Chromatography Handbook Of Hplc - Katz Eksteen Schocnmakers Miller wiley Sons -104

94\nTijssen\nif valid, it is to be used only for high Reynolds numbers much larger than 10 [15], whereas for LC, in particular, but also for GC, the velocity working range leads to Re<, 1. Better correlations, supported by theoretical experiments have since become available [15], and these would yield quite different velocity dependencies; namely, proportionally with v^2/3 [as found by Horvath; see Eq. (49)] rather than with v^(1/2). Hence, it is no surprise that based on careful experiments [see, e.g., Ref. 21] the Huber equation can be discarded. Only occasionally, over a relatively small velocity range, one can obtain a satisfactory fit of experimental data with a Huber-type equation of the form:\nH = A + B/(u_m) + C(u_m)^(1/2) (51)\nwhere it should be questioned whether the coefficients have any physical meaning in the light of the derivations given in the foregoing. The C\' term reflects contributions from mass transfer in the stationary phase and for porous materials incorporates stagnant mobile-phase regions inside the particles of the packing. Because the Horvath equation is very similar to the Huber equation [see Eq. (49)], except for the improved exponent of the power function of v, the same reasoning applies to Eq. (49). Yet, especially for nonretained solutes, or in the absence of a stationary phase and at not too elevated velocities, the Horvath equation is almost identical with the Knox equation. It has been used in several theoretical studies (e.g., in supercritical fluid chromatography: SFC [33]), and was in the practice of LC [26,28]. Because this equation is the most extended one, also offering a term for intraparticle stagnant mobile-phase phenomena and a kinetic term representing resistance for solute binding or adsorption, we will give the extended Horvath and Lin equation here for completeness and in the dimensional form to see the details:\nH = 2γD_m^3/(u_0)\n + 2AD_m^μ/(u_0^3) + αD_m^d/(u_0^3) (axial molecular diffusion)\n + β(u + k + ψ(k)D_m^2/(u_0))(1 + ψ)D_m/(convection dispersion)\n + (u + k + ψ)D_m^2/(u_0)(1 + ψ)(1 + k)D_m (resistance at interface of moving and stagnant mobile phase)\n + (u + k + ψ)D_m^2(u_0/30)D_m(1 + ψ)(1 + k) (intraparticle diffusion resistance)\n + 2k_h(1 + ψ)(1 + k)r_p/k_s (kinetic resistance for adsorptive solute binding)\n(49a)\nHere the new symbols have the following meaning:\nγ = θ(μ), where θ is an alternative tortuosity factor, and ω, α, and β are additional structural packing parameters.\n\nϕ = (e_e/e_g) = [(1/φ) - 1] is another porosity fraction representing the ratio of intraparticle and interstitial mobile phase volumes (V_m/V_o).\nϕ_s is the phase ratio defined as the concentration of surface adsorption sites onto the packing per unit volume of mobile phase.\nk_s is the rate constant for adsorption.\n\nNote further the correct use of the interstitial velocity u_0 = L/t_0 of actual moving mobile phase (see Sec. II.A).\nAccording to Horvath and Lin [25], for LC systems with particle sizes smaller than 3–5 μm, the kinetic spurion term will outweigh all the other terms. This conclusion is rather uncertain, however, for the mechanism and kinetics of solute sorption onto and from the stationary phase are largely unexplored. Giddings’ treatment of the underlying kinetics yields different results [15; see also Eq. (33e)] and experimental verification of the Horvath equation, on the whole, has no favorable outcome for this relation [21,31,35].\nEqually uncertain in their connection to the underlying physical phenomena, but far more practical, are the equations of Snyder and Knox [Eqs. (47) and (48), respectively], which