Compressibility equation, integral equations

The first tenn in the compressibility equation is the ideal gas temi and the second temi, the integral of g r)- ... 

The MS closure results from s = 2. The HNC closure results from s = 1. In the latter two expressions, additional adjustable parameters occur, namely ( for the RY closure and for the BPGG version of the MS approximation. However, even when adjustable, these parameters cannot be chosen at will, as they should be chosen such that they eliminate the so-called thermodynamic inconsistency that plagues many approximate integral equations. We recall that a manifestation of this inconsistency is that there is a difference between the pressure as computed from the virial equation (10) and as computed from the compressibility equation (20). Note that these equations have been applied to a very asymmetric mixture of hard spheres [53,54]. Some results of the MS closure are plotted in Fig. 4. The MS result for y d) = g d) is about the same as the MV result. However, the MS result for y(0) is rather poor. Using a value between 1 and 2 improves y(0) but makes y d) worse. Overall, we believe the MS/BPGG is less satisfactory than the MV closure. 

A calculation of the effect of pressure on the activity that does not involve the assumption of constant Vm usually starts with the compressibility k. Integration of equation (1.39) that relates k to V, while assuming that k is independent of pressure, gives the equation... 

The total work of compression from a pressure P to a pressure Pi is found by integrating equation 8.27. For an ideal gas undergoing an isothermal compression ... 

Knowledge of the sample pressure is essential in all high-pressure experiments. It is vital for determinations of equations of state, for comparisons with other experimental studies and for comparisons with theoretical calculations. Unfortunately, one cannot determine the sample pressure directly from the applied force on the anvils and their cross-sectional area, as losses due to friction and elastic deformation cannot be accurately accounted for. While an absolute pressure scale can be obtained from the volume and compressibility, by integration of the bulk modulus [109], the most commonly-employed methods to determine pressures in crystallographic experiments are to use a luminescent pressure sensor, or the known equation of state of a calibrant placed into the sample chamber with the sample. W.B. Holzapfel has recently reviewed both fluorescence and calibrant data with the aim of realising a practical pressure scale to 300 GPa [138]. 

In which subscript 1 indicates the solvent. We use the subscripts 1 and 2 to indicate solvent and solute, respectively, throughout this chapter. In order to relate ir to the concentration of the solution, then, we must find a way to integrate Equation (19). The easiest way of doing this is to assume that F, is constant. This approximation is justified because the solution is a condensed phase and shows negligible compressibility. Making this assumption and integrating Equation (19) gives... 

One of the limitations in the use of the compressibility equation of state to describe the behavior of gases is that the compressibility factor is not constant. Therefore, mathematical manipulations cannot be made directly but must be accomplished through graphical or numerical techniques. Most of the other commonly used equations of state were devised so that the coefficients which correct the ideal gas law for nonideality may be assumed constant. This permits the equations to be used in mathematical calculations involving differentiation or integration. 

In compressible gases, the molar volume changes with pressure. Using the ideal gas laws in integrating Equation (2.4) gives... 

Integral equation theories of g(r) do in general yield only an approximate estimate of this quantity, and hence they are, to more or less extent, thermodynamically inconsistent. In practice, instead of Eq. (16), one prefers to apply the pressure-compressibility (P — %T) condition expressed by... 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

Moreover the components of vector A change to become Aa = aA/vA, etc. An Ornstein-Zemike (OZ) approach (referred to as the integral equation theory) describing multicomponent compressible polymer blend mixtures has been extensively investigated [35]. The multicomponent OZ equation relates the direct correlations matrix C and the total (i.e., direct and indirect) correlations matrix H as ... 

By integrating Equation 5.8 over an isentropic path using Equation (5.9), it can be shown that the work of compression for an ideal gas,... 

The compressibility equation is a simple integral over g(R). It does not require explicit knowledge of U(R) (or higher order potentials). It is true that g(R) is a functional of U(R). However, once we have obtained g(R), we can use it directly to compute the compressibility by means of (3.109). This is not possible for the computation of, say, the energy. 

In applying the compressibility equation (3.109), care must be exercised to use the pair correlation function g(R) as obtained in the grand canonical ensemble, rather than the corresponding function g(R) obtained in a closed system. Whenever this distinction is important, we use the notation gQ (R) and gc(R) for open and closed systems, respectively. Although the difference between the two is in a term of the order of AT 1 this small difference becomes important when integration over the entire volume is performed as in the definition of the quantity G (equation 3.110). 

As we have expressed the compressibility equation in terms of the integral over the direct correlation function in (C.16), one can write the KB theory in terms of Cy instead of G / the two are equivalent formulations. O Connel (1971) has expressed the view that the formulation in terms of Cy might be more useful for numerical work since the direct correlation function is considered to be shorter range than the pair correlation function. For further applications of this approach, see O Connel (1971), Perry and O Connel (1984), and Hamad et al. (1987, 1989, 1990a, b, 1993, 1997, 1998). 

Here, the integrations extend over the entire volume of the system. The interpretation of (G.7) and (G.8) is straightforward. The quantity pG is the change in the number of particles in the entire system caused by placing one particle at some fixed point, say R0. When N is constant, this change is exactly — 1 the particle we have placed at. No such closure condition is imposed in an open system. Equation (G.8) is just the compressibility equation. 

Studies of the thermodynamic properties obtained from solutions of the SSOZ equation have been much more extensive but the success has been mixed. The first such calculations were those of Lowden and Chandler who obtained the pressure of hard diatomic fluids from the RISM (SSOZ-PY) equation. They used two routes to the equation of state a compressibility equation of state in which they integrated the bulk modulus calculated from the site-site correlation functions via... 

The pressure dependence of the melt viscosity (r ) can be estimated by using Equation 13.19 (derived from classical thermodynamics to relate the pressure and temperature coefficients of r [V]), where p is the hydrostatic pressure, K is the isothermal compressibility, a is the coefficient of volumetric thermal expansion, and d is a partial derivative. The sign of the pressure coefficient of the viscosity is opposite to the sign of the temperature coefficient. Consequently, since r decreases with increasing T, it increases with increasing p. Equation 13.20 is obtained by integrating Equation 13.19. 

Computer simulation of explosive fracture of rock can be carried out with finite difference stress wave propagation codes, such as the YAQUI code (2). YAQUI integrates in time the coupled partial differential equations for the conservation of mass, momentum, and energy. For a compressible fluid, these equations are... 

The first term in the compressibility equation is the ideal gas term and the second term, the integral of g r)- = h r), represents the non-ideal contribution due to the correlation or interaction between the particles. The correlation function h r) is zero for an ideal gas, leaving only the first term. The correlations between the particles in a fluid displaying a liquid-gas critical point are characterized by a correlation length that becomes infinitely large as the critical point is approached. This causes the integral in the compressibility equation and the compressibility Ky.to diverge. 

With regard to this latter point Brereton et al. point out that the integral equation predicts that the stress-maximum observed at strain-rates greater than ip, should be approximately linearly dependent on the logarithm of the applied strain-rate. Such a feature is in accord with yield-stress data at different temperatures for a wide range of isotropic polymers, in both tension and compression. Many workers have attempted to describe this linear dependence of the yield stress on log (strain-rate) in terms of various modifications of Eyring s theory of viscosity. For example the equation... 

As an example of how to determine compressibilities by mean of integral equations, we would like to quote here the already cited paper of Richardi et al. in which molecular Omstein-Zemike (MOZ) theory and site-site Omstein-Zemike (SSOZ) theory have been exploited. In particular, using the MOZ, the compressibility can be evaluated using the equation ... 

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