  We present the X-ray and optical properties of the galaxy groups selected in
the Chandra X-Bo\"otes survey. Our final sample comprises 32 systems at
\textbf{$z<1.75$}, with 14 below $z = 0.35$. For these 14 systems we estimate
velocity dispersions ($\sigma_{gr}$) and perform a virial analysis to obtain
the radii ($R_{200}$ and $R_{500}$) and total masses ($M_{200}$ and $M_{500}$)
for groups with at least five galaxy members. We use the Chandra X-ray
observations to derive the X-ray luminosity ($L_X$). We examine the performance
of the group properties $\sigma_{gr}$, $L_{opt}$ and $L_X$, as proxies for the
group mass. Understanding how well these observables measure the total mass is
important to estimate how precisely the cluster/group mass function is
determined. Exploring the scaling relations built with the X-Bo\"otes sample
and comparing these with samples from the literature, we find a break in the
$L_X$-$M_{500}$ relation at approximately $M_{500} = 5\times10^{13}$ M$_\odot$
(for $M_{500} > 5\times10^{13}$ M$_\odot$, $M_{500} \propto L_X^{0.61\pm0.02}$,
while for $M_{500} \leq 5\times10^{13}$ M$_\odot$, $M_{500} \propto
L_X^{0.44\pm0.05}$). Thus, the mass-luminosity relation for galaxy groups
cannot be described by the same power law as galaxy clusters. A possible
explanation for this break is the dynamical friction, tidal interactions and
projection effects which reduce the velocity dispersion values of the galaxy
groups. By extending the cluster luminosity function to the group regime, we
predict the number of groups that new X-ray surveys, particularly eROSITA, will
detect. Based on our cluster/group luminosity function estimates, eROSITA will
identify $\sim$1800 groups ($L_X = 10^{41}-10^{43}$ ergs s$^{-1}$) within a
distance of 200 Mpc. Since groups lie in large scale filaments, this group
sample will map the large scale structure of the local universe.
