DALEF. RGDD
1254
+
In [ ( Y 2 r)l(YA - r)l - 72. Computation of the value of T~ was performed by an iterative procedure using Kewtoii’s method 7,
=
7,’
s + -___(T1’)
(?‘(TI‘)
r,’
__
cot11 ~[ - , [ (~q _1 ) ~_2- qrl’ l ] } - tanh ‘ _ Y 2q csch2 { y [ ( q 1 ) r 2 - qrl’ -f 11 } - sech’ T ~ ’ .
-1
+
+
+
The criterion for accuracy 0 f - q was 7‘ - r l ’ < 1;inally the values of Y used for t’he theoretical curves were computed by means of the basic eq. lo.’* Some idea of the magnitude of the error resulting from neglect of the thermal rate of initiation as(14) The tab1.m of associated data and the thirty theoretical ciirves 1,lotted therefrom, as referred t o in this paper, have been deposited as Docrirnent number 6274 with the A D 1 Auxiliary Publication Project, Photoduplica,ion Service, Library of Consress, Washington 25 D. C. A copy m a y be secured by citing t h e Document number and b y remitting in advance $2.,50 for photoprints or $1.75 for 35 mm. microfilm payahle t o : C l i i ~ f ,Photodiiplication Service, Library of (‘ongrms.
Yol. G4
sociated with a typical sensitizer may be gleaned from the following case. Given a sensitizer with a dark rate which is 8% of the measured rate ill steady state illumination. (See the appropriate theoretical curve in the accompanying figure.) Given also a sector cut-out providing equal light and dark periods. The value of As, the average lifetime of growing chains a t the steady state, wai determined by the above procedures to be 1.12 seconds. By neglecting the dark rate and computing by the method of Swain and Bartlett5 we obtained a value of As = 1.30 seconds. Thus, by neglecting an 8% dark rate, an error of about 16% was experienced. Acknowledgment.-We are indebted to the Office of Naval Research for financial support during the courqe of this work, to Prof. P. D. Bartlett of I-Iarvard for much encouragement and to Miss 31. Mintzer arid Prof. It. D. Jackson a t the University of Delaware for very valuable assistance in bringing the latter phases of this work to conclusion.
OS THE PAPER .iDSORPTIOS CHROMATOGRAPHIC I’HESOPITENA BY DALEI;. RUDD Departttwtit of Chptniral and Ilrtalluryzcczl Enyzneerang, Cnziieiszty of dlzchiqcin, .inn A?bor, J l ~ htz i j c i i i Receaved February 22, 1000
d mathematical analysis of an equilibrium adsorption paper chromatography system is presented This dnaly& shows the gross features of the chromatographic phenomena siich as the movemrnt of the solvent front arid tlir Gepardtion of t h e Doliite bands The diffiision model for solvent penetration J ields an experimentally obtained correlation
Introduction The analysis of the phenomena of paper chromatography is by no means complete. The relative motion of the solute bands during the travel of the solvent through the paper is caused by a combination of solute adsorption, ion exchange, and solvent partition. The chromatographic phenomena have been analyzed in detail by Consden, Gordon and Martin’ when solvent partition is the controlling mechanism as in the separation of carbohydrates. This paper gives an analysis of the phenomena when equilibrium qolute adsorption on the paper fibers predominates as in the separation of dyestuffs. Also an analysis of the diffusion model for solvent penetration shows that a n experimentally obtained correlation between the height of solvent rise and the time of rise follows directly from the assumption that the solvent is transferred according to Fick’s law. Solvent Penetration.-The detailed experimental work of Muller and Clegg2 indicates that, over the range of interest, the relationship between the Yisual height of rise, X,, of a solvent in a strip of paper and time of rise, t , is quadratic in X,.. .In analysis of the diffusion model for solvent penetration shows that this empirical correlation follon s directly from the assumption that the solvent iq transferred according to Fick’s law, (1) R. Consden, A . H. Gordon a n d A . J. Martin, Baochem. J., 38,
2 2 4 (1944).
(2) R. Muller a n d D. I . Clegg, A n a l . Chem., 23, 8% (1951).
