= Solution
A conventional <Face-centered cubic lattice>[face-centered cubic] cell contains four <Chemical atom>[atoms]. With three <Conduction electron>[conduction electrons] per <Chemical atom>[atom],
$$
n=\frac{12}{(0.405\times10^{-9}\ \mathrm m)^3}=1.8064\times10^{29}\ \mathrm{m^{-3}},
\quad k_F=(3\pi^2n)^{1/3}=1.7488\times10^{10}\ \mathrm{m^{-1}}.
$$
Using the <Electron mass>[bare electron mass],
$$
\boxed{E_F=\frac{\hbar^2k_F^2}{2m_e}=11.65\ \mathrm{eV},\qquad
T_F=1.352\times10^5\ \mathrm K.}
$$
The <Drude model>[Drude] transport relation is $\rho=m_e/(ne^2\tau)$, giving
$$
\boxed{\tau=\frac{m_e}{ne^2\rho}=7.02\times10^{-15}\ \mathrm s.}
$$
For a <degenerate electron gas>, transport samples <Electron>[electrons] near the <Fermi surface>, so use $v_F=\hbar k_F/m_e=2.025\times10^6$ m/s:
$$
\boxed{\ell_F=v_F\tau=1.42\times10^{-8}\ \mathrm m=14.2\ \mathrm{nm}.}
$$
This is a <Momentum relaxation time>[transport relaxation time] and associated <Mean free path>[transport mean free path] inferred from the specified <Electrical resistivity>[resistivity]. It need not equal a single-particle quantum lifetime when scattering is strongly angle dependent. The free-electron valence count also neglects the detailed band structure of <aluminium>, as the stipulated model requires. A classical <Thermal speed>[thermal speed] is not the appropriate velocity scale for this strongly degenerate metal.
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