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The Gibbs Adsorption Isotherm at Charged Interfaces
Appendix 6
The Gibbs Adsorption Isotherm at Charged Interfaces
At a charged interface we have at constant temperature and pressure (Delahay, 1965):
- dy = γτ,άμ, + YJ^ '
(A6.1)
J
where Γ,, Γ, are the surface excesses of the various components, μ} is the chemical potential and μ, is the electrochemical potential, so that the sum mation over / is for charged components and that over j is for uncharged components. Consider an Agi crystal in contact with an aqueous solution containing A g N 0 3 and HNO3. Then: - dy = rAg+ · ΦΑ 8 * + Γ ι- dßr + ΓΗ+ · άμ^ + Γ ΝΟί
ΦΝΟ3-
+
ΓΗ 2 ΟΦΗ 2 Ο
(Α6.2) But /Ug+ + μ\- - MAgi = constant and so dßkg* = Also
μΑ8+ = ^Ag+ + H
-αμι-.
and
μ,- = μι- - F(/>.
Substituting these expressions in eq (2) and remembering that rAg* - Γ,- + ΓΗ+ - Γ ΝΟ - = 0
we find -dy=
(r Ag+ - Γ,- )άμκ%* + ΓΗ+ < W + Γ ΝΟ - . ΦΝΟ,370
(A6.3) (Α6.4)
APPENDIX 6
371
Substituting for ΓΗ+ from eq (3) we can write eq (4) as: - dy = (rAg+ - r r )(dßAg+ - φ Η + ) + ΓΝΟ- · ΦΗΝΟ3
(A6·5)
Now consider the electrochemical cell: Cu'|Ag|Agl|AgN03, HN0 3 |H 2 , Pt|Cu" 01 02 03 04 05 06 whose cell reaction is : Agi +^H 2 ->Ag + H + +I~. The e.m.f. of this cell is given by : r
RT
r
F
^
And, hence — FdE+
=ΦΗ
+
+ Ψ Ι - = ΦΗ+
—
(Α6.6)
ΦΑ 8 +
Combining eqs (5) and (6) ( Α6 · 7 )
- dy = F(rAg+ - r r )dE+ + ΓΝΟ-. Φ Η ΝΟ,
Normally, the concentrations of Ag* and I" are very low and most of the ions in the system are H + and NOJ (or some other ions with no special relationship to the crystal surface). We can then regard ^(r Ag+ — r r ) as the surface charge density σ, on the Agi crystal and eq (A6.7) becomes : - dy = σάΕ+ + ΓΝθ3- · ΦΗΝΟ3 (A6·8) Note that the subscript + on Prefers to the fact that the e.m.f., £+, is measur ed with reference to an electrode which is reversible to a cation (in this case H + ). The corresponding equation with £_ involves the surface excess of the cation (ΓΗ+)·
A6.1. The surface potential of the Agi crystal
The e.m.f. of the cell is measured by determining the difference in electro chemical potential of the electron μβ in the two pieces of copper wire (see Section 2.1). Since these are of the same composition we can write: /UCu") - /ie(Cu') = FE+ = Ε{φ6 - φ,) = F[(4>6 - φ5) + (φ5 - φ4) + (φ4 - φ3) + (03 - φ2) + (φ2 - φχ)] const
const
const
Some of these terms can be assumed to remain constant as the concentration of Ag+ or H + is altered, but the potential drop across the solid solution
372
APPENDIX 6
interfaces can be altered : dE+ = -^αμΗ+ +άφ F where φ is the surface potential on the Agi crystal. Substituting eq (9) in (8) and using (3) gives: - φ = σάφ + Σ Γ ιΦ«
(Α6.9)
( A6.10)
i
where the summation is now over all charged species other than the potential determining ions. Reference Delahay, P. (1965). 'The Double Layer and Electrode Kinetics." Interscience, New York and London.
The Gibbs Adsorption Isotherm at Charged Interfaces
At a charged interface we have at constant temperature and pressure (Delahay, 1965):
- dy = γτ,άμ, + YJ^ '
(A6.1)
J
where Γ,, Γ, are the surface excesses of the various components, μ} is the chemical potential and μ, is the electrochemical potential, so that the sum mation over / is for charged components and that over j is for uncharged components. Consider an Agi crystal in contact with an aqueous solution containing A g N 0 3 and HNO3. Then: - dy = rAg+ · ΦΑ 8 * + Γ ι- dßr + ΓΗ+ · άμ^ + Γ ΝΟί
ΦΝΟ3-
+
ΓΗ 2 ΟΦΗ 2 Ο
(Α6.2) But /Ug+ + μ\- - MAgi = constant and so dßkg* = Also
μΑ8+ = ^Ag+ + H
-αμι-.
and
μ,- = μι- - F(/>.
Substituting these expressions in eq (2) and remembering that rAg* - Γ,- + ΓΗ+ - Γ ΝΟ - = 0
we find -dy=
(r Ag+ - Γ,- )άμκ%* + ΓΗ+ < W + Γ ΝΟ - . ΦΝΟ,370
(A6.3) (Α6.4)
APPENDIX 6
371
Substituting for ΓΗ+ from eq (3) we can write eq (4) as: - dy = (rAg+ - r r )(dßAg+ - φ Η + ) + ΓΝΟ- · ΦΗΝΟ3
(A6·5)
Now consider the electrochemical cell: Cu'|Ag|Agl|AgN03, HN0 3 |H 2 , Pt|Cu" 01 02 03 04 05 06 whose cell reaction is : Agi +^H 2 ->Ag + H + +I~. The e.m.f. of this cell is given by : r
RT
r
F
^
And, hence — FdE+
=ΦΗ
+
+ Ψ Ι - = ΦΗ+
—
(Α6.6)
ΦΑ 8 +
Combining eqs (5) and (6) ( Α6 · 7 )
- dy = F(rAg+ - r r )dE+ + ΓΝΟ-. Φ Η ΝΟ,
Normally, the concentrations of Ag* and I" are very low and most of the ions in the system are H + and NOJ (or some other ions with no special relationship to the crystal surface). We can then regard ^(r Ag+ — r r ) as the surface charge density σ, on the Agi crystal and eq (A6.7) becomes : - dy = σάΕ+ + ΓΝθ3- · ΦΗΝΟ3 (A6·8) Note that the subscript + on Prefers to the fact that the e.m.f., £+, is measur ed with reference to an electrode which is reversible to a cation (in this case H + ). The corresponding equation with £_ involves the surface excess of the cation (ΓΗ+)·
A6.1. The surface potential of the Agi crystal
The e.m.f. of the cell is measured by determining the difference in electro chemical potential of the electron μβ in the two pieces of copper wire (see Section 2.1). Since these are of the same composition we can write: /UCu") - /ie(Cu') = FE+ = Ε{φ6 - φ,) = F[(4>6 - φ5) + (φ5 - φ4) + (φ4 - φ3) + (03 - φ2) + (φ2 - φχ)] const
const
const
Some of these terms can be assumed to remain constant as the concentration of Ag+ or H + is altered, but the potential drop across the solid solution
372
APPENDIX 6
interfaces can be altered : dE+ = -^αμΗ+ +άφ F where φ is the surface potential on the Agi crystal. Substituting eq (9) in (8) and using (3) gives: - φ = σάφ + Σ Γ ιΦ«
(Α6.9)
( A6.10)
i
where the summation is now over all charged species other than the potential determining ions. Reference Delahay, P. (1965). 'The Double Layer and Electrode Kinetics." Interscience, New York and London.