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Thermodynamics of an ideal generalized gas: I Thermodynamic laws

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.