THERMODYNAMIC PROPERTIES AND THE STRUCTURE OF FLUID PHASES JAMES C. M E L R O S E Field Research Laboratory, Socony Mobil Oil Co., Inc., Dallas, Tex.
The thermodynamic properties of one-component fluid phases conforming to the principle of corresponding states are considered. In the equations of state of such a fluid there appear integrals involving a oneparameter radial distribution function. These integrals define the so-called configurational properties of the fluid. The latter, therefore, are directly related to the structure of the fluid, as given by the radial distribution function and are also interrelated according to a one-component thermodynamic formalism fully analogous to the ordinary thermodynamic formalism for a fluid phase. The formalism in question i s examined in detail and is utilized to exhibit the experimentally measured properties of a simple fluid. Finally, it is shown that the configurational properties associated with the internal energy and with the pressure can b e used to derive a rough estimate of the coordination number of the fluid.
T IS
now recognized that the heavier rare gas substances obey
I a common reduced equation of state, a t least to a high de-
gree of approximation (5, 36). Such behavior implies that the molecular descriptions of these substances possess features which lead to the principle of corresponding states (7, 73, 32). Consequently, the thermodynamic properties of one-component fluid phases conforming to this principle are said to define a simple fluid (37). The structure of a simple fluid, as for any fluid phase, can be represented by a one-parameter radial distribution function, G(R). The thermal equation of state, which follows from the virial theorem, is then written in terms of an integral involving G(R) (6, 37). Similarly, the internal energy, given by the caloric equation of state, is dependent on this function. A third equation of state, for the chemical potential, is related to a function G ( R , [ ) ,where 4 is a coupling parameter (76, 27). Thus functions G(R) and G(R,[)define the structural information required for predicting the thermodynamic properties of a simple fluid. Each of the three equations of state can be regarded as defining a surface in a three-dimensional Gibbs space. If, for each case, density and temperature are regarded as the independent variables, the third dimensions defining these spaces are, respectively, the dependent variables: pressure, molar internal energy, and chemical potential. Since the three resulting surfaces, and the thermodynamic properties of the simple fluid which they describe, are related to the structural functions G ( R ) and G(R, recovered. According to the theorem of corresponding states. the intermolecular potential function which appears in Equations 1 is a spherically symmetric function with only two characteristic parameters. I t is therefore possible to express the configurational properties in rzduced or dimensionless form. T h e critical temperature and molar volume, T , and u,, suffice for this Thus, reduced properties and variables are denoted by Greek letters,
.\ = h/(2nmkT)"*
the equation of state for the chemical potential (molar Gibbs function) is written as p / - l T 0 = kT log [.13A\\;/VJ -
sParameter ( introduced in this expression, which is due to Kirkivood (27), has the effect of coupling a single particle, chosen a t random from the fluid, to the remaining -Y - 1 particles. T h e three equations of state given by Equations 1 must conform to the usual thermodynamic formalism represented by a fundamental Gibbsian equation and the Gibbs-Duhem equation. \Vhen s is chosen a s the molar entropy, these relations are du := Tds
dh
=:
-sdT
T h e fundamental integrated Equations 2 is
- Pdv
+ VdP
(24 (2b)
relation which accompanies
u = T s - P v + ~
(3)
I t folloxrs from Equation 3 that the reference state for the molar entropy must be the same as for the quantities given by Equations l-Le., the ideal gas a t the absolute zero of temperature. Thus a n equation of state for entropy may be formally written as
in which the quantity denoted by s* is clearly just the difference between the ideal gas entropy and the actual entropy. Definition of Configurational Properties. Of the two parts into which each of the equations of state separates, the first clearly arises from the motion of the particles because in each case it is identical with the perfect gas equation of state. The second part in Equations 1 and 4 is described here by the term "configurational property." Such a quantity, it is clear, represents only the contribution of the intermolecular forces. In Equations 1 the inregrals which represent the configurational properties are so defined as to be positive in sign over most of the fluid density range. This follows from the behavior of the functions G(R) and @(R),the general features of which may be accepted as known. Each of the configurational properties defined in Equations 1 and 4 , taken as a function of the independent variables, density and temperature, establishes a three-dimensional configurational Gibbs space. The four surfaces in these spaces are also interrelated by a formalism analogous to that of Equations 2 and 3. This may be seen by substituting Equations 1 and 4 in Equations 2 and 3. Since the kinetic or perfect gas parts of
=
i!ip
k Tcvr
L1JOm
47rR2G(R:~)[-@(R)]dRd( (6c)
l,7 =
r*,'.\-*k
(7)
Defining a reduced configurational pressure by 7r
= up
then permits the formalism of Equations 2 and 3 to be written as
de = Odv - adp-'
(84
+ p-'d7r = 07 + s'
d{ = -7dO E
(8b)
(9)
7rp-1
Other Definitions. The configurational quantities defined by Equations 6 and 7 are precisely (except for a change in sign) those Lvhich ha\;e been called internal quantities by Michels and coworkers (24, 25) and residual properties by Ro\vlinson (38). Roxvlinson (37) has also defined a set of properties Ti-hich are called configurational. These properties are based upon the formalism of the canonical ensemble of statistical mechanics. Thus the various properties are defined by the usual connections betiveen thermodynamic properties and the partition function, employ-ing in this case the so-called configurational partition function. T h e properties based on the configurational partition function, hoivever, have the disadvantage that. the quantities playing the roles of entropy and chemical potential have logarithmic infinities in the limit of zero density. Furthermore, these configurational properties fail to conform to the formalism of Equations 8. This is readily seen from the relationships betlveen these properties (denoted by the subscript) and those defined by Equations 6c and 7,
