Research: Science and Education
Thermodynamics of Mixing Real Gases Simeen Sattar Department of Chemistry, Bard College, Annandale-on-Hudson, NY 12504-5000;
[email protected] Undergraduate physical chemistry textbooks commonly treat the thermodynamics of mixing ideal gases but neglect that of mixing real gases. This often prompts students, who are accustomed to considering the behavior of ideal and real gases in tandem in a thermodynamics course, to ask about real gas mixtures. Although the thermodynamics of mixing real gases can be subtle, it is worth considering inclusion of this topic in the syllabus. Not only does it complete the parallel with ideal gases and anticipate the comparison between ideal and real solutions, but students will find study of real gas mixtures rewarding because it draws on their existing knowledge of gases and intermolecular forces and connects to newly learned topics such as the relationships among thermodynamic functions, fugacities, and virial coefficients. Moreover, the abundance of experimental data on real gas mixtures allows students to compare their expectations to actual behavior. An added benefit of the data is that it provides a rich and ready source of literature-based problems and classroom presentations. Although mixtures of real gases are addressed in advanced treatises on thermodynamics (e.g., refs 1–3), the following discussion is designed to be continuous with the coverage of ideal gas mixtures usually found in undergraduate texts. It draws heavily from recent experimental studies of gas mixtures, with the goal of impressing upon students that gas-phase thermodynamics is a vital and engaging subject that can test their understanding of molecular behavior.
The thermodynamics of mixing real gases at constant temperature and pressure is approached in exactly the same way as for the mixing of ideal gases. An expression for the Gibbs energy of mixing, ∆ mixG, is the fundamental result, and from this, expressions for the entropy, enthalpy, and volume of mixing easily follow. The Gibbs energy of a system of unmixed gases is given by (1)
where ni is the amount of component i, µi* is its chemical potential (an asterisk denotes a pure component), and the summation extends over all components of the system. For a real gas at a temperature T and pressure p, the chemical potential is stated in terms of its ideal gas standard state chemical potential µi° and fugacity fi *: µi*( p,T ) = µi°(T ) + RT ln[ fi *( p,T )/p°]
(2)
Once the gases are mixed, the Gibbs energy of the system becomes Gfinal = Σi ni µi (3) with µi( p,T ) = µi°(T ) + RT ln[ f i ( p,T )/p°]
∆ mixG = RT Σi ni ln (fi/fi*)
(4)
Here f i ( p,T ) refers to the fugacity of component i in a mixture whose total pressure is p.
(5)
One way to evaluate this expression is to appeal to the empirical rule of Lewis and Randall, which states that fi/fi * = yi, where yi is the mole fraction of component i. (Note that for an ideal gas mixture, this rule reduces to Dalton’s law.) Although extremely convenient, this rule is not generally correct. For instance, a pair of gases might obey it under some conditions of temperature and pressure but not others (4 ). Substituting the Lewis and Randall rule into eq 5 yields ∆ mixG = RT Σi ni ln yi
(6)
which is identical to the expression for mixing ideal gases— an uninformative result. Thus a more accurate means of evaluating the fugacities required by eq 5 must be found. Examining the definition of fugacity is helpful here. For a pure gas fugacity is given by the relation p
f i* = p exp
V m* RT
– 1 dp′ p′
(7)
0
and for a component of a mixture it is given by the analogous relation p
Gibbs Energy of Mixing Real Gases
Ginitial = Σi ni µi*
The change in the Gibbs energy on mixing is found by subtracting eq 1 from eq 3:
f i = p i exp 0
V i,m RT
– 1 dp′ p′
(8)
Note that eq 8 is very similar to eq 7, except that a component’s partial pressure p i ≡ y i p and partial molar volume Vi, m ≡ (∂V/ ∂ni )T, p, nj ≠ni replace the total pressure and molar volume. In both cases, the total pressure p is the upper limit of integration. Just as the fugacities in eq 7 are evaluated from p,V,T measurements on pure gases, those in eq 8 may be evaluated from similar measurements on gas mixtures, with composition as an added variable. These p,V,T data are then fitted to an equation of state such as the virial equation, in which the compression factor Z is expanded in powers of Vm1 : Z ≡ pVm/RT = 1 + BVm1 + CVm2 + …
(9)
