Physics:Fundamental resolution equation

The fundamental resolution equation is used in chromatography to help relate adjustable chromatographic parameters to resolution, and is as follows: R_s = [N^1/2/4][(α-1)/α][k₂'/(1+k₂')], where

N = Number of theoretical plates

α = Selectivity Term = k₂'/k₁'

The [N^1/2/4] term is the column factor, the [(α-1)/α] term is the thermodynamic factor, and the [k₂'/(1+k₂')] term is the retention factor. The 3 factors are not completely independent, but they are very close, and can be treated as such.

So what does this mean? It means that to increase resolution of two peaks on a chromatogram, one of the three terms of the equation need to be modified.

1) N can be increased by lengthening the column (least effective, as doubling the column will get a 2^1/2 or 1.44x increase in resolution).

2) Increasing k' also helps. This can be done by lowering the column temperature in G.C., or by choosing a weaker mobile phase in L.C. (moderately effective)

3) Changing α is the most effective way of increasing resolution. This can be done by choosing a stationary phase that has a greater difference between k₁' and k₂'. It can also be done in L.C. by using pH to invoke secondary equilibria (if applicable).

The fundamental resolution equation is derived as follows:

For two closely spaced peaks, ω₁ = ω₂, and σ₁ = σ₂

so R_s = (t_r2 - t_r1)/ω₂ = (t_r2 - t_r1)/4σ₂

Where t_r1 and t_r2 are the retention times of two separate peaks.

Since N = [(t_r2)/σ₂]², then σ = t_r2/ N^1/2

Using substitution, R_s = N^1/2[(t_r2 - t_r1)/4t_r2] = (N^1/2/4)(1 - t_r1/t_r2)

Now using the following equations and solving for t_r1 and t_r2

k₁' = (t_r1 - t₀)/t₀ ; t_r1 = t₀(k₁' + 1)

k₂' = (t_r2 - t₀)/t₀ ; t_r2 = t₀(k₂' + 1)

Substituting again and you get:

R_s = [N^1/2/4][1 - (k₁' + 1)/(k₂' + 1] = [N^1/2/4][(k₂' - k₁')/(1 + k₂')]

And finally substituting once more α = k₂'/k₁' and you get the Fundamental Resolution Equation:

R_s = [N^1/2/4][(α-1)/α][k₂'/(1+k₂')]

References

• Spring 2009 Class Notes, CHM 5154, Chemical Separations taught by Dr. John Dorsey, Ph.D, Florida State University

• "Fundamental Resolution Equation" (in en). LibreTexts. 29 December 2016. https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_(Analytical_Chemistry)/Analytical_Sciences_Digital_Library/Courseware/Separation_Science/02_Text/04_Fundamental_Resolution_Equation. 

• "Appendix 1: Derivation of the Fundamental Resolution Equation" (in en). LibreTexts. 30 December 2016. https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_(Analytical_Chemistry)/Analytical_Sciences_Digital_Library/Courseware/Separation_Science/02_Text/07_Appendix_1%3A__Derivation_of_the_Fundamental_Resolution_Equation.