cr = s" - O(1 - log ( S / V ) ] Of = 17
+ { 1 - log (.Y/V))
(loa) (1Ob)
A4notherexpression Tvhich has been used to denote the contribution of intermolecular forces to the thermodynamic properties of a simple fluid is "excess property." Such a set of properties has been defined by Hill (77). These properties, like those of Equations 6 and 7, all vanish in the limit of zero density. I n this case, hoivever, the disadvantage is retained that, although Equation 9 is obeyed, Equations 8 are not. T h e excess chemical potential and entropy defined by Hill, compared ivith those given by Equations 6c and 7, contain added terms, as shoivn by the folloJving relationships, VOL. 5
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225
L
=
r + e log { 1 - w / e i
qe = 1 - log
( 1 - ./e)
(114 (llb)
Thus the definitions of the configurational properties which have been adopted in the present discussion have two desirable features not possessed by other definitions which have been suggested. These arise as a result of choosing as the reference state for each of the configurational (or residual) properties the perfect gas a t the same temperature and density as the real fluid. In fact, this particular choice of reference state is essential for the compactness and the associated close connection with fluid structure, which folloh s from the use of the configurational Gibbs space formalism. Transforms a n d Intrinsic Properties. The properties defined by Equations 6 are those most directly related to the equations of state. However, since these properties conform to the formalism expressed by Equations 8 and 9, it is possible to consider them as thermodynamic potentials, by analogy with the usual definitions of such potentials. In fact, it is of interest to examine briefly the status of these potentials and to introduce some useful transforms. Following Callen ( 3 ) , transforms of both the energy function and the entropy function will be discussed. A consideration of such transforms is necessary in order to achieve the full advaiitages of the configurational Gibbs formalism. The Gibbs function (free energy, free enthalpy) for the configurational formalism is, of course, simply the quantity, {, defined by Equation 6c. Similarly, the grand canonical potential is the quantity, a, defined by Equation 6a. For both of these functions, the corresponding extensive quantities are Legendre transforms of the energy. The name “grand canonical potential” indicates the status of the quantity, -PV, as the fundamental thermodynamic function for the grand canonical ensemble of statistical mechanics, for which T , V , and p are independent variables. A relationship linking the derivatives of { and w can be obtained from Equation 8b,
This indicates that the isotherms for these potentials are simply related. Thus, provided that the vapor-liquid phase transition is not encountered, we have
t
= w
+
(f)
dP
Both of these potentials vanish in the limit of zero density, as do the energy, E , and the entropy, q . I t is of interest, however, to describe as precisely as possible the behavior of the isotherms of all of these quantities in the low density region. Hence it is useful to define what will be called intrinsic configurational properties. Such a property is obtained simply by dividing the corresponding configurational property by the density. Thus, for the intrinsic energy and entropy, we can write a = E/p;
u = q/p
(14)
I n the case of the potentials appearing in Equation 12, i t is advantageous, when defining intrinsic properties, to employ the corresponding Legendre transforms of the entropy. Such transforms are described by Callen (3) as generalized Massieu functions and are obtained by dividing the potential in question by the temperature. Thus, from the Gibbs function we obtain what will be called the intrinsic Planck function, = p/eP 226
I&EC FUNDAMENTALS
(15 4
The name associated with this function follows the usual terminology employed in thermodynamics. A change in sign has been introduced, in order to preserve the convenience of dealing with positive numerical values for this quantity. A similar sign choice is made for the intrinsic grand Massieu function, defined as =
w/ep
(15b)
The name adopted for this function reflects its status as a generalized Massieu function, as well as its relationship to the grand canonical potential. Again, the isotherms for h and 4 are simply related. Corresponding to Equations 12 and 13 we now have hip(% aP ) 0 = 2 4 + P ( $ ) 0
Configurational Thermodynamic Properties
T h e Simple Fluid. We turn now to the experimental data which are to be employed in representing the properties of the simple fluid. For this purpose the recent equation of state measurements by Michels, Levelt, and de Graaff (26) for argon appear to be most suitable. Tabulated values of the Pv product, as well as the internal-Le., configurationalenergy and entropy are given by these authors (26, 27). From thege values, the reduced configurational properties defined by Equations 6 and 7 were computed and tabulated as functions of the reduced temperature and density, as defined by Equations 5. Table I records the values of the reduced configurational properties a t the critical point of the simple fluid. I n Figures 1 and 2, complete sets of reduced configurational properties are plotted as isotherms. The curves of Figure 1 represent a reduced temperature considerably above the critical temperature, while Figure 2 illustrates the behavior of the isotherms only slightly above the critical temperature. At the lower temperature each property is seen to have increased in magnitude, but otherwise no marked change in the character of curves is observed. In particular the isotherms near the critical temperature show no exceptional features which can be characterized as critical behavior. In order to study the behavior of the isotherms more precisely, it is convenient to employ the properties defined above as intrinsic configurational