Transition entropy, rules

No general rules about the entropy of transitions, as were found for liquid and plastic crystal transitions, can be set up for condis crystals. Two typical examples may illustrate this point. Polytetrafluoroethylene has a relatively small room-temperature transition-entropy on its change to the condis state and a larger transition entropy for final melting. Polyethylene has, in contrast, a higher condis crystal transition entropy than melting entropy (see Sect. 5.3.2). 

Numerical Observations Figure 3.42 shows a schematic plot of H versus A for A = 8 Af = 5 two dimensional CA. The lattice size is 64 x 64 with periodic boundary conditions. In the figure, the evolution of the single-site entropy is traced for four different transition events. In each case, for a given A, a rule table consistent with that A is randomly chosen and the system is made to evolve for 500 steps to allow transients to die out before H is measured. 

In fact, the mean-field estimate for H, given by equation 3.64, predicts a second-order transition. While abrupt changes in entropy characterize sudden transitions between regions of periodic and chaotic rules, smooth changes in entropy as A is increased instead suggest that the rule path sometimes passes through a region of... 

Table 6.2 Number of nodes Ht, C = 0,.. . 4 in the minimal deterministic state transition graph (DSTG) representing the regular language r2t[0, where 0 is an elementary fc = 2, r = 1 CA rule Amax is the maximal eigenvalue of the adjacency matrix for the minimal DSTG and determines the entropy of the limit set in the infinite time limit (see text), Values are taken from Table 1 in [wolf84a. ...

This change in the behavior of the simulation is counterintuitive. One would expect the introduction of greater uncertainty in the transition rules to increase the entropy of the system, but it has the opposite effect, leading to... 

The reactions with which we are mostly concerned in chemistry take place in solution rather than in the gas phase. The majority of them moreover involve reactants, products, or transition states carrying electric charges. In such cases the entropies of solvation are extremely large, and these entropies cannot be estimated at present. This immediately rules out any possibility of estimating absolute values of equilibrium or rate constants for reactions of this kind. If we are concerned with absolute calculations of rates and equilibria, we must confine ourselves either to gas-phase reactions, or to reactions of non-polar type. Even here we will usually be forced to make estimates of entropies that are of dubious significance chemical theory has not yet progressed to a point where problems of this kind can usefully be discussed. 

Here Tx/2/Tm is the value of T/Tm at which half of the entropy of fusion of the liquid is lost on supercooling, where Tm is the melting temperature. The success of this type of scaling is due to the fact for a number of substances the rule of glass formers apply, specifically, the glass transition is approximately two-thirds of the melting temperature [37,88],... 

Plastic crystal phases in organic materials have been known since the time of Timmermans [1], and these phases are often reached via a solid-solid transition below the final melting point of the crystal. These transitions often represent the onset of rotational motions of the molecules within the crystalline lattice and the resultant phases are sometimes referred to as rotator phases [18-23]. Timmermans proposed a general rule that plastic phases have a low final entropy of fusion (A5f < 20 J K mol because the rotational component of the entropy of fusion of the fully ordered phase is already present in the plastic phase. The bulk of the... 

Intuitively, surface and interfacial tensions may be expected to be related to a number of physical characteristics of the liquid or the liquid-vapour transition. Two of these are the enthalpy and entropy of evaporation, discussed in sec. 2.9. Other parameters that come to mind are the molar volume V, the isothermal compressibility and the expansion coefficient. The combination of certain powers of such parameters and y sometimes leads to products with interesting properties, like temperature independence or additivity. Severed of such scaling rules have been proposed over the past century, mostly with limited quantitative success. A few of these wUl now be discussed. 

The probability of an entire spectrum will be determined by the procedure of maximum entropy.49 The purpose is to show that under the constraints considered so far, including the various sum rules, the distribution of intensities of different lines are independent of one another. When considered as a function of the ratio of the actual intensity to its mean, local value, all transitions have the same distribution, which is -square in the general case. Hence, one can group intensities together into bins, and consider the distribution of intensity irrespective of line position provided one takes proper account of the variation of the envelope with energy. The technical backup for these statements and the specific conditions under which they will fail are the subjects of this subsection. 

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