Heat capacity critical point divergence

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.)... 

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. 

On the basis of this brief summary of RPM criticality, one might be tempted to conclude that the problem has been solved all finite-size scaling analysis point towards the Ising universality class. There is, however, one critical phenomenon which does not seem to have been demonstrated unambiguously in the RPM. This is the critical divergence of the constant-volume heat capacity, Cy. Recall that on the critical isochore and close to the critical temperature where the parameter t = (T — Tc)/Tc is small,... 

However, just as is the case for many other kinds of critical phenomena (e.g. the one-component fluid, magnetism, order-disorder transitions in solids, etc.) such predictions do not agree either with the results of careful experimental measurements or with simple theoretical models that can be treated nearly exactly. The coexistence curve is more nearly cubic than parabolic, the critical isotherm is of distinctly higher order than cubic, and the heat capacity Cp,x,m diverges at the critical point. 

Similar problems will arise in the analysis of other properties. For example, the observed heat capacity Cj, will not display an actual divergence. Only one layer in the cell is at the true critical point and the total enthalpy is averaged over all heights and will not show singular behaviour. The phenomenon is entirely analogous to the gravitational perturbation of the measurement of Cv for a pure fluid. ... 

The thermal conductivity A of pure fluids diverges at the critical point about halt as strongly as the isobaric heat capacity (Figs. 5a and 5b), and therefore the thermal diffiisivity, A/pCf>, goes to zero. 

Contrary to the tail of the correlation function, which is directly connected with the strong divergence of the compressibility, the tirst peak of the correlation function, as pointed out by Stell and H0ye [11), bears a subtle relation to the internal energy. The latter property behaves smoothly at the critical point, but its first temperature derivative, the heat capacity Cv, has a weak divergence. 

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... 

The remaining bulk thermodynamic exponent a determines the rate of divergence of the constant-volume heat capacity Cv at the liquid-vapour critical point of a one-component fluid [or that of the constant-pressure heat capacity Cp, or of the literal mechanical (rather than osmotic) compressibility k, or of the coefficient of thermal expansion, of a liquid mixture near its consolute point]. At fixed p = p , the heat capacity Cv as a function of temperature has the shape shown schematically in Fig. 9.3(a) with... 

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. 

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