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2 THEORY AND PRINCIPLES 2.1 Distribution Constant, Retention Times and Efficiency of Separations The effectiveness of separation depends on the column and consequently the relative rates at which components are eluted. The differences on the extent at which a solute is distributed between the mobile and stationary phases can be expressed in terms of an equilibrium constant, K . Equilibrium is involved as a solute distributes itself into the mobile and stationary phases. This could be represented by the equation: A(mobile) ↔ A(stationary)
Accordingly, solute A is distributed into the mobile and stationary phases and in terms of equilibrium constant expression
Where Cs is the molar concentration of solute A in the stationary phase and C m is its molar concentration in the mobile phase. K is known as the distribution constant. In order to achieve separation in chromatography, the analytes must be retained in the column. The fundamental relationship between retention volume (V R) and, the quantity of the stationary phase (V S) and the distribution coefficient ( K ) is expressed in the equation: VR = VM + K VS where VM is the volume of mobile phase within the column. The retention volume can be determined directly from the chromatogram as VR= FC x tM where FC is the flow rate of the solvent, and t R is the retention time of the component; similarly V M= FC xtM, where tM is the time taken for an unretained compound to traverse the column. For this specific relationship, in order for K , which measures the extent of retention, to be evaluated, V M and VS must be known. A more practical equation for solute retention is given by k , the capacity factor:
Here, nS is the total number of moles solute component in the stationary phase and n M is the total number of moles of analyte in the mobile phase. It can be subsequently shown that VR= VM(1+k ) and that
Thus the capacity factor, k , is a measure of the sample retention by the column and can be determined from the chromatogram directly. A chromatogram is a plot of solute concentration as detected against elution time or elution volume (Skoog, 2007).
Figure 1. An example of a chromatogram with nomenclature (From Scott, Principles and P ractice of Chromatography,pp. 13.2003)
2.2 Column Efficiency and Theoretical Plates A measure of column performance and efficiency is expressed in terms of theoretical plates (N) and the height equivalent of a theoretical plate (HETP) denoted by H, and secondly, to measure the resolution (R S) attained in the chromatogram. The degree of separating efficiency is given by the number of theoretical plates the column is equivalent to, which is defined in terms of the retention time (t R) and the base width of the peak(wb):
() = 5.54 () 2
The efficiency of any column is best measured by the HETP which is expressed as H= L/N where L is the length of the column. The resolution, R S , of two compounds is defined as being equal to the peak separation divided by the mean base width of the peaks:
A more practical expression for R S in terms of the experimental factors α, k and N , can be arrived at by substitution for t R and wbin terms of N. Thus RS, as a function of α, k and N is given by
RS ()( ) where the selectivity factor α = k 2 /k 1 for components band 1 and 2 in the chromatogram. Plate theory is useful in monitoring and comparing column performance and in giving a measure of the resolution achieved in the chromatogram but has a number of limitations. This approach does not consider the influence of important chromatographic variables such as particle size, stationary phase loading, eluent viscosity and flow-rate of column performance (Robinson, 2005).
2.3 Band Broadening and Kinetic Effects: The van Deemter Equation Band broadening reflects a loss of column efficiency. The kinetic effects giving rise to band broadening during the development of the chromatogram are combined to
give a general expression for the overall plate height is contained in van Deemter equation: H = A + B/u + Cu
where A is a coefficient that describes multiple-path effects and B/u and Cu are the longitudinal diffusion term and mass transfer coefficients, respectively. The multipath term A applies to columns packed with support particles and becomes zero for open tubular columns when the mobile phase velocity is slow enough for the flow to be laminar. Longitudinal diffusion results in the migration of a solute from a concentrated center of a band to the more dilute regions on either toward or against the direction of the flow. Diffusion of the molecules from a region of high concentration to a region of low concentration occurs due to random thermal process. The longitudinal diffusion is of little significance in liquid chromatography. The faster the linear mobile phase velocity u, the less time the analytes have to spread in the mobile phase. Thus, band spreading and H are inversely proportional to u. The mass transfer C relates the diffusion in both the mobile and stationary phases. If the distance molecules travelled into and back out of each phase were infinitely small, and the mobile phase flow were infinitely slow, there would be time at every point along the column for the molecules to achieve perfect equilibration between the phases. This mechanism for band spreading is directly proportional to u. Additionally, under laminar flow conditions in small channels, the flow rate will actually vary from zero at the boundary between the two phases to a maximum at the channel center (Ahuja,2005). By collecting separated components as they exit the bottom of the column, one could isolate pure material. If a detector in the exiting fluid stream (called the effluent) can respond to some property of a separated component other than color, then neither a transparent column nor colored analytes are necessary to separate and measure materials by “chromatography”. The critical defining properties of a chromatographic process are immiscible stationary and mobile phase, an arrangement whereby a mixture is deposited at one end of the stationary phase, flow of the mobile phase toward the other end of the stationary phase, different rates or ratios of partitioning for each component of the mixture, and many cycles of this process during elution, a means of either visualizing
bands of separated components on or adjacent to the stationary phase, or of detecting the eluting bands as peaks in the mobile phase effluent (Robinson,2005). The longitudinal diffusion term (B/u) is more important in GLC, however, than in other chromatographic processes because of the much larger diffusion rates of gases (10
times greater than liquids). As a result, the minima in curves relating plate
height H to flow rate (van Deemter plots) are usually considerably broadened in GC (Skoog, 2007). Figure 2 illustrates this.
Figure 2. A comparison on the effect of mobile phase flow-rate on plate height in Liquid Chromatography and Gas Chromatography (From Skoog, pp 933, 2003). 2.4 Retention Volumes for GC
To take into account the effects of pressure and temperature in GC, it is often useful to use retention volumes rather than the retention times.The relationship between the two is given by VR↔ tRF
where F is the average volumetric flow rate within the column; V and t are retention volumes and times, respectively; and the subscripts Rand M refer to species that are retained and not retained on the column. The flow rate within the column is not directly measurable. Instead, the rate of gas flow as it exits the column is determined experimentally with a flow meter. For popular soap-bubble type flow meters, where the gas is saturated with water, the average flow rate F is related to the measured flow rate F m , by
where T is the column temperature in kelvins, T, is the temperature at the flow meter, and P is the gas pressure at the end of the column. Usually P and T are the ambient pressure
and temperature. The term involving the vapor pressure of water,
is a correction for
the pressure used when the gas is saturated with water. Both V R and VM depend on the average pressure within the column - a quantity that lies intermediate between the inlet pressure Pi and the outlet pressure P (atmospheric pressure). The pressure drop correction factor j, also known as the compressibility factor, accounts for the pressure within the column being a nonlinear function of the Pi / P ratio. Corrected retention volumes
which correspond to volumes at the average column pressure, are obtained from the relationships
where j can be calculated from
] ⁄ [ [ ⁄ ] The specific retention volume Vg is then defined as
where mS is the mass of the stationary phase. A quantity determined at the time of column preparation. The specific retention volume V, can be related to the distribution constant K. To do so, we substitute the expression relating t Rand t M to k into
Combining this expression yields
Substituting k gives
s is given by
The density of the liquid on the stationary phase
where Vs is the stationary-phase volume. Thus,
It is important to note that, Vg, at a given temperature depends only on the distribution constant of the solute and the density of the liquid making up the stationary phase (Skoog, 2007).