B and C are the second and third virial coefficients, and they are functions of temperature and composition. Virial coefficients for mixtures retain the same microscopic significance as for pure gases: B accounts for two-body interactions, C for threebody interactions, and so on. Therefore, for mixtures, B must be an average of the second virial coefficients for all possible pairs of like and unlike molecules, each weighted by the probability that the corresponding two-body interaction will occur: B = Σ Σ yi yj Bi j i j
(10)
A binary mixture, for instance, is described by one cross-term
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(B12) and two pure-component (B11, B 22) second virial coefficients. Especially if students have encountered some statistical mechanics, it is worth pointing out that virial coefficients are intimately related to molecular interactions through the equation
B ij = 2π NA
∞ 0
exp ui j r′ /kT – 1 r′ 2dr′
(11)
where the function uij (r) is the pair potential for molecules separated by a distance r. Negative values of Bi j indicate that attractive forces are predominant between species i and j, whereas positive values indicate that repulsive forces are predominant. Of course, temperature plays a decisive role in determining the sign of Bi j . By assuming a form for the potential energy function, it is sometimes possible to extract its parameters from knowledge of the second virial coefficients (5). Consequently, virial coefficients are quantities of both fundamental and practical import. It is not difficult to deduce the form of a mixture’s third virial coefficient from eq 10: C = Σ Σ Σ yi yj yk Ci jk i j k
(12)
A term in the sum such as yi2yjCiij accounts for the likelihood and strength of simultaneous interaction between two molecules of species i and one of species j. The relationship between Ci jk and pair potentials can be found in books on statistical mechanics (6 ). Returning to the task of determining f i , the next step would be to solve eq 9 for Vm and obtain the corresponding Vi, m. However, even when the series in eq 9 is truncated after the second term, it is still quadratic in Vm. To circumvent this difficulty, Z is instead expressed as a polynomial in p: Z ≡ pVm/RT = 1 + Bp/RT + (C – B 2)(p/RT ) 2 + … (13) Now, solving for Vm is straightforward. The series on the right can be simplified by assuming that the gas mixture is only slightly imperfect, a definition that omits all terms beyond Bp/RT. Because the parameter B accounts for pairwise interactions between the gas molecules, the approximation limits subsequent discussion to moderate pressures, a limit sensitive not only to the identity of the gases but also to the composition of the mixture. (From the familiar plot of Z( p), students might anticipate that this should be true. At 273 K for example, Z is fairly linear to 10 MPa for hydrogen but to only 1 MPa for nitrogen. Mixtures of the two should, and do, show intermediate behavior [7 ].) Truncating eq 13 after the second term, substituting for B using eq 10, and evaluating the partial derivative (∂V /∂ni)T, p, nj ≠ ni yields an expression for the partial molar volume of component i (8): Vi, m = RT /p + Σ Σ yj yk (2Bij – Bjk ) j k
(14)
As before, the summations extend over all components of the system, including species i. Substituting this intermediate result into eq 8 yields fi = pi exp[Σ Σ yj yk (2Bij – Bjk )p/RT ] j k
(15)
As a check, note that eq 15 reduces to the correct expression for the fugacity of a slightly imperfect pure gas: f i* = p exp(Bii p/RT ) 1362
(16)
Thus for slightly imperfect gas mixtures the problem of determining the fugacities of its components devolves to obtaining a complete set of second virial coefficients. Besides compression factor measurements, second virial coefficients may be derived from data such as pressure or enthalpy changes on mixing. Reference 5 is an invaluable compendium of virial coefficients and the methods by which they were obtained. In the absence of experimental results, second virial coefficients may be calculated from correlations based on the principle of corresponding states (1–3) and pair potential functions (2). Substituting eqs 15 and 16 into eq 5 yields a general expression for the Gibbs energy for mixing slightly imperfect gases: ∆ mixG = RT Σ ni ln yi + Σ ni[Σ Σ yj yk (2Bij – Bjk ) – Bii ]p (17) i
i
j k
