The Chemistry Maths Book, Second Edition

1.8 Units 25

EXAMPLES 1.19Molecular properties: wavelength, frequency, and energy

The wavelength λand frequency νof electromagnetic radiation are related to the

speed of light by

c 1 = 1 λν (1.14)

(see Example 3.7), wherec 1 = 1 2.99792 1 × 110

8

1 m 1 s

− 1

1 ≈ 131 × 110

8

1 m 1 s

− 1

. The energy of

a photon is related to the frequency of its associated wave via Planck’s constant

h 1 = 1 6.62608 1 × 110

− 34

Js:

E 1 = 1 hν (1.15)

In a spectroscopic observation of the transition between two states of an atom or

molecule, the frequency of the radiation emitted or absorbed is given byhν 1 =|∆E|,

where∆E 1 = 1 E

2

1 − 1 E

1

is the energy of transition between states with energies E

1

and E

2

.

Different spectroscopic techniques are used to study the properties of atoms and

molecules in different regions of the electromagnetic spectrum, and different units

are used to report the characteristics of the radiation in the different regions. The

values of frequency and wavelength are usually recorded in multiples of the SI units of

hertz (Hz 1 = 1 s

− 1

) and metre (m), respectively, but a variety of units is used for energy.

For example, the wavelength of one of the pair of yellow D lines in the electronic

spectrum of the sodium atom isλ 1 = 1 589.76 nm 1 = 1 5.8976 1 × 110

− 7

m. By equation (1.14),

this corresponds to frequency

and by equation (1.15), the corresponding energy of transition is

∆E 1 = 1 hν 1 = 1 (6.62608 1 × 110

− 34

J s) 1 × 1 (5.0833 1 × 110

14

s

− 1

) 1 = 1 3.368 1 × 110

− 19

J

Energies are often quoted in units of the electron volt, eV, or as molar energies in

units ofkJ mol

− 1

. The value of eV is the product of the protonic charge e(see Table 1.4)

and the SI unit of electric potentialV 1 = 1 J C

− 1

(Table 1.2): eV 1 = 1 1.60218 1 × 110

− 19

J. The

corresponding molar energy is

eV 1 × 1 N

A

1 = 1 (1.60218 1 × 110

− 19

J) 1 × 1 ( 6.02214 1 × 110

23

mol

− 1

)

= 1 96.486 kJ mol

− 1

(1.16)

whereN

A

is Avogadro’s constant. For the sodium example,

∆E 1 = 1 3.368 1 × 110

− 19

J 1 = 1 2.102 eV 1 ≡ 1 202.8 kJ mol

− 1

Very often, the characteristics of the radiation are given in terms of the wavenumber

(1.17)



ν

λ

ν

===

1

c

E

hc

∆

ν

λ

==

×

×

=×

−

−

c 2 99792 10

5 8975 10

5 0833

81

7

.

.

.

m s

m

110

14 1

s

−