Low Temperature Limit

Returning to the more general expression, in the low temperature limit we find that the rate... 

Corrosion occurs when the metallic iron in DRI is wetted with fresh or salt water and reacts with oxygen from air to form mst, Ee(OH)2- The corrosion reactions continue as long as water is present. Because water evaporates at approximately 100°C, corrosion reactions have a low temperature limit even though the reactions are exothermic. Small amounts of hydrogen may be generated when DRI reacts with water. However, this poses no safety problem as long as proper ventilation is provided. 

Some quite viscous oils in the 450 650 mm /s are employed for high temperatures. Less viscous oils, down to 25 mm /s and lower at 40°C, are used in special greases for low temperatures. The maximum oil viscosity in a grease for starting medium torque equipment is about 100, 000 mm /s(= cSt) (4). Extrapolations for various oils can be made on viscosity—temperature charts, as shown in Figure 8, to estimate this approximate low temperature limit. 

The low temperature limitation of homogeneous catalysis has been overcome with heterogeneous catalysts such as modified Ziegler-Natta (28) sohd-supported protonic acids (29,30) and metal oxides (31). Temperatures as high as 80°C in toluene can be employed to yield, for example, crystalline... 

The experimental studies of a large number of low-temperature solid-phase reactions undertaken by many groups in 70s and 80s have confirmed the two basic consequences of the Goldanskii model, the existence of the low-temperature limit and the cross-over temperature. The aforementioned difference between quantum-chemical and classical reactions has also been established, namely, the values of k turned out to vary over many orders of magnitude even for reactions with similar values of Vq and hence with similar Arrhenius dependence. For illustration, fig. 1 presents a number of typical experimental examples of k T) dependence. 

Exploration of the region 0 < T < requires numerical calculations using eqs. (2.5)-(2.7). Since the change in /cq is small compared to that in the leading exponential term [cf. (2.14) and (2.18)], the Arrhenius plot k(P) is often drawn simply by setting ko = coo/ln (fig. 5). Typical behavior of the prefactor k and activation energy E versus temperature is presented in fig. 6. The narrow intermediate region between the Arrhenius behavior and the low-temperature limit has width... 

Friction also changes the way k ) approaches its low-temperature limit and widens the intermediate region between the two asymptotes of k(P). At temperatures far below the cross-over point k T) behaves as... 

Adiabatic reactions, occurring on a single-sheet PES correspond to B = 1, and the adiabatic barrier height occurs instead of E. The low-temperature limit of a nonadiabatic-reaction rate constant equals... 

The transition is fully classical and it proceeds over the barrier which is lower than the static one, Vo = ntoColQl- Below but above the second cross-over temperature T 2 = hcoi/2k, the tunneling transition along Q is modulated by the classical low-frequency q vibration. The apparent activation energy is smaller than V. The rate constant levels off to its low-temperature limit k only at 7 < Tc2, when tunneling starts out from the ground state of the initial parabolic term. The effective barrier in this case is neither V nor Vo,... 

While being very attractive in view of their similarity to CLTST, on closer inspection (3.61)-(3.63) reveal their deficiency at low temperatures. When P -rcc, the characteristic length Ax from (3.60b) becomes large, and the expansion (3.58) as well as the gaussian approximation for the centroid density breaks down. In the test of ref. [Voth et al. 1989b], which has displayed the success of the centroid approximation for the Eckart barrier at T> T, the low-temperature limit has not been reached, so there is no ground to trust eq. (3.62) as an estimate for kc ... 

In fig. 26 the Arrhenius plot ln[k(r)/coo] versus TojT = Pl2n is shown for V /(Oo = 3, co = 0.1, C = 0.0357. The disconnected points are the data from Hontscha et al. [1990]. The solid line was obtained with the two-dimensional instanton method. One sees that the agreement between the instanton result and the exact quantal calculations is perfect. The low-temperature limit found with the use of the periodic-orbit theory expression for kio (dashed line) also excellently agrees with the exact result. Figure 27 presents the dependence ln(/Cc/( o) on the coupling strength defined as C fQ. The dashed line corresponds to the exact result from Hontscha et al. [1990], and the disconnected points are obtained with the instanton method. For most practical purposes the instanton results may be considered exact. 

