THE RANDOM DOWNSTREAM MIGRATION OF MOLECULES I N CHROMATOGRAPHY' 1. CALVIN GIDDINGS University of Utah, Salt Lake City
THEORIES of chromatography are notably abstruse to those not conversant with a rather intricate mathematics (1). This mathematical difficulty can be partially avoided by use of the theoretical plate model, hut even this model, especially in some of its recent ramifications, is exceedingly complex. The big disadvantage of the theoretical plate model, however, is that it sidesteps the kinetic processes that make chromatography what it is. This article outlines a theory of chromatography, based on the classic random-walk problem, that includes the various kinetic effects. The approach involves more physical intuition and less difficult mathematics than current theories. Two identical molecules, started a t the same position in a chromatogram (of the paper, gas, or adsorption type, etc.), would soon he found in different positions due t o random influences. I t is well known that, molecules are active in changing from the fast moving mobile phase (the carrier) to the stationary phase, and back again. Each transfer is a random incident depending upon energy fluctuations for its success. I t is as if a coin were being tossed and with "heads" the molecule would change to the other phase, and with "tails" it would remain as is. All that is needed, then, to separate these two molecules in identical places, is for one t o come up with "heads" and the other "tails." This would put them in different phases moving with respect to one another, and insuring a separation between their respective locations. Successive coin tosses would lead t o greater or lesser separation, but on the average, according to the statistical laws governing these things, the molecules would drift apart. The problem, then, is one of coin tossing and knowing how many tosses in the development of a chromatogram. Before solving this problem, we will outline other influences responsible for separating these molecules, along with a simple statistical approach to cope with these effects. The carrier substance, in chromatography, must percolate through a porous medium as part of the process. The medium, or part of it (perhaps just its surface in some cases), is the stationary phase. In all cases the carrier is forced to follow a tortuous path, veering from one direction to another in avoiding granules or paper fibers of the medium. Two molecules, taking separate and winding channels through the medium, will not, on the average, go the same distance (leaving out, for the moment, the transfer between phases) in a given time. Each molecule is acain following a random The results nresented here constitute nart of a momam sunported by a research grant, RG-5317, from the National Institute of Health, Public Health Service.
.-
path, and in passing by each granule it may be considered, by some random procedure, to be choosing either a fast or a slow channel. In addition to the above two effects, there is another more familiar influence in the separation of identical molecules. This is ordinary diffusion. This operates, also, as a random process on the molecular level. We will review some simple concepts and formulas of general use in statistics and diffusion that will make possible the calculatiou of zone spreading in chromatography. RANDOM-WALK MODEL
Although each molecule varies in the amount of time it spends sorbed in the stationary phase, there is an average value for this quantity when all the migrating molecules are considered. Typical of results determined by numerous random incidents, the sorption times of the molecules group around the average sorption time in the form of a "Gaussian," or "normal" density function. The measure of the spread of this normal curve is the standard deviation, usually denoted by a. The random-walk model involves a series of steps in either the positive or negative direction (8). The direction of each is determined by the outcome of a simple random experiment. If a coin is used, the random walk is symmetrical, since there is (supposedly) an equal chance for either outcome. After n steps, each of length 1, the particle has been displaced a distance X from the origin. The standard deviation in X is In a general way, 1 may represent not only a displace ment in distance, but in time, etc., as mill be seen. We will find the effective 1 and n in chromatographic processes in order to relate the width of zones to kinetic processes. There is another result in statistics stating that if several simultaneous and independent random processes are occurring, each haviug a value ar as in equation (I),then the final spread of the zone is determined by a, where this equation is stated by saying the separate variances, rr2,are additive. We will use this equation t o calculate the combined effect of the kinetic, flow, and diffusional influences on zone structure. An additional concept, related t o diffusion, is useful here. Ordinary diffusion is a result of the random JOURNAL O F CHEMICAL EDUCATION
