The van Deemter Equation

A chromatography column is a race. No winner is declared until every runner crosses the same line. But each runner finishes at a slightly different time. The spread in those finish times — band broadening — decides the outcome. It determines whether two compounds emerge as separate peaks or one smeared blob. In 1956, three Shell Oil researchers wrote down a single equation. They were J. J. van Deemter, F. J. Zuiderweg, and A. Klinkenberg. Their equation predicts how peak width per unit column length behaves. It depends on how fast the mobile phase is pushed through the column. That equation is plotted as height equivalent to a theoretical plate (H) against linear velocity (u). This plot produces a characteristic U-shaped curve with a genuine minimum. That minimum is a flow rate where the column is as efficient as possible. This matters because efficiency is not free to maximize by brute force. A chromatographer cannot simply run the mobile phase infinitely slowly. Molecules diffuse and bands spread with time regardless of flow. Nor can they run it arbitrarily fast. Mass transfer between phases becomes the bottleneck at high speed. Bands then spread with distance instead. The van Deemter equation resolves this tension quantitatively. It assigns each broadening mechanism its own term and velocity dependence. Then it sums them to locate the sweet spot. The big picture is that a chromatographic peak's width is not one phenomenon. It is three, superimposed: multipath eddy diffusion, longitudinal molecular diffusion, and finite-rate mass transfer. This transfer occurs between mobile and stationary phases. Each mechanism has a distinct physical origin. These are packing geometry, Fick's laws, and phase-equilibration kinetics, respectively. Each also responds differently to velocity. Only by separating them can an experimentalist diagnose why a column underperforms. Only then can they know whether to repack it, change particle size, lower the temperature, or simply turn a knob on the pump. This article builds the van Deemter equation term by term. It starts from the physics of Rate Theory and Band Broadening. It derives the optimum velocity and minimum plate height algebraically. It works three realistic numerical examples. These span packed GC, capillary GC, and modern HPLC columns. It closes with the practical limits of the model. It shows where the model fails. It shows where the related Golay equation takes over. It shows how the same curve underlies day-to-day method development.

Key takeaways

$H = A + B/u + Cu$ describes plate height as a sum of eddy diffusion, longitudinal diffusion, and mass-transfer resistance terms.
$A$ is flow-independent (packing irregularity); $B/u$ dominates at low velocity (diffusion has time to act); $Cu$ dominates at high velocity (equilibration lags behind the flow).
The minimum plate height $H_{min}$ occurs at the optimum velocity $u_{opt} = \sqrt{B/C}$ , found by setting $dH/du = 0$ .
At $u_{opt}$ , $H_{min} = A + 2\sqrt{BC}$ — a clean, testable relationship between the curve's minimum and its two velocity-dependent terms.
Real columns are usually run above $u_{opt}$ in practice (faster analysis, small efficiency penalty) because the $Cu$ branch rises gently compared to how steeply $B/u$ falls near the origin.
Capillary (open-tubular) columns replace the $A$ term with a modified mass-transfer expression (the Golay equation), since there is no packing to generate multiple flow paths.
Key idea: plate height is a competition between diffusion (which wants slow flow) and mass-transfer kinetics (which wants fast flow), and the van Deemter curve's minimum is where neither mechanism dominates.

Where this lives today

Every modern chromatography method-development workflow, whether in a pharmaceutical QC lab running USP-mandated HPLC assays or a petrochemical lab running detailed hydrocarbon GC, begins conceptually with a van Deemter curve, even when the software doing the optimization never shows the equation explicitly. Instrument vendors publish typical $A$ , $B$ , $C$ values (or, more often, plots of $H$ versus flow rate) for their commercial columns, letting a method developer pick a flow rate close to $u_{opt}$ as a starting point before adjusting for practical speed constraints — this is the direct, everyday link to Optimizing Efficiency and Flow Rate. In modern ultra-high-performance liquid chromatography (UHPLC), sub-2 μm fully porous or superficially porous ("core–shell") particles push $u_{opt}$ to much higher velocities while simultaneously lowering $H_{min}$ , exactly as illustrated in the applied worked example — this is why UHPLC columns can achieve GC-like efficiency in analysis times of a minute or less, at the engineering cost of operating at pressures exceeding 15,000 psi, a tradeoff the van Deemter curve itself does not capture but that governs HPLC Instrumentation and Overview pump and fitting design. In capillary gas chromatography, the elimination of the $A$ term via open-tubular geometry (no packing, hence no multipath dispersion) is the single biggest efficiency advance in the technique's history, routinely delivering $10^5$ – $10^6$ theoretical plates on a 30 m fused-silica column — orders of magnitude beyond what any packed column can achieve — a direct consequence of the modified van Deemter (Golay) relationship central to Gas Chromatography Instrumentation. Beyond routine analysis, the same rate-theory logic underlies fields as far afield as chromatographic bioseparations of proteins (where slow diffusion coefficients for large biomolecules shift $B$ dramatically downward, forcing very low optimum velocities and motivating monolithic or perfusion stationary phases that bypass diffusion-limited pore transport), and supercritical fluid chromatography (where the unusually high diffusivity of supercritical CO₂ mobile phases shifts the entire curve toward much higher optimum velocities than either GC or HPLC, enabling fast chiral separations in the pharmaceutical industry).

