Heat capacity divergence

Heat capacity measurements give information complementary to magnetic susceptibility and magnetization. This complementarity arises naturally from thermodynamics. This technique has also become much more available as instrumentation based on the relaxation method has been marketed widely in the past few years. The heat capacity diverges at Tc and thus provides a very precise measure of Tc, as seen in Figure 12. The shape of the anomaly resembles the Greek... 

Experimentally, it is well established that asymptotically close to the critical point all physical properties obey simple power laws. The universal powers in these laws are called critical exponents, the values of which can be calculated from RG recursion relations. The phenomenological approach that interrelates the critical power laws is called scaling theory. In particular, the isochoric heat capacity diverges at the vapour-liquid critical point of one-component fluids along the critical isochore as... 

At a eritieal point, the phase transition under consideration is the second-order transition. For the uniform system, in the critical region the correlation length of statistical fluctuations, the isothermal compressibility and the heat capacity diverge to the infinity, according to the well-known power laws [337]. Moreover, the order parameter, which, for the gas-Uquid transition, is defined as the difference between the densities of both coexisting phases, approaches zero at the critical temperature. 

Divergence of Heat Capacity at Phase Transitions. The heat capacity diverges at phase transitions. This divergence can be more easily understood by considering the definition of Cv in terms of the system entropy " ... 

In 1953 Scott [H] pointed out that, if the coexistence curve exponent was 1/3, the usual conclusion that the corresponding heat capacity remamed finite was invalid. As a result the heat capacity might diverge and he suggested an exponent a= 1/3. Although it is now known that the heat capacity does diverge, this suggestion attracted little attention at the time. 

However, the discovery in 1962 by Voronel and coworkers [H] that the constant-volume heat capacity of argon showed a weak divergence at the critical point, had a major impact on uniting fluid criticality widi that of other systems. They thought the divergence was logaritlnnic, but it is not quite that weak, satisfying equation (A2.5.21) with an exponent a now known to be about 0.11. The equation applies both above and... 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

We have seen in previous sections that the two-dimensional Ising model yields a syimnetrical heat capacity curve tliat is divergent, but with no discontinuity, and that the experimental heat capacity at the k-transition of helium is finite without a discontinuity. Thus, according to the Elirenfest-Pippard criterion these transitions might be called third-order. 

The mysteries of the helium phase diagram further deepen at the strange A-line that divides the two liquid phases. In certain respects, this coexistence curve (dashed line) exhibits characteristics of a line of critical points, with divergences of heat capacity and other properties that are normally associated with critical-point limits (so-called second-order transitions, in Ehrenfest s classification). Sidebar 7.5 explains some aspects of the Ehrenfest classification of phase transitions and the distinctive features of A-transitions (such as the characteristic lambda-shaped heat-capacity curve that gives the transition its name) that defy classification as either first-order or second-order. Such anomalies suggest that microscopic understanding of phase behavior remains woefully incomplete, even for the simplest imaginable atomic components. 

Another claim for an apparent mean-field behavior of ionic fluids came from measurements of heat capacities. The weak Ising-like divergences of the heat capacities Cv of the pure solvent and CPtx of mixtures should vanish in the mean-field case (cf. Table I). The divergence of Cv is firmly established for pure water. Accurate experiments for aqueous solutions of NaCl... 

Very few examples of heat capacity or compressibility behavior of the type shown in the second column have been observed experimentally, however. Instead, these two properties most often are observed to diverge to some very large number at Tt as shown in the third column of Figure 13.1.1 The shapes of these curves bear some resemblance to the Greek letter, A, and transitions that exhibit such behavior have historically been referred to as lambda transitions. 

An ideal gas with constant heat capacities enters a converging/ diverging nozzle with negligi velocity. If it expands isentropically within the nozzle, show that the throat velocity is given by 1... 

Figure 12 Divergence of the heat capacity at Tc for ferromagnetic GdCh. (Reprinted from Ref 13, 1966, with permission from Elsevier)...

---