Theory of Gradient Elution Liquid Chromatography with Linear Solvent Strength: Part 1. Migration and Elution Parameters of a Solute Band

Abstract

Based on the previous theoretical developments most notably by Snyder, this report offers the most complete theoretical framework of gradient elution LC with linear solvent strength (LSS). All statements of the theory are formulated as explicit mathematical expressions. The physics of chromatography in general and of the LSS model in particular were used only to justify the most basic mathematical expressions of the framework. Everything else was obtained by means of verifiable mathematical transformations. The framework was used for derivation of the largest systematic collection of mathematical expressions describing migration and elution parameters of a solute band. Majority of these expressions are new. They include not only the elution parameters of a band, but also previously unknown migration parameters as functions of distance and time traveled by the band. The set of the band parameters in this report was chosen on the basis of the needs for the study of the peak width formation (part 2 of this series) and for detailed study of performance of gradient LC similar to that recently published for temperature-programed GC. As an illustration of the utility of several parameters considered here, a simple way of prediction of a possibility of the reversal of a solute elution order due to the change in the gradient steepness has been found.

Appendix: Verification of Eq. ( 28 )

It was pointed out in the text following Eq. (28) that Eqs. (27) and (28) can be verified by their direct substitution in Eq. (22). Equation (27) is the simplest of the two and its equivalents are known from literature [3]. On the other hand, Eq. (28) is more complex and was previously unknown. Its verification is provided below. Similar approach can be used for verification of Eq. (27) if necessary.

It follows from the definition of function W(x) [see text following Eq. (28)] that its derivative W′(x) can be found from expression

$$ {\text{W}}^{\prime} \left(x \right) + \frac{{{\text{W}}^{\prime} \left(x \right)}}{{{\text{W}}\left(x \right)}} = \frac{1}{x}. $$

(63)

This yields

$$ {\text{W}}^{\prime} \left(x \right) = \frac{{{\text{W}}\left(x \right)}}{{\left({1 + {\text{W}}\left(x \right)} \right)x}}. $$

(64)

As a result, differentiation of Eq. (28) by τ yields

$$ \frac{{{\text{d}}\zeta}}{{{\text{d}}\tau}} = \left({\frac{{{\text{W}}\left({\frac{1}{{k_{\text{init}}}}\exp \left({\frac{1}{{k_{\text{init}}}} + r_{\phi} \tau} \right)} \right)}}{{1 + {\text{W}}\left({\frac{1}{{k_{\text{init}}}}\exp \left({\frac{1}{{k_{\text{init}}}} + r_{\phi} \tau} \right)} \right)}}} \right). $$

(65)

Inversion (dτ/dζ) of this derivative [i.e., the derivative of τ by ζ in the left-hand side of Eq. (22)] is

$$ \frac{{{\text{d}}\tau}}{{{\text{d}}\zeta}} = 1 + \frac{1}{{{\text{W}}\left({\frac{1}{{k_{\text{init}}}}\exp \left({\frac{1}{{k_{\text{init}}}} + r_{\phi} \tau} \right)} \right)}}. $$

(66)

On the other hand, substitution of Eq. (28) in the right-hand side of Eq. (22) yields

$$ 1 + k_{\text{init}} {\text{e}}^{{r_{\phi} (\zeta - \tau \,)}} = 1 + k_{\text{init}} \exp \left({{\text{W}}\left({\frac{1}{{k_{\text{init}}}}\exp \left({\frac{1}{{k_{\text{init}}}} + r_{\phi} \tau} \right)} \right) - \frac{1}{{k_{\text{init}}}} - r_{\phi} \tau} \right)^{\,}, $$

(67)

$$ 1 + k_{\text{init}} {\text{e}}^{{r_{\phi} ({\zeta - \tau})}} = 1 + \frac{{k_{\text{init}}}}{{{ \exp }\left({r_{\phi} \tau + \frac{1}{{k_{\text{init}}}}} \right)}}\exp \left({{\text{W}}\left({\frac{1}{{k_{\text{init}}}}\exp \left({\frac{1}{{k_{\text{init}}}} + r_{\phi} \tau} \right)} \right)} \right) $$

(68)

As e^W(x) = x/W(x) [see text following Eq. (28)], the last expression can be rearranged as

$$ 1 + k_{\text{init}} {\text{e}}^{{r_{\phi} ({\zeta - \tau})}} = 1 + \frac{1}{{{W}\left({\frac{1}{{k_{\text{init}}}}\exp \left({\frac{1}{{k_{\text{init}}}} + r_{\phi} \tau} \right)} \right)}}. $$

(69)

Comparison of Eqs. (66) and (69) confirms that the solution in Eq. (28) satisfies Eq. (22).