chapter 9 interfacial equilibria
9.1Introduction...... ….………………………..1
9.2Models of the Electrified Solid/Aqueous Interface…………………………………….2
9.2.1Origins of Surface Charge.……………………………………………………….. 2
9.2.2Determination of Surface Charge.………………………………………………..3
9.2.3The Point of Zero Charge and the Isoelectric Point…………………………….....4
9.2.4Electric Double Layer and Triple Layer Models………………………………….. 9
9.2.5The Space Charge Region...... …………………...19
9.3Surface Ionization Equilibria………………………………………………………….24
9.3.1One-site Amphoteric Surfaces…………………………………………………...24
9.3.2Zwitterionic Surfaces…………………………………………………………….36
9.3.3Two-site Amphoteric Surfaces…………………………………………………... 39
9.3.4Two-site Amphoteric/Basic Surfaces……………………………………………. 42
9.3.5Multi-site Surfaces: Bond Valence Model………………………………………43
9.4Specific Adsorption and Surface Complexation……………………………………...49
9.4.1Specific Adsorption Models…………………………………………………...... 49
9.4.2Adsorption of Inorganics: Surface Occupancy Models………………………..,,51
9.4.3Adsorption of Inorganics: Surface Complexation Models……………………...56
9.4.4Adsorption of Surfactants……………………………………………………...... 60
9.4.5Adsorption of Organic Polymers………………………………………………...67
9.5Interfacial Activity in Liquid/Liquid and Gas/Liquid Systems……………………..
9.5.1Surface Active Agents…………………………………………………………...
9.5.2Interfacial Tension……………………………………………………………….
9.5.3Adsorption and the Gibbs Equation…………………………………………….
9.5.4Aggregation and Micellization………………………………………………….
9.1 Introduction
Interfaces encountered in heterogeneous aqueous systems (solid/aqueous, oil/aqueous, gas/aqueous) are usually charged. The surface charge corresponds to a preferential accumulation (i.e., adsorption) of charged entities (electrons, cations, or anions) in the interfacial region, and it arises from interaction of the solid, oil or gas phase with its aqueous environment. A major concern in this chapter is a quantitative description of the equilibrium condition of electrified interfaces and the associated adsorption processes.
9.2Models of the Electrified Solid/Aqueous Interface
9.2.1Origins of Surface Charge
When materials such as oxides, sulfides, insoluble salts, and activated carbons are immersed in aqueous solution, they acquire a surface charge the presence of which controls much of their interfacial behavior. This charge arises from the dissociation of surface functional groups, and is akin to the charging of the metal/solution interface as discussed in Chapter 8. For example, when a metal oxide surface is exposed to water, adsorption of water molecules produces a hydroxylated surface:
MO (surf) + H2O = MOH2O (surf) = MOHOH (surf)(9.1)
Subsequent protonation and dissociation (i.e., deprotonation) of the surface hydroxyls produce a charged oxide surface:
MOH (surf) + H+ = MOH2+ (surf) (9.2)
MOH (surf) = MO-(surf) + H+ (9.3)
That is, the surface hydroxyl can act both as a basic site (Equation 9.2) and an acidic site (Equation 9.3). When the hydroxyl group acts as a basic group, adsorption of H+ yields a positively charged site. On the other hand, when the hydroxyl group behaves as an acidic group, it can deprotonate to give a negatively charge site.
A similar process is responsible for the ionization of carbon surfaces. Here the surface charge arises from the acid-base reactions of hydroxyl groups present on the original material or produced by interaction of oxygen-containing surface functional groups with water:
CO (surf) + H2O = COHOH (surf)(9.4)
The charge on a solid surface may also result from the selective desorption (or dissolution) of the constituent surface atoms:
MA (surf) = M+ (surf) + A-(9.5)
MA (surf) = A- (surf) + M+ (9.6)
Alternatively the surface charging of MA may result from the preferential adsorption of the constituent ions from the bulk aqueous phase:
MA (surf) + M+ = M2A+ (surf) (9.7)
MA (surf) + A- = MA2-(9.8)
The surface ionization processes described by Equations 9.5 - 9.8 are characteristic of the interfacial behavior of salt-type materials e.g. CaCO3, CaWO4, AgI etc.
In the case of sulfides, the surface ionization is attributable to a combination of oxide-type and salt-type behavior. The interaction of a sulfide MS with water may be depicted as:
MS (surf) + H2O = MSH2O (surf) = MOHSH (surf)(9.9)
That is, a surface metal atom binds a hydroxyl group provided by the adsorbed water molecule while the sulfur atom binds the corresponding proton to give a thiol group. The resulting MOH group can undergo acid-base ionization reactions as described by Equations 9.2 and 9.3. The thiol group can also participate in protonation/deprotonation reactions:
SH (surf) + H+ = SH2+ (surf) (9.10)
SH (surf) = S- (surf) + H+ (9.11)
The presence of charge on the solid surface results in a potential difference between the solid surface and the bulk aqueous phase. The charge and the potential at the solid surface are designated as o and o respectively. The ions which are responsible for the development of the surface charge and the surface potential are designated as potential-determining ions. For metal oxides and hydroxides, the potential determining ions are the proton and the hydroxyl ion. In the case of silver iodide, AgI, the potential determining ions are Ag+ and I-.
9.2.2Determination of Surface Charge
Consider a volume (V) of aqueous solution prepared by mixing certain amounts of strong acid and strong base solutions. In the absence of chemical reaction (including neutralization) the respective concentrations of added acid and base are [H+]T and [OH-]T respectively. Let an insoluble metal oxide be dispersed in the aqueous solution such that the solid/liquid interfacial area per unit volume is S. The solid surface interacts with H+ and OH- ions to give final bulk aqueous phase concentrations of [H+]a and [OH-]a respectively.
The following mass balance can be written for the equilibrated system:
[H+]T - [OH-]T = [H+]a - [OH-]a + SH+ - SOH-(9.12)
where H+ and OH- respectively represent the surface concentrations of adsorbed proton and hydroxide ions. Equation 9.12 can be rearranged to give the surface charge (o) as:
(o/F) = ([H+]T - [OH-]T + [OH-]a - [H+]a)/S = (H+ - OH-)(9.13)
9.2.3The Point of Zero Charge and the Isoelectric Point
The point of zero charge (pzc) is an important interfacial parameter which is used extensively in characterizing the ionization behavior of a surface. The pzc refers to the bulk aqueous phase concentration (or more precisely, activity) of potential-determining ions which gives zero surface charge. Also since the surface potential exists as a direct result of the presence of surface charge, zero surface charge implies zero surface potential. That is, at the pzc,
o = 0, o = 0(9.14)
EXAMPLE 9.1Acidimetric - Alkalimetric Titration of - Al2O3
Hohl and Stumm (J. Colloid Interface Sci., 55, 281 (1976)) obtained acid-base titration data for - Al2O3 as summarized in Table E9.1. Using these data, determine the pH-dependence of the surface charge density (o) of -Al2O3. The solid material has a specific surface area of 117 m2g-1. The experiments were conducted at a constant ionic strength of 0.1 M NaClO4, and a solids loading of 3.75 g/L.
Table E9.1 Acidimetric-alkalimetric titration data for -Al2O3
Acid added*(cm3) / Solution
pH / Base added+
(cm3) / Solution
pH
0.045 / 7.83 / 0.035 / 8.64
0.084 / 7.23 / 0.075 / 9.04
0.12 / 6.63 / 0.10 / 9.40
0.16 / 6.07 / 0.15 / 9.60
0.20 / 5.52 / 0.20 / 9.80
0.25 / 4.96 / 0.25 / 9.95
0.275 / 4.61 / 0.275 / 10.1
0.32 / 4.41 / 0.30 / 10.2
0.35 / 4.21 / 0.35 / 10.2
0.40 / 4.11 / 0.40 / 10.3
0.45 / 4.01
______
*0.1M HClO4 / ______
+0.1M NaOH
example 9.2 Estimation of the Point of Zero Charge of Ag2O
(a)It has been suggested that at the point of zero charge of a solid MA where M is a metal atom, the concentrations of the soluble M-containing species are such that these species contribute a net zero charge in the bulk aqueous phase. Table E9.2 presents chemical reactions and corresponding equilibrium constants for the Ag-H2O system. Estimate pHpzc.
Table E9.2
ReactionlogK
Ag2O(s) + 2H+ = 2Ag+ + H2O 12.7
AgOH(aq) + H+ = Ag+ +H2O 12.0
Ag(OH)+ 2H+ = Ag+ + 2H2O 24.0
(b)Show that the pHpzc of Ag2O as estimated with the isoelectric pH of the solution also corresponds to the pH of minimum solubility of this oxide.