Ask the tutor: How does the Golay equation for open-tubular columns modify the van Deemter picture when there's no packing at all?

The physical picture

Picture a packed chromatographic column as a bed of small, roughly spherical particles with mobile phase percolating through the interstitial spaces. An injected band of analyte molecules — sharp as a knife-edge at the inlet — broadens as it migrates, and by the time it reaches the detector it has become a roughly Gaussian peak with standard deviation $\sigma$ in length units. The plate height $H$ is defined operationally as

H = \frac{\sigma^2}{L}

where $L$ is the column length; it measures how much "smearing variance" accumulates per unit length traveled, and it is the natural bridge between the molecular-scale broadening mechanisms and the macroscopic peak widths measured at the detector, related to plate number by $N = L/H$ . Three independent processes contribute variance additively (variances of independent random processes sum), and each has a different physical story.

Eddy diffusion (the $A$ term). In a packed bed, molecules do not all take the same path. Some squeeze through wide gaps between particles and move quickly; others thread through narrow, tortuous channels and lag behind. This is a purely geometric effect — a consequence of how the particles happen to be packed — and it does not depend on how fast the mobile phase flows, at least not to first order. A more uniformly packed column with smaller, more monodisperse particles reduces $A$ .

Longitudinal molecular diffusion (the $B/u$ term). Even with a perfectly uniform flow path, analyte molecules diffuse randomly along the direction of flow, spreading out from the concentration gradient at the band's edges exactly as described by Fick's second law. The longer a molecule spends in the column, the more it diffuses — so this term scales inversely with velocity: slow flow means long residence time means more longitudinal spreading.

Resistance to mass transfer (the $Cu$ term). Analyte partitions between mobile and stationary phases, and that partitioning is not instantaneous. At high velocity, a molecule in the mobile phase is swept forward before its partner molecules still dissolved in (or adsorbed on) the stationary phase can catch up by diffusing back out. This lag broadens the band, and it gets worse as velocity increases — the faster the flow, the more the system runs behind local equilibrium.

Figure

The three van Deemter contributions and their velocity dependence

The sum of these three curves — a horizontal line, a falling hyperbola, and a rising straight line — is a U-shaped curve with one genuine minimum. That minimum is the operating point at which the column delivers the smallest possible plate height, hence the largest possible plate number $N=L/H$ , hence the sharpest possible peaks for a given column length.

Figure

Eddy diffusion: multiple flow paths through a packed bed

The core ideas

The van Deemter equation states plate height as the sum of three independent contributions:

H = A + \frac{B}{u} + Cu

where:

$H$ = plate height (length, typically mm or μm) — variance accumulated per unit column length,
$A$ = eddy-diffusion coefficient (length units), independent of velocity, governed by particle size $d_p$ and packing quality,
$B$ = longitudinal-diffusion coefficient (length²/time), proportional to the mobile-phase molecular diffusion coefficient $D_m$ ,
$C$ = mass-transfer resistance coefficient (time), combining contributions from the stationary phase ( $C_S$ ) and mobile phase ( $C_M$ ),
$u$ = linear velocity of the mobile phase (length/time), the variable actually under experimental control via flow rate.

Each coefficient has its own theoretical expression, developed from the microscopic kinetic theory of the column:

A = 2\lambda d_p

where $\lambda$ is a packing-irregularity factor (order unity) and $d_p$ is the average particle diameter — smaller, more uniform particles give smaller $A$ .

B = 2\gamma D_m

where $\gamma$ is an obstruction/tortuosity factor (typically 0.6–0.7 for packed columns, since the packing itself partially blocks straight-line diffusion) and $D_m$ is the analyte's diffusion coefficient in the mobile phase.

C = C_S + C_M = \frac{q\, k'\, d_f^2}{(1+k')^2 D_S} + \frac{f(k')\, d_p^2}{D_M}

where $k'$ is the retention factor, $d_f$ is the stationary-film (or liquid-phase) thickness, $D_S$ and $D_M$ are diffusion coefficients in stationary and mobile phases, $q$ is a geometric constant, and $f(k')$ is a function of retention factor describing mobile-phase mass-transfer resistance. In practice, $A$ , $B$ , and $C$ are usually treated as empirical fitting constants obtained by measuring $H$ at several velocities and fitting the resulting curve — the microscopic formulas above justify the functional form, not necessarily precise numerical prediction, since real packed beds have void distributions more complex than the idealized model assumes.

Reading the equation. At low $u$ , the $B/u$ term dominates: a molecule lingers in the column and Fickian diffusion spreads it substantially along the axis of flow. At high $u$ , the $Cu$ term dominates: the flow outruns the equilibration kinetics between phases, and the very act of moving fast smears the band. The $A$ term sits underneath both, a velocity-independent floor set by how well the column is packed (or, in open-tubular columns, absent altogether since there is no packing to generate multiple paths).

Derivation AFinding the optimum velocity and minimum plate height

Goal. Starting from $H(u) = A + B/u + Cu$ , find the velocity $u_{opt}$ that minimizes $H$ , and the corresponding minimum plate height $H_{min}$ .

Step 1 — Differentiate. Treat $A$ , $B$ , $C$ as constants and differentiate $H$ with respect to $u$ :

\frac{dH}{du} = -\frac{B}{u^2} + C

The $A$ term vanishes because it does not depend on $u$ ; the $B/u$ term differentiates to $-B/u^2$ ; the $Cu$ term differentiates to $C$ .

Step 2 — Set the derivative to zero. A minimum (rather than an inflection or maximum) requires $dH/du = 0$ at the critical point:

-\frac{B}{u_{opt}^2} + C = 0

u_{opt}^2 = \frac{B}{C}

u_{opt} = \sqrt{\frac{B}{C}}

This is the central practical result: the optimum linear velocity depends only on the ratio of the diffusion coefficient $B$ to the mass-transfer coefficient $C$ , not on $A$ at all. A poorly packed column ( $A$ large) still has the same optimum velocity as a well-packed one with the same $B$ and $C$ — it just sits higher on the curve everywhere.

Step 3 — Confirm it is a minimum. The second derivative is

\frac{d^2H}{du^2} = \frac{2B}{u^3}

which is positive for all $u>0$ (since $B>0$ , $u>0$ ), confirming the critical point is indeed a minimum, not a maximum or saddle.

Step 4 — Substitute back to find $H_{min}$ . Insert $u_{opt} = \sqrt{B/C}$ into the original equation:

H_{min} = A + \frac{B}{\sqrt{B/C}} + C\sqrt{\frac{B}{C}}

Simplify each velocity-dependent term separately. For the second term,

\frac{B}{\sqrt{B/C}} = B \cdot \sqrt{\frac{C}{B}} = \sqrt{B^2 \cdot \frac{C}{B}} = \sqrt{BC}

For the third term,

C\sqrt{\frac{B}{C}} = \sqrt{C^2 \cdot \frac{B}{C}} = \sqrt{BC}

Both velocity-dependent terms contribute equally at the optimum — a direct consequence of the $u^{-1}$ and $u^{+1}$ powers meeting at a geometric mean. Adding them:

H_{min} = A + \sqrt{BC} + \sqrt{BC} = A + 2\sqrt{BC}

Limiting-case check. If mass transfer were instantaneous, $C \to 0$ , then $u_{opt} = \sqrt{B/C} \to \infty$ and $H_{min} \to A$ — the column could be run arbitrarily fast with efficiency limited only by packing geometry, which is physically sensible since there would be no kinetic penalty for speed. Conversely, as $C$ grows large relative to $B$ (sluggish mass transfer, viscous stationary phase, thick films), $u_{opt}$ shrinks toward zero and the achievable $H_{min}$ rises, correctly predicting that poorly-equilibrating phases force painfully slow, less efficient separations.

Go deeper · Derivation B · 1 minRelating plate height to observed peak width and column resolution

Derivation BRelating plate height to observed peak width and column resolution

Goal. Show how the abstract quantity $H$ connects to something actually measured on a chromatogram — the peak width at the detector — and hence to resolving power.

Step 1 — Define plate number from a Gaussian peak. For a Gaussian elution peak with retention time $t_R$ and baseline width $w_b$ (the width where tangents to the inflection points meet the baseline), the standard result from Gaussian statistics is that $w_b = 4\sigma_t$ , where $\sigma_t$ is the temporal standard deviation. Since $\sigma_t = \sigma_L / u$ (converting a length-domain spread into a time-domain spread via the velocity), and $t_R = L/u$ for an idealized non-retained-time-corrected picture, the plate number is defined as

N = \left(\frac{t_R}{\sigma_t}\right)^2 = 16\left(\frac{t_R}{w_b}\right)^2

Step 2 — Connect $N$ to plate height. By definition $H = \sigma_L^2 / L$ and $N = L/H$ , so

N = \frac{L}{H} = \frac{L}{\sigma_L^2/L} = \frac{L^2}{\sigma_L^2}

Since $\sigma_L = u\,\sigma_t$ and $L = u\,t_R$ , substituting gives back $N = (t_R/\sigma_t)^2$ exactly, confirming the two definitions of $N$ (length-based and time-based) are the same dimensionless number, as they must be for a self-consistent theory.

Step 3 — Show why minimizing $H$ maximizes resolution. Resolution between two adjacent peaks is

R_s = \frac{\sqrt{N}}{4}\left(\frac{\alpha - 1}{\alpha}\right)\left(\frac{k_2'}{1+k_2'}\right)

where $\alpha$ is the selectivity factor and $k_2'$ the retention factor of the second peak. Since $N = L/H$ , resolution scales as $R_s \propto \sqrt{L/H}$ : for fixed column length, minimizing $H$ (operating at $u_{opt}$ ) directly maximizes $\sqrt{N}$ and therefore $R_s$ . This is the practical payoff of the entire van Deemter analysis — it is not an abstract curve-fitting exercise but a direct lever on whether two real peaks are baseline-resolved or not.

Limiting-case check. As $H \to A$ (the theoretical floor when $B$ and $C$ contributions vanish), $N \to L/A$ , its maximum possible value for a given packing quality — confirming that no amount of flow-rate optimization can push efficiency past the multipath-diffusion limit set by particle packing.

Worked examples

Example 1 — Foundational: fitting A, B, C from three data points and finding u_opt. A packed GC column is characterized by measuring plate height at three linear velocities:

$u$ (cm/s)	$H$ (mm)
5.0	1.65
15.0	1.15
40.0	1.90

Assume the van Deemter form $H = A + B/u + Cu$ applies. With three data points and three unknowns, solve exactly. Write the three equations:

1.65 = A + B/5.0 + 5.0C

1.15 = A + B/15.0 + 15.0C

1.90 = A + B/40.0 + 40.0C

Subtract equation 1 from equation 2:

1.15 - 1.65 = B\left(\frac{1}{15.0}-\frac{1}{5.0}\right) + C(15.0-5.0)

-0.50 = B(0.06667 - 0.20000) + 10.0C = -0.13333B + 10.0C

Subtract equation 2 from equation 3:

1.90 - 1.15 = B\left(\frac{1}{40.0}-\frac{1}{15.0}\right) + C(40.0-15.0)

0.75 = B(0.02500 - 0.06667) + 25.0C = -0.04167B + 25.0C

Now solve the two linear equations

-0.13333B + 10.0C = -0.50 \qquad (i)

-0.04167B + 25.0C = 0.75 \qquad (ii)

Multiply (i) by 2.5:

-0.33333B + 25.0C = -1.25

Subtract (ii):

(-0.33333B + 25.0C) - (-0.04167B + 25.0C) = -1.25 - 0.75

-0.29167B = -2.00 \ \Rightarrow\ B = 6.857 \ \text{mm}\,\cdot\,\text{cm/s}

Substitute into (i):

-0.13333(6.857) + 10.0C = -0.50

-0.9143 + 10.0C = -0.50 \ \Rightarrow\ 10.0C = 0.4143 \ \Rightarrow\ C = 0.04143 \ \text{mm}\,\cdot\,\text{s/cm}

Substitute both into equation 1:

1.65 = A + 6.857/5.0 + 5.0(0.04143) = A + 1.3714 + 0.2071

A = 1.65 - 1.5786 = 0.071 \ \text{mm}

So $A \approx 0.071\ \text{mm}$ , $B \approx 6.86\ \text{mm}\,\cdot\,\text{cm/s}$ , $C \approx 0.0414\ \text{mm}\,\cdot\,\text{s/cm}$ (three significant figures, limited by the precision of the input data). Now find the optimum velocity:

u_{opt} = \sqrt{\frac{B}{C}} = \sqrt{\frac{6.857}{0.04143}} = \sqrt{165.5} = 12.9\ \text{cm/s}

and the minimum plate height:

H_{min} = A + 2\sqrt{BC} = 0.071 + 2\sqrt{(6.857)(0.04143)}

= 0.071 + 2\sqrt{0.2841} = 0.071 + 2(0.5330) = 0.071+1.066 = 1.137\ \text{mm}

This is consistent with the measured minimum near $u=15$ cm/s ( $H=1.15$ mm) — the fitted optimum at 12.9 cm/s and $H_{min}=1.14$ mm land close to, but not exactly on, the middle data point, exactly as expected since real data has scatter and the true minimum generally falls between measured points.

Example 2 — Applied: comparing particle size and choosing a flow rate for an HPLC method. An analyst is developing an HPLC method and compares two columns of the same length and chemistry but different particle diameters: Column X (5 μm particles) and Column Y (3 μm particles). Reduced-plate-height theory and empirical calibration give, for this analyte and mobile phase: Column X: $A = 0.010$ mm, $B = 0.60 \times 10^{-2}\ \text{mm}^2/\text{s}$ , $C = 0.90\times10^{-3}\ \text{s}$ Column Y: $A = 0.006$ mm, $B = 0.60\times10^{-2}\ \text{mm}^2/\text{s}$ , $C = 0.35\times10^{-3}\ \text{s}$ (Both columns see the same mobile-phase diffusion coefficient, so $B$ is unchanged; smaller particles reduce $A$ via $A=2\lambda d_p$ and dramatically reduce $C$ because equilibration path lengths inside/around particles shrink with $d_p^2$ .) For Column X:

u_{opt,X} = \sqrt{\frac{B}{C}} = \sqrt{\frac{0.60\times10^{-2}}{0.90\times10^{-3}}} = \sqrt{6.667} = 2.582\ \text{mm/s}

H_{min,X} = A + 2\sqrt{BC} = 0.010 + 2\sqrt{(0.60\times10^{-2})(0.90\times10^{-3})}

= 0.010 + 2\sqrt{5.40\times10^{-6}} = 0.010 + 2(2.324\times10^{-3}) = 0.010+0.00465 = 0.01465\ \text{mm}

For Column Y:

u_{opt,Y} = \sqrt{\frac{0.60\times10^{-2}}{0.35\times10^{-3}}} = \sqrt{17.14} = 4.140\ \text{mm/s}

H_{min,Y} = 0.006 + 2\sqrt{(0.60\times10^{-2})(0.35\times10^{-3})} = 0.006 + 2\sqrt{2.10\times10^{-6}}

= 0.006 + 2(1.449\times10^{-3}) = 0.006 + 0.00290 = 0.00890\ \text{mm}

So Column Y (smaller particles) not only achieves a smaller $H_{min}$ (0.0089 mm vs 0.0147 mm — a 39% improvement in minimum plate height, hence roughly a 39% improvement in achievable $N$ for the same length) but does so at a higher optimum velocity (4.14 mm/s vs 2.58 mm/s). This is the central selling point of modern sub-3-μm and sub-2-μm HPLC particles: they push both axes of the van Deemter minimum in the favorable direction simultaneously — better efficiency and faster analysis — at the cost of requiring higher operating pressure (via the Kozeny–Carman/Darcy relation, pressure drop scales as $1/d_p^2$ , a pressure-versus-efficiency tradeoff not itself captured by the van Deemter equation but essential to the engineering decision of which column to run). For a 15.0 cm column, Column Y at $u_{opt}$ delivers

N = \frac{L}{H_{min}} = \frac{150\ \text{mm}}{0.00890\ \text{mm}} = 16{,}854 \approx 1.69\times10^4\ \text{plates}

compared with Column X's

N = \frac{150}{0.01465} = 10{,}239 \approx 1.02\times10^4\ \text{plates}

a substantial efficiency gain for the same column length, directly attributable to the reduced $A$ and $C$ terms.

Example 3 — Exam/competition level: working backward from resolution requirement to required flow rate, with a non-optimal operating point. A separation of two structurally similar steroids on a packed GC column requires a resolution of at least $R_s = 1.50$ to be considered baseline-resolved. The column is 2.00 m long, and van Deemter parameters measured for the more strongly retained analyte are

A = 0.65\ \text{mm}, \qquad B = 25.0\ \text{mm}\,\cdot\,\text{cm/s}, \qquad C = 0.0180\ \text{mm}\,\cdot\,\text{s/cm}

The selectivity factor is $\alpha = 1.060$ and the retention factor of the second peak is $k_2' = 8.40$ . The analyst wants the fastest possible analysis (highest $u$ ) that still meets $R_s \geq 1.50$ . Find the maximum allowable linear velocity.

Step 1 — Express $N$ required from the resolution equation. Rearranging

R_s = \frac{\sqrt{N}}{4}\left(\frac{\alpha-1}{\alpha}\right)\left(\frac{k_2'}{1+k_2'}\right)

for $N$ :

\sqrt{N} = \frac{4R_s}{\left(\dfrac{\alpha-1}{\alpha}\right)\left(\dfrac{k_2'}{1+k_2'}\right)}

Compute the two bracketed factors:

\frac{\alpha-1}{\alpha} = \frac{0.060}{1.060} = 0.05660

\frac{k_2'}{1+k_2'} = \frac{8.40}{9.40} = 0.8936

\sqrt{N} = \frac{4(1.50)}{(0.05660)(0.8936)} = \frac{6.00}{0.05058} = 118.6

N_{required} = 118.6^2 = 14{,}066 \approx 1.41\times10^4\ \text{plates}

Step 2 — Convert required $N$ into a maximum allowable plate height. With $L = 2.00\ \text{m} = 2000\ \text{mm}$ ,

H_{max} = \frac{L}{N_{required}} = \frac{2000\ \text{mm}}{14{,}066} = 0.1422\ \text{mm}

Any operating velocity giving $H \le 0.1422$ mm satisfies the resolution requirement; the analyst wants the largest $u$ still inside that window.

Step 3 — Solve $H(u) = H_{max}$ for $u$ , taking the larger root (since $H$ falls then rises, there are generally two velocities giving any $H$ above $H_{min}$ , and we want the one on the high- $u$ , mass-transfer-dominated branch). Set up

0.1422 = 0.65 + \frac{25.0}{u} + 0.0180\,u

Rearrange into standard quadratic form (multiply through by $u$ ):

0.0180\,u^2 + (0.65 - 0.1422)u + 25.0 = 0

0.0180\,u^2 + 0.5078\,u + 25.0 = 0

Apply the quadratic formula:

u = \frac{-0.5078 \pm \sqrt{(0.5078)^2 - 4(0.0180)(25.0)}}{2(0.0180)}

= \frac{-0.5078 \pm \sqrt{0.2579 - 1.800}}{0.0360}

The discriminant is negative ( $0.2579 - 1.800 = -1.542$ ), meaning there is no real solution — $H$ never reaches as low as 0.1422 mm for any velocity on this column with these parameters. This is an important, realistic outcome: check the achievable minimum first.

H_{min} = A + 2\sqrt{BC} = 0.65 + 2\sqrt{(25.0)(0.0180)} = 0.65 + 2\sqrt{0.450} = 0.65 + 2(0.6708) = 0.65+1.342 = 1.992\ \text{mm}

Since $H_{min} = 1.992\ \text{mm} \gg H_{max} = 0.1422\ \text{mm}$ , this column cannot achieve the required resolution at any flow rate — the required plate count of 14,066 vastly exceeds what a 2.00 m packed column with these $A$ , $B$ , $C$ values can deliver even at its very best operating point (which itself only gives $N_{achievable} = L/H_{min} = 2000/1.992 = 1004$ plates).

Conclusion and the real lesson. The analyst cannot simply "tune the flow rate" out of this problem — the fix must change $A$ , $B$ , or $C$ themselves: use a longer column (doubling $L$ doubles $N$ at fixed $H_{min}$ ), switch to smaller particles (lowering both $A$ and $C$ ), improve the stationary-phase film thickness (lowering $C_S$ ), or increase $\alpha$ by changing the stationary phase chemistry (which enters resolution outside the van Deemter framework entirely, often far more powerfully than any efficiency gain). This worked example demonstrates the discipline's most important practical habit: always check whether $H_{min}$ can meet the requirement before solving for a flow rate, because the van Deemter curve has a floor that plumbing alone cannot get under.

Limiting cases, scope & common traps

Regime	What happens to $H$	Physical interpretation
$u \to 0^+$	$H \to \infty$ (dominated by $B/u$ )	Molecules sit in the column so long that longitudinal diffusion smears the band without bound
$u \to \infty$	$H \to \infty$ (dominated by $Cu$ )	Flow outruns mass-transfer equilibration; band smears in proportion to velocity
$u = u_{opt} = \sqrt{B/C}$	$H = H_{min} = A + 2\sqrt{BC}$	True minimum; both velocity-dependent terms contribute equally ( $\sqrt{BC}$ each)
$C \to 0$ (instant equilibration)	$u_{opt}\to\infty$ , $H_{min}\to A$	No mass-transfer penalty; run as fast as desired, limited only by packing quality
$B \to 0$ (negligible diffusion, e.g. large/slow-diffusing analytes)	$u_{opt}\to 0$ , $H_{min}\to A$	Diffusion term vanishes; optimum shifts to very low velocity, again bottoming at $A$
Open-tubular (capillary) columns	$A \to 0$ identically	No packing, no multipath term; replaced by the Golay equation, $H = B/u + C_M u$ with a different $C_M$ expression for a wall-coated open tube
Very high-pressure UHPLC with sub-2 μm particles	$C$ term includes added contributions from extra-column band broadening and frictional heating	Van Deemter curve alone underestimates $H$ ; must add instrumental variance terms

The chemist behind it

Jan Jozef van Deemter worked as a chemical engineer at Koninklijke/Shell in Amsterdam during the 1950s, at a time when gas chromatography was newly emerging as a practical laboratory technique following the seminal 1952 work of Martin and James on gas–liquid partition chromatography. Van Deemter's expertise was not analytical chemistry in the classical sense but chemical engineering — specifically the theory of packed-bed reactors and mass transfer in flowing systems, a background that proves essential in the equation's derivation, which borrows directly from plate-theory and rate-theory descriptions of countercurrent extraction columns and catalytic reactors developed in industrial chemical engineering. Working with F. J. Zuiderweg and A. Klinkenberg, van Deemter recognized that the "theoretical plate" concept borrowed from distillation theory — a static equilibrium-stage picture — was inadequate for describing chromatographic band broadening as a dynamic, rate-dependent process. Their 1956 paper, published in Chemical Engineering Science, adapted the random-walk and diffusion arguments already used to model dispersion in packed catalytic beds, applying them to the specific problem of a solute partitioning between a flowing gas and a stationary liquid film. The equation's industrial origin — solving a chemical-engineering dispersion problem rather than a purely academic analytical-chemistry one — explains why its structure so closely resembles earlier axial-dispersion models for packed-bed reactors: van Deemter was applying engineering rate theory to a new instrument, not inventing chromatography theory from scratch.

Origins · 1 minJan Jozef van Deemter and how this was discovered

Jan Jozef van Deemter

1919–1985 · Shell Research, Amsterdam, 1950s gas chromatography theory

A chemical engineer, not a bench analytical chemist, van Deemter brought packed-bed reactor and mass-transfer theory to a young instrumental technique. His 1956 paper with Zuiderweg and Klinkenberg gave gas chromatography its first quantitative, testable theory of efficiency versus flow rate — turning column optimization from trial-and-error into a calculation.

History

- 1952 — A. J. P. Martin and A. T. James publish gas–liquid partition chromatography, launching practical GC.

- 1956 — van Deemter, Zuiderweg, and Klinkenberg publish the rate-theory equation

H=A+B/u+Cu

in Chemical Engineering Science.

- 1958 — Marcel Golay derives the analogous equation for open-tubular (capillary) columns, removing the

A

term entirely.

- 1960s–1970s — Giddings and others refine and generalize rate theory (coupled eddy-diffusion terms, reduced plate-height/reduced-velocity plots for cross-column comparison).

- 1970s–2000s — the same U-shaped curve is re-derived and applied for HPLC, guiding the industry-wide shift toward smaller particle diameters and higher operating pressures (UHPLC).

What are typical numerical values of A, B, and C for a real column?

