Thermodynamic properties near criticality

The form of equations (8.11) and (8.12) turns out to be general for properties near a critical point. In the vicinity of this point, the value of many thermodynamic properties at T becomes proportional to some power of (Tc - T). The exponents which appear in equations such as (8.11) and (8.12) are referred to as critical exponents. The exponent 6 = 0.32 0.01 describes the temperature behavior of molar volume and density as well as other properties, while other properties such as heat capacity and isothermal compressibility are described by other critical exponents. A significant scientific achievement of the 20th century was the observation of the nonanalytic behavior of thermodynamic properties near the critical point and the recognition that the various critical exponents are related to one another ... 

In recent years, it has been established experimentally and theoretically that the variation of many thermodynamic properties near a critical point can be described by the use of critical exponents In the case of fluids, the density difference p T) — pJT) is well represented by... 

As a result of the significant variation in thermodynamic properties near and at the critical point, it is difficult to use Computational Fluid Dynamics (CFD) when modeling supercritical flows. Also, since small changes in temperature and pressure can have large effects on the structure of a fluid near the critical point, local property values are very important. 

The universal singular behavior of the thermodynamic properties near the critical point is associated with the presence of long-range fluctuations of the order parameter (density in a one-component fluid). The size of the fluctuations is called the correlation length which diverges at the critical isochore in accordance with the asymptotic power law... 

A chart which correlates experimental P - V - T data for all gases is included as Figure 2.1 and this is known as the generalised compressibility-factor chart.(1) Use is made of reduced coordinates where the reduced temperature Tr, the reduced pressure Pr, and the reduced volume Vr are defined as the ratio of the actual temperature, pressure, and volume of the gas to the corresponding values of these properties at the critical state. It is found that, at a given value of Tr and Pr, nearly all gases have the same molar volume, compressibility factor, and other thermodynamic properties. This empirical relationship applies to within about 2 per cent for most gases the most important exception to the rule is ammonia. 

The thermodynamic properties of a number of compounds are shown in Appendix D as pressure-enthalpy diagrams with lines of constant temperature, entropy, and specific volume. The vapor, liquid, and two-phase regions are clearly evident on these plots. The conditions under which each compound may exhibit ideal gas properties are identified by the region on the plot where the enthalpy is independent of pressure at a given temperature (i.e., the lower the pressure and the higher the temperature relative to the critical conditions, the more nearly the properties can be described by the ideal gas law). 

The benefits from tuning the solvent system can be tremendous. Again, remarkable opportunities exist for the fruitful exploitation of the special properties of supercritical and near-critical fluids as solvents for chemical reactions. Solution properties may be tuned, with thermodynamic conditions or cosolvents, to modify rates, yields, and selectivities, and supercritical fluids offer greatly enhanced mass transfer for heterogeneous reactions. Also, both supercritical fluids and near-critical water can often replace environmentally undesirable solvents or catalysts, or avoid undesirable byproducts. Furthermore, rational design of solvent systems can also modify reactions to facilitate process separations (Eckert and Chandler, 1998). 

First-principle calculations of the thermodynamic properties are more or less hopeless enterprise. One of the most famous phenomenological approaches was suggested by van der Waals [6, 8, 9]. Using the dimensionless pressure it = p/pc, the density v = n/nc and the temperature r = T/Tc, the equation of state for the ideal gas reads it = 8zzr/(3 -u) — 3zA Its r.h.s. as a function of the parameter v has no singularities near u = 1 v = it = t = is the critical point) and could be expanded into a series in the small parameter 77 = [n — nc)/nc with temperature-dependent coefficients. Solving this... 

Not only do the thermodynamic properties follow similar power laws near the critical temperatures, but the exponents measured for a given property, such as heat capacity or the order parameter, are found to be the same within experimental error in a wide variety of substances. This can be seen in Table 13.3. It has been shown that the same set of exponents (a, (3, 7, v, etc.) are obtained for phase transitions that have the same spatial (d) and order parameter (n) dimensionalities. For example, (order + disorder) transitions, magnetic transitions with a single axis about which the magnetization orients, and the (liquid + gas) critical point have d= 3 and n — 1, and all have the same values for the critical exponents. Superconductors and the superfluid transition in 4He have d= 3 and n = 2, and they show different values for the set of exponents. Phase transitions are said to belong to different universality classes when their critical exponents belong to different sets. 

The transport properties of a near-critical system contain an enhancement or a reduction due to critical fluctuations in addition to the contributions of molecular transport processes which are strictly a function of the thermodynamic state. Therefore, the transport coefficients in the critical region are usually... 

Preliminary to such a search we examine several thermodynamic properties of fluids at or close to criticality, that clearly show why and how fluctuations dominate under such conditions, (i) Consider first the isothermal compressibility, kj = —(dV/dP)T/V. At the critical point the isotherm dP/dV)r has zero slope thus, Ki grows indefinitely as T —> Tc. (ii) Using Eq. (1.3.13) and the definition for K one finds that (dV/dT)p = -(dV/dP)TidPldT)v = KiV dP/dT)y, wherein (dP/dT)v does not vanish. Therefore, the coefficient of thermal expansion, = i /V) dV/BT)p also grows without limit as the critical point is approached, (iii) According to the Clausius-Clapeyron equation in the form AH = T(Vg — Vi)(dP/dT), the heat of vaporization of the fluid near the critical point becomes very small, since Vg — Vi 0, whereas dP/dT remains finite. 

The main aim of this contribution is to demonstrate the severe ensemble dependence of near-critical thermodynamic properties of the RPM and AHS fluid, and to examine the effects of ion pairing - or rather the lack thereof - on the criticality in ionic fluids. We will deal with these topics in turn, and then present a summary and discussion in Sec. 5. 

P. J. Smits, I. G. Economou, C. J. Peters and J. de Swaan Arons, Equation of State Description of Thermodynamic Properties of Near-critical and Supercritical Water, J. Phys. Chem., 98, 12080-12085 (1994). 

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