Isotherm data is typically measured using a series of batch experiments. A stock solution with initial concentration c_0 is prepared. Then, a volume V_0 of this stock is mixed with a volume V_g of gel (chromatography particles). After equilibrium is achieved, the liquid phase concentration c is measured in the supernatant. Finally, the corresponding stationary phase concentration q is determined from a mass balance:
c_0 V_0 = c V_0 + q V_g\tag{Ia}
\Leftrightarrow q = \frac{V_0}{V_g}(c_0 - c) \tag{Ib}
Assume the Langmuir isotherm is used to create grount truth data for some modeling purpose:
q = \frac{q_m k_{eq}c}{1+k_{eq}c} \tag{II}
The expected supernatant concentration c can be determined by eliminating q from eqs. I and II and some tedious algebra:
c = \frac{c_0 k_{eq} V_0 - k_{eq} V_g q_m - V_0 \pm \sqrt{c_0^2 k_{eq}^2 V_0^2 - 2 c_0 k_{eq}^2 V_0 V_g q_m + 2 c_0 k_{eq} V_0^2 + k_{eq}^2 V_g^2 q_m^2 + 2 k_{eq} V_0 V_g q_m + V_0^2}}{2 k_{eq} V_0} \tag{III}
Now for the error model. The volumes V_0 and V_g and the initial concentration c_0 are prone to pipetting and weighing errors. These errors propagate to c via eq. III. In addition, c is prone to absolute and relative measurement errors. The combined errors propagate to q via eq. I.
Important points to note:
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The result will be much different from just adding random noise to c and q.
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The same procedure can be applied to other isotherm models, but there might be no analytical solution for c. An iterative solver can be used instead.
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The procedure can further be extended to multi-component systems. For complex compositions, c_0 is further prone to measurement errors.
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In practice, a series of experiments is created either by successively diluting c_0 or by successively incrasing V_g. The error model captures both cases, but different systematic errors are inflicted on c_0 and V_g, i.e. their errors are not fully independent between individual experiments.
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Real or virtual experiments are prepared by computing nominal V_g values for pre-specified V_0 and c_0 values and a series of desired c values. Alternatively, nominal c_0 values can be computed for desired c values. Both computations are straightforward by inserting eq. II in eq. I. The results also depend on estimated (real experiments) or groud truth (virtual experiments) k_{eq} and q_m data. These calculations reflect the wanted experiment under ideal conditions. Various errors are only added in the reverse direction, when the real experiment is mimicked as described above.
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The above approach neglects the pore phase mass fraction in the particles. This can be justified for V_0 \gg V_g.
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Both c and q refer to the total gel volume V_g while in CADET, \bar{c} = \varepsilon_p c refers to the pore phase volume and \bar{q} = (1-\varepsilon_p)q to the stationary phase volume (\varepsilon_p denotes particle porosity). As a consequence, the isotherm parameters \bar{k}_{eq} = \frac{k_{eq}}{\varepsilon_p} and \bar{q}_m = (1-\varepsilon_p) q_m also have different values in CADET. This can easily be seen by inserting \bar{c} and \bar{q} in eq. II.