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(1.35)

(1.36)

Figure 1.8 Resolution factor. A simulation of chromatographic peaks by juxtaposition of two identical Gaussian curves to a greater or lesser extent. The visual aspects corresponding to the values of R are indicated on the diagrams. From a value of R = 1.5 the peaks can be considered to be baseline resolved, the valley between them being around 2%.

1.10 INFLUENCE OF SPEED OF THE MOBILE PHASE

      In all of the previous discussion and particularly in the various equations that characterize separations, the velocity (a function of flow rate) of the mobile phase in the column is not taken into account. However, if it becomes too high, the speed, which has an influence upon the progression of the analytes down the column, disturbs the equilibrium kinetics (Solute)_MP / (Solute)_SP, and hence it acts on their dispersion, in other words on the quality of the analysis undertaken (compare Figure 1.9).

      The influence of the speed of the mobile phase was demonstrated in the case of packed columns in gas chromatography and then modelled by Van Deemter, who proposed the first kinetic equation.

Figure 1.9 Effect of column length on the resolution. Chromatograms obtained with a GC instrument illustrating that by doubling the length of the capillary column, the resolution is multiplied by a factor of 1.41.

      (Source: Adapted from a document provided by the Waters company.)

      1.10.1 Van Deemter Equation

The simplified equation proposed by Van Deemter in 1956, is well known for packed GC columns (Eq. (1.37)). The expression links the plate height H (HETP) to the average linear velocity of the mobile phase ū in the column (Figure 1.10):

(1.37)

      This equation reveals that there exists an optimal flow rate for each column, corresponding to the minimum value of H, as shown by the curve of this equation. The loss in efficiency as the flow rate increases is obvious and represents what occurs when an attempt is made to rush the chromatographic separation by increasing the mobile phase flow rate. However, the loss in efficiency that occurs when the flow rate is too slow is less intuitive. To explain this phenomenon, the origins of the terms A, B, and C must be reviewed. Each of these parameters represents a domain of influence that can be perceived on the graph (Figure 1.10).

      The three basic experimental coefficients A, B, and C are related to diverse physico‐chemical parameters of the column and to the experimental conditions. If H is expressed in cm, A will also be in cm, B in cm²/s and C in s (where velocity is measured in cm/s). The curve of the Van Deemter equation is a hyperbola that goes through a minimum (H_min) when:

      (1.38)

Figure 1.10 Van Deemter curve in gas chromatography with the domains of parameters A, B, and C indicated. There exists an equation similar to Van Deemter’s that considers temperature: H = A + B/T + CT.

      Packing‐related term A = 2λ.d_p

      Term A is related to the flow profile of the mobile phase passing through the stationary phase. The size of the particles (diameter d_p), their size distribution, and the uniformity of the packing (packing factor λ) can all create preferential paths, potentially causing imperfect exchanges between the two phases. This is known as the turbulent or Eddy diffusion factor, which is considered to have little importance in liquid chromatography and to be absent by nature for wall‐coated open tubular (WCOT) capillary columns in GC (Golay equation without term A, see Section 1.10.2).

      Gas (mobile phase) diffusion term B = 2γD_G

      Term B, which can be expressed from D_G, the diffusion coefficient of the analyte in the mobile phase, and λ, the above packing factor, is taken into consideration above all when the mobile phase is a gas. The longitudinal diffusion in the column is in effect quite fast.

      This term is a consequence of entropy, which states that a system will tend spontaneously towards the greatest degree of disorder, just as a drop of ink diffuses into a glass of water into which it has fallen. Consequently, if the flow rate is too slow, the compounds undergoing separation will mix faster than they will migrate. This is why one must never interrupt a chromatographic analysis, even temporarily, once it is underway, as this could negatively impact efficiency.

      Liquid (stationary phase) term C = C_G + C_L

      Term C, which is related to the resistance to mass transfer of the solute between the two phases, becomes dominant when the flow rate is too high for an equilibrium to be attained. Local turbulence within the mobile phase and concentration gradients slow the equilibrium process (C_S ⇔ C_M). The diffusion of solute between the two phases is not instantaneous, and thus the solute will be carried along out of equilibrium. No simple formula exists which takes into account the different factors involved in term C. The parameter C_G is dependent upon the diffusion coefficient of the solute in the gaseous mobile phase, while

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