The HETP Equation An HETP equation quantitatively describes the peak dispersion (band spreading) by expressing the ‘variance per unit length’ of the column as a function of the linear mobile phase velocity and the various physical properties of the distribution system and the solute being eluted. The variance per unit length of the column is measured as the ratio of the column length to the column efficiency, i.e, the height of the theoretical plate or the HETP. There have been a number of different HETP equations developed, the first for packed columns was the Van Deemter equation which was followed by the Huber equation, the Giddings equation, the Horvath equation, the Knox equation and finally the Golay equation for capillary columns. All the equations, except the Knox equation, were developed from first principles, the Knox equation, however, was developed from experimentally observed relationships and subsequently rationalized on a first principle basis. All the HETP equations include functions that describe dispersion due to longitudinal diffusion and dispersion resulting from the resistance to mass transfer in both phases. All the HETP equations for packed columns include a function that describes dispersion that results from the tortuous paths taken by solute molecules as they wind their way through the intersticies of the packing. This dispersion process was given the term ‘eddy diffusion’ by Van Deemter With the exception of the Giddings equation, the function for eddy diffusion dispersion is expressed as a constant depending on the particle diameter of the packing and independent of the mobile phase linear velocity. Giddings however, introduced a coupling function to the eddy diffusion term so that the dispersion only became constant and independent of mobile phase velocity at relatively high velocities. The Golay equation, as there was no packing, only contained three terms, one for longitudinal diffusion and one each for the resistance to mass transfer in the mobile and stationary phases respectively. The Huber and Horvath equations contain an extra term that involves a power function of the mobile phase velocity, largely to account for the curving of the HETP graph towards the base line at very high mobile phase velocities. Subsequently, this curving effect was shown to be an artifact resulting from experimental problems that arose when measuring the elution curves of very fast eluting peaks. All the equations have hyperbolic characteristics relating variance per unit length to linear mobile phase velocity. This means that the variance per unit length-mobile phase velocity curve shows a minimum at a particular velocity which has been termed the optimum linear velocity. The column will exhibit a maximum efficiency for a specific solute when operated at the optimum velocity. In practice, it has been shown that the equation that best describes the experimental data from a packed LC column is probably the Van Deemter equation.
Author: RPW Scott
Book:Dispersion in Chromatography Columns
Section:Dispersion Compressibility-Effects-GC
Effect of Mobile Phase Compressibility On the HETP Equation for a Packed GC Column As the pressure falls along the column length, the velocity changes and, as the solute diffusivity depends on the pressure, the diffusivity of the solute will also change. The multi-path term, which contains no velocity or gas pressure dependent parameters, will be unaffected and the expression that describes it the same. The other terms in the HETP equation, however, all contain parameters that are affected by gas pressure (solute diffusivity and mobile phase velocity) and, therefore, need to be modified to accommodate the compressibility of the mobile phase. Reiterating the HETP equation for a packed column, where f1(k") = and f2(k&
Author: RPW Scott
Book:Dispersion in Chromatography Columns
Section:Dispersion Alternative-Equations Giddings
time constants etc.) than any inadequacies of the Van Deemter equation. Nevertheless, it was the poor agreement between theory and experiment at the time, that provoked a number of workers in the field to develop alternative HETP equations. This work was carried out in the hope that a more exact relationship between HETP and linear mobile phase velocity could be obtained that would be compatible with experimental data. The Giddings Equation In 1961, Giddings (16) developed an HETP equation of which the Van Deemter equation was shown to be a special case. Giddings work was not provoked by poor agreement between theory and experiment but because he was dissatisfied with the Van Deemter equation inasmuch that it predicted a finite contribution to dispersion, independent of the solute diffusivity, in the limit of zero mobile phase velocity. This concept, not surprisingly, appeared to him unreasonable and unacceptable. Giddings developed the following equation to avoid this
Author: RPW Scott
Book:Dispersion in Chromatography Columns
Section:Dispersion Alternative-Equations Huber
, that this was the reason 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. The Huber Equation The next HETP equation to be developed was that of Huber and Hulsman in 1967 (17). 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:- (47) It is seen that the first term differs from that in the Giddings equation, in that it now contains the mobile phase velocity to the power of one half. Nevertheless, again when u1/2 >> E, the first term reduces to a constant
Author: RPW Scott
Book:Principles and Practice of Chromatography
Section:Principles Peak-Dispersion Stationary-Phase
the Van Deemter equation that describes the variance per unit length of a column in terms of the physical properties of the column contents, the distribution system and the linear velocity of the mobile phase. Alternatively the Van Deemter equation can be expressed in the form, (10) where (H) is the Height of the Theoretical Plate. The relationship between 0 and (sx) is explained in The Plate Theory and Extensions . Hence the term "HETP equation" for equation (10). This form of the Van Deemter equation is very nearly correct for LC but, due to the compressibility of the gaseous mobile phase in GC, neither the linear velocity nor the pressure is constant along the column. Furthermore, as the diffusivity, (Dm), is a function of pressure, the above form of the equation can only be approximate. However, equation (10) generally gives the correct form of the relationship between (H) and the linear
Author: RPW Scott
Book:Dispersion in Chromatography Columns
Section:Dispersion Golay-Equation
Taking a value of 2.5 x10-5 for Dm (the diffusivity of benzyl acetate in n-heptane) equation (52) can be employed to calculate the curve relating (H) and (u) for an uncoated capillary tube. The results are shown in figure 17. It is seen that the Golay equation produces a curve identical to the Van Deemter equation but with no contribution from a multipath term. It is also seen that the value of (H) is solely dependent on the diffusivity of the solute in the mobile phase and the linear mobile phase velocity. It is clear that the capillary column can, therefore, provide a simple means of determining the diffusivity of a solute in any given liquid. The Golay equation (equation (52)) can be put in a simplified form in a similar manner to the equations for packed columns:- &
Author: RPW Scott
Book:Dispersion in Chromatography Columns
Section:Dispersion Experimental-Validation
these two particular solvent/stationary phase/solute systems. Overall, however, all the results in figures 23, 24 and 25 support the Van Deemter equation extremely well. J.Chromatogr.,270(1983)65. Figure 25. Graph of (C) Term against the Reciprocal of the Solute Diffusivity. Katz et al. (12) also examined the effect of particle diameter on the value of the overall resistance to mass transfer constant (C). They employed columns packed with 3.2 m, 4.4 m, 7.8 m, and 17.5 m, and obtained HETP curves for the solute benzyl acetate in 4.3%w/w of ethyl acetate in n-heptane on each column. The data was curve fitted to the Van Deemter equation and the values for the A, B and C terms for all four columns calculated. According to the Van Deemter equation the (C) term should be linearly related to the square of the particle diameter. A graph relating the value of the (C) term with the square of the particle diameter is shown in figure 26
