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Helioseismology opens new possibility to check EOS of weakly nonideal hydrogen plasmas with high precision, using reconstructed local sound velocities within 10 −4 accuracy. A comparison of different theoretical models with experiment permits to verify the existing methods of calculation bound states and continuum contribution to the second virial coefficient within the framework of physical nature.The regular way of the deduction expression for EOS is presented and generalization of the EOS for broad atomic states and two temperature non-equilibrium case is proposed. cannot be considered as a convincing proof -each step contains infinities for the pure Coulomb case.To calculate thermodynamic functions of weakly nonideal hydrogen plasmas it is convenient to use KadanoffBaym-Keldysh technique [8,9]. From [8] it is possible to obtain correction δp to the ideal gas pressure p 0 :where S a -spin value of the particle a, integration over λ parameter corresponds to integration over charge e 2 → e 2 λ, Σ > a -mass operator (Σ +− in Keldysh notations) represents decay rate of excited a-particle state with momentum p and frequency ω; G < a -kinetics Green's function (G −+ in Keldysh notations); β = T −1 , T -temperature in energy units.From (1) we obtain contribution from e − i system to thermodynamical potential in grand canonical ensembleHere µ a -chemical potential of a particle of a-kind,λ a -thermal de-Broigle wavelength,λ a =

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