Theory of Spectrodensitometry

In spectrophotometric measurements where the ab-sorbance is measured as a result of a beam of light of set wavelength passing through a fixed pathlength of solution, a direct relationship exists between the observed absorbance and the concentration of the solution. This is known as Beer's law. However, it should be pointed out that this relationship is not linear over the whole range of concentration, and it depends on the sample solution being transparent.

As TLC/HPTLC plates are opaque, a somewhat different approach is required. In the 1930s Kubelka and Munk investigated the relationship between ab-sorbance, transmission and reflectance, deriving mathematical expressions to explain the effects of absorb-ance and reflectance. When a ray of incident light comes into contact with the surface of the opaque TLC layer, some light is transmitted, some is reflected in all directions at the surface and some rays are propagated in all directions inside the adsorbent. The theory which explains to a large degree what is happening in this process is known as the Kubelka-Munk theory. Certain assumptions can be made which simplify the mathematical equations derived. The theory assumes that both the transmitted and reflected components of incident light are made up only of rays propagated inside the sorbent in a direction perpendicular to the plane of the surface. All other directions will lead to longer pathways and hence stronger absorption. These rays therefore contribute little to either the transmitted or reflected light and their contribution can be treated as negligible. When light exits from the sorbent at the layer-air boundary, light scattering occurs, and it is distributed over all possible angles with the surface.

The coefficient of light scatter (S), can therefore be proposed; this depends on the layer thickness. If we assume that this is unchanged in the presence of a chromatographic zone, the following equation can be derived for an infinitely thick opaque layer:

where Rx is the reflectance for an infinitely thick opaque layer, am is the molar absorptivity of the sample, c is the molar concentration of the sample and S is the coefficient of scatter per unit thickness.

This equation is clearly less than ideal as the layer has a finite thickness. More meaningful expressions for the intensity of the reflected light, IR, and the transmitted light, IT, for a layer of thickness (l) are given by the following hyperbolic solutions:

It"

a • sinh(b • S • l) + b • cosh(b • S • l)

b a • sinh(b • S • l) + b • cosh(b • S • l)

KA is the coefficient of absorption per unit thickness.

The application of the equations to quantitative analysis in TLC is quite complex, but it can be greatly simplified by making a number of reasonable assumptions that would hold true for TLC. One thing that eqn [2] immediately reveals is that the relationship between the reflected light and the concentration of the chromatographic zone is nonlinear. This is what is found in practice over the full range of concentrations. The data when graphically displayed fit a polynomial curve (eqn [4]).

However, over a narrow concentration range the relationship is seen to be linear. This means that if it is necessary to have a calibration curve over the whole range of concentrations, at least four but no more than six standards will be required for the determination of one separated analyte. Of course, only two standards may be needed if the concentrations are close to that of the analyte, because it can be assumed that the curve is linear over a small range.

Although it may seem that errors could easily creep into the determination procedure, this is not the case. The assumptions made have only a negligible effect on the final result. Hence, even including any errors which may originate from the scanning spectrodensitometer, the percentage relative standard deviation is normally below 2% and quite often well below 1%.

For a wide concentration range, the MichaelisMenten regression curve can be used. The calibration is calculated as a saturation curve:

data fit is not compromised when as few as three standards are used over the whole concentration range.

The mathematical treatment of the data for fluorescence intensity can be expressed according to the well-known Beer - Lambert law. The fluorescence emission (F) is given by the equation:

and is theoretically only permitted within the calibration range (between the largest and the smallest standard amounts applied). This regression always passes through the origin.

In some cases there is a better curve fit to the data if the Michaelis-Menten regression does not pass through the origin. Better resolution is therefore obtained if the data produce a function that does not tend towards zero:

where F is the fluorescence emission and d is the quantum yield.

For low sample concentrations the following assumption can be made:

Therefore:

As before, this is theoretically admissible only within the calibration range.

It is also possible to linearize the data graphically. The simplest transformation procedures involve converting the data on reflectance and concentration into reciprocals, logarithms or squared terms. The following equations can thus be proposed:

where Re is the reflectance signal and c is the sample concentration.

Eqns [7] and [9] result in linearization over the middle of the concentration range, whereas eqn [8] showed better linearization, but even this fails at very low concentration. None of these methods is able to linearize the data over the whole concentration range.

A solution to the above is to use nonlinear regression analysis based on second-order polynomials. These can be described by the following equations:

ln Re = a0 + a1 • ln c + a2 • (ln c)2 [10] Re = a0 + a1 • c + a2 • c2 [11]

Over the whole concentration range, eqn [10] gives the best results. In fact, it has been shown that the

It follows that, for low concentrations, the fluorescence emission is linearly dependent on the sample concentration. In practice this proves to be the case even though this equation was derived without taking into consideration the influence of absorption or scatter.

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