movement of molecules first in one direction and then in the other direction. This process resembles the random walk problem that we have discussed. A well-known formula (3) relates the spread due t o random fluctuations, u, t o the spreading influence of diffusion, the diffusion coefficient, D where t is the time that the process has been going on. From this simple equation, we see that it is possible either t o describe chromatography as a random process with a value of u, or a diffusion process with a value of D. However, aside from this, equation (3) will be used to find the u due to ordinary diffusion processes once the diffusion coefficient, D, for that process is known. The above equations can be used to find the spread of a zone, but the location of the zone must also be known. This is related t o the average sorption time that was mentioned. The carrier stream in chromatography is moving with an average speed which we will denote by u. While the molecules are in this moving phase, they too are moving with a velocity u. However, while they are in the stationary phase, their downstream velocity is zero. The average velocity of these molecules, fi, is somewhere between zero and u, depending upon the relative time spent in moving and stationary phases. If t2 is the average of the total time spent in the stationary phase (average sorption time), and ?I the average time spent in the moving phase, then the average velocity, 6,is
+
ti) is the fractional time that a molethe term tl/(tl cule spends in the moving phase. It can be denoted by R, and is usually equal t o the R,value so commonly used in chromatography. R is determined by the equilibrium distribution of molecules between the moving and stationary phases. The above relations were first obtained by LeRosen (4). ELUTION CHROMATOGRAPHY
We will now proceed with an analysis of a method known as elution chromatography. The solute zones, in this case, are washed completely through the sorbent strip or bed, and analyzed as they reappear on the downstream end. Experimentally one measures the concentration of the eluted material as a function of time and usually obtains a Gaussian peak for each substance. Interest is centered around the time of appearance of the peak and t,he width of the peak in seconds. It is possible, now, to give quantitative meaning to the terms in equation (1) for the elution case of chromatography. The number of steps, or coin tosses, n, in the random-walk problem, can be interpreted as the number of random incidences leading t o the separation of molecules in identical places. Every transfer of a molecule from the carrier to the stationary phase, and back, may be considered such a random incident. If we let s equal the total number of sorptions (carrier to stationary) before elution, then there must also be e desorptions (stationary to carrier), since every sorption must be followed by a desorption. The total VOLUME 35, NO. 12, DECEMBER, 1958
number of phase changes, then, is 2 e, and this may be considered as the number of steps, n, in the random walk. The length of a step, 1, in equation (1) is the distance traveled in the positive or negative direction with each coin toss. Thus if two molecules are in identical positions, and only one makes a random move, the molecules will as a result be separated by a distance 1. We are not interested in a distance separation here, but in a time separation. Molecules appear with different elution times rather than spatial distributions. Hence we are interested in the time separation of two molecules due to a single random move. If a random move were made such that one molecule were placed in the moving phase and the other in the stationary phase, the stationary phase molecule would be behind in the race by a time equal to the average time required for its desorption. Let this time equal At. This time is equivalent to the length of step in equation (1) since it is the separation due to a single random event. We are now able to put together the various parts of equation (1). The standard deviation has the dimensions of time, and is denoted by 7 . Thus substituting r for u, At for 1 and 2 c for n, we have This equation relates T to kinetic phase-transfer terms, At and e, but further refinements are desirable. First, let us consider a bed of length L through which the carrier is moving with velocity u. A given molecule of the carrier substance spends the residence time, = L/v before the exit of the column is reached. The solute molecules, on the other hand, spend only a fraction of their time moving with this velocity u. Nevertheless, each molecule must spend the time i, in the carrier before exit. An additional time, t, ,is spent in the stationary phase. The average elution time of these molecules is il t2. We have left until now a discussion of the rates of transition between the mobile and stationary phases. Let kl equal the transition rate from the mobile phase t o the stationary phase (k, can be interpreted simply as the average number of sorptions per second of existence in the mobile phase).' Similarly, k2 is defined as the transition rate from the stationary to mobile phase. If each molecule must spend a time i, in the mobile phase before elution, and in each second it sorbs kl times, then it sorbs a total of k,i, times before elution. Thus e = kltr. The value of At in equation (5) can be given in terms of kr. It is to he remembered that At is the average time required for desorption. Smce k2 can be interpreted as the average number of desorptions per second, then l/k2 is the average time required for one desorption, At. Substituting, then, l/ka for At and k,t, for r , equation (5) becomes
