The limit of low densities

We now turn to a microsropic treatment of the. Toule-Thomson effect and begin with the limit of vanishing density. The treatment below is very similar to the one presented in Section 3.2.2 where we derived molecular expressions for the first few virial coefficients of the one-dimensional hard-rod fluid. Here it is important to realize that a mechanical expression for the grand potential exists for a fluid confined to a slit-pore with chemically structured substrate surfaces as we demonstrated in Section 1.6.1 [see Eq. (1.65)). Combining this expression with the molecular expression given in Eq. (2.81) we may write 

Unfortunately, the original expansion of t in tenns of the activity z is somewhat awkward in practice. Instead we would prefer an expansion of ry in terms of the mean density p of the confined fluid. This can be accomplished 1 first noting from Eq. (5.142) that we may write 

We can now derive an expression for the inversion temperature that is valid in the limit of sufficiently low densities. Therefore we differentiate [see Eq. (5.149)1 

Tills cxprc ssion may be simplified even fmlher by noting that 

Comparing this expression with the thermodynamic one for 5 defined in Eq. (5.134) it is clear that the inversion temperature can be obtained here from the zero of 

This expression may be simplified even further by noting that /a(A/As ) N 

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In the limit of low densities, A2.4.15 shows that the zeroth-order approximation forg(r) has the fonn... 

Figure 12 illustrates these measurements for monovalent counterions and three values of B/cr = 1, 2, 3. Several things may be noted The osmotic coefficient from the simulations is always smaller than the PB prediction. In the limit of low density both values converge. This also illustrates that the Manning limiting law from the r.h.s of Eq. 39 becomes asymptotically... 

Finally, we resolve the long-standing problem of behavior of this equation of state in the limit of low densities, which arises in the integration of the thermodynamic equation of state, to obtain the change of internal energy as a function of density along isotherms. 

In previous reports we have shown graphically the behavior of functions (p,T), (p,T), B(p), and C(p) for Equation 4. In Ref. 3 we illustrated nonanalytic behavior in relation to the maximum in specific heats at the critical point. In this chapter we have given a solution for the long-standing problem of behavior in the limit of low densities, namely, a completely new type of formulation for the saturated-vapor densities, which extrapolates to Za = 1 at p = 0, T = 0. 

It is easily seen that this set of equations is in agreement with Eq. (19) in the limit of low densities, where only the first term in Eq. (39) is retained. But, of course, the equations are of interest only in the critical region. One remark on Eq. (39) should be made the virial expansion of is quite different from the virial expansion of G r), and probably converges to zero more quickly than G r). However, does not appear in every term of the expansion and it is not certain that the second moment of is finite at the critical point, which is a necessary condition for the validity of the Ornstein-Zernike theory. 

The correction term refers to the permanent dipole-induced dipole interaction. In the limit of low density in media 1 and 2 and when medium 3 is vacuum, this expression corresponds to the permanent dipole interaction of Keesom. ... 

Ideal gas behavior can be realized by either systems of hypothetical, strictly-non-interacting particles, or by real particles but at the limit of low densities p 0. Both of these give the same... 

In general, the equation of state for a substance depends specifically upon its composition. For example, the equation of state for liquid water is different than that for liquid ethanol. However, as we shall see, in the limit of low densities the properties of gases can be described by an equation of state that is independent of the nature and composition of the gas. 

In the limit of low density, v is the negative of the second virial coefficient for the effective monomer-monomer interactions w(r, — r ). 

We wish to obtain the first-order deviation from the ideal-gas expression for the chemical potential. This may be obtained either from (5.9.27) or from (5.9.38). We know from section 5.3.2 that at the limit of low density we have... 

It should be added that the determination of the interaction viscosity and the cross-section ratio in the way described allows the binary difhision coefficient in the limit of low density to be evaluated through equation (4.116) with an accuracy comparable with that of the best direct measurements. 

In the limit of low temperatures and high densities, the number of holes must tend towards zero. On the other hand in the limit of low densities we obtain... 

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