Van Deemter's equation

The Van Deemter equation in theoretical plates. The Van Deemter equation is a hyperbolic function that predicts that there is an optimum velocity at which there will be the minimum variance per unit column length and, thence, a maximum efficiency. The Van Deemter equation was the result of the first application of rate theory to the chromatography elution process.

Van Deemter equation

The Van Deemter equation for the plate height (H) is:

$H = A + \frac{B}{u} + C \cdot u$

Where

A = Eddy-diffusion
B = Longitudinal diffusion
C = mass transfer kinetics of the analyte between mobile and stationary phase
u = Linear Velocity.

A is equal to the multiple paths taken by the chemical compound, in open tubular capillaries this term will be zero as there are no multiple paths. The multiple paths occur in packed columns where several routes through the column packing, which results in band spreading.

B/u is equal to the longitudinal diffusion of the particles of the compound.

Cu is equal to the equilibration point. In a column, there is an interaction between the mobile and stationary phases, Cu accounts for this.

Plate count

The plate height given as:

$H = \frac{L}{N} \,$

with $L \,$ the column length and : $N\,$ the plate count can be estimated from a chromatogram by analysis of the retention time $t_R \,$ for each component and its standard deviation $\sigma \,$ as a measure for peak width, provided that the elution curve represents a Gaussian curve.

In this case the plate count is given by ^[3]:

$N = \left(\frac{t_R}{\sigma}\right)^2 \,$

By using the more practical peak width at half height $W_{1/2} \,$ the equation is:

$N = 5.55 \cdot \left(\frac{t_R}{W_{1/2}}\right)^2 \,$

or with the width at the base of the peak:

$N = 16 \cdot \left(\frac{t_R}{W_{base}}\right)^2 \,$

Expanded van Deemter

The Van Deemter equation can be further expanded to:

$H = 2\lambda d_p + 2GD_m/\mu + \omega(d_p \mbox{ or } d_c)^2\mu/D_m +Rd_f^2\mu/D_s$

Where:

H is plate height
λ is particle shape (with regard to the packing)
d_p is particle diameter
G, ω, and R are constants
D_m is the diffusion coefficient of the mobile phase
d_c is the capillary diameter
d_f is the film thickness
D_s is the diffusion coefficient of the stationary phase.

Van Deemter's equation indicates that band broadening mechanisms are proportionally dependent on flow rate, inversely proportional to flow rate and independent of flow rate.

References

^ Van Deemter, et al.: Chem. Eng. Sc., 5 (1956) 271
^ van Deemter JJ, Zuiderweg FJ and Klinkenberg A (1956). Longitudinal diffusion and resistance to mass transfer as causes of non ideality in chromatography. Chem. Eng. Sc. 5: 271–289.
^ Gold Book definition Link

Chromatography

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