  We study the thermodynamic properties of impure superconductors by explicitly
taking into consideration the finiteness of electronic bandwidths within the
phonon-mediated Eliashberg formalism. For a finite electronic bandwidth, the
superconducting transition temperature, $T_c$, is suppressed by nonmagnetic
impurity scatterings. This is a consequence of a reduction in the effective
electron-phonon coupling, $\lambda_{eff}$. The reduced $\lambda_{eff}$ is
reflected in the observation that the coherence peak in $1/(T_1 T)$, where
$T_1$ is the nuclear spin-lattice relaxation time and $T$ is the temperature,
is enhanced by impurity scatterings for a finite bandwidth. Calculations are
presented for $T_c$ and $1/(T_1 T)$ as bandwidths and impurity scattering rates
are varied. Implications for doped C$_{60}$ superconductors are discussed in
connection with $T_c$ and $1/T_1$ measurements.
