184
J. CALVINGIDDINGS
General Combination Law for C, Terms in Gas Chromatography
by J. Calvin Giddings Department of Chemistry, University of Utah, Salt Lake City 12, Utah (Received September 5, 2965)
The dependence of zone dispersion in gas-liquid chromatography on the diffusion of solute in and out of the liquid phase is investigated with the aid of the nonequilibrium theory of chromatography. Although explicit expressions are not obtained for zone dispersion in the (typical) presence of highly complex liquid masses, it is shown that the dispersion can be written as a sum of terms, each referring to a small, relatively simple unit of liquid. The differential equation (of the Poisson type) applicable to each such unit is given and previously obtained solutions are shown. It is demonstrated that the combination law reduces a problem of considerable complexity into its rather simple component parts, and that no practical difficulty stands in the way of solution providing only that the liquid configuration can be determined or estimated.
A basic problem in the study of zone spreading in chromatography, as well as in the broader field of dispersion in any porous material with fluid flow in it, is the manner in which the processes in neighboring regions combine in their over-all contribution to the dispersion of incorporated solute. If rules governing the interaction of such regions can be found, the dispersion problem reduces to the relatively simple one of treating isolated regions. These isolated regions may be complex, but individually they are far less formidable than when considered in total. The particular system considered here is that generally associated with gas-liquid chromatography in which a finely dispersed partitioning liquid is found on the surface and in the cavities and recesses of a porous material’ (usually a diatomaceous earth derivative). The over-all liquid configuration is enormously complex. In this paper it will be shown that such a liquid, insofar as its influence on zone spreading is concerned, can be broken into individual liquid units Qf comparatively simple geometry. The general combination law for the effect of these liquid units will then be elucidated. The dispersion of a chromatographic zone is usually expressed in terms of the column plate height, H . A definition of this quantity will be presented later. The plate height is composed of a number of additive terms, each representing a particular mechanism leading to zone dispersion. The so-called nonequilibrium processes (next section) lead to a term proportional to The Journal of Physical Chemistry
flow velocity, CO. The C is in turn divided between a C1 term’ and a C, term,2 referring to liquid diffusion and gas diffusion processes, respectively. The CI term, to be investigated here, is generally the most important source of zone dispersion in gas chromatography. In the following treatment the C1 term can be obtained as the derived value of H divided by the mean velocity O. For convenience, most results will be left in the H form, but it should be remembered that this is only a particular part of the total H , a part most commonly identified by reference to the CI term.
Theory At flow rates which are of practical importance in gas chromatography, zone dispersion is largely a result of various nonequilibrium processes. These arise as the zone’s changing concentration profile moves through a given segment and equilibrates a t a finite rate with each small region of the column (a qualitative view of nonequilibrium and its effects is given elsewhere3). The nonequilibrium may be controlled by liquid diffusion, gaseous diffusion, or both. The slowness of liquid diffusion makes this term rate-controlling in the largest number of cases. The liquid contribution, only, will be considered here. (1) J. C. Giddings, Anal. Chem., 34, 468 (1962). (2) J. C. Giddings, ibid., 34, 1186 (1962). (3) J. C. Giddings, ‘‘Chromatography,” E. Heftmann, Ed., Reinhold Publishing Corp.. New York, N. Y . , 1961, Chapter 3.