For a well-packed GC column with ~150–200 μm particles, typical values might be $A \approx 0.5$ –2 mm, $B \approx 10$ –40 mm·cm/s, and $C \approx 0.01$ –0.05 mm·s/cm, giving $u_{opt}$ in the range of roughly 10–30 cm/s and $H_{min}$ around 1–2 mm. For modern HPLC with sub-2 μm particles, all three constants shrink dramatically: $A$ can be under 0.005 mm, $C$ under $10^{-4}$ s, giving $H_{min}$ well under 0.01 mm and correspondingly very high plate counts per unit length. These numbers always depend on the specific analyte, mobile phase viscosity and diffusivity, and temperature, so published values should be treated as illustrative rather than universal.

Does the van Deemter equation apply to all types of chromatography?

The functional form — a sum of a flow-independent term, an inverse-velocity term, and a linear-velocity term — extends conceptually to gas, liquid, and supercritical-fluid chromatography, but the coefficients' physical origins differ. Open-tubular columns drop the $A$ term entirely (Golay equation). Very fast HPLC systems sometimes need additional terms for extra-column band broadening (in tubing, injector, detector) that are not part of the classical column-only theory. Ion-exchange and size-exclusion chromatography follow similar rate-theory logic but with $C$ terms dominated by pore diffusion rather than simple film mass transfer, sometimes requiring modified equations (e.g., the Knox equation) for accurate fitting.

How is the van Deemter curve actually measured in a lab?

An analyte with a well-characterized retention factor is injected repeatedly at a series of controlled flow rates (hence linear velocities, computed from flow rate, column cross-sectional area, and porosity). At each flow rate, the resulting peak's retention time and baseline width (or width at half-height) are measured, plate number $N$ is calculated from $N=16(t_R/w_b)^2$ or the half-height formula $N=5.54(t_R/w_{1/2})^2$ , and plate height is obtained as $H=L/N$ . Plotting $H$ against $u$ and fitting the resulting curve to $H=A+B/u+Cu$ (nonlinear least-squares, since it is not linear in $u$ ) yields the three coefficients.

Why is the C term sometimes split into C_S and C_M?

Mass-transfer resistance has two physically distinct sources: resistance to equilibration within the stationary phase ( $C_S$ , e.g. diffusion through a liquid film or into/out of pores) and resistance to equilibration within the mobile phase itself, including stagnant mobile phase trapped in particle pores ( $C_M$ ). Both broaden the band as velocity increases, but they respond to different engineering fixes — thinner stationary-phase films reduce $C_S$ ; smaller particles or less porous supports reduce $C_M$ . Lumping them into a single $C$ is convenient for curve-fitting but obscures which physical lever to pull when optimizing a column.

Is it ever possible for a column to have no measurable minimum in its H-vs-u curve?

In principle, if $C$ is extremely small relative to the accessible velocity range, the curve can appear nearly flat and monotonically decreasing over the range actually tested, with the true minimum occurring at velocities beyond practical flow rates. This is common for very small, highly efficient particles at moderate flow, where the $Cu$ term barely rises within the pressure limits of the pump. The mathematical minimum still exists — it is a property of the equation whenever $B>0$ and $C>0$ — but it may not be experimentally observable within achievable operating conditions.

How does temperature affect the van Deemter curve?

Raising temperature increases the diffusion coefficient $D_m$ in gases (and, to a lesser extent, $D_S$ within many stationary phases), which increases $B$ (more longitudinal diffusion per unit time) but can decrease $C$ (faster equilibration, less mass-transfer lag), shifting $u_{opt}$ upward and often changing $H_{min}$ only modestly. In practice, GC temperature programming is chosen primarily for selectivity and elution-time management, with van Deemter effects treated as a secondary consideration layered on top of the primary retention-time optimization.

What is the practical difference between minimizing H and maximizing resolution?

Minimizing $H$ maximizes plate number $N=L/H$ for a fixed column length, which does help resolution since $R_s \propto \sqrt{N}$ . But resolution also depends on selectivity $\alpha$ and retention factor $k'$ , both of which are controlled by chemistry (stationary phase, mobile phase composition, temperature) rather than flow rate. In practice, improving $\alpha$ from 1.02 to 1.05 often buys far more resolution than any achievable improvement from sitting exactly at $u_{opt}$ , which is why real method development spends most of its effort on selectivity before fine-tuning flow rate.