Chromatography Handbook Of Hplc - Katz Eksteen Schocnmakers Miller wiley Sons -100

relatively large because of small \\( \\Omega = R_d / d_p \\) values (R is small) and approximates that in open tubular columns (see Sec. II.E.4; i.e., \\( k_{inert} = 1/50 \\).\n\nThis is large compared with normal-sized (ID > 2-mm) columns but the reduction in plate height from the small diameters (factor 2, in the \\( C_{t,mm} \\) more than offsets the increased \\( k \\) value, which makes this type of column more efficient than conventional (up to 6-mm-ID) columns, and allows us to still attain good efficiencies at high speeds. Note, that for higher velocities, the convective diffusivity becomes ever more important and for very high velocities the velocity profile \\( C_v(u) \\) term does become independent of the velocity and levels off to a plateau value of \\( H(u \\to \\infty) \\to 2kR^2 \\cdot D_{z,app} \\cdot \\alpha_d \\). It is seen that zone-broadening by the velocity differences is entirely determined by the hydrodynamics of the packed bed, the typical condition in which the classic concept of eddy diffusion has its full physical meaning.\n\nThis is exceptional in GC (because of the fast molecular diffusion), but in LC it is easier to attain. Here, for typical LC conditions (\\( d_p = 5 \\ \\mu m = 5.10^{-4} \\) cm, and \\( \mu \\) = 0.2 cm/s), convective radial diffusivity will be about 2.5 \\( \\times 10^{-10} \\) cm²/s, not too different from the molecular diffusivity in liquids (see Table 2). Indeed, at the transition velocity of \\( u_{max} = \\gamma D_{z,app} / \\alpha_d \\) = 0.5 cm/s the two processes become equivalent, and the typically curved plate height plot is obtained. This is most clearly seen in the dimensionless representation of the mobile phase parameter \\( H_m \\) in Eq. (40):\n\n\\[ h_m = H_d / d_p = 2k(2/\\alpha_d + \\gamma / v) \\approx 0.11(\\alpha_d + \\gamma / v) \\] (42)\n\nThe total plate height can now be represented as:\n\n\\[ H = H_c + H_d + H_a + B + \\left(\\frac{1}{[(1/\\alpha_d) + (1/C_{t,mm})]}\\right) + C_t(u_m) \\] (43a)\n\nor in reduced parameters:\n\n\\[ h = H / d_p = 2\\alpha_{app} + 2\\gamma / v + f(k)2k^2\\alpha_{app}/\\gamma + (2/3)g(k)dD_v \\] (43b)\n\nwhere f(k) and g(k) are functions of k, replacing f(k) and g(k) = (3/2)g(k) in the foregoing. Figure 5 shows the reduced plate height plot and the different contributors to total plate height. In the normal working range for GC (\\( u_m \\)) is far smaller than 100 cm/s (i.e., \\( v << 100 \\)), whereas for LC (\\( u_m \\)) is in the range of 0.1–1 cm/s and 50 < v < 100. These are clearly different working ranges in the dimensionless plot. In LC the molecular diffusion is seldom significant, whereas in GC, the convective dispersion region is seldom in operation.\n\nThis does not mean, however, that in LC curved H versus \\( \mu \\) plots should necessarily occur. This very much depends on the choice of experimental conditions and parameters and, in actuality, a van Deemter type of efficiency plot can often be observed (see, e.g., Ref. 21).\n\nIn our derivation, the convective dispersion term is not dependent on the partition equilibrium and, thus, should not contain a function of k. However, this is only true if the stationary phase is distributed very evenly throughout the column cross section. This may not always be true in practice. Indeed, experimentally one may observe a slight k-dependence of the mobile phase convective dispersion term as predicted in the extended van Deemter equation. For that reason, we added the function f(k) in Eq. (43b), which is still to be determined. Presently we remark that it is not uncommon, not even in LC [21], to use the result for f(k)