Physical Chemistry Third Edition

28.5 The Structure of Liquids 1187

Using Eq. (28.5-3), the classical canonical partition function of a monatomic liquid

ofNatoms is

Zcl(2πmkBT)^3 N/^2

⎧

⎪⎨

⎪⎩

4

3

π

⎡

⎣

(√

2 Vm

NAv

) 1 / 3

−

(√

2 Vm,s

NAv

) 1 / 3 ⎤

⎦

3

e−u^0 /kBT

⎫

⎪⎬

⎪⎭

N

(28.5-4)

The pressure can be calculated from Eq. (27.5-7) using this partition function:

PkBT

(

∂ln(Z)

∂V

)

T



NkBT

V

1

1 −

(

Vm,s/Vm

) 1 / 3 (28.5-5)

Exercise 28.10

Carry out the differentiation to obtain Eq. (28.5-5).

The formula for the pressure given in Eq. (28.5-5) approaches the ideal gas value for

large molar volume, and diverges as the molar volume approaches the molar volume

of the solid at 0 K. This behavior is qualitatively correct, but the cell model does not

predict accurate values of the pressure. Lennard–Jones and Devonshire^25 developed an

improved version of the cell model, in which they explicitly summed up the potential

energy contributions for the nearest neighbors, obtaining better results.

Computer Simulations of Liquid Structure

With the advent of fast computers, numerical simulations of liquid structure have

become practical. There are two principal simulation methods, theMonte Carlomethod

andmolecular dynamics. The Monte Carlo method is so named because it uses a random

number generator, reminiscent of the six-sided random number generators (dice) used

in gambling casinos, such as those in Monte Carlo. This method was pioneered by

Metropolis.^26

In the Monte Carlo method a system of several hundred or a few thousand molecules

is considered. An initial set of coordinates for all of the molecules is generated in some

way, and further states are generated as follows: A random number generator is used

to pick a number,b, between−1 and 1. A particle is moved a distance∆xabin

thexdirection, whereais a predetermined maximum displacement. The change in

potential energy of the system,∆V, is then calculated. If∆V <0, the particle is left

at the new location. If∆V >0, the particle is assigned a probability of staying at the

new location. This is done by choosing a new random number,c, between 0 and 1. If

c>exp(−∆V/kBT), the particle is left at the new location. Otherwise, it is returned

to its old location. Similar displacements are taken in theyandzdirections for the first

particle, and then in all three directions for all other particles.

Each time a new set of locations is obtained (including a set obtained by returning

a particle to its old position), the value of the quantity to be averaged is calculated and

(^25) J. E. Lennard–Jones and A. F. Devonshire,Proc. Roy Soc. (London),A163, 53 (1937) andA165,
1 (1938).
(^26) N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller,J. Chem. Phys., 21 ,
1087 (1953).