Critical coordinates

The acentric factor is also dependent on the critical coordinates being used. 

More detailed calculations of the elastic properties of model networks have confirmed Phillips model. The coordination dependence of the elastic modulus is shown in Fig. 2.12 (He and Thorpe 1985). Both the modulus Cn and the number of zero frequency vibrational modes, /, drop to zero at the critical coordination of 2.4, as predicted by Eq. (2.17). The properties are explained in terms of percolation of rigidity. The coordination of 2.4 represents the lowest network coordination for which locally rigid structmes are fully connected, so that the entire network is rigid, but only just so. The elastic modulus is therefore non-zero and continues to increase as the network becomes more connected. The four-fold amorphous silicon network is far from the critical coordination and is very rigid. 

As before, the energy in the reaction coordinate, Ct is measured from the top of the barrier. Solutions to equation (7.67) for the barrier of figure 7.28, are plotted in figure 7.29 for three values of hv, 3000, 2122, and 1000 cm , but for the same curvature at the top of the barrier. These are chosen to correspond approximately to a reaction in which the critical coordinate involves an H atom transfer, a D atom transfer, and perhaps a C—C bond break. The marked differences show the well known effeet of the reduced mass on the transmission probabilities. Note that even at the top of the barrier, the transmission coefficient equals 0.5 exactly. Note also that while the Eckart equation is a function of the curvature, k, which depends upon the critical frequency and the reduced mass, the transmission coefficient depends only upon the critical frequency. 

Casting the equations of state into reduced variables implies commonality of behavior for all fluids for example, common PVT surface, or the common value of compressibility at the critical point P V /RT = 3/8. This commonality is known as the corresponding-states principle (CSP), discussed later. Since Eq. (6.3) is general, so is the critical point. Thus, originally for the low-molecular-weight gases and liquids, its experimental critical coordinates, Pc,Vc, and Tc, were used as the characteristic reducing parameters. 

Tritium specific activity determinations are usually carried out with larger uncertainties than mass spectrometric deuterium determinations. Calculations of k kj ratios with the use of equation 203 are therefore not very reliable. Nevertheless, the data obtained in the high-temperature region indicate clearly that the critical coordinate for reaction 197 is the C—H rather than the C—C distance and support the Slater assumption based on theoretical considerations concerning unimolecular decomposition of cyclopropane. 

The factor of accounts for the fact that half the A molecules revert to A molecules. The bracketed expression is the probability that a molecule with total energy c and vibrational energy has an energy concentrated in the critical coordinate. 

If the RRKM assumption of random access is invalid, different models must be formulated in order to obtain an expression for k(e), if (9.31) is to be used. Slater investigated the other extreme model and assumed no vibrational energy transfer between normal modes of a molecule. This approach is too restrictive. Recent work attempts to steer a middle (and more difficult) course.The dynamics of vibrational energy transfer are incorporated in the theory. Effects attributable to vibrational excitation of modes both strongly and weakly coupled to the critical coordinate can then be exhibited. 

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