Francis Rouessac - Chemical Analysis

The separation factor α ( Eq. (1.31)) enables the comparison of two adjacent peaks 1 and 2 present in the same chromatogram ( Figure 1.7). It is defined by Eqs. (1.31)and ( 1.32).

By definition α is greater than unity:

(1.31)

or

(1.32)

The expression, connecting α to the Nernst distribution coefficients of the two solutes, shows that selectivity is dependent only on the value of these constants (intensity of interactions, temperatures) and does not depend on the column’s geometry (length and diameter) or its packing (diameter of particles and quantity of stationary phase). For nonadjacent peaks, the relative retention factor r is calculated in a similar manner to α , and cannot be less than 1.

To quantify the separation between two compounds, the resolution factor R is calculated from the chromatogram ( Figure 1.8):

(1.33)

For two adjacent peaks, this relationship involves two parameters: firstly, the difference between their retention times, t R(2) − t R(1), which corresponds to the distance between the two peaks, and, secondly, their half‐width at the base, ½ ( ω 2 + ω 1) if we assume that each peak corresponds to an isosceles triangle ( Figure 1.8).

Other expressions derived from the preceding ones and established with a view to replacing one parameter by another or to accommodating simplifications may also be employed to express the resolution. Therefore, Eqs. (1.34)– (1.36) are used quite often.

Equation ( 1.35) shows how resolution is affected by the efficiency, capacity factor, and selectivity factor. The chromatograms in Figure 1.9present an experimental verification of this.

(1.34)

(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%.

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.)

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 cm 2/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 .

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).

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.