Equipotential volume

This expression is very similar to equation (9) where the term (Vf — V2) replaces E d. One can hope to produce a phase shift of the same order of magnitude if (Vf — V2) Ed, while the phase associated to the polarisability term will be considerably reduced. Moreover, if the construction is well symmetric and if the potentials V and V2 are opposite, with additional entrance and exit electrodes held at V = 0, phases associated to the polarisability term should cancel as a result of symmetry. However, these phases are not very easy to evaluate as the electric field is nonzero at the entrance and exit of the equipotential volumes and the geometry of this field is not simple. This arrangement is very nice from a theoretical point of view, but its alignment is more difficult than for the configuration using only one capacitor. 

A still different approach to multilayer adsorption considers that there is a potential field at the surface of a solid into which adsorbate molecules fall. The adsorbed layer thus resembles the atmosphere of a planet—it is most compressed at the surface of the solid and decreases in density outward. The general idea is quite old, but was first formalized by Polanyi in about 1914—see Brunauer [34]. As illustrated in Fig. XVII-12, one can draw surfaces of equipo-tential that appear as lines in a cross-sectional view of the surface region. The space between each set of equipotential surfaces corresponds to a definite volume, and there will thus be a relationship between potential U and volume 0. 

Here is a point inside the volume V. Hence, it is impossible to distinguish between the field caused by a volume distribution of masses and the field generated by masses on the equipotential surface S, provided that the condition (4.6) is met and the observation point is located outside S. As a rule, a three-dimensional body and... 

In this theory the adsorbed layers are considered to be contained in an adsorption space above the adsorbent surface. The space is composed of equipotential contours, the separation of the contours corresponding to a certain adsorbed volume, as shown in Figure 17.7. The theory was postulated in 1914 by Polanyi(18), who regarded the potential of a point in adsorption space as a measure of the work carried out by surface forces in bringing one mole of adsorbate to that point from infinity, or a point at such a distance from the surface that those forces exert no attraction. The work carried out depends on the phases involved. Polanyi considered three possibilities (a) that the temperature of the system was well below the critical temperature of the adsorbate and the adsorbed phase could be regarded as liquid, (b) that the temperature was just below the critical temperature and the adsorbed phase was a mixture of vapour and liquid, (c) that the temperature was above the critical temperature and the adsorbed phase was a gas. Only the first possibility, the simplest and most common, is considered here. 

The potential theory postulates a unique relationship between the adsorption potential ep and the volume of adsorbed phase contained between that equipotential surface and the solid. It is convenient to express the adsorbed volume as the corresponding volume in the gas phase. 

According to the potential theory the volume V, defined by the adsorbent s surface and the equipotential plane , can contain adsorbate in three different conditions depending upon the temperature. Above the critical temperature the adsorbate can not be liquified and the gas in the adsorption volume V simply becomes more dense near the surface. At temperatures near, but less than the critical temperature, the adsorbate is viewed as a liquid near the surface and a vapor of decreasing density away from the surface. Substantially below the critical temperature... 

Polanyi described the adsorption space as a series of equipotential surfaces, each one with a given adsorption potential (A) and each one enclosing a volume (W). As one moves away from the surface, the adsorption potential decreases until it goes to zero and the adsorption space increases up to a limiting value, W0 (where the potential becomes zero). On the surface, W= ) and A, = Amax. The building of the volume enclosed within the adsorption space is described by a function of the type A = flW). 

All elementary volumes are divided into three types A (anode), B (electrolyte) and C (cathode) blocks (see Fig.2). Eqns (18-19) may be written only for blocks B, because the electrodes (that is, blocks A and C) are equipotential and contain only carbon and aluminium, while blocks B contain a mixture of ions and molecules, A10F2- and C02 in particular. 

Polanyi [12] took a somewhat different approach to multilayer adsorption by assuming that dispersion forces play the determining role in adsorption, resulting in the existence of a potential field in the vicinity of the adsorbent surface. The adsorbed layer has the highest density at the solid surface and its density decreases as the distance from the surface increases. Thus, it is possible to draw equipotential surfaces as shown in Fig. 3.3. The space between each adjacent potential surface represents a defixute adsorption volume which is a function of the potential field. Mathematically, it can be represented as... 

In equilibrium, no current flows through the electrochemical chain and each conducting volume is equipotential. The voltage between the terminals is the algebraic sum of the... 

Each of the five conducting volumes inside the electrochemical chain is equipotential. However, there is a potential difference at each interface (in this simplified representation the thickness is assumed to be insignificant), including for ionic junctions. 

The laws of electrostatics show that a conductor that is in vacuum is equipotential in volume it can therefore only be charged on the surface. The electrostatic potential is constant throughout the conductor volume, but then it suddenly varies at the conductor surface, before finally seeing slower changes in vacuum further away from the conductor. The potential difference between the conductor volume and vacuum at infinite distance, [Pg.121]

As previously stated, any conducting phase in equilibrium is equipotential in its volume and electroneutrality applies at each point . ... 

For an ideahzed, energetically homogeneous surface, aU the points equidistant from the surface will have the same potential s and wiU, therefore, form an equipo-tential plane. Thus, the broken lines in the figure represent planes connecting points of equal potential. The value of the adsorption potential s of the parallel equipotential planes decreases as their distance from the surface increases and falls to zero at the maximum distance. Each equipotential surface encloses between itself and the surface of the adsorbent a volume W. The maximum volume is enclosed between the adsorbent surface and the limiting equipotential plane at which the potential has decreased to zero. Thus, the volumes enclosed between the adsorbent and the equipotential surfaces s= s . .. are W2 W3... The quantity represents the volume of the entire adsorption space. As W increases from zero to... 

The oscillator frequencies can be related to the principal semiaxes a, b and c (see, Eq. (20)) via the volume-conservation constraint and the requirement that the surface of the cluster is an equipotential one, namely... 

This choice makes kd k a differential increment of internal energy per unit actual volume and EkcIDk the corresponding increment of internal energy per unit reference volume. It is also worthwhile to note that - because of the properly l kdxk = ElcIXl, implied by (6.6) - for electrostatic situations the fields may be represented by the potentials (x) = equipotential surfaces that can be identified with material surfaces in the two configurations and we have as a consequence of S = —gradcp also... 

The polarization of an electrode triggers the movement of ionic species in the solution. The current lines represent the path of the ionic species which are perpendicular to the equipotential lines. When two macroelectrodes of comparable size are used, the equipotential lines are parallel to the electrode and the current lines are perpendicular to the electrodes. The current lines are confined within the volume defined by the area of the electrodes and the distance between them (Fig. 3c). 

The Potential Theory as originally conceived by Polanyi is well documented in the classic text by Brunauer (1943). Polanyi considered contours of equipotential energy above solid surfaces and ascribed a volume to the space between the ith equipotential surface of energy e and the adsorbent surface. The potential was assumed to be independent of temperature so that = /(0) is essentially an isotherm equation. The adsorption potential is defined as the work of compression of the gas from a pressure p to the saturation pressure ps. For one mole of a perfect gas of volume v in an open thermodynamic system the adsorption potential is therefore... 

This model for mixed adsorption (Grant and Manes 1966) is based upon the idea of equipotential energies among the components of the adsorbed mixture and is thus related to the Polanyi potential theory discussed in Section 3.3.5. As previously recorded, Dubinin and Radushkevich (1947) postulated a direct relation between the affinity coefficient Pi of a component i and the molar volume Vmt of the saturated pure liquid. The equipotential energy concept for two components is thus (eiiPi) = (ey/ft). Hence, by use of equation (3.18) for each component... 

---