Van Deemter equation

The van Deemter equation relates height equivalent to a theoretical plate (HETP) of a chromatographic column to the various flow and kinetic parameters which cause peak broadening, as follows:

${\displaystyle HETP=A+{\frac {B}{u}}+(C_{s}+C_{m})\cdot u}$

Where

• HETP = a measure of the resolving power of the column [m]
• A = Eddy-diffusion parameter, related to channeling through a non-ideal packing [m]
• B = diffusion coefficient of the eluting particles in the longitudinal direction, resulting in dispersion [m² s⁻¹]
• C = Resistance to mass transfer coefficient of the analyte between mobile and stationary phase [s]
• u = speed [m s⁻¹]

In open tubular capillaries, the A term will be zero as the lack of packing means channeling does not occur. In packed columns, however, multiple distinct routes ("channels") exist through the column packing, which results in band spreading. In the latter case, A will not be zero.

The form of the Van Deemter equation is such that HETP achieves a minimum value at a particular flow velocity. At this flow rate, the resolving power of the column is maximized, although in practice, the elution time is likely to be impractical. Differentiating the van Deemter equation with respect to velocity, setting the resulting expression equal to zero, and solving for the optimum velocity yields the following:

${\displaystyle u={\sqrt {\frac {B}{C}}}}$

The plate height given as:

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

with ${\displaystyle L\,}$ the column length and ${\displaystyle N\,}$ the number of theoretical plates can be estimated from a chromatogram by analysis of the retention time ${\displaystyle t_{R}\,}$ for each component and its standard deviation ${\displaystyle \sigma \,}$ as a measure for peak width, provided that the elution curve represents a Gaussian curve.

In this case the plate count is given by:[2]

${\displaystyle N=\left({\frac {t_{R}}{\sigma }}\right)^{2}\,}$

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

${\displaystyle N=8\ln(2)\cdot \left({\frac {t_{R}}{W_{1/2}}}\right)^{2}\,}$

or with the width at the base of the peak:

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

The Van Deemter equation can be further expanded to:[3]

${\displaystyle H=2\lambda d_{p}+{2\gamma D_{m} \over u}+{\omega (d_{p}{\mbox{ or }}d_{c})^{2}u \over D_{m}}+{Rd_{f}^{2}u \over D_{s}}}$

Where:

• H is plate height
• λ is particle shape (with regard to the packing)
• d_p is particle diameter
• γ, 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.
• u is the linear velocity

The Rodrigues equation, named for Alírio Rodrigues, is an extension of the Van Deemter equation used to describe the efficiency of a bed of permeable (large-pore) particles.[4]

The equation is:

${\displaystyle HETP=A+{\frac {B}{u}}+C\cdot f(\lambda )\cdot u}$

where

${\displaystyle f(\lambda )={\frac {3}{\lambda }}\left[{\frac {1}{\tanh(\lambda )}}-{\frac {1}{\lambda }}\right]}$

and ${\displaystyle \lambda }$ is the intraparticular Péclet number.

• Resolution (chromatography)
• Jan van Deemter