Plate Height Contributions in Gas Chromatography SIR: The nonequilibrium method, (1-5) used for calculating plate height
contributions when either single step or diffusion processes are rate controlling, can be applied immediately to determine the plate height contribution of a number of processes of current interest. Included are liquid phase diffusion in glass bead columns, narrow pores, and within solid support particles, gas phase diffusion within solid support particles, and sorption and desorption processes occurring a t the gas-liquid and liquid-solid interface. I n calculating the plate height of a column, the assumption is usually made that the various nonequilibrium contributions are additive. This is related to the high degree of symmetry of hypothetical models used to approximate column packing geometry. I n practice such symmetry rarely exists, and, as borne out by preliminary calculations on a two-dimensional lattice model in the author's laboratory, the additive assumption is somewhat in error. For complex processes, however, there is little recourse but to use the additive assumption and symmetrical models. This applies to many of the results below. The additive assumption has been discussed more fully in
(4).
Diffusion in Narrow Pores. It will be assumed t h a t t h e liquid phase has accumulated in pores sufficiently narrow (or with sufficient symmetry) t o prohibit significant concentration gradients laterally across t h e pore. If t h e cross-sectional area of t h e pore is written as a = constant.(d--r)", where d is the pore depth and z is the distance into the pore, then we obtain the plate height contribution H -
- (n
d2
2
+ I ) ( n + 3) R ( 1 - R ) v -
Dl
(1)
where n 2 0 by physical arguments. The liquid diffusion coefficient is Di,and the ratio of zone velocity, U, to average gas velocity, v, is defined as R ( R may be k ) where IC is the written as 1/(1 amount of solute in the liquid phase divided by that in the gas). This equation predicts the plate height for pores of a given taper, and is also in agreement with some previously derived (4) special cases. The coefficient equals */3 ( n = 0) for a uniform film or coating, (n = 1) for cylindrical rods (important in paper chromatography) , and (n = 2) for spheres (important in ion exchange chromatography). Khile these geometries are obviously not 'harrow pores," symmetry makes i t possible to consider them as such by
+
962 *
ANALYTICAL CHEMISTRY
eliminating the possibility of lateral gradients. Glass Beads. Glass bead columns have been employed recently for the rapid analysis of high boiling compounds ( 7 ) . No theory has been presented, however, for calculating the efficiency of these columns. If such columns are equilibrated at high temperature, as prescribed by Littlewood ( I I ) , then presumably the bulk of liquid is condensed to a uniform depth around the contact points betneen bcads. If small amounts of liquid (up to a few per cent) are used, the liquid rogions are mathematically equivalent t o narrow pores with n = 3. Thus H,
=
R ( l - R ) ~ d ~ / I)i l?
(2)
This equation should be subject to experimental verification. The quantity d 2 can be obtained as (2r2,'5) (y',py,' 3 r n p ~ ) ~ / where *, r is the bead radius, yo is the weight percentage of liquid loading, p a and p~ are the densities of the glass and liquid, and rn is the number of contact points per sphere, probably from 5 to 8. From this equation the plate height of glass bead and other columns can be theoretically compared. Of particular interest is a comparison with the ideal capillary column described quantitatively by Golay (6). The comparison is meaningful only if the two columns are operated a t equal R values with the same solute and liquid phases. Considering only the contribution of liquid phase diffusion, we obtain
The support particles are assumed here to be spherical with the pore diameter much smaller than the particle diameter, 2b. The interior of a particle may then be considered mathematically as a continuous medium in which diffusion occurs with the effective coefficient, D,/TZ, where Do is the gas phase diffusion coefficient and T is the tortuosity (13). The plate height contribution is obtained as
