= Solution
\b[The estimate is $\boxed{t_{10\%}=1.60\times10^{-5}\,\mathrm s\approx16\,\mathrm{\mu s}}$.] During the initial buildup, assume instantaneous mixing, initially zero ONOO, constant temperature, and negligible depletion of NO and oxygen. Their concentrations can then be held fixed while solving for $X=[\mathrm{ONOO}]$:
$$
\frac{dX}{dt}=f-\lambda X,\quad
f=5.67\times10^{-10}\,\mathrm{M\,s^{-1}},\quad
\lambda=6600\,\mathrm{s^{-1}}.
$$
The initial-value solution is $X(t)=X_{\rm ss}(1-e^{-\lambda t})$. Setting $X=0.1X_{\rm ss}$ gives
$$
t=-\frac{\ln(0.9)}{6600}=1.596\times10^{-5}\,\mathrm s.
$$
A constant-initial-rate estimate gives $0.1X_{\rm ss}/f=1.52\times10^{-5}\,\mathrm s$, consistent to about 5%. The intermediate concentration and total reaction progress over this interval are tiny relative to $10^{-7}\,\mathrm M$ NO, supporting the negligible-depletion approximation. The actual onset could be limited by mixing or instrumental response; a large vessel does not itself guarantee homogeneous mixing in microseconds.
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