Chromatography Handbook Of Hplc - Katz Eksteen Schocnmakers Miller wiley Sons -103

Mechanisms and Importance of Zone-Spreading 93\n\ning the potentially concave curvature of plate height plots, is equally well explained by Eq. (43). In a more simple representation we may write Eq. (43) in a dimensional form related to the van Deemter equation, still reflecting the velocity and partitioning dependence of the mobile-phase terms, now grouped together in the C\\_{(u_{m})k} term:\n\nH = A + B/(u_{m}) + C\\_{c}(u_{m})k + C\\_{(k)}(u_{m})\n\n(d) Pragmatic Approximate Equations for Plate Height. In view of the relative complexity mainly of the expressions for plate height stemming from mobile-phase effects, chromatographic workers, especially in the field of LC, have tried over the years to simplify matters with some success, often based on empirical relations for extrapartic mobile-phase mass transfer in technological and chromatographic sciences.\n\nTable 3 gives an overview of equations describing the mobile phase effects that appeared since 1956, including the ones discussed in the foregoing, in a dimensionless form for easy comparison.\n\nIdeally, a, b, and c are constants related to the reduced coefficients present in eddy diffusion, molecular diffusion, and mobile-phase dispersion, but in using these functions as fitting functions to experimental data, they loose their physical meaning.\n\nTo compute the corresponding equations, the stationary-phase contribution, to be represented by a C term proportional in v, so c\\_v, should be added to each of the equations to obtain total reduced plate height. In view of a potential C\\_{m} term, representing the effect of stagnant mobile phase within porous packings (see Sec. II.E.4.g), both the Huber equation and the Horvath equation are supplied with further terms to be represented as dv^2 (Huber) and dv^3 + v (Horvath). For example, the complete Huber equation reads:\n\nh = b + a/(1 + (a/e)(v^{-2}) + c\\_v + dv^2\n\nThis equation is largely based on correlations for mass transfer in packed bed systems as provided by physical chemistry literature [15]. The second term can be both a correlation after Hiby [19], which has been already cited in Sec. II.E.4.b:\n\n1/P_e = y_o/R_e Sc + λ_{1}/[1 + λ_{2}/(R_e Sc)^{1/2}] λ_{1} = 0.65, λ_{2} = 7.0\n\nWe have already mentioned that the plot 1/P_e versus Re,Sc, the analog of the H versus (u) plot in chromatography, is curved by virtue of the velocity-dependent denominator in the Hiby correlation. Qualitatively this corresponds to the observation of concave 1/ v versus (u) plots, notably in LC. Quantitatively, however, although Eq. (50) was obtained for both gases and liquids, later results question the accuracy especially for liquids [14]. In addition, a further dimensionless correlation for the mass-transfer coefficient in the process of equilibration between phases, leading to the dv^2 term, was used by Huber; namely, the Thoenes correlation. The latter is known to exhibit a large uncertainty (a factor of 4 is no exception) and, moreover,\n\nTable 3 Equations Describing Plate Height Relations for Mobile-Phase Effects\n\nAuthors | Equation (number) | Ref.\nvan Deemter et al. | h = a + b/v + c*v | (38a) 7.20\nGiddings | h = b/v + a/(1 + (a/e)(v^{-1}) | (39c) 1.5\nHuber and Hulsman | h = b/v + a/(1 + (a/e)(v^{-2}) | (46) 22\nSic and Rinders, Johnson, | h = a + b/v + a_{p}/(1 + (a_{p}/c)(v^{-1}) | (43c) 11.13\nand Littlewood\nSnyder | h = c*v, with 0.3 < n < 0.7 | (47) 23\nKnox et al. | h = b/v + e/v^3 | (48) 24\nHorvath and Lin | h = b/v + a/(1 + (a/e)(v^{-1}) | (49) 25