# Batch isotherm error model

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:

1. The result will be much different from just adding random noise to c and q.

2. 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.

3. The procedure can further be extended to multi-component systems. For complex compositions, c_0 is further prone to measurement errors.

4. 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.

5. 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.

6. The above approach neglects the pore phase mass fraction in the particles. This can be justified for V_0 \gg V_g.

7. 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.

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If you are interested, here is the above mentioned algebra for calcultaing c:

c V_0 + \frac{q_m k_{eq}c}{1+k_{eq}c} V_g - c_0 V_0 = 0

c V_0 (1+k_{eq}c) + q_m k_{eq}c V_g - c_0*V_0 (1+k_{eq}c)= 0

c V_0+c * V_0 k_{eq}c + q_m k_{eq}c V_g - c_0 V_0- c_0 V_0k_{eq}c= 0

V_0 k_{eq}c^2 + (V_0+ q_m k_{eq}V_g- c_0 V_0k_{eq})c - c_0 V_0= 0

c^2 + \frac{V_0+ q_m k_{eq}V_g- c_0 V_0k_{eq}}{V_0 k_{eq}}c - \frac{c_0}{k_{eq}}= 0

c = \frac{-p \pm \sqrt{p^2-4 q}}{2} with p = \frac{V_0+ q_m k_{eq}V_g- c_0 V_0k_{eq}}{V_0 k_{eq}} and q = - \frac{c_0}{k_{eq}}

c = \frac{-\frac{V_0+ q_m k_{eq}V_g- c_0 V_0k_{eq}}{V_0 k_{eq}} \pm \sqrt{(\frac{V_0+ q_m k_{eq}V_g- c_0 V_0k_{eq}}{V_0 k_{eq}})^2+4 \frac{c_0 V_0}{k_{eq}V_0}}}{2}

c = \frac{-V_0- q_m k_{eq}V_g+ c_0 V_0k_{eq} \pm \sqrt{(V_0+ q_m k_{eq}V_g- c_0 V_0k_{eq})^2 + 4 c_0 k_{eq} V_0^2}}{2 V_0 k_{eq}}

c = \frac{c_0 k_{eq} V_0 - k_{eq} V_g q_m - V_0 \pm \sqrt{(V_0+ q_m k_{eq}V_g- c_0 V_0k_{eq})^2 + 4 c_0 k_{eq} V_0^2}}{2 k_{eq} V_0}

c = \frac{c_0 k_{eq} V_0 - k_{eq} V_g q_m - V_0 \pm \sqrt{(V_0+ q_m k_{eq}V_g- c_0 V_0k_{eq})^2 + 4 c_0 k_{eq} V_0^2}}{2 k_{eq} V_0}

c = \frac{c_0 k_{eq} V_0 - k_{eq} V_g q_m - V_0 \pm \sqrt{V_0 V_0+ V_0 q_m k_{eq}V_g- V_0 c_0 V_0k_{eq}+ q_m k_{eq}V_g V_0+ q_m k_{eq}V_g q_m k_{eq}V_g - q_m k_{eq}V_g c_0 V_0k_{eq} - c_0 V_0k_{eq} V_0 - c_0 V_0k_{eq} q_m k_{eq}V_g + c_0 V_0k_{eq} c_0 V_0k_{eq} + 4 c_0 k_{eq} V_0^2}}{2 k_{eq} V_0}

c = \frac{c_0 k_{eq} V_0 - k_{eq} V_g q_m - V_0 \pm \sqrt{V_0^2 + k_{eq} V_0 V_g q_m - c_0 k_{eq} V_0^2 + k_{eq} V_0 V_g q_m + k_{eq}^2 V_g^2 q_m^2 - c_0 k_{eq}^2 V_0 V_g q_m- c_0 k_{eq} V_0^2 - c_0 k_{eq}^2 V_0 V_g q_m + c_0^2 k_{eq}^2 V_0^2 + 4 c_0 k_{eq} V_0^2}}{2 k_{eq} V_0}

c = \frac{c_0 k_{eq} V_0 - k_{eq} V_g q_m - V_0 \pm \sqrt{V_0^2 + 2 k_{eq} V_0 V_g q_m - 2 c_0 k_{eq} V_0^2 + k_{eq}^2 V_g^2 q_m^2 - 2 c_0 k_{eq}^2 V_0 V_g q_m + c_0^2 k_{eq}^2 V_0^2 + 4 c_0 k_{eq} V_0^2}}{2 k_{eq} V_0}

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 + 4 c_0 k_{eq} V_0^2}}{2 k_{eq} V_0}

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}}

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