Chromatography Handbook Of Hplc - Katz Eksteen Schocnmakers Miller wiley Sons -108

hms,cap = Hm,cap/2R, = 2/v,cap + (1/96)(f)(k)v,cap (542b)\n\nAn important assumption in the derivation of this result, is that mass transfer between and inside phases is sufficiently fast to reach equilibrium, as was expressed by the Fourier number requirement Fo > 0.8 (see Sec. II.E.2.c). Gill et al. [27], in a most thorough analysis of the situation, prove that not only the dimensionless time (or Fourier number) is limiting, but quite logically also the Peclet number Pe,m = (u,exp)(R)/D,m (i.e., our reduced velocity v). They solved the exact mass balance equations to find the concentration profile of a migrating zone in the absence of stationary phase (or k = 0) for all possible values of Pe,m = v, and Fo. Their solutions include the Taylor, Aris, and Golay result Eq. (54d), proved to be valid in the domain where Pe,m = v > 20 and Fo > 0.8. In addition they also provide solutions in all other v-Fo domains, which can be summarized in our notation as follows:\n\nhms,cap = 2/v,cap + 0.014Fo^0.5 \n for Fo < 0.6\nhms,cap = 2/v,cap (1/96)v,cap) \n for Fo > 0.8 and v,cap > 20\nhms,cap = 2/v,cap (54e) \n for v,cap^0.5 < 1\n\nHere, for completeness, Eq. (54d) is the Taylor–Aris–Golay solution, valid within a large, yet restricted v-Fo domain. Equation (54e) is the pure diffusion solution (i.e., only the B term), which is of no practical use for chromatography, just as the pure convection solution (sheatting) from the velocity profile in the absence of radial diffusion which results in a very broad and tailing peak.\n\nTo summarize the domains in which Eqs. (54c–e) are valid we can construct the approximate graph as seen in Figure 6. The figure clearly shows the region where the Taylor–Aris– Golay approximation is valid, the usual domain of use for chromatography.\n\nThe last contribution to the plate height in capillary columns is the mass-transfer term Ct for stationary phase. Ideally, this phase is distributed as a uniform film of thickness d on the inside column wall. If this is true and if the column radius is not too small, the stationary phase can be treated as a film which, for the classical Ct term after van Deemter et al. is already available, Eq. (34c2). We thus arrive at an expression for the total plate height for capillary columns, named after Golay:\n\nHcap = ΣH,i = Bm,cap/(u,m) + C,m,cap(u,m) + C,(u,m) \n = 2D,m(u,m) + (1/24)(K)(R^2)D,m(u,m) + (2/3)(k)(8)D,(u,m) (55a)\n\nor\n\nhms,cap = Hcap/2R, = 2/v,cap + (1/96)(f)(k)v,cap + (2/3)(k)(d^2xD,mcap\n\nwhere, as before, d,m = (6/d,p) = (6/2R) and D = (D,m/D,m).\n\nJust as for packed columns, plate height first decreases hyperbolically at (very) low velocities owing to molecular diffusion in the mobile phase (for LLC, to be extended with that for the stationary phase as well, see Sec. II.E.1). In contrast, at higher velocities, plate height increases linearly with velocity, caused by both Ct terms. Somewhere in between, at moderate velocities in GC practice and at very low velocities in LC practice, plate height obtains a minimum value, indicating maximum efficiency at a specific optimal velocity. This behavior is analogous to that of the van Deemter equation.\n\nEspecially for capillary columns in GC, one of the considerable advantages in practice is that the contribution from the stationary phase is almost negligible because very thin films are used: δ, is often chosen to be <1/300, which in actual practice means that δ = 0.1– 1 µm. If we also try to apply capacity factors as large as possible, the stationary phase contribution term Ct is further reduced by the function g(k) = k/(1 + k^2), which has its maximum at k = 1 and tends to zero for both k → 0 and large k. This situation is strongly\n\n