A new method to obtain the thermodynamic coefficients in an analytic and exact form for applications in radiative and convective transport in thermodynamic equilibrium for stellar atmosphere and plasma spectroscopy is developed. The resulting exact expressions are formed by sums of the degrees of ionization of the components of the system. Therefore, they are easy to calculate and are numerically stable. The method is developed initially for two elements: hydrogen and helium, with constant partition functions, to show the simplicity of the procedure. The method is very easy to generalize to any number of elements and for partition functions dependent on temperature and pressure as well as for the first negative ions. The thermodynamic coefficients derived are the adiabatic gradient, the specific heats for constant volume and constant pressure, the dilatation coefficient, and the velocity of sound in the given medium. The derivation is based on the perfect gas and Saha ionization equations. Therefore, the results are valid for the regime where these equations are valid. These results will be of value to astrophysics in stellar structure and atmospheres, in geophysics, in planetary atmospheres, and in plasma physics in spectroscopic analysis and diagnostics.