The first term on the right is the Gibbs energy for mixing ideal gases ∆ mixG (0); it is necessarily negative. The second term accounts for deviations; it may be positive or negative. Discussion is greatly simplified by considering two-component mixtures. In this case, equation 17 reduces to ∆mixG = nRT ( y1 ln y1 + y 2 ln y 2) + ny1 y 2(2B12 – B11 – B22 )p
(18)
where n is the total amount of gas. All the quantities on the right can be determined experimentally. Within the slightly imperfect gas approximation, for (2B12 – B11 – B22)p/RT ≤ 2, ∆ mixG attains its minimum value when an equal number of moles of the two components are mixed, just as is true for binary mixtures of ideal gases. For values of this ratio greater than 2, more complex behavior occurs, but such values are unlikely within this approximation. Before leaving this section, it is worth noting that in developing an expression for ∆ mixG, only deriving eq 14 might present a substantive difficulty to students; even this difficulty vanishes if discussion is confined to binary systems at the outset. Excess Thermodynamic Functions for Binary Mixtures It is natural at this point to introduce excess thermodynamic functions, which focus attention on deviations from ideality. From eq 18, the excess molar Gibbs energy is given by G Em(1) = y1 y 2(2B12 – B11 – B22 )p
(19)
where the superscript (1) serves as a reminder that this is just a first-order correction. The term on the right merits close examination. Its magnitude increases with the pressure of the mixture, since increasing pressure at a fixed temperature increases interactions, whether favorable or not, by bringing molecules closer together. (Of course, this linear relationship is not true at high pressures, where the slightly imperfect gas approximation fails.) As p → 0, gas mixtures, like pure gases, behave ideally. The factor y1 y 2 weights deviations by the probability of interaction between unlike molecules. The term 2B12 – B11 – B22 (henceforth the excess second virial coefficient B E ) can be negative, zero, or positive, depending on whether, loosely speaking, interactions between unlike molecules (1–2) are stronger than, identical to, or weaker than the average interactions between like molecules (1–1 and 2–2). (Care must be taken not to confuse this statement with averages of pair potentials.) When B E < 0, there is an additional driving
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force that favors mixing beyond the larger volume available to each component, and conversely for B E > 0. Whether enthalpy or entropy effects are predominant is a question to be examined shortly. When BE = 0, the mixture is said to be ideal (9, p 312), even though the gas molecules may be interacting with one another. The concept that ideality for a gas mixture includes molecular interactions aids students in developing a definition of ideal solutions that is applicable to any phase. Of the mixtures included in Figures 1 and 2 (10–16 ), B E is most nearly zero for N2/CO, a system in which the components are well matched in size, dipole moment, and polarizability. A second such example is the system CH3Cl/C2H5Cl. Even though both component molecules are polar, their similar sizes and dipole moments (1.87 and 2.02 D, respectively) give rise to fairly small values of B E. Systems consisting of disparate species tend to have large, positive excess second virial coefficients. Negative values of BE are indicative of a specific interaction between unlike species, such as complex formation or hydrogen bonding. For instance, the strongly negative values of B E for the system CH3OH/(CH3)3N arise from formation of a hydrogen-bonded alcohol–amine complex. It is reasonable to expect that molecular interactions will become less significant at high temperatures, causing the magnitude of BE to decrease with increasing temperature. Figures 1 and 2 bear out this expectation. Small discrepancies attributable to uncertainties in B11, B22, and B12 may be noted. An alternative way of gauging the significance of deviations from ideality is to compare the quantity G Em(1) to ∆ mixGm(0). Except for CH3OH/(CH3)3N, the magnitudes of G Em(1) given in Table 1 are not more than 2% of ∆ mixGm(0) and they are frequently much smaller. It is important to bear in mind that G Em(1) is merely the first term in the complete expression for G Em. Phase separation (gas–gas immiscibility) has been observed in some binary systems above the critical point of the pure components (17 ), implying that the excess Gibbs energy can become significant in comparison to ∆ mixG(0).
