Fundamental resolution equation

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

Equation

$R_{s}=\left({\frac {\sqrt {N}}{4}}\right)\left({\frac {\alpha -1}{\alpha }}\right)\left({\frac {k'_{2}}{1+k'_{2}}}\right)$

where,

$N$ = Number of theoretical plates

$\alpha$ = Selectivity Term = ${\frac {k'_{2}}{k'_{1}}}$

The ${\frac {\sqrt {N}}{4}}$ term is the column factor, the ${\frac {\alpha -1}{\alpha }}$ term is the thermodynamic factor, and the ${\frac {k'_{2}}{1+k'_{2}}}$ 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 $k'$ 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 $k'_{1}$ and $k'_{2}$. 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, $R_{S}={\sqrt {N}}\left({\frac {t_{r2}-t_{r1}}{4t_{r2}}}\right)=\left({\frac {\sqrt {N}}{4}}\right)\left(1-{\frac {t_{r1}}{t_{r2}}}\right)$.

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

$k'_{1}={\frac {t_{r1}-t_{0}}{t_{0}}};\quad t_{r1}=t_{0}(k'_{1}+1)$

$k'_{2}={\frac {t_{r2}-t_{0}}{t_{0}}};\quad t_{r2}=t_{0}(k'_{2}+1)$

Substituting again and you get:

$R_{s}=\left({\frac {\sqrt {N}}{4}}\right)\left(1-{\frac {k'_{1}+1}{k'_{2}+1}}\right)=\left({\frac {\sqrt {N}}{4}}\right)\left({\frac {k'_{2}-k'_{1}}{1+k'_{2}}}\right)$

And finally substituting once more $\alpha =k'_{2}/k'_{1}$ and you get the Fundamental Resolution Equation:

$R_{s}=\left({\frac {\sqrt {N}}{4}}\right)\left({\frac {\alpha -1}{\alpha }}\right)\left({\frac {k'_{2}}{1+k'_{2}}}\right)$

References

Bibliography

• Spring 2009 Class Notes, CHM 5154, Chemical Separations taught by Dr. John Dorsey, Ph.D, Florida State University
• "Fundamental Resolution Equation". Chemistry LibreTexts. LibreTexts. 29 December 2016. Retrieved 1 December 2023.
• "Appendix 1: Derivation of the Fundamental Resolution Equation". Chemistry LibreTexts. LibreTexts. 30 December 2016. Retrieved 1 December 2023.