Ideal gas except for the specification of these constants. The results of the quantum Boltzmann gas are used in a number of cases including the Sackur–Tetrode equation for the entropy of an ideal gas and the Saha ionization equation for a weakly ionized plasma. The classical thermodynamic properties of an ideal gas can be described by two equations of state: The ideal gas law is the equation of state for an ideal gas, given by: where The ideal gas law is an extension of experimentally discovered gas laws. It can also be derived from microscopic considerations. Real fluids at low density and
Positive and negative parts be expressed in terms of "f" and "f" as Also note that Using these two equations one may express the positive and negative parts as Another representation, using the Iverson bracket is One may define the positive and negative part of any function with values in a linearly ordered group. Given a measurable space ("X",Σ), an extended real-valued function "f" is measurable if and only if its positive and negative parts are. Therefore, if such a function "f" is measurable, so is its absolute value |"f"|, being the sum of two measurable functions. The converse, though, does not necessarily hold: