Nitrogen dioxide fluorescence quenching and lifetime Nitrogen dioxide fluorescence quenching:

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

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:

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

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?
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.