romatography of A iffusion (Mass Transfer) Mechanisms PAUL
B. HAMILTON, DONALD C. BOGUE,’
and ROBERTA A. ANDERSON
Alfred 1. do Pont Instifufe, Wifmingfon, Del., and Deparfmenf o f Chemical Engineering, University of Delaware, Newark, Del.
b in this study, a theoretical approach to the ion exchange chromatography of
amino acids was followed. Equations were derived which explicitly relate the current “plate” and “rate” theories and which combine the key phenomena and the column operating variables in compact algebraic form. The experimental results from the chromatography of cysteic acid, taurine, aspartic acid, ssrine, glutamic acid, glycine, alanine, glucosamine, and valine on columns packed with spherical beads of sulfonated polystyrene-divinylbenzene cationic exchange resin were used to test the equations. Resin particle diameter, column diameter, column length, flow rate, temperature, pH, and sodium ion concentration were studied as variables. The role of mass transfer phenomena (solid, liquid, and axial diffusion) was examined. Tentative numerical values of coefficients of solid diffusion, D,, for some of the solutes in the resin were obtained. The resolution of two components was expressed b y a simple algebraical formula which could be developed to show clearly the fundamental phenomena upon which resolution depends.
is concerned with the chromatographic separation of some amino acids by elution from ion exchange columns of Dowex 50-X8. The experiments were designed to study underlying mechanisms to extend the usefulness of the theoretical approach for substances of biological interest. hiore precise delineation of the effects of the column variables on the form of the chromatogram has been established and the precision with which certain variables need to be controlled to obtain reproducible results has been determined, HIS PAPER
THEORY
General. Chromatographic separations have been treated theoretically with r a r y i n g degrees of rigor, as recent revielm testify (5,6, 16, 18, 20). The most generally ueeful theories are Present address, Department of Chemical Engineering, University of Delaware, Kevark, Del. 1
1782 *
ANALYTICAL CHEMISTRY
those of van Deemter, Zuiderweg, and Klinkenberg (36), and Glueckauf (7, 8), which express the key principles in compact form. Both theories include all possible diffusional effects-namely, axial diffusion, and the mass transfer resistances in the solid and liquid phases. Both theories assume linear equilibrium isotherms, a small volume of teat solution (hereafter referred to as a feed pulse), long columns, and operating conditions which do not cause gross departures from equilibrium states. The criteria which must be met to make the two latter assumptions valid are given by van Deemter el al. (35) in their appendix. These criteria are satisfied in most liquid-resin chromatographic separations and probably in many gas-liquid chromatographic separations. The volume of a feed pulse must be small compared with the volume oi’ effluent which contains the bulk of a n eluted component; the criterion is discussed by Glueckauf ( 6 ) . The assumption of a linear isotherm, while not universally applicable, is realistic where trace components are considered, as, for example, in most liquid elution chromatography. The present studies fall within this category. With nonlinear isotherms, the distribution coefficient, Kd, lvould be a function of concentration and those factors ivvhich affect the concentration of the pulse while in the columncolumn length, column diameter, flow rate, and particle diameter. Such variation of Kd, which could be noted experimentally, was not observed in the present work. van Deemter, Zuiderweg, and Klinkenberg (S6) simplified the rigorous treatment of Lapidus and Amundson (,?I), who started with general differential equations describing a mass balance in the bed. Glueckauf ( 7 ) , with essentially the same approach, maintained a discreteness in his equations (the HETP) to account for the particulate nature of the resin bed. The formally dissimilar equations of these two theories may be rearranged to the same form with the Substitution HETP [Glueckauf (7)l =
2DFia’’ [van Deemter et al. (%)I,
where HETP is the height equivalent of a theoretical plate, D(axlai)is the axial diffusion coefficient, and U is the interstitial fluid velocity. Initially Glueckauf (8) did not incorporate the mass transfer resistances, but later ( 7 ) extended his analysis to include them (requiring in the final treatment, additional terms in the above identity). His final result [Equation 14 ( 7 ) ] is essentially the same as that derived b y van Deemter et al. (So’), except that the solution of the latter authors gave a somewhat better insight into the construction of the axial diffusion term (Appendix, this paper). For the analysis of experimentai data, relatively simple algebraic expressions are the most useful. The rate of movement along the column of the peak of each band is, for any given srt of operating conditions, a constant for each species and has been given e\plicitly b y Glueckauf (7, 8), or implicit:y by van Deemter et al. (56) (Appendix, this paper) as B = AZ(K,j
+ Fr)
(1)
where 8 is the peak elution volumc, d is the cross-sectional area of the column, Z is the column length, and F I is the void fraction. I n practice, this equation is used to calculated Kd from the experimental value of 1. Furthermore, from the result of van Deemter et al. (36) the following expression may be derived (Appendix, this paper)
c
d = 2AZZ Fr(& -k FI)’
u,
D(aaia~)
where, besides the synibols defined above, uz is the variance, G,is the superficial or linear velocity, Kd is the distribution coefficient, and K I ,is the orerall mass transfer coefficient. The spread of the puke (more precisely the variance uz, in volume units) is described in terms of the contributions of axial diffusion and of mass transfer resietances. Analyzing column behavior directly in terms of Equations 1 and 2, elucidates the underlying mechanisms in as fundamental a manner as possible without
the introduction of empirical parameters. Axial Diffusion. T h e apparent axial diffusion is t h e s u m of ordinary molecular diffusion and eddy diffusion (that caused b y t h e irregular flow patterns). As stated by v a n Deemter et al. (36),the axial diffusion coefficient can be expressed as D
=
YDL -I- DE (2a) (molecular) (eddy)
The labyrinth factor, y, is a n empirical correction of the order 0.5 to 1.0 t o account for the meandering path of the fluid. In the present w x k molecular diffusion is unimportant and the first term can be neglected without significant error. The eddy diffusion coefficient, DE, is usually defined by a n expression of the type
