Ind. Eng. Chem. Res. 2010, 49, 7687–7692
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Critical Analysis of Maxwell’s Equal Area Rule: Implications for Phase Equilibrium Calculations Karthikeyan Rajendran and R. Ravi* Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai-600036, India
The fundamental flaws inherent in Maxwell’s equal area rule are analyzed. Using an alternative principle that overcomes the limitations of Maxwell’s rule, general equations are derived for determining the saturation pressure and other properties associated with the liquid-vapor phase transition given an arbitrary equation of state. The equivalence of the equation for saturation pressure to those available in the literature is proved. It is found that the alternative principle predicts saturation pressures that are significantly different from those calculated using Maxwell’s rule. These results are based on numerical studies on three different equations of state and fifteen substances. Similar results are obtained for the vaporization entropy as well. Introduction The common procedure for calculating the saturation pressure (psat) of a substance at a given temperature from an equation of state (EOS) is based on the equality of chemical potential of the liquid and vapor phases. The geometric representation of the criterion arising out of this procedure is the well-known Maxwell’s equal area rule according to which the p ) psat line cuts the p-V isotherm in such a way that the magnitudes of the areas between the line and the curve above and below the line are equal (Figure 1). This rule may be represented analytically as psat(VV1 - VL1 ) -
∫
V1V
V1L
p(V;T1) dV ) 0
(1)
where VL1 and VV 1 are the molar specific volumes of the saturated liquid (L) and vapor (V), respectively, at a temperature T1 at which psat is to be determined. In a typical derivation of the rule,1 the chemical potential of the possible vapor state is calculated by integrating from the ideal gas state to VV 1 . To calculate the chemical potential of the corresponding liquid state, the integration is further continued up to VL1 . In this process, the unstable portion of the EOS, referred to as the “spinodal” region (the portion of the isotherm along which ∂p/∂V > 0), is traversed. Implicit in this procedure is the assumption that thermodynamic relations are valid in the spinodal region. Davis and Scriven,2 in their review of the theory of interfaces, point out that “it is generally accepted that Maxwell’s tie-line constructions of coexisting phases are correct and that thermodynamic functions can be computed by integrating the equation of state over any desired density domain”. However, Aifantis and Serrin3,4 have expressed doubts over this claim. They adopt a continuum approach to interfaces, thereby obviating the need for using thermodynamic arguments in the spinodal region. Their approach has a molecular component as well. Using such a framework, they prove that the molecular theory of interfaces discussed in the work of Davis and Scriven “can be inconsistent with Maxwell’s rule unless certain very special conditions are satisfied”.3 Questions about the validity of Maxwell’s rule have been raised within the context of classical thermodynamics as well. For instance, Tisza5 objects to the application of the rule on * To whom correspondence should be addressed. Phone: 91-4422574167. Fax: 91-44-22570509. E-mail address:
[email protected].
the grounds that it involves integration along the unstable branch of the isotherm that is not accessible to experimental observation. Further, he argues that the equation of state p ) p(V, T) is not a fundamental relation and hence by itself cannot yield the saturation pressure. Energetic information in terms of the specific heats must be provided. These ideas are reflected in the work of Kahl6 who uses an integration path that avoids the spinodal region and instead includes the critical point. By doing so, Kahl arrives at a governing equation for psat for the van der Waals (vdW) EOS that reduces to Maxwell’s rule when the specific heats of the liquid and the Vapor are equal. The fact that the specific heats of liquids can sometimes be even an order of magnitude greater than that of gases gives an indication of the extent to which Maxwell’s rule could be in error. Application of Kahl’s principle to phase equilibrium calculations have been few and far between. Nitsche,7 who has derived a generalized Kahl’s equation valid for an arbitrary EOS, argues against using experimental psat data to adjust parameters in the equation of state according to the flawed Maxwell’s rule. However, Nitsche does not apply Kahl’s rule for predicting psat; this requires specific heat data which the author does not use. Instead, he calculates a factor ∆(T) that represents the correction to the liquid phase Helmholtz free energy as implied by Kahl’s rule. A similar viewpoint is adopted by Chen et al.8 More recently, Serrin,9 using a rigorous mathematical framework, also obtains a generalized Kahl’s equation valid for an arbitrary EOS. From this equation, Serrin obtains conditions
Figure 1. Maxwell’s tie-line construction.
