Carnahan-Starling equation for

One drawback of the MF1V2 model is the inability of UNIFAC to predict (vapour + liquid) equilibria (VLB) and (liquid + liquid) equilibria (LLE) conditions using the same set of group-interaction parameters. In general, cubic equations of state do not provide precise predictions of the phase equilibria when the mixture is asymmetric in size that is attributed to the large differences in the pure-component co-volumes. The Carnahan -Starling equation for hard spheres is a more realistic model for the repulsive contribution than that proposed by van der Waals. Mansoori et al. proposed an equation for mixtures of hard spheres that has been found to correlate the phase behaviour of non-polar mixtures with large molecular size differences. 

In their paper on mixture properties, Huang and Radosz also use the full mixture version of the Carnahan-Starling equation for the hard-sphere mixtures reference system ... 

Figure 17.18. Variation of the excess Rayleigh ratio. A/ , extrapolated to zero scattering vector, as a function of the droplet volume fraction, 0. The solid line is the best two-parameter fit using the Carnahan-Starling equation for the hard-sphere equation of state. The two fitted parameters were rHs = 86 A and rhc = 76 A (data taken from ref. (17))...

Modified Carnahan-Starling Equation for Dispersions of Deformable Droplets... 

Now, let us consider a model in which the association site is located at a distance slightly larger than the hard-core diameter a. The excess free energy for a hard sphere fluid is given by the Carnahan-Starling equation [113]... 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

Fig. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan-Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC configuration. Spring stiffness is K = 70,000, granular temperature is 9 = 1.0, and coefficient of normal restitution is e = 1.0. The system is driven by rescaling.

In eq 3.1, the activity coefficients appear as a result of the hard-sphere repulsions among the droplets. Since the calculations focus on the most populous aggregates, the hard-sphere repulsions will be expressed in terms of a single droplet size corresponding to the most populous aggregates. One can derive expressions for the activity coefficients y ko of a component k in the continuous phase O starting from an equation for the osmotic pressure of a hard-sphere fluid,3-4 such as that based on the Carnahan—Starling equation of state (see Appendix B for the derivation) ... 

In obtaining the expression for the activity coefficient part of the chemical potential, we have considered droplets of a single size represented by the most populous size (corresponding to the maximum in the size distribution). A more formal equation allowing for droplets of various sizes can be written according to the Mansoori—Carnahan—Starling equation of state for mixtures of hard spheres.26 The results based on such an expression are not expected to be essentially different from those obtained on the basis of a single droplet size. 

Table 1. Parameters for the Carnahan - Starling equation of state at 298. IS K...

Here p(r) is the smoothed density and A is the thermal de Broglie wavelength. The repulsive part of the Helmholtz free energy is usually calculated by the Carnahan-Starling equation derived for the hard sphere fluid [80] ... 

Estimate the packing fraction for a hard-sphere liquid with a density of 21.25 atoms nm and a hard-sphere diameter of 350 pm. Use this result to calculate the Percus-Yevick product for the system at 85 K using the Carnahan-Starling equation of state (equation (2.9.11)). 

Specifically, we have used the Carnahan-Starling Equation (9) for rigid spheres. Equations 1 and 9-18 of Ref. 9 were used to obtain Equation 8 here. 

Figure 4.8 Phase diagram for a pure substance composed of hard spheres. The fluid-phase Z was computed from the Carnahan-Starling equation (4.5.4) the solid-phase Z was taken from the computer simulation data of Alder et al. [14]. The broken horizontal line at Zt = 6.124 connects fluid (T = 0.494) and solid (t = 0.545) phases that can coexist in equilibrium, as computed by Hoover and Ree [12].

At least at low order, this expansion is reproduced to good accuracy by the well-known Carnahan-Starling equation [6], z = l + q + rf — r/ )/(l — 7), which yields 6 = n -(-n—2 for n> 1.) The expansion in terms of the packing fraction rj is related to the more familiar virial expansion in terms of the particle density p,... 

