Temkin equations

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). 

The adsorption on a solid surface, the types of adsorption, the energetics of adsorption, the theories of adsorption, and the adsorption isotherm equations (e.g., the Langmuir equation, BET equation, Dubinin equation, Temkin equation, and the Freundlich equation) are the subject matter of Chapter 2. The validity of each adsorption isotherm equation to the adsorption data has been examined. The theory of capillary condensation, the adsorption-desorption hysteresis, and the Dubinin theory of volume fllhng of micropores (TVFM) for microporous activated carbons are also discussed in this chapter. 

An adsorption isotherm known as the Temkin equation [149] has the form tt = ofF /F where a is a constant and F" is the limiting surface excess for a close-packed... 

The Temkin-Pyzhev equation deseribes the net rate of synthesis over promoted iron eatalyst as ... 

The low-coverage parts of the adsorption isotherms evaluated at different temperatures have shown a remarkable feature of linear dependence between the adsorption and the logarithm of gas pressure. This sort of behavior corresponds to the well-known Temkin equation of adsorption... 

Since most equations of state are pressure-explicit, Eqs. (6) and (99) are often more convenient than Eqs. (5) and (98). With these equations, basing his calculations on van der Waals equation of state, Temkin (Tl) showed that gas-gas immiscibility may occur if the van der Waals constants a and b... 

This isotherm is consistent with the modified electrochemical Langmuir isotherm, the Nemst equation and the potential-work function equivalence. For intermediate 0j and Pj values the isotherm of Eq. (6.58) is well approximated both by the Fowler-Guggenheim and by the Temkin isotherms. 

On inhomogeneous surfaces where adsorption obeys the Temkin isotherm, an exponential factor will appear in the kinetic equation ... 

Regarding submerged plants, sorption of Cu(II) by Myriophyllum spicatum L. (Eurasian water milfoil) has been shown to be fast and fits isotherm models such as Langmuir, Temkin, and Redlich-Peterson. The maximum sorption capacity (c/lll l j ) of copper onto M. spicatum L. was 10.80 mg/g, while the overall sorption process was best described by the pseudo-second-order equation.115 Likewise, Hydrilla verticillata has been described as an excellent biosorbent for Cd(II). In batch conditions, the qmsx calculated was 15.0 mg/g. Additionally, II. verticillata biomass was capable of decreasing Cd(II) concentration from 10 to a value below the detection limit of 0.02 mg/L in continuous flow studies (fixed-bed column). It was also found that the Zn ions affected Cd(II) biosorption.116... 

Droplet suspensions (gas-liquid, two-component system) Since the inertia of a liquid suspended in the gas phase is higher than the inertia of the gas, the time for the displacement of liquid under the pressure waves should be considered. Temkin (1966) proposed a model to account for the response of suspension with pressure and temperature changes by considering the suspensions to move with the pressure waves according to the Stokes s law. The oscillatory state equation is thereby approximated by a steady-state equation with the oscillatory terms neglected, which is valid if the ratio of the relaxation time to the wave period is small, or... 

Develop the rate equation of Temkin Pyzhev for the rate of the reaction, N2 (A) + 3H2(B) > 2NH3(C), on these assumptions ... 

Annable, D., Application of the Temkin kinetic equation to ammonia synthesis in large-scale reactors, Chem. Eng. Sci. 1(4), 145-153 (1952). 

Equation 3.156 follows the general treatment proposed by Temkin (1945) for fused salts. 

This quasi-lattice formulation of fused salts is known as Temkin s equation. Its application to silicate melts was provided by Richardson (1956), but it is inadequate for the compositional complexity of natural melts, mainly because, in a compositionally complex melt, the types of anions and consequently the entity of the anion matrix vary in a complicated way with composition. 

The constants of equations 6.16 and 6.17 are identical, on the basis of the principle of equal reactivity of cocondensing functional groups. Adopting Temkin s model for fused salts, and thus assuming that molar fractions represent activities over the appropriate matrix, we obtain... 

Temkin, 1963). From the various equations it follows that... 

Figure 13.6 Step feed and shut-off of 700 ppm NH3 in He -I- 700 ppm NO -I- l%v/v Oj over VjO., WO,/liO, model catalyst at 220 C. Dashed lines, inlet NH, concentration solid lines, model fit with Temkin-type coverage dependence and modified Langmuir kinetics, Equation (13.10). (Adapted from ref. [52]).

Whereas the Langmuir equation assumes a uniform energy of adsorption, the Temkin equation uses a logarithmic relationship... 

According to the Temkin s classification, this mechanism is non-linear, because in the first step oxygen reacts with two catalyst sites K. Numbers to the right of stoichiometric equations are Horiuti numbers vi = 1 and V2 — 2. An analysis of data by Boreskov showed that the power M in Equations (13) and (14) is... 

Introduction of stoichiometric number concept and linear transformation of the "conventional" QSSA equations (16) to the equivalent system (20) was essentially the major (and, possibly, only) result of theory of steady reactions developed independently by J. Horiuti in 1950s and M. I. Temkin in 1960s. 

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. 

If the ratio p/cr remains constant, the last equation can be expressed with the ratio as a selectivity so that only a single free parameter remains, as recently suggested by Temkin (23). 

Information on the steps in a reaction mechanism can be extended significantly by isotopic tracer measurements, especially by transient tracing [see Happel et al. (54,55)]. Studies by Temkin and Horiuti previously referenced here have been confined to steady-state isotopic transfer techniques. Modeling with transient isotope data is often more useful since it enables direct determination of concentrations of intermediates as well as elementary step velocities. When kinetic rate equations alone are used for modeling, determination of these parameters is more indirect. 

Tafel equation, 1054,1066,1106,1115,1133, 1249,1404,1440, 1456,1507,1528 applications, 1508 and distribution of electronic states, importance, 1466 importance, 1508 in quantum calculations, 1495 in semiconductors, 1085 tunneling, 1495 Tafel, Julius, 1106 Tafel lines, oxygen reduction, 1207 Tamm states, 1082 Tarasevich, 1495 Taylor, electrodeposition, 1303 Temkin isotherm, 927, 938, 1195... 

This is the complete Temkin isotherm. In the second part of this equation, the term pj represents the value of p when 5 = 1, and Po> the value of p when s = 0. If the appropriate conditions exist such that pjC 1 p0c, i.e., all the patches with large p are full and the others are empty independently of the concentration, then,... 

As to when to use Frumkin s approach or that of Temkin, as long as one is willing to accept the approximation leading to Eq. (7.156), it does not matter as far as the resulting kinetic equations (see below) are concerned. However, the thought process behind Frumkin s equation is to take account of the interaction between the adsorbed entities and that behind Temkin s is to allow for the difference in adsorption energies in different sites on the surface equation. 

Consider, now the dependence of 0 upon potential under the condition that AGe varies with 0. It will he less dramatic (i.e., d dV will be much smaller) than in the situation represented by the original, simpler, corresponding, Langmuir equation [Eq. (7.153)]. If the latter isotherm is applicable to a variation of 0 with potential at constant concentration, the surface is effectively either empty of intermediate (0 1) or near to 0 = 1. With Frumkin-Temkin in control, 0 varies linearly and more slowly with V than it does with the Langmuir equation. Thus, from (7.156), at constant cp... 

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