10.1021/ie100571m 2010 American Chemical Society Published on Web 07/06/2010
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Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010
for Maxwell’s rule to hold. The equality of liquid and vapor specific heats obtained by Kahl for the vdW EOS follows as a special case. In addition, a second-order differential equation for psat is derived. Serrin also proposes a four-parameter modified vdW EOS which in combination with the generalized Kahl’s equation and approximate specific heat data is shown to predict psat to a remarkable degree of accuracy for the four substances studied. The focus in the present work differs from the abovementioned contributions in the literature and is motivated by Kahl’s conclusion6 that Maxwell’s rule does not result in much error for saturation pressures but does so for vaporization entropies. However, Kahl uses a value of 3R/2 for the gas specific heat and 5R/2 for the liquid specific heat. It is an open question as to how his conclusions would be modified if actual specific heats (especially for the liquid) are used. More importantly, the flaws underlying Maxwell’s rule put a question mark on the conclusions reached so far regarding the predictive capabilities of the various equations of state. Typically, in a two-parameter EOS, the two parameters are determined for a particular substance by matching the actual critical pressure and temperature of the substance with those corresponding to the model substance represented by the EOS. Alternately, a third parameter can be incorporated into the EOS for greater flexibility. For instance, in the Peng-Robinson (PR) EOS,10 the parameter representing the attractive interaction is given by aR(T;ω) where a is a constant and ω is the acentric factor. The function R is determined by forcing the saturation pressure predicted by the EOS to match the experimental data. The above procedure is implemented based on the assumption that Maxwell’s rule is valid. A natural question would be: How would these results be modified if Kahl’s rule is applied? Further, one may ask: how good are the two parameter equations of state in predicting phase change properties? So far, their performance has been assessed using the flawed Maxwell’s rule. Clearly, to assess an EOS accurately, the correct rule must be applied. In this work, we address these questions in a systematic manner. First, we derive a set of equations for calculating phase change properties using Kahl’s principle, given an arbitrary EOS. We adopt a thermodynamic path that includes the ideal gas state so that the resulting equations involve the ideal gas specific heats which in turn are readily available. We then prove the equivalence of our equation for determining psat with those available in the literature.7,9 In addition, we sketch the derivation of the corresponding equations implied by Maxwell’s recipe so that the fundamental differences between the two approaches may be easily seen and the rationale behind the objections to Maxwell’s rule understood. Then the prediction of the two rules are compared for three different equations of state: the vdW EOS, the PR EOS, and the “uncorrected”, two-parameter, PR EOS which is obtained from the PR EOS when R ) 1. Both the saturation pressure and the vaporization entropy are calculated. Three groups of substances are chosen for this study: (a) alkanes, (b) alkenes/alkynes, and (c) a group of alcohols/ inorganic substances. Our calculations reveal that in many instances the predictions of the two rules are quite different. In fact, in many cases, the uncorrected PR EOS as well as the vdW EOS yield better predictions for the saturation pressure and vaporization entropy as compared to the PR EOS when Kahl’s rule is employed. This provides us with a motivation to investigate the possibility of carrying out a fitting procedure analogous to that employed to obtain the original PR EOS but
Figure 2. Thermodynamic path associated with Maxwell’s rule and that employed in this work. (inset) Kahl’s path on a T-V diagram.
