Chromatography Handbook Of Hplc - Katz Eksteen Schocnmakers Miller wiley Sons -115

Mechanisms and Importance of Zone-Spreading 105\n\nWe will now proceed to apply the exact mass-balance equations to yield an exact solution for the concentration profile of the mobile zone [7, and references therein]. Subsequent application of correlations between dimensionless mass-transfer numbers [15] for spherical particles then yields the contribution of slow and obstructed diffusion in the tortuous stationary particle region (γD/p,can) as an additional C term, which we call the C,m term because of the partitioning between two regions of mobile phase. This contribution reads:\n\nH,wm = (1/30)kel/(e, + 1 k)jkl0(aud2p,/γD,p\n\n(59a)\n\nThe diffusion factor γD,p,can can be assumed to be determined by diffusion through the mobile phase in the pores along γD, p,can = γD,p,De, which applies to GC packings where De is large and renders the C, term small for all practical purposes. In LC, however, γD,p,can will most likely be determined by diffusion in both phases, which may lead to a distinct contribution to plate height. Notably in RPLC, this term is present and is smaller than anticipated by the diffusion in the stationary phase alone.\n\nWe can express the C,m contribution in our usual notation with (u,m) and k as:\n\nH,wm = (c,m,u) = (1/30)kel/(e, + 1 k)(1 - φ + k/) \n\n(1 + k)(γD (ad2p,/γD,pD,p)\n\n(59b)\n\nwhere, as before, φ is the fraction of total mobile phase that occupies interparticle space: φ = e,/(e, + e) = φ/(e/e) and (u,m) is the usual mean mobile-phase velocity (u,m) = L/tm. Note, that for lightly loaded porous particles e = 0, and the factor [e,(e, + e)] = = φ/(1 - φ), and Eq. (59b) becomes:\n\nH = c,m(u,m) = (1/30)kel/(1 + k)(1 - φ)φ/(1 + k)(ud2p,/γD,pD,p)\n\n(59c)\n\nThis equation has also been derived within the framework of the nonequilibrium theory after (1) considering the dimensionless parameters (i.e., 1/2(R/φ/(1 - φ) + k)/(1 - φ) + k)/(1 - φ) with R = 1/(1 + k).\n\nThe problem of describing combined internal partitioning and resistance to mass transfer in porous particles has been exhaustedly treated by Gruber [36]. A common assumption in flow processes is that the flowing fluid does reach the wall (zero velocity and so-called no-slip). Thus, for the mobile phase at the particles surfaces, Grubner introduced a stagnant film coating, with mobile phase around the porous particle, which is used as the interface between moving and stationary regions. Partitioning at this interface is governed by the partition coefficient K,u = c,m/e,0 over the volumes V,m and V,o, which approaches 1 for small molecules, as the interparticle solute concentration in the moving phase c,m equals that inside the porous particles in the intraparticle stagnant phase c,m. This is not true for larger molecules, which, as a result of their discrete sizes, will be partly excluded from intraparticle pore space by the principle of size exclusion chromatography (SEC) [2,37]. Hence, it is more general to allow explicitly for partitioning governed by size exclusion at the interface of the particle and provided much less than volume provided by a stationary fluid at the walls of the intraparticle pores, or by adsorption onto those walls. If we denote the associated partition coefficient with K*, the resulting plate height for the total diffusion-controlled process for spherical particles, as obtained by Gruber, using the method of statistical moments, as introduced by Aries [17], reads in our notation:\n\nH = A + B/u0 * (1/30)k0/(1 + k0)2[φ/K,m/(1 - φ)](ud2p/D,k) + kinetic terms (59d)\n\nwhich is identical with the foregoing results for the C term (see Eqs. (59b), (c) for K,m = 1. However, if exclusion occurs and K,m = K, SEC < 1, while at the same time K, = 1 is assumed (no sorption at the porous particles, as is the ideal case for SEC), we have\n\nH,SEC = c,SEC(u,m) = (1/30)k0/(1 + k0)(1 - φ)φ/(ud2p/D,k)\n\n(59e)