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:

$$T = \frac{I}{I_0}$$ (15)

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

$$I = I_0 e^{-\sigma\, N\, l},$$ (16)

where $\sigma$ is an absorption cross section and $N$ is the number of molecules per volume (concentration). Adequate units are: $cm^2$ for $\sigma$, $cm^{-3}$ for $N$, and and $cm$ for $l$.

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

$$I = I_0 10^{-\epsilon\, c\, l},$$ (17)

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

$$c = \frac{n}{V} = \frac{\mbox{number of molecules}}{N_A V}= N/N_A,$$ (18)

where $N_A$ is the Avogadro number, $N_A \approx 6.022\,10^{-23} mol^{-1}$. 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

$$\mbox{ln}(\frac{I}{I_0}) = - \epsilon\, c\, l\, \mbox{ln(10)} = -\sigma\, N\, l,$$ (19)

where the $I$, $\sigma$ and $\epsilon$ are function of the light frequency $\nu$.

In case if the exponent factor $\alpha = \sigma\, N\, l$ in eq.(16) is small compare to unity $\alpha \ll 1$ 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:

$$I = I_0 (1-\sigma\, N\, l).$$ (20)

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

$$<\sigma> = \int \sigma d\nu = B_{mn}\frac{h\nu_0}{c},$$ (21)

where $B_{mn}$ is the Einstein absorption coefficient and $\nu_0$ is the center of the molecular absorption line.

Thus the Einstein coefficient $B_{mn}$ can be directly determined from experiment.

Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.