Virial expansion
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The virial expansion is a model of thermodynamic equations of state. It expresses the pressure P of a gas in local equilibrium as a power series of the density. This equation may be represented in terms of the compressibility factor, Z, as
Second and third virial coefficients
The second, B, and third, C, virial coefficients have been studied extensively and tabulated for many fluids for more than a century. Two of the most extensive compilations are in the books by Dymond[2][3] and the National Institute of Standards and Technology's Thermo Data Engine Database[4] and its Web Thermo Tables.[5] Tables of second and third virial coefficients of many fluids are included in these compilations.
Casting equations of the state into virial form
Most equations of state can be reformulated and cast in virial equations to evaluate and compare their implicit second and third virial coefficients. The seminal van der Waals equation of state[6] was proposed in 1873:
In the van der Waals equation, the second virial coefficient has roughly the correct behavior, as it decreases monotonically when the temperature is lowered. The third and higher virial coefficients are independent of temperature, and are not correct, especially at low temperatures.
Almost all subsequent equations of state are derived from the van der Waals equation, like those from Dieterici,[7] Berthelot,[8] Redlich-Kwong,[9] and Peng-Robinson[10] suffer from the singularity introduced by 1/(v - b).
Other equations of state, started by Beattie and Bridgeman,[11] are more closely related to virial equations, and show to be more accurate in representing behavior of fluids in both gaseous and liquid phases.[citation needed] The Beattie-Bridgeman equation of state, proposed in 1928,
can be rearranged as
More improvements were achieved by Starling[13] in 1972:
Following are plots of reduced second and third virial coefficients against reduced temperature according to Starling:[13]
The exponential terms in the last two equations correct the third virial coefficient so that the isotherms in the liquid phase can be represented correctly. The exponential term converges rapidly as ρ increases, and if only the first two terms in its Taylor expansion series are taken, , and multiplied with , the result is , which contributes a term to the third virial coefficient, and one term to the eighth virial coefficient, which can be ignored.[original research?]
After the expansion of the exponential terms, the Benedict-Webb-Rubin and Starling equations of state have this form:
Cubic virial equation of state
The three-term virial equation or a cubic virial equation of state
With this cubic virial equation, the coefficients B and C can be solved in closed form. Imposing the critical conditions:
Between the critical point and the triple point is the saturation region of fluids. In this region, the gaseous phase coexists with the liquid phase under saturation pressure , and the saturation temperature . Under the saturation pressure, the liquid phase has a molar volume of , and the gaseous phase has a molar volume of . The corresponding molar densities are and . These are the saturation properties needed to compute second and third virial coefficients.
A valid equation of state must produce an isotherm which crosses the horizontal line of at and , on .[citation needed] Under and , gas is in equilibrium with liquid. This means that the PρT isotherm has three roots at . The cubic virial equation of state at is:
See also
References
- ^ Kamerlingh Onnes H., Expression of state of gases and liquids by means of series, KNAW Proceedings, 4, 1901-1902, Amsterdam, 125-147 (1902).
- ^ Dymond J. D., Wilhoit R. C., Virial coefficients of pure gases and mixtures, Springer (2003).
- ^ Dymond J. H., Smith E. B., Virial coefficients of pure gases and mixtures. A critical compilation, Oxford University Press, 1st Edition (1969), 2nd Edition (1980).
- ^ "ThermoData Engine".
- ^ "NIST/TRC Web Thermo Tables (WTT): Critically Evaluated Thermophysical Property Data".
- ^ van der Waals J. D., On the continuity of the gaseous and liquid states (Doctoral dissertation). Universiteit Leiden (1873).
- ^ Dieterici(7), C. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69, 685 (1899).
- ^ D. Berthelot, D., in Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII (Paris: Gauthier-Villars, 1907).
- ^ Redlich, Otto; Kwong, J. N. S. On The Thermodynamics of Solutions, Chem. Rev. 44 (1): 233–244 (1949).
- ^ Peng, D. Y.; Robinson, D. B., A New Two-Constant Equation of State. Industrial and Engineering Chemistry: Fundamentals. 15: 59–64 (1976).
- ^ Beattie, J. A., and Bridgeman, O. C., A new equation of state for fluids, Proc. Am. Acad. Art Sci., 63, 229-308 (1928).
- ^ Benedict, Manson; Webb, George B.; Rubin, Louis C., An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures: I. Methane, Ethane, Propane, and n-Butane, Journal of Chemical Physics, 8 (4): 334–345 (1940).
- ^ a b Starling, Kenneth E., Fluid Properties for Light Petroleum Systems, Gulf Publishing Company, p. 270 (1973).
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