The quantitative description of nonequilibriurn will procced i n terms of tJhe equilibrium departure tcrm, e, dcfiricd by the equation1
m
=
m*(l
+ e)
(1)
where m is thc actual value of thc local concentration of solute at a given point and m* is the local concentration assuming that complete cquilibrium has been rcachcd within the cross section of column containing that point. The magnitude of e van genrrally be ass1imc.d as small compared to unity. This condition greatly simplifies the mathematical t rcatment of noncquilibriurn4and the consequent zonc spreading. The generalized noncquilibrium indicates that the rate of accumulation of the sample component pcr unit volume in rrgion i due to noneyuilibrium gradients is given by si
= (ut
- ti)bm,*:dz
(2)
where zit is thc downstream velocity of phase i at the point under cwnsideration, ti the mean downstrcam velocity of the sample zone, and z the distance along the axis of the column. The velocity, u, for the liquid 1s of course zero and eq. 2 simplifies to st = -zibm,*/bz
rium. The summation rovers all regions of the column, liquid arid gas. IIowevcr, with u t = 0 in the liquid, the summation, effectivcly, covers only the gas phase. Furthermore, when investigating solely the liquid contribution to zone dispersion, 6 in the gas phase may hc assumed locally constant, e,. (This assumption and some later ones depends on the additive contribution of gas arid liyriid terms to 9. The additive condition, discussed elsrwhcreJ7 is certainly valid in this case because of the much larger difyusivity of gases compared to liquids, a feature which ensures that each part of a liquid droplet’s surface will be bathed at the same concentration). Crider these circumstances eq. 5 becomes
where m,* is the equilibrium concentration in the gas, fi the mean downstream fluid (gas in this case) velocity, A , is the fraction of cross section occupied by gas. Since m,*A, is the gas phase contribution to c*, the ratio ma*Aa/c* = R, a commonly used term denoting the fraction of solute in the mobile phase. Thus the last equation becomes
(3)
where ml* is the equilibrium liquid concentration. For diffusion controlled processes, s t may conveniently be written as 1)lV2m1,which, with eq. 1 arid tlie condition that m* is locally constant, becomes .Dlml*V2et. In this equation, D I is the diffusion coefficient of solute in the liquid. The substitution of this into eq. 3 gives one form of the basic differential equation for et
(4) This equation must be integrated over thc entire liquid mass in ordcr to obtain point-by-point e-values. Once e has been obtained at each point within the liquid, it is not difficult to formulate the effective diffusion coefficient for dispersion or the theoretical plate height which is likewise a measure of dispersion. This is done, as shown elsewhere, by obtaining an cxprcssion for the material flux through various cross sections. The effective dispersion coefficient, D, is shown to be7
where A , is the fraction of total cross-sectional area occupied by thc ith phase or region and c* is the over-all concentration (per unit volume of column rather than per unit volume of a particular phase) at local equilib-
(7) and the plate height, which can be defined as 29/RO, is simply
H
=
- 2 ea/d I n c*/bz
(8)
The ea and et terms are assured by the nature of eq. 4 of being proportional to b In c*/dz (mi* and ma* are proportional to c*) ; hence the final 3 or H expression is not dependent on the concentration gradient. The e t values from eq. 4 arc corivcrted to an el( value by the local equilibration condition e,&
=
-a(l - R )
where ZI is the value of e (or total liquid content.
et)
(9)
averaged over tlie
The Liquid Phase The bulk of thc liquid phasc in gas chromatography may be considered to occupy the cavities or pores of the solid support. Bctween these isolated liquid pools is a thin film of liyiiid held to the solid by adsorp(4) J. (5) .J. (6) *J. (7) J. (8) J.
C. Giddings, J . Chem. l’hyy., 3 1 , 1482 (1959) C. Giddirigs, J . Chromatog., 3, 443 (lW$O). C. Giddings, .Yature, 188, 847 (19W). C. Giddings, J . Chromatog., 5 , 46 (1081). (’. Giddings. Anal. (,“hem.,35, 439 (1963).
tion forces. ' As the chromatographic zone passes through each region the liquid pools first absorb solute, then release it back into the gas stream.3 At any given moment there is a certain flux of solute across t,he boundaries of a particular pool. AIost of this flux typically occurs through the gas-liquid interface. Very little of it occurs through the thin film connecting one pool to another because of the film's dimensions. Virtually none of the flux occurs through the liquidsolid interface because of the solid's impermeability. Thus the diffusion of solute into such pools is governed, in part, by the boundary condition bmI/bw = 0 nonflux boundaries
(10)
where w is the distance along the outwardly directed normal. This can to good approximation be applied to all such nonflux boundaries, but not to the gas-liquid interface. Somewhat more generally, we shall define a unit of liquid as being any liquid mass whose principal gain or loss of solute is through the gas-liquid interface, and whose other boundaries are therefore subject to eq. 10. A nonflux boundary may arise because the liquid is in contact with a solid, or as a result of certain symmetry characteristics. An example of the latter arises in connection with the contact point liquid in glass bead columns. The ring of liquid can be divided by numerous planes of symmetry, and each plane may be regarded as a nonflux boundary. Thus a unit of liquid, as defined above, may be a very small part of the liquid pool. RIost pools of the type described above are, however, sufficiently isolated to qualify as a unit. If two nearby cavities are filled in such a way that an exchange of solute takes place between them, the two together must be considered as a unit. At this point it is necessary to consider the application of eq. 4 to the liquid. If +t is defined by 4t =
Then, since 4 becomes
ti =
€1
Di Red In c*/bz
Re and b In c*/bz
=
b In ml*/bz, eq.
V%${ = -1
(12)
The boundary conditions applying to (where the subscript now refers to unit i) are 4, = constant at the gas-liquid interface (due to the constancy of solute concentration at this interface) and b+,/bw = 0 a t all nonflux boundaries. The solutlon to this equation can be written in the form I#J
