The Beer-Lambert law states that the absorbance of a sample at a wavenumber
\(\$ is proportional to the molar concentration \)[1]\( of the absorbing species
\)\mathrm{J}\( and to the length 1 of the sample (eqn \)13.4)\(. In this problem
you will show that the intensity of fluorescence emission from a sample of
\)\mathrm{J}\( is also proportional to \)[\mathrm{J}]\( and \)l\(. Consider a sample
of \)\mathrm{J}\( that is illuminated with a beam of intensity \){ }^{I_{0}(0)}\(
at the wavenumber \){ }^{0 .}\( Before fluorescence can occur, a fraction of
\)\mathrm{I}_{0}\left({ }^{0)}\right.\( must be absorbed and an intensity \){
}^{\prime(0)}\( 'will be transmitted. However, not all of the absorbed
intensity is emitted and the intensity of fluorescence depends on the
fluorescence quantum yield, \)\varphi_{\mathrm{f}}\(, the efficiency of photon
emission. The fluorescence quantum yield ranges from 0 to 1 and is
proportional to the ratio of the integral of the fluorescence spectrum over
the integrated absorption coefficient. Because of a Stokes shift of magnitude
\)\Delta \mathrm{v}_{\text {Slokes }}\(, fluorescence occurs at a wavenumber
\)\mathrm{v}_{\mathrm{f}}\(, with \)\mathrm{v}_{\mathrm{f}}+\Delta
\mathrm{v}_{\text {stokes }}=\mathrm{v} .\( It follows that the fluorescence
intensity at \)\mathrm{v}_{\mathrm{f}},
\mathrm{I}_{\mathrm{f}}\left(\mathrm{v}_{\mathrm{f}}\right)\(, is proportional
to \)\varphi_{\mathrm{f}}\( and to the intensity of exciting radiation that is
absorbed by \)\mathrm{J},
\mathrm{I}_{\mathrm{abs}}(\mathrm{v})=\mathrm{I}_{0}(\mathrm{v})-\mathrm{I}(\mathrm{v})\(.
(a) Use the Beer-Lambert law to express
\)\mathrm{I}_{\mathrm{abs}}(\mathrm{v})\( in terms of
\)\mathrm{I}_{0}(\mathrm{v}),[\mathrm{J}], \mathrm{I}\(, and
\)\varepsilon(\mathrm{v})\(, the molar absorption coefficient of \)\mathrm{J}\( at
\)\mathrm{V}\(. (b) Use your result from part (a) to show that
\)\left.\mathrm{I}_{\mathrm{f}}\left(\mathrm{v}_{\mathrm{f}}\right) \infty
\mathrm{I}_{0}(\mathrm{v}) \varepsilon(\mathrm{v}) \varphi_{\mathrm{f}}
\mathrm{f}\right] \mathrm{I}$.