Ideal Thermochromatography

The formulae of the above section give the retention time as a function of desorption energy or, vice versa, they allow evaluating the energy from the measured time, provided that to is known. We emphasize that it holds for an imaginary ideal situation, when the input spike and the resulting chromatographic zone are infinitely narrow. 

In the TC processing the average molecular speed and the flow rate vary with z because of changing temperature. The retention time and the coordinate for any temperature profile of the column are related by the equations  

Here zs is the coordinate of the starting point with the temperature 7s and za is the position of the zone with the temperature 7a. In the ideal conditions, the latter is identical to in Fig. 4.1. 

Suppose that the TC column temperature profile is linear, that is Tz = 7s - gz and dz = -dTz/g Then the net retention time equals  

The pertinent indefinite integrals and approximating series were listed in Sect. 2.1.6. In practice, the value of the definite integral for 7s is usually much smaller than that for l and can be neglected. Taking only two terms in the asymptotic series, we obtain for the case of a constant temperature gradient that  

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The theory of ideal thermochromatography suggests that, at any point, there exists thermodynamical (adsorption) equilibrium between the gas-phase and surface concentrations of the adsorbable molecules. The ideal elution TC, like the ideal IC of nonradioactive species, would yield a narrow band at some 7 d - a function of A ads H and Aac s.S, as well as of the processing time and temperature profile. Figure 4.1 illustrates different situations. 

The proposed approach may also be useful in simulating thermochromatography in vacuum columns, chromathermography and other separation techniques in open columns. Moreover, repeated Monte Carlo experiments with small number of molecules serve to visualize the uncertainties imposed by poor statistics. They are very helpful in evaluating Bayesian confidence intervals for the parameters measured in the experiment performed in non-ideal conditions, when any attempt to obtain an analytical solution fails completely. This will be discussed and illustrated in Sect. 6.2. 

In the ideal frontal thermochromatography, the high temperature branch of the TC peak must obviously approach the adsorption isobar... 

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