liquids. I n the absence of more extensive experimental data a t low Reynolds numbers, recourse to estimating X is as satisfactory a method as any; a value of X of the order of 0.5 to 4 is reasonable. Approximate values of 1 deduced experimentally in the present research are within the limits of the data reported b y Carberry and Bretton (5)and Klinkenberg (19). Neglecting niolecular diffusion and substituting XdpU for DB,Equation 2 thus takes the form
+ ]K*
(3)
M a s s Transfer Resistances. T h e absorption onto or desorption from a resin of a particular ion must be considered in five steps, a n y combinations of which may be rate-limiting. These DE = XdpU steps as enumerated b y Boyd, Adamson, where d, is the particle diameter and and Myers (2) are: 1, diffusion of the is assumed constant-that is, X = ion to be absorbed from the bulk liquid to the resin surface; 2, diffusion of this 1'(dpUIDf). The ratio d,ujDX is a dimensionless ion through the resin to the active site; group, called the Peclet number, N P ~ , 3, reaction a t the site; 4,diffusion of the which has been found experimentally displaced ion back through the solid; constant with particle diameters within and 5, diffusion of this ion from the certain flow regimes studied to date in resin surface to the bulk fluid. In the packed beds. LIcHenry and Wilhelm present experiments, where the sodium (B') using , a cell mixing model, preion being displaced is present in great dicted A-pe = 2-i.e., X = 0.5-for high excess over the ion which is absorbed Reynolds numbers, and they report (amino acid), steps 4 and 5 are clearly substantiating data for gases. Hownot important. [If the displacing ion ever, the axial diffusion of liquids in is a macro component of the bulk fluid, flow through fixed beds is apparently a steps 4 and 5 may be important, a more complicated event. Carberry fact which has sometimes been overand Bretton ( 5 ) report data which looked.] In view of the ionic nature of indicate that the Peclet number a t the exchange, step 3 is probably very low iiquid flow rates is about ' / t fast and can also be neglected. This to 1, the theoretical limiting value of 2. has been verified for the N a + - HT X void cell mixing efficiencv model is exchange b y Boyd, Adamson, and invoked b y these workers to suggest Myers ( 2 ) . Thus it is assumed here, that at low Reynolds numbers, IVP. as in other theoretical developments, should indeed be much less than 2. that steps 1 and 2 only need be conI n terms of 1,the data of Carberry and sidered. Bretton (6) and those of Klinkenberg The theories require linear mass (19) indicate X = 2 to 4 a t lorn Reynolds transfer equations of the form numbers region. The Reynolds number, N = kL(C - c,) = kkq, - g) (4) L Y R ~defined , by a dimensionless group of the form d,u, p ' p is generaily the \?here .V is the total molar flux, kL most useful measure of fluid flow and k , are the liquid and solid phase through packed beds; in addition to mass transfer coefficients, respectively, the quantities defined previously, p c and c1 are the concentrations in the is the density of the fluid and p is the bulk liquid phase and in the liquid viscosity (poise). It is unlikely that a n phase at the solid-liquid interface, unmodified cell mixing model of axial respectively, y i is the concentration in diffusion suffices for the very low the solid phase at the solid-liquid Reynolds numbers (< 0.1) of the present interface, and p is the mean concenwork since distinct mixing cells are tration in the solid phase. Introducing difficult to visualize a t low flow rates. the over-all mass transfer coefficient, In such cases, a more complicated K L ,and eliminating cI.and qt, model is indicated, perhaps patterned after the development of Taylor (34) for flow in a tube, where D Eis a function of the molecular diflusion coefficient, where F I I is the fraction of the column DL. No doubt imperfect cell mixing occupied by the solid resin phaseand dead space retention (capacitance) i.e.,FII = (1 - F r ) . Hence best characterize axial diffusion of
Solid Phase. T h e usefulness of a simple linearized equation in t h e solid phase was indicated b y Glueckauf and Coates (IO) and confirmed by Glueckauf (9). I n connection with the present work, the validity of this approach was further examined theoretically by Bogue ( I ) , who concluded that for long pulses, the flux can be given very accurately by:
N' = (15D,/T2) (9s - 4) where S'is the total solid phase flux, D , is the solid diffusion coefficient, r is the particle radius, and y L and 4 are as in Equation 4, or IC, = (ISFIID,/+) c- 60F11D,/dp2 ( 6 ) The assumption of a concentrationindependent diffusion coefficient is probably satisfactory in view of the microconcentrations of the diffusion species, but cannot be explicitly justified. Experimental values of D. and its variation with temperature are discussed in a later section. Liquid Phase. T h e liquid phase coefficient is, in general, a function of t h e diffusing species and t h e flow pattern. Considerable d a t a for mass transfer in packed beds are available at Reynolds numbers higher t h a n those of the present work; these data have been recently brought together by Carberry (4). Extrapolation to low Reynolds numbers in the absence of experimental data is unwarranted. Mass transfer to a single sphere has been studied extensively and the data are summarized by Sherwood and Pigford (32). The single sphere correlation is useful as a limiting case since it is probable that k L for a packed bed is greater than that for a single sphere, due to the disturbed flow caused by abutting spheres. Such a n increase in mass transfer in the packed bed is not a certainty, hoIvever. It could be opposed by the formation of stagnant regions which would reduce the effective area for transfer. Furthermore a t very low Reynolds numbers the mass transfer problem in packed beds is not well represented by a linear equation of the form of Equation 4, since k~ becomes a function of the history of the fluid element. The packed bed correlation of Carberry (&) and the single sphere correlation of Sherwood and Pigford (31) are shown (see Figure 12). The effective area for transfer per unit of column volume was taken as GF~rld,, which assumes complete accessibility of the surface. The introduction of a superficial velocity, Lro, into the single sphere correlation is a n artifice for comparative purposes and is somewhat arbitrary. The relation U = U,'0.39 VOL. 32, NO. 13, DECEMBER 1960
"1783
was used, where 0.39 is an average void fraction. A crude but sometimes useful model is that of a n “effective” stagnant layer surrounding the particle (the so-called Wernst layer). Using simple diffusion theory (SI) and taking the effective area of the sphere as 4rr2,one obtains for k~ in the present units:
DL
k L = Frr -
(7)
dd
where DL is the liquid diffusion coefficient and 8 is the thickness of the stagnant fluid layer. Comparison with the limiting value of kL = 12171Jl~/dp2 of the single sphere correlation (see Figure 11) s h o w that 6 = d p / 2 a t very low Reynolds numbers. I n the case of a packed bed, it is difficult to visualize
Table I.
