Surfactants (Soaps and Detergents) and Physicochemical Properties - Surface and Colloid Chemistry: Principles and Applications - page 60

and one obtains

Γ SDS = − 1/2(R T) d γ/(d ln C SDS ) (3.39)

In the case where the ion strength is kept constant, that is, in the presence of added
NaCl, then the equation becomes

Γ SDS = − 1/(R T) d γ/(d ln C SDS ) (3.40)

Comparing Equation 3.39 with Equation 3.40, it will be seen that they differ by a
factor of 2, and that the appropriate form will need to be used in the experimental
test of the Gibbs equation. It is also quite clear that any partial ionization would lead
to considerable difficulty in applying the Gibbs equation. Finally, while on this topic,
if SDS were investigated in a solution using a large excess of sodium ions, produced
by the addition, say, of NaCl, then the sodium ion term in Equation 3.39 will vanish
and will arrive back at an equation equivalent to Equation 3.40.
The adsorption of detergents at the surface of the solution can be estimated by the
Gibbs equation. It is convenient to plot γ versus the log (C detergent ).
From the γ versus C alkyl sulfate data, the following data is obtained from the Gibbs
equation:

concentration
mol/l

Γ Salkyl Sulfate
10 −12 mol/cm 2

a
(area/molecule)

NaC 10 sulfate (0.03 mol/L) 3.3 50 Å 2

NaC 12 sulfate (0.008 mol/L) 3.4 50 Å 2

NaC 14 sulfate (0.002 mol/L) 3.3 50 Å 2

From the plots of γ versus concentration, the slope is related to the surface excess,
Γ Salkyl sulfate . The area/molecule values indicate that the molecules are aligned vertically
on the surface, irrespective of the alkyl chain length. If the molecules were oriented
flat, then the value of area/molecule would be much larger (approximately 100 Å 2 ).
Further, the fact that the alkyl chain length has no effect on the area also proves this
assumption. These conclusions have been verified from spread monolayer studies.
Further, it is also found that the polar group, that is, −SO 4− , would occupy something
like 50 Å 2 . Later, it will be shown that other studies confirm that the area per mol-
ecule is approximately 50 Å 2 .
The Gibbs adsorption equation is a relation about the solvent and a solute (or
many solutes). The solute is present either as excess (if there is an excess surface
concentration) if the solute decreases the γ, or as a deficient solute concentration (if
the surface tension is increased by the addition of the solute).
It can be further explained that, if we consider the system, water, to which a sur-
factant such as SDS is added, the molecules at the surface will change as follows:

pure water ( w ) surface:
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