The Many Faces of van der Waals's Equation of State J. G. Ebethart University of Colorado, Colorado Springs. CO 80933
Despite their acknowledged quantitative limitations, twoparameter equations of state continue to enjoy great popularitv because of the relative simnlicitv with which thev represent thep-u-T (pressure-molar volume-temperature) hehavior of a fluid. Some of the more familiar two-varumeter equations are shown in Table 1.In all these equaGons band a are the two parameters, with b providing a measure of the size of the molecules composing the fluid and a providing a measure of the attractive force between the molecules. Although the two parameters are abbreviated with the same two symbols in each equation in Table 1, b and a do not necessarily have the same dimensions, numerical values, or theoretical interpretation. Despite the availability of theoretical means for calculating b and a for some of these two-parameter equations of state, all the equations provide better agreement with experimental p-u-T hehavior if b and a are calculated from experimental data ( I ) . The most commonly used experimental data are the critical constants of the fluid of interest, p,, u,, and T,. These three properties are used in conjunction with the three critical conditions for a fluid as a means of determining the constants in a two- or three-parameter equation of state, or as constraints on the constants in an equation of state with four or more parameters. In the case of a two-parameter equation of state these critical conditions produce an unusual state of affairs-the three critical conditions (equations) overdetermine the two adjustable parameters. Thus, there exists no set of values of b and a that can simultaneously satisfy all three critical conditions (and is thus also consistent with all three critical constants). There are two approaches that have been used to remedy this situation. The first is to discard one of the three equations, while the second involves retaining all three equations
. -
906
Journal of Chemical Education
Table 1. Two-Parameter Equatlons of State van der Waais: p = RT/(v- b)
- a/?
Bemelot: p = RT/(v- b) Redlich-Kwmg: p = RTIW- b)
IT*
- dT1"2r(v+
b)
Percus-Yevick: p =~77?
@)/(v- b)'
Camahan-Starling: p = R77? bvt
ff - b3/v)l(v-
+ bv+
+
- E& b)'-
a/?
and treating the molar gas constant R as a third adjustable parameter. Either approach leads to a system of equations where the number of equations equals (rather than exceeds) the numher of unknowns. Thus, there are a variety of formulas that can he developed for finding band a fromp,, u,, and T,. Guggenheim (2) appears to he the first to have noted that there are various ways of calculating b and a from the critical constants. He also explicitly pointed out three methods based on the three possible combinations of two critical constants. (These will later he referred to as methods 1-3.) Levine (3) . . also has a very concise description of this same situation followed by a recommendation to use what is here called method 3. Martin (4) explored these three possibilities and their consequences in more detail and briefly commented on two other approaches (referred to later here as methods 5 and 6). The purpose of this paper is todescribe six different methods for determining b and a from the critical constants of a fluid. These six procedures will he illustrated here with van der Waals's equation. The same six choices are available for
all two-parameter equations of state, hut for many of them the mathematical details are far more complicated. The consequences of the choice one makes in how to deal with the overdetermination of b and a will he illustrated with a calculation of these two parameters and of the second virial coefficient for the simple fluid argon. It will be seen that in many cases the choice made has a dramatic effect on quantitative predictions of the equation of state. In addition it will he demonstrated that only one of the six methods is consistent with the common texthookderivation of the principle of corresponding states (PCS) from van der Waals's equation. The Crnlcal Condnlons The relationships that are most commonly used to evaluate the constants in a two-parameter equation of state are the three critical conditions. These three equations state that the critical isotherm has a horizontal inflection point at the critical point. Thus, if the equation of state is written as p = f(u, T),the three critical conditions are
The prime on the critical pressure indicates that this DroDerty is calculated or predGted value based on the equation of state, rather than the experimental value that is denoted hy PC. This result is often expressed in terms of the compressihility factor, Z = pulRT. The van der Waals equation of state thus predicts, from eq 12, a critical eompressihility factor of
a
where the prime on the left-hand side again indicates a prediction. The predicted value can be compared with the actual experimental value, which is
-
For most fluids Z, = 0.25-0.30 (5, 6), while for the simple fluids (like argon) which ohey the PCS, Z, 0.29 (5). Dividing eq 13 by eq 14 we obtain Z,'/Z, = 3/8Z, = p,'lp,, so that the predicted and the actual critical pressures ohey the relationships P: = (3/8Z,)p,
fUCuc,T J = 0
(2)
fsu(~e.T J = 0
(3)
where fdu, T ) = (aplav)~ and fzu(u,T ) = ( a 2 p / a ~ 2 ) ~ . The equation of state of van der Wads is, as indicated in Table 1, so that the first two derivatives are
--
For simple fluids, then, it follows that p,' 1.3 p,, i.e., the predicted critical pressure exceeds the actual value by about 30%. To summarize these results, then, we see that it is not possihle to find values of b and a for a fluid that will provide a horizontal inflection point at the experimental critical point (u,, T,,p,). However, method 1does provide a horizontal inflection noint at what mieht he called the nseudocritical point (u,, T,,p:). This point has the correct experimental value3 of u and T. hut a value of D that i* considerahlv- lareer than the correct experimental value. ~~~~
