About the Structure of a General Equation of State for Liquids and Gases
About the Structure of a General Equation of State for Liquids and Gases I.H. Umirzakov
Keywords: thermodynamics, equilibrium, equation of state, compressibility factor, virial coefficient, virial equation of state.
Abstract
It is shown that the functional form of the equation of state of [4-8] is not correct in general case.
The equation of state (the dependence of a pressure ),( vTp or compressibility factor ),( vTz on temperature T and molar volume v ) can be obtained from the partition function known [1]. However the partition function of bulk system is known only for the ideal gas and for the system of harmonic oscillators [2-3]. Therefore in order to establish the equation of state (EOS) it is necessary to use experimental ),,( Tvp -data and other thermodynamic data for real substances [4], and the functional form of the EOS based on equilibrium statistical mechanics. The functional form of the EOS of [5-8] is not correct in general case. The equation v vTURTvT vTz ∂∂⋅=∂∂ ),(),( , (1) where R is the universal gas constant, RT vvTpvTz ⋅≡ ),(),( is the compressibility factor, and ),( vTU is the internal energy, can be obtained [5-8] from thermodynamic equation [1] ),(),(),( vTpT vTpTv vTU −∂∂⋅=∂∂ . We integrate the equation (1) and obtain )),((),(),( vvTzTdv vTUTR vvTz TvT +′∂ ′∂⋅′= ∫ , (2) where )( vT and )),(( vvTz are the arbitrary functions. We have [9] ))(,(),(),( vTvFTvFTdv vTUTR v TvT −=′∂ ′∂⋅′ ∫ , where vTURTvv TvF ∂∂⋅=∂∂ ),(),( . The Eq. (2) can be presented as )(),(),( vzvThvTz += , (3) where ))(,(),(),( vTvFTvFvTh −= , )),(()( vvTzvz ≡ . It is implicitly assumed in [5-8] that in (3) ≡ vTvF , (4) ≠ vz . (5) However there is no proofs in [5-8] that the ≡ vTvF for arbitrary functions )( vT and ),( vTU , and the identity ≡ vz is not correct. If ∞<′∂ ′∂⋅′ ∫ ∞ T Tdv vTUTR v ),( , (6) then the equation (2) can be presented as )(),(),()(),(),( vzvFTvFvzTdv vTUTR vvTz T +∞−=+′∂ ′∂⋅′= ∫ ∞ , (7) where ∞→ = T vTzvz ),()( . However there is no proofs in [5-8] that the ≡∞ vF for arbitrary function ),( vTU , and the identity ≡ vz is not correct in (7). One can easily see that the inequality (6) is not valid for the EOS ),()(),( vTqvfkTvTp ⋅⋅= , (8) where ),( vTq is an arbitrary function, )(),( vvTq χ≠ for ∞<< T , where )( v χ is an arbitrary function, and = ∞→ v vTq . Particularly, )/(1)( bvvf −= , )/exp(),( vkTavTq −= , where >= consta , >= constb , for the Dieterici equation of state [10,11], and )(/][)( bvvbvbbvvvf −−++= , bb = , )/exp(),( vkTavTq −= for the Carnahan-Starling-Dieterici equation of state [12]. There is no proof in [5-8] that the EOS cannot have the form (8). The virial equation of state [1,11,13] gives ∑ ∞= − += /)(1),( n nn vTBvTz , (9) here )( TB n is the n -th virial coefficient. One can put in (3) that ∑ ∞= − += /)(1),( n nn vTBvTh and = vz . Therefore the equation of state in the form (3)-(5) is not valid. There is no proof in [5-8] that the EOS cannot have the form of the virial EOS (9). One can show that the inequality (6) is not valid for the special form of the EOS ),()(),( )( vTgvfTvTU v +⋅= λ , (10) where )( v λ , )( vf and ),( vTg are arbitrary functions, if ≥ v λ . There is no proof in [5-8] that the EOS cannot have the special form (10). There is no proof in [5-8] that the inequality (6) takes place in general case. We see from above consideration that the form (3)-(5) of the equation of state is not correct for the general case. The validity of the form of the equation of state (3)-(5) cannot be proved in the framework of the equilibrium thermodynamics. The functional form (structure) of the EOS can be established in the framework of the equilibrium statistical mechanics. REFERENCES 1. Landau L.D., Lifschitz E.M. Statistical physics. P.1. Pergamon Press. Oxford. 1980. Feynman R. P. Statistical Mechanics. W.A. Benjamin, Inc. Massachusetts. 1972. 407p. 3.
Mayer J. E., Goeppert-Mayer M. Statistical Mechanics. John Wley & Sons, New York. 1977. 544p. 4.
Tegeler Ch., Span R., Wagner W. A new equation of state for argon covering the fluid region for temperatures from the melting line to 700K at pressures up to 1000 MPa. J. Phys. Chem. Ref. Data. V. 28. N.3. PP. 779-850. 1999. 5.
Kaplun A.B., Meshalkin A.B. On the Structure of a General Equation of State for Liquids and Gases. Doklady Physics. V. 46. N 2. 2001. PP. 92–96. 6.
Kaplun A.B., Meshalkin A.B. Thermodynamic Validation of the Form of Unified Equation of State for Liquid and Gas. High Temperature. V. 41. N 3. 2003. PP. 319–326. 7.
Kaplun A.B., Meshalkin A.B. Approximate and High-Accuracy Equations of State of One-Component Normal Substances. Russian Journal of Physical Chemistry. 2006. V. 80. N 11. PP. 1868–1873. 8.
Kaplun A.B., Meshalkin A.B. Phenomenological Method for Construction of the Liquid and Gas Equation of State. J. Chem. Eng. Data. 2010. V. 55. PP. 4285–4289. 9.
Korn 10.
Dieterici C. Ann. Phys. Chem. Wiedemanns Ann. 1899. V. 69. P. 685. 11.
Hirschfelder J.O., Curtiss Ch. F. Bird R.B. Molecular theory of gases and liquids. John Wiley & Sons, Inc. London, 1954. 930p. 12.