the dead time for the column is random. (Call this random variable 7.) In other words, the total amount of free time necessary to travel through the column equals r, a random variable. This is probably a good model to account for the effect of longitudinal diffusion. In this case we can derive the following results for mean and variance of the exit time: E(T) = E(7)[1 + ( p w / b F ) l
(374
out that the usual approach for such a stochastic model is to assume a specific mechanism (such as exponential free and stuck times) and derive from this, rigorously, the exit (or position) density. One can then derive a limiting approximation to this for long time intervals. The strength of this approach is also the cause of its biggest weakness. A specific mechanism must be assumed, but only for a very few mechanisms is it possible to derive the exact distributions of exit time and position. In contrast, our approach, while it does not give the precise distributions, does give the asymptotic or limiting distributions provided, only, that we know the means and variances of the stuck times and free distances (or free times). In addition, the assumptions of our model are reasonable and easily understood. The renewal model fits into current approaches to chromatographic theory as a refinement of rate theory and an extension of the random walk model.
ACKNOWLEDGMENT We thank Krishna Athreya for his helpful suggestions.
LITERATURE CITED Giddings, J. C. "Dynamics of Chromatography"; M. Dekker: New York, 1965; Part I, p 15-26. Horvath, C; Lln, H. J. J. Chromatogr. 1978, 749, 43. Kaiser, R. E.; Oeirich, E. "Optimierung in der HPLC"; Huthig: Heiderberg, 1979. Niisson, 0. HRC CC , J High Resoluf Chromatogr , Chromatogr , Commun. 1982, 5,38, 143. Sald, A. S. HRC CC, J. High Resolut. Chromatogr. Chromatogr. Commun. 1979, 2 , 193. Knox, J. H.; Scott, H. P. J. Chromatogr. 1983, 282, 297. Smuts, T. W.; Buys, T. S.; du Toit, 0.; du Toit, J. W. HRC CC , J . High Resoiut . Chromatogr ChfOmatOQf. Commun . 1981, 4 , 363. Wright, N. A.; Vilialanti. D. C.; Burke, M. Anal. Chem. 1982, 54, 1735. Chen, J. C.; Weber, S. G. Anal. Chem. 1983, 55, 127. Fritz, J. S.; Scott, D. M. J. Chromatogr. 1983, 277, 193. Giddings, J. C. I n "Chromatography"; Heftman, E., Ed.: Relnhoid: New York, 1961; Chapter 3. Giddlngs, J. C. "Dynamics of Chromatography"; Marcel Dekker: New York, 1965; Part I, pp 29-33, 65-66. Cox, D. R. "Renewal Theory"; Methuen: London, 1962. Giddings, J. C. "Dynamics of Chromatography"; Marcel Dekker: New York, 1965; Part I , pp 66-72. Beynon, J. H.;Clough, S.; Crooks, D. A.: Lester, G. R. Trans. Faraday SOC. 1958, 54, 705. Giddings, J. C. J. Chem. fhys. 1957, 26, 169. McQuarrie, D. A. J. Chem. fhys. 1963, 38, 437. Takac, L. Ann. frobability 1974, 2 , 420.
.
Compare this with eq 15 using E(r) = l/u. If the variation in dead time is due to longitudinal diffusion of the sample molecules in the mobile phase, then it can be shown that
E(7) = l / v Var
(7)=
la2/v3
(384 (3%)
where a2is the variance parameter of this diffusion. Using these results in eq 37b we see that the expected value of the exit time is unchanged from the result without diffusion. The variance, on the other hand, is increased by the term
DISCUSSION Our model falls under the general category that Giddings has referred to as the "stochastic approach" (14). He points
.
.
RECEIVED for review January 17, 1983. Resubmitted November 3,1983. Accepted March 26,1984. The Ames Laboratory is operated for the US.Department of Energy by Iowa State University under Contract W-7405-ENG-84. This work was supported in part by the Director of Energy Research, Office of Basic Energy Sciences.
