Van Deemter equation

The van Deemter equation, named after Jan van Deemter, describes how the variance per unit length of a chromatographic column depends on the linear velocity of the mobile phase. It accounts for physical, kinetic, and thermodynamic factors such as the pathways through the column, axial and longitudinal diffusion, and the mass transfer kinetics between the stationary and mobile phases. This equation was the first application of rate theory to the elution process in chromatography.

In liquid chromatography, the mobile phase velocity is defined as the exit velocity—the flow rate in mL/s divided by the cross-sectional area of the column-exit flow path. For packed columns, this area is typically taken as 0.6 times the column’s cross-sectional area. Alternatively, linear velocity can be calculated as the column length divided by the dead time. For gas mobile phases, a pressure correction is necessary. The variance per unit length is the ratio of column length to column efficiency, measured in theoretical plates.

The van Deemter equation is a hyperbolic function, predicting an optimal velocity that minimizes variance per unit length and thus maximizes column efficiency. The equation is expressed as HETP = A + B/u + C·u, where HETP is the height equivalent to a theoretical plate (a measure of resolving power), A is the eddy-diffusion parameter (related to channeling in non-ideal packing), B is the longitudinal diffusion coefficient, C is the resistance to mass transfer coefficient, and u is the linear velocity. In open tubular capillaries, the A term is zero because no packing exists, so channeling does not occur. For capillary columns, the Golay equation applies: HETP = B/u + (C_s + C_m)·u.

The minimum HETP occurs at the optimum velocity, found by differentiating the equation and setting the result to zero, yielding u = √(B/C). Plate count N is related to HETP by H = L/N, where L is column length. N can be estimated from a chromatogram using retention time t_R and peak standard deviation σ: N = (t_R/σ)². More practically, using peak width at half height W₁/₂ gives N = 8 ln(2) · (t_R/W₁/₂)², or with baseline width, N = 16 · (t_R/W_base)².