Table 1. Selected Thermodynamic Data for Equimolar Mixtures of Various Gases BE / G E m (1 )(p°)/ ∆m ix Gm (0 )/ 1 J mol 1 cm mol J mol 1 a
Mixture
T/K
CO/N2
303.2
1.5 b
CH3 Cl/C2 H5 Cl
290.2
71 c
3
.
CH3 CN/n-C4 H1 0
373.1
1822 d
C2 H5 OH/C6 H1 2
363.2
970 c
.
.
0.038
Ref
1746
10
1.8
1672
11
45.56
2150
12
24.3
2093
13
CH2 FCF3 /C3 H8
299.9
248.8
6.22
1729
14
CHCl3 /C6 H1 2
353.2
118 e
2.94
2036
15a
CHCl3 /C6 H6
353.2
274 e
6.85
2036
15a
CH3 OH/(CH3 )3 N
308
1775
16
.
b
.
.
11,980 c
299.5
the 1st-order correction to the excess Gibbs function, p° = 1 MPa. 2.1 cm3 mol 1. c Uncertainty not reported. d Minimum uncertainty is ± 57 cm3 mol 1, as error in B 22 is not available. e Maximum uncertainty is ± 58 cm3 mol 1. a For b±
Since the excess molar entropy for a slightly imperfect gas mixture is given by S Em(1) = (∂G Em(1)/∂T )p, y 1 = y1 y 2(dBE/dT )p p S Em(1)
(20)
E
then > 0 when dB /dT < 0. This suggests that mixtures such as CH3CN/n-C4H10 have a greater entropy than they would if they were ideal. When dBE/dT > 0, as is true for CHCl3/C6H6, then S Em(1) < 0. To what can the excess entropy be attributed? For ideal gases, it is the larger volume available to the individual components that accounts for all the entropy of mixing at constant T and p, even though the net volume of the system does not change as a result of mixing (18). For real gases, then, one contribution to the excess entropy must come from any net volume change on mixing; it may be positive or negative. A second (negative) contribution might arise when unlike species form a complex that constrains the -8
2000
-10
B E / (dm3 mol −1)
B E / (cm3 mol −1)
1500
1000
500
-12
-14
-16
0
-500
-18
250
300
350
400
450
500
550
295
T/K BE
Figure 1. vs T for the systems ( 䊊) CH3CN/n-C4H10; (ⵧ) C2H5OH/ C6H12); (䉫) CH2FCF3/C3H8; (䉭) CHCl3/C6H12); (䊉) CH3Cl/C2H5Cl; (䉱) N2/CO; (䉬) CHCl3/C6H6. See Table 1 for references.
300
305
310
315
320
T/K BE
Figure 2. versus T for CH3OH/(CH3)3N. B11, B22, and B12 are read from Fig. 5-28 in ref 2, which is based, in part, on ref 16. Note the change in units from Fig. 1.
