edited bv
computer series, 144 adsorbed partly desorbed moved along with the flow adsorbed again
Column Operations: A Spreadsheet Model Benson R. ~undheim' New York University New York, NY 10003
Many important laboratory operations are based on an equilibration between two phases that leads to a separation of components. For example, distillation is essentially the establishment of a succession of equilibria between the vapor phase and the liquid phase of a mixture, combined with stepwise separation of the two phases and subsequent condensation. (Sublimation is a similar process involving a solid phase and a vapor phase.) The vapor composition differs from that of the liquid (or solid), as shown by the coexistence diagram. Thus, the condensate is progressively modified ( I ) . Equilibration and Separation in Chromatography Other examples of such operations, which use successive eqilibration followed by separation, include many varieties of chromatography. Adsorption equilibrium causes the amount of material on a substrate to differ from the amount in the lluid that is in contact with thc substrate Then senal separation and eauilibration lead LO cllicient separation (21.-other examples include
purification by repeated recrystallization use of the Craig countercurrent extraction processor (3) u s e of the related flow-injection analysis system (4) For each of these, the partition of some substance between the two phases is described by a function that gives the composition of one phase in terms of the composition of the other phase (and perhaps in terms of temperature and pressure). The stepwise character of such processes, which are generically called "column operations" regardless of the specific physical layout, is well-suited to modeling by spreadsheet ( 5 )representation. The fundamental process can be easily represented, and a variety of experimentally significant complications can be readily treated. Detailed numerical modeling provides valuable insight in many situations, such as studying the dependence on various parameters. However. this method can provide much more, such as uscful rcsults in situations that arc too complex for con\,entional mcans of calculation. To make my kxample concrete, I will use the language of column chromatography in developing a model. A Chromatographic Column A chromatographic column consists of a long tube that is packed with an adsorbent through which a solvent is percolated. One mode of operation involves inserting some adsorbate into the flowing stream a t one end of the column. Then the adsorbate undergoes a cycle many times before emerging a t the far end of the column. The adsorbate is 'Complete listings of both spreadsheets can be supplied upon a p plication to the author.
JAMESP. BlRK Arizona State University Tempe, AZ 85281
Although individual molecules may follow quite different paths, the average behavior is well-defined. Equilibration in Steps
I n this description, the column may be thought of as a sequence of separate cells, in which the basic processes occur. The adsorbate is fed from one cell to the next. In the same way, a distillation column is often modelled as comprising a definite number of "theoretical plates" on which a series of separate condensations and evaporations occur. Any of the column processes considered here may be represented as a finite sequence of individual steps involving perfect equilibration that is followed by perfect separation. The Partition Coefficient
The mathematical function needed to describe the adsorption equilibrium depends on the particular chemical system, but approximately linear relations are quite common. The wncentrations of adsorbate in the two phases remain in a constant ratio, called the partition coefficient (61,which is usually a good approximation over some concentration range. In the same way, vapor pressures are often well-represented by Henry's Law over some range and solubilities. Consider a long column that is packed with V, mL of a stationary phase (71,such as alumina or an ion exchange resin. A constant stream of mobile phase, such as a solvent or gas, flows through the column. At a given time there are V,, mL of mobile phase in the column. Assume that the column may be divided into N theoretical plates. Although the solute is equilibrated between the mobile and stationary phases within each theoretical plate, this equilibrium is not maintained between theoretical plates. Within a given theoretical plate the equilibrium is governed by a partition coefficient. The general expression may be rather complex (see below). For now, we will assume the most straightforward form for this equilibrium relationship, as stated below. moles of solute in the stationary phase
K - volume of solvent in the statwcgyh.+se moles of svlute rn the rnuldt_ph~\evolume of solvent in the mobile phase The Distribution of Solute
The relative molarities of solute in these phases are
and
wherep is the fraction of solute within a theoretical plate that is found in its mobile phase; 1 - p is the fraction of Volume 69 Number 12 December 1992
1003
solute in the stationary phase; and VJN and VJN are the volumes of these two phases. To satisfy the partitioning equilibrium we must have the following.
