by Jan Ståhlberg Academy of Chromatography www.academyofchromatography.com. The Theory for Gradient Chromatography Revisited. Objective of the presentation. Discuss the background of the traditional theory for gradient chromatography.
The traditional derivation starts with the velocity of the migrating
zone as a function of the local retention factor.
Mobile phase velocity
Zone velocity Local retention factor as a
function of mobile phase
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:
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.
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
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)).
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.
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.
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.
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:
Assume that the solute is injected at x=0 as a profile according to
The solution of the differential equation is:
Assume that the solute concentration is constant and independent of time. The solution of the
differential equation is: