virial expansion

The second, {{mvar|B}}, and third, {{mvar|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 DymondDymond 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). and the National Institute of Standards and Technology's Thermo Data Engine Database{{Cite web|url=https://trc.nist.gov/tde.html|title = ThermoData Engine}} and its Web Thermo Tables.{{Cite web|url=https://wtt-lite.nist.gov/wtt-lite/|title = NIST/TRC Web Thermo Tables (WTT): Critically Evaluated Thermophysical Property Data}} Tables of second and third virial coefficients of many fluids are included in these compilations.

File:Argon b c.jpg

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 statevan der Waals J. D., On the continuity of the gaseous and liquid states (Doctoral dissertation). Universiteit Leiden (1873). was proposed in 1873:

$$P\; =\; \backslash frac\{RT\}\{\backslash left(v-b\backslash right)\}\; -\; \backslash frac\{a\}\{v^2\}$$

where {{math|1=v = 1/ρ}} is molar volume. It can be rearranged by expanding {{math|1/(v − b)}} into a Taylor series:

$$Z\; =\; 1\; +\; \backslash left(b-\backslash frac\{a\}\{RT\}\backslash right)\backslash rho\; +\; b^2\backslash rho^2\; +\; b^3\backslash rho^3\; +\; \backslash cdots$$

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,Dieterici(7), C. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69, 685 (1899). Berthelot,D. Berthelot, D., in Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII (Paris: Gauthier-Villars, 1907). Redlich-Kwong,Redlich, Otto; Kwong, J. N. S. On The Thermodynamics of Solutions, Chem. Rev. 44 (1): 233–244 (1949). and Peng-RobinsonPeng, D. Y.; Robinson, D. B., A New Two-Constant Equation of State. Industrial and Engineering Chemistry: Fundamentals. 15: 59–64 (1976). suffer from the singularity introduced by {{math|1/(v - b)}}.

Other equations of state, started by Beattie and Bridgeman,Beattie, J. A., and Bridgeman, O. C., A new equation of state for fluids, Proc. Am. Acad. Art Sci., 63, 229-308 (1928). 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|date=September 2021}} The Beattie-Bridgeman equation of state, proposed in 1928,

$$p=\backslash frac\{RT\}\{v^2\}\backslash left(1-\backslash frac\{c\}\{vT^3\}\backslash right)(v+B)-\backslash frac\{A\}\{v^2\}$$

where

• $A\; =\; A\_0\; \backslash left(1\; -\; \backslash frac\{a\}\{v\}\; \backslash right)$
• $B\; =\; B\_0\; \backslash left(1\; -\; \backslash frac\{b\}\{v\}\; \backslash right)$

can be rearranged as

$$Z=1\; +\; \backslash left(B\_0\; -\backslash frac\{A\_0\}\{RT\}\; -\; \backslash frac\{c\}\{T^3\}\backslash right)\; \backslash rho\; -\; \backslash left(B\_0\; b-\backslash frac\{A\_0\; a\}\{RT\}\; +\; \backslash frac\{B\_0\; c\}\{T^3\}\backslash right)\; \backslash rho^2\; +\; \backslash left(\backslash frac\{B\_0\; b\; c\}\{T^3\}\backslash right)\; \backslash rho^3$$The Benedict-Webb-Rubin equation of stateBenedict, 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). of 1940 represents better isotherms below the critical temperature:

$$Z\; =\; 1\; +\; \backslash left(B\_0\; -\backslash frac\{A\_0\}\{RT\}\; -\; \backslash frac\{C\_0\}\{RT^3\}\backslash right)\; \backslash rho\; +\; \backslash left(b-\backslash frac\{a\}\{RT\}\backslash right)\; \backslash rho^2\; +\; \backslash left(\backslash frac\{\backslash alpha\; a\}\{RT\}\backslash right)\; \backslash rho^5\; +\; \backslash frac\{c\backslash rho^2\}\{RT^3\}\backslash left(1\; +\; \backslash gamma\backslash rho^2\backslash right)\backslash exp\backslash left(-\backslash gamma\backslash rho^2\backslash right)$$