E’ick’s law stat’es that the mass rate of mlrent, transfer through the paper in the direction x is - D A (dc/bz), where D is the diffusion transfer coefficient, A is the cross-sectional area of the paper, and c is the solvent concentration i n m a s per unit, volume of the paper. Consider a long strip of chromatographic paper initially dry. One end of the paper is placed in a solvent’ reservoir and a solvent front then progresses through t8he paper. The motion of this solvent front is obtained from t’he solution of t’he partial differential equation form of t’he t8raiisient8conservation of mass law. Consider now a t’ransient mass balance on the int’erval of paper (x, x 6z). The net rat’e of solvent influx into the interval by diffusion equal? the rate of change of the solvent ma5:s. Hence
+
-D$
ax
x
+ D A ):
x+sx
= .4
62
dc 31
which becomes as 6x + 0
The boundary conditions on the partial differential equation 1 are c(z,O) = 0 paper is initially dry c(0,t) = co paper end is always saturated Lim cix,t) = 0 paper is dry far ahead of the wlvent X-+ m front
The solution of equation 1 subject to the boundary conditions is, by Laplace transform theory
Sept., 1960
1255
P A P E R LiDSORPTIOK CHROMATOGRAPHIC PHENOMESA 50
I
-_'' - 0 ' 0
The motion of t'he solvent front can be represeiit,ed by the constant solvent coiicent'ration cont>oursin the x,t plane. The ratio e , ' ~is~ const'ant, when the group x/2 is constant,. The visually observed solvent front,al motion corresponds to the mot'ion of a constantt solvent concentration front. Hence if e,,,' eo is t'he solvent concentrat'ion ratio of the visually observed front and E is the constant whose complement'ary error funct'ion is cv/co then, by equation 2, the relationship between the height of Polvent, rise, X , , and t'he time of rise, t , is S,,?= 4EDt
1 5 1
1
SOLVENT FRONT
(3)
Equation 3 is of the same form as the empirical relationship obtained by Jluller and Clegg. Thus, over the mnge of interest, the diffusion model adequately represent's the solvent' penetration phenomenon. It should be uot'ed tha,t Washburn's analysis3 of flow through horizont'al capillaries yields an equation of t'he same form as equation 3. This indicates that the penetration of solvent through the chromatographic paper may be thought of as either a diffusion or capillary phenomenon. The diffusion model is used in the further discussion. Solute Penetration.-The solut'e materials are carried along with the advancing solvent and are adsorbed o n t'he fihers of the chromatographic paper. The solute adsorption causes the solute frontal velocities t'o differ, allowing the chromatographic separation of the solutes. The solute adsorption phenomenon has been analyzed mat'hematically for chromatographic columns by h u n d son and L a p i d ~ s . ~ The diffusion model for solvent penetration and the ideas of equilibrium solute adsorption on t,he fibers are now combined to give a model for paper chromatography. Consider tthe chromatographic separation of m solutes. Each of these solutes is transferred through t8he paper along with the advancing solvent and is in equilibrium adsorption with t'he paper fibers. The equilibrium relationship is 7ji = + ( z . ) i = 1,2,. . ., m (4) where yi is the inass of solute adsorbed on the paper fibers per unit volume of paper and i i is the mass of solute in the solvent phase per unit mass of solvent. Equation 4 is a general form of the equilibrium adsorption isotherm but does not allow solute interference. ,Itransient mass balance on t'he ith kind of solute in the soh-ent phase over the interyal of paper (z.x ax) yields
+
The first, two terms on the left are the rates of solute influx iiito t'he interval along with t'he advancing solvent'. The third term is the rate of loss of solute from t'he solvent phase by adsorpt,ion on the paper fibers. These terms equal the mass rate (:3) E. I\-. \I.ashb;irn, l ' k w , Rei,., 17, 276 (1921). (4) h-. R. .\rniiniison and L. T,nlJidiis Tmn JOCRNAL, 54, 821 (1950).
0'
0
"
I 10 0
50 TIME
t
,
HOURS,
Fig. 1.-Chromntogrnphic separation: D = 178 cm.?/sec.; C,, = 0.8 g.lcm.3; a1 = 0.1: N? = i .O.
of change of solute in the solvent phase. This m a s balance hecomeq 6x + 0 and by uqe of equations I and4
The movement of the solute bands through the paper can be represented by the constant solute concentration contours in the x,t plane. These contours can be obtained immediately from equation 5 . Quasi-linear partial differential equations of the first order such as equation 5 possess characteristics in the x,t plane. I n this case these characteristics correspond physically to the constant solute coilcentration contours. From the theory of characteristics5 the constant concentration contours are given by the solution of the ordinary differential equation
In general the solution of equation 6 can be achieved nunierically.5 To illustrate these methods consider the chromatographic separation of two solutes with linear equilibrium adsorption isotherms. g, = z l m ,
+ p,
a = 1,2
At t = 0 the end of the dry paper is placed in the solvent reservoir. At t = 18 min. a sample of the solute mixture is placed on the paper in the interval (0,3.0 em). The solvent penetration is characterized by equation 2 or 3. The chromatographic separation of the two solutes is characterized by their constant concentration contours obtained from the solution of equation 6, which is, in this case ( 5 ) A . Acrivos, Ind. Eng. Chem., 48, 703 (1956).
