Chromatography Handbook Of Hplc - Katz Eksteen Schocnmakers Miller wiley Sons -116

106 Tijssen\n\nwhere k0 = KSECψ(1 + ϕ = εs/εo = Vrn/V0 is the alternative void fraction introduced before in the Horvath equation [see Eq. (49a)] and representing the ratio of intraparticle and extraparticulate (interstitial) pore volume: = (1/φ) - 1 or φ = 1/(1 + ψ). As a result we may write:\n\nHSEC = CSECψ(1 + KSECψ)1b*(<u>md2/Drp)\n(59f)\n\na well-known relation for SEC zone dispersion, generally cited in the literature [37]. For completeness, we note that the Horvath equation (see Eq. (49b)) which contains the Cmr term expressed in terms of ψ rather than in φ is exactly identical with the present results, [see Eqs. (59a-f)], and it shows the same dependence on particle size, velocity, and diffusivity as all earlier expressions, including the numerical factor 1/30.\n\nWithout going into the mathematical details of the derivation of Eqs. (59), we can obtain some insight into its character, by looking at the very basis of the phenomenon of partitioning between inter- and intraparticle void space. So much is clear, that slow (strongly hindered) intraparticle diffusion with a diffusion coefficient Deff = μy,yD,op occurs in the series of particles, which are assumed to be spherical. In a three-dimensional sphericalgeometry, the random-walk approach for diffusion, leading to the relation in Eq. (7a), which is in the one-dimensional case, is slightly altered because three-dimensional diffusion takes place in rather than two directions; namely, in the x, the y, and the z directions. For a number of N = τ steps of length λ, with step time t and occurring with equal probabilities (1/6) for all directions, the mean square displacement should be equal for all directions, so (x2) = (y2) = (z2) = Nλ2/3. The diffusion coefficient D remains by definition (see Sec. II.C.1) the same for all directions; for example (x2) = 2Dt and so the mean square spatial distance (r2) = (x2) + (y2) + (z2) = Nλ2 = 6Dt, and the three-dimensional diffusion coefficient can be factorized to instead 2 and reads: D = (r2)/6t = Nλ2 = λ2/67 [21].\n\nHence, for a step length of the order of the radius of a spherical particle, the characteristic time τm = R2/10yD,ip = 3/2Dg,y,yD,ip\nwhich is to be considered a characteristic equilibrium time between moving and stagnant regions of the mobile phase. The associated time τ0, the step time spent in the moving interstitial region, is then found from the partitioning of solute molecules via a partition coefficient of 1 over the interstitial and intraparticle mobile-phases (i.e., τ0/τm = φ/(1 - φ)). The associated particle height contribution from this zone-broadening effect is found from using Eq. (11) for a general velocity difference. In the present situation, the latter is determined by the difference between the interstitial (maximum) velocity u0 = (φuM) and the zone velocity, and reads uL = u0 - u. Hence, the step length of molecules in mobile mobile phase ℓm = w0/(1 + k) = ℓ(1 + k)φ/(1 + ϕ) Following the same reasoning that led to Eq. (33), a molecule chooses in N = 2L/m1 − φ/(1 + k) forward or backward relative to the zone center, and the associated spreading after the random-walk rule Eq. (7) is σ2: = Nλ2 = 2L(m1 − 1 - φ/(1 + k)), from which:\n\nHmn = 2ε(m1) - {φ(1 + k)2 = 2[1 − φ/(1 + k)]}u0τ0\n = 2[1 - (1 + k)φ](uMmd/τm - (1 - ε)\n = {1 - φ/(1 + k)}(uMmd/2y0yD,ip = cm,μm)\n(61)\n\nwithin a factor of 4/3 (1/40 instead of 1/30) equal to the exact result of Eq (59c). The difference in the constant may be attributed to the fact that the constant 1/40 has been derived