properties. In Figures 3 and 4, isotherms for the intrinsic configurational energy and entropy are shown. I n addition to several isotherms above the critical temperature, an isotherm below this temperature is included for each property. Also shown is a portion of the vapor-liquid coexistence line in a region near the critical point. T h e latter, of course, is located a t a value of unity on the reduced density scale. For both energy and entropy the two branches of the isotherm below the critical temperature are seen to be different in character. Hence, the precise nature of an interpolation to form a continuous isotherm is not a t all obvious in these cases. This is in marked contrast to the predictions of approximate equations of state of simple analytical form, such as the van der Waals equation of state (see below). These intrinsic configurational properties, in fact, provide a highly sensitive test for any proposed equation of state. I n the case of the intrinsic properties corresponding to the Planck and grand Massieu functions, the isotherms are some-
Table 1.
Comparison of Critical Properties Simple clan der Waals Properties Fluid” Fluid wc 0.7089 0.625 ec 1 .918b 1.125 rc 1 .639b 1.34453 ’7c 0.988b 0.40547 a T , = 750.86‘K.; D, = 74.54cc./mole; p c = 48.34atm. b Graphical interpolation, using intrinsicprope~ties.
2.25
hi
Figure 3.
I
-1
5
10
I5
2.0
W
Isotherms for intrinsic configurational energy
-P 2.5
Figure 1. Configurational properties above the critical temperature
I
I
I
---P
I
\
Figure 2. Configurational properties near the critical temperature
I -+
what simpler in form. For the latter function this is shown in Figure 5. Now, the vapor and liquid branches of the isotherm below the critical temperature appear to be very nearly parallel to the correspo’ndingparts of the isotherms above the critical temperature. Hence it seems reasonable to suppose that the two branches could be connected by a monotonically decreasing curve. A t least the possibility of making such a n interpolation appears inviting. I n support of this conjecture, it may be remarked that in adopting a n intrinsic configurational property as a dependent variable, the isotherm “loop” associated with the van der Waals equation in its usual form is eliminated. The isotherms for the intrinsic configurational Planck function, A, are somewhat Zimilar in character to those for the intrinsic grand Massieu function. This, of course, is a direct
P
I
-5
5
10
20
I5
25
Figure 5 . Isotherms for intrinsic configurational Massieu function
grand
consequence of the relationship between the two functions, expressed by Equation 16b. For the corresponding Gibbs and grand canonical potential, w, Equation 13 function, expresses this relationship, and Figures 1 and 2 illustrate the similarity in isotherms. Such similarities are not maintained for other definitions of the configurational Gibbs function, such as those given in Equations 10a and l l a . T h e similarity in character between the isotherms for the intrinsic Planck and grand Massieu functions is preserved for isotherms below the critical temperature. This suggests
r,
VOL. 5
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227
strongly that an interpolation for the function X can be based on the interpolation for +-that is, Equation 16b, or alternatively Equation 13, can be invoked even in the unstable region which separates the liquid and vapor branches of such isotherms. The question of the proper form for the interpolations of the thermodynamic properties of a fluid in the unstable region, and indeed the question of whether any such interpolation is valid, represent long debated problems of fundamental importance. The difficulties of a theoretical nature associated with these questions have been discussed in several treatises ( 7 7 , 75, 23, 40). The numerous studies devoted to these problems in recent years have been reviewed by Fisher ( 9 )and Fixman (10). Despite intensive efforts, no clearly established conclusions have been forthcoming. The formalism utilized in Figures 1 to 5 permits the experimental evidence bearing on these questions to be assessed from a somewhat different viewpoint than in the usual discussions. The type of interpolation indicated by Figure 5 suggests that such interpolations are not so difficult as has been supposed. T h e van der Waals Fluid. The thermodynamic behavior predicted by the van der Waals equation of state may be said to define a van der Waals fluid. As indicated above, the expressions for the intrinsic configurational properties of such a fluid have a simple analytical form. Because of this simplicity it is possible to use the van der M'aals fluid as a model to study some of the general aspects of the configurational formalism so far developed. At the same time, the deficiencies of the van der Waals fluid in approximating the behavior of the simple fluid are most clearly exhibited. The expressions for intrinsic energy and entropy predicted by the van der Waals equation of state have the forms, a = 9/8
" = -log 1 P
(1)
(174
3-
Equations 17 clearly fail to reproduce the behavior shown in Figures 3 and 4. For the intrinsic properties corresponding to the Planck and grand Massieu functions, the van der, Waals expressions are also simple in form,
The curves representing Equations 18 are, in contrast to the predictions of Equations 17, not very different in character from those of the real fluid. This may be seen from Figure 6, which gives isotherms for the intrinsic grand Massieu function. T h e sections of the isotherms corresponding to the unstable states within the vapor-liquid coexistence line are shown as dashed lines. The characteristic loops are now absent. T h e same is true for isotherms representing the intrinsic Planck function. As indicated by Equation 18a, these isotherms are similar in general form to those of Figure 6. Equations 18 satisfy the integral relationship of Equation 16b. T h e latter, as well as Equation 13, can clearly be applied even through the vapor-liquid phase transition, because the continuity of states assumption of the van der Waals equation means simply that Equations 17 and 18 refer to both vapor and liquid branches of the isotherms. T h e compactness in the representation of the thermodynamic properties of fluids, which results from the use of the configura228
