Oct., 1959
CHROMATOGRAPHIC FRONTAL ADVANCE OF AN ADSORBATE
present measurements show that the surface structure of calcite is unaffected by the outgassing temperature. Conclusions The heat of immersion of kaolinite exhibits a discontinuity corresponding to dehydration between surface hydroxyl groups. This occurs concurrently with the loss of bulk lattice water. The heat of immersion of the kaolinite surface free of physically adsorbed water is 660 5 ergs/cm.2 and that of the high temperature phase (metakaolinite) is approximately 400 ergs/cm.2. The heat of immersion of calcite is 762 i 2 ergs/cm. and is independent of the outgassing temperature from 120 to 300". Noticeable conversion of CaC03 t o CaO occurred only a t the highest outgassing temperatures investigated. Acknowledgment.-This work is a contribution from Project 47d of the American Petroleum Institute and the Department of Chemistry, The University of Texas. The authors wish to express their appreciation to the American Petroleum Institute for their unfailing support. They also wish to express their appreciation to Mr. Richard Every for his continued assistance throughout the investigation and to Mr. Clarence Willia.ms, the departmental glassblower, for his effective help in building the apparatus.
*
1641
DISCUSSION A. C. ZETTLEMOYER (Lehigh University).-Couldn't
the water lost from the kaolinite be from the exchange ions rather than from silanol groups? N. HAcKERMAN.--Not completely. The discontinuity in the kaolinite data comes where hhe lattice loses its bulk water. This was seen in the outgassing procedure where unless the temperature was raised slowly through this point the samples exploded.
L. -4.ROMO(du Pont Company).-In addition to the adsorbed water which varies from 1 to 595, there is the water from dehydroxylation. For this infrared data would be useful. It is a well known fact that dehydroxylation takes place between 500 and 600" as shown by D.T.A. curves. Thus the dehydroxylation which may have taken place a t 300" must be very small. With reference to edge effects, English workers have shown that the edges of silica tetrahedra contribute about 5% of the total cation exchange capacity of kaolinite.
N. HACKERMAN.-D.T.A. curves are definitely nonequilibrium in character (see Ref. 4) and are generally not carried out in evacuated samples. Our samples were outgassed a t mm. and this lowers the temperature a t which bulk lattice water is lost. A. C. ZETTLEMOYER.-~should like to comment that the bulb breaking correction depends upon the liquid in the calorimeter. Your reported value, it should be emphasized, is for water; for some organics we now believe the correction is negligible. N. HACKERMAN.-oUr correction for bulb breaking is based on water. Corrections based on heats of vaporization of approximately 8 kcal./mole, such as for a typical non-polar hydrocarbon, are negligible.
CHROMATOGRAPHIC FRONTAL ADVANCE OF AN ADSORBATE THAT HAS AN ADSORPTION MAXIMUM BY IRVING FATT AND MOHAMED A. SELIM Department of Mineral Technology, University of California, Berkeley 4 , California Received June 80, 1969
Laboratory studies of surfactants of potential use as aids in waterflood recovery of petroleum often have showed those surfactants to have a maximum in the adsorption isotherm. Recently Vold and Sivaramakrishnan have given a plausible explanation of such adsorption behavior. This paper shows the effect of an adsorption maximum on the frontal advance of an adsorbate through a chromatographic column. Using an initid concentration distribution it is shown that the adsorption maximum leads to a concentration front that is stepped rather than a single vertical line on a concentration versus distance plot as would be obtained from a Langmuir type isotherm. Continued injection of a solution more concentrated than that a t the adsorption maximum may lead to a concentration front that moves backward. The effect on frontal movement of a discontinuity in the slope of the adsorption isotherm a t the c.m.c., as shown by Vold and Sivaramakrishnan, is also demonstrated. Frontal advance in a system in which the input concentration increases with time also is shown.
