On the thermodynamic derivation of the Saha equation modified to a two-temperature plasma

So far in the literature there mainly exist two quite different forms of the mass action law or the Saha equation modified to a two-temperature plasma. It has been shown in the present paper that the modified Saha equation deduced from the minimum Gibbs or Helmholtz free-energy principle is incorrect. The minimum Gibbs or Helmholtz free-energy principle, which is valid for a single-temperature system as a substitute of the maximum entropy principle in an isolated system, cannot be extended to the two-temperature plasma. A more rigorous thermodynamic derivation of the Saha equation modified to a two-temperature plasma system is presented based on the basic thermodynamic principle that the entropy of an isolated system will assume its maximum at the equilibrium state. It is pointed out that one has to pay close attention to the chemical potential expressions for the case of the two-temperature plasma, since any negligence may cause mistakes in the derivation of the two-temperature Saha equation.