Mobile phase mass transfer

In lithium-ion batteries substances should be used as cathode material which can intercalate and discharge lithium ions at a highly positive potential - compared to the intercalation into the carbon anode - and with only low kinetic hindrance, i.e. at low over-voltage or nearly reversible. The first requirement is fulfilled especially by transition metal oxides and halides and also, to a lesser extent, by sulfides. The second requirement of low kinetic hindrance for insertion and release of lithium ions is meant as a requirement of high mobility of lithium ions and electrons within the cathodic lattice and of unhindered mass transfer across phase boundaries as far as phase transitions happen in the host lattice during in- and excorporation of lithium. As the transition metal halides are poorer electronic conductors than oxides, only the latter are used in practice. 

Parameter in Knox Eqn (11) relates to packed bed Parameter in Knox Eqn relates to diffusion in mobile phase Parameter in Knox Eqn relates to mass transfer between phases Mobile phase concentration at the kih. time step and /th distance step Mobile phase concentration of solute Mobile phase concentration of solute /... 

To determine how the height of a theoretical plate can be decreased, it is necessary to understand the experimental factors contributing to the broadening of a solute s chromatographic band. Several theoretical treatments of band broadening have been proposed. We will consider one approach in which the height of a theoretical plate is determined by four contributions multiple paths, longitudinal diffusion, mass transfer in the stationary phase, and mass transfer in the mobile phase. 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

Kovat s retention index (p. 575) liquid-solid adsorption chromatography (p. 590) longitudinal diffusion (p. 560) loop injector (p. 584) mass spectrum (p. 571) mass transfer (p. 561) micellar electrokinetic capillary chromatography (p. 606) micelle (p. 606) mobile phase (p. 546) normal-phase chromatography (p. 580) on-column injection (p. 568) open tubular column (p. 564) packed column (p. 564) peak capacity (p. 554)... 

The dispersion of a solute band in a packed column was originally treated comprehensively by Van Deemter et al. [4] who postulated that there were four first-order effect, spreading processes that were responsible for peak dispersion. These the authors designated as multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase and resistance to mass transfer in the stationary phase. Van Deemter derived an expression for the variance contribution of each dispersion process to the overall variance per unit length of the column. Consequently, as the individual dispersion processes can be assumed to be random and non-interacting, the total variance per unit length of the column was obtained from a sum of the individual variance contributions. 

Dispersion Due to Resistance to Mass Transfer Resistance to Mass Transfer in the Mobile Phase... 

Dispersion caused by the resistance to mass transfer in the stationary phase is exactly analogous to that in the mobile phase. Solute molecules close to the surface will leave the stationary phase and enter the mobile phase before those that have diffused further into the stationary phase and have a longer distance to diffuse back to the surface. Thus, as those molecules that were close to the surface will be swept along in the moving phase, they will be dispersed from those molecules still diffusing to the surface. The dispersion resulting from the resistance to mass transfer in the stationary phase is depicted in Figure 8. 

When u E, this interstitial mixing effect was considered complete, and the resistance to mass transfer in the mobile phase between the particles becomes very small and the equation again reduces to the Van Deemter equation. However, under these circumstances, the C term in the Van Deemter equation now only describes the resistance to mass transfer in the mobile phase contained in the pores of the particles and, thus, would constitute an additional resistance to mass transfer in the stationary (static mobile) phase. It will be shown later that there is experimental evidence to support this. It is possible, and likely, that this was the rationale that explains why Van Deemter et al. did not include a resistance to mass transfer term for the mobile phase in their original form of the equation. 

In 1967, Huber and Hulsman [2] introduced yet another HETP equation having a very similar form to that of Giddings. Their equation included a modified multipath term somewhat similar in form to that of Giddings and a separate term describing the resistance to mass transfer in the mobile phase contained between the particles. The form of their equation was as follows ... 

The ratio of the resistance to mass transfer in the mobile phase to that in the stationary phase (Rms) will indicate whether the expressions can be simplified or not. Now, (Rms) will be given by. 

It should be pointed out, however, that the diffusivity of the solute in the mobile phase can be changed in two ways. The solute that is chromatographed can be changed, in which case the above assumptions are clearly valid, as (Ds) is likely to change linearly with (Dm)- However, the solute diffusivity can also be changed by the employing a different mobile phase. In this case, (Dm) will be changed but (Ds) will remain the same. In the second case, the above assumptions are not likely to be precisely correct. Nevertheless, if the resistance to mass transfer in the stationary phase makes only a small contribution to the overall value of (H) (i.e., because df dp (see equation (l)),then the assumption Dm = eDg will still be approximately... 

It is seen that the Van Deemter equation predicts that the total resistance to mass transfer term must also be linearly related to the reciprocal of the solute diffusivity, either in the mobile phase or the stationary phase. Furthermore, it is seen that if the value of (C) is plotted against 1/Dni, the result will be a straight line and if there is a... 

Now, it is of interest to determine if either the resistance to mass transfer term for the mobile phase or, the resistance to mass transfer term in the stationary phase dominate in the equation for the variance per unit length of a GC packed column. Consequently, taking the ratio of the two resistance to mass transfer terms (G)... 

Thus as (y) will always be greater than unity, the resistance to mass transfer term in the mobile phase will be, at a minimum, about forty times greater than that in the stationary phase. Consequently, the contribution from the resistance to mass transfer in the stationary phase to the overall variance per unit length of the column, relative to that in the mobile phase, can be ignored. It is now possible to obtain a new expression for the optimum particle diameter (dp(opt)) by eliminating the resistance to mass transfer function for the liquid phase from equation (14). 

A liquid mobile phase is far denser than a gas and, therefore, carries more momentum. Thus, in its progress through the interstices of the packing, violent eddies are formed in the inter-particular spaces which provides rapid solute transfer and, in effect, greatly increases the effective diffusivity. Thus, the resistance to mass transfer in that mobile phase which is situated in the interstices of the column is virtually zero. However, assuming the particles of packing are porous (i.e., silica based) the particles of packing will be filled with the mobile phase and so there will... 

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