The Theory for Gradient Chromatography Revisited
by
Jan Ståhlberg
Academy of Chromatography
www.academyofchromatography.com
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.
(c) Academy of Chromatography 2007
Version: 05/20/07
2
Brief review of the traditional theory (1)
The traditional derivation starts with the velocity of the migrating
zone as a function of the local retention factor.
F(x,t)
us
Mobile phase velocity
Zone velocity
Version: 05/20/07
Local retention factor as a
function of mobile phase
composition F
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:
Version: 05/20/07
Brief review of the traditional theory (3)
The retention time is found from the integral:
In many cases the retention factor of a solute decreases
exponentially with F, i.e.:
Where S is a constant characteristic of the solute.
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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)
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
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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)
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.
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Mass balance approach(4)
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.
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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
•
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
Version: 05/20/07
Gradient equation; Gaussian injection;S*G=1
•
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
Version: 05/20/07
Gradient equation; Gaussian injection;S*G=0.1
•
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
Version: 05/20/07
Gradien equation; Gaussian injection;S*G=0.05
•
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
Version: 05/20/07
Gradient equation; Gaussian injection: S*G=0.01
•
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
Version: 05/20/07
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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