Isothermal compressibility density fluctuations

For polymers the density fluctuation background is high. This is a result of its relation to the isothermal compressibility (and the velocity of sound) in the material. 

The approach to the critical point, from above or below, is accompanied by spectacular changes in optical, thermal, and mechanical properties. These include critical opalescence (a bright milky shimmering flash, as incident light refracts through intense density fluctuations) and infinite values of heat capacity, thermal expansion coefficient aP, isothermal compressibility /3r, and other properties. Truly, such a confused state of matter finds itself at a critical juncture as it transforms spontaneously from a uniform and isotropic form to a symmetry-broken (nonuniform and anisotropically separated) pair of distinct phases as (Tc, Pc) is approached from above. Similarly, as (Tc, Pc) is approached from below along the L + G coexistence line, the densities and other phase properties are forced to become identical, erasing what appears to be a fundamental physical distinction between liquid and gas at all lower temperatures and pressures. 

A wide variety of density- and temperature-dependent input parameters are required. These include p, the number density of the solvent, kj, the isothermal compressibility, f, the correlation length of density fluctuations, y = Cp/Cv, the ratio of specific heats, and 37, the viscosity. Very accurate equations of state for ethane (74,75,99) and CO2 (74) are available that provide the necessary input information. The necessary input parameters for fluoroform were obtained by combining information from a variety of sources (76,100,101). There is somewhat greater uncertainty in the fluoroform parameters. 

These regions provoke anomalous fluctuations of density and entropy, from which, at least at a qualitative level, an explanation can be derived for the increase of the isothermal compressibility and specific heat in the metastable region. 

It. is also instructive to consider fluctuations in the one-dimensional hard-rod fluid. Focusing on density fluctuations one realizes from Eqs. (1.81) and (2.75) that the isothermal compressibility is a quantitative measure of such fluctuations. For the ono-dimonsional fluid considered in this section, we may dcfiiu ... 

Despite the absence of capillary condensation, the one-dimensional hard-rod fluid is still so useful because we have an analytic expression for its partition function [see Eq. (3.12)] that permits us to derive closed expressions for any thormophysical property of interest. One such quantity that is closely related to the Isothermal compressibility discussed in the preceding section is the particle-number distribution P (N), whidi one may also employ to compute thermomechanical properties [see, for example, Eqs. (3.65) and (3.68)]. Moreover, in a three-dimensional system P ) is useful to investigate the sj stem-size dependence of density fluctuations as we shall demonstrate in Section 5.4.2 [see Eq. (5.80)]. 

The supercooled liquid catastrophe, if it exists, would necessarily be associated with diverging fluctuations in the structural order parameter F. This stems from the fact that the Y surface develops a vanishing curvature in the F direction as this endpoint is approached. Because the bicyclic octamer elements are bulky, fluctuations in their coiKentration amount to density fluctuations. Diverging density fluctuations then imply diverging isothermal compressibility. Furthermore the infinite slope of the metastable liquid locus at its endpoint implies the divergence of thermal expansion. Potential energy fluctuations remain essentially normal, so constant-volume heat capacity remains small. But the volumetric divergence creates an unbounded constant-pressure heat capacity. 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response functions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where N is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothermal compressibility Kj, which is another response function. Temperature-dependent fluctuations characterize the dynamic equilibrium of thermodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

No system is exactly uniform even a crystal lattice will have fluctuations in density, and even the Ising model must permit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothermal compressibility Kj, and the number of molecules N in volume V ... 

The three interrelated phenomena that are observed near the critical point are (a) increase in the density fluctuation, (b) increase in the isothermal compressibility, and (c) increase in the correlation length of G(R). 

There are several circumstances in which the quadratic fluctuation theory presented here breaks down. When derivatives of any of the intensive parameters with respect to the extensive parameters are very small, the corresponding fluctuations are very large. The Taylor expansion of SE in the fluctuations are then very large and the Taylor expansion of SE in the fluctuations cannot be truncated at the second-order term. For example, the mean-square density fluctuation is given by Eq. (10.C.28), where the isothermal compressibility and correspondingly (Sp2y diverges when (dP/d V)t - 0. This happens at the gas liquid critical point. Likewise at the critical consolute point... 

We are particularly interested at this point in the density fluctuations present on a macroscopic scale. As stated in any textbook on statistical mechanics, the fluctuation in the number N of atoms, contained in an open system under constant volume and temperature, can be calculated by means of the grand canonical ensemble formalism. The result shows that the mean square fluctuation ((AN)2) in N is on the order of N itself and is related, in a system of macroscopic size, to the isothermal compressibility fa by... 

An important consequence of the above assumption is the presence of density fluctuations with a non-zero correlation length. That is because a molecule with a larger than average number of HBs is more likely to be surrounded by other molecules also with a larger than average number of HBs. In this way, it is possible to justify the anomalous increase of compressibility with decreasing temperature. At low temperatures, the number of bonds increases and the density fluctuations increase as well. These correlated fluctuations are superimposed on the normal thermally driven density fluctuations present in other non-associated liquids. The combination of the two competing behaviors yields the compressibflity minimum of the temperature dependence of isothermal compressibility. 

To calculate the masses of the liquids injected it is necessary to know the densities of the liquids at the temperature and pressure in the injectors. The injectors are not thermostatted but set above the bath in an air-conditioned room. Because the injectors themselves have a large heat capacity, rapid fluctuations in the room temperature will not have a major effect. Long-term drifts in the temperature would be more serious. Compression of the liquids to 500 kPa will cause an increase in density of about 0.05 per cent. Van Ness points out that there will be a negligible error in the mole fraction calculated from atmospheric data provided all volume measurements on both liquids are made at the same applied pressure and the liquids have similar isothermal compressibilities. When the isothermal compressibilities of the two components differ appreciably a more detailed calculation is necessary to determine the mole fraction. 

Fluctuations in thermodynamics automatically imply the existence of an underlying structure that has created them. We know that such structure is comprised of molecules, and that their large number allows statistical studies, which, in turn, allow one to relate various statistical moments to macroscopic thermodynamic quantities. One of the purposes of the statistical theory of liquids (STL) is to provide such relations for liquids (Frisch and Lebowitz 1964 Gray and Gubbins 1984 Hansen and McDonald 2006). In such theories, many macroscopic quantities appear as limits at zero wave number of the Fourier transforms of statistical correlation functions. For example, the Kirkwood-Buff theory allows one to relate integrals of the pair density correlation functions to various thermo-physical properties such as the isothermal compressibility, the partial molar volumes, and the density derivatives of the chemical potentials (Kirkwood and Buff 1951). If one wants a connection between detailed correlations and integrated moments, one may ask about the nature of the wave-number dependence of these quantities. It turns out that the statistical theory of liquids allows an answer to such a question very precisely, which leads to new types of questions. The Ornstein-Zemike equation (Hansen and McDonald 2006), which is an exact equation of the STL, introduces the concept of correlation length which relates to the spatial extension of the density and/or concentration (the latter in the case of mixtures) fluctuations. This quantity cannot be accessed from pure... 

Thus, the light scattering intensity is immediately related to fluctuations in the number of particles in the system (Equation 105) and the isothermal compressibility (Equation 107). If the radial function of density distribution is permitted to be spherically... 

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