Physical Chemistry Third Edition

23.1 Emission/Absorption Spectroscopy and Energy Levels 957

time is proportional to the intensity of light and to the concentration of the absorbing

substance, so the change inIfrom one side of the thin slab to the other is

−dIk(λ)Icdx (23.1-12)

wherecis the concentration of the absorbing substance. The proportionality factork(λ)

depends on the identity of the absorbing substance and on the identity of the solvent. It

is a function of wavelength, but if the Beer–Lambert law is obeyed, it does not depend

on the concentration.

Equation (23.1-12) is a differential equation that can be solved by separation of

variables. Division byIgives

−

dI

I

k(λ)cdx (23.1-13)

We carry out a definite integration, lettingx0 be the front of the cell andxbbe

the back of the cell. Assuming that the concentration is uniform,

−

I∫(b)

I(0)

dI

I

k(λ)c

∫b

0

dx (23.1-14a)

−ln

(

I(λ,b)

I(λ,0)

)

ln

(

I(λ,0)

I(λ,b)

)

k(λ)cb (23.1-14b)

TheabsorbanceA(λ) (formerly called theoptical density) is defined as the common

logarithm of the same ratio as in the second natural logarithm in Eq. (23.1-14b):

A(λ)log 10

(

I(λ,0)

I(λ,b)

)

log 10

(

100%

T

)(

definition

of absorbance

)

(23.1-14c)

The Beer–Lambert law is

A(λ)a(λ)bc (23.1-15)

wherea(λ)istheabsorptivity(formerly called theextinction coefficient):

a(λ)

k(λ)

ln(10)



k(λ)

2. 302585

(23.1-16)

The absorptivity depends on the wavelength of light as well as on the identity of the

absorbing substance and the identity of the solvent. If the concentration is measured

in mol L−^1 , the absorptivity is called themolar absorptivity. The Beer–Lambert law is

well obeyed by many substances at low to moderate concentrations. Deviations occur at

higher concentrations, corresponding to an absorptivity that depends on concentration.

These deviations can be caused by specific chemical effects such as association of the

molecules of the substance.

600

5000 10,000

Reciprocal wavelength/cm^21

Absorptivity/L mol

21

cm

21

15,000 20,000 25,000

500

400

300

CoCI

200

100

0

2

4

Figure 23.5 The Absorption Spectrum
of 0.001 mol L–1CoCl2– 4 in 10 mol L–1
HCl Solution.This continuous spectrum
arises because of the broadening of the
spectral lines due to solvent interaction.
From Russell S. Drago,Physical Methods
inChemistry, W. B. Saunders Co., Philadel-
phia, 1977, p. 392.

Figure 23.5 shows an absorption spectrum for a sample in liquid solution, with

the absorptivity plotted as a function 1/λ, measured in cm−^1. This spectrum shows

absorption over broad ranges of wavelength, not just at a few wavelengths. This is

common for absorbing substances dissolved in liquid solvents. The spectral lines are