The Van Deemter Equation

The three components of plate height, Hg, and Hy may simply be added together to give a total plate height Hs, which describes the total peak broadening of the column  

The reason is that the plate height H stands for variances and that variances are additive. By substituting the individual parameters into Equation 2.17 and combining some of the elements, we may simplify as follows  

This is the simplest form of the van Deemter equation. It is quite clear from the graph (Fig. 48) that the total plate height H5 passes through a minimum the smallest possible plate height Hmipj, and hence minimum peak 

Deemter curve is much steeper than in the range where V l opt hereas if the carrier gas flow is too low, separation quickly deteriorates until it is no longer useful, V may frequently be increased to a value considerably greater than l opt. while still attaining the required value for resolution. It should be remembered that doubling i results in a halving of analysis time. 

In addition to the peak-broadening effects in the separation column, there are also those which may arise in dead volumes and turbulences in column 

Concentration profile of the band broadens due to longitudinal diffusion 

For these reasons, smaller-particle columns (i.e., 3 pm) are becoming popular in modern FIPLC because of their inherent higher efficiency.17 They are particularly useful in Fast LC and in high-speed applications such as high-throughput screening. In the last decade, sub-3-pm particles (i.e., 1.5-2.5pm) were developed for even higher column performance, as discussed in Chapter 3. 

The Van Deemter equation (1) was the first rate equation to be developed and this took place as long ago as 1956. However, it is only relatively recently that the equation has been validated by careful experimental measurement (2). As a result, the Van Deemter equation has been shown to be the most appropriate equation for the accurate prediction of dispersion in liquid chromatography columns, The Van Deemter equation is particularly pertinent at mobile phase velocities around the optimum velocity (a concept that will shortly be explained). Furthermore, as all LC columns should be operated at, or close to, the optimum velocity for maximum efficiency, the Van Deemter equation is particularly important in column design. Other rate equations that have been developed for liquid chromatography will be discussed in the next chapter and compared with the Van Deemter equation 

In fact, in the original form, equation (1), was introduced by van Deemter for packed 6C columns and consequently, the longitudinal diffusion term for 

Furthermore, as the equation was developed for GC, where the diffusivity of the solute in the gas was four to five orders of magnitude greater than in a liquid, Van Deemter considered the resistance to mass transfer in the 

The equation actually developed by Van Deemter took the form, 

Equation (3), however, was developed for a gas chromatographic column and in the case of a liquid chromatographic column, the resistance to mass transfer in the mobile phase should be taken into account. Van Deemter et al did not derive an expression for fi(k ) for the mobile phase and it was left to Purnell (3) to suggest that the function of (k ), employed by Golay (4) for the resistance to mass transfer in the mobile phase in his rate equation for capillary columns, would also be appropriate for a packed column in LC. The form of f (k ) derived by Golay was as follows, 

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An alternative form of this equation is the van Deemter equation... 

Plot of the height of a theoretical plate as a function of mobile-phase velocity using the van Deemter equation. The contributions to the terms A B/u, and Cu also are shown. 

There is some disagreement on the correct equation for describing the relationship between plate height and mobile-phase velocity. In addition to the van Deemter equation (equation 12.28), another equation is that proposed by Hawkes... 

Equation 16-183 is qualitatively the same as the van Deemter equation [van Deemter and Zuiderweg, Chem. Eng. Sci., 5, 271 (1956)] and is equivalent to other empiric reduced HETP expressions such as the Knox equation [Knox,y. Chromatogi Set., 15, 352 (1977)]. 

The silica dispersion showed the smallest retention volume. It should be noted, however, that the authors reported that the silica dispersion required sonicating for 5 hours before the silica was sufficiently dispersed to be used as "pseudo-solute". The retention volume of the silica dispersion gave the value of the kinetic dead volume, /.e., the volume of the moving portion of the mobile phase. It is clear that the difference between the retention volume of sodium nitroprusside and that of the silica dispersion is very small, and so the sodium nitroprusside can be used to measure the kinetic dead volume of a packed column. From such data, the mean kinetic linear velocity and the kinetic capacity ratio can be calculated for use with the Van Deemter equation [12] or the Golay equation [13]. 

The first alternative HETP equation to be developed was that of Giddings in 1961 [1] of which the Van Deemter equation appeared to be a special case. Giddings did not develop his equation because the Van Deemter equation did not fit experimental data. 

It is seen that when u E, equation (1) reduces to the Van Deemter equation. 

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. 

The curves represent a plot of log (h ) (reduced plate height) against log (v) (reduced velocity) for two very different columns. The lower the curve, the better the column is packed (the lower the minimum reduced plate height). At low velocities, the (B) term (longitudinal diffusion) dominates, and at high velocities the (C) term (resistance to mass transfer in the stationary phase) dominates, as in the Van Deemter equation. The best column efficiency is achieved when the minimum is about 2 particle diameters and thus, log (h ) is about 0.35. The optimum reduced velocity is in the range of 3 to 5 cm/sec., that is log (v) takes values between 0.3 and 0.5. The Knox... 

