Temkin relation

Consider a case where A depends on surface coverage S. The simplest form of this dependence is the linear Temkin relation... 

If there exists a rate determining step, an isotope tracer method can be used for the determination of a or Urds at equilibrium or near equilibrium conditions. Its basic principle has been expounded systematically by Horiuti, Temkin and Happel. If the exchange rate or rate at equilibrium defined by Wangner is introduced, when the reaction approaches equilibrium, A is close to zero for any elementary step. Consequently, Horiuti-Temkin relation can be deduced ... 

This ratio is readily obtained from the quasi-stationary state approximation, noting that V = v = V2- The use of Temkin relation illustrates how the unknowns can be eliminated by alternation of the indices. Thus the rate can always be obtained, at least in principle, although it can be difficult when the rates of elementary steps are not linear with the concentration of active intermediates, e.g., for dissociated adsorption.The turnover rate is then ... 

Temkin S. I., Abdrakhmanov B. M. Does the Hubbard relation hold in the liquid cage model Phys. Lett. A155, 43-8 (1991). 

However, with an inhomogeneous electrode surface and adsorption energies that are different at different sites, the reaction rate constant and the related parameter will also assume different values for different sites. In this case the idea that the reaction rate might be proportional to surface concentration is no longer correct. It was shown by M. Temkin that when the logarithmic adsorption isotherm (10.15) is obeyed, the reaction rate will be an exponential function of the degree of surface coverage by the reactant ... 

In section 2.12 we recalled the significance of the overall reaction progress variable I which represents the extent of advancement of simultaneous irreversible processes in heterogeneous systems (Temkin, 1963). The overall progress variable is related to progress variables of individual reactions by... 

Horiuti calls H the number of independent intermediates. Temkin (10) describes the equation P = S - H as Horiuti s rule, and the equation R = Q — H as expressing the number of basic overall equations. To avoid confusion, let us confine the term basis and the concept of linear independence to sets of vectors, and let numbers such as H, P, Q, R, S be understood as dimensions of vector spaces. This makes it simple to determine their values and the relations among them, as will be done in Section III. 

In the following discussion, an example is given that serves to show that introducing a Frumkin-Temkin isotherm does affect the kinetic relation between the current density, i, and the corresponding overpotential. The example chosen will use the hydrogen evolution reaction once more because it is relatively simple but at the same time involves consecutive steps and alternative pathways thus it has characteristics of many practical electrode reactions likely to be met in practice.68... 

In the Temkin adsorption isotherm, the amount adsorbed is related to the logarithm of the pressure of the adsorptive... 

The easiest test concerns Eq. (22a) describing the LJ-type dissociation. The equation establishes the linear correlation between the dissociation barrier A for a homonuclear admolecule A2 and the atomic heat of chemisorption Qh with the slope of k = 3/2. As seen from Table V, for H2, 02, and N2 dissociated on surfaces of metals as varied as Fe, Ni, Cu, W, and Pt, the experimental values of k lie within the range k = 1.4-1.7, that is, within 10-15% of the theoretical LJ value of k = 1.5. It should be stressed that, unlike similar linear relations between the activation barriers and the heats of reactions (Brpnsted, Polanyi, Frumkin-Temkin-Semye-nov, etc.), Eq. (22a) is not a postulate but a corollary of the general principle (BOC-MP) applied to the one-dimensional dissociation ABS As + Bs. 

When reactants or intermediates are adsorbed, the rate of the reaction may no longer be related to the concentration by a simple law. This situation is best understood where a reactant is nonspecifically adsorbed in the outer -> Helmholtz plane. The effect of such adsorption on the electrode kinetics is usually termed the -> Frumkin effect. Physical and chemical adsorption on the electrode surface is usually described by means of an -> adsorption isotherm and kinetic equations compatible with various isotherms such as the - Langmuir, -> Temkin, -> Frumkin isotherms are known. 

