  The analysis of X-ray and Sunyaev-Zel\'dovich measurements of the
intracluster medium (ICM) assumes that electrons are in thermal equilibrium
with ions in the plasma. However, electron-ion equilibration timescales can be
comparable to the Hubble time in the low density galaxy cluster outskirts,
leading to differences between the electron and ion temperatures. This
temperature difference can lead to systematic biases in cluster mass estimates
and mass-observable scaling relations. To quantify the impact of
non-equilibrium electrons on the ICM profiles in cluster outskirts, we use a
high resolution cosmological simulation with a two-temperature model assuming
the Spitzer equilibration timescale for the electrons. First, we show how the
radial profile of this temperature bias depends on both the mass and mass
accretion rate of the cluster; the bias is most pronounced in the most massive
and most rapidly accreting clusters. For the most extreme case in our sample,
we find that the bias is of order 10% at half of the cluster virial radius and
increases to 40% at the edge of the cluster. We also find that gas in filaments
is less susceptible to the non-equilibrium effect, leading to azimuthal
variations at large cluster-centric radii. By analyzing mock Chandra
observations of simulated clusters, we show that such azimuthal variations can
be probed with deep X-ray observations. Finally, the mass-dependent temperature
bias introduces biases in hydrostatic mass and cluster temperature, which has
implications for cluster-based cosmological inferences. We provide a
mass-dependent model for the temperature bias profile which can be useful for
modeling the effect of electron-ion equilibration in galaxy clusters.
