Rate theory

• 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 ChromatographyThe 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 of Chromatography

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 rate theory has resulted in a number of different equations

The Van Deemter Equation

H = A + B/u + u [CM + CS]

The Rate Theory of Chromatography

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 plotA 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

The Rate Theory of Chromatography

H = A + B/u + u [CM +CS]

Van Deemter model

u = L/ tM

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 diffusionMP 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.

Term B

H = A + B/u + u [CM +CS]

Van Deemter model

Longitudinal diffusion

B = 2γ DM

γ: Impedance factor due to packing DM: molecular diffusion coefficientB 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 DIFFUSIONThe diffusion coefficient measures the ratio at which a substance moves randomly from a region of higher concentration to a region of lower concentration

Term B

H = A + B/u + u [CM +CS]

Van Deemter model

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)

B - Longitudinal diffusionThe 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 then the analyte spends less time in the column, which decreases the effects of longitudinal diffusion.

Cs: stationary phase-mass transferCs = [(df)2]/Ds

df: stationary phase film thicknessDs: diffusion coefficient of analyte in SP

CM: mobile phase–mass transfer

CM = [(dP)2]/DM packed columnsCM = [(dC)2]/DM open columns

H = A + B/u + u [CM +CS]

Van Deemter model

Term C

dP: particle diameterdC: column diameter

BandwidthStationary

phase

Mobilephase

Elution

Broadened bandwidth

Slow equilibration

H = A + B/u + u [CM +CS]

Van Deemter model

Term C (Resistance to mass transfer) BandwidthStationary

phase

Mobilephase

Elution

Broadened bandwidth

Slow equilibration

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