Introduction with an adsorption maximum can be obtained by Movement of the adsorbate concentration front using a method of solving differential equations when a single solute solution is flowed through a known as the method of characteristics. De Vault's Equation.-De Vault has shown that chromatographic column can, in most cases, be calculated using methods given by De Vau1t.l in a linear column with instantaneous adsorption Difficulties arise, however, when De Vault's treat- equilibrium and no diffusional effects, the position ment is applied to a solute that has an adsorption of any concentration at the front will be given by maximum. Laboratory studies of surfactants of z = Wc) [ V / ( , Mf'(C))l (1) potential use as aids in water-flood recovery of petroleum often have showed these surfactants to the distance of any point in the column have. an adsorption maximum. Recently Vold the initial distribution of solute concentration in and Sivaramakrishnan2 have given a plausible the column explanation of such adsorption behavior. The purvolume of solution that has been pumped into the column pose of this paper is to show that the shape and pore volume er unit length of column position of the concentration front in a system amount of agorbing material per unit length of
+
( 1 ) D. De Vault, J . A m . Chem. Soc., 66, 532 (1943). (2) R. D. Vold and N. H. Sivaramakrishnen, THISJOURNAL, 62, 984 (1958).
+
column first derivative of the adsorption isotherm concentration of solution
IRVING FATTAND MOHAMED A. SELIM
1642 I
-t
Fig. 1.-Isotherma: A, idealized Vold and Sivaramakrishnan isotherm; B, typical Langmuir isotherm. I.,,,,
the method of characteristics can be integrated to give equations (1) and (2). An important difference between the methods of De Vault and of Fayers and Perrine arises when the adsorption isotherm has a maximum, an inflection point, a discontinuity in the first derivative or any combination of these. For such systems the method of characteristics predicts a stepped front during the early part of the injection history whereas De Vault’s equation, equation (2) above, if used without caution, may lead to a single front. To demonstrate the application of the method of characteristics an idealized form of one of the isotherms given by Vold and Sivaramakrishnan was used. This isotherm is shown as curve A in Fig. 1. Curve B in Fig. 1 shows a Langmuir type isotherm. The rate of advance of any concentration is given by
($)o
Fig. 2.-Characteristic plot for isotherm of Fig. 1A: A, initial concentration profile; B, characteristics; C, concentration profile a8 a function of time.
When equation (1) is applied to an isotherm of the Langmuir type the plot of C versus X may become double valued a t the front and therefore physically unreal. To circumvent this difficulty, De Vault presents what he calls a discontinuous solution. This is simply a method whereby a material balance is used to obtain the position of the vertical line on the C v e r m s X plot behind which must be all of the solute pumped in up to that time. The position of this line is given by
+ lc;[a -k Mf’(C)lS(c)dC) (2) - ca) + MIf(cb)
v(cb - c.1 Xa
=
m(cb
-f(ca)l
where Ca is the concentration in the column ahead of the front and c b is the concentration of the injected solution. Fayers Method of Characteristics.-Recently and Perrine3 have shown that the rate of frontal advance in a chromatographic column can be obtained by the method of Characteristics from the same basic equations used by De Vault. When applied to the usual Langmuir type isotherm the method of characteristics leads to the time derivatives of equations (1) and (2). If the injection rate is constant, then the equations from (3) F. J. Fayers and R. L. Perrine, Trans, Am. Znsl. Mining, Met. ond Pet. E n p s . , in press,
Vol. 63
Q
=
CY
+ Mf’(C)
(3)
where q = V/t. Equation (3) is the time derivative of equation (2) for p = constant. A plot in the X,t plane will be a series of lines of constant concentration, known as the characteristics, with slope equal to the rate of advance of that concentration. The initial concentration distribution is arbitrary and, for reasons to be discussed later, will be taken as shown in Fig. 2A. Figure 2B shows the constant concentration lines in the X,t plane based on the isotherm of Fig. 1A. This isotherm has a discontinuity in the first derivative at CD and a maximum at CM. All characteristic and front profile diagrams in this paper are for demonstration only and are not intended to show quantitative behavior of any particular system. When two characteristics intersect, the physically impossible situation of two concentrations a t the same point and same time is predicted. This situation is interpreted to mean that a concentration shock is formed. At the shock front the concentration rises abruptly. Fayers and Perrine show that the velocity of propagation of the shock front is given by &3= dt
a(Cb
q(Cb
- c.1
- ca)
+ M[f(cb)
-.f(ca)l
(4)
Equation (4)gives the slope of the line that passes through the intersection of the characteristics for Ca and C b . If the concentration increments are small, then the characteristics are close together and there is no problem in determining the shock front path from the slope of the path a t the intersections. Note that as Cb - Ca, the “strength” of the shock front, vanishes then equation (4) goes over into the characteristic equation (3). Equation (4)can be used t o trace the development and movement of a shock. Shock formation occurs a t that instant on the time axis when two infinitely close characteristics cross. For example, this may be the point 2 in Fig. 2B. The slope of the shock path a t Z is the same as the slope of the characteristic through Z. When the shock path is curved, step by step methods must be used to trace it. Given the known
Oct., 1959
CHROMATOGRAPHIC FRONTAL ADVANCE OF
location and slope of a shock path, a t Z for example, the path can be continued as a straight line for a small time interval At to determine its position a t t At. The characteristics which meet the shock At can be found by trial and error. These at 1 characteristics carry the concentrations Ca and Cb and the slope of the shock path at t At is then given by equation (4). If the time intervals are chosen small enough, this method allows the shock path to be traced as accurately as desired. The use of a finite number of characteristics in the X,t plane is equivalent to replacing the continuous front of Fig. 2A by a number of steps equal to the number of characteristics. At this point the reason caution is needed in using equation (2) can be seen. If S(c), the initial concentration, is not zero then the integral in equation (2) must be evaluated. Also, for a n isotherm that has a maximum the quantity rf(cb)- f(Ca)] will be negative in certain regions. T o use equation (2) the region from zero concentration to cb must be divided into several increments. If these increments are sufficiently small, then equation (2) will show the development of several fronts just as the method of characteristics does. I n fact, the method of characteristics is simply a convenient way of solving equation (2) in increments of concentration. To avoid the possibility that a shock front will form at negative values of the x coordinate the initial concentration distribution was taken as shown in Fig. 2A. This causes no difficulty because the x = 0 point can be put a t any point on the x coordinate. An interpretation of the frontal advance from Fig. 2B is presented. Given the initial concentration profile of Fig. 2A, all characteristics for concentrations less than CD will collide to form the front marked Fl in Fig. 2B. The position of this front a t any time is read directly from the F1trajectory. Characteristics of concentrations between CD and CMwill collide to form the front marked Fz. From time T I to time T z there are two fronts advancing. After time Tz fronts F1 and Fz advance as the single front Flzwith concentration limits of zero and CM. These concentration profiles are shown in Fig. 2C. To avoid confusing the characteristic diagram several characteristics in the range from Cg to Clo are shown only partially. It is to be understood that the trajectory of fronts F12and Fa is composed of collision between characteristics for Co and CI and those eminating from T = 0 in the region on the 2 axis between Cg and G o . During the time the aforementioned two fronts were forming and coalescing, another front, FB, was forming between the injection concentration, CI, and CM. This front is formed by the intersection of the characteristic for CI and a characteristic for a lower concentration, which has a negative slope. The negative slope arises from the portion of the isotherm which has negative slope. The trajectory of the front, as shown in Fig. 2B, is such that for a time the front is moving counter to the direction of injected solution. Depending upon the isotherm, this front may move to x = 0 and temporarily disappear from the column. continued injection at
AN
ADSORBATE
1643
++
+
1Fig. 3.-Characteristic plot for isotherm of Fig. 1 A : A, input concentration as a function of time; B, characteristics.
Fig. 4.-Concentration
I
profiles obtained from Fig. 3B.
A
Fig. 5.-Characteristic plot for isotherm of Fig. 1A: A, input concentration as a function of time; B, characteristics; C, concentration profiles.
concentration CI will bring the Fa front back into the column and it will eventually merge with front Flz to form front F123 which has the concentration limits zero and CI. The method of characteristics can be used to show frontal advance for a system in which the injected concentration increases with time. At the input concentration the characteristics have positive slope. The isotherm of Fig. 1A and the input concentration versus time curves of Fig. 3A are used. Figure 3B shows the development of the two forward moving fronts F1 and Fz and the backward moving front Fa.
IRVING FATTA N D MOHAMED A. SELIM
1644
Vol. 63
TFig. 6.-Chara.cteristic plot for isotherm of Fig. 1A: input switched from solution to pure solvent.
cl
Fig. 8.-Characteristic plot for isotherm of Fig. 1A: input slowly changed from solution to pure solvent.
-
t
C 0
0 Fig. 7.-Concentration
X-
profiles obtained from Fig. 6 .