l&EC FUNDAMENTALS
"-1
-5
5
,
10
15
20
25
Figure 6. van der Waals isotherms tional formalism, is illustrated by the van der IYaals fluid, as hell as by the simple fluid. Table I compares the values for the configurational properties a t the critical points of the two fluids. This comparison indicates that the well known deficiency of the van der Waals expression in predicting quantitatively the critical compressibility factor of real fluids extends to each of the configurational properties. The similarity of the isotherms of Figure 6 to those of Figure 5 is, of course, deceptive. The configurational energy is obtained, since the Gibbs formalism must be obeyed, by taking the temperature derivative of the integral of the function over the appropriate density range. Thus it is the temperature dependence specified by Equation 18b which is responsible for the overly simplified result represented by Equation 17a. I t is perhaps worthwhile to emphasize a t this point an important feature of the configurational properties used in the present discussion. Since these properties have been defined in such a way as to obey the Gibbs formalism, it follows that these functions represent completely, and at the same time exclusively, the contributions of the intermolecular forces to the thermodynamic properties of the fluid. Hence the deficiencies of any proposed equation of state may be expected to be most clearly exhibited by considering the set of configurational properties shonn in Figures 1 and 2 and listed in Table I. The behavior of the simple fluid which is defined by Figures 1 through 5 thus includes all of the information which, together with the ideal gas functions, determines the thermodynamic properties of such fluids within the specified temperature and density ranges. Discussion of Related Work. The configurational property which is called here the intrinsic grand Massieu function is, of course, that which is most directly derived from the experimental Pu isotherms. A function identical with q5 (except for sign) has been employed by Pings and Sage (29) as a basis for their development of a n orthogonal polynomial equation of state. Experimental data for propane were considered (30), and the general behavior of the function in this case was similar to that presented in Figure 5. Also, one of the residual properties discussed by Rowlinson (38) is equivalent to -04. Isotherms showing the behavior of this quantity for the case of xenon are again similar in nature to those of Figure 5 . When expressed in terms of reduced variables, the behavior of the xenon isotherms should be, of course, very nearly identical to the behavior of the argon isotherms. The slight differences which have been noted (5, 24) indicate a very small departure from the principle of corresponding states. Since nonspherically symmetric intermolecular potentials give rise to much
larger deviations from this principle, it would be of considerable interest to compare the complete set of configurational properlies for a substance such as propane with those of the simple fluid. Carrying out such a comparison over a range of temperatures and densities Ivould then provide a comprehensive description of the effects arising from the lack of spherical symmetry in the potential. I t may be anticipated that such a description would supplement, with a n increase in sensitivity and perhaps finer detail, the use of the acentric factor which was introduced by Pitzer ( 4 , 33) for this purpose. I n addition to the work just cited, the literature contains a wealth of contributions concerned with the systematic and compact representation of the thermodynamic properties of the simple fluid. For example, the program carried out by Pitzer and coworkers ( 4 , 33) includes a complete representation of the properties of the simple fluid. T h e excess functions employed in this work, however, differ considerably in both magnitude and behavior I-dativeto each other from the properties represented in Figures 1 to 5. Another important treatment is that of Hirschfelder and coworkers (78), who use excess functions which are subject to the same disadvantage as those defined by Equations 10. Structural Interpretation
Limitations. The co:nfigurational properties defined in terms of the equations of state, Equations 1 or 6, are directly related to the radial distribution function, G ( R ) , and the associated function, G(R,E). These functions, by virtue of their definitions (6, 76, 37), are able to reflect long range structure, if it exists, as well as short range structure. T h e configurational quantities, however, do not have this property, because the intermolecular potential function, @ ( R ) ,or the virial function, (R/S)d@(R)/dR,also appears in each of the integrals which define the configurational properties. These functions are of short range, and since they enter the integrals in question as factors multiplying the functions G(R) or G(R,(), the structural information associated with the possible long range character of the latter is effectively lost. I n fact, the entirely nosrmal behavior of the configurational properties in the neighborhood of the critical point is explained by this limitation on the range of the structural information available. I t is only a t or very near the critical point that the function G ( R ) reflects long range correlations in the positions of the molecules constituting the fluid. T h e configurational properties, however, are insensitive to this type of information. Only the short range stru.cture contributes to these properties. Hence, the critical point in configurational Gibbs spaces fails to exhibit any unusual character. I n contrast, the ordinary thermodynamic properties show a distinctive and special behavior a t the critical point. Since these properties reflect the contributions of the kinetic or motional effects of the mole:cules, as well as the interactions, it follows that the critical point corresponds to a highly sensitive state of balance between these contributions. This balance is characterized by the equality between the kinetic pressure, which in reduced form is equal to PO, and the quantity, p(bn/bp)s, which is dependent on the short range structure. Under such conditions, the fluid compressibility becomes infinite, as do fluctuations in pressure. Associated with these phenomena is the appearance of long range structure. At this point reference should be made to the behavior of the configurational heat capa.city a t constant volume, which was observed by Michels and coworkers (27). Over a temperature range of the order of 50" above the critical temperature, maxima in the specific heat isotherms occur for densities in the
