LIQUIDS AND DENSE GASES 1. Introduction

A liquid possesses some short range order in the instantaneous distribution of near neighbors about an atom or molecule, but in general the geometry of this loose packing differs radically from that in the solid phase: as well as the loss of long

138 Chapter 4 Table 4.2. Values ofev (J.mol-1.K-1) for some gases at pressure of 0.1 MPa

(=1 bar) [Jac97] (see also ITS-90 for values of Tt [Qui90))

range order when a crystal melts, there is a discontinuous change in the nature of the short-range order. This was first illustrated by Bernal in an experimental study of the random packing of uniform hard spheres. In the close-packed crystalline form (e.g., fcc) each sphere is surrounded by twelve nearest neighbors, and 74% of the space is occupied. In various experiments Bernal found that when spheres were shaken together in a random arrangement only 64% of the space was occupied, and the coordination patterns were quite different from those in a solid (for details of this and later work see, e.g., [Cus87]).

Computer simulations (MC and MD as described in Section 2.3.3) on random assemblies of atoms, have substantially confirmed Bernal's conclusions about liquid structure, and have provided quantitative results for simple models that also enable approximate analytic theories to be tested. Such work is beyond the scope of this monograph, but some references have already been given in Section 2.3.3. To these can be added The Physics of Structurally Disordered Matter by Cusack [Cus87], which discusses also the electronic properties of liquids and glasses; the student text Gases, Liquids and Solids by Tabor [Tab91]; Liquids and Liquid Mixtures by the phys-ical chemist Rowlinson [Row69]; Theory of Simple Liquids, 2nd. edition, by Hansen and McDonald [Han86]; and Theory of Molecular Fluids. Vol. 1: Fundamentals by Gray and Gubbins [Gra84].

4.3.2. Changes in Thenoodynamic Properties on Melting

Except for helium at low temperatures (Section 4.4), entropy increases on melting.

The molar entropy !l.fS of fusion is of the order R: about O.8R for many metals, about

Fluids

Pig. 4.4. Heat capacities Cp and Cv of solid and liquid nitrogen along the equilibrium vapor pressure curve (Man97]. and of gaseous nitrogen at a pressure of 0.1 MPa (l bar) [Jac97].

1.5R for the rare gas solids, hydrogen and hydrogen halides, about 2.5R for the semi-metals Ga, Sb and Bi, 3.8R for Si and Ge, and distributed in this range for various crystals ofsmall molecules [Moe57, Wal91]. For the same groups the molar volumes increase by a few percent for most metals, by about 15% for the rare gas solids, and by 6-12% for various molecular crystals. Molar volumes decrease by a few per cent for Ga, Sb, and Bi, and by about 8% for Si and Ge' These values contrast with those for vaporization - typically about 12R for I1vS and 2 x 10-² m ³ for l1v V.

Thus l1vV /l1fV

>

UP, whereas I1vS/l1fS ~ 10; by the Clapeyron Eq. (2.26) this explains why solid-liquid equilibrium lines in P-T phase diagrams are much steeper than liquid-vapor lines (Fig. 4.1). Similarly, solid-liquid lines for substances whose volumes decrease on melting have a negative slope. These include the semimetals and semi-conductors mentioned above, in which the closer packing in the liquid is associated with radical change in the electronic structure [WaI91]; and also ice, where on melting there is a partial breakdown of the tetrahedral hydrogen bonding between the oxygen atoms.

For the heat capacity, we must distinguish between Cy and Cpo Since each translational or rotational degree of freedom contributes only tR to Cy when the molecule is free in an ideal gas, but double that amount for classical harmonic vibrations, we might expect Cy to decrease as the molecules become freer on melting.

This is seen to happen in all three phase transitions shown for N2 in Fig. 4.4. In the a solid phase there is l()ng ~ge orientational order; this is lost in the

f3

solid phase, where however there is believed to be correlated rotation retaining considerable short range order. Cy decreases also when there is further loosening of the structure both on melting and on vaporization. In contrast, the difference in heat capacities C p - C y increases on melting, presumably due to an increase in compressibility and hence in thermal expansion.

140 Chapter 4

4.3.3. The Critical Region

Both liquid and gas have random structures. At temperatures just above the triple point the molecules of a liquid cohere because of the attractive part of the intermolecular potential. At the saturated vapor pressure the liquid has the same Gibbs free energy H - T S as the vapor, whose greater enthalpy is compensated by its greater entropy. But at temperatures high enough for kT to be much greater than the depth of the potential well, only the repulsive part of the potential is important, and there is no tendency for the molecules to cohere. Then, as the pressure is increased, the random structure of the fluid changes continuously from dilute gas to dense fluid, and there is no phase change until the Gibbs free energy of the ordered crystalline structure becomes less than that of the fluid. There is therefore a critical temperature Tc above which the distinction between gas and liquid disappears. The fluid can change continuously from low temperature gas to low temperature liquid by passing above Tc (Fig. 4.1).

It was J. D. van der Waals who produced the first and simplest theoretical model which demonstrates this behavior. The ideal equation of state is modified to give

(4.16)

where the effective reduction in molar volume to (V m - b) simulates the effect of the hard core inner repulsion, and a/V~ simulates the effect of the attractive forces.

The success of this beautifully simple approximation in the interpretation of fluid behavior is described in countless books; for a full and clear discussion see [Dom96, Ch. 2]. The critical point is given by

RTc = 27b' 8a Vc = 3b, (4.17)

The second virial coefficient is B(T)

=

b - (a/RT), illustrating the general depen-dence on temperature discussed in Section 4.2.4, and the Boyle temperature is thus T8

=

a/Rb. Cv is finite but has a discontinuity at the critical point and Cp and XT diverge.

Such a simple model does not of course represent quantitatively the behavior of even the simplest real fluids. For example. Eq. (4.17) gives a value of 8/3

=

2.67 for the dimensionless ratio RTc/Pc Vc , compared with values for Ar and n-H2 of 3.43 and 3.28 respectively. Moreover, the nature of the singularity at the critical point (for example. how XT and Cp approach infinity) is different from that observed.

Experimental study of the singularity demands high precision as the critical point is approached, and theoretical study demands extensive calculations using either series expansions or computer simulation. As an introduction to the vast literature on this subject, see [Sta71. Dom96]; for critical phenomena in liquid mixtures. see [Row69].

Fluids 141

4.4. QUANTUM FLUIDS; LIQUID HELIUM

In document at Low Temperatures (pagina 144-148)