Capacity: differential diffuse layer

FIG. 8 Inverse differential capacity at the zero surface charge vs. inverse capacity Cj of the diffuse double layer for the water-nitrobenzene (O) and water-1,2-dichloroethane (, ), interface. The diffuse layer capacity was evaluated by the GC ( ) or the MPB (0,)> theory. (From Ref. 22.)... 

On the basis of this model, the overall differential capacity C for a system without specific adsorption, i.e. if the compact layer does not contain ions, is divided into two capacities in series, one corresponding to the compact layer Cc and the other to the diffuse layer Cd ... 

The differential capacity of the diffuse layer is defined by the relationship Cd = -dad/d02- According to this definition we obtain, from Eq. (4.3.11),... 

The charge density on the electrode a(m) is mostly found from Eq. (4.2.24) or (4.2.26) or measured directly (see Section 4.4). The differential capacity of the compact layer Cc can be calculated from Eq. (4.3.1) for known values of C and Cd. It follows from experiments that the quantity Cc for surface inactive electrolytes is a function of the potential applied to the electrode, but is not a function of the concentration of the electrolyte. Thus, if the value of Cc is known for a single concentration, it can be used to calculate the total differential capacity C at an arbitrary concentration of the surface-inactive electrolyte and the calculated values can be compared with experiment. This comparison is a test of the validity of the diffuse layer theory. Figure 4.5 provides examples of theoretical and experimental capacity curves for the non-adsorbing electrolyte NaF. Even at a concentration of 0.916 mol dm-3, the Cd value is not sufficient to permit us to set C Cc. 

According to Eq. (4.3.13) the differential capacity of the diffuse layer Cd has a minimum at 2 = 0, i.e. at E = Epzc. It follows from Eq. (4.3.1) and Fig. 4.5 that the differential capacity of the diffuse layer Cd has a significant effect on the value of the total differential capacity C at low electrolyte concentrations. Under these conditions, a capacity minimum appears on the experimentally measured C-E curve at E — Epzc. The value of Epzc can thus be determined from the minimum of C at low electrolyte concentrations (millimolar or lower). 

Differentiating Eqn. 5—3 with respect to the potential, we obtain the differential electric capacity, Ci = (3oM/d oHP), of the diffuse layer in an aqueous ionic solution of z-z valence as shown in Eqn. 5-4 ... 

According to this model, the experimental electrode differential capacity of the interface, which is potential-dependent, can be described in terms of the capacity of the inner layer CH and the capacity of the diffuse layer Cd. 

Here P and y are the thickness of the hydrocarbon tail and of the polar head region of the lipid monolayer, Sp and eY are the corresponding distortional dielectric constants, %e and Xm are the surface dipole potentials due to the electron spillover and to the oriented polar heads, and fa is the potential difference across the diffuse layer. At ion concentrations that are not exceedingly low, fa can be disregarded as a good approximation. Moreover, the orientation of the polar heads of the lipid film is hardly affected by changes in aM. The differential capacity C of the electrode can, therefore, be written ... 

Our modeling approach was first used to describe the EDL properties of well-characterized, crystalline oxides ( 1). It was shown that the model accounts for many of the experimentally observed phenomena reported in the literature, e.g. the effect of supporting electrolyte on the development of surface charge, estimates of differential capacity for oxide surfaces, and measurements of diffuse layer potential. It is important to note that a Nernstian dependence of surface potential (iIJq) as a function of pH was not assumed. The interfacial potentials (4>q9 4> 9 in Figure 1) are... 

Figure 3.10 illustrates the same trends in terms of capacitances. In this example the asymmetry of the electrolyte has been varied at fixed concentration. In this plot the trends are more pronounced than in fig. 3.9. The new feature is that the capacity minimum no longer coincides with the zero point of the diffuse layer potential, but is shifted in the direction where the multivalent ion is the co-ion. (In fig. 3.9 the same can be seild of the position of the minimum slope.) The value y (min) where the capacitance minimum is located can be obtained by differentiating [3.5.34] with respect to y leading to the condition... 

Van Hal et al. [48] used the 2-pK and MUSIC models combined with diffuse layer and Stern electrostatic models (with pre-assumed site-density and surface acidity constants) to calculate the surface potential, the intrinsic buffer capacity -(d(To/dpHs)/e where pHs is the pH at the surface, the sensitivity factor -(d o/dpH) x [e/(kTln 10)], which equals unity for Nernstian response, and the differential capacitance for three ionic strengths as a function of pH. The calculated surface potentials were compared with the experimentally measured ISFET response. 

