( PV ) t* = R × 273.16 K (3.30) Division of Eq. (3.29) by Eq. (3.30) gives:
T K = 273.16 ______ ( PV ) *
(PV) t* (3.31)
This equation provides the experimental basis for the ideal-gas temperature scale throughout the temperature range for which values of (PV)* are experimentally accessible. The Kelvin temperature scale is defined so as to be in as close agreement as possible with this scale.
The proportionality constant R in Eqs. (3.29) and (3.30) is the universal gas constant. Its numerical value is found from Eq. (3.30):
R = ________ ( PV ) t*
273.16 K
The accepted experimental value of ( PV ) t* is 22,711.8 bar·cm3 · mol−1, from which6 R = 22,711.8 bar· cm ___________________3 ·mol −1 273.16 K = 83.1446 bar· cm 3 ·mol −1 ·K −1
Through the use of conversion factors, R may be expressed in various units. Commonly used values are given in Table A.2 of App. A.
Two Forms of the Virial Equation
A useful auxiliary thermodynamic property is defined by the equation:
Z ≡ _PV
RT = _V
V ig (3.32)
3.4. Virial Equations of State 91 This dimensionless ratio is called the compressibility factor. It is a measure of the deviation of the real-gas molar volume from its ideal-gas value. For the ideal-gas state, Z = 1. At moderate temperatures its value is usually <1, though at elevated temperatures it may be >1. Figure 3.6 shows the compressibility factor of carbon dioxide as a function of T and P. This figure presents the same information as Fig. 3.4, except that it is plotted in terms of Z rather than V. It shows that at low pressure, Z approaches 1, and at moderate pressures, Z decreases roughly linearly with pressure.
Figure 3.6: PZT surface for carbon dioxide, with isotherms shown in black and the vapor/liquid equilibrium curve in white.
200 250
300
350 0
20 40 60 80 100 0
0.2 0.4 0.6 0.8 1
Z
T (K)
P (bar)
With Z defined by Eq. (3.32) and with a = RT [Eq. (3.29)], Eq. (3.28) becomes:
Z = 1 + B′P + C′ P 2 + D′ P 3 + . . . (3.33) An alternative expression for Z is also in common use:7
Z = 1 + B_
V + _C V 2 + _D
V 3 + . . . (3.34) Both of these equations are known as virial expansions, and the parameters B′, C′, D′, etc., and B, C, D, etc., are called virial coefficients. Parameters B′ and B are second virial coeffi-cients; C′ and C are third virial coefficients, and so on. For a given gas the virial coefficients are functions of temperature only.
7Proposed by H. Kamerlingh Onnes, “Expression of the Equation of State of Gases and Liquids by Means of Series,” Communications from the Physical Laboratory of the University of Leiden, no. 71, 1901.
The two sets of coefficients in Eqs. (3.33) and (3.34) are related as follows:
B′= ___B
RT (3.35a) C′=
C − B 2
_____
( RT ) 2 (3.35b) D′= __________D − 3BC + 2 B ( RT ) 3 3 (3.35c)
To derive these relations, we set Z = PV/RT in Eq. (3.34) and solve for P. This allows elimina-tion of P on the right side of Eq. (3.33). The resulting equaelimina-tion reduces to a power series in 1/V which may be compared term by term with Eq. (3.34) to yield the given relations. They hold exactly only for the two virial expansions as infinite series, but they are acceptable approxima-tions for the truncated forms used in practice.
Many other equations of state have been proposed for gases, but the virial equations are the only ones firmly based on statistical mechanics, which provides physical significance to the virial coefficients. Thus, for the expansion in 1/V, the term B/V arises on account of interactions between pairs of molecules; the C/V2 term, on account of three-body interactions;
etc. Because, at gas-like densities, two-body interactions are many times more common than three-body interactions, and three-body interactions are many times more numerous than four-body interactions, the contributions to Z of the successively higher-ordered terms decrease rapidly.
3.5 APPLICATION OF THE VIRIAL EQUATIONS
The two forms of the virial expansion given by Eqs. (3.33) and (3.34) are infinite series. For engineering purposes their use is practical only where convergence is very rapid, that is, where two or three terms suffice for reasonably close approximations to the values of the series. This is realized for gases and vapors at low to moderate pressures.
Figure 3.7 shows a compressibility-factor graph for methane. All isotherms originate at Z = 1 and P = 0, and are nearly straight lines at low pressures. Thus the tangent to an isotherm at P = 0 is a good approximation of the isotherm from P → 0 to some finite pressure. Differ-entiation of Eq. (3.33) for a given temperature gives:
( ___∂ Z
∂ P )
T
= B′+2C′P + 3D′ P 2 + . . . from which,
( ___∂ Z
∂ P ) T;P=0 = B′
Thus the equation of the tangent line is Z = 1 + B′P, a result also given by truncating Eq. (3.33) to two terms.