More improvements were achieved by StarlingStarling, Kenneth E., Fluid Properties for Light Petroleum Systems, Gulf Publishing Company, p. 270 (1973). in 1972:

$$Z\; =\; 1\; +\; \backslash left(B\_0\; -\backslash frac\{A\_0\}\{RT\}\; -\; \backslash frac\{C\_0\}\{RT^3\}\; +\; \backslash frac\{D\_0\}\{RT^4\}\; -\; \backslash frac\{E\_0\}\{RT^5\}\backslash right)\; \backslash rho\; +\; \backslash left(b-\backslash frac\{a\}\{RT\}-\backslash frac\{d\}\{RT^2\}\backslash right)\; \backslash rho^2\; +\; \backslash alpha\backslash left(\backslash frac\{a\}\{RT\}+\backslash frac\{d\}\{RT^2\}\backslash right)\; \backslash rho^5\; +\; \backslash frac\{c\backslash rho^2\}\{RT^3\}\backslash left(1\; +\; \backslash gamma\backslash rho^2\backslash right)\backslash exp\backslash left(-\backslash gamma\backslash rho^2\backslash right)$$

Following are plots of reduced second and third virial coefficients against reduced temperature according to Starling:

File: BWRS-reduced 1.jpg

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, $1-\backslash gamma\backslash rho^2$, and multiplied with $1\; +\; \backslash gamma\backslash rho^2$, the result is $1\; -\; \backslash gamma^2\backslash rho^4$, which contributes a $c\; /\; RT^3$ term to the third virial coefficient, and one term to the eighth virial coefficient, which can be ignored.{{Original research inline|date=May 2024}}

After the expansion of the exponential terms, the Benedict-Webb-Rubin and Starling equations of state have this form:

$$Z\; =\; 1\; +\; b\backslash rho\_r\; +\; c\backslash rho\_r^2\; +\; f\backslash rho\_r^5$$

The three-term virial equation or a cubic virial equation of state

$$Z\; =\; 1+B\backslash rho+C\backslash rho^2$$

has the simplicity of the Van der Waals equation of state without its singularity at {{math|1=v = b}}. Theoretically, the second virial coefficient represents bimolecular attraction forces, and the third virial term represents the repulsive forces among three molecules in close contact.{{Citation needed|date=September 2021}}

With this cubic virial equation, the coefficients B and C can be solved in closed form. Imposing the critical conditions:

$$\backslash frac\{\backslash mathrm\{d\}P\}\{\backslash mathrm\{d\}v\}=0\; \backslash qquad\; \backslash text\{and\}\; \backslash qquad\; \backslash frac\{\backslash mathrm\{d\}^2P\}\{\backslash mathrm\{d\}v^2\}=0$$

the cubic virial equation can be solved to yield:

$$B\; =\; -v\_c\; ,$$ $$C\; =\; \backslash frac\{v\_c^2\}\{3\},$$ and $$Z\_c\; =\; \backslash frac\{P\_c\; v\_c\}\{RT\_c\}\; =\; \backslash frac\; 1\; 3.$$

$Z\_c$ is therefore 0.333, compared to 0.375 from the Van der Waals equation.

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 $P\_\backslash text\{sat\}$, and the saturation temperature $T\_\backslash text\{sat\}$. Under the saturation pressure, the liquid phase has a molar volume of $v\_\backslash text\{l\}$, and the gaseous phase has a molar volume of $v\_\backslash text\{g\}$. The corresponding molar densities are $\backslash rho\_\backslash text\{l\}$ and $\backslash rho\_\backslash text\{g\}$. 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 $P\_\backslash text\{sat\}$ at $v\_\backslash text\{l\}$ and $v\_\backslash text\{g\}$, on $T\_\backslash text\{sat\}$.{{Citation needed|date=September 2021}} Under $P\_\backslash text\{sat\}$ and $T\_\backslash text\{sat\}$, gas is in equilibrium with liquid. This means that the PρT isotherm has three roots at $P\_\backslash text\{sat\}$. The cubic virial equation of state at $T\_\backslash text\{sat\}$ is:

$$P\_\backslash text\{sat\}\; =\; RT\_\backslash text\{sat\}\; \backslash left(1\; +\; B\backslash rho\; +\; C\backslash rho^2\backslash right)\; \backslash rho$$

It can be rearranged as:

$$1\; -\; \backslash frac\{RT\_\backslash text\{sat\}\}\{P\_\backslash text\{sat\}\}\; \backslash left(1\; +\; B\backslash rho\; +\; C\backslash rho^2\backslash right)\; \backslash rho\; =\; 0$$

The factor $RT\_\backslash text\{sat\}/P\_\backslash text\{sat\}$ is the volume of saturated gas according to the ideal gas law, and can be given a unique name $v^\backslash text\{id\}$:

$$v^\backslash text\{id\}\; =\; \backslash frac\{RT\_\backslash text\{sat\}\}\{P\_\backslash text\{sat\}\}$$

In the saturation region, the cubic equation has three roots, and can be written alternatively as:

$$\backslash left(1\; -\; v\_\backslash text\{l\}\; \backslash rho\; \backslash right)\; \backslash left(1\; -\; v\_\backslash text\{m\}\; \backslash rho\; \backslash right)\; \backslash left(1\; -\; v\_\backslash text\{g\}\; \backslash rho\; \backslash right)\; =\; 0$$

which can be expanded as:

$$1\; -\; \backslash left(v\_\backslash text\{l\}\; +\; v\_\backslash text\{g\}\; +\; v\_m\backslash right)\backslash rho\; +\; \backslash left(v\_\backslash text\{l\}\; v\_\backslash text\{g\}\; +\; v\_\backslash text\{g\}\; v\_\backslash text\{m\}\; +\; v\_\backslash text\{m\}\; v\_\backslash text\{l\}\backslash right)\backslash rho^2\; -\; v\_\backslash text\{l\}\; v\_\backslash text\{g\}\; v\_\backslash text\{m\}\; \backslash rho^3\; =\; 0$$

$v\_\backslash text\{m\}$ is a volume of an unstable state between $v\_\backslash text\{l\}$ and $v\_\backslash text\{g\}$. The cubic equations are identical. Therefore, from the linear terms in these equations, $v\_m$ can be solved:

$$v\_\backslash text\{m\}\; =\; v^\backslash text\{id\}\; -\; v\_\backslash text\{l\}\; -\; v\_\backslash text\{g\}$$

From the quadratic terms, B can be solved:

$$B\; =\; -\backslash frac\{\backslash left(v\_\backslash text\{l\}\; v\_\backslash text\{g\}\; +\; v\_\backslash text\{g\}\; v\_\backslash text\{m\}\; +\; v\_\backslash text\{m\}\; v\_\backslash text\{l\}\backslash right)\}\{v^\backslash text\{id\}\}$$

And from the cubic terms, C can be solved:

$$C\; =\; \backslash frac\{v\_\backslash text\{l\}\; v\_\backslash text\{g\}\; v\_\backslash text\{m\}\}\{v^\backslash text\{id\}\}$$

Since $v\_\backslash text\{l\}$, $v\_\backslash text\{g\}$ and $P\_\backslash text\{sat\}$ have been tabulated for many fluids with $T\_\backslash text\{sat\}$ as a parameter, B and C can be computed in the saturation region of these fluids. The results are generally in agreement with those computed from Benedict-Webb-Rubin and Starling equations of state.{{Citation needed|date=September 2021}}

• Virial theorem
• Statistical mechanics
• Equation of state

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Category:Statistical mechanics