Analytical Chemistry & Chromatography Engineering
Van Deemter HETP & Optimal Velocity Formula
Calculate Optimal Mobile-Phase Velocity u_opt = SQRT(B / C), Van Deemter Height Equivalent to a Theoretical Plate HETP = A + B/u + C*u, Actual Column Plates N, and Peak Theoretical Plates N_max.
Balance longitudinal diffusion B/u against C*u mass-transfer band broadening in HPLC & GC!
J.J. van Deemter, F.J. Zuiderweg, and A. Klinkenberg (1956, Chemical Engineering Science; 1952 Nobel-linked partition chromatography foundations) explained why running an HPLC or gas-chromatography column either TOO SLOWLY or TOO FAST widens elution peaks! Height Equivalent to a Theoretical Plate is HETP = A + B / u + C * u, combining (A) velocity-independent multi-path Eddy Diffusion around packing particles, (B / u) axial Longitudinal Molecular Diffusion that dominates at slow mobile-phase linear velocity u, and (C * u) Stationary/Mobile Mass-Transfer Resistance that dominates at fast velocity u! Differentiating d(HETP)/du = -B / u^2 + C = 0 yields Optimal Linear Velocity u_opt = SQRT(B / C) and Minimum Plate Height H_min = A + 2 * SQRT(B * C), maximizing column efficiency N = L / HETP!
Verify on a Sub-2-Micron UHPLC Column—Eddy_Term A = 0.004 mm (4 um, B2), Longitudinal_Diff B = 0.016 mm^2/s (B3), Mass_Transfer C = 0.001 s (B4 -> u_opt = SQRT(0.016 / 0.001) = 4.00 mm/s in E2 and H_min = 0.004 + 2 * 0.004 = 0.012 mm), Operating_Velocity u = 8.00 mm/s (fast throughput mode, B5), and Column_Length L = 140 mm (B6): Operating HETP is 0.004 + 0.016 / 8 + 0.001 * 8 = 0.0140 mm (E3); Operating Plate Count N is 140 / 0.014 = 10,000 plates (E4); and Peak Possible Plates at u_opt = 4.00 mm/s is ROUND(140 / 0.012, 0) = 11,667 plates (E5)!
| Van Deemter UHPLC Metric (A=0.004mm, B=0.016, C=0.001s, u=8 mm/s, L=140 mm) | Formula | Verified Output |
|---|---|---|
| Optimal Linear Velocity u_opt | SQRT(B / C) | 4.00 mm/s (H_min = 0.0120 mm) |
| Operating Plate Height HETP | A + B / u + C * u | 0.0140 mm (0.0120 mm @ u = 4) |
| Operating Theoretical Plates N | ROUND(Column_Length_mm / HETP, 0) | 10,000 plates (11,667 @ u = 4) |
| Maximum Column Plates N_max | ROUND(L / (A + 2 * SQRT(B * C)), 0) | 11,667 plates |
Van Deemter HETP & Theoretical Plate Count Formula
Validate A >= 0 (capillary open-tubular GC columns have A = 0: the Golay equation!), B > 0, C > 0, u > 0, and L > 0!
=LET(ed_a, B2, df_b, B3, mt_c, B4, vl_u, B5, cl_l, B6, IF(OR(ed_a < 0, df_b <= 0, mt_c <= 0, vl_u <= 0, cl_l <= 0), "Invalid HPLC Velocity", LET(ht_p, ed_a + df_b / vl_u + mt_c * vl_u, ROUND(cl_l / ht_p, 0))))Optimal Linear Velocity Verification (u = u_opt = 4.00 mm/s)
Slow mobile-phase velocity from 8.00 mm/s to u_opt = 4.00 mm/s in cell B5: HETP drops to H_min = 0.0120 mm and Operating Plates E4 rises to match N_max = 11,667!
=IF(ISTEXT(E3), E3, ROUND(B6 / (B2 + 2 * SQRT(B3 * B4)), 0))Try the example
Use the original values or change them to test your understanding.