Fundamentals of Rate theory in Chromatography

• In the rate theory, a number of different
peak dispersion processes were
proposed and expressions were
developed that described
•
• the contribution of each of the
processes to the total variance of the
eluted peak
the final equation that gave an
expression for the variance per unit
length of the column
The Rate Theory of Chromatography

The processes proposed were
•Eddy diffusion
•Longitudinal diffusion
•Resistance to mass transfer in the
mobile phase
•Resistance to mass transfer in the
stationary phase

This Theory
• Gives more realistic description of the
processes that work inside a column
• Takes account of the time taken for the
solute to equilibrate between the
stationary and mobile phase (unlike the
plate model, which assumes that
equilibration is infinitely fast)
• The resulting band shape or a
chromatographic peak is therefore
affected by the rate of elution

• It is also affected by the different paths
available to solute molecules as they travel
between particles of the stationary phase
• If we consider the various mechanisms which
contribute to band broadening, we arrive at the
Van Deemter equation:
HETP = A + B / u + C u
where u is the average velocity of the mobile
phase. A, B, and C are factors which contribute
to band broadening

The rate theory has resulted in a number of different
equations
All such equations give a type of hyperbolic function
that predicts a minimum plate height at an optimum
velocity and, thus, a maximum efficiency. At normal
operating velocities it has been demonstrated that the
Van Deemter equation gives the best fit to
experimental data
The Van Deemter Equation
H = A + B/u + u [CM + CS]

The rate theory provides another equation that allows
the calculation of the variance per unit length of a
column (the height of the theoretical plate, HETP) in
terms of the mobile phase velocity and other
physicochemical properties of the solute and
distribution system
H = σ2/L
σ = Standard deviation of the band
H = plate height, which is equal to H/dP
dP = particle diameter

Van Deemter plot
A plot of plate height vs average linear velocity of mobile
phase
Such plot is of considerable use in determining the optimum
mobile phase flow rate

Van Deemter model
u = L/ tM
H = A + B/u + u [CM +CS]
A: random movement through stationary phase
B: diffusion in mobile phase
C: interaction with stationary phase
H: plate height
u: average linear velocity

Term A
-molecules may travel
unequal distances
- independent of u
-depends on size of
stationary particles or
coating (TLC)
H = A + B/u + u [CM +CS]
Van Deemter model
time
Eddy diffusion
MP moves through the column
which is packed with stationary
phase. Solute molecules will take
different paths through the
stationary phase at random. This
will cause broadening of the
solute band, because different
paths are of different lengths.

Van Deemter model
H = A + B/u + u [CM +CS]
Term B
Longitudinal diffusion
B = 2γ DM
γ: Impedance factor due to
packing
DM: molecular diffusion
coefficient
B term dominates at low u, and
is more important in GC than LC
since DM(gas) > 104 DM(liquid)
One of the main causes of
band spreading is
DIFFUSION
The diffusion
coefficient measures
the ratio at which a
substance moves
randomly from a region
of higher concentration
to a region of lower
concentration

Van Deemter model
H = A + B/u + u [CM +CS]
Term B
Longitudinal diffusion
B = 2γ DM
γ: Impedance factor due to
packing
DM: molecular diffusion
coefficient
more important in GC than LC
since DM(gas) > 104 DM(liquid)
B - Longitudinal diffusion
The concentration of analyte is less
at the edges of the band than at
the centre. Analyte diffuses out
from the centre to the edges. This
causes band broadening. If the
velocity of the mobile phase is high
B term dominates at low u and is then the analyte spends less time
in the column, which decreases the
effects of longitudinal diffusion.

C : stationary phase-mass transfer
s
Cs = [(df)2]/Ds
f
d : stationary phase film thickness
s
D : diffusion coefficient of analyte in SP
CM: mobile phase–mass transfer
CM = [(dP)2]/DM
CM = [(dC)2]/DM
packed columns
open columns
Van Deemter model
H = A + B/u + u [CM +CS]
Term C
dP: particle diameter
dC: column diameter
Bandwidth
Stationary
phase
Mobile
phase
Elution
Broadened bandwidth
Slow
equilibration

H = A + B/u + u [CM +CS]
Van Deemter model
Term C (Resistance to mass transfer) Bandwidth
Stationary
phase
Mobile
phase
Elution
Slow
equilibration
Broadened bandwidth
The analyte takes a certain amount of time to equilibrate between the
stationary and mobile phase. If the velocity of the mobile phase is high,
and the analyte has a strong affinity for the stationary phase, then the
analyte in the mobile phase will move ahead of the analyte in the
stationary phase. The band of analyte is broadened. The higher the
velocity of mobile phase, the worse the broadening becomes.

• Figure 1 illustrates the effect of these terms,
both individually and accumulatively. Eddy
diffusion, the A term, is caused by a turbulence
in the solute flow path and is mainly unaffected
by flow rate. Longitudinal diffusion, the B term, is
the movement of an analyte molecule outward
from the center to the edges of its band. Higher
column velocities will limit this outward
distribution, keeping the band tighter. Mass
transfer, the C term, is the movement of analyte,
or transfer of its mass, between the mobile and
stationary phases. Increased flow has been
observed to widen analyte bands, or lower peak
efficiencies.

H = A + B/u + u [CM +CS]
Van Deemter model Figure 1

Decreasing particle size has been observed to limit
the effect of flow rate on peak efficiency—smaller
particles have shorter diffusion path lengths,
allowing a solute to travel in and out of the particle
faster. Therefore the analyte spends less time
inside the particle where peak diffusion can occur.
Figure 2 illustrates the Van Deemter plots for
various particle sizes. It is clear that as the particle
size decreases, the curve becomes flatter, or less
affected by higher column flow rates. Smaller
particle sizes yield better overall efficiencies, or
less peak dispersion, across a much wider range
of usable flow rates.

Smaller particle sizes yield higher overall peak
efficiencies and a much wider range of usable flow
rates (Figure 2)

Resolution
•Ideal chromatogram exhibits a distinct
separate peak for each solute
in reality: chromatographic peaks often
overlap
•We call the degree of separation of two
peaks: resolution which is given as
resolution = peak separation/average
peak width

Resolution
•Resolution =∆ tr / wavg
•let’s take a closer look at the significance of
the problem:

Resolution
•So, separation of mixtures depends on:
–width of solute peaks (want narrow)
•efficiency
–spacing between peaks (want large
spacing)
•selectivity

Example
•What is the resolution of two Gaussian
peaks of identical width (3.27 s) and height
eluting at 67.3 s and 74.9 s, respectively?
•ANS: Resolution = 2.32

Fundamentals of Rate theory in Chromatography

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