Analytical Chemistry & Pharmaceutical QC
Purnell Chromatography Peak Resolution Rs Formula
Calculate Retention Factor k_2 = (t_R2 - t_0) / t_0, Selectivity Factor alpha = k_2 / k_1, Purnell Master Resolution R_s = (SQRT(N)/4) * ((alpha - 1)/alpha) * (k_2 / (1 + k_2)), and USP R_s >= 1.5 Separation Gate.
Optimize HPLC method development across efficiency SQRT(N), selectivity alpha, and retention k!
In United States Pharmacopeia (USP <621>) impurity and chiral assay validation, adjacent chromatographic peaks achieve quantitative baseline separation (99.7% non-overlap for equal Gaussian peaks) only when Resolution R_s >= 1.50! J.H. Purnell (1960) factored peak resolution into three orthogonal physical levers: R_s = (SQRT(N) / 4) * ((alpha - 1) / alpha) * (k_2 / (1 + k_2)), where t_0 is unretained void time, k_1 = (t_R1 - t_0) / t_0 and k_2 = (t_R2 - t_0) / t_0 are Retention Factors, alpha = k_2 / k_1 >= 1 is Stationary-Phase Selectivity, and N is Column Theoretical Plates! Notice why doubling column length (doubling N and doubling run time!) raises R_s by only SQRT(2) = +41.4%, whereas nudging selectivity alpha from 1.05 to 1.20 nearly quadruples resolution!
Verify on Void_Time t_0 = 1.00 min (B2), Impurity_Peak1 t_R1 = 3.50 min (k_1 = 2.50, B3), Active_API_Peak2 t_R2 = 4.00 min (B4 -> k_2 = 3.00 in E2, Selectivity alpha = 3.00 / 2.50 = 1.2000 in E3), and Column_Plates N = 3,600 (B5 -> SQRT(3,600)/4 = 15): Purnell Resolution R_s equals 15 * (0.20 / 1.20) * (3.0 / 4.0) = 15 * (1/6) * (3/4) = 1.875 (E4), clearing "USP Baseline Separated (Rs >= 1.5)" (E5)!
| USP <621> Purnell Metric (t0=1.0, tR1=3.5, tR2=4.0 min, N=3,600 Plates) | Formula | Verified Output |
|---|---|---|
| Peak 2 Retention Factor k_2 | (t_R2 - t_0) / t_0 | 3.00 (k_2 / (1 + k_2) = 0.75) |
| Selectivity Factor alpha | k_2 / k_1 = (t_R2 - t_0)/(t_R1 - t_0) | 1.2000 ((alpha - 1)/alpha = 0.1667) |
| Purnell Peak Resolution R_s | (SQRT(N)/4)*((a-1)/a)*(k2/(1+k2)) | 1.875 (0.938 @ Aged N=900) |
| USP Assay Suitability Gate | R_s >= 1.50 Baseline Criterion | USP Baseline Separated (Rs >= 1.5) |
Purnell Master Chromatographic Resolution Formula
Enforce strict elution order 0 < t_0 < t_R1 < t_R2 (if alpha = 1 because t_R1 = t_R2, peaks co-elute identically with zero thermodynamic selectivity) and N > 0!
=LET(t_vd, B2, t_r1, B3, t_r2, B4, pl_n, B5, IF(OR(t_vd <= 0, t_r1 <= t_vd, t_r2 <= t_r1, pl_n <= 0), "Invalid Retention Order", LET(k_p1, (t_r1 - t_vd) / t_vd, k_p2, (t_r2 - t_vd) / t_vd, al_s, k_p2 / k_p1, ROUND((SQRT(pl_n) / 4) * ((al_s - 1) / al_s) * (k_p2 / (1 + k_p2)), 3))))Aged-Column System Suitability Failure Check (N = 900)
Simulate silica bed voiding that cuts column plates 4x from 3,600 to 900 in cell B5: R_s halves to 0.938 and fails USP system suitability ("Co-Eluting Overlap (Rs < 1.5)")!
=IF(ISTEXT(E4), E4, IF(E4 >= 1.5, "USP Baseline Separated (Rs >= 1.5)", "Co-Eluting Overlap (Rs < 1.5)"))Try the example
Use the original values or change them to test your understanding.