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freedom of motion of the individual components. Figures 1 and 2 show that because the rate of change of B E becomes smaller with increasing temperature, the excess entropy becomes smaller as well. The temperature dependence of B E also has implications for the excess enthalpy. (In practice, excess enthalpy measurements may themselves be the source of second virial coefficients.) For a slightly imperfect binary gas mixture
Table 2. Selected Thermodynamic Data for Equimolar Mixtures of CHCl3 /C6 H1 2 and CHCl3 /C6 H6 at pⴗ = 0.1 MPa T/K
dBE /dT/ H E m ( p°)/ H E m ( p°) (1 )/ S E m ( p°) (1 )/ B E/ cm3 mol 1 a , b cm3 K1 mol 1 a J mol 1 c J mol 1 J K 1 mol 1
CHCl3 /C6 H1 2 353
114
1.49
17.5
16.0
0.0372
363
90
1.19
14.6
13.1
0.0299
373
89
1.13
13.2
12.8
0.0283
(21)
383
92
1.07
13.2
12.6
0.0267
Since the ideal enthalpy of mixing is zero, H Em is identical to ∆ mixHm. It is reasonable that H Em(1) depends on the pressure, since increasing pressure is expected to increase interactions, whether favorable or not. Plots of H Em(p) for equimolar mixtures of H2O/C6H12 up to p = 12.5 MPa (see Fig. 3) confirm this behavior only to an extent; at sufficiently high pressures, the excess enthalpy actually decreases with increasing pressure (19). Two additional features of Figure 3 are instructive: (i) at the highest temperature, not only is H Em smallest, but it is also most nearly linear, implying once again that the slightly imperfect gas approximation holds best at high temperatures; and (ii) at all temperatures, H Em→0 as p →0. Equation 21 predicts that the magnitude of H Em(1) will be largest for an equimolar mixture. Though this is apparently correct near atmospheric pressure, a plot of H Em( yCO2) for CO2/C2H6 at a near-critical pressure exhibits a slight asymmetry, as shown in Figure 4 (20). Figures 3 and 4 together demonstrate that the slightly imperfect gas approximation fails at high densities. Properties of gas mixtures at high pressures are discussed in ref 1. Even near atmospheric pressure, values of H Em(1) generated from eq 21 are not identical to measured values of the enthalpy of mixing (see Table 2). This discrepancy arises, of course, because eqs 19–21 are based on a truncated virial equation of state. At the next level of approximation, the term (Bp/RT )2 in eq 13 is included in the derivation of an expression for f i , adding a term proportional to p 2 to the right
393
66
0.84
10.8
9.9
0.0210
403
39
0.62
8.6
7.2
0.0155
413
64
0.76
9.0
9.4
0.0189
423
31
0.50
7.1
6.1
0.0125
353
274
3.55
40.9
38.2
0.0888
363
240
2.87
32.7
32.1
0.0718
373
206
2.30
28.0
26.6
0.0575
383
189
1.94
24.2
23.4
0.0486
393
172
1.66
19.9
20.6
0.0414
403
165
1.45
18.5
18.8
0.0363
413
136
1.11
16.4
14.9
0.0278
423
145
1.09
9.4
15.2
0.0274
H Em(1) = G Em(1) + TS Em(1) = y1 y 2[BE – T(dBE/dT )p ]p
CHCl3 /C6 H6
a Calculated
from data in refs 15a (mixtures), 15b (chloroform), and 15c (hydrocarbons). b Maximum uncertainty is ± 58 cm3 mol 1 . c ± (0.7–2.0) J mol 1.
side of eq 21. Further correction involves the term C(p/RT )2 and introduces third virial coefficients into expressions for fugacity and excess thermodynamic functions (21). Like excess entropy, whether the excess enthalpy is negative or positive, its magnitude diminishes as temperature is raised, reflecting the fact that the kinetic energy (~kT ) of the molecules begins to overwhelm their interaction energy (ui j (r)). From the observation that B E(dB E/dT )p < 0 (see 500
6000
5000
400
HmE / (J mol −1)
HmE / (J mol −1)
4000
3000
2000
300
200
100 1000
0
0 0
2
4
6
8
10
12
14
p / MPa Figure 3. Measured values of H Em vs p for H2O/C6H12 at different temperatures: ( 䊊) 498.2 K; (ⵧ) 598.2 K; ( 䉭) 698.2 K. Points are connected solely to guide the eye. Adapted from ref 19.