Fig. 26. Arrhenius plot [ln(fc/a>o) against a>o /2it] for the PES (4.28) with Q = 0.1, C = 0.0357, = 1, F /a>o = 3. Solid line shows instanton result separate points, numerical calculation data from Hontscha et al. [1990] and dashed line, low-temperature limit using (3.32) for fc,D.

Difoggio and Corner [1982] and Wang and Comer [1985] have discovered tunneling diffusion of H and D atoms on the (110) face of tungsten. They saw that the Arrhenius dependence of the diffusion coefficient D sharpy levels-off to the low-temperature limit (D = D ) at 130-140 K (fig. 47) the values of depend but slightly on the mass of the tunneling particle for the D and... 

The low-temperature limit of the rate constant for the isomerization of the biradical... 

The chain polymerization of formaldehyde CH2O was the first example of a chemical conversion for which the low-temperature limit of the rate constant was discovered (see reviews by Goldanskii [1976, 1979]). As found by Mansueto et al. [1989] and Mansueto and Wight [1989], the chain growth is driven by proton transfer at each step of adding a new link... 

In these equations A and np are the molar fractions of A and P (8 v) = v - vp is the difference between the resonance frequencies of the nuclei in positions A and P, usually determined from the low-temperature limit A is the full-width at half height in the absence of exchange (r - °°) and v is the variable radio frequency of the NMR experiment. 

We see immediately that the reaction orders are = 0.5 and Wco = iti the low temperature limit. The negative order in CO shows that the surface is completely covered by CO. Any further increase in CO pressure will reduce the rate because free sites are blocked, and consequently oxygen cannot adsorb and react. 

In the low-temperature limit we find K = 0 and qA= indicating that the system is entirely in the ground state of A, as this yields the lowest possible (free) energy. At... 

Appropriate approximations of the integral yield a dependence for/(T) in the low temperature limit (T d). 

Thus, the allowance for the dependence of the resonance integral on qsk may not be reduced in general to averaging the transition probability over the distribution function in Eq. (102). The function s(qk) plays the role of the distribution function for the coordinates qk in the case of the symmetric transition. In the classical limit, the results of Flynn and Stoneham62 can be obtained from Eq. (103), and in the low-temperature limit, the result of Kagan and Klinger64 can be obtained. 

Low temperature limit of range (°C) Upper temperature limit of range (°C) Maximum allowable change in capacitance from 25°C (at 0 VDC, over entire operating temperature range)... 

The low-temperature limit of magnetic thermometry with paramagnetic salts (see Section 9.9) is given by the ordering temperature of the electronic magnetic moments. Such ordering temperature is around 1 mK (example the CMN thermometer). 

Equation 1 describes the radiationless decay rate for a single-frequency model with weak electron-vibration coupling in the low temperature limit as derived by Englman and Jortner. 

As an alternative numerical experiment, I suppress all ice formation by modifying subroutine SWALBEDO. The modifications are to set the fraction of the ocean covered by sea ice to zero, regardless of temperature, and to set an unattainably low temperature limit in the IF statement that branches to temperature-dependent albedo on land. The program is listed as DAV11. 

We draw attention to the fact that in the first-order perturbation theory we have = 1 and B %) = 0 in Eq. (4.2.41) for any N, and the expression for the leaving rate of the molecule from the subbarrier states reduces to two well-known special cases. The first of these corresponds to the low-temperature limit - 0 for which166... 

Eq. (A3.19) corresponds to compact diagrams in Fig. A3.la. The expression obtained allows the line shape to be determined in the approximation of the small anharmonic coefficient y, as well as in the low-temperature limit. 

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