= Solution
For a cell of length $\ell$ aligned with the capillary, the steady linear <gradient> produces the front-to-back concentration difference
$$
\Delta c=\ell\left|\frac{dc}{dx}\right|=\frac{c_0\ell}{L}.
$$
At the given initial length, $L=5\times10^{-3}\ \mathrm m$, this is $100(20\times10^{-6})/(5\times10^{-3})=0.4\ \mathrm{nM}$, above the stated threshold. Requiring $\Delta c\geq0.1\ \mathrm{nM}$ gives
$$
L\leq\frac{(100\ \mathrm{nM})(20\times10^{-6}\ \mathrm m)}{0.1\ \mathrm{nM}},\qquad\boxed{L_{\max}=2.0\times10^{-2}\ \mathrm m=2.0\ \mathrm{cm}}.
$$
The <diffusion coefficient> does not enter this steady-gradient threshold. It does constrain experimental waiting time: for fixed reservoir concentrations, the slowest decaying mode of the <diffusion equation> has timescale $L^2/(\pi^2D)$. With the supplied $D$, this is about $7.04$ hours at $0.5$ cm and $4.69$ days at $2$ cm; several such times may be required for an accurate <steady state>. Thus the maximum from the detection threshold is usable only if the reservoirs and cells remain suitable while the profile develops.
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