Richardson’s equation

Millikan s experiment did not prove, of course, that (he charge on the cathode ray. beta ray, photoelectric, or Zeeman particle was e. But if we call all such particles electrons, and assume that they have e/m = 1.76 x Hi" coulombs/kg. and e = 1.60 x 10" coulomb (and hence m =9.1 x 10 " kg), we find that they fit very well into Bohr s theory of the hydrogen atom and successive, more comprehensive atomic theories, into Richardson s equations for thermionic emission, into Fermi s theory of beta decay, and so on. In other words, a whole web of modem theory and experiment defines the electron. The best current value of e = (1.60206 0.00003) x 10 g coulomb. 

The flrst attempts to describe the ionization of a flame containing particles was made by Sugden and Thrush (7) who modified Richardson s equation (Equation 1) for the vapor pressure of electrons over an infinite plane surface ... 

According to Tammann and Dreyer the velocity of evaporation of crystals reaches a maximum with rise of temperature, and is thus different from that of liquids ( 6.VII H). They concluded that it is proportional to V(T). and thus follows the old form of Richardson s equation ( 11.VIII K) for the emission of electrons from a heated metal. 

Since the transfer process involves no mechanical work against an external pressure, —AF may be replaced by —A G, as was done in the deduction of Richardson s equation. 

Broughton and Squires (45) later modified Richardson s equation to... 

The phenomenon of compensation is not unique to heterogeneous catalysis it is also seen in homogeneous catalysts, in organic reactions where the solvent is varied and in numerous physical processes such as solid-state diffusion, semiconduction (where it is known as the Meyer-Neldel Rule), and thermionic emission (governed by Richardson s equation ). Indeed it appears that kinetic parameters of any activated process, physical or chemical, are quite liable to exhibit compensation it even applies to the mortality rates of bacteria, as these also obey the Arrhenius equation. It connects with parallel effects in thermodynamics, where entropy and enthalpy terms describing the temperature dependence of equilibrium constants also show compensation. This brings us the area of linear free-energy relationships (LFER), discussion of which is fully covered in the literature, but which need not detain us now. 

As a consequence, when Sieverts-Fick law is valid, the hydrogen flux is written in terms of the so-called Richardson s equation ... 

The hydrogen permeation rate through the palladium membranes follows Richardson s equation, where the values of the apparent activation energy Ea and pre-exponential factor Pe are 12 540 J mol and 2.21 x 10 ° mol s m Pa °, respectively (experimentally determined). 

In diodes, the conducting polymer (usually a p-type semiconductor) is in contact with an n-type semiconductor or a metal. In the former case a heterojunction can form at the interface, while in the latter case a Schottky barrier can be created. The relation between the current density and the voltage is described by Richardson s equation [217] ... 

In order to obtain a quantitative relationship between the gradient of pressure set across the membrane and the flux of molecular hydrogen, it is necessary to relate the hydrogen pressure in the gas phase and the hydrogen concentration in the metal. Combining Sieverts equation (which pertains to equilibrium) and Rck s first law of diffusion (which pertains to the transport of atomic hydrogen by diffusion), the following relationship, known as Richardson s equation, is obtained (Lewis, 1967 Buxbaum and Marker, 1993) ... 

The exponent n, instead, varies in the range 0.5—1, and it allows recognition of the limiting step if it is 0.5 the Eqn (4.7) becomes the Sieverts-Fick law, so the diffusion is the limiting step if it is bigger than 0.5, the reactions of dissociation or recombination are slower than the bulk diffusion. The dependency from temperature is expressed as an Arrhenius law for the permeabihty, which can be substituted inside the Sieverts-Fick law for obtaining the so-caUed Richardson s equation, reported below. 

Riemann(78) has suggested another variant of Richardson s equation based upon thermodynamic considerations and Tick s law. His equation took the form... 

It is noteworthy that Sieverts law - Eq. (4.5) - is valid only at low H2 pressures, otherwise deviations may occur. In addition, if the Pd membrane is defective and allows for the passage of other gases, or the Pd membrane surface has low activity due to the adsorption of contaminants, the permeation behavior of the membrane also deviates from Sieverts law. In a general form, the hydrogen permeation flux in Pd-based membranes can be expressed by Richardson s equation ... 

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