TI here Q is the fraction of gas occupying interparticle space and 1- 0 is the fraction in intraparticle space. The average gas velocity z is, operationally, more significant (and less in magnitude) than the interparticle average velocity. Liquid Phase Diffusion within Support Particles. While liquid phase diffusion is not likely to contribute much to the over-all intraparticle mass flux, i t determines the extent of nonequilibrium nithin the liquid. T h e folloaing model, significantly different from previous ones, accounts for pore structure and the presence of isolated units of liquid in the smaller pores. Each disconnected unit of liquid occupies a pore of a certain effective diameter, 6, and will h a r e an effective depth also approximately equal to 6. The plate height contribution of an assortment of unequal pores has been obtained as (4)
H,
where, arbitrarily, the capillary (cap.) and glass bead (g.b.) radii have been assumed equal, m = 6, p a I p I = 2, and the porosity = 0.4. Thus the glass bead column is only about lyc as efficient as the capillary in typical circumstances a t high flow rates. Both capillary and conventional (packed) columns have an inherent advantage over glass beads because of the better distribution of the liquid phase. Gas Phase Diffusion within Support Particles. Support particles of diatomaceous earth and firebrick are highly porous. Neglecting hysteresis effects the liquid phase establishes itself in the smallest pores of the support. Except when the liquid loading is excessive the liquid will be found in small disconnected units. Solute molecules must diffuse through the interconnected open pores to partition with thc liquid. The possibility of this diffusion being a contributing step to the plate height has been emphasized b> Jones (8)and Kieselbach 110,.
=
?R(1 - R ) vd2/3 Dr
(5)
d.
where is the square of the depth averaged over the liquid volume. With the above assumption regarding pore depth, this quantity can be obtained from the pore size distribution ( I S ) , a(6), by means of the equation
nhere 6, is the masinium diameter of filled pores. The pore distribution can be obtained experimentally in a number of ways. Thus H, can be cdculated in terms of experinental par miet'ers. Adsorption and Desorptior, Processes. Attention has beer, fncuseo recently (12) on adsorption a i the gas liquid and liquid-solid interfa-m. .A: any given moment a significant frL'.tion of the solute may be adsorbed 2:; one 0:' thes- interfaces. This is i t < dicated by an increase in the rtit8entivrA volume (I,?;,. Ir? order to i,rex: thi: phenonienoI& i:) t!,., li:,car r:tx;P'- t h ~
following sequence considered
DO A1
gas phase vi=v
F!
of
processes
is
D1 * A3 F! Ai gas liquid Liquid A?
liquid phase solid interface interface 17) v p = o
2 . 3 = 0 v 4 = 0
The fraction of solute occupying the respective states a t equilibrium is given by R , x2*, x*3, and x4*. These can be obtained from equilibrium measurements. The rate constant between Ai is given as any two states, Ai k,,, and is so defined as t o have the dimension, time-’. The model assumed above is different than t h a t proposed b y Khan (9) in which adsorption does not contribute t o the retention volume, i.e., x2 = 2 4 = 0. His result may be obtained as a special case of that presented below. Assuming t h a t the contributions at the gas-liquid and liquid-solid interfaces are additive, the contribution of adsorption t o the plate height may be deduced from Equation 34 of (2).
-
T o this must be added the plate height terms for gas phase and liquid phase diffusion. If the liquid forms a uniform layer of depth d, its contribution is H,
=
2Rvd2 (23*/3
+
24*
+
24
* ‘/XS* ) / D I (9
It is important to note t h a t this is functionally different from the previous expressions for the liquid term. This is no longer true when x3* = x4* = 0 . The gas phase term (4) is not changed in form by adsorption, and consequently need not be considered here. LITERATURE CITED
(1) Giddings, J. C., J . Chem Phys. 31,
1462 (1959).
(2) Giddings, J. C., J . Chromatog. 3, 443
(1960).
(3j zbid., 3,520 (1960). (4) Zbid., 5 , 45 (1961).
