# Bonded Phases - The Derivation of the Langmuir Adsorption Isotherm for Reverse Phases

For example, if the intercept and slope of a curve
relating (1/V') to (C_{m}) is (A) and (B) respectively,

Then, _{}

or, _{}
.......................................... (6)

and _{}
................................................. (7)

Scott and Simpson (29) took some
accurate measurements of (V') and (C_{m}) for several
different aliphatic alcohols and two aliphatic acids in an attempt to
measure their adsorption-desorption coefficients and the unique surface
area of the reverse phase that was available to each solvent.
Unfortunately, in practice, the method suggested above suffers from one
severe drawback. In order to measure (V') the corrected retention
volume accurately, it is necessary to obtain a precise value for the
effective dead volume. Dead volume measurement has been studied
extensively (32-37) and the problems associated with the accurate
measurement of the dead volume has been discussed generally in the
literature. There is still much disagreement on the best method of
measurement and in their work, Scott and Simpson, employed sodium
chloride as the dead volume marker. In subsequent review by other
workers in the field, this solute was not considered appropriate and
thus, many of their conclusions were considered in error. However,
their data can be used very effectively without knowing the column dead
volume. The following method, circumvents the uncertainties that arise
from dead volume measurement but still provides values for the
adsorption-desorption coefficients and the surface area of the reverse
phase that is available to each solvent.

Restating equation (5)

_{}

Now, as, V' = V_{r} - V_{o ,}
then, substituting for (V') in equation (5)

_{} ................
(8)

To aid in algebraic manipulation equation (8) can be put in the simple form,

_{} ..................... (9)

where, as defined before _{} and _{}

Now, when the solvent is chromatographed as the
solute in **pure** water as the mobile
phase, (i.e. C_{m} = 0, and
V_{r} = V_{r(0)} ), then,

_{}

Re-arranging, _{} ...................................... (10)

Substituting for (V_{o})
from equation (10) into equation (9),

_{} ..... . .
(11)

Where (V_{r(n)}) is the
retention volume of the solvent when eluted as a solute in a mobile
phase containing the solvent at a concentration (C_{m(n)}).
Equation (11) relates the *retention
volume* of the solvent (as opposed to the *corrected retention volume*) to the solvent
concentration in the mobile phase and thus, *the
need to determine the column dead volume is
eliminated*.