The Constant-Capacity Region

The Capacitance Hump and the Capacity Minimum. A discussion of the hump should really begin with a consideration of the question Why does the differential capacity of the interface increase when the electrode charge becomes positive with respect to the constant-capacity region Why doesn t the... 

Each region can be considered as an individual condenser and the effective capacity is obtained by treating these three condensers to be in series. This model has been treated in detail and has several satisfactory features. For instance, it could explain the magnitude of the constant capacity region independently of the radius of the ions present in the electrolyte. There was, however, some difficulty in predicting quantitative capacity data over the entire capacity-potential region. 

From the above argument and Eq. (16) we instantaneously find that the isosteric heat of adsorption cannot be constant within the two-phase region but must also show changes with the surface coverage. In the case of heat capacity we also observe important effects due to the surface heterogeneity. 

In the region of supercritical point, most properties of supercritical water vary widely. The most prominent of these is the heat capacity at constant pressure, which approaches infinity at the critical point. Even 25°C above Tc, at 80 bar away from Pc, the heat capacity of water is an order of magnitude greater than its value at higher or lower pressure. 

Consider a plane metal electrode situated at z = 0, with the metal occupying the half-space z < 0, the solution the region z > 0. In a simple model the excess surface charge density a in the metal is balanced by a space charge density p(z) in the solution, which takes the form p(z) = Aexp(—kz), where k depends on the properties of the solution. Determine the constant A from the charge balance condition. Calculate the interfacial capacity assuming that k is independent of a. 

The thermal expansivity of a solid is in general low at low temperatures and the anharmonic contribution to the heat capacity is therefore small in this temperature region and Cv m Cpm. At high temperatures the difference between the heat capacity at constant pressure and at constant volume must be taken into consideration. 

Would Eq. (6.261) for the total differential capacity be able to reproduce the experimental capacity curves Let us have a look again at one of the complete capacity-potential curves shown in Fig. 6.65(b) and illustrated in Fig. 6.106 in this section. This is not a simple curve. It breaks out into breaks and flats, and it has a complicated fine structure that depends upon the ions that populate the interphasial region. Whereas there is a region of constant capacity at potentials more negative than the electrocapillary maximum, there is also a "hump in the capacity-potential... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

For a system with n components (including nonad-sorbable inert species) there are n — 1 differential mass balance equations of type (17) and n — 1 rate equations [Eq. (18)]. The solution to this set of equations is a set of n — 1 concentration fronts or mass transfer zones separated by plateau regions and with each mass transfer zone propagating through the column at its characteristic velocity as determined by the equilibrium relationship. In addition, if the system is nonisothermal, there will be the differential column heat balance and the particle heat balance equations, which are coupled to the adsorption rate equation through the temperature dependence of the rate and equilibrium constants. The solution for a nonisothermal system will therefore contain an additional mass transfer zone traveling with the characteristic velocity of the temperature front, which is determined by the heat capacities of adsorbent and fluid and the heat of adsorption. A nonisothermal or adiabatic system with n components will therefore have n transitions or mass transfer zones and as such can be considered formally similar to an (n + 1)-component isothermal system. 

Compared with ambient values, the specific heat capacity of water approaches infinity at the critical point and remains an order of magnitude higher in the critical region [26], making supercritical water an excellent thermal energy carrier. As an example, direct measurements of the heat capacity of dilute solutions of argon in water from room temperature to 300 °C have shown that the heat capacities are fairly constant up to about 175-200 °C, but begin to increase rapidly at around 225 °C and appear to reach infinity at the critical temperature of water [29]. 

The proportionality constant between the applied potential and the charge due to the species ordering in the solution interfacial region is the double layer capacity. The study of the double layer capacity at different applied potentials can be done by various methods. One much used is the impedance technique, which is applicable to any type of electrode, solid or liquid, and is described in Chapter 11. Another method uses electrocapillary measurements. It was developed for the mercury electrode, being only applicable to liquid electrodes, and is based on measurement of surface tension. 

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