= Solution
For the <tropospheric OH budget>, let $P$ be <hydroxyl radical> production per unit volume and $K$ its total first-order loss frequency. The present <hydroxyl radical> steady state has
$$
P=\frac{10^{-6}(2.5\times10^{19})}{1\ \mathrm{yr}}
=7.922\times10^5\ \mathrm{molecules\,cm^{-3}\,s^{-1}},
\qquad[\mathrm{OH}]_0=\frac P{1.3}=6.094\times10^5\ \mathrm{molecules\,cm^{-3}}.
$$
Here one year is $365.25$ days; using it consistently makes the reported lifetimes in years independent of that conversion convention. <Methane>'s chemical lifetime is $\tau_0=1/(k_2[\mathrm{OH}]_0)$, giving
$$
\boxed{\tau_0=8.254\ \mathrm{yr}.}
$$
The current combined loss frequency already includes existing hydrogen, so \b[add only the incremental hydrogen sink]:
$$
\Delta K=k_1(2.0-0.5)10^{-6}[M]=0.25125\ \mathrm{s^{-1}},
\qquad K_1=1.55125\ \mathrm{s^{-1}}.
$$
Holding production and the other losses fixed then gives $[\mathrm{OH}]_1=5.107\times10^5\ \mathrm{molecules\,cm^{-3}}$ and
$$
\boxed{\tau_1=9.849\ \mathrm{yr},\qquad\tau_1/\tau_0=1.19327.}
$$
The rapid <hydroxyl radical> adjustment time, $1/K$ of order a second, supports its <steady-state approximation> in this box model. Subsequent <methane> growth would itself change <hydroxyl radical> consumption, so the computed increase is not a fully coupled long-term atmospheric prediction. Detailed <molecular hydrogen> modelling treats changes in <methane>, <ozone> and stratospheric <water> alongside reductions in other precursor emissions.<Atmospheric composition and climate impacts of a future hydrogen economy by Nicola J. Warwick and others>[Atmospheric Chemistry and Physics 23 (2023), §3.1 and Abstract: indirect methane, ozone and water-vapor responses]{ref}
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