A Rational Basis for the Thermodynamic Treatment of Real Gases and Mixtures of Real Gases

From the general laws of thermodynamics and two isothermal assumptions, one of which states that at constant temperature the energy of a pure gas approaches a function of the temperature in the following manner: $U_1=f_1\mathrm{}\left(T\right)+{\zeta }_{1}p$ and the other that at constant temperature and composition the ratio of the equilibrium pressure of a gas in a mixture to the product of the mole fraction of the gas in the mixture multiplied by the total pressure of the mixture approaches unity in the following manner: $\frac{p_{\mathrm{ei}}}{p{x}_i}=1+{\xi }_{i}p$ where ${\zeta }_1$ and ${\xi }_i$ are bounded parameters, it has been shown that the following relations hold for real gases or mixtures of real gases at infinitely low pressures: