The virial expansion is a model of thermodynamic
equations of state
In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most mod ...
. It expresses the
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
of a gas in local
equilibrium
Equilibrium may refer to:
Film and television
* ''Equilibrium'' (film), a 2002 science fiction film
* '' The Story of Three Loves'', also known as ''Equilibrium'', a 1953 romantic anthology film
* "Equilibrium" (''seaQuest 2032'')
* ''Equilibr ...
as a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
of the
density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
. This equation may be represented in terms of the
compressibility factor
In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas ...
, , as
This equation was first proposed by
Kamerlingh Onnes.
[Kamerlingh Onnes, H.]
"Expression of the equation of state of gases and liquids by means of series"
''KNAW, Proceedings'', 4, 1901-1902, Amsterdam, 125-147 (1902). The terms , , and represent the
virial coefficients. The leading coefficient is defined as the constant value of 1, which ensures that the equation reduces to the
ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
expression as the gas density approaches zero.
Second and third virial coefficients
The second, , and third, , 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 and the
National Institute of Standards and Technology
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into Outline of p ...
's Thermo Data Engine Database and its Web Thermo Tables. 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
The van der Waals equation is a mathematical formula that describes the behavior of real gases. It is an equation of state that relates the pressure, volume, Avogadro's law, number of molecules, and temperature in a fluid. The equation modifies ...
of state was proposed in 1873:
where is
molar volume
In chemistry and related fields, the molar volume, symbol ''V''m, or \tilde V of a substance is the ratio of the volume (''V'') occupied by a substance to the amount of substance (''n''), usually at a given temperature and pressure. It is also eq ...
. It can be rearranged by expanding into a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
:
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, Berthelot, Redlich-Kwong, and Peng-Robinson suffer from the singularity introduced by .
Other equations of state, started by Beattie and Bridgeman, are more closely related to virial equations, and show to be more accurate in representing behavior of fluids in both gaseous and liquid phases. The Beattie-Bridgeman equation of state, proposed in 1928,
where
*
*
can be rearranged as
The Benedict-Webb-Rubin equation of state
[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).] of 1940 represents better isotherms below the critical temperature:
More improvements were achieved by Starling
[Starling, Kenneth E., Fluid Properties for Light Petroleum Systems, Gulf Publishing Company, p. 270 (1973).] in 1972:
Following are plots of reduced second and third virial coefficients against reduced temperature according to Starling:
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.
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
has the simplicity of the Van der Waals equation of state without its singularity at . Theoretically, the second virial coefficient represents bimolecular attraction forces, and the third virial term represents the repulsive forces among three molecules in close contact.
With this cubic virial equation, the coefficients B and C can be solved in closed form. Imposing the critical conditions:
the cubic virial equation can be solved to yield:
and
is therefore 0.333, compared to 0.375 from the Van der Waals equation.
Between the critical point and the
triple point
In thermodynamics, the triple point of a substance is the temperature and pressure at which the three Phase (matter), phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at ...
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
. 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:
It can be rearranged as:
The factor
is the volume of saturated gas according to the
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
, and can be given a unique name
:
In the saturation region, the cubic equation has three roots, and can be written alternatively as:
which can be expanded as:
is a volume of an unstable state between
and
. The cubic equations are identical. Therefore, from the linear terms in these equations,
can be solved:
From the quadratic terms, ''B'' can be solved:
And from the cubic terms, ''C'' can be solved:
Since
,
and
have been tabulated for many fluids with
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.
See also
*
Virial theorem
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by a conservative force (where the work done is independent of path), with ...
*
Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
*
Equation of state
In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most mo ...
References
{{Reflist
Statistical mechanics