Phase transitions thermodynamic functions

In a first order phase transition, thermodynamic functions by definition discon-tinuously change as one cools the system along a path crossing the equilibrium coexistence line (Fig. 5a, path j8). In a rea/experiment, however, this discontinuous change may not occur at the coexistence line because a substance can remain in a supercooled metastable phase until a limit of stability (a spinodal) is reached [2] (Fig. 5b, path fi). 

There are many more first- versus second-order phase transitions, state functions versus path-dependent functions, and so forth. However interwoven, the subject can be divided roughly into two parts as presented in Figure 3.1. One part concentrates on the heat and work transferred between a system and its surroundings. The other part attends to the relationships between a system s state variables and functions. There are quite a number of these beginning with temperature (I), pressure (p), and volume (V), as introduced in Chapter 1. If the chemist chooses a quantity such as enthalpy (H), there is quite a story to tell about its relation to other system properties such as compressibility, heat capacity, and so on. Suffice to say that the variables and functions form the infrastructure for thermodynamics under the umbrella of physical laws. 

Besides shear-induced phase transitions, Uquid-gas equilibria in confined phases have been extensively studied in recent years, both experimentally [149-155] and theoretically [156-163]. For example, using a volumetric technique, Thommes et al. [149,150] have measured the excess coverage T of SF in controlled pore glasses (CPG) as a function of T along subcritical isochoric paths in bulk SF. The experimental apparatus, fully described in Ref. 149, consists of a reference cell filled with pure SF and a sorption cell containing the adsorbent in thermodynamic equilibrium with bulk SF gas at a given initial temperature T,- of the fluid in both cells. The pressure P in the reference cell and the pressure difference AP between sorption and reference cell are measured. The density of (pure) SF at T, is calculated from P via an equation of state. 

The uncertainties in the condensed-phase thermodynamic functions arise from (1) the possible existence of a solid-solid phase transition in the temperature range 2160 to 2370 K and (2) the uncertainty in the estimated value of the liquid heat capacity which is on the order of 40%. While these uncertainties affect the partial pressures of plutonium oxides by a factor of 10 at 4000 K, they are not limiting because, at that temperature, the total pressure is due essentially entirely to O2 and 0. 

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

We shall now discuss the phase transition from the viewpoint of statistical thermodynamics. " The total free energy G can be expressed as a function of N (total number of cation sites = total number of anion sites), (total number of anions), (number of cations on the A sites), Ag, A,-, and Aq as G = G(A,Ax,Aa,Ab,Ac,Ad) (1.234)... 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

The function describing the change in entropy, as a function of temperature, involves the use of a prescription that contains a formula specific to a particular phase. At each phase transition temperature the function suffers a finite jump in value because of the sudden change in thermodynamic properties. For example, at the boiling point 7b the sudden change in entropy is due to the latent heat of evaporation (see Figure 2.8). 

In Fig. 3 c the schematic volume-temperature curve of a non crystallizing polymer is shown. The bend in the V(T) curve at the glass transition indicates, that the extensive thermodynamic functions, like volume V, enthalpy H and entropy S show (in an idealized representation) a break. Consequently the first derivatives of these functions, i.e. the isobaric specific volume expansion coefficient a, the isothermal specific compressibility X, and the specific heat at constant pressure c, have a jump at this point, if the curves are drawn in an idealized form. This observation of breaks for the thermodynamic functions V, H and S in past led to the conclusion that there must be an internal phase transition, which could be a true thermodynamic transformation of the second or higher order. In contrast to this statement, most authors... 

The key aspects of the modern understanding of phase transitions and the development of renormalization group theory can be summarized as follows. First was the observation of power-law behavior and the realization that critical exponents were, to some extent, universal for all kinds of phase transitions. Then it became clear that theories that only treated the average value of the order parameter failed to account for the observed exponents. The recognition that power-law behavior could arise from functions that were homogeneous in the thermodynamic variables and the scale-invariant behavior of such functions... 

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