Evaluating the right-hand sides of eqs 4-6 at u = u, and T = T. yields, according to eqs 1-3,
As indicated previously, eqs 7-9 overdetermine b and a. Six possihle remedies for this situation will he described. They are designated as methods 1-6 for determining b and a from p,, T,, and u,. Method 1 It is eqs 2 and 3 or 8 and 9 that guarantee a horizontal inflection point, while eqs 1or 7 place the horizontal inflection point at the point (u,, T,, p,) in the p-u-T coordinates. The most common remedy for the overdetermination of eqs 7-9 is the discarding of eq 7. If eq 8 is divided by eq 9, the result is u, - b = 2uJ3, which yields Substitution of eq 10 into eq 8 then provides, after simplification, It is important to note that eqs 10 and 11allow b and a to he calculated from only two of the experimental critical constants-~, and T,. The experimental value of p, is not required. The question naturally arises, then, as to what value the equation of state predicts for the critical pressure, using eqs 1,4, or 7 (whichwere discarded). Setting u = u, and T = T, in eq 4, and substituting eqs 10 and 11 into eq 4 gives, after simplification, p: = 3RTJ8uC
(12)
(15)
~
~~
~
.
Method 2 In method 1 u, and T, are selected as the two critical constants that are assigned their correct experimental values. Clearly two other choices are possihle. In method 2 u, and p, are given their experimental values and T,' is calculated from eq 12, which is rewritten as Z,' = p,u,lRT,' = 318. Solving for T,' gives T: = 8pgJ3R
(17)
Comparison of eqs 17 and 14 then yields T i = (8Z,l3)Tc< T ,
(18)
Eqs 10 and 11 can he expressed in terms of u, and p, by replacing T, with the expression for T,' in eq 17, yielding
Thus, method 2 provides values of b and a that yield a horizontal inflection point at the pseudocritical point (u,, T,,, p,), where T,' is considerably smaller than T,. Method 3 The last choice of a pair of critical constants for experimental assignment is T, and p, with u,' ohtained from the eq 12 modification Z,' = p,u,'lRT, = 318. Thus, in method 3 while comparison with eq 14 yields Eqs 10 and 11 can he expressed in terms of T, and p, by replacing u, with the expression for u,' in eq 21, which gives b = RT,I8p,
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907
Method 3 then provides values of b and a that yield a horizontal inflection point at the pseudocritical point (u,', T,, p,), where u,' is considerably larger than u,.
Then division by p. and rearrangement gives
Method 4
There is one other procedure based on the van der Waals prediction of Z: = 318, and it involves the radical departure of using the correct values of all three critical constants and requiring the molar gas constant to satisfy the modified eq 12, Z,' = p,uJRIT, = 3/8. Here van der Waals's equation is considered to have three adjustable parameters rather than two. Ordinarily R is not considered an adjustahle parameter, but rather is defined as the constant
All fluids approach this ideal behavior in the limit of very low pressure. Clearly van der Waals eq 4 obeys this limiting condition because a s p 0, u m, u - b u, and aIu2 0. However, if it is desired to have the horizontal inflection point at the correct experimental value of all three critical constants (u,, T.,p,), this can be accomplished by sacrificing the usual limiting behavior and assigning a value to the pseudomolar gas constant of
- -
-
-
which yields eq 33 via the reduced variable definitions of eqs 3CL32. Without defining R through eq 26, it is impossible to derive eq 33. It should be pointed out that a working statement of the PCS is really much broader than eq 33, because no particular equation of state is actually required. A version the author favors is as follows: For each fluid a different p-u-T relationship exists. (The van der Wads equation serves as an example because b and a have different values for each substance.) However, if reduced variables are used, there is a class of suhstances, called simple fluids, that obey the same r r 4 relationship. Thus, it is possible to predict the p o - T hehavior of any simple fluid from that of any other simple fluid, provided the critical constants of both are known. Method 5
Methods 1-4 exhaust all the possihle ways of determining b and a hased using eqs 8 and 9 and discarding eq 7. All four approaches provided a horizontal inflection at either a pseudocritical point or the actual critical point. Next we will consider using eqs 7 and 8 and disregarding eq 9. This choice resembles the s~inodalooints used to calculate a limit of -~ superheating for a liquid or supercooling for a vapor, because ( a.~ l ~. -u ~ ~ v a n while i s h e (d201au%does s not 17). Examination of eqs 7 Ad 8 shows clearly that in method 5 all three critical constants are required for the calculation of b and a. Solving eq 9 for a yields ~
Comparison with eq 14 yields Thus, R'is considerably less than R. Equations 10 and 11can be expressed in terms of the critical constants by replacingR with the expression for R' in eq 25, giving
~~
~~~~
~
~~~~~~~
Substitution of eq 34 into eq 8 gives, after rearrangement, a quadratic in u, - b, This approach to calculating b and a probably strikes the reader as highly unconventional. However, it is exactly this procedure of giving up the correct limiting hehavior of a fluid that is an unstated assumntion in the usual textbook derivations of the PCS from vanher Waals's equation. This derivation will be considered next asan adiunct to the discussion of method 4. The Principle of Corresponding States
The usual derivation of the PCS begins with van der Waals's equation of state, eq 4, introduces the reduced volume, temperature, and pressure,
6 = ulu,
(30)
0 = TIT,
.
