Equation hard-sphere

Kapustinskii equation For an ionic crystal composed of cations and anions, of respective charge and z, which behave as hard spheres, the lattice energy (U) may be obtained from the expression... 

The hard-sphere treatment also suggested a relationship between surface tension and the compressibility of the liquid. In a more classic approach [48], the equation... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

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...

These equations provide a convenient and accurate representation of the themrodynamic properties of hard spheres, especially as a reference system in perturbation theories for fluids. 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

The virial pressure equation for hard spheres has a simple fomr detemiined by the density p, the hard sphere diameter a and the distribution fimction at contact g(c+). The derivative of the hard sphere potential is discontinuous at r = o, and... 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

Figure A2.3.10 compares the virial and pressure equations for hard spheres with the pressure calculated fonu the CS equations and also with the pressures detemiined in computer simulations.

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]. 

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... 

The high-temperatiire expansion, truncated at first order, reduces to van der Waals equation, when the reference system is a fluid of hard spheres. 

Wertheim M S 1963 Exact solution of the Percus-Yevick equation for hard spheres Phys. Rev. Lett. 10 321... 

Thiele E 1963 Equation of state for hard spheres J. Chem. Phys. 39 474... 

Waisman E and Lebowitz J K 1972 Mean spherical model integral equation for charged hard spheres... 

Ebeling W and Grigoro M 1980 Analytical calculation of the equation of state and the critical point in a dense classical fluid of charged hard spheres Phys. (Leipzig) 37 21... 

Baxter R J 1968 Percus-Yevick equation for hard spheres with surface adhesion J. Chem. Phys. 49 2770... 

The Boltzmaim equation for hard spheres is given then as... 

For dilute dispersions of hard spheres, Einstein s viscosity equation predicts... 

Figure C2.6.5. Examples of tire AO potential, equation (C2.6.12). The values of are indicated next to tire curves. The hard-sphere repulsion at r = 7 has not been drawn.

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

Piazza R, Bellini T and Degiorgio V 1993 Equilibrium sedimentation profiles of screened charged colloids a test of the hard-sphere equation of state Rhys. Rev. Lett. 71 4267-70... 

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