Where 2 = can be obtained from the Carnahan-Starling equation. Calculate the EOS parameters for the PHCT EOS. Obtain the expression for volume expansivity. 

The Carnahan-Starling equation is used for both pure components and mixtures to give... 

Similarly to the fluid-fluid intermolecular potential, we split the solid-fluid intermolecular potential into repulsive hard-sphere and attractive interactions. Here Fhs Ps P is the excess free energy of the solid-fluid HS mixture, for which we employ Rosenfeld fundamental m ure functional [26] with the recent modifications that mve an accurate Carnahan-Starling equation of state in the bulk limit [27,28] r-r ) is the attractive part of the solid-fluid intermolecular potential. Since the iM>lid-soIid attraction interaction is not included, the solid is effectively modeled as a compound of... 

As in the Wertheim equation of state, Chiew also made use of the Carnahan-Starling expression for the pressure of the corresponding hard-sphere liquid. Similar agreement between TPTl and the Chiew theory was found with Monte Carlo simulations of Dickman and Hall for the pressure. 

Nitta et. al. ( 7) extended the group interaction model to thermodynamic properties of pure polar and non-polar liquids and their solutions, including energy of vaporization, pvT relations, excess properties and activity coefficients. The model is based on the cell theory with a cell partition function derived from the Carnahan-Starling equation of state for hard spheres. The lattice energy is made up of group interaction contributions. 

Points 1 and 2 can be incorporated straightforwardly using the ideas presented earlier in this chapter. For example, we could use the analytic Percus-Yevick equations of state for hard spheres (Eqs. 47a and b) or the Carnahan-Starling equation of state (Eq. 49) for p. Furthermore, we could use the hard-sphere radial distribution function obtained numerically from one of the integral equations or even that calculated from computer simulation. Points 3 and 4 are less straightforward and represent contributions that were made around 1970 by Barker and Henderson (1976) and by Weeks, Chandler and Andersen (1971). The results of these two approaches are comparable and are illustrated in Figs. 10 and 11 and Table 3. 

Figure 16. The counterion density near a charged wall for the DFT scheme. Conditions as in Fig. 14. DFT denotes the density functional theory based on the fundamental measure theory [78], and WDA denotes the weighted density approximation based on the generalized Carnahan-Starling equation [75].

It was later modified to include an attractive contribution, " regarding the latter as a small perturbation. It is therefore tempting to modify the Carnahan-Starling equation of state in such a way that it would become applicable to deformable fluid droplets. Unfortunately the perturbation approach is not relevant to the case of deformable fluid droplets. This becomes clear if one writes the perturbation term, Pqs, for the osmotic pressure ... 

Estimate the value of PV/ Nk- T) for a hard-sphere fluid using the Carnahan and Starling equation of state assuming a concentration of 10 M and a hard-sphere diameter of 300 pm. 

In this relation, N is the number density of the scattering microemulsion droplets and S(q) is the static structure factor. Equation (2.12) is only strictly valid for the case of monodisperse spheres. However, for the case of low polydispersities the occurring error is small [63, 64]. S(q) describes the interactions between and the spatial correlations of the droplets. These are in general well approximated by hard sphere interactions in microemulsion systems [65], The influence of inter-particle interactions as described by S(q) canbe estimated at least for S(0) using the Carnahan-Starling expression [52,64,66]... 

We now consider modifications to the repulsive term in the van der Waals equation. Although the van der Waals hard-sphere term is correct at low densities. Figure 4.13 shows that it quickly becomes erroneous as the density is increased the excluded volume is not constant, but depends on density in some complicated way. Therefore we can improve the equation of state by using the Carnahan-Starling form (4.5.4) for Z/jg. Our modified Redlich-Kwong (mRK) equation of state is then [29]... 

N. F. Carnahan and K. E. Starling, "Equation of State for Nonattracting Rigid... 

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