  The equations of state for an ideal generalized gas, like an ideal quantum
gas, are expressed in terms of power laws of the temperature. The reduction of
an ideal generalized gas to an ideal classical case occurs when the
characteristic empirical temperature exponents in the thermal equation of state
and in the absolute temperature coincide in contrast to the merger of an ideal
quantum gas with an ideal classical gas in the high temperature limit. A
corollary to Carnot's theorem is proved asserting that the ratio of the work
done over a cycle to the heat absorbed to increase the temperature at constant
volume is the same for all bodies at the same volume. As power means, the
energy and entropy are incomparable and a new adiabatic potential is introduced
by showing that the volume raised to a characteristic exponent is also the
integrating factor for the quantity of heat so that the second law can be based
on the property that power means are monotonically increasing functions of
their order.