+
7
=
-\/amz
(6)
This value of T has been obtained by more rigorous methods (6). This equation relates the spreading of a zone directly t o the rates of the kinetic processes involved. Finally, without introducing any new terms, it is possible to calculate the average elution time of a com-
ponent. It was shown that the total number of sorptions is k,il, and it may similarly be shown that the number of desorptions = k&. Since these are equal, we have
want the time separation in a step, we again divide the length of step by C. Thus using equation (1) with the l/C in place of 1, we have
The mean elution time, i (the time to reach the peak maximum), is t, tz
The last expression is obtained with the help of equation (12). Substituting zZ from equation (10) into the last equation yields
+
and, as has been shown, h can be written in terms of column length, L, and flow velocity v . The term (1 kl/kz) is equal to the reciprocal of the R value mentioned earlier. The elution time can, with equal success, be described in terms of R or kl and kn A process similar to the above, as we have mentioned earlier, is the spreadmg of the elution peak due to the random and tortuous path of molecules through the porous medium. Siice the streamlines must continuously bend, each molecule spends a good deal of time moving laterally instead of in the over-dl direction of flow. At any instant the solute molecule may take a lateral path or a flow-direction path, in addition to having paths of different velocities. We can simplify the picture by assuming that each path is equal in length to the average particle diameter, d, of the medium. Random influences determine, a t the conclusion of every step, whether the next step will he lateral or in the flow direction. Each choice involves a distance approximately equal to d, so that this may be considered the length of step; the number of steps is simply the number of paths of length d needed to pass through the column length, L. Thus n = L/d. Although d is the distance of separation with a given step, information is needed concerning the difference in elution time caused by a single step. Remembering that the velocity of the average molecule is zi, the time separation of molecules with one step is the distance separation, d , divided by the velocity, zi. Substituting 1 = d/C, and n = L/d into equation (1) yields
+
Equation (4) shows that zi is the product of u and the ratio, ill(& i2) = R. Using equation (7) for is, we have
+
Substituting this into equation (9) we have
This is an approximate value for the standard deviation in elution time due to the porous nature of the medium. This result can be obtained by other methods (6). Ordinary diffusion is usually written in terms of D rather than 1 and n. These can be related by equating the value of u in equation (1) with that in (3)
The time i = t, since appreciable diffusion occurs only when the molecule is in the mobile phase. Since we
This equation, then, serves to relate an additional spread of the elution curve t o the diffusion coefficient of the solute in the carrier substance. We now have three contributions to the over-all spread in the elution peak. These are given by equation (6), ( l l ) , and (14). Since the variances add as shown in equation (2), we have for the over-all variance,
where we have substituted L/u for 4. Thus the overall variance can be expressed in terms of the fundamental molecular parameters and operating conditions, kI, k2,D, d, v, and L. ANALYSIS OF ZONE SPREAD
It often happens that zones are stopped somewhere before elution, and cut from the chromatogram. This is especially common with .paper chromatography. The zone in this case has a certain thickness with the standard deviation measured in terms of distance. This distance is shown as u. We exclude both here and in the elution case a discussion of chromatography when tailing is present. Zone asymmetry of this type does not yield to such simple theory as presented here. A simple relationship can be expressed between u and 7. The time r is required for the zone to move forward the distance u. Since the mean velocity of the zone is zi, (In this same way we found. that the relationship between the length of step in distance and in time involved zi.) If the value of from equation (10) is used in (16), and this substituted into (15), it is found that
