  We present a 368 ks deep Chandra observation of Abell~1240, a binary merging
galaxy cluster at a redshift of 0.195 with two Brightest Cluster Galaxies
(BCGs) may have passed each other 0.3 Gyr ago. Building upon previous
investigations involving GMRT, VLA, and LOFAR data, our study focuses on two
prominent extended radio relics at the north-west (NW) and south-east (SE) of
the cluster core. By leveraging the high-resolution Chandra imaging, we have
identified two distinct surface brightness edges at $\sim$ 1 Mpc and 1.2 Mpc NW
and SE of the cluster center, respectively, coinciding with the outer edges of
both relics. Our temperature measurements hint the edges to be shock front
edges. The Mach numbers, derived from the gas density jumps, yield
$\cal{M}_{\rm SE}$ = 1.49$^{+0.22}_{-0.24}$ for the South Eastern shock and
$\cal{M}_{\rm NW}$ = 1.41$^{+0.17}_{-0.19}$ for the North Western shock. Our
estimated Mach numbers are remarkably smaller compared to those derived from
radio observations ($\cal{M}_{\rm SE}$ = 2.3 and $\cal{M}_{\rm NW}$ = 2.4),
highlighting the prevalence of a re-acceleration scenario over direct
acceleration of electrons from the thermal pool. Furthermore, we compare the
observed temperature profiles across both shocks with that of predictions from
collisional vs. collisionless models. Both shocks favor the Coulomb collisional
model, but we could not rule out a purely collisionless model due to pre-shock
temperature uncertainties.
