  To study the relativistic thermodynamic properties of a Fermi gas in a strong
magnetic field, we construct the relativistic thermodynamic potential by the
relativistic Fermi distribution function taking into account that the motion of
particles in a plane perpendicular to the magnetic field is quantized. With
this general potential at hand, we investigate all the thermodynamic quantities
as a function of densities, temperatures and the magnetic field. We obtain a
novel set of adiabatic equations. Having the expression of the pressure and
adiabatic state equations, we determine the sound velocity for several cases
revealing a new type of sound velocity. Finally, we disclose the magnetic
cooling in the quantized electron Fermi gas, which is based on an adiabatic
magnetization in contrast to the known adiabatic demagnetization.