The term L now represents the distance the zone has moved along the bed rather than the entire bed length. The last equation may seem to be derived by a somewhat roundabout method, since, for the last two terms, the length of step was converted to the time of step using 6, and now we have used zi in the inverse sense to convert a time, r, to a distance, c. Thus the term, Ld, for random flow effects, may be obtained immediately by considering the length of the step as d, the number of steps as L/d, and uz as the square of the first multiplied by the second. The present method was used, however, since the contribution due to pbasetransfer reactions, equation (6), is most simply obtained in terms of time increments. This term, the first on the right hand side of both (15) and (17), is the JOURNAL OF CHEMICAL EDUCATION
most interesting term from a chemical point of view. An interesting point concerns the relative importance of the terms on the right of equation (17). This depends upon the operating conditions, particularly on the flow velocity, u, of the carrier in the chromatoaram. At large flow rakes the first term (phase-transfer effects) predominates, while a t small u, ordinary diffusion is the large term. The middle term (random flow effects) is intermediate between the others. It is evident that the flow rate can be adjusted to give the smallest value of a. At this optimum flow rate, the three terms, a t least for gas chromatography, are the same order of magnitude. THEORY AND SEPARATION
The zone spread given by equation (17) becomes very significant for borderline separation problems. While the center of a zone may have migrated away from the center of another, the spread is often sufficient to cause the overlap of the two. I n this case the effectiveness of the separation is impaired. This situation can he remedied either by effecting a greater separation of the centers or by reducing the spread of one or both of the components. The first of these is tantamount to altering R values. The separation of two zones is proportional to the difference in their separate R values, other things being constant. The value of R can he altered by chwging the stationary or mobile phases, and it is only necessary to find a combination such that the R's are changed in opposite directions. A change in R may be considered either as a kinetic effect or an equilibrium effect. Equation (10) can be used to evaluate R in terms of the transition rates, k, and k2, but R is more often considered as the equilibrium fraction of molecules in the mobile phase. The two values are identical; the outlook, only, is different. The second remedy to incomplete separations proceed by reducing the spread of either or both components. We have already indicated that this can be done by altering flow rates in accord with the theoretical equations. A more fundamental study, just beginning, involves kl and lip. These two kinetic parameters are fixed by the chemical nature of the mobile and the stationary phases, and the interface between them. When this dependence is better understood, a certsin degree of control on k, and k2, and hence on o and T , will be possible. It should be noted that the height equivalent to a theoretical plate (H.E.T.P.), which is
VOLUME 35, NO. 12, DECEMBER, 1958
itself a measure of spread, is related to the more fundamental kinetic parameters kl and kz. The relation between them is (H.E.T.P.) =
2k,v ( k , Ic,)*
+
(18)
We see from this that any change in the kinetics of the system will be reflected in the value of (H.E.T.P.). In summary, separability depends upon relative zone movement and upon zone spreading. The common index of the former is the R value and of the latter is (H.E.T.P.). Both of these, and thus the over-all performance of a chromatogram, depend upon k, and k2. The prediction of the latter from first principles shows a great deal of promise by way of leading to improved separations. CONCLUSION
The use of random variables can be extended to other migration problems in addition to chromatography. The migration in an electrical field of an ion that changes its charge has been considered by the more rigorous method of random flights (7, a), but could have been considered here. Chromatography itself could he more exactly treated as a problem in random flights since the length of step, often considered in this paper, is not constant. For this reason (among others), the result obtained in equation (6) may be considered correct only by accident. We may say, to its advantage, that such a treatment would not be exnected to be in error bv more than a constant multi~lving term the order ofeunity. Also it is doubtful ihat such simplicity as obtained here could be introduced into an exact, rigorous treatment. Chromatography has been treated rigorously as one involving random processes, but the methods are not simple (9, 10). The results, however, agree with equation (6). LITERATURE CITED (1) See, for example, T a o ~ A s ,H. C., Ann. N. Y. Acad. Sd., 49, 161 (1948); also see ref. ( 8 ) . S., Re". Mod. Phys., 15, 1 (1943). (2) CHANDRASEKHAP: G. E., AND L. S. ORNSTEIN, Phys. Rev., 36, (3) UHLENBECK, 93 (1930). A. L., J. Am. dhem. Soc., 67, 1693 (1945). (4) LEROSEN, J. C., AND H. EYEING,J. PhW. Chem., 59, 416 (5) GIDDINGS, (1955). M. J . , J. Chem. Phys., 27, 270 (1957). (6) BERAN, (7) MYSELS,K . J., J . Chem. Phys., 24, 371 (1956). J. C . . J. Chem. Phvs.. (8) GIDDINGS. . . 26.. 175.5 11957). . . igj laid., p. i69. ' (10) BEYNON, J. H . , ET AL.,Trans. Faraday Soe., 54, 705 (1958).
