The Theory for Gradient Chromatography Revisited: The Theory for Gradient Chromatography Revisited by Jan Ståhlberg Academy of Chromatography www.academyofchromatography.com


Objective of the presentation: Version: 05/20/07 (c) Academy of Chromatography 2007 2 Objective of the presentation Discuss the background of the traditional theory for gradient chromatography. Show how a more fundamental and general theory for gradient chromatography can be obtained. Show some applications of the general theory.


Brief review of the traditional theory (1): Brief review of the traditional theory (1) Version: 05/20/07 The traditional derivation starts with the velocity of the migrating zone as a function of the local retention factor. Zone velocity Local retention factor as a function of mobile phase composition F Mobile phase velocity F(x,t) us


Brief review of the traditional theory (2): Brief review of the traditional theory (2) Version: 05/20/07


Brief review of the traditional theory (3): Brief review of the traditional theory (3) Version: 05/20/07 In many cases the retention factor of a solute decreases exponentially with F, i.e.: Where S is a constant characteristic of the solute.


Brief discussion of the traditional theory (4): Brief discussion of the traditional theory (4) Version: 05/20/07


Mass balance approach(1): Mass balance approach(1) Version: 05/20/07 A fundamental starting point for an alternative gradient theory is the mass balance equation for chromatography: c= solute concentration in the mobile phase n= solute concentration on the stationary phase F= column phase ratio D= diffusion coefficient of the solute x= axial column coordinate t= time


Mass balance approach(2): Mass balance approach(2) Version: 05/20/07


Mass balance approach(3): Mass balance approach(3) Version: 05/20/07 The mass balance equation becomes:. Here, the diffusive term has been omitted. The equation is the analogue of the ideal model for chromatography. The term ∂n/∂Φ is a function of c, i.e. In the limit c→0, the traditional representation of gradient chromatography theory is obtained.


Mass balance approach(4): Mass balance approach(4) Version: 05/20/07 For a solute it is often found that: Where c is the concentration of the solute in the mobile phase and k0 the retention factor of the solute when Ф =0. The function ∂Ф/∂t is known and determined by the experimenter. For a linear gradient it is equal to the slope, G, of the gradient.  


Mass balance approach(5): Mass balance approach(5) Version: 05/20/07


Mass balance approach(6): Mass balance approach(6) Version: 05/20/07


Gradient equation; Gaussian injection;S*G=5: Gradient equation; Gaussian injection;S*G=5 Version: 05/20/07 Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10mm, for the same input parameters. c0=10 mmol, t0=50,s ki=10, ,ti=10s


Gradient equation; Gaussian injection;S*G=1: Gradient equation; Gaussian injection;S*G=1 Version: 05/20/07 Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10mm, for the same input parameters. c0=10mmol, t0=50,s ki=10, ,ti=10s


Gradient equation; Gaussian injection;S*G=0.1: Gradient equation; Gaussian injection;S*G=0.1 Version: 05/20/07 Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10mm, for the same input parameters. c0=10 mmol, t0=50,s ki=10, ,ti=10s


Gradien equation; Gaussian injection;S*G=0.05: Gradien equation; Gaussian injection;S*G=0.05 Version: 05/20/07 Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10mm, for the same input parameters. c0=10 mmol, t0=50,s ki=10, ,ti=10s


Gradient equation; Gaussian injection: S*G=0.01: Gradient equation; Gaussian injection: S*G=0.01 Version: 05/20/07 Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10mm, for the same input parameters. c0=10 mmol , t0=50,s ki=10, ,ti=10s


Mass balance approach(7): Mass balance approach(7) Version: 05/20/07


Mass balance approach(8): Mass balance approach(8) Version: 05/20/07


Conclusions: Conclusions Version: 05/20/07