Physical Chemistry Third Edition

28.5 The Structure of Liquids 1185

The radial distribution function can be determined experimentally by neutron

diffraction, because neutrons can exhibit de Broglie wavelengths roughly equal to inter-

molecular spacings in liquids. Figure 28.16 shows the experimental radial distribution

functions of liquid and solid mercury.

EXAMPLE28.12

a.Find the speed of a neutron such that its de Broglie wavelength is 1.50× 10 −^10 m.

b.In gas kinetic theory the root-mean-square speed of gas molecules of massmis given by

vrms

√

3 kBT

m



√

3 RT

M

whereRis the ideal gas constant,kBis Boltzmann’s constant,Tis the absolute tempera-

ture, andMis the molar mass. Find the temperature such that the root-mean-square speed

of thermally equilibrated neutrons is equal to the speed of part a.

Solution

a.

v

h

mλ



6. 6261 × 10 −^34 Js

(

1. 6749 × 10 −^27 kg

)(

1. 50 × 10 −^10 m

) 2. 637 × 103 ms−^1

b.

vrms 2. 637 × 103 ms−^1 

√

3 kBT

m

T

mv^2 rms

3 kB



(

1. 6749 × 10 −^27 kg

)(

2. 637 × 103 ms−^1

) 2

3

(

1. 3807 × 10 −^23 JK−^1

) 281 K

1

01

Average

2345

r/Å

6789

2

Radial distribution function

g

(r

) 3

Liquid Hg (normalized height units)

Solid Hg (arbitrary height units)

Figure 28.16 The Radial Distribution Function of Solid and Liquid Mercury.Since the

solid has a lattice structure, the positions of neighboring atoms give narrow “blips” in the

radial distribution function. In the liquid, the disorder that is present makes the function into

a smooth curve, which shows vestiges of the crystal lattice. From D. Tabor,Gases,Liquids

andSolids, 2nd ed., Cambridge University Press, Cambridge, England, 1979, p. 197.