page_title: Virial expansion
section_title: Gas-liquid-solid equilibrium
hierarchical_section_title: Virial expansion / Gas-liquid-solid equilibrium
caption_attribution_description: English: PVT isotherm of argon
context_page_description: The classical virial expansion expresses the pressure of a many-particle system in equilibrium as a power series in the number density:
  Here the quantity is the compressibility factor. This is the virial equation of state, the most general function relating PρT properties of fluids, first proposed by Kamerlingh Onnes. The compressibility factor is a dimensionless quantity, indicating how much a real fluid deviates from an ideal gas. A is the first virial coefficient, which has a constant value of 1 and it makes the statement that at low molar density, all fluids behave like ideal gases. Virial coefficients B, C, D, etc., are temperature dependent, and are generally presented as Taylor series in terms of 1/T.
context_section_description: The cubic virial equation of state accurately represents the gas-liquid equilibrium of most substance from the critical point down to the triple point, where solid phase starts to appear. It is a simple matter to extend it to account for the gas-liquid- solid equilibrium:
  In this virial equation, the first term  represents the pressure generated by kinetic energy of molecules. The second term  represents long range bimolecular attraction, and the third term  represents short range tri-molecular repulsion. The second term pulls the PVT isotherm down as volume is reduced, while the third term pushes the isotherm up. When the temperature is below critical point, the PVT isotherm thus has an S shaped bent which allows a liquid phase to coexist with the prevalent gaseous phase.
  Now, if we had a  term to pull the PVT isotherm down in the liquid phase, and a  terms to push it back up, a solid phase could be created, as these two terms producing another S shaped bend between liquid and solid. It was demonstrated  that such an S shaped bend could be synthesized using a -function like Lorentzian function over a van der Waals equation of state. Such an equation of state was ugly, and very difficult to manipulate mathematically. A virial equation shown above is much cleaner and easier to handle.
  Argon is used to evaluate realistically this extended virial equation for gas-liquid-solid equilibrium. Data will be analyzed in the reduced forms. All PVT variables are scaled by their respective critical values. It is expect from the principle of corresponding states that the results would apply to other well behaved fluids. The relevant data of argon are summarized in the following table:
  When the variables P, V, and T are replaced by their reduced equivalents, , , and , the virial equation takes the following form:
  where , , , , and . We will be concerned mostly with condition at the triple point of argon, where b = 3.424 and c = 1.152 from an earlier study.
   must be slightly larger than the volume of solid argon, 0.33, and  must be between the volumes of liquid and solid argon. Initially,  is set to the volume of solid, to produce the last sharply rising edge of the isotherm where solid phase appears at very low volume. The exponential n must be then determined, so that the valley in the n-2n potential must fit between the volumes of solid (0.33) and liquid (0.378). After the exponential n is determined, the value of  can be adjusted to satisfy the Gibbs Rule, which requires that the Gibbs free energy of liquid phase and that of solid phase must be equal under the triple point temperature and pressure.
  To produce a solid phase in argon, the exponential value n must be very large, larger than 20; otherwise, the PVT isotherm would not bend to an S-shape between liquid and solid. The best estimation is that n = 30, , , and . The isotherm is shown in the right figure, in which three virial terms are plotted separately for clarity:
  In this figure,  represents the sum of the first three virial terms, of the cubic virial equation, and it shows the behavior of argon in its gaseous and liquid phases.  represents the potential contributed from the  term, and  represents the contributions from the  term. When n < 30,  would interfere with  and lower significantly the volume of liquid.
  It was surprising that n has to be greater than 20 to produce a solid phase. It was well known that the Lenard-Jones 6-12 potential was used to compute the second virial coefficient from quantum mechanical principles. Quantum mechanical reasoning relates the second virial coefficient to bimolecular attraction, and the third coefficients to tri-molecular repulsion, etc. In the liquid phase of argon, one atom is surrounded by 12 nearest neighbors, and up to 32 next-to-nearest neighbors. In the solid phase, all atoms are locked in place, and the number of interacting neighbors is infinite. Therefore, n = 30, or even greater, is entirely reasonable, though surprising.
  For the first time in the history of thermodynamics, we have a virial equation of state to describe quan
caption: PVT isotherm of argon with n=30.
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