Physical Chemistry Third Edition

1186 28 The Structure of Solids, Liquids, and Polymers

Solid-Like Liquid Models

There are several model theories that treat a liquid like a disordered solid. In thecell

model^24 each atom of a monatomic fluid such as liquid argon is assumed to be confined

in a cell whose walls are made up of its nearest neighbors. In the simplest version,

this cell is approximated as a spherical cavity inside which the potential energy of

the moving atom is constant and outside of which the potential energy is infinite.

Because each atom moves independently, the classical canonical partition function can

be written as a product of molecular partition functions. The classical canonical partition

function is

Zcl(2πmkBT)^3 N/^2

(

Vfe−u^0 /kBT

)N

(28.5-1)

whereVfis the “free volume” in which the center of the atom can move, and whereu 0

is the constant potential energy of an atom in the cell. Since the atoms are confined in

distinguishable cells, there is no correction for indistinguishability.

At 0 K the substance is a solid, and we assume that the atoms are hard spheres of

diameterdin contact with each other, so that the centers of nearest neighbor atoms

are at a distancedfrom each other (dequals twice the radius). In the liquid at nonzero

temperature, the thermal expansion of the lattice moves the centers of nearest neighbors

to an average distance that is larger thand, which we denote byb. The center of the

atom can now move in a small region at the center of the cell, approximated as a sphere

with radius equal tob−d.

The free volume is

Vf

4

3

π(b−d)^3 (28.5-2)

We can relatedandbto the molar volumes. We assume a face-centered close-packed

lattice and assume that the spheres are in contact in the solid. For spheres in contact

in a face-centered cubic lattice, the diagonal of a unit cell face is equal to 4 times

the radius of the spheres, equal to 2d, wheredis their diameter. The edge of the unit

cell is equal to

√

2 d. Each unit cell contains 4 spheres, so the molar volume of the

solid is

Vm,s

NAv

4

(√

2 d

) 3

NAv

d^3

√

2

d^3 

√

2 Vm,s

NAv

whereVm,sis the molar volume of the solid andNAvis Avogadro’s constant. Similarly,

for the liquid

b^3 

√

2 Vm

NAv

(28.5-3a)

whereVmis the molar volume of the liquid.

(^24) T. L. Hill,Statistical Thermodynamics, Addison-Wesley, Reading, MA., 1960, Ch. 16. H. Eyring and
M. S. Jhon,Significant Liquid Structures, Wiley, New York 1969, Ch. 2.