a n “effective liquid layer” as large as d p / 2 ; a emaller fraction of the diameter would be more reasonable. This strengthens the intuitive feeling that a packed bed would give a higher mass transfer coefficient than would a single sphere. Patterning the equation after the single sphere correlation, a likely form for mass transfer in packed beds a t all Reynolds numbers is, therefore,
ivhere, besides the symbols defined previously, Nsc is the Schmidt number defined by the dimensionless group, p l D L p , C1 is unspecified but most probably greater than 12, and where the function 9 is also unspecified, but 4-
Temperature Series 0.0983 2.4
2.93
0.200
0.3
54
Length Series 0.097 2.4
2.93
0.200
1 0.6 0.4
50
Area Series 0.097 2.4
2.95
0.237
2 1 0.5
0.636
107.5
35 45 54 65 75 85
17 18 19
0.636
67.0 108 156
20 21 22 23
0.218 0.286 0.414 0.636
150
24
0.636
F!ow and Particle Diameter Series 2.4 2.93 107.5 0.00783 51
0.636
108
54
0.636
108
54
12 13 14 15 16
1
25 26 27 28
29 30 31 32 33 34 35 36 37
1784 *
1
KL
60D&
+
I n the present work, the single sphere correlation has been considered as a limiting case and the results related to i t and to some of the correlations of packed bed data in a manner such t h a t reasonable conclusions could be drawn. Upon substituting ~ / K (Equation L 5) into Equation 3 with some rearrangement, the final form in which variance is described in known or determinable quantities is obtained:
Summary of Column Dimensions, Column Packing, and Experimental Operating Conditions
Particle DiamBuffer Pulse a6 Fraction Sodium Chromaeter, Column, Cm. ion of Col. togram T:mp., Flow Rate, Cm. X concn., M NO. Diam. Length C. Cm. Set.-' VOL % PH pH Series 0.200N 0.6 2.85 1 108 50 0.098 2.4 0.636 3 2.93 2 3 3.15 3 3 3.25 4 Sodium Series 2.93 0.170 0.3 5 50 0.0972 2.4 0.636 107.5 0.180 6 0.197 7 0.226 8 0,256 9 0.295 10 11
( N R ~N,s , ) is probably small conlpared with unity at Reynolds numbers of the order of 0.01. I n summary the over-all mass transfer coefficient K Lwill be expressed by
0.0335 0.0685 O.Og83 0.132 0.173 0.00783 0.0335 0.0685
0.200N
0.0
4.6
2.93
0.200N
0.6
8.0
2.93
0.200N
0.6
0.0983
ANALYTICAL CHEMISTRY
0.00783 0.0335 0.0685 0.0983
EXPERIMENTAL
Materials and Procedures. ION EXCHANGE RESIN. Dowex 50-XS in spherical bead form was classified hydraulically (18) and particles in the diameter range 20 to 30 microns were collected. This cut was reclassified and a very narrow particle diameter range cut was obtained. The mean diameter, determined by photomicrogcm. (24 microne) raphy was 2.4 X with approximately 80% of the particle diameters zkO.0002 cm. of the mean. Also 4.6 X 10-8 and 8.0 X cm. particle diameter cuts were prepared similarly, the former with 75% of the particle diameters =tO.OOOl cm. of the mean and the latter with 83% j=0.0005 cm. of the mean. The capacities, determined by standard procedures (20) of the 2.4, 4.6, and 8.0 X lo-* cm. diameter particles were 5.22, 5.27, and 5.28 nieq. per gram (dry). The wet densities were 1.275, 1.264, and 1.278 grams per cc., respectively, which correspond to 8.3, 7.7, and 8.3% cross linking, respectively. The resin m-as cleaned with acid and alkali plus Versene ( I d ) . COLUMNS. Columns suitable f o r operation a t pressures of approximately 600 p.s.i. were made of precision bore, borosilicate tubing. The details of their construction are described separately (IS). The internal diameter and length of each resin column are noted in Table I. With the very narrow range of particle diameter employed, i t was demonstrated in preliminary runs that it was immaterial whether packing was done in multiple or single sections. After every run the column was reconditioned by 50 ml. of 0.2N sodium hydroxide solution containing 1%Versene and e uilibrated with buffer until the pH of %e efluent was the same as the influent. The columns were jacketed and the temperature was.
