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 real numbers excess has been reciprocated. For formula_15 the sequence terminates with an exact fractional expression of formula_16, and for quadratic irrational formula_16 the sequence becomes a periodic continued fraction. In the study of classical groups, for every formula_18, the determinant gives a map from matrices over the reals to the real numbers formula_19 Restricting to invertible matrices gives a map from the general linear group to non-zero real numbers: formula_20. Restricting to matrices with a positive determinant gives the map formula_21; interpreting the image as a quotient group by the normal subgroup relation expresses the positive reals as a Lie group.