Solution
(a)Consider the reactions
Ag2O(s) + 2H+ = 2Ag+ + H2OlogK1 = 12.7 (1)
AgOH(aq) + H+ = Ag+ + H2OlogK2 = 12.0 (2)
Ag(OH) + 2H+ = Ag+ + 2H2OlogK3 = 24.0 (3)
It follows from Equations 1, 2, and 3 respectively that:
[Ag+] = K11/2[H+] (4)
[AgOH(aq)] = [Ag+]/[H+]K2 (5)
[Ag(OH)] = [Ag+]/[H+]2K3 (6)
Let C+ and C- respectively represent the concentrations of positive and negative charges due to the soluble silver species. Then
C+ = [Ag+] and C- = [Ag(OH)] (7)
At the isoelectric conditions in the solution C+ = C- and therefore Equations 4, 6 and 7 may be combined to give
pH = 1/2 log K3 = 1/2 (24.0) = 12.0 (8)
(b)Let [Ag] represent the total concentration of dissolved silver. Then,
[Ag] = [Ag+] + [AgOH(aq)] + [Ag(OH)] (9)
Substitution of Equations 4-6 into Equation 9 gives
[Ag] = K11/2 (1/K2 + [H+] + 1/[H+]K3) (10)
The condition for minimum solubility requires that
d[Ag]/d[H+] = 0 (11)
If follows from Equations 10 and 11 that the pH of minimum solubility is given by
[H+]2 = 1/K3 (12)
That is,
2log[H+] = -logK3 = -24 (13)
Thus it follows that:
pHpzc = -log[H+] = 24/2 = 12 (14)
example 9.3 Estimation of the Point of Zero Charge of Calcite
Determine the pHpzc of calcite on the basis of the isoelectric pH of a solution in equilibrium with CaCO3 and atmospheric CO2 (PCO2) = 10-3.5 atm). The relevant thermodynamic data are presented in Table E9.3
Table E9.3
CaCO3 (s) = Ca2+ + CO32-logK1 = -8.36(1)
CaOH+ + H+ = Ca2+ + H2OlogK2 = 12.75(2)
HCO3- = CO32-+ H+logK3 = -10.33(3)
H2CO3 (aq) = CO32- + 2H+logK4 = -16.69(4)
CO2 (g) + H2O = CO32- + 2H+logK5 = 29.93(5)
CaHCO3+ = Ca2+ + H+ + CO32-logK6 = -11.43(6)
CaCO3 (aq) = Ca2+ + CO32-logK7 = -3.15(7)
Solution
It follows from the above equations that
[Ca2+] = K1/[CO32-](8)
[CaOH+] = K2[Ca2+]/[H+](9)
[HCO3-] = K3[CO32-][H+](10)
[H2CO3 (aq)] = K4[CO32-][H+]2(11)
[CO32-] = K5PCO2/[H+]2(12)
[CaHCO3+] = K6[Ca2+][H+][CO32-](13)
[CaCO3 (aq) = K7[Ca2+][CO32-](14)
With the aid of Equations 8 and 12, [Ca2+] can be re-expressed as:
[Ca2+] = K1[H+]2/K5PCO2(15)
From Equations 9 and 15,
[CaOH+] = K1K2[H+]/K5PCO2(16)
From Equations 12, 13 and 15,
[CaHCO3+] = K1K6 [H+](17)
Also, from Equations 10 and 12,
[HCO3-] = K3K5PCO2/[H+](18)
Let C+ and C- respectively represent the concentrations of positive and negative charges associated with Ca - and CO3 - containing species. Then:
C+ = 2[Ca2+] + [CaOH+] + [CaHCO3+](19)
C- = 2[CO32-] + [HCO3-](20)
At the isoelectric pH of the solution, C+ = C- and therefore it follows from Equations 12 and 15 - 20.
2K1[H+]2/K5PCO2 + K1K2[H+]/K5PCO2 + K1K6[H+] = K5PCO2/[H+]2 + K3K5PCO2/[H+](21)
That is,
[H+]4 + a[H+]3 + c[H+] + d = 0(22)
where:
a = (K2 + K5K6PCO2)/2(23)
b = -K3K52PCO22/2K1(24)
d = -K52PCO22/2K1(25)
______
As illustrated in Ex 9.1, the point of zero charge can be determined via potentiometric titration of colloidal dispersions with the appropriate potential-determining ions. That is, in the case of oxides or hydroxides, plotting the net uptake of hydrogen and hydroxide ions vs. pH gives curves for different constant electrolyte concentrations which intercept at the pzc.
Table 9.1 The point of zero charge (pzc) and the isoelectric point (iep) of selected materials (see Parks, 1965, 1967, 1975, 1982).
Type of MaterialMaterialspzc/iep
HalideAg ClpAg 4.1 - 4.6 (iep)
AgBrp Ag 5.6 - 5.9 (pzc); pAg 4.2 - 5.4 (iep)
AgIPAg 4.8- 6.05 (pzc); pAg 5.1 - 6.2 (iep)
CaF2pCa 2.6 - 7.7 (iep)
SulfateBaSO4pBa 3.9 - 7.0 (iep)
PbSO4pPb 5.1 (iep)
PhosphateAlPO4 • 2H2OpH 4 (pzc)
FePO4 • 2H2OpH 2.8 (pzc) pH 3.4 (iep)
(Ce, La, Th) PO4pH 3.4 (iep)
Ca5(PO4)3 (F, OH)pH 6.9 - 8.5 (pzc), pH 4-6 (iep)
CarbonateCaCO3(calcite)pH 9.5 - 10.8 (iep)
MgCO3(magnesite)pH 5.2 (iep)
CaMg(CO3)2 (dolomite)<pH 8.5 (iep)
TungstateCaWO4pCa 4.8 (iep)
Oxide-Al2O3pH 9.1 (pzc)
-Al2O3pH 8.5 (pzc)
-Fe2O3pH 9.3 (pzc)
-FeOOHpH 7.6 - 8.3 (pzc)
-MnO2pH 2.8 - 4 (pzc)
-MnO2pH 4.5 (iep)
-MnO2pH 4.6 - 7.3 (iep)
-MnO2pH 5.5 (iep)
SiO2pH 3.0 (pzc)
SnO2pH 5.5 (pzc)
TiO2pH 6.0 (pzc)
SilicateAl2SiO5 (kyanite,pH 5.2 - 7.9 (iep)
andalusite, sillimanite)
NaAlSi3O8-KAlSi3O8pH 2.0 - 2.4 (iep)
(albite-orthoclase)
Mg3Si2O5 (OH)8
ChrysotilepH 10 - 12 (iep)
LizarditepH 9.6 (iep)
SerpentinepH 6.6 - 8.9 (iep)
Mg2SiO4 (Forsterite)≤ pH 8.4 (iep)
ClayKaolinitepH 2 - 4.6 (iep)
Montmorillonite≤ pH 2.5 (iep)
The presence of a charge at the solid/aqueous interface means that in the presence of an applied electrical field, there is relative motion between solid and liquid. This process is termed electrokinetics, and depending on the particular experimental situation, the process may involve the movement of colloidal particles in a stationary fluid (electrophoresis) or the movement of liquid along a stationary surface (electroosmosis). In either case, a thin immobile film of water adheres to the solid surface. The boundary between the immobilized water layer and the bulk water is termed the shear plane and the potential difference between the shear plane and the bulk liquid is denoted as the zeta potential (). The activity of potential determining ions that gives zero zeta potential is designated the isoelectric point (iep). Table 9.1 presents pzc and iep data for selected materials.
9.2.4Electric Double Layer and Triple Layer Models
The Gouy-Chapman Model. The charge on a surface will attract ions of opposite charge in the aqueous phase to the surface. The resulting system of a surface charge and a corresponding counter-charge on the aqueous side of the interface has been given the name of electric double layer. Various models of the electric double layer have been presented, the main differences in the models being dependent on the manner in which the spatial distribution of the counter-charge is visualized. The simplest model, the Gouy-Chapman model, considers that the electric double layer consists of a surface layer of charge on the solid surface () balanced by a diffuse layer of counter ion charge (d) located at the aqueous side of the interface, as illustrated schematically in Figure 9.3a. The surface charge results in a surface potential ( and a potential distribution ((x)) in the adjacent aqueous phase. The resulting electroneutrality condition is:
od(9.15)
Figure 9.3 The Gouy-Chapman model: (a) Charge distribution, (b) Potential distribution
Let us consider a flat surface adjacent to an aqueous electrolyte solution. Let the solution contain a single symmetrical electrolyte. That is, the electrolyte consists of cations and anions of equal and opposite charge. Following the approach in Chapter 6, the ionic concentrations in the diffuse layer may be related to the corresponding bulk aqueous phase concentrations through the Boltzmann relation:
ni = nio exp -zie/kT(9.16)
For a single symmetrical electrolyte, the cations are only of one type with charge z+ and the anions are of one type with charge z-; also, z+ = | z- | = z. In the bulk aqueous phase, electron entrality requires that n+o = n-o = no. Thus,
n+ = no exp - ze/kT(9.17)
n- = no exp -ze/kT(9.18)
Accordingly, at any point within the diffuse layer, the net charge density (can be expressed as:
= z+en+ + z-en- = ez(n+ - n-)(9.19)
Using Equations 9.17 and 9.18 in Equation 9.19,
= ezno[exp (- ze/kT) - exp(ze/kT)](9.20)
Recall,
sinh = (exp - exp - )/2(9.21)
It follows from Equations 9.20 and 9.21 that:
= -2ezno sinh(ze/kT)(9.22)
Again, as was done in Chapter 6, the charge density at any point x in the diffuse layer is related to the electric potential at that point through the Poisson equation:
(x) = d2(x)/dx2 = -(x)/o(9.23)
where = dielectric constant of water, and o = the permittivity of free space (o = 8.85 x 10-14 CV cm-1). Combining Equations 9.22 and 9.23 gives
d2/dx2 = (2ezno/o) sinh(ze/kT)(9.24)
Equation 9.24 can be solved with the following boundary conditions:
= o at x = 0(9.25)
= 0, d/dx = 0 at x = )
Using the above boundary conditions, Equation 9.24 gives:
x) = (2kT/ze) ln[(1 + exp (-x))/(1 - exp(-x))](9.27)
where
= exp [(zeo/kT) - 1]/exp[zeo/kT) + 1](9.28)
and
= (2z2e2no/okT)1/2 = 3.29 x 109 z C1/2 m-1 at 298K(9.29)
where C is the concentration of the electrolyte (mol dm-3) in the bulk aqueous phase.
The total charge density in the diffuse layer is given by:
d = (9.30)
Combining Equations 9.23 and 9.30,
d = - [d/dx]
Using the boundary condition d/dx = 0 at x = ∞, we get
d = o (d/dx)x=0(9.32)
It follows from Equations 9.27 and 9.32 that
d = -(8nookT)1/2 sinh(zeo/2kT)(9.33)
If is given in volts and the electrolyte concentration (C) in mol dm-3, Equation 16 becomes:
d (C cm-2) = - 11.74C1/2 sinh(19.46zo) at 298K(9.34)
EXAMPLE 9.4 Diffuse layer potentials for very low and very high surface potentials
(a)Derive an expression for the diffuse layer potential for the case where o < 25mV.
(b)Repeat (a) for the case where o > 25mV.
Solution
At 298K, kT = 25.7 mV. Thus, o < 25mV implies (ze/kT) < 1. Thus, under these circumstances, Equation 12 reduces to:
= exp (-1)/exp (1) =
The Stern Model. The Gouy-Chapman model of the electric double layer breaks down at small values of x and high o. This difficulty arises because the model assumes that the ions in the aqueous side of the interface are point charges. The Stern model accounts for ionic size by locating the centers of the first layer of ions at a mean distance d from the solid surface, as illustrated in Figure 9.4. Beyond this first layer, the ionic distribution follows the Gouy-Chapman picture of a diffuse layer based on point charges.
Figure 9.4 The Stern Model
The first layer of ions is designated the Stern plane with a potential d. The electroneutrality condition for the Stern model is given by:
o + d = 0(9.45)
In the region 0 < x < d, there is no charge and therefore the Poisson equation can be written as:
(9.46)
This means that
(9.47)
where A is a constant. Taking advantage of Equation 9.32, we can write:
(9.48)
where d- and d+ respectively represent the left-hand side and the right-hand side of the x = d line.
Equation 9.47 is valid for all parts of the region 0 < x < d and in particular for x = d. It follows therefore that A = d. That is,
It follows from Equation 9.49 that:
(9.50)
(9.51)
That is,
K1 = C1 = -d/(o - d) = o/(o - d) = 1/d(9.52)
where K1 and C1 are respectively the integral capacitance and the differential capacitance of the inner region.
Equation 9.52 may be rearranged to give:
o = o/C1 + d(9.53)
According to Equation 9.52, the inner region corresponds to a capacitor consisting of two parrallel plates distance d apart, with an average permittivity of 1.
