On an ideal gas related to the law of corresponding states
Using old ideas of A. I. Bachinsky and a new mathematical apparatus, the author develops a method for the construction of the equation of state for real gases consisting of one or two atoms or molecules. This new method is based on the correspondence between the thermodynamics of real nonideal gases and thermodynamics of ideal Bose gases but in a space of fractional dimension. The thermodynamic potential of the ideal Bose gas is used in the form \[ \Omega^{\mathrm{real}}(V(\gamma) V^{-1}_0- C_\gamma)[\pi^{1+\gamma} T^{2+\gamma}/\Gamma(2+ \gamma)] \int^\infty_0 \Biggl[{1\over e^{\zeta-k}-1}- {N\over e^{(\zeta-k)/N}-1}\Biggr]\,\zeta^{1+ \gamma}\,d\zeta. \] Here, \(T\) is the temperature, \(N\) is the real full number of particles in a gas, \(\zeta= \varepsilon/k_B T\), \(\varepsilon\) is the energy, \(k= \mu/k_BT\), \(\Gamma(x)\) is the Euler gamma function, \(\gamma= d/2- 1\), \(d\) is the fractional dimension, \(V_0\) is the normalization volume. Thus, the new thermodynamic variable \(d\) or more conveniently \(\gamma\), is introduced The fractional dimension \(d\) is unknown and it is unknown how the volume \(V_\gamma\) depends on \(\gamma\), whereas the character of this dependence is not known at present. Now, all thermodynamical functions (the equation of state, the compressibility factor) can be expressed using this new thermodynamic variable. In this paper, the method for obtaining the new thermodynamical variable \(\gamma\) is presented. The fractional dimension can be determined using the hypothesis that the compressibility factor of the ideal Bose gas is equal to the compressibility factor of the real gas and the isotherms of pressure of the ideal Bose gas, in units reduced to the critical one, can be equal to the reduced critical isotherms of a real gas.
scientific article; zbMATH DE number 6097042