Chromatographic Efficiency (peak width)
Concentration profiles of analytes A and B at two
points in time in the course of a separation:
where t2 > t1
Two phenomena are apparent:
1. The concentration profiles become increasingly
separated in space (separation in time at the end of
the column)
2. The concentration profiles become increasingly
broad
As long as the peak separation increases at a faster rate than peak width,
increased separation is obtained as column length increases.)
Band separation and band broadening are both kinetic
processes:
if rate of band separation > rate of band broadening,
a useful separation can be achieved
if rate of band separation < rate of band broading,
a useful separation is not obtained
Therefore, there are two general approaches to
improving a separation:
1. increase rate of separation – via α, selectivity
factor (stationary, mobile phases) (b)
2. decrease band broadening (reduce peak widths) (c)
We now examine the issue of peak shapes and
widths:
At the beginning of a chromatographic experiment,
the solution injected onto the column is a narrow
rectangular plug. This plug does not move through
the column without any changes in its shape. The
random nature of transfer between phases gives rise
to a gaussian peak shape (ideally)…
Note that the width of a chromatographic peak is
ultimately limited by the width of the sample injected
onto the column.
In cases in which the distribution constant for an
analyte, KA, is not actually constant (as in the case in
which KA is analyte concentration dependent) a nongaussian peak can be observed:
An example of a case where KA is concentration
dependent: column overloading – too few sites on
the stationary phase for the quantity of analyte
injected
In the overloaded condition (b), there are too few
sites for binding and some analyte species rush
onwards through the column so that they elute
slightly faster. The large number of bound analyte at
the trailing edge gives rise to a relatively high
concentration in this region thereby increasing the
likelihood that they will bind again. This tends to
prolong the retention of this population.
The concentration dependence of KA can give rise to
either tailing or fronting:
From this point, we concentrate on gaussian shapes...
A major goal in chromatography is to separate the
analyte species in the minimal time. Hence, it is
desirable to maintain narrow peak widths (minimize
broadening) so that short columns (short elution
times) can be used
A parameter that is used to quantify the width of a
chromatographic peak is called chromatographic
efficiency, N. Chromatographic efficiency is
inversely related to the rate of band broadening per
unit time.
L
N = where L is the length of the column and H is
H
referred to as the height equivalent of a theoretical
plate (HETP)
H=
σ2
L
something like the height equivalent of a separatory funnel
− ( x − μ )2
y=
e
2σ 2
σ 2π σ is the familiar standard deviation
associated with a gaussian function
N=
L2
σ2
note that the units cancel, N is dimensionless
Determination of N from experimental data:
Neither L nor σ are directly obtained from a
chromatogram. However, these parameters are
related to retention time and peak width, which are
obtained directly from the chromatogram.
in units of time, σ (units of length) translates to:
τ=
σ
L
tR
where L/tR is the average linear velocity
The area of the triangle drawn from the peak contains
roughly 96% of the peak area…
96% of the area of a gaussian peak is contained
within ±2σ.
In the time domain, this translates to ±2τ.
Therefore, the base of the triangle,
W, is +2τ-(-2τ) ≈ 4τ
Hence τ ≈ W/4
Solving for σ in terms of W gives:
σ=
LW
4t R
Using this relationship for σ to determine N in terms
of W gives:
2
2
L
L
N= 2 = 2 2
LW
σ
16t R2
⎛ tR ⎞
= 16⎜ ⎟
⎝W ⎠
2
2
⎛ tR ⎞
N = 16⎜ ⎟
⎝ W ⎠ the well-known relationship for
column efficiency
σ2
LW 2
H=
=
L
16t R2 need to know L to determine H
Variables that affect N:
Effect of mobile phase flow rate:
Note that the plate heights that apply to liquid
chromatography are smaller than those associated
with gas chromatography:
H=
σ2
L
However, L is limited in LC due to large pressures
required for long columns (relative to GC). Hence, N
values for GC are typically much greater than those
for LC because GC can more than compensate for
large H by use of much longer L.
N=
L2
σ2
σ2
LW 2
H=
=
L
16t R2 how to determine HETP from a peak
How to determine H from theory? How can theory
help us design improved separations?
Band broadening:
No single model accounts well for band broadening
over the wide range of conditions used in
chromatography. However, the most widely used
model is based on the van Deemter equation:
B
H = A + + (CS + CM )u
u
where u is the linear flow rate for the mobile phase,
A is an eddy diffusion coefficient, B is a longitudinal
diffusion coefficient, CS is a mass transfer coefficient
in the stationary phase, and CM is a mass transfer
coefficient in the mobile phase.
(Note that this relationship has a term that is
independent of u (A), a term that is inversely
related to u (B/u), and a term that is directly
related to u, ((CS+Cu)u).)
Eddy diffusion, “A” term in van Deemter equation:
Arises from the possibility of different path lengths
for different analyte species
indicated to be independent of flow rate but at very
low flow rates, diffusion can give rise to an averaging
effect that minimizes broadening from this source.
Eddy diffusion is minimized by use of small and
highly uniform packing materials. Capillary columns
have only one channel so that eddy diffusion is not an
issue with capillary column chromatography.
Longitudinal Diffusion, B/u:
Mass transfer due to a concentration gradient (see
also our discussion of concentration polarization in
electrochemistry)
Note that the contribution due to longitudinal
diffusion is inversely related to u, as one would
expect. The less time in the column, the less time
there is for longitudinal diffusion.
The C terms are diffusion terms for motion
orthogonal to mobile phase flow:
CMu: The mobile phase mass transfer term.
If u is large relative to the rate at which analyte can
move in the mobile phase to the surface, a
broadening of the analyte distribution occurs.
CSu: The stationary phase mass transfer term
Time spent in a liquid stationary phase is dependent
upon the thickness of the phase and the diffusion
coefficient of the analyte in the stationary phase.
Low mobility of the analyte and thick stationary
phase increase time spent in the stationary phase. If
the desorption rate from the stationary phase is slow
relative to u, broadening occurs.
The rate of diffusion to the surface and the rate of
adsorption/desorption from the surface relative to u
determine broadening due to the C terms in the van
Deemter equation.
B
H = A + + (CS + CM )u
u
We are now in a position to understand the shape of
the H versus u plots shown previously: