Fundamental resolution equation

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

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

References