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

2. 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.

3. 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  Mobile phase velocity  (x,t) usus

4. Brief review of the traditional theory (2) Introduce the coordinate z where : Assume that a given composition of the mobile phase migrates through the column with the same velocity as the mobile phase, i.e. u 0. Let the solute be injected at x=0 and t=0. The equation for the migrating zone can now be written: Version: 05/20/07

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

6. Brief discussion of the traditional theory (4) For a linear gradient with slope G and for a solute with retention factor k i at t=0, integration gives: Version: 05/20/07

7. 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

8. Mass balance approach(2) The stationary phase concentration is a function of the mobile phase composition, Φ, i.e. n=n(c,Ф(x,t)). This means that: For a linear adsorption isotherm F*δn/ δ c is equal to the retention factor k(Ф(x,t)). Version: 05/20/07

9. 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.

10. 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 k 0 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.

11. Mass balance approach(5) For this particular case the mass balance equation is: Where k i is the initial retention factor at t=0. The solution of this equation is of the form: where f(x,t) is determined by the boundary and initial conditions. Version: 05/20/07

12. Mass balance approach(6) Example: Assume that the solute is injected at x=0 as a Gaussian profile according to The solution of the differential equation is found to be: Version: 05/20/07

13. 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=10  m, for the same input parameters. c 0 =10 mmol, t 0 =50,s k i =10,,t i =10s

14. 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=10  m, for the same input parameters. c 0 =10mmol, t 0 =50,s k i =10,,t i =10s

15. 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=10  m, for the same input parameters. c 0 =10 mmol, t 0 =50,s k i =10,,t i =10s

16. 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=10  m, for the same input parameters. c 0 =10 mmol, t 0 =50,s k i =10,,t i =10s

17. 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=10  m, for the same input parameters. c 0 =10 mmol, t 0 =50,s k i =10,,t i =10s

18. Mass balance approach(7) Example: Assume that the solute is injected at x=0 as a profile according to The solution of the differential equation is: Version: 05/20/07

19. Mass balance approach(8) Example: Assume that the solute concentration is constant and independent of time. The solution of the differential equation is: Version: 05/20/07

20. Conclusions A fundamental and general theory for gradient chromatography can be obtained from the mass balance equation for chromatography. The traditional theory for gradient chromatography is a special case of a more general theory, it is valid in the limit c(solute) 0. By neglecting the dispersive term in the mass balance equation, algebraic solutions are easily found. Practical consequences: By comparing experimental data with the exact solution, the effect of dispersion can be quantified. …….. Version: 05/20/07