Appendix 2 Derivation of the general resolution equation

372 Appendix 2

Derivation of the general resolution equation (Referring to Fig. 2.5 in Chapter 2) Resolution is defined by expression:

R =

2[(tRb - tRa)/(Wb

+ Wa)l

(1)

Assuming that for two peaks which are c!ose in retention time the peak widths are approximately the same, i.e., W, = Wb , then eqn. 1reduces to: =

(2)

(tRb - tR,)/Wb

The efficiency equation relates retention times with peak widths by the expression (3)

N = 16(tRb/Wb)'

where Wb is the base, width of peak b. Substituting this expression in eqn. 2 to eliminate peak widths gives:

- t R a ) / t R b l /4 (4) The capacity factor, k', relates retention time of peaks relative t o the void time of a column, i.e.: =

'k

dN[(tRb

to)/to Rearranging this gives: =

(tRb -

= to(kk

tRb

+ 1)

Substituting for tR in the denominator of eqn. 4 gives:

- tRa)/tOl [l/(kb + Multiplying numerator and denominator by tR =

'4

=

(5)

dR[(tRb

[(tRb

- to gwes:

- tRa)/(tRb

- t o ) ] [ ( t R b - t O ) / t o ][ l / ( k b -t l)]

- tRa)/(tRb

- to)] [kb/(kb

/4

(6)

This reduces to: =

d'[(tRb

+ 1)1/4

(7)

Rearranging gives:

fi {[(tRb

- to) - (tR, /(tRb - tO)}[kb/(kb + '11 The selectivity factor, a,is defined by =

a =

[ ( t R b - tO)/(tRa

-

I4

(8)

APPENDICES

373

i.e., resolution is a function of the square root of the column efficiency, yet is directly related t o the selectivity and capacity of the chromatographic system. Note: It is very common for the general resolution equation to be applied to pairs of closely eluting peaks. In this case, it is normally assumed that 0: x 1.This assumption leads to a simplified form of the equation. Thus: R =

fi(a- l ) ( k b / k b + 1)/4