Analytical solutions of the ideal model for gradient liquid chromatography

The analytical solutions of the ideal model for gradient elution that ignores the influence of the solute concentration on the retention factor (k) were studied by using the method of characteristics for solving partial differential equations. It is found for any gradient profiles and solvent strength models used that the concentration of the solute will be discontinuous where the mobile-phase composition is. On a given characteristic curve, the product of the concentration and the retention factor is kept constant at the point where the concentration is continuous. At the point where the concentration is discontinuous, the product on the left side of this point is equal to that on the right side. We also discussed the basic equations to predict the retention time in gradient elution and introduced the injection time into them. For linear solvent strength stepwise and linear gradient elution, general expressions were proposed for the prediction and they can be used as the basis to derive others for specific gradient modes such as single linear, stepwise, and ladderlike gradients. For these modes, simple expressions to account for the band compression and the concentration change during the elution were also given.