Critical index

It is interesting that the main thermodynamic functions can be approximated near the Iran.sition point as power functions 

Te being the phase transition temperature or the critical temperature. The dimensionless pau-ameter characterizes the temperature distance of the system from the phase transition or critical temperature. The indices A and X are called critical indices and vary within 0.3... 0.5 in most cases. 

The interest in critical indices is caused by their easiness of experimental determination from the plot In /( ) vs ln . On the other hand, the modern theories of phase transitions 

In the region T Tc, the order parameter is equal to zero only on the average, but its local value differs from zero owing to the liuctuations. The characteristic magnitude of such fluctuations (the correlation length f) obeys Equations 2-4 with the critical indices 1/ and 1/ as well (see Table 1.2). 

When approaching the critical points in a binary system (generally, approaching the spinodal), dpijdx — 0. From Equation 1.4-33, the mean square concentration fluctuation ((Axq)) — 00. In terms of the order parameter, it means 00 in accordance with ik e and (- ) .  

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Here, x and x" are the mole fractions in the coexisting phases Tc. pc and Xc are the temperature, the pressure and the mole fraction at the critical point PP and Pt are the critical indices Bp and Bt are the amplitudes of the equilibrium curves. For the T,x binodal at atmospheric pressure the exponent pp has a universal value close to 1/3. An analysis made for several isobars has shown that within the experimental error the quantity Pp retains its value along the line of critical points including the double critical point. The critical exponent pT for isotherms does not considerably change its value at pressures up to 70-80 MPa, with px=pp. In the vicinity of the DCP, one can observe an anomalous increase of pT. The behaviour of the exponents Pp and pT along the line of critical points at pressures from atmospheric to 200 MPa is shown in figure 3. 

Process safety metrics are critical indicators for evaluating a process safety management system s performance. Tracking the number of process safety incidents is one common measure of performance, but merely tracking the number of incidents after the fact is insufficient to understand the system failure that allowed the incident to occur and what can be done to prevent a recurrence. More than one metric and more than one type of metric are needed to monitor performance of a process safety management system. A comprehensive process safety management system should contain a variety of different metrics that monitor different dimensions of the system and the performance of all critical elements. 

The critical indices v, ( . y for the percolation cluster are given in Table V. 

Critical indices S and x, obtained numerically, are shown in Table VIII (according to Ref. 76). 

Here, the critical indices for the connecting set correlation length and density are related via the fractal dimension df(lo) and (3((o), v(fo) as... 

Probability functions Y lx,ly,p) for fractal ensembles grown on several lattices (of the generating cells lx x ly where 2 < lx < 4,1 < ly < 4) are presented in the Appendix, while calculated values of the percolation threshold Pc, fractal dimension of the ensemble at p = 1, d (lxIy ), mean fractal dimension at p = pc df), and critical indices p(/v, ly) and v(/v, ly) are listed in Table IX. The index ai in this table is calculated from... 

The results of calculations of the effective Poisson s ratio vp dependence on the bulk concentration of a rigid phase p at various values of a = log i/C/Au) are shown in Fig. 53. The calculations were made for Poisson s ratios of the phases ranging from 0.1 to 0.4. It can be seen that at percolation threshold Poisson s ratio of the isotropic fractal composite is vp = 0.2, when K jK > 0 it is also independent of the Poisson s ratios of the individual components of the composite. The Poisson s ratio obtained by us near the percolation threshold is in agreement with computer simulation results and the conjecture of Arbabi and Sahimi [161]. It has been shown that an approximate theoretical treatment of percolation on a cubic lattice exactly reproduces the Poisson s ratio obtained in computer simulation at the percolation threshold. This result may encourage one to use this approximation to describe various elastic properties of composites. It is worth noting that some critical indices have been calculated recently with a high degree of accuracy in the context of the present model. 

Inequalities for Critical Indices near Gas-Liquid Critical Point ... 

In this paper we present an approach to describe the equilibrium behavior of a classical fluid in the neighborhood of its critical point. The approach which we use here is based on a simplified version of work described earlier. Some physical consequences, especially those which relate the critical indices of the various thermodynamic and statistical-mechanical quantities, can be drawn by properly interpreting the equations derived. 

INEQUALITIES OF CRITICAL INDICES NEAR GAS-LIQUID CRITICAL POINT... 

Therefore, the quantity f is usually a small number. In the following section, the terms containing 0 s in (42) are investigated. Study of this term yields various inequalities relating the critical indices for both the thermodynamic and the statistical-mechanical quantities. 

This new relationship can be used to obtain further inequalities relating the critical indices. 

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