= Solution
Model the target as a perfectly absorbing sphere of reaction radius $r_0$, with a fixed target and a dilute far-field polymerase density $c_\infty$. Steady radial <diffusion> satisfies $\nabla^2c=0$, $c(r_0)=0$, $c(\infty)=c_\infty$, hence
$$
c(r)=c_\infty(1-r_0/r),\qquad
J=4\pi r_0^2D_A\left.\frac{dc}{dr}\right|_{r_0}=4\pi D_Ar_0c_\infty.
$$
Therefore, for a rate law multiplying <Number density>[number densities],
$$
\boxed{k_a=4\pi D_Ar_0,\qquad [k_a]=\mathrm{m^3\,s^{-1}}.}
$$
For the <earliest diffusive target encounter among independent searchers>, with $N_p$ independent well-mixed searchers in <volume> $V$, the approximate first-arrival rate is $N_pk_a/V$, giving $\tau_3\simeq V/(N_pk_a)$. The target's effective reaction radius includes encounter geometry and is not uniquely determined by its <Polymer contour length>[contour length].
A numerical estimate needs <temperature>, cell <volume> and a capture-geometry convention, none of which is fixed numerically in the source. State representative estimates $T=300\,\mathrm K$ and $V=1\,\mathrm{\mu m^3}$. The <Stokes–Einstein relation>[Stokes–Einstein] estimate for a sphere of radius $15\,\mathrm{nm}$ in the stipulated <viscosity> is
$$
D_A=\frac{k_BT}{6\pi\eta(15\,\mathrm{nm})}
\simeq1.46\times10^{-12}\,\mathrm{m^2\,s^{-1}}.
$$
Using the target contour scale $40\times0.34\,\mathrm{nm}=13.6\,\mathrm{nm}$ as an order-of-magnitude $r_0$ gives $k_a\simeq2.50\times10^{-19}\,\mathrm{m^3\,s^{-1}}$ and
$$
\boxed{\tau_3\simeq1.33\times10^{-2}\,\mathrm s\,
\frac{V}{1\,\mathrm{\mu m^3}}\frac{300}{N_p}
\frac{13.6\,\mathrm{nm}}{r_0}\frac{300\,\mathrm K}{T}.}
$$
This is an idealized $10^{-2}$-second-scale estimate, not a unique geometric prediction. Finite cell shape, crowded motion, accessibility and orientational restrictions can reduce the ideal capture rate.
If nonspecific binding merely sequesters searchers and their free fraction is $0.2$, the three-dimensional capture hazard is reduced fivefold: $\tau\simeq\tau_3/0.2\simeq0.067\,\mathrm s$ for those estimates. If bound molecules instead search productively by <Facilitated target search>[sliding], a free-time fraction alone cannot determine the total arrival time or its sign of change. That is the distinct <Facilitated target search>[facilitated-search] mechanism treated in <5/c>.
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