The Theory for Gradient Chromatography Revisited

by

Jan Ståhlberg

Academy of Chromatography

www.academyofchromatography.com

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)

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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)

us

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. u0. Let the solute be injected at x=0 and t=0.

The equation for the migrating zone can now be written:

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Brief review of the traditional theory (3)

The retention time is found from the integral:

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In many cases the retention factor of a solute decreases exponentially with , i.e.:

Where S is a constant characteristic of the solute.

Brief discussion of the traditional theory (4)

• For a linear gradient with slope G and for a solute with retention factor ki at t=0, integration gives:

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Mass balance approach(1)

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A fundamental starting point for an alternative gradient theory is the mass balance equation for chromatography:

c= solute concentration in the mobile phasen= solute concentration on the stationary phaseF= column phase ratioD= diffusion coefficient of the solute x= axial column coordinatet= time

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)).

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Mass balance approach(3)

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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 traditionalrepresentation of gradient chromatography theory is obtained.

Mass balance approach(4)

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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)

For this particular case the mass balance equation is:

Where ki 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.

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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:

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Gradient equation; Gaussian injection;S*G=5

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• Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10m, for the same input parameters. c0=10 mmol, t0=50,s ki=10, ,ti=10s

Gradient equation; Gaussian injection;S*G=1

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• Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10m, for the same input parameters. c0=10mmol, t0=50,s ki=10, ,ti=10s

Gradient equation; Gaussian injection;S*G=0.1

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• Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10m, for the same input parameters. c0=10 mmol, t0=50,s ki=10, ,ti=10s

Gradien equation; Gaussian injection;S*G=0.05

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• Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10m, for the same input parameters. c0=10 mmol, t0=50,s ki=10, ,ti=10s

Gradient equation; Gaussian injection: S*G=0.01

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• Solution of the gradient equation for a Gaussian profile, red line. Numeric simulation of the complete mass balance equation, H=10m, for the same input parameters. c0=10 mmol , t0=50,s ki=10, ,ti=10s

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:

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Mass balance approach(8)

Example:

Assume that the solute concentration is constant and independent of time. The solution of the

differential equation is:

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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.

• ……..

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