t a k e n to be that prevailing in the circulating bath. SOLUTIONSFOR COLUMNDEVELOPMENT. The citrate buffer solutions xere prepared as previously described (14, 26). The 2.00N sodium hydroxide used in their preparation was standardized against sulfamic acid. The buffers also contained Brij 35 (Atlas Powder Co., Wilmington, Del.) as wetting agent a n d thiodiglycol as antioxidant (66). The p H of the buffers was checked at t h e glass electrode. AMINO ACIDS. Analytical grade amino acids (California Corp. for Biochemical Research, Los Angeles, Calif.), which had been previously subjected to critical examination for purity (14) were used. Weighed portions were dissolved together in 0.1N hydrochloric acid. COLUMN OPERATION AND AMINOACID ANALYSIS. The technique of column operation has been described (14). Buffer solutions were forced through the columns b y a piston-type pump (Minipump, Milton Roy Co., Mermaid Lane, Philadelphia, Pa.) operated a t 29 Etrokes per minute. The volumetric input was determined by measuring with a stop watch the time necessary to collect the effluent from the column to the mark of a calibrated volumetric flask. A volume of amino acid test solution (feed pulse) was placed carefully on the drained resin surface of the column and allowed to enter by gravity or forced in under air pressure of 10 p.s.i., care being taken to introduce no air into the resin. The column effluent was collected in a time-flow fraction collector and the fractions were analyzed for amino acids by standard procedures (14, 66, 26). The fraction volume for the 0.636cm. diameter column was 1.0 ml.; for smaller diameter columns it was reduced in proportion to the crosssectional area. Experimental details are summarized in Table I.
(16)] were plotted and the true gauss curve so derived was compared with the experimental curve. A number of elution peaks selected a t random were found to compare closely with the true gauss form; it was assumed, therefore, t h a t this was generally true for all the curves. $ssuming a gaussian form for the experimental curves, i t follows t h a t the peak width a t the inflection points is 2 u. It also follows, from other properties of the normal curve of error, that tangents through the inflection points cut the t axis a t 1 2 u from the mean, corresponding to * u t on the u axis. Band width (to) on the abscissa scales is, therefore, given by
w = 4 u (sigma units, t scale) = 212 - 01 (volume units, 1, scale)
Hence u =
GR.4PHIC ANALYSISO F CHROMATOCRAMS. The net absorbance readings (absorbance minus blank) obtained from analysis of the effluent fractions by the ninhydrin method were plotted as ordinates against the fraction number as abscissa and a smoothed curve was drawn through the points. It was first established t h a t the experimental curves had a gaussian form. Making use of the well known property of the normal curve of error that the ordinate values of the inflection points are equal to iCmaxe-1'2 = 1 0 . 6 0 7 C,,, the highest point on the experimental curve (Cmax) was determined by inspection and the inflection point ordinates were calculated. These points, which were also assumed to lie on the experimental curve, are at t = 0 and t = =ku on the abscissa scale of the gauss curve and correspond to (mean) and v, on the fraction volume (v) axis. These relationships are shown in Figure 1. Other ordinates corresponding to different values of t [the values of the ordinates for various values of t are given in
w/4 =
(02
- vi)/4 and ~2 =
= (02
(w/4)a
- v1/4)2
In practice, to allow for minor discrepancies in drawing the peak and placing tangents on the experimental curve, the peak width a t Cmaxe-l'* was multiplied by 2 and averaged with the band width w. The variance \$-as calculated as uz
= (tE/4)2 where
was the average band width expressed in volume units.
It has been pointed out (27) that the area under the gauss curve is 1.033 times the area of the triangle formed by the tangents, but this is not a serious error from a practical point of view. The distribution coefficient Kd was calculated from Equation 1 in the form Kd = (ii/AZ) - F I . The determination of F I is described below. COLUMNVOIDSPACE,F I . This was determined by measuring the volume of fluid necessary to move a n anion which takes no part in the exchange process through the column.
An 0.636 X 108 em. column, equilibrated with 0.2N sodium hydroxide solution a t 50' C., received a feed pulse of 0.2 ml. of 0.2N sodium chloride solution on the drained resin surface, which entered by gravity. The column was filled with 0.2N sodium hydroxide and the solution was pumped through at a flow rate of 0.097 em. per second. EfHuent fractions of 0.251 ml. were collected. Each fraction was tested with silver nitrate and nitric acid. The pulse was identified by the precipitated chloride, the bulk of which was located in one, two, or three fractions. The mid-point of the chloride breakthrough was assumed to be the mean of those fractions containing precipitate. The ratio of the volume of solution passed up to the mid-point of the chloride breakthrough, to the total column volume, was taken as the void fraction. For 67, 108, and 156 X 0.636 em. columns packed with 2.4 X em. diameter particles, the void fractions were 0.38, 0.37, and 0.36, respectively. For 108 X 0.638 cm. columns packed with 4.6 X 10-3 cm. or 8.0 X cm. diameter particles, the void fractions n'ere 0.39 and 0.41, respectively. These results are in accord with those of hianalo, Turse, and Rieman (64), and those obtained in this laboratory by standard density procedures (20). The operating pressures of the five columns, in order of their enumeration above were 225, 385, 550, 97, and 48 p.s.i. The inverse relationship between void fraction and pressure seemed qualitatively consistent with the expectation that at higher pressures, packing would be greater and that this would be reflected by a slightly smaller void fraction. These results also suggest that the columns were free of packing defects and that channeled sections were absent. I n subsequent calculations, the chloride-pulse void fraction, appropriate for the particular column being considered, was used. VOL. 32, NO. 13, DECEMBER 1960
o
1785
necessary to be without effect on the s-ariance. Substitution of N, (number of theoretical plates) = ( O / U ) ~ (Appendix this paper), enables Glueckauf's criterion to be expressed in terms of variance rather than theoretical plates. I n the present work feed pulse volumes of 3% or less of the total colunin volunie were eniployed in all runs SO that disturbance of band width due to large pulse volumes was probably avoided for all solutes. Also the variance was the same for amino acids dissolved in pH 2.93 buffer or in 0.1S hydrochloric acid; for convenience, the amino acids were dissoived in the latter.
