  We calculate and discuss the effects on the thermodynamic properties of a 3D
Bose gas caused by a gap $\Delta$ in the energy of the particles constituting
the gas that without the gap behaves like an ideal Bose gas. Explicit formulae
with arbitrary $\Delta$ values are discussed for: the critical temperature
which increases as the gap grows; the condensate fraction; the internal energy;
and the constant-volume specific heat found to possess a jump-discontinuity for
any $\Delta$ different from zero. Three dimensional infinite ideal Bose gas
results are recovered when we the energy gap goes to zero.
