Temperature Dependency — Limiting Cases

In this section we consider the temperature dependency of the rates of catalysed reactions, since a typical behaviour can be expected for catalysed reactions, due to the effect of (competitive) adsorption. This will be demonstrated with some simplified cases, starting with a consideration of the rate determining step. 

According to the transition state theory, the reaction rate of the rate determining step can be computed from expression (3.30). 

It is assumed that the reacting complex is in equilibrium with the transition state complex (T.S.) and that the number of molecules in the transition state that react to give the product per unit of time is given by the frequency barrier The rate of this step is assumed to be rate limiting. It implies that energy exchange is fast compared to the overall reaction rate. This is depicted in Fig. 3.5. Because K is an equilibrium constant it can be written as in Eqn. (3.31), where AG, AH and AS are the free enthalpy, the enthalpy and the entropy differences between the transition state and the ground state, respectively. 

For the rate constant of the rate determining step this results in kT 

Starting with Eqn. (3.14) and assuming that the reverse reaction can be neglected, the rate expression for this single site reaction, with the surface reaction being rate determining, can be written as 

---

As the feed-to-steam ratio is increased in the flow sheet of Fig. 11-125 7, a point is reached where all the vapor is needed to preheat the feed and none is available for the evaporator tubes. This limiting case is the multistage flash evaporator, shown in its simplest form in Fig. 11-125 7. Seawater is treated as before and then pumped through a number of feed heaters in series. It is given a final boost in temperature with prime steam in a brine heater before it is flashed down in series to provide the vapor needed by the feed heaters. The amount of steam required depends on the approach-temperature difference in the feed heaters and the flash range per stage. Condensate from the feed heaters is flashed down in the same manner as the brine. 

Figures 11.2-11.6 show how the room temperature microstructure of carbon steels depends on the carbon content. The limiting case of pure iron (Fig. 11.2) is straightforward when yiron cools below 914°C a grains nucleate at y grain boundaries and the microstructure transforms to a. If we cool a steel of eutectoid composition (0.80 wt% C) below 723°C pearlite nodules nucleate at grain boundaries (Fig. 11.3) and the microstructure transforms to pearlite. If the steel contains less than 0.80% C (a hypoeutectoid steel) then the ystarts to transform as soon as the alloy enters the a+ yfield (Fig. 11.4). "Primary" a nucleates at y grain boundaries and grows as the steel is cooled from A3...

Instantaneous boiling takes place only if the temperature of a liquid is higher than its supeiheat-limit temperature (also called the homogeneous-nucleation temperature), in which case, boiling occurs throughout the bulk of the liquid. This temperature is only weakly dependent on the initial pressure of the liquid and the pressure to which it depressurizes. As stated in Section 6.1., T has a value of about 0.89T,., where is the (absolute) critical temperature of the fluid. 

The magnitude on the left is the heat absorbed in the isothermal change, and of the two expressions on the right the first is dependent only on the initial and final states, and may be called the compensated heat, whilst the second depends on the path, is always negative, except in the limiting case of reversibility, and may be called the uncompensated heat. From (3) we can derive the necessary and sufficient condition of equilibrium in a system at constant temperature. 

Although a wide choice for the other parameters occurring in Eqs. (3.64) and (3.65) is possible, their temperature dependence is small in the vicinity of T°. In practice either G/p or log (G/P) is normally plotted as some function of temperature which necessarily entails some choice for these parameters. Each case should be examined individually to ascertain the change a different choice would make, and to only rely on the results within these limits. 

In this chapter it is of interest to discuss the dependence of BEo=0/BT on AX. Data for a number of faces of Ag and Au are available and constitute the basis for some correlations. In particular, Trasatti and Doubova32 have shown that a common correlation exists (Fig. 25) between BEas0/BTmd AX for single-crystal faces of Ag and Au in the sense that BEg /BT becomes less positive as AX increases. As a limiting case, a negative temperature coefficient has been found393 for Ag(110), which exhibits the highest AX. 

For a range of simple substitutional solid solutions to form, certain requirements must be met. First, the ions that replace each other must be isovalent. If this were not the case, other structural changes (e.g., vacancies or interstitials) would be required to maintain electroneutrality. Second, the ions that replace each other must be fairly similar in size. From a review of the experimental results on metal alloy formation, it has been suggested that 15% size difference can be tolerated for the formation of a substantial range of substitutional solid solutions. For solid solutions in nomnetal-lic systems, the limiting difference in size appears to be somewhat larger than 15%, although it is very difficult to quantify this. To a certain extent, this is because it is difficult to quantify the sizes of the ions themselves, but also because solid solution formation is very temperature dependent. 

In studying interfacial electrochemical behavior, especially in aqueous electrolytes, a variation of the temperature is not a common means of experimentation. When a temperature dependence is investigated, the temperature range is usually limited to 0-80°C. This corresponds to a temperature variation on the absolute temperature scale of less than 30%, a value that compares poorly with other areas of interfacial studies such as surface science where the temperature can easily be changed by several hundred K. This "deficiency" in electrochemical studies is commonly believed to be compensated by the unique ability of electrochemistry to vary the electrode potential and thus, in case of a charge transfer controlled reaction, to vary the energy barrier at the interface. There exist, however, a number of examples where this situation is obviously not so. 

Solution of Equation (4.14) takes two forms (a) the case where Cp is considered not to depend on temperature (i.e. determining the value of A S over a limited range of temperatures) and (b) the more realistic case where Cp is recognized as having a finite temperature dependence. 

---