= Solution
The <free electron model> replaces the detailed periodic potential by a constant background, treating conduction <electrons> as mobile particles with approximately parabolic dispersion $E=\hbar^2k^2/(2m)$. It is useful to distinguish the classical <Drude model> from the quantum <free electron gas>. They share a simple transport picture, but differ fundamentally in occupation statistics.
With a momentum-relaxation time $\tau$, both lead in a simple isotropic one-band approximation to
$$
\boxed{\sigma=\frac{nq^2\tau}{m},\qquad R_H=-\frac1{nq}.}
$$
This explains approximately Ohmic longitudinal conduction and a carrier-density scale from the <Hall effect>. The relaxation time is phenomenological: introducing it does not itself predict scattering from defects or <phonons>. A four-probe resistance measurement separates the sample's longitudinal response from current-contact voltage drops; field and current reversals isolate the transverse signal.
The classical model incorrectly assigns roughly $3Nk_B/2$ of electronic <heat capacity>, far too large in a degenerate <metal>. <Fermi-Dirac statistics> fixes this: most occupied states cannot change occupation after a small thermal perturbation because nearby low-energy states are already filled. Only a fraction of order $k_BT/E_F$ near the <Fermi energy> participates. For $T\ll T_F$,
$$
C_e=\frac{\pi^2}{2}Nk_B\frac{T}{T_F}.
$$
The quantum model also gives a finite <Fermi velocity> even at zero <temperature> and <Pauli paramagnetism>. Under predominantly elastic scattering, its electronic heat and charge conductivities obey the <Wiedemann-Franz law>,
$$
\frac{\kappa_e}{\sigma T}=L_0,\qquad
L_0=\frac{\pi^2k_B^2}{3q^2}.
$$
This is a quantum success; classical equipartition does not give the correct small <heat capacity> and <Lorenz number> together.
Its limitations are equally instructive. Without the periodic potential it has no <band gaps>, so it cannot distinguish a filled-band insulator from a <metal> or explain <semiconductor doping>. The simplest single-electron Hall formula cannot explain positive <Hall coefficients>; band curvature and <electron holes>, or a multiband transport treatment, are needed. It neglects much Fermi-surface anisotropy and detailed optical interband <absorption>. Noninteracting <electrons> alone do not explain <ferromagnetism> or <superconductivity>, and <lattice> thermal behavior needs <phonons>. The <nearly-free electron model>, which allows reciprocal-lattice Fourier components of the potential to mix <plane waves>, provides a controlled first correction near Bragg planes.
\b[Its main success is a simple quantitative baseline for degenerate conduction <electrons>; its main failure is treating the <crystal> and interactions as too featureless.] A fit of conductivity alone is not a test of the whole model: compare <heat capacity>, Hall sign, optical response and <temperature> dependence using independently constrained parameters.
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