Chemistry Reference
In-Depth Information
−d γ = S
s,x
dT + Γ
x,1
dμ
1
+ Γ
x,2
dμ
2
+ … + Γ
ix
dμ
i
(3.24)
This equation is similar to the Gibbs-Duhem equations for the bulk liquid system.
To make this relationship more meaningful, Gibbs further pointed out that the
position of the plane may be shifted parallel to GG′ along the x-direction and fixed
in a particular location, when (n
it
) becomes equal to (n
1
+ n
i
). Under this condition,
n
ix
(n
1
by convention) becomes zero. The relation in Equation 3.24 can be rewrit-
ten as
−d γ = S
s,1
dT + Γ
2
dμ
2
+ … + Γ
i
dμ
i
(3.25)
= S
s,1
dT + 3 Γ
i
dμ
i
(3.26)
At constant T and p, for a two-component system (say, water(1) + alcohol(2)), we thus
obtain the classical Gibbs adsorption equation as
Γ
2
= − (d γ/dμ
2
)
T,p
(3.27)
The chemical potential μ
2
is related to the activity of alcohol by the equation
μ
2
= μ
2
o
+ R T ln(a
2
)
(3.28)
If the activity coefficient can be assumed to be equal to unity, then
μ
2
= μ
o
2
+ R T ln(C
2
)
(3.29)
where C
2
is the bulk concentration of solute 2. Then, the Gibbs adsorption can be
written as
Γ
2
= −1/RT(d γ/d ln(C
2
))
= −C
2
/RT (d γ/dC
2
)
(3.30)
This shows that the
surface excess
quantity on the left-hand side is proportional to
the change in surface tension with concentration of the solute (d γ/d(ln(C
surfaceactivesubs
tance
). A plot of ln (C
2
) versus γ gives a slope equal to
Γ
2
(
RT
)
(3.31)
from which the value of Γ
2
(moles/area) can be estimated.
This shows that all surface-active substances will always have a higher concentra-
tion at the surface than in the bulk of the solution.
This relation has been verified using radioactive tracers. Further, as will be shown
later under spread monolayers, there is very convincing support to this relation and
the magnitudes of Γ for various systems.
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