  In this paper, we focus on the limb-brightened, prolate spheroidal cavity of
the radio galaxy Cygnus A, as revealed by the Chandra X-ray Observatory. We use
the shock heated, thermal intracluster medium around the expanding cavity to
infer the properties of the radio synchrotron-emitting gas inside the cavity.
The gas along the N and S edges of the cavity is found to have an average
temperature of 6.0 keV, which is hotter than the temperature (4.6 keV) of the
adjacent intracluster gas. It is proposed that this hotter gas is intracluster
gas shocked by the expanding cavity. The shock is thus inferred to be weak
(Mach number 1.3, a value also inferred from the density jump at the cavity
edge) and its velocity 1,460 km/s. The total kinetic power of the expansion is
found to be 1.2 x 10^{46} erg/s, which is larger than both the total radio
power and the power emitted by the entire intracluster medium in the 2 -- 10
keV band. It appears that most of the power of the jets in Cygnus A is
currently going into heating the intracluster medium. From the derived pressure
inside the cavity, there is no conclusive evidence for a component contributing
pressure additional to the magnetic fields and relativistic particles
responsible for the synchrotron radio emission. Further, the ratio of energy
densities in positive to negative cosmic rays in Cygnus A is between 1 and 100
(the value in our Galaxy).
