Pressure virial expansion

Hill, T. L. 1959. Theory of solutions. II. Osmotic pressure virial expansion and hght scattering in two component solutions. Journal of Chemical Physics. 30, 93. 

In the thennodynamic limit (N x, F -> oo withA7F= p), this is just the virial expansion for the pressure, with 7,(7) identified as the second virial coefficient... 

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

The PirialExpansion. Many equations of state have been proposed for gases, but the virial equation is the only one having a firm basis in theory (1,3). The pressure-expHcit form of the virial expansion is... 

For supercritical temperatures, it is satisfactory to ever-higher pressures as the temperature increases. For pressures above the range where Eq. (4-190) is useful, but below the critical pressure, the virial expansion in density truncated to three terms is usually suitable ... 

Coefficients in the virial expansion of the osmotic pressure as a power series in the concentration c (Chap. XII et seq.). 

Coefficients in the alternative virial expansion of the osmotic pressure (see Eqs. VII-13 and XII-76). 

The macroscopic properties of a material are related intimately to the interactions between its constituent particles, be they atoms, ions, molecules, or colloids suspended in a solvent. Such relationships are fairly well understood for cases where the particles are present in low concentration and interparticle interactions occur primarily in isolated clusters (pairs, triplets, etc.). For example, the pressure of a low-density vapor can be accurately described by the virial expansion,1 whereas its transport coefficients can be estimated from kinetic theory.2,3 On the other hand, using microscopic information to predict the properties, and in particular the dynamics, of condensed phases such as liquids and solids remains a far more challenging task. In these states... 

This is a simple and important result. It equates VPIE to the isotopic difference of standard state free energies on phase change, plus a small correction for vapor phase nonideality, here approximated through the second virial coefficient. Therefore Equation 5.8 is limited to relatively low pressure. As T and P increase third and higher virial corrections may be needed, and at even higher pressures the virial expansion must be abandoned for a more accurate equation of state. 

A quite different approach to the detonation product state has been to treat it as solidlike. Jones and Miller6 performed equilibrium calculations on TNT with this idea in mind. They used an equation in which the volume was a virial expansion in the pressure. Other solidlike equations are cited in Ref. 2, but these have mostly been used for computing the state parameters with an assumed product state. The modified Kistia-kowsky-Wilson equation of interest to us liere appears to be one of several possible compromises between the hard-sphere molecule approach and the solid state approach. 

Equation (2.1) is written in a virial expansion form by introducing the density p = 1/V and the reduced variables for density, temperature, and pressure... 

This is a virial expansion form of the osmotic pressure analogous to the van der Waals fluid. Dusek and Patterson examined this equation and predicted the presence of two phases, i.e. collapsed and swollen phases. % is temperature dependent and is given by,... 

In the broad vicinity of the low-pressure (or ideal gas) limit, induced spectra may also be represented in the form of of a virial expansion,... 

In these equations ns is the solvent refractive index, dn/dc the refractive index increment, c the polymer concentration in g/ml, T the temperature in K, R the gas constant, NA Avogadro s number, and n the osmotic pressure. Equation (B.8) follows from Eq. (B.7) by using the familiar virial expansion of the osmotic pressure... 

What does this picture predict for the scaling functions Let us first consider the osmotic pressure. Clearly for small overlap the virial expansion holds... 

Some authors [59] call the region c < c the virial regime. Probably they consider that the virial expansion for osmotic pressure would diverge as c approaches c. ... 

At low concentration region, colligative properties such as surface pressure are best described by a virial expansion of the surface concentration T [59,64],... 

Although the virial expansions might seem very complicated, because they contain an infinite number of terms, their power lies in the fact that usually only a few terms must be considered. As pressure is reduced and molar volume gets very large, the higher terms in the expansion become negligible and only the first two terms need be considered. Equation (26) then becomes... 

Derive an equation for the work of mechanically reversible, isothermal compression of 1 mol of a gas from an initial pressure P, to a final pressure P2 when the equation of state is the virial expansion [Eq. (3.10)] truncated to... 

The two forms of the virial expansion given by Eqs. (3.10) and (3.11) are infinite series. For engineering purposes their use is practical only where convergence is very rapid, that is, where no more than two or three terms are required to yield reasonably close approximations to the values of the series. This is realized for gases and vapors at low to moderate pressures. 

In the expressions (184) and (184b) the second, temperature-dependent term defines the Born effect due to superposition of the two non-linear processes of second-order distortion and reorientation of permanent dipole moments in the electric field. Buckingham et al. determined nonlinear polarizabflities If and c for numerous molecules by Kerr effect measurements in gases as a function of temperature and pressure. It is here convenient to use the virial expansion of the molar Kerr constant, when the first and second virial coefficients Ak and Bk result immediately from equations (177), (178), and (184). Meeten et al. determined nonlinear molecular polarizabilities by measuring K in liquids as a function of temperature. 

Restriction to moderate pressures allows calculation of thefugacity coefficients m Eq. (14.2) to be based on Eq. (3.37), the two-term virial expansion m P. They are then given by Eq. (11.61), here written ... 

We shall now derive the virial expansion of the osmotic pressure following McMillan and Mayer and Hill (P). but simplifying the derivation. The virial expansion plays an imjwrtant role in the theory of solutions. For our purpose we introduce the grand partition function of... 

Each polymer coil in a solution contributes to viscosity. In very dilute solutions, the contribution from different coils is additive and solution viscosity r] increases above the solvent viscosity t/s linearly with polymer concentration c. The effective virial expansion for viscosity at low concentration is of the same form as Eq. (1.76) for osmotic pressure and Eq. (1.96) for light scattering ... 

This virial expansion is analogous to that used for the osmotic pressure in Chapter 1 [Eq. (1.74)] and we will see in Section 3.3.4, how the excluded volume is related to the second virial coefficient. 

This expression of osmotic pressure can be written in the form of the virial expansion in terms of number density of A monomers c = 4>jb [see Eq. (3.8)]... 

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