now using Kahl’s rule rather than Maxwell’s rule. The problems associated with this fitting procedure are identified. In view of the continued widespread acceptance of the Maxwell’s rule in the chemical engineering literature, we hope that the theoretical and numerical results presented in this article would be of value from both a fundamental and a practical perspective. Phase Equilibrium Properties Implied by Kahl’s Principle. The conditions of equilibrium for two-phase coexistence of a pure substance imply the equality of temperature, pressure, and chemical potential. Given that the chemical potential of a homogeneous part of a substance is equal to its specific molar Gibbs free energy (g), we may express the condition of equilibrium as g(T1, VL1 ) ) g(T1, VV1 )
(2)
Since g ) ε + pV - Ts, eq 2 reduces to ∆ε - T1∆s + psat∆V ) 0
(3)
L V L V where ∆ε ) ε(T1,VV 1 ) - ε(T1,V1 ) ≡ ε - ε , ∆s ) s(T1,V1 ) L s(T1,VL1 ) ≡ sV - sL, ∆V ) VV V , ε, and s are the specific 1 1 internal energy and specific entropy respectively. We now calculate ∆ε and ∆s using suitable paths (Figure 2) that include the critical point specified by Tc and Vc. We illustrate the procedure for ∆ε. First, we rewrite ∆ε as
∆ε ) [εV - εc] + [εc - εL]
(4)
where εc ) ε(Tc,Vc). The first term in eq 4 is calculated using path “1” in Figure 2 while the second term is calculated using path “2”. While path “2” is that employed by Kahl, path “1” is a slight modification of the corresponding path employed by Kahl (shown in the inset of Figure 2) in that it includes the ideal gas (IG) states at T1 and Tc. Thus, we get εV - εc )
∫
∞
Vc
∂ε (V;Tc) dV + ∂V
∫
T1 IG cV (T)
Tc
dT -
∫
∞
V1V
∂ε (V;T1) dV (5) ∂V
and εc - εL )
∫
Tc L cV (T;VL1 )
T1
dT +
∫
Vc
V1L
∂ε (V;Tc) dV ∂V
(6)
Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010 IG
where cV is the specific heat at constant volume of the ideal gas associated with the given substance. Adding eqs 5 and 6, we get ε -ε ) V
L
∫
∞
∂ε (V;Tc) dV ∂V
V1L
∫ ∫
which is a consequence of eqs 9 and the definition of cV. Integration of eq 16 leads to
2
Tc
1
1
∂s (V;T ) dV + ∫ ∂V ∫ T1 [c (T;V ) - c
∞
L V
T1
L 1
IG V (T)]
dT (8)
Noting that )T
( ∂T∂p )
V
( ∂V∂s ) ) ( ∂T∂p )
- p and
T
(9)
V
we obtain εV - εL )
∫
∞
V1V
∫
∞
V1L
[
∂p Tc (V;Tc) - p(V;Tc) dV ∂T
]
∂p T1 (V;T1) - p(V;T1) dV + ∂T
[
]
∫
Tc
T1
[cVL(T;VL1 ) cVIG(T)] dT (10)
sV - sL )
∫
∞
V1L
∂p (V;Tc) dV ∂T
∫
∫
∞
V1V
Tc
T1
∫
∂p (V;T1) dV + ∂T
V1V
V1L
p
(VV1
-
∫
[
∞
V1L
VL1 )
-
∫
V1V
V1L
(Tc - T1)
cVV(T;VV1 ) + T
∫
V1V
V1L
1 L [c (T;VL1 ) - cVIG(T)] dT (11) T V
where
∫
T1
(
p
(VV1
∫
-
V1V
V1L
VL1 )
-
∫
)
]
where
∫
Tc
T1
(
[cVL(T;VL1 ) - cVV(T;VV1 )] 1 -
)
T1 dT (15) T
Equation 14 corresponds to eq 2 in Nitsche’s7 article. To establish the equivalence between eqs 12 and 14, we note that ∂cV /∂V ) T ∂2p/∂T2
∫
V1V
V1L
∂2p (V;T) dV ) 0 ∂T2
(20)
(21)
cVL(T;VL1 ) ) cVV(T;VV1 )
(22)
cVL(T;VL1 ) ) cVV(T;VV1 ) ) cVIG(T)
(16)
(23)
This is not surprising because the vdW EOS, the one considered by Kahl, satisfies eq 21. So does the uncorrected PR EOS. Equation 21 implies that (∂p/∂T)V is independent of temperature. This condition can be shown equivalent to the conditions that T(∂p/∂T)V - p, (∂ε/∂V)T, and (∂s/∂V)T are each independent of temperature. When these conditions are satisfied, eqs 10-12 reduce to εV - εL )