-
45 !
I - = 2-
"
K d 3I -
280
Figure
300 310 pH nF INFLUE'IT
290
2.
320
Distribution coefficients influent pH
(Kd)as function of A
Glucosamine
0 Valine
+ Alanine
Glutamic acid 'J Glycine Serine 0 Aspartic acid 0 Cysteic acid X Taurine
FEEDPULSEVOLUMESAND CONFor feed pulse volumes up to approximately 6yoof the colunin volume, and with amounts from 0.5 to 2.5 pmoles of each solute. the variance was constant. Feed pulse volumes of 15% of the column volume gave a demonstrable increase in variance. These results are in accord n i t h the criterion of Glueckauf [Equation 29, (S)] which can be applied to judge tlie smallness of the feed pulse volume CENTRATIONS.
Equations 1 and 10 were tested in detail, the former by varying conditions such as pH, sodium ion concentration, and temperature which shifted the equilibrium and the latter by varying column diameter, length, rate of flow, and particle diameter as well as the equilibrium conditions. I n the treatment that follows, Equation 10 has been considered correct and D, has been treated as a n unknown. Liquid phase diffusion was neglected, an assumption that is discussed later in detail. Effect of pH. A column v a s equilibrated with citrate buffer of t h e same
Table II. Variance as Function of pH
Chromatogram
KO."
1 2 3 4 2.85 2.93 3.15 3.23 U2
0.20 0.56 0.25 Cyeteicacid 0.22 0 . 3 3 0.33 Taurine Aspartic acid 2.22 1.66 1.44 2.13 1.32 1.38 Serine Glutamic acid 6.05 3.96 3.80 4.24 2.56 2.89 Glycine 5.64 4.20 3 . 8 0 Alanine 9.24 10.16 11.0 Glucosamine 15.65 12.95 11.0 Valine a 24-micron diameter particles.
0.56 0.36 1.50 1.26 2.80 1.63 2.90 12.4 8.26
016 018 0 2 0 022 024 026 028 030 032 SODIUM ION CONC -M
3. Distribution coefficients function of influent sodium ion concentration Figure
(Kd)as
Table Ill. Variance as Function of Sodium Ion Concentration n Chromatogram No., 3 6 8 9 Sodium concn. 0,295 0.256 0,226 0,197 0.180
10
0.170
V2
1786
ANALYTICAL CHEMISTRY
"
'
"
"
"
I
3 2
I30
40
50
60 T "C
70
80
90
Figure 4. Distribution coefficients (Kd)as function of column temperature See legend Figure 2
0.42 0.46 1.26 1.05 3.38 2.25 3.24 7.24 10.40
pH as t h a t used subsequently t o elute t h e amino acids. T h e expeiimental details are in Table I, pH series. Ka was calculated by Equation l . T h e results are s h o u n i n Figure 2 . K d varied inversely as the pH over the range studied. hforeover, for the accurate determination of Kd fiom Equation 1. rigid control of pH is necessary to obtain reproducible values; for present conditions, a change of 0.01 pH unit shifted the position of a peak approximately one fraction volume. Variance bore a n inverse relationship to pH (Table 11) and within the limits of experimental error and over the pH range studied the change was accountPd for by the change in Kd. Using values of calculated from tlie flow runs, D, was calculated from the smoothed data, using Equation 10. The values of D, are shown in Table IX. Sodium Ion Concentration. Chiomatograms were obtained with citrate buffers of different sodiuni ion coiicentrations (Table I, sodium seiiea). T h e effect on Kd of sodium as a vaiiable is s h o a n in Figure 3. Examination of the slope of the glycine curve a t O.20OAV,for example, s h o w that a change of 0.002 in the normality of sodium concentration is sufficient to cause a shift in the positon of the peak approximately one fraction. Standardized sodium ion concentrations are, therefore, necessary to obtain reproducible results. The variance n a s found to bear a n inverse relationship to sodium ion concentration, and the change in JTariance was accounted for by the change in Kd. The results are shown in Table 111. D, was determined in the same manner as that described under pH. The results are shown (see Table
u
0
See legend Figure 2
0.22 Cysteic acid 0.28 Taurine 1.05 Aspartic acid 0.88 Serine 2.69 Glutamic acid 1.63 Glycine 2.59 Alanine 6.38 Glucosamine 9.30 Valine a 24-micron diameter particlee.
"
RESULTS
A
PH
Kdl I
0.36 0.33 1.32 1.21 3.46 2.33 3.84 9.55 11.14
0.22 0.56 1.76 1.47 4.30 2.12 4.45 10.40 14.74
0.64 0.30 2.03 1.62 5.02 2.97 5.29 12.8 16.8
0.19 0.31 1.85 1.69 4.66 3.33 5.40
15.0 16.6
IX).