neighborhood of the critical density. This behavior is sometimes regarded as anomalous, in the sense of being related to critical phenomena. However, these isotherms reflect only the temperature dependence of the configurational energy. Hence, the interpretation of these maxima as a type of critical point anomaly is inconsistent with the conclusions reached above concerning the limited range of the fluid structure to which the configurational properties are sensitive. More probably, the explanation for the specific heat maxima is to be sought in relatively minor variations in the short range structure. When the configurational energy is plotted as in Figure 3, the isotherm behavior in the region of the critical density no longer appears anomalous in character. These considerations, of course, are not in conflict with the interpretation of the specific heat a t constant pressure data discussed by Jones and LValker (79). These specific heats cannot be converted to configurational properties simply by subtracting the corresponding perfect gas values. Instead, such properties inherently reflect the sensitive balance between the kinetic and interaction contributions of the constituent molecules, in a fashion similar to that of the fluid compressibility, Hence, the very much larger specific heat maxima reported by Jones and Walker represent a type of behavior which is related to critical phenomena. In considering the limitations regarding the structural information associated with the configurational properties, there remains the question of the relative importance of the functions G ( R ) and G(R,E). Since it is not immediately apparent from its definition (27) that G(R,E) provides any further information of a structural nature, the following discussion of a principal feature of the fluid structure relies on the configurational properties directly related to G ( R ) . Coordination Number Estimate. The dominant feature of the distribution function, G ( R ) ,in the region of short range structure is the first peak. The space integral over this first peak gives directly the average number of nearest neighbors. This, by definition, is the coordination number of the fluid, denoted as -VI. This quantity, then, should be the principal structural feature which is reflected by the magnitude and behavior of the configurational properties. T o develop an approach to this quantity, it is necessary to employ a model for the intermolecular potential function. For this purpose the Lennard-Jones 6-12 potential is particularly useful. According to this model the potential and virial functions both separate into two parts, one of which represents the inverse sixth power attraction, while the other expresses the inverse twelfth power repulsion. These expressions may be written as
where E and Ro represent the depth of the potential well and its position, respectively. If Equations 19 are inserted in the integrals appearing in Equations 6a and 6b, the latter can then be regarded as providing two equations, each containing two unknown factors. These factors both have a purely structural meaning, since they may be defined, independently of the Lennard-Jones model, as 17
"rn
VOL. 5
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229
From Equations 6a and 6b and assuming the model represented by Equations 19,
-
I
N I - 6.5
A Thus the configurational properties E and w can be used to evaluate El’and ivl”, which will be referred to as LennardJones structural parameters. Except for numerical factors, the extensive quantities corresponding to these parameters are identical with the functions introduced by Rice and Kirkwood (35) for the purpose of relating shear viscosity and thermal conductivity to the thermodynamic properties of the simple fluid. Parameter E,’ may be regarded as a rather crude estimate of the coordination number of the fluid, because the factor (R,/R)6 has the effect of eliminating the part of the function G ( R ) which extends beyond the first peak. Also, this factor is unity a t the first peak of G(R)and has no effect a t low values of R, where G ( R ) vanishes. This behavior is illustrated in Figure 7 , in which a typical radial distribution function is plotted, along with the function which results when the heighting factor, (R,/R)6, is applied. The distribution function used in Figure 7 is not based on well established results, but rather is estimated from experiments 0:’ limited accuracy, together with approximate theoretical calculations. What is significant, however, is the general effect of the weighting factor which is indicated. The first peak of G ( R ) is shifted to somewhat smaller distances and becomes appreciably sharper as well as slightly higher. I n spite of this rather marked change in character, it is seen from Figure 7 that the net change in area under the first peak of the distribution function is not sufficient to prevent from serving as an estimate of the coordination number. I t is also clear that the quantity fvlr‘will correspond to a weighted distribution function which is shifted to smaller distances in even a more drastic manner. However, mlr’is also of the same order of magnitude as the true coordination numThus the low distance side of G ( R ) predominates in ber, establishing the magnitude of both RI’and m l r f . I n Figure 8, the Lennard-Jones structural parameter, m l t ,is plotted against density for two temperatures. The vaporliquid coexistence line is not shown, since it coincides to within less than 0.01 unit of m1’with the curve for the temperature near the critical. T h e value of E used in calculating mIt is that quoted by Michels and coworkers (26) and is based on fitting second virial coefficient data to the Lennard-Jones 6-12 potential (see Table 11). T h e extent to which n’,‘ and vary in estimating the magis indicated by the fact that a t most the two nitude of quantities differ by about 0.5 unit of 171’. Thus the structural
Figure 7.