(5) Giddings, J. C., Aature 188, 847 (1960). (6) Golay, M. J. E., “Gas Chromatography,” p. 36, D. H. Desty, ed., Academic Press, Xew Ycrk, 1958. (7) Hishta, C., Messerly, J. P., Reschke, R. F.,ANAL.CHEM.32, 1730 (1960). (8) Jones, W. L., Gas Chromatography Symposium, U. S.Public Health Service, Cincinnati, Ohio, Feb. 21, 1957. (9) Khan, M. A., lVature 186, 800 (1960). (10) Kieselbach, R., ASAL. CHEM. 33, 23 (1961). (11) Littlewood, A. B., “Gas Chromatography,” p. 23, D. H. Desty, ed., Academic Press, New York, 1958. (12) Martin, R. L., ANAL.CHEM.33, 347 (1961). (13) Scheidegger, A. E., “The Physics of Flow Through Porous Media,” p. 7, Macmillan, Xew York, 1957. J. CALVISGIDDIKGS
Department of Chemistry University of Utah Salt Lake City 12, Utah RECEIVEDfor review March 20, 1961. Accepted April 10, 1961. Work supported by research grant, A-2402 (C3), from the National Institutes of Health, Public Health Service. Presented in part Division of Analytical Chemistry, 139th Meeting, ACS, St. Louis, Mo., September 1961.
Loss of Carbon-14 Radioactivity by Counting (Hexadecyl-1-C”1trimethylammonium Bromide in Glass Planchets SIR: I n counting radioactive samples one does not usually consider the material of which the planchet is constructed to be critical. Occasionally, however, errors can be caused b y the use of planchet material which is later found to be unsuitable [Blair, D. G. R., Potter, V. R., J . Am. Chem. SOC.82, 3223 (1960)l. Concave glass planchets are advantageous for drying and counting radioactive solutions, particularly of low-energy beta emitters such as carbon14. Much better counting efficiency may be obtained than with flat planchets, presumably because of better geometry and perhaps also because of lower self-absorption losses. The use of such planchets (of borosilicate glass) in counting (hexadecyl1-C14)trimethylammonium bromide led, however, to loss of radioactivity, which could be avoided b y using nickelplated planchets. Counting rates, measured with a thin-end window, gasflow Geiger counter are given for a solution of the bromide (Table I). The solution had been dried in glass and
nickel-plated planchets, respectively, under a 250-watt infrared heating lamp before the first count was made, and then heated successively for l/phour
Table 1.
Effect of Heating on Count
Planchet Type Flat, cupped nickelplated Concave elass I
C.P.M. after Successive 1/2-Hour Heating Periodsapb 1086, 1096,‘ 1085, 1094, 1095, 1078, 1083. 676dv; 1871. 1859. 1837. 1820. 1809, 1817,c lj47, ’ 1743, 93Odva
1 ml. of an aqueous solution of (hexadecyl - 1 - C14)trimethylammonium bromide containing 1.55 mg./ml. b 10-minute counting period. Counted after standing a t room temperature over the weekend in the chamber of a thin-end window, gas-flow counter (without a flow of gas). After addition of 1 ml. of 0.2% NazCOS solution and heating. 30-minute counting period. (1
0
intervals before each additional count. (Similar decreases in count upon heating were obtained with several other glass planchets.) One milliliter of 0.2% sodium carbonate solution was finally added to the contents of the nickel-plated planchet, after the prolonged drying without loss of radioactivity, and the contents were dried again under the heating lamp. The marked drop in count seemed t o confirm the explanation that the loss in radioactivity in the glass planchet is due t o reaction with the alkali of the glass, presumably by volatilization (under heat of the lamp) of active hexadecene formed b y a Hofmann degradation. Addition of sodium carbonate solution to a sample in a glass planchet caused a similar loss of activity. The loss of activity could not be accounted for by increased self-absorption due t o the alkali added. WILLIAMSEAMAN JR. DAVIDSTEWART, American Cyanamid Co. Organic Chemicals Division Bound Brook, N. J. VOL. 33, NO. 7, JUNE 1961
963
