c
:
ertamly, thls field has not suffered from a lack of ’ treatment in eitherjournals or textbooks. Dozens of equations have been proposed in the past hundred years and many reviews and chapters of books have been written to compare their relative merits. Despite this prolific outpouring on the subject, some facts and approaches are either misunderstood or not fully appreciated and these justify discussion. In view of the large number of equations of state which have been suggested, it is natural to search for the motivation. There are two rather obvious reasons for the widespread activity in this field over an extended period of time. The first is the fact that the problem of developing an equation of state is mathematically fascinating and particularly tantalizing because it seems so simple, at least at the start. Examination of the data in regular tabular or graphical form can lead one to believe that it should not be difficult to obtain a satisfactory algebraic representation. The second reason concerns the exceptional power and utility of an equation of state. When combined with appropriate thermodynamic relations, a well behaved equation can predict
I
APPLIED THERMODYNAMICS SYMPOSIUM
EQUATIONS OF STATE JOSEPH J. MARTIN
with highprecision isothermal changes in heat capacity,
.. , :. .e&dpy, entropy and fugacity, vapor pressure, latent heat ‘bf vaporization, activity coefficients, and vapor’
Development of on equation of stote is mathematicolly foscinoting ond particularly tontalizing because it opparently seems so simple, at leost of the start, and the exception01 power and utility answer the question,
“Why another orticle on equotions of state?’’
.
...
.
liquid equilibria in mixtures, not to mention the assistance it offers in transport property correlations. Unfortunately, even though the useful applications of a n equation of state are so extensive and attractive, the development of a high performance equation proves to be so involved that to date no one has come close to dwovering a single relation which is truly good over a wide range of density. Because of the apparent simplicity of the task, it is not surprising that many have rushed in where more experienced workers have feared to tread. Thin is not meant to frighten away interested but uninitiated workers; it is primarily to emphasize the frontier challenge and opportunity for further r e search. Since the phrase “equation of state” has more than a single connotation in thermodynamics, it is well to emphasize that here it refers to the equilibrium relation, in the absence of special force fields, among pressure, volume, temperature, and composition of a substance whether it be quite pure or in a uniform mixture. In functional form this relation is f(P,V,T,x) = 0, though in this paper only pure chemical species will be treated ( x = unity), and reference will be made simply to P V T equations and P V T data. Equations of state may be applied to gases, liquids, and solids, but application to solids will not be considered. Some persons are understandably interested only in the final comparison of the predictions of an equation of state and the experimental data. What is needed, however, is thorough comprehension of the inherent properties of certain algebraic expressions that are or may be used in equations of state, and the characteristic behavior of P V T data. Three previous papers (27, 23, 24) have covered in some detail the characteristics of P V T data that should be kept in mind when seeking to produce an equation to represent the data with high precision. I t is not the objective here to review all such characteristics or to evaluate all of the equations that have been proposed. Rather it is to look at the objectives sought in the more recent equations and to seek insight into how an equation of state is designed and what must be accounted for if it is successfully to carry out its job. Two approaches to the development of an equation of state may be followed. One is the theoretical a p proach based on either kinetic theory or statistical mechanics involving intermolecular forces. This is exemplified by such equations as those of van der Waals (37), Lennard-Jones and Devonshire (79), Hirschfelder, Bird, and Spotz (73), and Flory, Orwall, and Vrij (9). The other approach is empirical or at best only semitheoretical. This is exemplified by equations such as those of Clausius (6), Berthelot (3), Dieterici (S), Wohl (38), Keyes (73, the NBS (%), Beattie and Bridgeman (I), Keyes, Smith, and Gerry (78), Benedict, Webb, and Rubin (2), Redlich and Kwong (33),Martin and Hou (23),Pings and Sage (a), Hirschfelder, Buehler, McGee, V O L 5 9 NO. 1 2 D E C E M B E R 1 9 6 7
35
and Sutton (14), Strobridge (35),Costolnick and Thodos (7), McCarty and Stewart (26),and Goodwin (70). This paper focuses attention on the empirical or semitheoretical approach since this is the one which has had the greatest success in representing data with high precision over a wide range of density. A companion paper (20) is currently being prepared on the determination of closed equations of state (as contrasted to open ended series or “virial” equations as they are called) through a statistical analysis of intermolecular force potentials. I t suffices to say, however, that such theoretical equations as have been developed to date are definitely more limited in range of application than the empirical equations. This is not to infer that the new developments of tomorrow will not come from statistical mechanics; it is merely a realistic evaluation of the field at this moment. For example, if a table of thermodynamic properties were to be prepared today for the gas and liquid phases of a pure substance for which the P V T data were available, it would be based on an empirical equation of state or even numerical analysis rather than on an equation from kinetic theory or statistical mechanics. I n the search for an equation of state, one must make three decisions at the beginning. The first concerns the amount and kind of data that will be required to obtain the parameters in the equation, the second concerns the range of density to be covered, and the third concerns the precision with which the PVT data are to be