Temperature. Chromatogi a m s were obtained a t 35', 45O, 54", 6 5 O ,
7 3 O , and 85' C. (Table I, temperature seiies). T h e effect of temperature on Kd is shown in Figure 4, a n d t h e change of variance with temperature is shou-n in Table IV. D,was calculated directly from Equation 10, the liquid diffusion term again being neglected. Values of X were used from the flow series. The 54' results are collected in Table IX and the values a t other temperatures are shown in Figure 5 . Activation energies, E(,,t), at 54' C. were obtained b y the relation D, = Cz exp [--E(,,,)IRT], where CZ is a constant, and R is the gas constant. Log D ewas plotted us, 1/T, where T is the absolute temperature and E(,,t) mas computed from the slope of the line a t 54' C. The data are displayed in Figure 5 . For comparison, the liquid diffusion temperature dependence line was also plotted, being derived from the Stokes-Einstein relation DLW1) -=-
TlPm
DL(Td
T2PW1)
Since only the slope was significant for present considerations, the liquid line was plotted, without regard to its absolute position, on a convenient part of the graph. The computed values of E(,,t) are collected in Table V. Column Length. Equation 10 predicts that u2 us. Z should yield a straight line through the origin, providing the starting feed pulse band is zero: A finite band would be revealed as a n intercept on the variance axis. I n the present experiments the starting bands were considered small enough for the intercepts to be aithin the limits of experimental error. Three columns were used (Table I, length series), three runs were made with each column, and the results were averaged. The dependence of uZ on 2 is summarized in Figure 6. Kithin the limits of experimental error and for the column lengths studied, the prediction of Equation 10 was verified. A slightly better fit would be obtained from some solutes b y using a small intercept, but the improvement is statistically negligible, The data of Table VI showed that
Table V.
Glucosamine
11 35
Variance as Function of Temperature 12 13 14 15 45 54 65 75
0.25
0.64
5.30 2.56 12.75 6.71 17.8 17.8 35.8
0.25 0.25 2.81
2.07
7.02
4.80
7.34 17.20 24.4
0.42
0.50 0 25 1.54
0.53 1 86 1 69 4 00 3 28 4 95 10.0
1.44
2.76 2.48 3.25 6.72 10.17
-
14.24
0.25 0.56 1 .00 1.13 1,35
0.25
0.49 I .35
1.18 1,50 2.10 2.64 6.00 6.00
1.96
2.44 5.02 5.52
particles.
b Figure 5. Solid diffusion coefficients (D,) as function of I / T o K Activation energies €> 2-i.e., t o obtain narrow peaks as widely separated as possible. Time here is a secondary consideration. However, no less important is the proper manipulation of all variables and this can be achieved more readily with the help of the algebraic expres-ions 5' presented in this paper.
with the present definition of k, and reveals
- Reynolds
kL
Koting the relationship M = 16(0/wj2 u2z = (Op/u2) and the result HETP = ==
vz
U22
+ Fr)2
i12Z2(Kd and the definition ~ / K L= 1/k1'
+ Fzi/ksKd
+
one finally obtains from Equation 3, the following:
APPENDIX
Rearrangement of Equations of van Deemter, Zuiderweg, and Klinkenberg. The solution to the basic equation pf Lapidus and rlmundson (21) as given in Equation 33 of the paper by van Deemter, Zuidermg, and Xlinkenberg (56) is C/C,
This is identical to the previous reeult, Equation 3, if we take D(,,,,I- = 0.82r~iO/Fi = 0.41dPU,whirhis equivalent to neglecting molecular diffusion and taking X = 0.41 in Equation 2a of this paper.
=
NOMENCLATURE
+
In the present woik K L and Kd are the same as a and FiilK, respectively, in the terminology of van Deemter el al. (56). The folloming relations are also noted: v = UA1Fltand CFi = Po
The peak of the eluted pulse occurs when the exponent in Equation 1 is zero. Equating 2 / L r and t, and denoting the volume eluted a t the appearance of the peak by 6, after Eeveral substitutions one ObtainE: i A2 (Ki F i )
+
+
Introducing the definition a' = 0,: rearranging gives finally a ~ ~ / ( p / U A F z ) 2and
area of the column, sq. cm. A2 = total column volume, cc. [X of Gluecltauf ( 7 ) ] a, b = subscripts to denote components a and b CI, C,, C3 = constants C = conrentration in liquid phase, moles cm.-3 C. = concentration in liquid phase a t the solid-liquid interface, moles 0 1 1 1 . ~ 3 CO = concentration of niicrocomponent in the pulse, moles cm.-S D/sxlai) = effective axial diffusion coefficient, sq. cm. sec.-l DE = eddy diffusion coefficient, sq. cm. eec.-l DL = liquid phase diffusion coefficient, sq. cm. set.-' D, = solid phase diffusion coefficient, sq. cm. set.-' d, = paiticle diameter, cm. E(,,t, = activation energy, cal. mole-' Fi = void fraction, dimensionless [ a of Glueckauf ( 7 ) ] FZI = (1 - F z ) = fraction of column occupied by solid, dimensionless HETP = height equivalent of a theoretical plate, cm., Glueckauf A
where u12 = (22, P ) D ( a x , 8 ~ ) and 6 2 2 = 2p2ZKa2,( K L F I C ) and 1/p = 1 K,'FI
This is a gauseian pulse with variance given by
