CS equation of state

This is Camalian and Starling s (CS) equation of state for hard spheres it agrees well with the computer simulations of hard spheres in the fluid region. The excess Hehnholtz free energy... 

Figure A2.3.4 compares PIpkT- 1, calculated from the CS equation of state for hard spheres, as a fiinction of...

As pointed out earlier, the contributions of the hard cores to the thennodynamic properties of the solution at high concentrations are not negligible. Using the CS equation of state, the osmotic coefficient of an uncharged hard sphere solute (in a continuum solvent) is given by... 

FIG. 3 The functions g r) and y r) for a hard sphere fluid. The broken curve gives PY results and the sohd curve gives the results of a fit of the simulation data. The circle gives the simulation results. The point at r = 0 gives the result obtained from Eq. (36), using the CS equation of state. 

Figure 11. The HS excess chemical potential p tex calculated with ZSEP closure relation (squares) and with BB approximation. Comparison with the CS equation of state results (open circles). Taken from Ref. [96],...

A commonly used reference fluid is the hard-sphere fluid expressed by the Camahan-Starling (CS) equation of state ... 

Figure 3. VE vs. P for equimolar N2 + CHk as calculated by the CS equation of state (parameters from Tables /, //, and III)...

Figure A2.3.10 Equation of state for hard spheres from the PY and FfNC approximations compared with the CS equation (-,-,-). C and V refer to the compressibility and virial routes to the pressure (after [6]).

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

FIG. 2 The equation of state for hard spheres, obtained from the HNC equation (part a) and the PY equation (part b). The dot-dashed and dotted curves and the circles have the same meaning as in Fig. 1. The solid curves give the results of the CS equation. 

The g r) that results from the modified Verlet (MV) closure is very close to the simulation results in Figs. 2 and 3. The MV results for g d), or equivalently, y d), are plotted in Fig. 4(a). The resulting equation of state is similar to the CS expression. An even more demanding test is an examination of the MV results for y r) for r < d. As is seen in Fig. 4(b), the MV results for y(0) are quite good [25], and are better than the PY and HNC results. Some results have also indicated that the MV closure gives quite accurate results for a mixture of hard spheres [26]. 

In CS one selects an appropriate equation of state (EOS), expresses the parameters in terms of critical properties so far as possible, and fits the result to experimental data to define a minimum set of system specific parameters. A recent example used a modified form of the reduced Van der Waals equation... 

Critical Properties The critical temperature T, pressure P , and volume V of a compound are important, widely used constants. They are important in determining the phase boundaries of a compound and (particularly T and P ) are required input parameters for most thermal and volumetric property calculations of the equilibrium phases using CS or analytical equations of state. Most estimation methods employ weighted group, atom, or bond contributions. 

Mixtures Application of CS methods and analytical equations of state (EoS) to mixtures typically requires pseudocritical temperatures and pressures. These values are obtained from the pure-component critical properties by applying mixing rules specific to the method under consideration, mile pseudocritical values should be used for mixture CS and EoS applications, the values are usually quite different from the mixtures true critical properties. A variety of methods are available for estimating true critical properties for mixtures (see PGL4), but only modest accuracy can be expected. 

Bithas, S. et al.. Correlation and prediction of Henry constants for liquids and gases in five industrially important polymers using a CS-type correlation based on the van der Waals equation of state. Comparison with other predictive models. Fluid Phase Equilibria, 113, 79-102, 1995. 

In Equation 1 XnSM is the value of X for a hard-sphere mixture of diameters dn, d22. . . etc., and XIIS is the value of X for a pure hard-sphere fluid with diameter d. The value of XHs is calculated from the Camahan-Starling (CS) equation (4). XREf represents the value of X as obtained from a reduced equation of state for the pure reference fluid evaluated at T and p made dimensionless by the pseudo parameters c and d3. 

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