The two forward moving fronts collide after a time and proceed as one front. The backward moving front, FB, forms between concentration characteristics for which the slope is negative. This front moves out of the column entrance. Front F12now begins to collide with positive slope characteristics of saturations above CM and begins to form front F124 which extends between zero concentration and CI. The front then advances at a constant rate with concentration limits of zero and CI
.
Concentration profiles resulting from the characteristics in Fig. 3B are shown in Fig. 4. Figure 5 shows the characteristics solution for a system in which the input concentration is on the negative slope of the isotherm. On cursory examination the formation of front Fa in Fig. 3B seems to violate the law of physics which states that no event of the future can influence present behavior. That this law is not violated can be shown. The negative slope of concentration characteristics above CM seems to indicate that these concentrations appear before they are injected. This is actually true because solutions of concentration greater than CMcause desorption and thereby carry forward a more concentrated solution than was injected. In this connection it should be pointed out that concentration CM moves a t the velocity of injection and this is the maximum possi-
Fig. 9.-Concentration
profiles obtained from Fig. 8.
ble velocity. This is true despite the fact that equation (2) predicts an infinite velocity for the concentration CM when cy = Mf’(C) in the region where f’(C) is negative. The method of characteristics also can be used to determine the profile of the trailing edge when injection of solution is discontinued and pure solvent injection begins. At Tz in Fig. 6 solution injection ended and solvent injection began. This means that characteristics for all concentrations from CIto CM must radiate from X = 0, T = Tz. Concentration CM will move a t maximum velocity and build up a concentration bank ahead of itself. This causes concentrations between CI and CM to move backward. The resultant trailing edge profile is shown in Fig. 7 . As the distance on the time coordinate between the last CI characteristic and the first Ca characteristic is brought to zero, it is seen that front Fs becomes more vertical until the characteristic plot shows that F1 has only an instantaneous existence. Figure 7 shows the concentration profiles resulting from the characteristic plot of Fig. 6. In practice the transition from CI to CO cannot be made instantaneously, therefore Ff may exist as shown in Fig. 8. The concentration profiles are shown in Fig. 9. One problem that may arise when using the
I
I
,
Oct., 1959
CHROMATOGRAPHIC FRONTAL ADVANCE OF AN ADSORBATE
method of characteristics in chromatography is scaling of the characteristic plots. Some isotherm and column properties require that different scales be used for the several parts of the characteristic plot. For example, in Fig. 2B the characteristics that develop front Ff may have, if plotted on the same scale as those that develop front F1, such high slope as t o make the point of their intersection difficult to determine. I n such case the F1and Fz fronts can be developed on different plots and only their trajectory transferred to a common plot. Summary.-Solutions which have an adsorption maximum or discontinuities in the first derivative of the isotherm will have a stepped concentration profile in a chromatographic column. The method of characteristics provides a convenient graphical procedure for obtaining the concentration profiles. Acknowledgment.-The authors wish t o thank R. L. Perrine, J. F. Fayers, and W. T. Cardwell of California Research Corporation, and J. W. Sheldon and B. Zondek of Computer Usage Company for providing manuscripts of unpublished papers on the method of characteristics.
1645
DISCUSSION M. J. VOLD(University of Southern California).-Dr.
Fatt’s paper is very interesting and important since it offers a means of verifying the existence of an adsorption maximum. Dr. Robert Vold and his stpdents have verified their experimental results repeatedly with carefully purified materials and many similar isotherms have been obtained by others. But Dr. K. J. Mysels has devised a perpetual motion machine based on an adsorption isotherm having a maximum in it! Also we are painfully aware of the many years of effort that went into explaining the “minimum” in surface tension-concentration curves which turned out to be due to impurities after all. Dr. Fatt’s method can be used to show that a distinctive pattern of concentration of effluent from a column of given length results from this type of isotherm. Do you have in mind undertaking experiments to verify this? I. FATT.-we do plan some experiments to verify the predictions from the method of characteristics. There are, however, several complicating features. If the adsorption isotherm with a maximum is a non-equilibrium situation but is reversible then an experiment carried out over a period that is short, relative to the time needed for complete equilibrium, will verify the existence of the maximum. If, however, the maximum is part of the irreversible behavior of the system then a chromatographic experiment will not clarify the situation. The method I have proposed for determining frontal advance requires that the isotherm be completely reversible. This is also the requirement of DeVault’s method.
