Capacity: differential integral

It should be noted that the capacity as given by C, = a/E, where a is obtained from the current flow at the dropping electrode or from Eq. V-49, is an integral capacity (E is the potential relative to the electrocapillary maximum (ecm), and an assumption is involved here in identifying this with the potential difference across the interface). The differential capacity C given by Eq. V-50 is also then given by... 

Fig. V-12. Variation of the integral capacity of the double layer with potential for 1 N sodium sulfate , from differential capacity measurements 0, from the electrocapillary curves O, from direct measurements. (From Ref. 113.)...

The variation of the integral capacity with E is illustrated in Fig. V-12, as determined both by surface tension and by direct capacitance measurements the agreement confrrms the general correctness of the thermodynamic relationships. The differential capacity C shows a general decrease as E is made more negative but may include maxima and minima the case of nonelectrolytes is mentioned in the next subsection. 

The first mtegral is the energy needed to move electrons from fp to orbitals with energy t> tp, and the second integral is the energy needed to bring electrons to p from orbitals below p. The heat capacity of the electron gas can be found by differentiating AU with respect to T. The only J-dependent quantity is/(e). So one obtains... 

Equation (2.18) is another example of a line integral, demonstrating that 6q is not an exact differential. To calculate q, one must know the heat capacity as a function of temperature. If one graphs C against T as shown in Figure 2.8, the area under the curve is q. The dependence of C upon T is determined by the path followed. The calculation of q thus requires that we specify the path. Heat is often calculated for an isobaric or an isochoric process in which the heat capacity is represented as Cp or Cy, respectively. If molar quantities are involved, the heat capacities are C/)m or CY.m. Isobaric heat capacities are more... 

The differential capacity is given by the slope of the tangent to the curve of the dependence of the electrode charge on the potential, while the integral capacity at a certain point on this dependence is given by the slope of the radius vector of this point drawn from the point Ep = pzc. 

A computer can do only three things add, subtract, and decide whether some value is positive, negative, or zero. The last capacity allows the computer to decide which of two alternatives is best when some quantitative objective function has been selected. The ability to add and subtract permits multiplication and division, plus the approximation of integration and differentiation. 

Figure 1. Integral (aHx) and differential (d(aHx)/dx) enthalpies of Ca - 2K ( ) and 2K + Ca (0) exchange on Wyoming bentonite as a function of the fractional K saturation of the exchange capacity.

The procedure for numerical integration (7) is analogous to that for differentiation. Again we will cite an example of its use in thermodynamic problems, the integration of heat capacity data. Let us consider the heat capacity data for solid n-heptane listed in Table A.4. A graph of these data (Fig. A.3) shows a curve for which it may not be convenient to use an analytical equation. Nevertheless, in connection with determinations of certain thermodynamic functions, it may be desirable to evaluate the integral... 

This is the integral capacitance, and it is generally used for electrical capacitors where the capacity is constant and independent of the potential. This constancy of capacity may not be the case with electrified interfaces and in order to be prepared for this eventuality, it is best to define a differential capacity C thus... 

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... 

The term (9U/9T)y is defined from thermodynamics as the heat capacity at constant volume (Cy). The second term on the right hand side (RHS) of Equation 2.20, (9U/9V)T, is much less than Cv and can be neglected. By taking the integral of our differential expression we obtain a relation for internal energy ... 

Most chemical processing plants include pure or multicomponent streams which change phase or which have strongly temperature-dependent heat capacities. Under these conditions the minimum approach temperature in a network can occur anywhere inside an exchanger. Therefore integral or differential equations are generally required to locate A Tm. 

Cd differential capacity of double layer Cj integral capacity of double layer Cs capacity in RC series combination... 

The charge density of a -> double layer is Q = C(E - Ezc)> where C is the differential capacity and Ezc is a -+ potential of zero charge, i.e., the potential of electro-capillary maximum. By integrating the Lippmann equation a relationship between the surface tension and the electrode potential is obtained y = ymax -C(E - Ezc)212. This is a parabola symmetrical about ymax> if C is independent of E [ii]. 

---