(31)
=PIP,
(32)
and using consequences of the three critical conditions (developed here previously) proceeds to the reduced van der Waals equation
.
- 113) - 31@
= (8813)/(6
(33)
It can immediately be noted, however, that the molar gas constant R is nowhere to be seen in eq 33, and, thus, the reduced van der Wads equation does not have the perfect gas limiting behavior at low pressure. In fact, it is easily shown from eqs 33 and 26 that as p approaches zero pulT approaches R' rather than R. Looking at the details of the derivation, we see that it is invariably hased on substituting eq 26 for R in eq 4. Thus, eq 33 is obtained via the substitutions of eqs 26,28, and 29 908
Journal of Chemical Education
+
pJu, - bI2 - RTJu, - b) RTcuJ2 = 0 which has the solution b = u,
(35)
- [RT. * (R~T: - 2RTcu~c)112]12~c
or simplifying with eq 14 The choice of the plus sign in the quadratic formula gives b < 0, while the minus sign yields a physically realistic b > 0. Thus, the plus sign was eliminated in eq 36. Method 5 then requires first the use of eq 36 for b, followed by eq 34, which gives a in terms of b. The isotherm that is assigned the critical temperature has a minimum and a maximum (it has the van der Wads loop associated with a subcritical isotherm), and the above values of b and a place the critical point at the minimum. Method 5 thus provides values of b and a that yield a horizontal tangent (but not an inflection point) at the true critical point ( ~ cTc, , PC). Method 8
Finally, the last choice possible for dealing with the overdetermination of b and a is the use of eqs 7 and 9 and the disregarding of eq 8. Again it is apparent that all three critical constants are required for the calculation of b and a. Solving eq 9 for a gives a = RTCu,'I3(u, - b)3 (37) Substitution of eq 36 into eq 7 yields a cubic equation in u, b,
Table 2.
Experimental
and Predicted Crltlcal Constants for Argon
Table I Vlrlal Coettlclents and Boyle Temperatures Method
Fmpenies experimental
100 K
300 K
B/cm3mol-' 500 K
700K
TB/K
predicted 1 2 3 4 5 6
T; = 117.18K
T, = 150.75 K p, = 48.1 am
v. = 7 4 . cm3nm-' ~ ~ R = 82.0575 am cm3~-'mol-' 2 . = 0.291,
p; = 61.8, am v: = 96.4, cm3mol-'
R = 63.7, am crn3~-'ml-' 2 , ' = 0.375
Exper
Table 9. Comparlson of van der Waals Constants lor Argon
The equation can be solved numerically for b and has three real roots. The smallest value of u, - b gives b > u,, which is physically unrealistic. The largestvalueof u, - b yields b < 0, which is also unacceptable. The middle root, however, yields b values in the range 0 < b < u,, which is physically realistic. The isotherm that is assigned the critical temperature in method 6 has (as with method 5) a minimum and a maximum. The values of b and a provide an inflection point (but not a horizontal tangent) between the two extrema at the ~ positive at true critical point (u,, T,, p,). Since ( a p l a u ) is this inflection point, method 6 does have the disadvantage of providing a mechanically unstable (7)critical point. A Comparlson of the SIX Methods
Having described the six methods of finding the two van der Waals constants from the three critical constants of a fluid, it is now possible to calculate b and a for the simple fluid argon and to see the effect the six choices have on the predicted second virial coefficient of the fluid and its Boyle temperature. t he critical properties of argon are, according to Sengers ( 8 ) .t , = -122.3 OC. D, = 48.1 atm. and the critical densitv D, = 0.533 g em3. ~ a s e d on these results and an argon mil; mass of 39.948 e mol-I, the required experimental and predicted constan& are given in 'l'ible 2. using these const& and the formulas derived previously, b and o values are calculated for the six methods and are summarized in Tahle 3. The variability of the resulting van der Waalsconstants is 29% forb and 66% for a. Clearlv t h choice ~ of method has a large impact on the values of tde two parameters. The various sets of values of b and a yield different predictions for the properties of argon. The second virial coefficient, B, (in the series expansion of Z in powers of 110)is used hereto illustrate. If Z = g(o, T),where a = l l u is the molarity of the fluid, then B is defined as B = gJ0, T )