p(V;T1) dV + φ (T1 ;Tc) +
∂p (Tc - T1) (V;Tc) + p(V;T1) - p(V;Tc) dV ) 0 (14) ∂T
φV(T1 ;Tc) )
)
we recover Kahl’s condition, namely,
V
[
](
cVL(T;VL1 ) -
In combination with eq 17, we obtain
VV
sat
[
∂2p )0 ∂T2
T1 [cVL(T;VL1 ) - cVIG(T)] 1 dT (13) T
and the term ∫VL11 ) dV has been added and subtracted to bring out the difference with Maxwell’s rule (eq 1) clearly. Equation 12 is an expression of Kahl’s principle to calculate psat for a general EOS. Equations 10 and 11 are corresponding equations that yield the change in internal energy and in entropy across the phase transition. If Kahl’s path had been used (this is the path used by Nitsche7 and Serrin9 as well), the resultant equation would be V1V V1L
Tc
T1
In the special case when
p(V;T1) dV + φ (T1 ;Tc) +
]
∫
T1 ∂2p (V;T) dV 1 dT ) 0 (19) 2 T ∂T
cVL(T;VL1 ) - cVV(T;VV1 ) + T
∂p (V;Tc) + p(V;T1) - p(V;Tc) dV ) 0 (12) ∂T
φIG(T1 ;Tc) )
∂p (V;Tc) + p(V;T1) ∂T p(V;Tc) (18)
which is the equation obtained by Serrin9 (eq 4.1 in that article). While eqs 12, 14, and 19 are all equivalent representations of Kahl’s basic idea, eq 19 stands out when viewed in relation to eq 1. It immediately leads to a generalization of Kahl’s condition for Maxwell’s equal area rule to hold (Corollary 4.1 of the work of Serrin), namely
IG
Tc
- T1)
p(V;T1) dV +
Substitution of eqs 10 and 11 into the criterion, eq 3, leads to sat
(17)
one obtains eq 12. On the other hand, replacing the integrand in the last term of eq 14 by the left-hand side of eq 18 leads to psat(VV1 - VL1 ) -
T
∂2p (V;T) dV ∂T2
1
V1V
Tc
( ∂V∂ε )
c
2
∞
∂s (V;Tc) dV ∂V
V1L
T
Substituting eq 17 in eqs 15 and 14 and noting that ∂p (T;V) dT ) (T ∫T (T - T ) ∂T
∫
V1V
∞
1
Similarly, sV - sL )
∫
cVV(T;VV1 ) - cVIG(T) )
∞
∂ε (V;T1) dV + V1V ∂V Tc L [cV (T;VL1 ) - cVIG(T)] dT (7) T
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∫
V1V
V1L
[T ∂T∂p (V;T ) - p(V;T )] dV + 1
1
1
∫
Tc
T1
sV - sL )
∫
V1V
V1L
∂p (V;T1) dV + ∂T
psat(VV1 - VL1 ) -
∫
V1V
V1L
[cVL(T;VL1 )
∫
Tc
T1
- cVIG(T)] dT (24)
1 L [c (T;VL1 ) - cVIG(T)] dT T V (25)
p(V;T1) dV + φIG(T1 ;Tc) ) 0 (26)
Fundamental Flaws in Maxwell’s Rule: A Critical Analysis. It is instructive to compare eqs 10-12 or eqs 24-26 with the corresponding equations implied by Maxwell’s approach. Typically, they are derived from formulas for departure
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Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010
functions. For instance, the departure function for the internal energy at a given temperature (T1) and pressure (p1) is given by
Table 1. Errors in Saturation Pressures for Different Substances Using the Three Equations of State with the Rules of Maxwell and Kahl EOS
∫
(ε - ε )T1,p1 ) IG
V1
∞
[
∂p T1 (V;T1) - p(V;T1) dV ∂T
]
(27)
where p1 ) p(V1,T1). To evaluate εV - εL, one evaluates εV εIG by evaluating the above integral at V1 ) VV 1 and subtract from it εL - εIG which in turn is evaluated at V1 ) VL1 . This latter integral involves integrating all along the vapor segment of the subcritical isotherm, then through the unstable or spinodal region and finally along the liquid segment right up to VL1 (Figure 2). The resulting expression is εV - εL )
∫
V1V
V1L
[T ∂T∂p (V;T ) - p(V;T )] dV 1
1
1
(28)
van der Waals
uncorrected PR
Peng-Robinson
rule
Maxwell
Kahl
Maxwell
Kahl
Maxwell
Kahl
methane ethane propane butane cyclohexane ethene propylene butene acetylene methylacetylene ammonia carbon dioxide water methanol ethanol
96.2% 120.3% 132.8% 142.6% 280.9% 127.2% 174.5% 211.1% 178.3% 493.6% 623.8% 77.3% 463.5% 562.5% 675.6%
55.9% 5.6% 27.4% 7.6% 134.4% 9.7% 39.0% 93.9% 43.1% 108.8% 45.0% 5.2% 60.4% 148.8% 142.9%
73.6% 96.2% 108.3% 118.3% 231.4% 101.1% 141.4% 174.4% 147.5% 398.7% 488.9% 65.5% 383.9% 485.9% 586.8%
46.3% 6.4% 33.6% 15.6% 132.5% 15.3% 43.6% 93.3% 52.5% 130.6% 11.5% 11.6% 118.1% 222.3% 221.4%
0.6% 0.2% 0.5% 0.3% 0.2% 0.2% 0.6% 0.2% 0.3% 0.6% 0.7% 0.7% 1.4% 2.0% 2.7%
61.6% 33.4% 22.5% 33.8% 13.4% 30.8% 28.4% 17.2% 26.1% 36.6% 60.5% 24.2% 31.5% 18.5% 27.7%
A similar procedure leads to s -s ) V
L
∫
V1V
V1L
∂p (V;T1) dV ∂T
(29)
Substituting eqs 28 and 29 into eq 3 can be shown to lead to Maxwell’s rule for calculation of psat, namely, eq 1. It is clear from the above steps that the central feature of eqs 28, 29, and 1 is the integration along the unstable part of the isotherm to calculate liquid state properties. This is to be contrasted with eqs 10-12, 14, 19, or 24-26 which are obtained using a set of paths that bypass the unstable region. We now consider the special case represented by eq 21 in some detail. In this case, integration of eq 16 along the unstable portion of the isotherm from the vapor state to the liquid state leads to eq 23, the equality of liquid and vapor specific heats. Substitution of eq 23 into eqs 24-26 leads to the corresponding equations implied by Maxwell’s rule, namely, eqs 28, 29, and 1. Could this then be taken as justification for Maxwell’s prescription? The crucial point to note is that obtaining eq 23 from eq 16 again involves integrating all along the unstable part of the isotherm. However, the unreasonableness of this feature is starkly revealed in the “restrictive consequence”6 eq 23. The important point to note is that, while Kahl’s rule allows cVL to differ from cVIG, Maxwell’s rule requires them to be equal. Another striking feature of eqs 1, 28, and 29 is that, from them, all properties associated with phase change can be calculated with only the EOS as input. This is to be contrasted with the fact that even to calculate the property change between two states in which there is no change of phase, specific heat information is required apart from the EOS as input, in general. This is understandable because the EOS is not a fundamental equation.5 Thus it seems unreasonable to expect the EOS alone to yield psat and the other property changes associated with phase change. That specific heat information must be provided here also is clear from eqs 24-26. This is avoided in Maxwell’s scheme only because integration is extended into the liquid region through the unstable portion of the isotherm. As Tisza5 points out, “a spurious interpolation through the instable range is substituted for the missing information”. In summary, the main objection to Maxwell’s criterion stems from two factors. One, it leads to the obViously unphysical consequence eq 23 for an EOS that satisfies eq 21. Second, it results in the unreasonable situation that quantities associated with phase change can be calculated from EOS information alone without any specific heat inputs.