According to the Stern model, d is given by the Gouy-Chapman expression (Equations 9.33 and 9.34) with the Stern potential (d) replacing the surface potential (o):
d = - (8nookT)1/2 sinh(zed/2kT)(9.54a)
d (C cm-2) = -11.74C1/2 sinh (19.46 zd)(9.54b)
Equation 9.54a may be expressed as
d = (2kT/ze) sinh-1 [-d/(8nookT)1/2](9.55)
The Stern-Grahame Model. The Stern model can be further improved by ascribing sizes to the ions right next to the Stern plane. The ions in the Stern plane are considered to have lost their waters of hydration whereas the ions in the diffuse layer retain their hydration shells. As a result, as illustrated in Figure 9.5a, the Stern layer can be subdivided into the first layer of dehydrated ions, termed the Inner Helmholtz Plane (IHP) and the first layer of hydrated ions, termed the Outer Helmholtz Plane (OHP). The OHP marks the beginning of the diffuse layer. The potential and charge of the IHP are represented by and respectively. The resulting potential distribution is illustrated in Figure 9.5b.
Figure 9.5 The Gouy-Chapman-Stern-Grahame(GCSG) electric double layer.
The electroneutrality condition is:
o + + d = 0(9.56)
The region between the surface plane (x = 0) and the IHP may be treated as a condenser with an integral capacity per unit area of C1. Similarly, the region between the IHP and the OHP may be treated as a condenser with an integral capacity per unit area of C2. Thus, the following potential-charge relationships can be obtained:
o - = o /C1(9.57)
- d = (o + )/C2 = - d/C2(9.58)
Equations 9.57 and 9.58 may be derived by applying the Poisson equation to the regions 0 < x < b and b < x < d:
1d2/dx2 = 00 < x < b(9.59)
2 d2/dx2 = 0b < x < d(9.60)
From Equations 9.59 and 9.60,
1 d/dx = A0 < x < b(9.61)
2 d/dx = Bb < x < d(9.62)
From Equation 9.32,
1(d/dx)b- = 2(d/dx)b+ = (+ d)(9.63)
2(d/dx)d- = (d/dx)d+ = d(9.64)
2(d/dx)d- = d(9.65)
(9.66)
(9.67)
Figure 9.6 Models of the electrified interface. (After Westall)
Depending on the nature of the solid material and the composition of the adjacent aqueous phase, several variations of the above models may be formulated, as summarized schematically in Figure 9.6. The most general situation is that of the triple layer model presented schematically for a planar surface in Figure 9.6a. Here the solid/liquid interface is visualized in terms of three layers of charge. The innermost layer, called the surface layer, consists of the solid surface itself and it is the locale of the primary potential determining ions (i.e. H+, OH- for a metal oxide, and Mz+, Az- for a general solid metal compound MA(s)). The charge and potential associated with this layer are designated as o and o respectively. The next charge region is the inner Helmholtz plane (IHP), which is a compact layer of counter-charge typically consisting of relatively strongly bound (i.e. specifically adsorbed) ions. The corresponding charge and potential are identified as 1 and 1 respectively. The third charge region is the diffuse layer. This is the location of ions, termed indifferent ions, which are only weakly attracted to the solid surface. The diffuse layer outside a charged macroscopic surface is equivalent to the ionic atmosphere that surrounds a central ion in an aqueous solution (see Chapter 6). The plane of the diffuse layer closest to the solid surface is designated the outer Helmholtz plane (OHP). The potential at the OHP is 2 and the charge associated with the diffuse layer is denoted by 2.
From an electrostatic standpoint, the electrical interface based on the triple layer model involves three capacitances in series, i.e., C1, representing the region bounded by the surface plane and the IHP, C2, representing the region bounded by the IHP and the OHP, and Cd, representing the diffuse layer capacitance. Thus the total interfacial capacitance CT is given by:
1/CT = 1/C1 + 1/C2 + 1/Cd(9.70)
The relationship between charge and potential in the first two layers is:
o = C1(o -1)(9.71)
1 = C2(1 -2)(9.72)
Based on the analogy between the ionic atmosphere and the diffuse layer, it can be shown that the following relationship exists between the diffuse layer charge (2), the potential (2) at the OHP, and the concentration (Cb) of the electrolyte in the bulk solution:
2 = -(8RTCb)1/2 sinh (F2/2RT)(9.73a)
= -0.1174 Cb1/2 sinh (F/2RT), at 25°C(9.73b)
The diffuse layer capacitance is then given by
Cd = d2/d2(9.74)
The electrified interface must satisfy the electroneutrality condition. Thus
o + 1 + 2 = 0(9.75)