The Intensity of Spectral Lines: Absorption

In absorption spectroscopy the ratio of the transmitted light intensity to the incident light intensity at a given frequency is called transmittance, T of the sample:

T = I/I₀

According to the Lambert-Beer law, the transmitted light intensity varies with the sample length l as

I = I₀ e^{-sigma Nl},

where sigma is an absorption cross section and N is the number of molecules per volume (concentration). Adequate units are: cm² for sigma, cm^-3 for N, and and cm for l.

Another form of eq. (16) which is widely used in laboratory practice is

I = I₀ 10^{-epsilon cl},

where epsilon is the extinction coefficient and c is a molar concentration :

c = n/V = number of molecules/(N_A V) = N/N_A,

where N_A is the Avogadro number, N_A ~ 6.022*10^-23 mol^-1 . The appropriate units are mole/liter ([mol L^-1]) for c and [L mol^-1 cm^-1] for epsilon.

Each of the coefficient sigma and epsilon can be determined from experimental data

ln(I/I₀) = - epsilon cl ln(10) = -sigma Nl,

where the I, sigma and epsilon are function of the light frequency v.

In case if the exponent factor alpha = sigma, N, l in eq.(16) is small compare to unity alpha << 1 the exponential function can be expanded over alpha. Keeping in this expansion only first two terms one comes to the important for practice particular case called low optical density of the sample:

I = I₀ (1-sigma Nl). (20)

Integrating the expression in eq.(20) over the light frequency within the absorption peak, one obtains the integrated cross section

(sigma)_aver = int sigma dv = B_mn hv₀/c,

where B_mn is the Einstein absorption coefficient and v₀ is the center of the molecular absorption line.

Thus the Einstein coefficient B_mn can be directly determined from experiment.

Molecular Spectroscopy, Oleg Vasyutinskii>>