HIROSHI FUJITA AND LOUISJ. GOSTING
1236 exp -
[L]'
.'k = V B co 2 d z i tit 4' erfc [A+ ] ai
Vol. G4
Discussion = 1,J
2dDt
The solution was obtained using the Euler method of numerical integration. Figure 1 shows this chromatographic separation. The solvent front precedes the solute bands. The movement of the solute bands is characterized by the characteristics (constant concentration contours) emanating from the front and back of the initial solute pulse. The chromatographic separation is caused by the difference in the adsorption isotherm constants ai for the t ~ solutes. o
The analysis presented shows that the diffusion model for solvent penetration does represent the physical phenomena and that the paper chromatographic effect can be explained when equilibrium solute adsorption predominates. The secondary effect of solute diffusion has been neglected. Attempts to include this led to equations which preclude the use of the simple method of characteristics. Solvent evaporation from the paper surface also has been neglected since in practice chromatography is accomplished in a saturated atmosphere. This simple analysis gives further insight into the fundamental mechanism of adsorption paper chromatography.
A NEW PROCEDURE FOR CALCULATING THE FOUR DIFFUSION COEFFICIENTS OF THREE-COMPOSEKT SYSTEMS FROPI1 GOUY DIFFUSIOMETER DATA' BY HIROSHI FUJITA~ AND LOUISJ. GOSTIXG Contribution from the Department of Chemistry and the Enzyme Institute, University o j Wisconsin, Madison 6 , Wisconsin Received February 22, 1960
An exact procedure is developed which permits calculation of the four diffusion coefficients at a given composition of a ternary system from suitable free-diffusion experiments performed with the Gouy diffusiometer. This new procedure can be applied regardless of the relative magnitudes of the diffusion coeficients and should yield somewhat more accurate results than do previous methods. The data required are the reduced height-area ratios of the refractive index gradient curves and the areas of graphs of Gouy fringe deviations which summarize deviations of the refrartive index gradient curves from Gaussian shape. Data for the system NaC1-IICl-H20 a t 25" which were reported previously from this Laboratory are reanalyzed to illustrate the use of this proceure. It is found that the new values so obtained for the diffusion coefficients satisfy Onsager's reciprocal relation somewhat better than did the values which were calculated by a former method.
The thermodynamics of irreversible processes requires four diffusion coefficients for complete description of isothermal diffusion in a ternary system. During the past few years several procedures have been published for calculating four such diffusion coefficients, Djj, from experimental data. We review briefly those methods here to show their relation to the new method which is presented in this paper. Baldwin, et aL13presented a method which uses d a b for the reduced second and fourth moments, Szm and Dim,of the refra,ctive index gradient, curves obtained from suitable free-diffusion experiments; for given mean concentrations of both solut.es, 1 and 2, a t least t'mo experiments are required with different values of el,the fractional refract,ive index increment of solute 1 across the sharp initial boundary. Subsequently a procedure was developed by Fujit,a and Gosting4 who used data for t'he 92, and the reduced height-area ratios, 5 ) ~ ,of the refractive index gradient curves for these experiments. The second procedure yields more accurate results than the first because experimental determinations of Dzm a.re quite inaccurate com(1) Presented at t h e 138th meeting of the American Chemicnl Society, Atlantic City, K e w Jersey, September, 1959. (2) On leave from t h e Physical Chemistry Lahoratory, Departm e n t of Fisheries, University of Kyoto, hlaiauru, J a p a n . (3) R. L. Baldwin. P. J. Diinlop and L. J. Gosting. J . A m . Chem. Soc., 7 7 , 5235 (1955). (4) H. Fujita and Id.J. Gosting, ibid.. 78, 1099 (1950).
pared to those of 3 ~ Homver, . results from the second procedure may still contain appreciable errors because the Dz, are somewhat less accurate than the DA (which can be measured to 0.1% or better with interferometric instruments such as the Gouy diffusiometer). Both of these methods are general, being applicable regardless of the values of the diffusion coefficients. A modification by Dunlopj of the second procedure achieves improved accuracy for some systcms by determining experimentally the tn-o values of a1 for which the refractive index gradient curves become Gaussian. These values of cy1, \Then combined with the data for 3.4, permit calculation of aZ, for any value of ai. Unfortunately the applicability of this method is limited in practice because inverted density gradients may be encountered for some systems in determining the required values of ai. A different method4 developed by Fujita and Gosting uses data for the DA and for reduced Gouy fringe deviatioiq6 Q((), corresponding to a particular value of the reduced fringe number, f (f) ( = f ( 4 2 ) = 0.73854) ; experiments for a t least two values of cy1 are required. When applicable thiq method can give more accurate resultq than the first two methods; however it depends on a w i e s expansion which converges satisfactorily only for suitable ( 5 ) P. J. Diinlop, TITIS JOURUAL, 61, 994 (1957). ( G ) D. F. Akeley a n d I . J. Gosting, J . A m . Chem. Soc., 7 5 , 5685 (1953).
