Solvent velocity

HETP of a TLC plate is taken as the ratio of the distance traveled by the spot to the plate efficiency. The same three processes cause spot dispersion in TLC as do cause band dispersion in GC and LC. Namely, they are multipath dispersion, longitudinal diffusion and resistance to mass transfer between the two phases. Due to the aforementioned solvent frontal analysis, however, neither the capacity ratio, the solute diffusivity or the solvent velocity are constant throughout the elution of the solute along the plate and thus the conventional dispersion equations used in GC and LC have no pertinence to the thin layer plate. 

HPLC columns with reduced diameters (microbore columns) are now available. The flow rate from such columns required to give a desired flow rate at the same linear solvent velocity (and thus retention time) as a 4.6 mm i.d. column operating at 1 mlmin is given by the following equation ... 

Flow rate in 4.6 mm Flow rate in small bore column at column, cm3 min-1 the same solvent velocity, p min-1... 

This expresses tR as a function of the fundamental column parameters t0 and k tR can vary between t0 (for k = 0) and any larger value (for k > 0). Since to varies inversely with solvent velocity u, so does tR. For a given column, mobile phase, temperature, and sample component X, k is normally constant for sufficiently small samples. Thus, tR is defined for a given compound X by the chromatographic system, and tR can be used to identify a compound tentatively by comparison with a tR value of a known compound. 

For example, we showed in Chapter 2 that that the solvent velocity, V, was proportional to the acoustic pressure, P, (Eq. 2.11), and that the acoustic pressure was proportional to the ultrasonic intensity, I (Eq. 2.13). 

If we further assume that the solute motion can be neglected and that the solvent velocity autocorrelations are independent of the presence of the solute, we get in the case of (quasi)linear solvent molecules... 

Let the solvent move such that its velocity is v°(r) if there are no particles present, though v° (r) need not be constant but could, for instance, be produced by bulk flow of the solvent in tubes or around obstacles. Now the particle is located at r - and moves with a velocity r, — v(r ) relative to the solvent velocity, v(r ), when the particle is absent. In a Newtonian liquid, this velocity difference between particle and solvent leads to a force between liquid and particle which is given by... 

Since the complications due to solvent structure have already been discussed, the remainder of this chapter is mainly devoted to a discussion of the complications introduced into the theory of reaction rates when the collision of solvent molecules does not lead to a complete loss of memory of the molecules about their former velocity. Nevertheless, while such effects are undoubtedly important over some time scale, the differences noted by Kapral and co-workers [37, 285, 286] between the rate kernel for reaction estimated from the diffusion and reaction Green s function and their extended analysis were rather small over times of 10 ps or more (see Chap. 8, Sect. 3.3 and Fig. 40). At this stage, it is a moot point whether the correlation of solvent velocity before collision with that after collision has a significant and experimentally measurable effect on the rate of reaction. The time scale of the loss of velocity correlation is typically less than 1 ps, while even rapid recombination of radicals formed in close proximity to each other occurs over times of 10 ps or more (see Chap. 6, Sect. 3.3). 

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

The solvent velocity profiles are shown in Fig. 6. Away from the brush, the velocity profiles are linear as expected. From this region, the shear rate y can easily be determined from the slope of v(z). In the vicinity of the brush, the solvent velocity decays as expected from the discussion above. Although the chains are... 

Fig. 6. Solvent velocity profile (points) and brush density profile (curves) for a polymer brush under shear flow for two values of pa. The chain length of the brush is AT=100, while that of the solvent is Nj=2. Results are shown for vw=0.02ct/t (solid line and triangles) and 0.2cr/r (dashed line and squares). p(z) has been multipled by a factor of 5 and 2.5 respectively, to put the data on the same scale. From ref. [63].

This identity means that the reference velocity co is equal to the free solvent velocities of both components. 

For dense suspensions of spherieal particles, an espeeially aecurate method ealled Stokesian dynamics has been developed by Bossis and Brady (1989). In Stokesian dynamics, one solves a generalized form of Eq. (1-40), in which the simple Stokes law for the drag force on sphere i, = — (x/ — v ), is replaced by a more accurate tensor expression that accounts for the hydrodynamic interactions—that is, the disturbances to the solvent velocity field produced by the relative motions of the other spheres. The Stokesian dynamics method accounts for hydrodynamic interactions among widely separated spheres by a multipole expansion, as well as for closely spaced ones by a lubrication approximation. Results from this method appear in Figs. 6-8 and 8-8. 

Suppose a force, F, is exerted by a bead on the Newtonian solvent at the origin. This force sets the surrounding solvent in motion away from the origin at a point r the solvent velocity, calculated from the Stokes equation, reaches the steady-state value... 

The dynamic viscosity r](cai) is conveniently defined under alternating shear-stress conditions. Assuming the fluid velocity along x and the velocity gradient along z, the solvent velocity at the position of the hth atom in the absence of the chain is... 

To reduce eddy diffusion, the size of the resin particles, d can be varied. In high resolution work small resin particles are used. This has the disadvantage of restricting the solvent velocity, V, which can. be obtained with pressures that do not distort the particles. So, when high resolution is not needed larger particles may be used. With all columns, better results are obtained if a uniform particle size is used, and it is necessary to remove the very fine particles sometimes produced during the washing procedures so that a reasonable flow rate may be obtained and a narrow particle size distribution used. 

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