p - 2
I
Weighted radial distribution function (schematic)
‘-1
1Y1’
ivl.
iv,
ivlr’
Table II. Parameters for Effective Pairwise Potential Potential Form tiuggenLennardheimJones Kihara McGlashan 6-72, (26) (28) ( 74) Ro, A; 3.822 3.678 3.818 E l k , K. 119.8 146.1 137.5 (Coeff. of R-6)/R06k, O K. 239.5 160.4 150 (Coeff. of R-e)),erg cm.6 x 1060 103.0 54.8 64.1
230
I&EC FUNDAMENTALS
Figure 8. Coordination number estimated from LennardJones structural parameter
parameter ,!TI” has a dependence on temperature and density which is similar to that of the parameter m 1 ’ . These two parameters d o not seem to have been considered heretofore in studying the relationship between thermodynamic properties and fluid structure. A related function has, however, been defined and utilized by Brown and Rowlinson (2). T h e function in question is a configurational property which bears the same relation to w that w bears to e. I t may be easily shown that, assuming again the applicability of the LennardJones 6-1 2 potential, this quantity is proportional to the difference, 2 lTlr‘ - N,‘. Discussion. A test of the approximation to the coordination number represented by the function iTl’ requires, of course, a comparison with direct measurements of the function G(R) as derived from x-ray and neutron scattering studies. Such measurements as have been published are of limited accuracy, particularly in the density and temperature ranges shown in Figures 1 and 2. However, the dashed line in Figure 8 represents the general trend of the coordination numbers as obtained from scattering measurements on several of the rare gases (72, 22). Since the precision of these data is not great, no individual points are shown. These dat2 in nearly every case represent temperature conditions corresponding to saturated liquid. The dashed line in Figure 8 thus constitutes a corresponding states correlation for coordination number. T h a t such a correlation for the various rare gases must exist is implied
by the correlation for the function G ( R ) , which has been discussed by Ricci (34). Highly accurate x-ray diffraction studies on argon in the critical region have recently been carried out in the laboratory of C. J . Pings a t the California Institute of Technology. T h e values of AV1’ given in Figure 8 are in approximate agreement ktith these results over the reduced density range of 0.5 to 1.3; a t higher densities the argon data data lie below the Rl’ values and are in good agreement with a smooth extension of the dashed line of Figure 8. The author is indebted to Pings for the communication of these results in advance of publication. T h e agreement of the curves shown in Figure 8 for the parameter iT1’ with the general trend of the measured coordination numbers appears to be reasonable. Better agreement, however, xvould result if the assumed value for the depth of the potential well: E? were adjusted to a value about 20y0 higher. Evidence derived from a number of reported efforts to obtain a n improved intermolecular potential function supports such a n adjustment. I n Table I1 the parameters of the Lennard-Jones 6-12 potential are compared with two of the improved potentials which have been suggested. As pointed out by Guggenheim and McGlashan ( 7 4 , the coefficient of R-6 in the LennardJones attractive term is considerably higher than most quantum mechanical calculations would indicate. T h e Guggenheim-McGlashan choice for this coefficient, on the other hand, is consistent with a somewhat deeper potential well. At the same time their potential is capable of fitting second virial coefficient data over a wider temperature range ( 8 ) . A still deeper well results from fitting second virial coefficient and viscosity data to the Kihara core potential ( 7 , 28). As Table I1 shsws, however, the Kihara potential requires a value of the R-6 coefficient even lower than that adopted by Guggenheim and McGlashan. I t might be argued that this is to be expected from a consideration of three-body forces, as recently discussed by K.estner and Sinanoglu (20). A second criticism of the quantum mechanical results employed by Guggenheim and McGlashan arises, however, from an analysis due to Salem (39). This work sdggests a n increased value of the R-6 coefficient. Consequently, there may well be a substantial cancellation of fairly large corrections of opposite sign. These more detailed representations of the intermolecular potential, therefore, ali.hough far from conclusive in nature, strongly indicate a value for the parameter, E, which is somewhat larger than the Lennard-Jones value. Turning now to the question of the temperature dependence of fluid structure, it appears from Figure 8 that the relationship between density and coordination number is somewhat insensitive to temperature. At low densities, the coordination number, a t least as estimated by the Lennard-Jones structure parameter, Ayl’, seems to decrease slightly with increasing temperature. There is a tendency for this effect to be reversed a t high densities. This reversal in temperature dependence is even more pronounced in the case of parameter Another feature of the isotherm shown in Figure 8 for the temperature slightly aklove critical is the insensitivity of the curve as the critical density is traversed. This again suggests, as in the case of the isotherms for the various configurational properties shown in Figure 2, that the local or short range structure is in no way unusual in the neighborhood of the critical point. The magnitude of the coordination number a t this point appears tcs be about 4. It is now possible to summarize briefly and in qualitative terms the relationship of the configurational properties e and ~3 to the fluid structure. First of all, each of these properties, as
ivl“.
