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Problem 4. The absorption spectrum and the $ \mathrm{S}_{1} \rightarrow \mathrm{~S}_{0} $ fluorescence spectrum Rhodamine B dissolved in ethanol (refractive index, 1.35) is given on the OMLC website: http://omlc.org/spectra/PhotochemCAD/index.html Find the value of the radiative time constant ( $ \tau_{\mathrm{rad}} $ ) associated with the $ \mathrm{S}_{1} \rightarrow \mathrm{~S}_{0} $ fluorescence. Use the Strickler-Berg equation (below) to solve this problem. You can read more about Strickler-Berg analysis in: B. Cohen, et al, Faraday Discuss., 2004, 127, 137-147. \[ \begin{aligned} 1 / \tau_{\mathrm{rad}}= & 8 \times 2303 \pi c n^{2} N_{\mathrm{a}}^{-1}\left(\left\langle\bar{v} \bar{f}^{-3}>_{\mathrm{Av}}\right)^{-1} \times \frac{\mathrm{g}_{\mathrm{s} 0}}{\mathrm{~g}_{\mathrm{s} 1}} \int \varepsilon \mathrm{~d} \ln \bar{v}\right. \\ & =2.880 \times 10^{-9} \mathrm{n}^{2}\left(<\bar{v} \bar{f}^{-3}>_{\mathrm{Av}}\right)^{-1} \times \frac{\mathrm{g}_{\mathrm{s} 0}}{\mathrm{~g}_{\mathrm{s} 1}} \int \frac{\varepsilon}{\bar{v}} \mathrm{~d} \bar{v} \end{aligned} \] \[ \left(<\bar{v}_{f}^{-3}>_{\mathrm{Av}}\right)^{-1}=\frac{\int \mathrm{I}(\bar{v}) \mathrm{d} \bar{v}}{\int(\bar{v})^{-3} \mathrm{I}(\bar{v}) \mathrm{d} \bar{v}} \]

          Problem 4.
The absorption spectrum and the \( \mathrm{S}_{1} \rightarrow \mathrm{~S}_{0} \) fluorescence spectrum Rhodamine B dissolved in ethanol (refractive index, 1.35) is given on the OMLC website: http://omlc.org/spectra/PhotochemCAD/index.html
Find the value of the radiative time constant ( \( \tau_{\mathrm{rad}} \) ) associated with the \( \mathrm{S}_{1} \rightarrow \mathrm{~S}_{0} \) fluorescence.
Use the Strickler-Berg equation (below) to solve this problem. You can read more about Strickler-Berg analysis in: B. Cohen, et al, Faraday Discuss., 2004, 127, 137-147.
\[
\begin{aligned}
1 / \tau_{\mathrm{rad}}= & 8 \times 2303 \pi c n^{2} N_{\mathrm{a}}^{-1}\left(\left\langle\bar{v} \bar{f}^{-3}>_{\mathrm{Av}}\right)^{-1} \times \frac{\mathrm{g}_{\mathrm{s} 0}}{\mathrm{~g}_{\mathrm{s} 1}} \int \varepsilon \mathrm{~d} \ln \bar{v}\right. \\
& =2.880 \times 10^{-9} \mathrm{n}^{2}\left(<\bar{v} \bar{f}^{-3}>_{\mathrm{Av}}\right)^{-1} \times \frac{\mathrm{g}_{\mathrm{s} 0}}{\mathrm{~g}_{\mathrm{s} 1}} \int \frac{\varepsilon}{\bar{v}} \mathrm{~d} \bar{v}
\end{aligned}
\]
\[
\left(<\bar{v}_{f}^{-3}>_{\mathrm{Av}}\right)^{-1}=\frac{\int \mathrm{I}(\bar{v}) \mathrm{d} \bar{v}}{\int(\bar{v})^{-3} \mathrm{I}(\bar{v}) \mathrm{d} \bar{v}}
\]

$Problem 4. The absorption spectrum and the S1→ S0 fluorescence spectrum Rhodamine B dissolved in ethanol (refractive index, 1.35) is given on the OMLC website: http://omlc.org/spectra/PhotochemCAD/index.html Find the value of the radiative time constant ( τrad ) associated with the S1→ S0 fluorescence. Use the Strickler-Berg equation (below) to solve this problem. You can read more about Strickler-Berg analysis in: B. Cohen, et al, Faraday Discuss., 2004, 127, 137-147. 1 / τrad= 8 × 2303 π c n^2 Na^-1(⟨v̅f̅^-3>Av)^-1×(gs 0)/( gs 1)∫ε dlnv̅. =2.880 × 10^-9n^2(<v̅f̅^-3>Av)^-1×(gs 0)/( gs 1)∫(ε)/(v̅) dv̅ (<v̅f^-3>Av)^-1=(∫I(v̅) dv̅)/(∫(v̅)^-3I(v̅) dv̅)$

Added by Keerthy P.

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