The Chemistry Maths Book, Second Edition

24 Chapter 1Numbers, variables, and units

EXAMPLES 1.18Molecular properties: mass, length and moment of inertia

(i)mass. Atomic and molecular masses are often given as relative masses:A

r

for an

atom andM

r

for a molecule, on a scale on whichA

r

(

12

C) 1 = 112. On this scale,A

r

(

1

H) 1 =

1.0078andA

r

(

16

O) 1 = 1 15.9948. The corresponding relative molar mass of water is

M

r

(

1

H

2

16

O) 1 = 121 × 1 A

r

(

1

H) 1 + 1 A

r

(

16

O) 1 = 1 18.0105,

the molar mass is

M(

1

H

2

16

O) 1 = 1 18.0105 g mol

− 1

1 = 1 0.01801 kg mol

− 1

,

and the mass of the individual molecule is

m(

1

H

2

16

O) 1 = 1 M

r

(

1

H

2

16

O) 1 × 1 u 1 = 1 2.9907 1 × 110

− 26

kg

(ii)length. The bond length of the oxygen molecule is R

e

1 = 1 1.2075 1 × 110

− 10

m, and

molecular dimensions are usually quoted in appropriate units such as the picometre

pm 1 = 110

− 12

m or the nanometre nm 1 = 110

− 9

min spectroscopy, and the Ångström

Å 1 = 110

− 10

mor the Bohr radiusa

0

1 = 1 0.529177 1 × 110

− 10

m 1 = 1 0.529177 Åin theoretical

chemistry. Thus, for O

2

,R

e

1 = 1 1.2075 Å 1 = 1 120.75 pm.

(iii)reduced mass and moment of inertia. The moment of inertia of a system of two

masses,m

A

andm

B

, separated by distanceRisI 1 = 1 μR

2

, where μis the reduced

mass, given by

Relative atomic masses can be used to calculate the reduced mass of a diatomic

molecule. Thus for CO,A

r

(

12

C) 1 = 112 andA

r

(

16

O) 1 = 1 15.9948, and these are the atomic

masses in units of the unified atomic mass unit u. Then

=6.8562 1 × 1 1.66054 1 × 110

− 27

kg 1 = 1 1.1385 1 × 110

− 26

kg

The bond length of CO is 112.81 1 pm 1 = 1 1.1281 1 × 110

− 10

1 m, so that the moment of

inertia of the molecule is

I 1 = 1 μR

2

1 = 1 (1.1385 1 × 110

− 26

kg) 1 × 1 (1.1281 1 × 110

− 10

m)

2

= 1 1.4489 1 × 110

− 46

kg m

2

The reduced mass and moment of inertia are of importance in discussions of

vibrational and rotational properties of molecules.

0 Exercises 104, 105

μ(

.

.

12 16

12 15 9948

27 9948

CO)

u

u

2

=

×





























=6 8562. u

11 1

μ

=+ μ=

mm +

mm

mm

AB

AB

AB

,