A more common form of this equation results from substitution for B′ by Eq. (3.35a):
Z = PV_
RT = 1 + BP_
RT (3.36)
3.5. Application of the Virial Equations 93
This equation expresses direct linearity between Z and P and is often applied to vapors at subcritical temperatures up to their saturation pressures. At higher temperatures it often pro-vides a reasonable approximation for gases up to a pressure of several bars, with the pressure range increasing as the temperature increases.
Equation (3.34) as well may be truncated to two terms for application at low pressures:
Z = ___PV
RT = 1 +
__B
V (3.37)
However, Eq. (3.36) is more convenient in application and is normally at least as accurate as Eq. (3.37). Thus when the virial equation is truncated to two terms, Eq. (3.36) is preferred.
The second virial coefficient B is substance dependent and a function of temperature.
Experimental values are available for a number of gases.8 Moreover, estimation of second virial coefficients is possible where no data are available, as discussed in Sec. 3.7.
For pressures above the range of applicability of Eq. (3.36) but below the critical pres-sure, the virial equation truncated to three terms often provides excellent results. In this case Eq. (3.34), the expansion in 1/V, is far superior to Eq. (3.33). Thus when the virial equation is truncated to three terms, the appropriate form is:
Z = _PV
RT = 1 + _B V + _C
V 2 (3.38)
This equation is explicit in pressure but is cubic in volume. Analytic solution for V is possible, but solution by an iterative scheme, as illustrated in Ex. 3.8, is often more convenient.
Values of C, like those of B, depend on the gas and on temperature. Much less is known about third virial coefficients than about second virial coefficients, though data for a num-ber of gases are found in the literature. Because virial coefficients beyond the third are rarely known and because the virial expansion with more than three terms becomes unwieldy, its use is uncommon.
Figure 3.7: Compressibility-factor graph for methane. Shown are isotherms of the compressibility factor Z, as calculated from PVT data for methane by the defining equation Z = PV/RT. They are plotted vs.
pressure for a number of constant temperatures, and they show graphically what the virial expansion in P represents analytically.
8J. H. Dymond and E. B. Smith, The Virial Coefficients of Pure Gases and Mixtures, Clarendon Press, Oxford, 1980.
Figure 3.8 illustrates the effect of temperature on the virial coefficients B and C for nitrogen; although numerical values are different for other gases, the trends are similar. The curve of Fig. 3.8 suggests that B increases monotonically with T; however, at temperatures much higher than shown, B reaches a maximum and then slowly decreases. The temperature dependence of C is more difficult to establish experimentally, but its main features are clear: C is negative at low temperatures, passes through a maximum at a temperature near the critical temperature, and thereafter decreases slowly with increasing T.
Example 3.8
Reported values for the virial coefficients of isopropanol vapor at 200°C are:
B = − 388 cm 3 · mol −1 C = − 26,000 cm 6 · mol −2 Calculate V and Z for isopropanol vapor at 200°C and 10 bar:
(a) For the ideal-gas state; (b) By Eq. (3.36); (c) By Eq. (3.38).
Solution 3.8
The absolute temperature is T = 473.15 K, and the appropriate value of the gas constant is R = 83.14 bar·cm3·mol−1·K−1.
(a) For the ideal-gas state, Z = 1, and V ig = ___RT
P =
( 83.14 ) ( 473.15 )
____________
10 = 3934 cm 3 ·mol −1 (b) From the second equality of Eq. (3.36), we have
V = ___RT
B = 3934 − 388 = 3546 cm 3 ·mol −1
Figure 3.8: Virial coefficients B and C for nitrogen.
100
0
100
200
3000 100 200 300 400
0
4000 2000 2000 4000
T/K
B/cm3mol1 C/cm6mol2B
C
3.6. Cubic Equations of State 95 and
Z = ___PV
RT =
_____V RT / P =
___V V ig = _____3546
3934 = 0.9014 (c) If solution by iteration is intended, Eq. (3.38) may be written:
V i+1 = RT___
P ( 1 +
_B V i +
_C V i2 )
where i is the iteration number. Iteration is initiated with the ideal-gas state value Vig. Solution yields:
V = 3488 cm 3 ·mol −1
from which Z = 0.8866. In comparison with this result, the ideal-gas-state value is 13%
too high, and Eq. (3.36) gives a value 1.7% too high.
3.6 CUBIC EQUATIONS OF STATE
If an equation of state is to represent the PVT behavior of both liquids and vapors, it must encompass a wide range of temperatures, pressures, and molar volumes. Yet it must not be so complex as to present excessive numerical or analytical difficulties in application. Poly-nomial equations that are cubic in molar volume offer a compromise between generality and simplicity that is suitable to many purposes. Cubic equations are in fact the simplest equations capable of representing both liquid and vapor behavior.