Physics:Fundamental resolution equation

The fundamental resolution equation is used in chromatography to help relate adjustable chromatographic parameters to resolution. a

Equation

$R_s=\left(\frac{\sqrt{N}}{4}\right)\left(\frac{\alpha -1}{\alpha }\right)\left(\frac{k{\text{'}}_2}{1+k{\text{'}}_2}\right)$

where,

$N$ = Number of theoretical plates

$\alpha$ = Selectivity Term = $\frac{k{\text{'}}_2}{k{\text{'}}_1}$

The $\frac{\sqrt{N}}{4}$ term is the column factor, the $\frac{\alpha -1}{\alpha }$ term is the thermodynamic factor, and the $\frac{k{\text{'}}_2}{1+k{\text{'}}_2}$ term is the retention factor. The 3 factors are not completely independent, but can be treated as such.

Intervention

To increase resolution of two peaks on a chromatogram, one of the three terms of the equation need to be modified.

• N can be increased by lengthening the column (least effective, as doubling the column will get a $\sqrt{2}$ or 1.44x increase in resolution).
• Increasing $k^{\prime }$ 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)
• 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 $k{\text{'}}_1$ and $k{\text{'}}_2$. It can also be done in L.C. by using pH to invoke secondary equilibria (if applicable).

Resolution

The fundamental resolution equation is derived as follows:

For two closely spaced peaks, ${\omega }_1={\omega }_2$ , and ${\sigma }_1={\sigma }_2$,

so,

$R_s=\frac{t_{r2}-t_{r1}}{{\omega }_2}=\frac{t_{r2}-t_{r1}}{4{\sigma }_2}$

Where $t_{r1}$ and $t_{r2}$ are the retention times of two separate peaks.

Since $N={\left(\frac{t_{r2}}{{\sigma }_2}\right)}^2$ , then $\sigma =\frac{t_{r2}}{\sqrt{N}}$

Using substitution, $R_S=\sqrt{N}\left(\frac{t_{r2}-t_{r1}}{4t_{r2}}\right)=\left(\frac{\sqrt{N}}{4}\right)(1-\frac{t_{r1}}{t_{r2}})$.

Now using the following equations and solving for $t_{r1}$ and $t_{r2}$

$k{\text{'}}_1=\frac{t_{r1}-t_0}{t_0};t_{r1}=t_0(k{\text{'}}_1+1)$

$k{\text{'}}_2=\frac{t_{r2}-t_0}{t_0};t_{r2}=t_0(k{\text{'}}_2+1)$

Substituting again and you get:

$R_s=\left(\frac{\sqrt{N}}{4}\right)(1-\frac{k{\text{'}}_1+1}{k{\text{'}}_2+1})=\left(\frac{\sqrt{N}}{4}\right)\left(\frac{k{\text{'}}_2-k{\text{'}}_1}{1+k{\text{'}}_2}\right)$

And finally substituting once more $\alpha =k{\text{'}}_2/k{\text{'}}_1$ and you get the Fundamental Resolution Equation:

$R_s=\left(\frac{\sqrt{N}}{4}\right)\left(\frac{\alpha -1}{\alpha }\right)\left(\frac{k{\text{'}}_2}{1+k{\text{'}}_2}\right)$

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.