  The Rare-RI Ring (R3) is a recently commissioned cyclotron-like storage ring
mass spectrometer dedicated to mass measurements of exotic nuclei far from
stability at Radioactive Isotope Beam Factory (RIBF) in RIKEN. The first
application of mass measurement using the R3 mass spectrometer at RIBF is
reported. Rare isotopes produced at RIBF, $^{127}$Sn, $^{126}$In, $^{125}$Cd,
$^{124}$Ag, $^{123}$Pd, were injected in R3. Masses of $^{126}$In, $^{125}$Cd,
and $^{123}$Pd were measured whereby the mass uncertainty of $^{123}$Pd was
improved. This is the first reported measurement with a new storage ring mass
spectrometery technique realized at a heavy-ion cyclotron and employing
individual injection of the pre-identified rare nuclei. The latter is essential
for the future mass measurements of the rarest isotopes produced at RIBF. The
impact of the new $^{123}$Pd result on the solar $r$-process abundances in a
neutron star merger event is investigated by performing reaction network
calculations of 20 trajectories with varying electron fraction $Y_e$. It is
found that the neutron capture cross section on $^{123}$Pd increases by a
factor of 2.2 and $\beta$-delayed neutron emission probability,
$P_\mathrm{1n}$, of $^{123}$Rh increases by 14\%. The neutron capture cross
section on $^{122}$Pd decreases by a factor of 2.6 leading to pileup of
material at $A=122$, thus reproducing the trend of the solar $r$-process
abundances. The trend of the two-neutron separation energies (S$_\mathrm{2n}$)
was investigated for the Pd isotopic chain. The new mass measurement with
improved uncertainty excludes large changes of the S$_\mathrm{2n}$ value at
$N=77$. Such large increase of the S$_\mathrm{2n}$ values before $N=82$ was
proposed as an alternative to the quenching of the $N=82$ shell gap to
reproduce $r$-process abundances in the mass region of $A=112-124$.