represented. For some applications, it is desirable to have an equation which can be obtained from a minimum of data-i.e., the critical temperature, pressure, and volume, or the boiling point, or some molecular parameter. In other instances an equation is sought which will represent a large amount of experimental P V T data within the precision of the experiment. Density is paramount in all pressure-explicit equations. If the same high order of precision is to be obtained at all times, a relatively simple short equation will suffice for low densities, whereas a long complicated equation will be required if coverage is to include both low and high densities. From our present knowledge high precision equations have many arbitrary constants whose number depends primarily upon the density range and in a minor way upon the temperature range. For example, as the situation exists today, an equation representing the data precisely up to a fiftieth of the critical density may need only two constants, while four or five constants will be necessary to go to one half the critical density. At least a half dozen or more constants will be required to continue up to the critical density, and twice that many will be needed if the goal is one-and-a-half to two times the critical density. This 36
INDUSTRIAL AND ENGINEERING CHEMISTRY
may be a pessimistic view of the status of the field, but it should be considered more of a challenge than a limitation on what may be discovered in the future. As examples of the objectives of various investigators, Keyes (77) and Beattie and Bridgeman (7) designed their equations to reproduce the data within the experimental precision for densities up to a little over half the critical density. Benedict, Webb, and Rubin (BWR) (2) essentially modified the Beattie-Bridgeman equation to extend the density range well past the critical and still fit the data with high precision, while Bloomer and Rao (4) modified the temperature coefficients of the BWR equation to fit nitrogen data. A previous equation by the author (23) had essentially the same goals of precision and density range as the Benedict, Webb, and Rubin equation. Keyes, Smith, and Gerry (78) carried their search for high precision to an extreme in calculating thermodynamic properties of steam (76) by developing an equation of state with nine constants including one term with kmperature to the forty-ninth power; yet even with this complexity the equation is valid only to a third of the critical density. Many years ago Onnes (29) suggested that high precision could be obtained by calculating pressure from a power expansion in density with the coefficients being temperature functions. This is the virial equation and a number of authors such as Hirschfelder, Bird, and Spotz (73), Gyorog and Obert (721, and Joffe and Pate1 (75) have devised theoretical and empirical procedures to obtain the virial coefficients for low densities. Michels (28) has long employed the isothermal virial equation to fit precise experimental data to two-and-ahalf times the critical density using terms as high as the ninth power of density. Hirschfelder, Buehler, McGee, and Sutton and Costolnick and Thodos covered a wide range of density and simultaneously maintained fair precision by dividing the P V T plateau of data into regions and writing separate equations for the different regions. Flory, Orwall, and Vrij and Lennard- Jones and Devonshire sought high precision in their equations but limited the range of applicability to the liquid region where the densities are always significantly greater than the critical density. Van der Waals and others that followed him, such as Clausius, Dieterici, Berthelot, Wohl, and Redlich and Kwong, endeavored to find equations covering the whole range of density from infinitely attenuated gas to compressed liquid, but were willing to accept fairly large deviations from the experimental results. I t is interesting to note, however, that Redlich in later papers with other collaborators (37, 32) focused attention on higher precision representation and increased the number of arbitrary constants in his equation from 2 to 44. With the improvement in experimental P V T
rigwe 7.
l'fesswc-wlwneplol
techniques it is natural that the trend in recent years has been toward more complicated equations which can fit the data within the experimental precision. Design of Simple Equation of S l o b
When one is developing an equation of state empirically, there are six graphs that it pays well to keep in mind. These are Figure 1, the P-V plot; Figure 2, the P - T plot; Figure 3, the generalized compressibility or I plot; Figure 4, the generalized vapor pressure plot; Figure 5, the generalized second virial Coefficient plot; and Figure 6, the generalized third virial coefficient plot. These will be discussed as the occasion demands their use. We shall start with the critical isotherm on the P-V plot, just as van der Waals did almost 100 yean ago, and seek to fit it with his equation:
Figure 2. Pressu
A ~ ~ a plot lu~e
4
p = - RT V - b
_
_a
VP
Other two-constant (exclusive of R ) equations could be used, but they offer no particular advantages. The standard procedure given in a multitude of books will be followed, but it is necessary tn repeat it here to demonstrate the thread of logic one must carry beyond the normal end point. Because the critical point is a point of horizontal slope and inflection, the usual first two pressure-volume derivatives are set to zero:
(g)
To,vo =
-RT,
(V. - b)'
2 a
+-=o V,"
Figure 3. G m a l i u d compressibility chmt
(2)
and
Joseph J . Martin is Professor of Chemical Engineering at the University of Michigan, Ann Arbor, Muh. The author acknowulcdges thejnancial assistance and p a t i m e over a 5-year period of the NationaI Science Foundation which made the s t d y of many kinds of equations of stakpossibIe. AUTHOR
Since Equation 1 holds in general, it holds at the critical point, so there are three equations available to solve for the two arbitrary comtants, a and b. If
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Figure 4. Gem '. I vapor pressure chart
Figure 5.
! Figure 6. Generalirad' third vinal
~
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
