The fundamental resolution equation or Purnell equation is used in chromatography to help relate adjustable chromatographic parameters to resolution.
Equation
R s = ( N 4 ) ( α − 1 α ) ( k 2 ′ 1 + k 2 ′ ) {\displaystyle R_{s}=\left({\frac {\sqrt {N}}{4}}\right)\left({\frac {\alpha -1}{\alpha }}\right)\left({\frac {k'_{2}}{1+k'_{2}}}\right)}
where,
N {\displaystyle N} = Number of theoretical plates
α {\displaystyle \alpha } = Selectivity Term = k 2 ′ k 1 ′ {\displaystyle {\frac {k'_{2}}{k'_{1}}}}
The N 4 {\displaystyle {\frac {\sqrt {N}}{4}}} term is the column factor, the α − 1 α {\displaystyle {\frac {\alpha -1}{\alpha }}} term is the thermodynamic factor, and the k 2 ′ 1 + k 2 ′ {\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 2 {\displaystyle {\sqrt {2}}} or 1.44x increase in resolution). Increasing k ′ {\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 k 1 ′ {\displaystyle k'_{1}} and k 2 ′ {\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, ω 1 = ω 2 {\displaystyle \omega _{1}=\omega _{2}} , and σ 1 = σ 2 {\displaystyle \sigma _{1}=\sigma _{2}} , so,
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