Rearranging and solving for p gives
Thus, we may solve for the fraction of solute in the mobile phase in terms of the characteristics of the column. Equilibrium a t Plate 0 Let Mo rnol of solute be initially applied a t plate 0.When equilibrium has been established (Fig. la), there will be M@ mol of solute in the mobile phase Mo(l- p ) mol in the stationary phase Next let us transfer the VAN mL of mobile phase from plate 0 to plate 1,along with the M@ rnol of solute it contains. Equilibrium a t Plate 1 When equilibrium (Fig. lb) is again established in plate 1,we have MG2mol of solute in the mobile phase M@(l- p ) rnol of solute in the stationary phase Equilibration a t Plate 0 Then the Mo(l - p ) rnol of solute that remain in plate 0 equilibrate and we get M@(l-p) rnol of solute in the mobile phase Mo(l -p12 rnol of solute in the stationary phase Next let us transfer the VJN mL of mobile phase, with its solute, from plate 1to plate 2. Then the total contents (Fig. lc) of the three plates are
.
plate 0: Mo(l-p)' plate 1: 2M@-p) plate 2: M@
MOP
a
1
Total
I
2
-f'
&(t-P)
The distribution of solute over the column a t any given time can be viewed by plotting a given column of the calculation versus plate number (see Fig. 2). Alternatively, the contents of the last plate may be plotted versus the accumulated time to see the eluate concentration distribution. Any given distribution may be wpied to the zero time column and used as input for a new calculation. Then, by chanfine the value of D, the effect of eluting with a different solv&t or column can be simulated. .. Mixtures of noninteraminn materials mav be treated bv duplicate computations using appropriate parameters and plotting the results together on a single graph. (Interactions can be treated as described below.) Awmmon complication in wlumn operations is mixing due to such factors as diffusion, turbulence, and channneling. In the spreadsheet representation, a term may be added to allow for a small amount of mixing between adjacent compartments and then varied to see its effect. Adjusting Parameters In the preceding case, due to the form of the partition coefficient used (without the mixing term), it is possible to treat this mathematical problem in closed form (7).By experimenting with the values of the parameters in the spreadsheet program, we can see that the retention volume is large when the solute is strongly adsorbed on the column, and the peaks are more narrow when there are many theoretical plates. When the length of the column is increased (by doubling the number of cells), both V , and V . are doubled, and so is the separation between the peaks of two different solutes. However, N is also doubled, so the net result is an increase Although both peaks in the peak widths by a factor of become broader, lengthening the column also separates them more than it broadens them, thus giving better separation. We use this model in an Exoerimental Methods laboratoiy in which the students adjust the variables to eive a reasonable fit of the calculated elution curve to one ihtained in a "wet" experiment. Using the Langmuir Isotherm
Continuing this process in time steps generates an array of cell contents as a function of time and position in the 0
Calculations
a.
An Array of Cell Contents
Plate
column. (See the appendix for details of a speadsheet realization.)
The partition equilibrium expression may be quite different from that used above. For example, the Langmuir "'"I
I
Mobile
I
Station.
Ma
Plate N u m k
Total
Mo(1 - p)'
ZMop(1
- p)
A40p2
Figure 1. Plate contents at successive time steps. 1004'
Journal of Chemical Education
I
.......b6
---.,=I2
..
t=18-
t&4
Figure 2. Column profiles at several time intervals for a column with a linear adsorption isotherm. ..
isotherm (81, which allows for saturation of the substrate, may be appropriate for some systems.
where M,I(M,),, denotes the fractional coverage of the substrate by the solute; MmlVmis the solute concentration in the mobile phase; and K is a constant. Making a small change in notation and rearranging, we get the following.