Pyzhov Equation. Temkin is also known for the theory of complex steady-state reactions. His model of the surface electronic gas related to the nature of adlay-ers presents one of the earliest attempts to go from physical chemistry to chemical physics. A number of these findings were introduced to electrochemistry, often in close cooperation with -> Frumkin. In particular, Temkin clarified a problem of the -> activation energy of the electrode process, and introduced the notions of ideal and real activation energies. His studies of gas ionization reactions on partly submerged electrodes are important for the theory of -> fuel cell processes. Temkin is also known for his activities in chemical -> thermodynamics. He proposed the technique to calculate the -> activities of the perfect solution components and worked out the approach to computing the -> equilibrium constants of chemical reactions (named Temkin-Swartsman method). 

For some conditions, the inclusion of other pools in the model may be important, and this can be carried out in a straightforward manner using the numerical methods discussed previously (107). The overall rates and the forward and backward rales are related by the identities formulated by Temkin (105, 109). One can hope to represent the kinetics over a wide range by such an elementary-step model. This way of treating the experimental data seems clearer and more powerful than linking the analysis to a parameter such as the response time obtained from certain isotopic experiments. 

The relationship of 0 or In 0 to V follows from the adsorption isotherm that applies to the chemisorption of the intermediate a Langmuir relation does not always apply, as we indicate in Section X on reaction order of elec-trocatalytic processes. For example, the Temkin isotherm for the condition 0.1 < 0 < 0.9, in the form... 

Several authors have derived rate expressions for non-uniform catalyst surfaces. Boudart and Djega-Mariadassou [2] show that relations are obtained with a mathematical similarity to those obtained for a uniform surface. In the rate expression for ammonia production, the Temkin isotherm has been used for a long time. This isotherm accoimted for a, supposedly, heterogeneous adsorption... 

The Temkin s relation for the Ui = J Xi) dependence represents the limiting relation and thus enables the rational introduction of the Stortenbeker s correction factor. 

The relation between Temkin s energy of activation quantity, i.e., at zero metal-solution p.d. (V), and the apparent one AH (referred to by Temkin as the real heat of activation) that is both experimentally measured and practically significant, i.e., as obtained from the reciprocal temperature coefficient of In/o, d nio/d /T), is conveniently derived (cf. Refs. 25, 81) as follows. Ln io is written as... 

The condition for equilibrium is that the rates of adsorption and desorption are equal. Isotherms may be obtained by equating these rates. Three theoretical isotherms, those of Langmuir (1918), Freun-dlich (1926) and Temkin (Brunauer et al. 1942) are important. Only the Langmuir isotherm is presented here because it is the one most widely used in work related to gas-solid catalytic systems. 

Temkin and co-workers have investigated the thermodynamic properties of the soluble complexes of unsaturated hydrocarbons with various metal salts with particular reference to their role in catalytic reactions. Using a potentio-metric technique, they were able to calculate the thermodynamic data shown in Table 6 for the silver(I)-acetylene complexes 30) and the silver(I)-ethylene complex 31). The results obtained for acetylene have been related to the low activity of silver salts as catalysts for the hydration of acetylene. For the sil-ver(I)-ethylene complex, the relationship between the ionic concentrations and... 

According to the Horiuti-Temkin theorem [9,10], the number of linearly independent RRs is equal to I = p - ranka = p - q. Now, let Ju Ji be the corresponding rates (or, the RR fluxes) along the arbitrarily selected set of linearly independent RRs, namely RRu RR2,..., RRi. Then, within the QSS approximation the following relation between RR fluxes and the rates of individual elementary reaction steps is valid... 

There are 3 steps in scheme 7.270, two intermediates (adsorbed hydrogen and vacant sites), one balance equation which relates these two intermediates, and then respectively two independent routes according to the Horiuti-Temkin rule. Steps 1, 2 and 3 are usually referred as Volmer, Tafel and Heirovsky reactions respectively acknowledging the names of researches who emphasized the importance of these processes. 

For slurry reactors several correlations were proposed in the literature, which relate the dimensionless Sherwood number to the Reynolds and Schmidt numbers. Data collected by Temkin for various slurry reactors (Kinetika i kataliz, 28 (1977) 493) show that the Sherwood number can be described by the following equation... 

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