(39)
where g,(a, T)= (aZ/ao)~.Equation 4 can be written as Z = U/(U- b) - aIRTu, so that
Z = g(v, TI = l l ( 1 -
b r ) - odRT
(40)
Then, since setting a = 0 in eq 41 gives Values of B for a range of temperatures were calculated from
-102.2 -73.8 -131.4 -102.1 -123.6 -116.6
-17.4 -7.9 -23.6 -17.4 -21.6 -20.9
-0.4 5.2 -0.6 -0.4 -1.2 -1.7
6.8 10.9 8.8 6.8 7.5 6.5
508.8 395.4 508.8 508.8 520.3 532.3
-187.9
-15.4
6.8
15.4
411.8
eq 42 for methods 1-6. The results are given in Table 4, along with the predicted Boyle temperature, TB,which is the temperature at whichBvanishes. From eq42, b - a/RTe = Oand Experimental values of B and TBare also provided here for comparison (91, even though eq 42 does not provide a temperature dependence that is adequate for representing B. Just as with the values of b and a themselves, the predictions of B are dramatically influenced by the method used to determine b and a. Which Method To Use? After observing the variety of ways that the two parameters in an equation of state can he determined from some or all of the critical constants, it is only natural to ask which method is best. The answer appears not to be simple and depends on what property wilibe calculated from the equation of state and what portion of the p-u-T space will he used. In comparing what here has been called methods 1-3, Levine (3) observes that u, is generally known with less accuracy than T. and p,. Thus, he favors method 3 over methods 1 or 2. If it was desired to calculate the vapor pressure curve of the fluid up to the critical point, then method 3 would again be superior to methods 1 and 2 because the curve would terminate at the correct temperature and pressure. However, if a coexistence curve were to be computed up to the critical point, then method 1 with its correct values of T, and u, (or p,) would be superior to methods 2 and 3. The regions of the p-WT space selected for a calculation are also of great importance in the choice of a method. If an equation of state were employed in a region of sufficient fluid dilution that ideal eas behavior was to be exoected. then methods 1-3 would probably be superior to method 4. However. in the neirhhorhood of thecritical -point.. method 4 is probably superioito methods 1-3. The advantages and disadvantages of the various approaches can be summarized with respect to the various features that are desirable for a simple equation of state: (1) ideal gas limiting behavior-methods 1 3 , 5 , and 6; (2) horizontal inflection point at the experimental critical p o i n t method 4; and ( 3 jwhen two critical constants are substituted into the equation of state, the third critical constant is obtained-methods 4 4 . Ultimately the user of a two-parameter equation of state must carefullv analvze the intended a~olicatiouof the eauation and try t o weigh this purpose G i n s t the advantages and disadvantages inherent in each method. Llterafure CMed 1. Barker, J. A.; Hend-n. D . J Chom. Educ. 1968.45.2-6. 2. Guggsnbim.E. A. Th~rmodvnomics,3rd ed.;North Holland: Amsterdam, 1957:p 165. 3. Levine, I. N. Physical Chembtry. 3rd ed.; McGraw-Hill:New York, 1988;p 217. I . Martin. J. J. I" Applied Thsrmodynomics: Guahee. D. E.. Ed: Americao Chemical Soeiety: Washington, OC. 1968, pp61-82. 5. Mathews, J. F..Chem.Rou. 1972.72,71-1W. 6. Kudchadker, A. P.; Alani, G. H.; Zwolinski, B. J. Chem. Rou. 1968,68,659-785. 7. Ebevhart. J. G. J. ColloidInlerloee Sei. 1976.56.262-9. 8. Sengors, J. M. H. L., privatemmmunicatian. ref. 5. 9. McClashen, M. L.. Chemical Th~rmodynamica:Academic: London. 1919:p 203.
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