Numerical Results and Discussion. Three different equations of state are studied: the vdW EOS, the uncorrected PR EOS, and the PR EOS. The PR EOS10 is given by p)
aR RT - 2 V-b V + 2bV - b2
(30)
where
[ ( )]
R) 1+κ 1-
T Tc
2
and κ ) 0.37464 + 1.54226ω - 0.26992ω2 (31)
ω being the acentric factor. When κ ) 0, (which implies that R ) 1), we get the uncorrected PR EOS. Three different classes of substances are modeled using the above equations of state: (a) alkanes, (b) alkenes/alkynes, and (c) alcohols/inorganic substances. The saturation pressures as well as the vaporization entropies are calculated for these substances using the three equations of state and the two approaches, namely, Maxwell’s rule and Kahl’s rule. For calculation of saturation pressure, an iterative solution scheme is required. For implementing Maxwell’s rule, eq 1 is used. For implementing Kahl’s idea, eq 26 is used for the vdW EOS and the uncorrected PR EOS while eq 12 is used for the PR EOS. For calculating the entropy of vaporization, eqs 29, 25, and 11 are used. For the purpose of calculations, the specific heat data are taken from Tables 2-196 (liquid) and 2-198 (ideal gas state) and the critical constants are from Table 2-164 of ref 11. The relative error of the predicted values with respect to the experimental values is calculated at selected values of temperature between the normal boiling point and the critical temperature of each substance. This range is also the one adopted by Peng and Robinson.10 The average of these errors is what is tabulated for each substance in columns 1-6 (the column that contains the names of substances is not included in this numbering) of Tables 1 and 2. That is, the entries in the tables N correspond to (1/N)∑i)1 εi expressed as percent, where εi ) |xcalc - xexpt|/xexpt and x is either the saturation pressure or the vaporization entropy. The quantity N is the number of temperatures at which calculations are carried out for each substance. We first analyze the predictions of psat and entropy of vaporization based on Kahl’s rule. A comparison of columns 2, 4, and 6 in Table 1 reveals that for most of the alkanes, ethene, ammonia, and carbon dioxide, the predictions of the vdW and the uncorrected PR EOS are actually superior to that of the PR EOS. In the case of the vaporization entropy (columns 2, 4,
Ind. Eng. Chem. Res., Vol. 49, No. 16, 2010 Table 2. Errors in Vaporization Entropies for Different Substances Using the Three Equations of State with the Rules of Maxwell and Kahl EOS
van der Waals
uncorrected PR
Peng-Robinson
rule
Maxwell
Kahl
Maxwell
Kahl
Maxwell
Kahl
methane ethane propane butane cyclohexane ethene propylene butene acetylene methylacetylene ammonia carbon dioxide water methanol ethanol
40.0% 47.0% 50.3% 52.8% 53.5% 45.5% 174.5% 52.0% 52.1% 54.4% 56.8% 54.1% 58.3% 64.1% 68.2%
48.1% 17.2% 15.4% 6.2% 27.8% 5.2% 39.0% 25.4% 16.6% 18.4% 14.0% 8.0% 20.5% 31.2% 29.9%
34.0% 41.6% 45.2% 48.0% 49.1% 40.0% 141.4% 47.3% 47.3% 50.1% 52.6% 49.2% 54.2% 60.6% 65.1%
48.6% 14.7% 12.4% 4.7% 24.9% 3.9% 43.6% 22.4% 14.3% 17.0% 11.0% 5.4% 20.8% 31.4% 30.3%
1.0% 1.0% 0.6% 0.7% 0.6% 1.3% 0.6% 0.9% 0.7% 0.7% 3.1% 1.4% 3.5% 7.7% 1.2%
77.4% 20.5% 25.5% 37.9% 13.7% 33.2% 28.4% 16.0% 24.5% 19.9% 50.3% 38.7% 23.8% 20.6% 19.2%
and 6 of Table 2), except for a handful of substances, the vdW and the uncorrected PR EOS do better than the PR EOS. This occurs despite the latter having an additional parameter. The above results are to be contrasted with the predictions made using Maxwell’s rule (columns 1, 3, and 5 of Tables 1 and 2) in which case the PR EOS does uniformly better. With respect to psat, this is not surprising because the additional parameter (R) in the PR EOS has been determined10 so that the psat predicted by the equation (using Maxwell’s rule) matches the experimental data. Our results are an indirect verification of the accuracy of our calculation procedure with respect to Maxwell’s rule. The above conclusions should not be construed as recommendations for the use of the vdW or the uncorrected PR EOS for practical calculations. It is well-known that the vdW EOS, for instance, does not predict p-V-T data very well. Even in cases reported here where the vdW and the uncorrected PR EOS do better than the PR EOS, the percent error could still be in excess of acceptable limits. Our objective here is not so much to evaluate the predictive capabilities of EOS as to bring out the difference between the predictions of Kahl’s rule and Maxwell’s