The variation of solute concentration in the stationary phase with solute concentration in the mobile phase, at constant temperature, is known as the sorption isotherm. Simple chromatographic theory assumes a linear isotherm relationship, i.e. the distribution coefficient is constant. Under these conditions the retention time is independent of sample concentration and the peak moves with a constant speed. Given a peak profile with plug-shape distribution on injection, this shape should be maintained as the peak passes through the column to emerge at the exit. However, because of longitudinal diffusion in the direction of flow, the peak takes on a Gaussian distribution. If the isotherm relationship is nonlinear (e.g. Langmuir or anti-Langmuir), the distribution coefficient is not constant but varies with solute concentration and there is a distribution of solute velocities across the peak, which is described as tailing or fronting. This relationship between isotherm shape and peak shape is illustrated in Figure 3.

The width of a chromatographic peak is a function of the column efficiency, expressed as the plate number (N), calculated from the following equations depending on the value used for the peak width (Figure 2):

where o is the standard deviation of the Gaussian peak.

The column length divided by the plate number gives the plate height or height equivalent to a theoretical plate (H) and normalizes the plate number for column length: H = L/N. The concept of plates in chromatographic theory (the plate theory) is by analogy with the distillation process and represents a notional length of the column in which the solute molecules reach a distribution equilibrium. Thus,

Figure 3 Isotherm shape and their effect on peak shape and retention times. (A) Linear; (B) Langmuir; (C) anti-Langmuir; (D) Gaussian; (E) tailing; (F) fronting.

a large number of theoretical plates corresponds to an efficient column.

Consideration of the chromatographic process as controlled by equilibrium gives a satisfactory explanation of chromatographic retention in term of the distribution coefficients, but in considering band broadening a different approach is required - the rate theory of chromatography. This was first applied by van Deemter, Klinkenberg and Zuiderweg to gas chromatography, but has been extended to include LC. As the solute band passes through the column the band width increases and the solute is diluted by the mobile phase. Although the process of fluid flow is complex, three main contributions to band broadening (i.e. to the variance (c2) of the Gaussian peak) may be recognized: the multipath effect (formally called eddy diffusion), molecular diffusion and mass transfer.

The Multipath Effect (the A term)

Molecules flowing through a packed bed of stationary phase will take paths of different lengths resulting in a small difference in retention times. This has the effect of broadening the band by an amount dependent on the particle diameter (dp), such that:

band broadening in the stationary phase (stationary phase mass transfer: Cs). The first of these processes involves the finite rate of mass transfer across the mobile-phase/stationary-phase interface. At the head of the column the solute is distributed between the stationary and mobile phases according to the value of the distribution coefficient. As the band moves down the column, solute at the leading edge of the band is continually meeting new stationary phase, into which it dissolves. To maintain the equilibrium, solute will dissolve from the trailing edge of the band out of the stationary phase back into the mobile phase. Because this process is not instantaneous the band is broadened. The second process involves the statistical distribution of the rates of diffusion of individual molecules in the stationary phase, resulting in small differences in the time that individual molecules spend in the stationary phase. A fast-moving mobile mass sweeps the zone more rapidly through the column and accentuates the band broadening as does a greater film thickness (df) of stationary phase. A higher rate of solute diffusion in the stationary phase (DS) will decrease the band broadening so that:

k d2

The packing constant (A) is an empirical term depending on the shape (spherical or irregular) of the packing material and the packing efficiency, and reaches a minimum value =0.5. For open tubular columns there is no A term.

Longitudinal Molecular Diffusion (the B term)

Solute molecules diffuse in a longitudinal direction (i.e. along the column axis) according to Fick's law of diffusion. The amount of band spreading is directly proportional to the coefficient of diffusion (DM) of the solute molecules in the mobile phase, and inversely proportional to the mobile-phase flow rate. An obstruction factor is introduced to account for the restricted diffusion in a packed bed.

Hence:

where q is a configuration factor depending on the nature of the stationary phase.

In adsorption chromatography, the Cs term is expressed in terms of the adsorption/desorption kinetics of the solute molecules on the stationary phase.

Band broadening in the mobile phase also results from two different processes. Moving mobile phase mass transfer (CM) results from frictional forces which modify the laminar flow profile across a channel between two particles, resulting in a higher flow velocity in the centre of the channel. Stagnant mobilephase mass transfer (CM) is the result of slow diffusion of solute molecules in and out of the pores of a porous stationary phase. The overall mobile-phase mass transfer can be represented by the expression:

Longitudinal molecular diffusion is only significant in LC if small (< 10 |im) stationary-phase particles are used at low mobile phase velocities and with a relatively high solute diffusion coefficient.

Mass Transfer (the Cterms)

In LC, band broadening due to mass transfer is a complex process involving both the stationary and mobile phases. Two processes are responsible for where dp is the particle diameter and dc is the column diameter, DM the solute diffusion coefficient in the mobile phase and u is the linear velocity.

Giddings, recognizing that molecules are free to diffuse from one flow path into another, introduced the idea of 'coupling' the multipath term (A) and the mobile-phase mass transfer (CM) so that the variation of the plate height (H) with u is then given by:

The contribution of the various terms to the total plate height is illustrated in Figure 4.

So far, we have only considered band-broadening processes within the chromatographic column itself but, in assessing the overall performance of the system, the instrument as a whole is important. Thus, the injection system, detector and connecting tubing all contribute to the overall peak shape. The objective for injection is to get the sample on to the column in as narrow a plug as possible. Slow transfer of the sample from the injector to the column causes peak broadening and peak tailing. Large dead volumes in the detector can lead to remixing of components and deterioration of the separation as well as dilution of the sample peaks, reducing detection limits. The peak broadening in an open tube (radius r and length L), volume flow rate F, and for a solute with diffusion coefficient DM is given by:

nr4FL/24DM

bonding forces) between the sample molecules and the stationary phase are sufficiently different. More fundamentally it is the free energies of distribution A(AG") of the components of a mixture which must differ. It can be shown that:

A stationary phase which produces a large degree of separation is said to have high selectivity. The separation of two components (1 and 2) is expressed by the relative retention (a):

a = tR(2)/tR(l) = Vr(2)/Vr(1) = k(2)/k(l) = KC(2)/KC(1)

If one of the pair is a standard substance, the symbol used for relative retention is r. Having achieved a separation, it is necessary to prevent remixing of the components and the ability to achieve this is a function of the column efficiency, as measured by the plate number. The combined effects of stationary-phase selectivity and column efficiency are expressed in the peak resolution (Rs) of the column:

In particular, short lengths ( < 30 cm) of narrowbore ( & 0.01 in) connecting tubing should be used.

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