A number of HETP equations were developed other than that of Van Deemter. Giddings developed an alternative form that eliminated the condition predicted by the Van Deemter equation that there was a finite dispersion at zero velocity. However, the Giddings equation reduced to the Van Deemter equation at velocities approaching the optimum velocity. Due to extra-column dispersion, the magnitude of which was originally unknown, experimental data were found not to fit the Van Deemter... 

In the next chapter, experimental data supporting the Van Deemter equation will be described and discussed. Only data that has been acquired with equipment that has been specifically designed to eliminate, or reduce to an insignificant level, the dispersion sources described in this chapter can be used reliably for such a purpose. 

The results obtained were probably as accurate and precise as any available and, consequently, were unique at the time of publication and probably unique even today. Data were reported for different columns, different mobile phases, packings of different particle size and for different solutes. Consequently, such data can be used in many ways to evaluate existing equations and also any developed in the future. For this reason, the full data are reproduced in Tables 1 and 2 in Appendix 1. It should be noted that in the curve fitting procedure, the true linear velocity calculated using the retention time of the totally excluded solute was employed. An example of an HETP curve obtained for benzyl acetate using 4.86%v/v ethyl acetate in hexane as the mobile phase and fitted to the Van Deemter equation is shown in Figure 1. 

Stating the Van Deemter and Knox equations in the explicit form, the Van Deemter equation is... 

Now, equations (1) and (2) indicate that both the Knox equation and the Van Deemter equation predict a linear relationship between the value of the (B) term (the longitudinal diffusion term) and solute diffusivity. 

Curves relating the optimum velocity to the solute diffusivity are shown in Figure 6. It is seen that the straight lines predicted by the Van Deemter equation are realized for both solutes. 

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... 

In Figure 7, the resistance to mass transfer term (the (C) term from the Van Deemter curve fit) is plotted against the reciprocal of the diffusivity for both solutes. It is seen that the expected linear curves are realized and there is a small, but significant, intercept for both solutes. This shows that there is a small but, nevertheless, significant contribution from the resistance to mass transfer in the stationary phase for these two particular solvent/stationary phase/solute systems. Overall, however, all the results in Figures 5, 6 and 7 support the Van Deemter equation extremely well. 

Katz et al. [1] also examined the effect of particle diameter on resistance to mass transfer constant (C). They employed columns packed with 3.2 im, 4.4 p,m, 7.8 pm, and 17.5 pm, 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 were curve fitted to the Van Deemter equation and the values for the A, B and C terms for all four columns extracted. A graph relating the value of the (C) term with the square of the particle diameter is shown in Figure 8. 

It follows, that if the Van Deemter equation is correct, a graph relating. 

In summary, it can be said that all the dispersion equations that have been developed will give a good fit to experimental data, but only the Van Deemter equation, the Giddings equation and the Knox equation give positive and real values for the constants in the respective equations. 

The Van Deem ter equation appears to be a special case of the Giddings equation. The form of the Van Deemter equation and, in particular, the individual functions contained in it are well substantiated by experiment. The Knox equation is obtained... 

It would appear, from the data available at this time, that the Van Deemter equation would be the most appropriate to use in column design. 

For convenience the Van Deemter equation for the LC column will be generally used in the following argument. However, when GC columns are considered the equations will be appropriately modified, where... 

Dispersion equations, typically the van Deemter equation (2), have been often applied to the TLC plate. Qualitatively, this use of dispersion equations derived for GC and LC can be useful, but any quantitative relationship between such equations and the actual thin layer plate are likely to be fraught with en or. In general, there will be the three similar dispersion terms representing the main sources of spot dispersion, namely, multipath dispersion, longitudinal diffusion and dispersion due to resistance to mass transfer between the two phases. 

The multipath dispersion on a thin layer plate is the process most likely to be described by a function similar to that in the van Deemter equation. However, the actual mobile phase velocity is likely to enter that range where the Giddings function (3) applies. In addition, as the solvent composition is continually changing (at least in the vast majority of practical applications) the solute diffusivity is also altered and thus, the mobile phase velocity at which the Giddings function applies will vary. 

The band spreading of peaks in SEC is described by a form of the van Deemter equation (5) ... 

Each of the PLgel individual pore sizes is produced hy suspension polymerization, which yields a fairly diverse range of particle sizes. For optimum performance in a chromatographic column the particle size distribution of the beads should be narrow this is achieved by air classification after the cross-linked beads have been washed and dried thoroughly. Similarly, for consistent column performance, the particle size distribution is critical and is another quality control aspect where both the median particle size and the width of the distribution are specified. The efficiency of the packed column is extremely sensitive to the median particle size, as predicted by the van Deemter equation (4), whereas the width of the particle size distribution can affect column operating pressure and packed bed stability. 

Resolution is strongly dependent on flow rate, as indicated by the van Deemter equation (13). This was an important criterion because we wanted to be able to operate our GPCs at a flow rate of about 1.0 ml/min in order to minimize sample run time. 

Equation (9) is the basic form of the Van Deemter equation that describes the variance per unit length of a column in terms of the physical properties of the column contents and the distribution system. 

When the mobile phase is a liquid a variety of equations can be used in addition to the van Deemter equation (1.31) to describe band broadening as a function of the mobile phase velocity, equations (1.36) to (1.39) [49,53,63,85-88]. 

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