= cross-sectional
(7) = ( p k i ~ ) / ( c )f e q u , l ) =
Rearrangement of Equation 14 of Glueckauf ( 7 ) . Equation 14 of Glueckauf is HETP = 1.64 r
+
except that in the present inetance, the factor FZZ has been added to the denominator of the last term. This factor should have been included in Equation 7 of Glueckauf for dimensional consistency but from a practical viewpoint it is not of particular importance, inasmuch as Fir, being nearly constant, can be lumped with other empirical constants. Comparison of Equations 4 and 7 of Glueckauf
distribution coefficient a t equilibrium, dimensionless = over-all mass transfer coefficient, set.-'; defined by (moles/cc. of column volume)/ (moles/cc. liquid) (see.) [a of van Deemter. Zuiderweg, and Klinkenberg I)%( = liquid phase mass transfer coefficient, sec.-l; defined by (moles/cc. column volume)/ (moles/cc. liquid) (sec.) = solid phase mass transfer coeficient, sec -l; defined by (moles/cc. column volume)/ (moles/cc. of resin solid) (SPC.) = milliequivalents of solute = molar flux, (moles)/(cc. column volume) X sec.-l = molar flux, (moles)/(cc. of solid phase) X sec.-l = number of theoretical plates of column, dimeneionless = Peclet number = d, V I D E , dimensionless
number = d p U o p / p , dimensionless Schmidt number = ~ / D L P , dimensionless concentration in solid phase, moles. cm.-3 mean concentration in solid phaee, moles ~ m . - ~ concentration in solid phase a t the solid-liquid interface, moles cm.? g a constant, ~ cal. mole-1 paiticle radius, cm resolution ratio, defined by RR = (0, - fib)/(Ua Ub), dimensionless timr. sec. duration of feed piilse, calculated a t rate of elution used in development, sec. temperature (absolute) K." superficial velocity (volumetric inout divided bv column area) em. sec.-1 [ F of Glueckauf ( 7 ) ] interstitial velocity, cm. see.-' = U"/FZ total volume of column effluent at time t , cc. volume of effluent, a t peak of an eluted pulse, cc. band width of eluted peak, measured on v axis, in v units, cc length of column, cm. FI/(FI Kd), dimensionless labi-rinth factor, a constant, dimensionlesE (Equation 2a) = effective stagnant film of liquid surrounding particle, cni. = ~ / N P .dimensionless (Equation 2a 1 = density of fluid, g. cm.+ = 42, dispersion of eluted piilPe in volume units, cc. = see Equation 1, appendix = variance of eluted pulse. defined by Equation 2, cm.O = viscosity of fluid, g. cm.-'l set.-' (poise) = functional symbol, Equation
+
11 LITERATURE CITED
(1) Bogue, D. C., AKAL. CHEX 32, 1777
(1960).
(2) Boyd, G. E., Adamson, A. W., Myers, L. S.,Jr., J . Am. Chem. Xoc. 69,2836 (1947). (3) Calmon, C., Kressman, T. R. E.,
"Ion Exchangers in Organic and Biochemistry," pp. 86-103, Interscience, Wew York, 1957. (4) Carberry, J. J., A. I . Ch. E . Journal, in press. (5) Carberry, J. J., Bretton, R. H., Ibid., 4 , 367 (1958). (6) Destx, D., H., "Gas Chromatography, Section I, pp. 3-141, Academic Press, h-ew Tork, 1958. ( 7 ) Glueckauf, ,E.,"Ion Exchange and Its Applicaticns, pp. 34-46, 1955. Published by SOC.Chem. Ind., 14 Belgrave Sq., 8. W.1, London. Reprinted 1958. ( 8 ) Glueckauf, E., Trans. Faraday SOC. 51, 34 (1955); (9) Ibid., 51, 1040 (1955). (10) Glueckacf, E., Coates, J. I., J . Chem. SOC.1947, 1315. (11) Gregor, H. P., Collins, F. C., Pope, AI., J . Colloid X c i . 6,304 (1951). (12) Hamilton, P. B., AXAL. CHEX 30, 914 11958). (13) Idid., 3 2 , 1779 (1960). (14) Hamilton, P. B., Anderson, R. A,, Ibid., 31,1504 (1959). (15) "Handbook of Chemistry and VOL. 32, N O . 13, DECEMBER 1960
e
1791
Physics,” 40th ed., p. 209, Chemical Rubber CO., 1959. (16) Helfferich, F., “Ionenau6tauscher,” Band I, Chap. 9, pp. 385-435, Verlag Chemie, 1959. ( 1 7 ) Hogfeldt, E., Arkiv Kemi 13, 491
( 1959). (18) Keulemans, A. I. M., “GaB Chromatography,” pp. 96-129, Reinhold, Xew York, 1957. (19) Klinkenberg, A., unpublished work
quoted by Kiinkenberg and Sjenitzer, Chem. Eng. Sci. 5 , 2 5 8 (1956). (20) Kunin, R., “Ion Exchange Resins,” 2nd ed., pp. 5-33, Wiley, ]Sew York, 1958. (21) Lapidus, L., Amundson, N. R., J . Phys. Chem. 5 6 , 9 8 4 (1952). (22) Longsvorth, L. G., “American In-
383 (1959). (25) Moore, S., Stein, W. H,, J . Biol. Chem. 176,367 (1948). (26) Ibid., 192, 663 (1951). (27) Moseley, Walter, Jr., private communication, E. I. du Pont de Nemours &. Co., Wilmington, Del., December 10, 1958. (28) Pepper, K. W., “Chemistry Re-
(29) Pepper, K. W.,Reichenberg, DJ 2, Eleklrochem. 57, 183 (1953). (30) Reichenberg, D., J . Am. Chem. Soe. 75,589 (1953). (31) Sherwood, T. K., Pigford. R. L., “Absorption and Extraction,” pp. 72-5, McGraw-Hill, New York, 1952. (32) Soldano, B. A., Ann. iV. Y . Acad. Sca. 57. 116 il953l. (33) Soldan& B.-4.,‘Boyd, G. E., J. Am. Chem. Sac. 75,6107 (1953). (34) Taylor, G., Proc. Roy. 8oc. A219, 186 (1953). (35) van Deemter, J. J., Zuidermeg, F.J., Klinkenberg A., Chem. Eng. Sci. 5 , 2 7 1 (1956j. (36) .Weiss, D. E., Australian J. A p p l . Scz. 4, 510 (1953).
