Dimensionless potential

Figure 4.16, Effect of catalyst potential, dimensionless catalyst potential n(=FUWR/RT), corresponding linearized51 Na coverage 0ns and pCo on the rate of CO oxidation on Pt/(T-A1203. T=350°C, po2=6 kPa.51 Reprinted with permission from Academic Press.

Thermal diffusivity Temperature sensitivity Temperature difference Thickness of tube Aspect ratio, relation of Cp/Cy Fluid dielectric constant Wall zeta potential Dimensionless temperature Friction factor, Debye length Mean free path Dynamic viscosity Kinematic viscosity Bejan number Density... 

Vickers hardness number corrected for microbrittleness —scratch depth, indent depth, height of cylinder —thickness of material —corrected height of cylinder —ionization potential —dimensionless constants —temperature in Kelvin... 

Fig. 57. Convolutive and deconvolutive transformations of current transients for reversible systems. Solution of 1 x 10 M Cd in 0.11 M KCl 1 and 1 forward and backward linear sweep curves, 0.2Vs" 2 and 2 semiintegral curves (convolution vs. time) 3 and 3 semidifferential curves (deconvolution vs. potential). Dimensionless variable y results in particulate operations. Adapted according to [123].

The potential has a spurious maximum at r where the r ° tenn again starts to dominate. The dimensionless parameter a is a measure of the steepness of the repulsion and is often assigned a value of 14 or 15. The ideas... 

The force constants kj,kc and the dimensionless Renner parameters r, c ate defined by the adiabatic potentials for the components of the II state at pure trans (Vj, Vj) and pure cis (V, V ) bending vibrations,... 

The dimensionless parameters Ot, , Ctc appearing in the last expression are connected with the sums and differences of the adiabatic potentials as shown elsewhere [149,150]. This effective Hamiltonian acts onto the basis functions (A.l) with A = 2. 

In this section we shall expand upon the problem of one-dimensional motion in a potential V x). Although it is a textbook example, we use here the less traditional Feynman path-integral formalism, the advantage of which is a possibility of straightforward extension to many dimensions. In the following portion of the theory we shall use dimensionless units, in which h = i,k = 1 and the particle has unit mass. 

For convenience of notation we accept from here on, that each frequency of the problem co has a dimensionless counterpart denoted by a capital Greek letter, so that co,- = coofl,. The model (4.28) may be thought of as a particle in a one-dimensional cubic parabola potential coupled to the q vibration. The saddle-point coordinates, defined by dVjdQ = dVjdq = 0, are... 

The dimensionless upside-down barrier frequency equals = 2(1 — and the transverse frequency Qf = Q. The instanton action at = oo in the one-dimensional potential (4.41) equals [cf. eq. (3.68)]... 

To illustrate the relationship between the microscopic structure and experimentally accessible information, we compute pseudo-experimental solvation-force curves F h)/R [see Eq. (22)] as they would be determined in SEA experiments from computer-simulation data for T z [see Eqs. (93), (94), (97)]. Numerical values indicated by an asterisk are given in the customary dimensionless (i.e., reduced) units (see [33,75,78] for definitions in various model systems). Results are correlated with the microscopic structure of a thin film confined between plane parallel substrates separated by a distance = h. Here the focus is specifically on a simple fluid in which the interaction between a pair of film molecules is governed by the Lennard-Jones (12,6) potential [33,58,59,77,79-84]. A confined simple fluid serves as a suitable model for approximately spherical OMCTS molecules confined... 

Here r is the distance between the centers of two atoms in dimensionless units r = R/a, where R is the actual distance and a defines the effective range of the potential. Uq sets the energy scale of the pair-interaction. A number of crystal growth processes have been investigated by this type of potential, for example [28-31]. An alternative way of calculating solid-liquid interface structures on an atomic level is via classical density-functional methods [32,33]. 

Pourbaix has evaluated all possible equilibria between a metal M and HjO (see Table 1.7) and has consolidated the data into a single potential-pH diagram, which provides a pictorial summary of the anions and cations (nature and activity) and solid oxides (hydroxides, hydrated oxides and oxides) that are at equilibrium at any given pH and potential a similar approach has been adopted for certain M-H2O-X systems where A" is a non-metal, e.g. Cr, CN , CO, SOj , POj", etc. at a defined concentration. These diagrams give the activities of the metal cations and anions at any specified E and pH, and in order to define corrosion in terms of an equilibrium activity, Pourbaix has selected the arbitrary value of 10 ° g ion/1, i.e. corrosion of a metal is defined in terms of the pH and potential that give an equilibrium activity of metal cations or anions > 10 g ion/1 conversely, passivity and immunity are defined in terms of an equilibrium activity of < 10 g ion/1. (Note that g ion/1 is used here because this is the unit used by Pourbaix in the S.I, the relative activity is dimensionless.)... 

Here, x and y are the dimensionless distance and potential defined by x = xr and y = e

elementary charge, T the absolute temperature, k the Boltzmann constant, and x the Debye screening parameter defined by x = (8jtne2/skT)1/2. 

Figure 6.3. Examples for the four types of global electrochemical promotion behaviour (a) electrophobic, (b) electrophilic, (c) volcano-type, (d) inverted volcano-type, (a) Effect of catalyst potential and work function change (vs I = 0) for high (20 1) and (40 1) CH4 to 02 feed ratios, Pt/YSZH (b) Effect of catalyst potential on the rate enhancement ratio for the rate of NO reduction by C2H4 consumption on Pt/YSZ15 (c) NEMCA generated volcano plots during CO oxidation on Pt/YSZ16 (d) Effect of dimensionless catalyst potential on the rate constant of H2CO formation, Pt/YSZ.17 n=FUWR/RT (=A(D/kbT).

Figure 6.23. Effect of partial charge transfer coefficient XD on catalyst performance for fixed X.A depending on dimensionless potential n, (a) electrophobic, (b) electrophilic, (c) volcano-type, (d) inverted volcano-type.

Figure 6.25, Experimental71 (left) and modelled simulated" (right) dependence of the rate of CO oxidation on Pt deposited on J3"-A1203 as a function of pco, catalyst potential UWR and dimensionless catalyst work function Il(=A

Figure 8.47 shows the effect of the dimensionless potential THFUwr/RT, on product selectivity, S, under constant feed conditions. The selectivity to h2co can be varied deliberately between 0.35 and 0.60 by varying the catalyst potential. 

Figure 8.46. Effect of Pt catalyst dimensionless potential ]1=FUwr/RT on the kinetic constants of formation of formaldehyde (a) and CO2 (b) during CH3OH oxidation on Pt/YSZ Conditions as in Fig. 8.45.50 Reprinted with permission from Academic Press.

Figure 8.51. Effect of dimensionless catalyst potential n=FUWR/RT on the rates of formation of H2CO, CO and CH4 during CH3OH dehydrogenation and decomposition on Ag/YSZ Pchsoh S kPa A, Ts=620°C, I, T=643°C, , T=663°C.56 Reprinted with permission from Academic Press.

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