This presents a complication because determining Mm and M, now requires solution of a quadratic equation. (Even more awkward forms may be encountered in other isotherms.) Nevertheless, the application of the quadratic formula and selection of the positive root leads to the following tractable expression, which can be used in the formula for updating each cell.
(See the appendix for details of a spreadsheet realization.! The results of such a calculatim. as shown in Fimre 3. can be used to examine the effect of surface saturation modelled by the Langmuir isotherm as reflected in the shape of the chromatographic peaks. Other possible explorations include interference between two adsorbed species. In other words, each adsorption isotherm depends on the concentration of both species. Distillation Columns For a distillation column, further complexities appear. The temperature, and hence the composition diagrams, change at each step and at each plate. The corresponding computation can always be made. However, further information is required in addition to the complete phase diagram data: heats of vaporization, the temperature dependence of the heats of vaporization, and heat losses in the column. The procedure used earlier can be applied again: Use the local values of the distribution ~arameters. . couded . with the conservation of materials fl&ng from cell to cell. (No specific example is given here (91.1 Along the same lines, competitive adsorption can be dealt with by calculating the local equilibrium. Then use the result to determine the appropriate partitioning of the several species. Conclusions A simple and physically transparent model for a number of column operations, such as chromatography, can be usefully modeled using a spreadsheet representation. By varying the relevant parameters, students can get a feel for their importance, and they can gain an understanding of the principles involved in the design and use of various column separation processes. When the adsorption isotherms are not of the simplest form, numerical simulation methods must be used, and not only for instruction. When the forms are complex, the analytical results will be unavailable or far too clumsy to be used. This occurs when complicated initial conditions are encountered or when other problems occur, such as complicated elutions, changes in solvent, and mixed adsorption.
Figure 3. Column profiles at several time intervals for a column with a Langmuir adsorption isotherm. Appendix 1. The column with a linear adsorption isotherm may be modelled as follows.
Define cells to contain the quantities p and FLAG. Fill column Afrom A10 downwards with the desired number of plates. Fill the corresponding cells in column B with the initial eonditians (e.g., B10 = 1and all others = 0). Enter the following expression in cell C10. +@IF($FLA&O,+RlO*(l-$p),O)
I n cell C11, enter +@IF($FLA&O,+BlO*$p+Bll*(l-$p),O) Capy eell C10 to the right as many time steps as desired Capy eell C11 to the right in the same way. Then copy row C downwards as required.
2. The column with a Langmuir adsorption isotherm may be modelled by the following steps. Add delinitlons of KAY1 and KAY2 Change the contents uf cells C1U and C11 Copy as before In ClO, enter
In cell C11, enter
References
Vopor-Liquid EquilP~ium,2nd d.; 2. Skwg, D. A,; Leery,;1. J.Principla ofinstrumenid Analysis, 4th ed.;Saunders: Fort Worth, 1991; Cha]3ter 24. 3. Schenk.0. . H:.nahn..R. B :.HertLoof. . .A. VI"fmductlo" to A"olvfim1 chamistn...2nd 4.;Allyn and Bamn: Boston, 1977;p 429ff. 4. Ruriek, J.; Hansen. E. H. Flow I"j'"~ctionilnolysis, 2nd ad.: Wiley: New York. 1988. 5. Omis, W J. 1 2 ~for 3 Sclentrsts and Engimrs: Sybex: San Francism, 1987. 6 . Skooe D. A,: Learn J. J. Princiola ofInsfrumenlolAnalvsis. . .4th ed.: Saundera: Fan worth, 1991:~hspfer21. p 582. 1. Hala, E.; Pick, R. E .: Fried, Y, Cob-, Pergamon: New York, 1987.
A P
7. Perrin. C. Mathamoflcs for Chemists: Why-lnterstience: N w York. 197%p229. 8. AWis,P. W Physiml Chamistry, 4th ed.: W.H.Flwman:San Raneiaeo, 1 9 9 %885. ~
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