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 non-gaussian 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.

HLN = where L is the length of the column and H is

referred to as the height equivalent of a theoretical plate (HETP)

LH

2σ= something like the height equivalent of a separatory funnel

( )

πσ

σμ

2

2

2

2−−

=

x

eyσ is the familiar standard deviation

associated with a gaussian function

2

2

σLN = 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:

RtLστ = 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:

RtLW4

=σ

Using this relationship for σ to determine N in terms of W gives:

2

222

2

2

2

16

16⎟⎠⎞

⎜⎝⎛===Wt

tWL

LLN R

R

σ

2

16 ⎟⎠⎞

⎜⎝⎛=WtN R

the well-known relationship for column efficiency

2

22

16 RtLW

LH ==

σ 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:

LH

2σ=

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.

2

2

σLN =

2

22

16 RtLW

LH ==

σ 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:

( )uCCuBAH MS +++=

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.

( )uCCuBAH MS +++=

We are now in a position to understand the shape of the H versus u plots shown previously: