  The spectroscopy of electron capture neutrinos emitted from nearby
pre-supernova collapsing stars before the neutrino trapping sets in, can yield
useful information on the physical conditions and on the nuclear composition of
the core. The neutrino spectrum depends on the thermodynamic conditions of the
core, the nuclear abundances, the lepton fractions and relevant nuclear
properties. In the pre-trapping core of a core-collapse supernova, the density
ranges from $0.1 - 100~10^{10}$ g/cm$^3$ and the temperature from $0.2 - 1.5$
MeV. The nuclear abundances as well as the electron capture rates are thus
determined, among other things, by the nuclear binding energies and the free
nucleon chemical potentials. Because shell and pairing effects persist strongly
up to temperatures of $\simeq 0.5$ MeV, any equation of state (EOS) relevant to
this phase of the collapse must reproduce well the zero temperature nuclear
properties and it must show a smooth transition to the known high temperature
and high density limits. In this work we use the microscopic Relativistic Mean
Field (RMF) theory based on a Lagrangian with non-linear self-interactions of
the $\sigma$-field for the neutron-rich nuclei of interest in the $f-p$ shell
to determine nuclear chemical potentials. We compare these results with those
computed from an EOS calculated with the macroscopic liquid drop model. We also
discuss extensions to finite temperature and we incorporate nuclear lattice
effects into the microscopic calculations.