search,” Her Majesty’s Stationery Office, London, England, 1962.
RECEIVEDfor review July 11, 1960. Accepted September 14, 1960.
stitute of Physics Handbook,” Tables 2s-4, pp, 2-193, McGraw-Hill, Xew Yo&, 1957. (23) McHenry, K. TV., Jr., Wiiheim, R. H., A. I . Ch. E. Journal 3 , 8 3 (1957). (24) Manalo, G. D., Turse, R., Riemati, Wiliiam, 111, Anal. Chim. Acta 21,
f
Photometric Titr qweous
Weak Acids
etesrnination of Phenols in Isopropyl Alcohol L. E. 1. HUMMELSTEDT and DAVID N. HUM€ Deparfmenf of Chemisfry and Laborofory for Nuclear Science, Massachusetfs Instifufe o f Technology, Cambridge 39, Mass.
k~ Photometric titrations of 16 weak acids, mainly phenols, were performed in commercial isopropyl alcohol using tetra-n-butylammonium hydroxide in isopropyl alcohol as titrant. In favorable cases, up to four components in a mixture were resolved if a suitable change in wave length setting was made during the titration. Nonlinear phoiometric titration curves for phenols provide new evidence for complex formation between phenols and phenolate ions. many of the characteristics of photometric titration commend it for use in the determination of small amounts of weak acids and weak acid mixtures, relatively little work seems to h a r e been done on this application. Goddu and Hume (8, 9) investigated the method in aqueous medium, and recently, McKinney and Reynolds (14) titrated phenolic compounds photometrically in butylamine using ethanolic sodium hydroxide as 3 titrant. Although an excellent titration medium for very weak acids, the widespread use of butylamine as a solvent is deterred by its unpleasant odor. The results of the present study show that the inherent advantages of the photometric titration method make it possible to use an essentially neutral solvent for many compounds which would require a strongly basic solvent if titrated potentiometrically. Ketonic solvents, which have been LTHOUGH
1792
0
ANALYTICAL CHEMISTRY
used extensively in potentiometric titrations of weak acids, absorb strongly i n the ultraviolet and are, therefore, useful photometrically essentially only in the ~ i s i b l eregion. Alcohols, on the other hand, can be used throughout nearly all the ultraviolet region. Isopropyl alcohol mas selected for this study because it is less acidic than methanol or ethanol (II), has good soIvent properties, and is readily available in sufficient purity at low cost. For comparison a number of titrations were performed in benzene, absolute ethyl alcohol, and mixtures of these solvents with isopropyl alcohol. EXPERIMENTAL
Apparatus. Photometric titrations were performed using a Becknian Model DU spectrophotometer adapted as described before (IS). The titration technique was the same a 8 used previously (IS) with the exception that the samples were protected from atmospheric carbon dioxide with a blanket of nitrogen. The titrant was delivered from a closed system consisting of a 5ml. Fischer and Porter Lab-Crest microburet equipped with a three-way Teflon stopcock for filling from a polyethylene storage battle connected to the buret with polyethylene tubing. The reagent was protected from the atmosphere with an Ascarite tube and a similar tube was attached to the top of the buret. The titration cell was a 150-ml. beaker made of Vycor grade 7910 (Corning Glass Works, Corning, N. Y . ) . Potentiometric titrations were per-
formed using a Leeds & Northrup line-operated pH meter equipped with a Leeds & Northrup 1199-30 glass electrode and a sleeve-type calomel electrode. Solvents and Chemicals. Commercial isopropyl alcohol (99%) is satisfactory as a titration medium. Reagent grade isopropyl alcohol was used in making up titrant. Most of t h e acids determined were of Eastman White Label grade and they mere in general titrated as received. The phenol was Merck reagent grade. Anhydrous diphenyl phosphate was prepared from the dihydrate (The Dov: Chemical Co.) according to a method described b y Davis and Hetzer (4). Titrant. T h e titrant was prepared from t h e 1.OM methanolic tetra-nbutylammonium hydroxide (Southwestern Analytical Chemicals, Austin, Tex.). Most of t h e methanol was evaporated under vacuum at room temperature using a rotating flask evaporator; then the solution was diluted with reagent grade isopropyl alcohol to give a 0.4M solution containing no more than 10 to 15% methanol by volume. The reduction in methanol content was desirable in view of the findings of Harlow and Wyld (11) in potentiometric titrations n i t h tetraalkylammonium bases in different solvents. The 0.4N titrant was stsndardized potentiometrically by titrating portions, diluted with water, with aqueous perchloric acid. RESULTS
The analytical characteristics of weak acids when titrated photometrically in