rule for a given EOS using three EOS as examples. The conclusions highlighted in the preceding paragraphs bring out this difference clearly. More insight in this regard is obtained by comparing columns 1 and 2, 3 and 4, and 5 and 6 in the two tables. With respect to Table 1, it can be seen that in a majority of the cases, there is a decrease in the percent error of more than 100% for the vdW EOS. For instance, in the case of ethane, butane, and ethene, the percent error in the predictions decreases from more than 100% to less than 10% with the vdW EOS, while for CO2, the error decreases from 77% to 5%. Thus, it is clear that the error in prediction of psat when using the vdW EOS is grossly exaggerated on account of use of the flawed Maxwell’s rule. Similar results are obtained with the uncorrected PR EOS. Thus, the inadequacy of the vdW EOS and the uncorrected PR EOS with respect to psat predictions is partly due to the inadequacy of the rule that is employed to calculate psat, namely, Maxwell’s rule. These results also indicate the necessity of using the correct values of the liquid specific heats. Kahl,6 using a value of 5R/2 found that there was not much difference between the predictions of the two rules for psat using vdW EOS. Here we find a significant difference in predictions when the same EOS is used to model real substances and the appropriate values of the specific heats are used. The specific heats of different substances
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Table 3. Constant Volume Specific Heats of Liquid and Gas at Normal Boiling Points substance methane ethane propane butane cyclohexane ethene propene butene acetylene methylacetylene ammonia carbon dioxide water methanol ethanol
normal boiling point (K) ideal gas cV (R) liquid cV (R) 111.6 184.4 230.8 272.4 353.0 169.1 225.0 291.0 189.0 249.8 239.7 216.0 373.0 337.7 351.4
3.01 3.17 6.29 10.02 14.58 3.09 5.35 9.07 3.19 5.54 3.11 2.94 3.09 4.63 4.75
7.07 11.66 12.93 18.21 21.20 8.16 11.16 15.29 9.72 12.29 23.31 8.90 9.14 10.88 11.34
at their normal boiling points, in units of R, are given in Table 3. The tabulated values clearly show that the liquid and ideal gas specific heats for all the substances are significantly different (sometimes much larger than R, in contrast to Kahl’s assumption), in clear violation of the condition given in eq 23. As far as the vaporization entropy is concerned, again, there is a consistent reduction in error when Kahl’s rule is applied (when compared with Maxwell’s rule) to the vdW EOS and the uncorrected PR EOS. However, on an average, the reduction is not as high as in the calculation of psat. The best predictions result when the PR EOS is used in combination with Maxwell’s rule. Thus, from an empirical point of view, the superiority of the three parameter PR EOS when used with Maxwell’s rule is clear. A distinct advantage is that this scheme does not require specific heat data as input. However, as is clear from Table 2, when the correct rule, namely Kahl’s rule is applied, the same EOS does not fare as well in predicting saturation entropies. Thus from a fundamental viewpoint, the PR EOS is not as good a model. The above discussion leads to the following question: If Kahl’s rule had been applied to find R(T) by fitting to experimental data on psat, would the resulting corrected EOS have been able to predict vaporization entropies more accurately? Such an EOS would not only have better predictive capabilities but also be fundamentally sound. It is worthwhile to recall the procedure adopted by Peng and Robinson10 in order to arrive at the correction factor R in eq 30. For each temperature in the range of interest, the corresponding value of R was found so that Maxwell’s rule is obeyed for the experimental value of psat. The resulting set of R’s was found to follow the temperature dependence given in eq 31. This procedure cannot be applied as such when using Kahl’s rule. This is because the additional term involving the equation of state (the first term in the volume integral in eq 12) would result in a term containing (dR/dT) evaluated at Tc. Thus a functional form for R(T) is required a priori. As a first choice, the form obtained by Peng and Robinson was chosen. This would mean that a parameter analogous to κ would have to be found for each substance. This parameter, however, varied with the temperature range studied effectively implying that the assumed form is not quite valid. Further, even within a narrow range of temperature, the best estimate of κ itself was not sufficiently accurate. Several other forms for R(T) also did not yield satisfactory results. In this context, the success achieved by Serrin9 is noteworthy and promising. Rather than try to modify parameter(s) in the equation of state so as to obtain a fit to experimental psat data, Serrin proposes a modified vdW EOS with two additional parameters. Serrin finds that saturation pressures are predicted