1364
0.0
0.2
0.4
0.6
0.8
1.0
y CO2 Figure 4. Measured values of H Em vs yCO2 for the system CO2/C2H6 at 3.10 MPa and 291.6 K compared with H Em(1) calculated from a fit to the data using eq 21. Note that the data are skewed toward ethane-rich mixtures. Adapted from ref 20.
Journal of Chemical Education • Vol. 77 No. 10 October 2000 • JChemEd.chem.wisc.edu
Research: Science and Education
Figs. 1 and 2) and the relation H Em(1)S Em(1)
= ( y1 y 2 p) [B (dB /dT )p – T(dB /dT 2
S Em(1)
E
E
E
)p2 ]
(22)
H Em(1)
it follows that and have the same sign. Thus when mixing is endothermic, the excess entropy is positive (owing to a net increase in volume) and vice versa. Positive excess enthalpies occur when components are ill matched with respect to the forces that dominate interactions between like components. An instance of this situation is C2H5OH/C6H12, which has a relatively large excess enthalpy: H Em( p°) = 182 ± 9 J mol1 at 363.2 K (13). A smaller excess enthalpy is observed for CHCl3/C6H12, with H Em( p°) = 17.5 ± 1.4 J mol 1 at 353.2 K (15a). It would seem that CHCl3/C6H6 ought to behave similarly, but here the enthalpy of mixing is exothermic: H Em( p°) = 40.9 ± 2.0 J mol 1 at 353.2 K (15a). Indeed, in spite of the resemblance between the hydrocarbons (their enthalpy of mixing is less than 0.5 J mol 1 at 363 K (22)), H Em and S Em are both positive for the first system and both negative for the second. Calculated values of H Em(1) and S Em(1) for equimolar mixtures are reported in Table 2. Although the magnitudes of S Em(1) appear to be negligible, those of T S Em(1), which are the apt comparison to H Em(1), are not. Their comparable sizes indicate that enthalpy and entropy effects make roughly equal contributions to deviations from ideality. Interestingly, for nearly a dozen gas mixtures of the type polar molecule/cyclohexane or benzene, the excess enthalpy is markedly less endothermic with benzene than with cyclohexane (23). It is conjectured that this is due to formation of a charge transfer complex involving the π system in benzene, which is, of course, not possible with cyclohexane (15a). This specific interaction compensates for the mismatch in intermolecular forces that predominate between the pure components. It is a simple matter to derive an expression for the excess molar volume (which is also the molar volume of mixing ∆ mixVm) for slightly imperfect gas mixtures: V Em(1) = (∂G Em(1)/∂p)T, y1 = y1 y 2 B E
(23)
For systems with BE > 0 (weaker 1–2 interactions), the excess molar volume is positive, and vice versa for BE < 0. Of course, zero excess molar volume does not necessarily imply an absence of molecular interactions. Together with the data in Table 1, eq 23 predicts that acetonitrile and n-butane will expand by about 450 cm3 mol1 when equal amounts are combined at 373 K and constant pressure. The mixture CH3OH/(CH3)3N may be expected to show striking negative changes in V Em, but unfortunately corroborative volumetric data are not available. It is at first puzzling that V Em(1) is independent of pressure. However, the relative change in volume on mixing does depend on pressure, because the higher the pressure at which mixing occurs, the lower the initial volume of the system. Measured values of V Em may depend on pressure, and the next level of approximation introduces a term proportional to p. Like the excess enthalpy, the excess volume for CO2/C2H6 near the critical region is not perfectly parabolic, attaining a maximum value of V Em ≈ 33 cm3 mol1 a little below yCO2 = 0.5 (20). Note that for this system, H Em and V Em have the same sign, which is physically reasonable. In fact, by the same argument used to show that H Em(1)S Em(1) > 0, it is also true that H Em(1)V Em(1) > 0. Thus for slightly imperfect gas mixtures, excess entropy, enthalpy, and volume are all positive or all negative.
