Number of theoretical plates (N), Calculation Formulas

Number of theoretical plates (N):

Often used to quantify the efficiency (performance) of a column (HPLC or GC). "Plates" are expressed per meter of column length and should be calculated based on a retained peak with ideal peak shape or symmetry. 

   N = Plates; tr = Retention Time of Peak; w = Peak width;  w_0.5 = Peak width measured at half height.

Two popular formulas are:

Tangent: USP (United States Pharmacopeia / ASTM)

Best for Gaussian peaks. Peak width is often determined at 13.4% of the peak height (w). Inaccurate for peaks which are non-Gaussian, poorly resolved or tail.

   N = 16 (tr / w)²

Half Peak Height: (European Pharmacopeia)

For peaks which are less Gaussian in appearance, using a slightly different formula with the peak width measurement made at the half-height (W_0.5). Less accurate for peaks which are poorly resolved or tail.

   N = 5.54 (tr / w_0.5)²

 

• Other formulas, not included, for calculating Plate numbers include: Half Width, Variance Method, Area / Height & Exponential Modified Gaussian (EMG).
• Caution. HPLC column "Plate" values should not be used for a final determination of efficiency unless you are comparing all results on the same exact HPLC system, which is setup and run under identical conditions each time. Since the result is based on many possible variables, including how your HPLC system is plumbed (dwell volume & tubing ID), the peak's Kprime and symmetry, the detector used, sampling rate, integration quality, flow cell volume, flow rate, actual column used (to name a few), it can easily be manipulated to be very large or small.

Common HPLC Calculations:

Capacity Factor / Retention Factor / Capacity Ratio:  k1 (K Prime)

k1 = T(R) - T(0) / T(0)
where T(R) equals the retention time of the peak in minutes and T(0) is
the retention time of an unretained peak. *For chromatography to take place, K Prime must be > 1.00 and for most modes of chromatography, should be greater than 1.5 or 2.0 for all samples !


Tailing Factor: USP: 't'

t = W(5.0)/tw/2

where tw equals the distance between peak front and T(R) at 5% of peak height units. W(5.0) equals peak width at 5% height, in minutes.


Theoretical Plates: USP and ASTM, 'N'

N = 5.54 x (T(R)/W(50))2          

Assumes width at peak half height (50)
* More info can be found at this link.


Resolution: USP, ASTM, 'R'

R = (T(R)(b)-T(R)(a)) x 2.35/(W(50)(b) + W(50)(a))/2

Assumes width at half height (50%) with peaks (a) and (b).

*Notes: Visually, "Baseline" resolution is R = 1.5. Your goal should be R = or > 2.0. ** R of 1.5 provides 99.8% separation which means you cannot accurately quantify a 0.1% impurity so develop the method to have a resolution value of at least 2.0.


Note: The appropriate formula(s) for use with your samples may depend on which of the many pharmaceutical guidelines and regulations apply in your country. Always consult the appropriate guidelines.