Inspirals and mergers of black hole (BHs) and/or neutron star (NSs) binaries are expected to be abundant sources for ground-based gravitational-wave (GW) detectors. We assess the capabilities of Advanced LIGO and Virgo to measure component masses using inspiral waveform models including spin-precession effects using a large ensemble of GW sources {bf randomly oriented and distributed uniformly in volume. For 1000 sources this yields signal-to-noise ratios between 7 and 200}. We make quantitative predictions for how well LIGO and Virgo will distinguish between BHs and NSs and appraise the prospect of using LIGO/Virgo observations to definitively confirm, or reject, the existence of a putative mass gap between NSs ($mleq3 M_odot$) and BHs ($mgeq 5 M_odot$). We find sources with the smaller mass component satisfying $m_2 lesssim1.5 M_odot$ to be unambiguously identified as containing at least one NS, while systems with $m_2gtrsim6 M_odot$ will be confirmed binary BHs. Binary BHs with $m_2<5 M_odot$ (i.e., in the gap) cannot generically be distinguished from NSBH binaries. High-mass NSs ($2<m<3$ $M_odot$) are often consistent with low-mass BH ($m<5 M_odot$), posing a challenge for determining the maximum NS mass from LIGO/Virgo observations alone. Individual sources will seldom be measured well enough to confirm objects in the mass gap and statistical inferences drawn from the detected population will be strongly dependent on the underlying distribution. If nature happens to provide a mass distribution with the populations relatively cleanly separated in chirp mass space, as some population synthesis models suggest, then NSs and BHs are more easily distinguishable.
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