implied by Equations 21, is determined by the magnitude of the two Lennard-Jones structural parameters a t each specific temperature and density condition. T h e factor which determines the behavior of the configurational energy, e, is to a very rough first approximation simply the increase in coordination number with increasing density. O n the other hand, the configurational function, w , which expresses the contribution of the intermolecular virial function (R/S)&(R)/dR to the P: product, depends on the difference between S l ’ and So far as nearest neighbors are concerned, the virial function is approximately zero. This implies ’ i ! 1 ” should be very nearly balanced. That this that i ! ~and balance is not exactly achieved is reflected in the fact that w goes through a maximum and eventually changes sign as the density increases. This behavior in turn can be attributed to the noncrystalline nature of the fluid structure. Thus, all of the nearest neighbors are not found a t precisely the same intermolecular distance. For particles contributing to the low distance side of the distribution function G ( R ) ,the intermolecular virial function is, in fact, negative. T h e effect of increasing density is, therefore, to alter the relative weights of the low and high distance sides of G ( R ) . For the larger values of the coordination number, the low distance side of G ( R ) becomes dominant. Conclusions
From the point of view of the theorem of corresponding states and the molecular model which it implies, the experimentally determined thermodynamic properties of argon can be taken as defining a simple fluid. iVriting the equations of state for such a fluid in terms of the radial distribution function then permits the thermodynamic properties to be related to the structure of the fluid. An examination of these concepts and the available data has led to the following conclusions: A set of configurational (residual) thermodynamic properties can be so defined as to vanish in the limit of zero density and conform to a Gibbsian formalism completely analogous to the ordinary thermodynamic formalism of a fluid phase. From temperatures considerably above the critical to very near the critical, the isotherms for the configurational properties show no marked change in character in the region of the critical density. I n particular, the inflection points typical of Pv isotherms are absent. If, further, intrinsic configurational properties are defined by dividing the configurational properties by the density, the isotherms corresponding to the Planck and grand Massieu functions are especially simple in form. Below the critical temperature (reduced temperatures between 0.9 and l . O ) , the vapor and liquid branches of the isotherms for these intrinsic functions are of the same shape and approximately parallel to the sections of the isotherms above the critical which have corresponding densities. This behavior is also exhibited by isotherms derived from quasitheoretical equations of state, such as that of van der Waals. By employing configurational properties, the “loop” characteristic of the unstable region lying within the coexistent line is eliminated in these cases and presumably also for real fluids. The normal behavior of the configurational properties in the neighborhood of the critical point demonstrates that only the short range structure of the fluid contributes to these properties. This is consistent with the expressions which relate those properties to integrals involving the intermolecular potential and virial functions. T h e configurational properties associated with the internal VOL. 5
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M A Y 1966
231
energy and the Pt product can be combined, under the assumption of a Lennard- Jones 6-1 2 intermolecular potential function, to give a rough estimate of the coordination number of the fluid. The agreement with evidence derived from x-ray and neutron scattering measurements is reasonable. The behavior of the estimated coordination number as a function of temperature and density suggests that the true coordination number is somewhat insensitive to temperature. At the critical density the coordination number appears to be about 4. Acknowledgment
I t is a pleasure to acknowledge the contributions of George
