Theory of Band Broadening Enormous theoretical and experimental effort was developed to improve quantitative relationships describing the effects of the experimental variables (Table 31-5) on plate heights for various types of columns. Perhaps a dozen or more expressions for calculating plate height have been put forward and applied with various degrees of success.

None of these models is entirely adequate to explain the complex physical interactions and effects that lead to zone broadening and thus lower column efficiencies. Some of the equations, though imperfect, have been very useful, however, in pointing the way toward improved column performance.

The efficiency of capillary chromatographic columns and packed chromatographic columns at low flow velocities can be approximated by the expression

Where H is the plate height in centimeters and u is the linear velocity of mobile phase in centimeters per second. B is the longitudinal diffusion coefficient, CS and CM are mass-transfer coefficients for stationary and mobile phases, respectively. At high flow velocities in packed columns where flow effects dominate diffusion.

Theoretical studies of zone broadening in the 1950s by Dutch chemical engineers led to the van Deemter equation, which can be written in the form where the constants A, B, and C are coefficients of multiple path effects, longitudinal diffusion, and mass transfer, respectively.

Today, van Deemter equation is considered to be appropriate only for packed columns at high flow velocities. For other cases, modified van Deemter Equation (31-27) is usually a better description or better

The Van Deemter equation can be further expanded to Where: H: is plate height λ : is particle shape (with regard to the packing) Dp: is particle diameter Dm:  is the diffusion coefficient of the mobile phase Dc:  is the capillary diameter Df:  is the film thickness Ds:  is the diffusion coefficient of the stationary phase. U: is the linear velocity γ, ω, and R are constants

The Longitudinal Diffusion Term, B/u Diffusion is a process in which species migrate from a more concentrated part of a medium to a more dilute region. The rate of migration is proportional to the concentration difference between the regions and to the diffusion coefficient DM of the species. DM is a measure of the mobility of a substance in a given medium, It is a constant for a given species equal to the velocity of migration under a unit concentration gradient.

In chromatography, longitudinal diffusion results in the migration of a solute from the concentrated center of a band to the more dilute regions on either side (toward and opposed to the direction of flow). Longitudinal diffusion is a common source of band broadening in gas chromatography where the rate at which molecules diffuse is high.

The phenomenon is of little significance in liquid chromatography where diffusion rates are much smaller. The magnitude of the B term in Equation 31-27 is largely determined by the diffusion coefficient DM of the analyte in the mobile phase and is directly proportional to this constant.

As shown by Equation 31-27, the contribution of longitudinal diffusion to plate height is inversely proportional to the linear velocity of the eluent. Such a relationship is not surprising inasmuch as the analyte is in the column for a briefer period when the flow rate is high. Thus, diffusion from the center of the band to the two edges has less time to occur.

The initial decreases in H shown in both curves in Figure 31-13 are a direct result of longitudinal diffusion. Note that the effect is much less pronounced in liquid chromatography because of the much lower diffusion rates in the liquid mobile phase. The striking difference in plate heights shown by the two curves in Fig. 31-13 can also be explained by considering the relative rates of longitudinal diffusion in the two mobile phases.

Figure 31-13 Effect of mobile-phase flow rate on plate height for (a) liquid chromatography and (b) gas chromatography.

In other words, diffusion coefficients in gaseous media are orders of magnitude larger than in liquids. Therefore, band broadening occurs to a much greater extent in gas chromatography than in liquid chromatography.

Diffusion coefficients in gases are usually about 1000 times larger than diffusion coefficients in liquids. The question is why still sharper peaks are concluded in GC than HPLC?

Linear and volumetric flow rate of mobile phase are higher in GC than in HPLC

The Stationary Phase Mass-Transfer Term, CSu. When the stationary phase is an immobilized liquid, the mass-transfer coefficient is directly proportional to the square of the thickness of the film on the support particles df2 (ds2 in equation) and inversely proportional to the diffusion coefficient DS (Dm in equation) of the solute in the film.

These effects can be understood by realizing that both of these quantities reduce the average frequency at which analyte molecules reach the interface where transfer to the mobile phase can occur.

That is, with thick films, molecules must on the average travel farther to reach the surface, and with smaller diffusion coefficients, they travel slower. The result is a slower rate of mass transfer and an increase in plate height. When the stationary phase is a solid surface, the mass-transfer coefficient CS is directly proportional to the time required for a species to be adsorbed or desorbed, which in turn is inversely proportional to the first-order rate constant for the processes.

The Mobile Phase Mass-Transfer Term, CMu The Mobile Phase Mass-Transfer Term, CMu. The mass-transfer processes that occur in the mobile phase are sufficiently complex.

