\(\mathrm{f}_{\mathrm{4}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{A}_{\mathrm{4}}\mathrm{+}\mathrm{B}_{\mathrm{4}}\mathrm{T}\)(18)
Figure 1 Pressure-Volume diagram12
Due to the introduction of B 4, a new initial value condition should be introduced. Hou proposed a new initial value condition possessing a universal significance. Based on the thermodynamic equilibrium that the Gibbs molar free enthalpy of pure substances is equal at vapor-liquid equilibrium under a constant temperature and pressure, the new initial value condition finally can be deduced as12:
\(\int_{\mathrm{V}_{\mathrm{l}}}^{\mathrm{V}_{\mathrm{v}}}\mathrm{P}\mathrm{\text{dV}}\mathrm{=}\mathrm{P}_{\mathrm{o}}\left(\mathrm{V}_{\mathrm{v}}\mathrm{-}\mathrm{V}_{\mathrm{l}}\right)\)(19)
The establishment of the novel M-H EOS
After rigorous experiment calculations and a large number of data analysis about substances in liquid-phase state and on the consideration of the simplicity of calculation, the novel M-H EOS is put forward as follows:
\(\mathrm{P=}\)\(\sum_{\mathrm{i}\mathrm{=1}}^{\mathrm{5}}\frac{\mathrm{f}_{\mathrm{i}}\left(\mathrm{T}\right)}{\left[\left(\mathrm{V-b}\right)\mathrm{h}\right]^{\mathrm{i}}}\)(20)
where\({\mathrm{h=}\ \left[\frac{\mathrm{\ln}\left(\mathrm{1}\mathrm{+}\mathrm{Z}_{\mathrm{c}}\right)}{\mathrm{Z}_{\mathrm{c}}}\right]}^{\mathrm{Z}_{\mathrm{c}}}\)(21)
The revision factor h , which is the function of Z c, is mainly proposed to correct the molar volume in liquid-phase state. The structure ofh is mainly established on the consideration that the derivation of the unknown characteristic constants is closely related to the property of critical point, which can be appropriately conveyed by a function of Z c.
After analyzing the solving method of unknown characteristic constants in Hou’ M-H EOS and the novel M-H EOS, we obtain the solving method of the unknown characteristic constants in the novel M-H EOS, which has a great improvement compared with Hou’ method. Refer to the appendixes for the details.
DISCUSSION ABOUT THE VERIFICATION AND APPLICATION OF THE NOVEL M-H EOS
The analysis of the derivation about the unknown c haracteristic constants
The solving details of the unknown characteristic constants in the novel M-H EOS are presented in the Appendix Ⅰ and Appendix II.
Through the analysis of the detailed derivation of the novel EOS amended byh , all the initial value conditions and the simplifiedfi (T ) of the novel M-H EOS are the same as those in Hou’s modified M-H EOS12. Meanwhile,h is a constant for a given substance, it avoids the difficulty of the discussion about variables. However, after analyzing the derivation process of the novel M-H EOS, we found a defect that over-revision about some characteristic constants maybe exist when the state is beyond liquid-phase state because the space between molecules will be obviously affected by temperature and pressure and the influence may be amplified by h because the solutions of all the characteristic constants are affected by h in some degree, although h is mainly designed to revise the molar volume in liquid-phase state.
In the novel M-H EOS, only a minimum amount of information is necessary to characterize a given substance. In this work, the novel M-H EOS has been applied to six representative substances: argon, methane, nitrogen, propane, benzene and water to verify its generality and calculation precision in liquid-phase state. The physical constants of these substances are listed in Table 1 except\(\frac{\mathrm{R}\mathrm{=\ 82.055}\left(\mathrm{\text{atm}}\mathrm{.}\mathrm{\text{cm}}^{\mathrm{3}}\right)}{\left(\mathrm{\text{K.mol}}\right)}\)and its calculated characteristic constants are listed in Table2 37-40.
Table 1 The physical constants of given substances