C.Wallick, who performed the analysis and programming for the computations of the van der Waals coexistence curve. Appreciation is also expressed to the Socony Mobil Oil Co., Inc., for permission to publish this paper. Nomenclature
E
minimum value of intermolecular potential function G(R) = radial distribution function G(R, 5 ) = radial distribution function with one molecule incompletely coupled h = Planck’s constant k = Boltzmann‘s constant m = molecular mass ‘V = number of molecules IV, = Avogadro’s number = average number of nearest neighbors (coordination number) R1’’ = Lennard- Jones structural parameters defined by Equations 21 P = fluid pressure = distance between molecular centers R = distance for minimum in intermolecular potential Ro S = molar entropy S* = molar configurational entropy T = temperature U = molar energy (internal) V = volume U = molar volume = mean thermal de Broglie wave length A @(R) = intermolecular potential function = chemical potential (molar Gibbs function) M 5 = coupling parameter =
n., nl’,
REDUCED PROPERTIES = = = = = = = = = =
232
intrinsic configurational energy (internal) configurational energy (internal) configurational chemical potential configurational entropy temperature intrinsic configurational Planck function intrinsic configurational entropy configurational pressure density intrinsic configurational grand Massieu function
l&EC FUNDAMENTALS
W
= configurational grand canonical potential
SUBSCRIPTS C
= values pertaining to critical point
e
= configurational properties defined as in (77)
f
=
configurational properties defined as in (37)
literature Cited
(1) Barker, J . A., Fock, W., Smith, F., Phys. Fluids 7 , 897 (1964). (2) Brown, \V.B., Rowlinson, J. S., Mol. Phys. 3, 35 (1960). (3) Callen, H. B., “Thermodynamics,” Chap. 5, It’iley, New
York, 1960. (4) Curl, R. F., .Tr., Pitzer, K. S.,Ind. Eng. Chem. 50,265 (1958). (5) Danon, F.: Pitzer, K. S.,J . Phys. Chem. 66, 583 (1962). (6) DeBoer, J., Rrpts. Progr. Phvs. 12, 305 (1949). (7) DeBoer, J., Michels, A., Physica 5, 945 (1938). (8) Fender, B. E. F., Halsey, G. D., Jr., J . Chem. Phys. 36, 1881 (1962). ( 9 ) Fisher, M. E., J . Math. Phys. 5 , 944 (1964). (10) Fixman, M., Adcan. Chem. Phys. 6 , 175 (1964). (11) Fowler, R. H., Guggenheim, E. A,, “Statistical Thermodynamics,” Chap. 7, Cambridge University Press, Cambridge, 1939. (12) Furukawa, K.? Rrpts. Progr. Phys. 25, 395 (1962). (13) Guggenheim, E. A , , J . Chem. Phys. 13,253 (1945). (14) Guggenheim, E. A . , McGlashan, M. L., Proc. Roy. Soc. (London) A255,456 (1960). (15) Hill, T. L., “Statistical Mechanics,” Chap. 5, Sec. 28, McGraw-Hill, Kew York, 1956. (16) Ibid., Chap. 6, Sec. 30. (17) Ibid., Chap. 6, Sec. 35. (18) Hirschfelder, J. O., Buehler, R. J., McGee, H. A , , Jr., Sutton, J. R., Ind. Eng. Chem. 50, 375, 386 (1958). (19) Jones, G. O., rYalker, P. A . , Proc. Phys. Soc. B69, 1348 (1956). (20) Kestner, N. R., Sinanoilu, O., J . Chem. Phys. 38, 1730 (1963). (21) Kirkwood, J. G.?Ibid., 3, 300 (1935). 122) Kruh. R. F.. Chem. Rem. 62. 319 11962). (23j Landau, L.’ D., Lifschitz: E. ’M., ’“Statistical Physics,” Chap. 8, Pergamon Press, London, 1958. (24) Levelt, J. M. H., Cohen, E. G. D., “Studies in Statistical Mechanics,” Vol. 11, Part B, North-Holland, Amsterdam, 1964. (25) Michels, .4.,Geldermans, M., DeGroot, S. R., Physica 12. 105 (1946). (26) Michels, A , , Levelt, J. M., Graaff, W. de, Ibid., 24, 659 (1958). (27) Michels, A., Levelt, J. M., LVolkers, G. J., Ibtd., 24, 769 (1958). (28) Myers, A. L., Prausnitz, J. M., Ibid.,28, 303 (1962). (29) Pings, C. J., Jr., Sage, B. H., Ind. Eng. Chem. 49, 1315 (1957). (30) Ibid., p. 1321. (31) Pitzer, K. S., J . A m . Chem. SOC. 7 7 , 3427 (1955). (32) Pitzer, K. S., J . Chem. Phys. 7 , 583 (1939). (33) Pitzer, K. S., Lippmann, D. Z., Curl, R. F., Jr., Huggins, C. M., Petersen, D. E., J . A m . Chem. SOC. 7 7 , 3433 (1955). (34) Ricci, F. P., Nuovo Cimento 16, 532 (1960). (35) Rice, S.A,, Kirkwood, J. G., J . Chem. Phys. 31, 901 (1959). (36) Rowlinson, J. S., Trans. Faraday SOC. 51, 1317 (1955). (37) Rowlinson, J. S., “Liquids and Liquid Mixtures,” Chap. 8, Butterworths, London, 1959. (38) Rowlinson, J. S.,“Properties of Real Gases,” in “Handbuch der Physik,” Vol. 12, Springer-Verlag, Berlin, 1958. (39) Salem, L., Mol. Phys. 3, 441 (1960). (40) Temperley, H. N. V., “Changes of State,” Chap. 4, CleaverHume Press, London, 1956. RECEIVED for review July 1, 1965 ACCEPTEDDecember 23, 1965
