Longitudinal molecular diffusion

Despite such complications of detail, the random-walk model describes the essence of chromatographic zone spreading. It properly accounts for the way in which all major experimental parameters influence the broadening process. 

The symmetric random walk used here is one in which forward and backward steps are equally probable. All positive and negative steps are considered as displacements with respect to the zone center. Therefore in chromatography where the zone as a whole is in a state of motion, we must consider all displacements with respect to a coordinate system moving with the zone center at velocity Y = Rv. 

We will return now to the three independent random processes that underlie migration and induce zone spreading molecular diffusion, sorption-desorption, and flow-diffusion processes in the mobile phase. 

The ceaseless thermal motion of molecules will cause each of them to move in a series of random steps up and down the flow axis. It is possible to characterize this motion in terms of step length and step number, but since in this case the process is nothing but molecular diffusion, a more universal reference parameter is the diffusion coefficient D. Following this line of attack the variance can be obtained directly from Eq. 5.35 in which overall diffusion coefficient DT is replaced by D 

Diffusion in the mobile phase, particularly in GC, dominates the longitudinal diffusion process. In this case the molecular diffusion coefficient D will be replaced by the specific mobile phase value Dm. The diffusion time t will be replaced by the time spent in the mobile phase in the process of migrating through the column, traveling distance L. Since the average velocity while in the mobile phase is u, this time is equal to LIv. With these substitutions, the variance becomes 

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Ordinary diffusion along the length of the path taken by a packet of molecules (called Longitudinal Molecular Diffusion). 

If a band of molecules or atoms are placed in a container such as a tube at the center of its length, the molecules will diffuse in the direction of lower concentration. Clearly this occurs in all directions but in a tube the walls are a limit and so our concern is along the axis of movement caused by the mobile phase and this is referred to as longitudinal molecular diffusion. 

Here, A is the random path, B the longitudinal molecular diffusion and C the RTMT contributions with the velocity of the mobile phase u shown separately. H is referred to as the Height Equivalent to a Theoretical Plate and is terminology borrowed from distillation. While the distillation HETP is not truly applicable, the terminology has persisted. It can be shown that the H in this expression is the equivalent to variance/unit length. This is the expression introduced by Van Deemter and co-workers in 1956 in a discussion of band broadening. 

B = Longitudinal molecular diffusion in both mobile and stationary phases, and C = Kinetic or mass transfer term originating in the stationary phase. 

Three main independent contributions to band spreading inside the column have been identified (32) as longitudinal molecular diffusion, the unevenness of flow through the nonhomogeneous packing, and the resistances to mass transfer in the mobile and stationary phases. 

The term B in equation (1.27) is related to the longitudinal molecular diffusion in the column. It is especially important when the mobile phase is a gas. This term is a consequence of entropy, telling us that a system will tend towards the maximum degrees of freedom as demonstrated by a drop of ink that diffuses after falling into a glass of water. Hence, if the flow rate is too slow, compounds being separated will mix faster than they will migrate. This is why one must never interrupt the separation process, even momentarily. 

Longitudinal molecular diffusion. Solute molecules are engaged in ceaseless Brownian motion, which is responsible for diffusion. The component of this erratic motion along the column axis, superimposed on the downstream displacement caused by flow, is one source of zone broadening. 

We have already seen the solution for the variance of the zone profile in open isothermal columns - cf. Eq. 4.26. Usually, its right-hand side term which accounts for longitudinal molecular diffusion can be neglected. Equating of the right-hand sides of Eqs. 4.26 and 4.27 provides the effective value of / to serve the Monte Carlo simulations ... 

The A term corresponds to the eddy diffusion which describes the irregular flow through the packed particles in a column causing different pathways and different exit times for the solute molecules. The B term is the longitudinal molecular diffusion or random diffusion along the column. The last term C, corresponds to the mass transfer in the stationary phase. This mass transfer occurs between the mobile and stationary phase of the chromatographic system and is dependant on several factors such as particle size, column diameter and diffusion coefficient. 

Term 5, which can be expressed from Dq, the diffusion coefficient of the analyte in the gas phase and A, the above packing factor, is related to the longitudinal molecular diffusion in the column. It is especially important when the mobile phase is a gas. 

Axial or longitudinal molecular diffusion, and (2) mass transfer processes. 

Axial or longitudinal molecular diffusion It is the migration of solute from regions of higher to those of lower concentrations in the direction of the column. Ordinary diffusion occurs when solute molecules randomly jump back and forth between successive collisions or equilibrium positions. 

The condition for applicability of this equation follows from the condition that Taylor s longitudinal diffusion by far exceeds the longitudinal molecular diffusion, that is, D. This condition, with account taken of (6.131) and (6.127), gives the final condition for applicability of the considered solution ... 

The A, B and C terms are related to flow anisotropy, molecular longitudinal diffusion and mass transfer processes, respectively. The theoretical support for the Knox equation was derived by Horvath [12]. The A term cannot be expressed simply. The theoretical treatment links A to structural parameters of the column packing, porosity, pore volume, pore diameter and tortuosity [12]. A is related to the flow pattern and the general band spreading due to "eddy" diffusion [13]. The B term (longitudinal molecular diffusion) was written as [13] ... 

When the term longitudinal diffusion is applied to the chromatographic band, we include the true longitudinal molecular diffusion (Section 2.3.2) and... 

A is related to eddy diffusion, B to longitudinal molecular diffusion (in the mobile phase), and C to mass-transfer resistance (lateral diffusion in the stationary phase). 

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