Environmental Engineering Reference
In-Depth Information
2.6.5 G IBBS A DSORPTION E QUATION
In environmental chemistry, we encounter systems composed of more than one
species. If, for example, we consider a two-component system (solvent
1 and
solute
2), the Gibbs equation can be rewritten using the Gibbs convention that
the surface concentration of solvent 1 with respect to solute 2,
Γ 1 ( 2 ) =
0, is
Γ 2 ( 1 ) =
d
σ
.
(2.72)
d
μ 2
Note that in most cases it is simply designated
. This is called the Gibbs adsorp-
tion equation . It is the analogue of the Gibbs-Duhem equation for bulk phases. The
Gibbs adsorption equation is used to calculate the surface excess concentration of a
solute by determining the change in surface tension of the solvent with the chemi-
cal potential of the solute in the solvent. This equation is of particular significance
in environmental engineering, since it is the basis for the calculation of the amount
of material adsorbed at air-water, soil (sediment)-water, and water-organic solvent
interfaces.
Γ
E XAMPLE 2.7 S URFACE C ONCENTRATION FROM S URFACE T ENSION D ATA
Problem : The following data were obtained for the surface tension of a natural plant-
based surfactant in water.
Aqueous Concentration (mol/cm 3 )
Surface Tension (erg/cm 2 )
10 8
1.66
×
62
3.33 × 10 8
60
1.66 × 10 7
50
3.33 × 10 7
45
1.66 × 10 6
39
The molecular weight of the surfactant is 300. Obtain the surface concentration at an
aqueous concentration of 1.66 × 10 7 mol/cm 3 .
Solution : Since Γ 2 ( 1 ) =− d σ / d μ 2 and, for a dilute aqueous solution of solute
concentration C w ,d μ 2 = RT ln C w , we have surface concentration Γ = Γ 2 ( 1 ) =
( 1 /RT )( d σ / dln C w ). A plot of σ versus ln C w gives a slope of d σ / dln C w =
6.2 erg / cm 2 at C w = 1.66 × 10 7 mol/cm 3 . Hence the surface concentration is
Γ =− ( 6.2 )/( 8.31 × 10 7
× 298 ) = 2.5 × 10 10 mol/cm 2 .
PROBLEMS
2.1 1 Classify the following properties as intensive or extensive. Give appro-
priate explanations: temperature, entropy, pressure, volume, number of
moles, density, internal energy, enthalpy, molar volume, mass, chemical
potential, and Helmholtz free energy.
 
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