Download (direct link):
The Huber Equation
The next HETP equation to be developed was that of Huber and Hulsman in 1967 (2). These authors introduced 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:-
H - —V + - + Cu + Du1/2 ......................................(2)
It is seen that the first term differs from that in the Giddings equation, in that it now contains the mobile pnase velocity to the power of one half. Nevertheless, again when ul/2 » E, the first term reduces to a constant similar to the Van Deemter equation. The additional term for the resistance to mass transfer in the mobile phase is an attempt to take into account the
'turbulent mixing' that takes place between the particles. Huber's equation implies but, in fact, was not explicitly stated by the authors, that the mixing effect between the particles (that reduces the magnitude of the resistance to mass transfer in the mobile phase) does not commence until the mobile phase velocity approaches the optimum velocity (as defined by the Van Deemter equation) Furthermore, it is not complete until the mobile phase velocity is well above the optimum velocity. Thus, the shape of the HETP/u curve will be a little different from that predicted by the Van Deemter equation.
The form of the HETP curve that is produced by the Huber equation is shown in figure (I).
H versus u Curves Resulting from the Huber Equation
Linear Velocity (cm/sec)
It is seen that the composite curve obtained from the Huber equation is indeed similar to that obtained from that of Van Deemter but the individual contributions to the overall variance are different. Although the contributions from the resistance to mass transfer in the mobile phase and longitudinal diffusion are common to both equations, the (A) term from the Huber equation increases with mobile phase flow-rate and only becomes a constant value, similar to the multipath term in the Van Deemter equation, when the mobile velocity is sufficiently large. In practice, however, it
would seem that the magnitude of the mobile velocity, where the (A) term gives a constant value, Is quite low, relative to the normal range of operating velocities employed In practical LC. The portion of the composite curve shown at the higher velocities is not quite linear due to the non-linear form of the term for the resistance to mass transfer in the mobile phase and this becomes more apparent at higher mobile phase velocities. At, and around the optimum velocity, however, the form of the two curves differ only slightly.
The Knox Equation
During 1972 and 1973 Knox and his co-workers (3), (4), and (5) carried out a considerable amount of work on different packing materials with particular reference to the effect of particle size on the reduced plate height of a column. The concept of reduced plate height (/?) and reduced velocity (v) was introduced by 6iddings(6) and (7) in 1965 in an attempt to form a basis for the comparison of different columns packed with particles of different diameter. The reduced plate height is defined as,
Ë-? ................................................................ (3)
In fact the reduced plate height merely measures the normal plate height in units of particle diameters. It is also seen that the reduced plate height is dimensionless.
The reduced velocity Is defined as,
v = —^ ................................................................ (4)
The reduced velocity compares the mobile phase velocity with the velocity of the solute diffusion through the pores of the particle. In fact, the mobile phase velocity is measured in units of the Intraparticle diffusion velocity. As the reduced velocity is a ratio of velocities, like the reduced plate height, it also is dimensionless.
Employing the reduced parameters the equation of Knox takes the following form,
h = - + Av'/3+ Cv ......................................... (5)
It should be noted that the constants of the equation were arrived at by a curve fitting procedure and not derived theoretically from a basic dispersion model; as a consequence the Knox equation has limited use in column design. It is, however, extremely valuable in accessing the quality of the packing. This can be seen from the diagram shown in figure 2.
Graph or Log. Reduced Plate height against Log. Reduced Velocity Ãîã Poor and Well Packed Columns
The curves represent a plot of Log.(/?),(Reduced Plate height)against Log.(v), (Reduced Velocity). The lower the Þä.(Ë) curve versus the Log.(v) curve the better the column is packed. At low velocities the (B) term dominates and at high velocities the (C) term dominates as In the Van Deemter equation. The best column efficiency is achieved when the minimum is about 2 particle diameters and thus, Log (/?) is about 0.35. The minimum value for (H) as predicted by the Van Deemter equation has also been shown to be about two particle diameters. The optimum reduced velocity is in the range of 3 to 5 that is Log.(v ) takes values between 0.3 and 0.5. The Knox equation is a simple and effective method of examining the quality of a given column but, as stated before, is not nearly so useful In column design due to the empirical nature of the constants.