The mobile-phase mass-transfer coefficient CM is known to be inversely proportional to the diffusion coefficient of the analyte in the mobile phase DM. For packed columns, CM is proportional to the square of the particle diameter of the packing material, dp2. For capillary columns, CM is proportional to the square of the column diameter, dc2, and a function of the flow rate.

The contribution of mobile-phase mass transfer to plate height is the product of the mass-transfer coefficient CM as well as the velocity of the solvent itself. Thus, the net contribution to plate height is not linear in u (see the curve labeled CMu in Figure 31-15) but bears a complex dependency on solvent velocity.

Fig. 31-15 Contribution of various mass-transfer terms to plate height Fig. 31-15 Contribution of various mass-transfer terms to plate height. CSu arises from the rate of mass transfer to and from the stationary phase, CMu comes from a limitation in the rate of mass transfer in the mobile phase, and B/u is associated with longitudinal diffusion.

Zone broadening in the mobile phase is due in part to the multitude of pathways by which a molecule (or ion) makes its way through a packed column. As shown in Fig. 31-14, the lengths of these pathways can differ significantly. This difference means that the residence times in the column for molecules of the same species vary. Solute molecules then reach the end of the column over a range of times, leading to a broadened band.

Figure 31-14 Typical pathways of two molecules during elution Figure 31-14 Typical pathways of two molecules during elution. Note that the distance traveled by molecule 2 is greater than that traveled by molecule 1. Therefore, molecule 2 will arrive at B later than molecule 1.

This multiple path effect, which is sometimes called eddy diffusion, would be independent of solvent velocity if it were not partially offset by ordinary diffusion, which results in molecules being transferred from a stream following one pathway to a stream following another. If the velocity of flow is very low, a large number of these transfers will occur, and each molecule in its movement down the column will sample numerous flow paths, spending a brief time in each. As a result, the rate at which each molecule moves down the column tends to approach that of the average.

Thus, at low mobile-phase velocities, the molecules are not significantly dispersed by the multiple path effect. At moderate or high velocities, however, sufficient time is not available for diffusion averaging to occur, and band broadening due to the different path lengths is observed. At sufficiently high velocities, the effect of eddy diffusion becomes independent of flow rate.

Pathways for the mobile phase through the column are numerous and have different lengths. Stagnant pools of solvent contribute to increases in H. For packed columns, band broadening is minimized by small particle diameters. For capillary columns, small column diameters reduce band broadening.

Pools of the mobile phase retained in the stationary phase Thus, when a solid serves as the stationary phase, its pores are filled with static volumes of mobile phase. Solute molecules must then diffuse through these stagnant pools before transfer can occur between the moving mobile phase and the stationary phase. This situation applies not only to solid stationary phases but also to liquid stationary phases immobilized on porous solids because the immobilized liquid does not usually fully fill the pores.

The presence of stagnant pools of mobile phase slows the exchange process and results in a contribution to the plate height that is directly proportional to the mobile-phase velocity and inversely proportional to the diffusion coefficient for the solute in the mobile phase. An increase in internal volume then accompanies increases in particle size.

Effect of Mobile-Phase Velocity on Terms in Equation 31-27. Fig. 31-15 shows the variation of the three terms in Equation 31-27 as a function of mobile phase velocity. The top curve is the summation of these various effects. Note that there is an optimum flow rate at which the plate height is a minimum and the separation efficiency is a maximum.

Summary of Methods for Reducing Band Broadening. For packed columns, one variable that affects column efficiency is the diameter of the particles making up the packing. For capillary columns, the diameter of the column itself is an important variable. The effect of particle diameter is demonstrated by the data shown in Figure 31-16 for gas chromatography.

Fig. 31-16 Effect of particle size on plate height for a packed gas chromatography column. The numbers to the right of each curve are particle diameters. (From J. Boheman and J. H. Purnell, in Gas Chromatography 1958, D. H. Desty, ed., New York: Academic Press, 1958.

A similar plot for liquid chromatography is shown in Figure 33-1. To take advantage of the effect of column diameter, narrower and narrower columns have been used in recent years. With gaseous mobile phases, the rate of longitudinal diffusion can be reduced appreciably by lowering the temperature and thus the diffusion coefficient. The result is significantly smaller plate heights at lower temperatures.

This effect is usually not noticeable in liquid chromatography because diffusion is slow enough that the longitudinal diffusion term has little effect on overall plate height. With liquid stationary phases, the thickness of the layer of adsorbed liquid should be minimized since CS in Equation 31-27 is proportional to the square of this variable.

The diffusion coefficient DM has a greater effect in gas chromatography than in liquid chromatography.