Fundamental resolution equation

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

Equation

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

where,

${\displaystyle N}$  = Number of theoretical plates

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

The ${\displaystyle {\frac {\sqrt {N}}{4}}}$  term is the column factor, the ${\displaystyle {\frac {\alpha -1}{\alpha }}}$  term is the thermodynamic factor, and the ${\displaystyle {\frac {k'_{2}}{1+k'_{2}}}}$  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 ${\displaystyle {\sqrt {2}}}$  or 1.44x increase in resolution).
• Increasing ${\displaystyle k'}$  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 ${\displaystyle k'_{1}}$  and ${\displaystyle k'_{2}}$ . 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, ${\displaystyle \omega _{1}=\omega _{2}}$  , and ${\displaystyle \sigma _{1}=\sigma _{2}}$ ,

so,

${\displaystyle R_{s}={\frac {t_{r2}-t_{r1}}{\omega _{2}}}={\frac {t_{r2}-t_{r1}}{4\sigma _{2}}}}$ 

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

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

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

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

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

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

Substituting again and you get:

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

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

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

References

• 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.