In light of the preceding discussion, under what conditions do two slightly imperfect gases form an ideal mixture? Using the criterion that for an ideal mixture G Em(1) = 0, eq 19 requires that at least one of the factors y1 (or y 2), BE, or p must be zero, provided the remaining two are finite. That is, the mixture should be very dilute in one component or the temperature very high or the pressure very low, where the precise meaning of “very” depends on the particular system. All these conditions minimize interactions between unlike molecules. When the mixture is ideal, the excess volume will be found to be zero. However, this statement is limited to slightly imperfect gases, because V Em(1) is independent of pressure. More generally, for a gas mixture to be ideal at any given composition and temperature, the excess volume must be zero throughout the pressure range [0, p] (1): G Em p,T, y 1 =
p
0
V Em p′,T, y 1 dp′
(24)
Acknowledgment I thank Robert J. Olsen for a critical reading of this manuscript. Literature Cited 1. Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill: New York, 1995. 2. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; P T R Prentice Hall: Englewood Cliffs, NJ, 1986. 3. Guggenheim, E. A. Mixtures; Clarendon: Oxford, 1952. 4. Lewis, G. N.; Randall, M. Thermodynamics, 2nd ed.; McGraw-Hill: New York, 1961; revised by Pitzer, K. S.; Brewer, L.; p 295. 5. Dymond, J. H.; Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures; Clarendon: Oxford, 1980. 6. McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976. 7. Verschoyle, T. T. H. Proc. R. Soc. A 1926, 111, 552. 8. Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry; Wiley: New York, 1980. 9. Haase, R. In Physical Chemistry: An Advanced Treatise; Jost, W., Ed.; Academic: New York, 1971; Vol. I, pp 293–365. 10. McElroy, P. J.; Buchanan, S. J. Chem. Thermodyn. 1995, 27, 755. 11. Massucci, M.; Wormald, C. J. J. Chem. Thermodyn. 1998, 30, 919. 12. Warowny, W. J. Chem. Thermodyn. 1998, 30, 167. 13. Wormald, C. J.; Sowden, C. J. J. Chem. Thermodyn. 1997, 27, 1223. 14. McElroy, P. J. J. Chem. Thermodyn. 1995, 27, 1047. 15. (a) Wormald, C. J.; Johnson, P. W. J. Chem. Soc., Faraday Trans. 1998, 94, 1267. (b) Doyle, J. A.; Hutchings, D. J.; Lancaster, N. M.; Wormald, C. J. Chem. Soc., Faraday Trans. 1998, 94, 1263. (c) Wormald, C. J.; Lewis, E. J.; Terry, A. J. J. Chem. Thermodyn. 1996, 28, 17. 16. Millen, D. J.; Mines, G. W. J. Chem. Soc., Faraday Trans. II 1974, 79, 693. 17. Gordon, R. P. J. Chem. Educ. 1972, 49, 249. 18. Meyer, E. F. J. Chem. Educ. 1987, 64, 676. 19. Colling, C. N.; Lancaster, N. M.; Lloyd, M. J.; Massucci, M.; Wormald, C. J. J. Chem. Soc., Faraday Trans. 1993, 89, 77. 20. Wormald, C. J.; Hodgetts, R. W. J. Chem. Thermodyn. 1997, 29, 75. 21. Wormald, C. J.; Lewis, K. L.; Mosedale, S. E. J. Chem. Thermodyn. 1977, 9, 27. 22. Wormald, C. J. J. Chem. Thermodyn. 1977, 9, 901. 23. Wormald, C. J.; Johnson, P. W. J. Chem. Thermodyn. 1998, 30, 1243; see also references therein.
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