Chromatographic Separations
Theory
Chromatographic separations of mixtures of various compounds are based on their distribution between a stationary and a mobile phase, which are present in a chromatographic column. The mobile phase moves across the column, in effect washing (eluting) compounds at a different rate. These differences are based in properties such as the boiling point, the polarity, the electric charge (for ionic compounds), the size of the molecule, and so forth. If at the column outlet there is a system for detecting and measuring the quantity of each component, then a quantitative determination of the separated components is achieved.
Theoretical Plates
The movement of substance A through a chromatographic column can be considered as a movement through successive equilibration chambers (see applet on Countercurrent Extraction - Craig Apparatus), called theoretical plates. A theoretical plate is the volume of the column required for a complete equilibration between stationary (S) and mobile (M) phases, described by the distribution ratio or distribution coefficient
For a given chromatographic column with a particular type of stationary phase the number of theoretical plates is not a fixed number (as the number of tubes in a Craig apparatus) but depends on a number of factors, most important of them being the natuere and the velocity of the mobile phase, kinetic factors involved in the distribution equilibrium and everything which can affect it (e.g. temperature, column packing) and the nature of A itself.
For example, in gas chromatography, where the mobile phase is usually an inert gas (carrier), the length of column behaving like an individual theoretical plate, known as height equivalent to a theoretical plate h is closely described by the van Deemter equation
where õ is the velocity of the carrier gas and A, B, and C are constants characteristic of a given column, which depend on the nature of the static and gaseous phases, and on the filling material and the packing of the chromatographic column.
Approximate model for chromatographic curves
In this applet a model approximating the behavior of chromatographic column is just an extension of the equations described with Craig apparatus based on the binomial expansion (p+q)^{n}
where D is the inverse of D_{A} (as defined above), f_{n,r} is the fraction of compound present in tube (theoretical plate) r after n transfers. For large n (typically for n>25) the curve describing fn,r as a function of r and n approximates the normal distribution (Gauss' curve) and the following equation can be used instead:
where r_{max} is the theoretical plate of maximum f_{n,r} (peak position), given by equation:
In this applet the last two equations were used for drawing the chromatographic curves across the column. However, it should be stressed that this model is rather crude since no tailing effects have been taken into account, mainly due to incomplete equilibration between the mobile and the stationary phase. In the picture below are shown the distribution profiles for an ideal and a real-world chromatographic column.
The profiles at the left represent the ideal case, where the distribution equilibrium is instantaneously achieved, hence the distribution coefficient applies at any point (slice) of the column. The profiles at the right represent a more realistic case, whereas incomplete equilibration gives rise to tailing of the chromatographic peak. Tailing depends strongly on the quality of the chromatographic column (geometry and packing) and the actual conditions (e.g. carrier velocity, temperature).
Approximate model for detector signal
The composition of the mixture in the last theoretical plate (at the outlet of the chromatographic column) can be used for approximating the detector signal. The composite signal S (as a function of n) of the detector, for a mixture of k components can be approximated by the following equation:
where N is the effective plate number of the chromatographic column considered. N is taken to be the same for all k components, and R_{i} is the relative response for compound i (i = 1, 2, …, k), which is proportional to its initially injected amount and to the relative sensitivity of the detector toward the same component.
Applet
The applet demonstrates the chromatographic separation of five components (A, B, C, D, and E) and it is quite simple to use.
There are default sets of distribution ratios, and of relative response factors as well. There is also a default number of theoretical plates. The user can change all these values by typing new values in the corresponding edit boxes. The ranges of acceptable values for distribution ratios are: 0.1-50.0, for response factors: 0.1-10.0, and for theoretical plates: 50-50000.
By clicking on the START (becoming RESTART after the first run) button the system is initialized and by clicking on the INJECT button the chromatogram starts running.
On the upper plotting area are shown the moving zones (as gaussian peaks) across the chromatographic column. Each component is identified by its own particular color. On the lower plotting area is shown the recording of the resulting composite signal.
You can experiment with different values of distribution ratios, response factors and theoretical plates. It is of interest to note how the number of theoretical plates affects the resolution of the recorded chromatographic peaks of components with almost similar distribution ratios.
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