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with very good accuracy for water, nitrogen, carbon dioxide, and Freon-13 with reasonable models for the specific heats. To determine the parameters in the modified EOS, Serrin uses, apart from the critical constants, the experimental value of the slope of the L-V coexistence curve at the critical point. In any case, the real objective of this work was not to find a substitute for the existing PR EOS but to estimate the error that one would incur in assessing an EOS for its predictive capabilities when one uses the fundamentally flawed Maxwell’s rule. Our results amply demonstrate the necessity of using the correct rule based on Kahl’s principle if one were to arrive at an accurate assessment of an EOS. Summary and Conclusions The fundamental flaws inherent in Maxwell’s equal area rule have been analyzed. Using the idea provided in an earlier work by Kahl,6 explicit equations are derived for obtaining properties such as saturation pressure and entropy of vaporization from an arbitrary equation of state. The equivalence of the defining equation for saturation pressure derived here to other such equations available in the literature is proved. Numerical results are provided for the estimation of these properties for three equations of state and for a variety of substances. Our results show significant differences in the predictions of Maxwell’s rule as opposed to Kahl’s rule for a given equation of state. Kahl’s conclusion that there is not much difference between the two rules for saturation pressure calculations is not borne out by our calculations. The discrepancy in conclusion arises due to our use of realistic values for the specific heats. From an empirical viewpoint, Maxwell’s rule is convenient to use and does not require specific heat data as input. However, if one were to assess an EOS for its predictive properties from a fundamental perspective, our results show that Maxwell’s rule could be seriously in error. Nomenclature AbbreViations EOS ) equation of state PR ) Peng-Robinson vdW ) van der Waals Notation cLV ) specific heat at constant volume of the substance in the liquid state
cVIG ) specific heat at constant volume of the substance in its ideal gas state g ) specific Gibbs free energy p ) pressure pc ) critical pressure psat ) saturation pressure R ) universal gas constant s ) specific entropy T ) temperature Tc ) critical temperature V ) specific volume Vc ) critical specific volume VL1 ) molar specific volume of saturated liquid VV 1 ) molar specific volume of saturated vapor ε ) specific internal energy ω ) acentric factor
Literature Cited (1) Callen, H. B. Thermodynamics and an Introduction to Thermostatistics; John Wiley&Sons: New York, 1985. (2) Davis, H. T.; Scriven, L. E. Stress and Structure in fluid interfaces. AdV. Chem. Phys. 1982, 49, 357–454. (3) Aifantis, E. C.; Serrin, J. B. Mechanical theory of fluid interfaces. J. Colloid Interface Sci. 1983, 96, 517–529. (4) Aifantis, E. C.; Serrin, J. B. Equilibrium Solutions in the Mechanical Theory. J. Colloid Interface Sci. 1983, 96, 530–547. (5) Tisza, L. Generalized Thermodynamics; MIT Press: Cambridge, Mass., 1966. (6) Kahl, G. D. Generalization of the Maxwell’s Criterion for van der Waals Equation. Phys. ReV. 1967, 155, 78–80. (7) Nitsche, J. M. New applications of Kahl’s VLE analysis to engineering phase behavior calculations. Fluid Phase Equilib. 1992, 78, 157–190. (8) Chen, G.-J.; Sun, C.-Y.; Guo, T.-M. A theoretical revision of the derivation of liquid property expressions from an equation of state and its application. Chem. Eng. Sci. 2000, 55 (21), 4913–4923. (9) Serrin, J. The Area Rule for Simple Fluid Phase Transitions. J. Elasticity 2008, 90 (2), 129–159. (10) Peng, D.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59–64. (11) Perry, R. H.; Green, D. W.; Maloney, J. O. Perry’s chemical engineers’ handbook, 7th ed.; McGraw-Hill: New York, 1997.
ReceiVed for reView March 10, 2010 ReVised manuscript receiVed June 3, 2010 Accepted June 23, 2010 IE100571M