The purely electrostatic diffuse layer model often underestimates the affinity of the counterions to the surface. In the Stem model, the surface charge is partially balanced by chemisorbed counterions (the Stem layer), and the rest of the surface charge is balanced by a diffuse layer. In the Stern model, the interface is modeled as two capacitors in series. One capacitor has a constant capacitance (independent of pH and ionic strength), which represents the affinity of the surface to chemisorbed counterions, and which is an adjustable parameter the relationship between a, and Vd in the other capacitor (the diffuse layer) is expressed by Equation 2.18. A version of the Stern model with two different values of C (below and above pHg) has also been used. The capacitance of the Stem layer reflects the size of the hydrated counterion and varies from one salt to another. The correlation between cation size and Stern layer thickness was studied for a silica-alkali chloride system in [733]. Ion specificity of adsorption on titania was discussed in terms of differential capacity as a function of pH in [545]. The Stern model with the shear plane set at the end of the diffuse layer overestimated the absolute values of the potential of titania [734]. A better fit was obtained with the location of the shear plane as an additional adjustable parameter (fitted separately for each ionic strength). Chemisorption of counterions can also be quantified within the chemical model in terms of expressions similar to the mass law (Section 2.9.3.3). 

In this case, it is useful to apply the so-called Parsons-Zobel plot [58] that shows the dependence of the reciprocal measured differential capacity on the reciprocal of the calculated diffuse layer capacity at a constant charge density covering wide spectra of electrolyte concentrations. The slope of this plot must be a straight line and equal to the R when we have a non-adsorbing electrolyte. However, it has been observed that at a low electrolyte concentration the linearity is lost and the slope becomes lower [57] (electrolyte concentrations below 0.01 M). 

Area, constant, optical absorption Activity, absorption coefficient Debye length Cyclic voltammogram Capacitance / j,F Double layer capacity / 0.F Differential double layer capacity Integral double layer capacity Concentration / M Surface concentration Bulk concentration / M Diffusion coefficient / cm s ... 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

If qM is evaluated experimentally, e.g. from the integration of differential capacity curves, A02 can be calculated using eqn. (44) of the diffuse double layer theory. Figure 3 shows the variation of A02 with... 

An important quantity with respect to experimental verification is the differential capacitance of the total electric double layer. In the Stern picture it is composed of two capacitors in series the capacity of the Stem layer, Cgt, and the capacitance of the diffuse Gouy-Chapman layer. The total capacitance per unit area is given by... 

Another factor augmectting the heat conductivuty of plastic foams under conditions is the absorbed moisture. For example, for CCljF-foamed polyurethane at 25 °C and a relative humidity of 65%, the ambient moisture diffusion rate is 10-20 g/m for 24 h. Especially strong is the effect of moisture on heat conductivity if the temperature differential across the sample is considerable. For example, in plastic foams used in cryogenic technology, the inner layers are exposed to low temperatures the water vapor first condenses and is then convected into ice. Since the thermoconductivity of water and ice are 0.5 and 1.5 kcal/m x h °C, respectively, even minor tunounts have a considerable detrimental effect of the heat insulating capacity of a foam material... 

The PZC may also be seen on differential capacity curves when specific adsorption is absent and the electrolyte concentration is low (< 0.01 M). At this point the capacity of the diffuse part of the double layer is a minimum and can fall below that of the compact or inner layer. As a result the total double layer capacity may... 

It should be noticed that the Galvani potential difference, A(p2, cannot be measured directly but its value can be derived from the Gouy-Chapman theory of the diffuse double layer. If the excess charge in the metal, q , is determined experimentally, that is, from the integration of differential capacity curves, then for a 1 1 electrolyte of concentration c, then A(p2 can be calculated ... 

Since the capacity of the diffuse double layer is in series with the Helmholtz double layer, the best chance to see it in the measurements requires Cd < Ch. Equations (2.50) and (2.54) indicate that Cd increases rapidly with Q. In accordance with this conclusion, a minimum of the differential capacity was found in measurements on metals in very dilute electrolyte solutions (<10 M), which is an indication for the absence of excess charge on the electrode. 

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