# Nitrogen dioxide fluorescence quenching and lifetime

Nitrogen dioxide fluorescence quenching:

\begin{align} \ce{NO2 + h\nu &->[\varphi_\mathrm{Ia}] NO2^{\ast}}\tag{I}\\ \ce{NO2^{\ast} &->[k_2] NO2 + h\nu'}\tag{II}\\ \ce{NO2^{\ast} + NO2 &->[k_3] 2 NO2}\tag{III}\\ \ce{NO2^{\ast} + Xe &->[k_4] NO2 + Xe}\tag{IV}\\ \ce{NO2^{\ast} + NO2 &->[k_5] 2 NO + O2}\tag{V}\\ \end{align}

The fluorescence lifetime of $$\ce{NO2^{\ast}}$$ in the presence of all the reactions, $$\tau$$, measured at $$\pu{298 K}$$ at different concentrations (molecules per litre) of the reactants:

$$\begin{array}{crrr} \hline \text{Experiment} & \tau/\pu{μs} & [\ce{Xe}]/\pu{L-1} & [\ce{NO2}]/\pu{L-1} \\ \hline 1 & 3.38 & \pu{1.6E19} & \pu{3.2E18} \\ 2 & 1.89 & \pu{1.6E19} & \pu{6.4E18} \\ 3 & 3.64 & \pu{0.8E19} & \pu{3.2E18} \\ \hline \end{array}$$

What is the real fluorescence lifetime of $$\ce{NO2^{\ast}}?$$

Is this question valid? Isn't the lifetime vs the concentration of the quencher supposed to be linear?

• Please visit this page, this page and this one on how to format your future posts better with MathJax and Markdown. It might also help if add some more details as to where this data is taken from: is this a textbook exercise, or an experiment conducted by you? Commented May 19, 2020 at 13:04
• Thanks! it's a HW assignment by my tutor. I'm not sure if this is solvable or not. Commented May 19, 2020 at 13:14
• @andselisk why did you use \phi instead of \varphi? Commented May 19, 2020 at 13:28
• @Zenix No reason, both are equally usable to me. Probably an old habit to avoid \var prefix from using $\mathrm\LaTeX.$ If you are interested in the whole var-thing, have a look at TeX.SE. But you are right, since the source image had φ and not ɸ, I changed my edit to \varphi. Thank you for pointing this out. Commented May 19, 2020 at 13:36
• Given that the question was constructed by your tutor, I suspect the question is really about half-life (a clue is a halving of the concentration). Commented May 19, 2020 at 17:30

You are looking for the rate constant $$1/\tau^0=k_2$$. The rate equation for the $$\ce{NO2^*}$$ is (using $$N^*$$ for $$\ce{[NO2^*]}$$ and N for ground state NO2), $$\displaystyle \frac{dN^*}{dt}=-k_2N^*$$ and when quenched $$\displaystyle \frac{dN^*}{dt}=-k_2N^* -N^*((k_3+k_5)N+k_4[Xe])$$ which can both be integrated to give exponentials, i.e $$N^*=N^*_0e^{-k_2t}$$ and similarly for the other equation. Writing the rate constants as $$1/\tau^0=k_2$$ and $$1/\tau_Q=k_2+(k_3+k_5)N+k_4[Xe]$$ you can find the values after looking at the pattern of the quencher concentrations in the question.