Water–sulphur dioxide association. Second virial cross coefficients for water–sulphur dioxide derived from gas phase excess enthalpy measurements

J. Chem. Thermodynamics 35 (2003) 91–100 www.elsevier.com/locate/jct

Water–sulphur dioxide association. Second virial cross coefficients for water–sulphur dioxide derived from gas phase excess enthalpy measurements C.J. Wormald

*

School of Chemistry, University of Bristol, Bristol, BS8 1TS, UK Received 13 June 2002; accepted 27 September 2002

Abstract A flow mixing calorimeter has been used to measure the excess molar enthalpy HmE of gaseous (water + sulphur dioxide) at the mole fraction y ¼ 0:5, at standard atmospheric pressure, and over the temperature range (383.2 to 483.2 K). Information about the strength of the water–sulphur dioxide interaction was obtained by analysing the measurements using a quasi-chemical association model. The second virial coefficient B11 of water was written ns B11 ¼ Bns 11  RTK11 . The non-specific interaction B11 between water molecules was calculated from the Stockmayer potential with parameters appropriate to a water-nonpolar fluid interaction, and the specific (hydrogen bonding) forces were described by the association model in terms of an equilibrium constant K11 ð298:15 KÞ ¼ 0:36 MPa1 and an enthalpy of formation of DH11 ¼ ð16:2  2Þ kJ  mol1 for the water–water hydrogen bond. The second virial cross coefficient was written B12 ¼ Bns 12  ðRTK12 Þ=2, and from the temperature dependence of ln K12 the enthalpy of formation of the water–sulphur dioxide interaction was found to be DH12 ¼ ð14:9  2Þ kJ  mol1 , and K12 ð298:15 KÞ ¼ 0:18 MPa1 . Values of the second virial cross coefficient are fitted by the equation B12 =ðcm3  mol1 Þ ¼ 29:6  39290  ðK=T Þ 3:0973  expf1493  ðK=T Þg. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Water; Sulphur dioxide; Second virial cross coefficient; Excess enthalpy

*

Tel.: +44-117-928-8161; fax: +44-117-925-1925. E-mail address: [email protected] (C.J. Wormald).

0021-9614/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 6 1 4 ( 0 2 ) 0 0 3 0 3 - 8

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1. Introduction Li and McKee [1] have used the Gaussian 2 method to compute the energies, enthalpies and Gibbs energies of OH, SO2 , and SO3 , and have studied the hydroxyl radical oxidation of SO2 to SO3 , and the hydrolysis of SO2 to H2 SO3 . They detail the series of steps by which it is believed that SO2 in the atmosphere is converted to H2 SO4 . In this context they choose to focus on a less well studied interaction, the hydrolysis of SO2 to form H2 SO3 , and point out that while hydrolysis reactions are not usually considered in atmospheric cycles, they are likely to be relevant when the water vapour concentration is high and the OH radical concentration is low. The enthalpy of formation DH (298 K) for the H2 O–SO2 complex, computed at the Gaussian 2 level, was found to be 14:6 kJ  mol1 . The interaction in H2 O–SO2 is not dominated by hydrogen bonding, the hydrogen atoms in the water molecule do not point towards the oxygen atoms in SO2 , the structure of the complex is one in which the molecular planes of the two molecules are not far from parallel. In the minimum energy configuration the oxygen atom in the water molecule is 0.2818 nm from the sulphur atom in the sulphur dioxide molecule. Information about the water–polar fluid pair potential can be extracted from measurements of the second virial cross coefficient B12 . Second virial coefficients are of course related to the Boltzmann-weighted average of the pair potential integrated over all angles and distances from zero to infinite separation. The measurement of virial coefficients for mixtures containing water by (p, V, T) methods is made difficult by adsorption errors, particularly at low temperatures. An alternative technique is the measurement of the excess molar enthalpy HmE by flow mixing calorimetry. Any molecules which adsorb onto the apparatus can be regarded as part of the calorimeter itself, and once the adsorption process has reached a steady state, the calorimeter gives results which are free from adsorption errors. In previous publications measurements of the excess molar enthalpy HmE of 24 binary mixtures containing steam as a component have been reported, and three methods for the calculation of thermodynamic properties of steam mixtures have been developed. This work is summarised in a review paper [2] and two subsequent papers [3,4]. Excess enthalpy measurements at pressures in the region of atmospheric and at temperatures in the range (353.15 to 453.15 K) were made using a glass flow mixing calorimeter surrounded by a vacuum jacket. A differential flow mixing calorimeter was used initially [5], but more recent measurements were made using a single stage ‘‘plug in’’ calorimeter [6] which could be quickly removed from the apparatus for modification or repair. Measurements at pressures up to 26 MPa and temperatures up to 698 K have been made using two stainless steel flow mixing calorimeters of different design [7,8], each contained in a pressure vessel. In this paper vapour phase measurements of the excess molar enthalpy of (water + sulphur dioxide) are reported, second virial cross coefficients B12 are extracted, and a quasi-chemical association model is used to obtain a value of the enthalpy of formation of H2 O–SO2 .

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2. Experimental A ‘‘plug in’’ flow mixing calorimeter suitable for measurements of the excess enthalpy HmE of low pressure gases has been previously described [6]. The advantage of the design is that the calorimeter can be easily removed from the apparatus and dismantled for modification or repair, or another calorimeter can be plugged into the flow system. The reverse flow labyrinth calorimeter, the glass heat exchange coils, and the ancilliary apparatus was the same as that used previously [6] for measurements on (steam + ammonia) [4]. The sulphur dioxide used for the measurements, supplied by the British Oxygen Company, was mole fraction 0.996 SO2 and was used as supplied 0.996 and was used as supplied. Steam was generated from ordinary distilled water supplied from a reservoir purged with nitrogen. Runs were begun by passing nitrogen at a flow rate of approximately 1:2 mmol  s1 through the calorimeter for about 2 h, to obtain a stable baseline on the chart recorder linked to a platinum resistance thermometer in the calorimeter. Oxygen-free water at the same molar flow rate was then vaporised in a flash boiler, and mixed with the stream of nitrogen. The mixing of (water + nitrogen)(g) is an endothermic process, and the chart recorder pen deflected to the left. The calorimeter heater was then switched on, and the power was adjusted to restore the pen to the baseline obtained on pure nitrogen. Flow rates were measured and the heat of mixing of (water + nitrogen) was calculated. This procedure checked the performance of the apparatus. Previous measurements [9] of HmE for (0:5H2 O þ 0:5N2 ) at p0 ¼ 0:101325 MPa, accurate to 1 J  mol1 , obtained using a differential flow mixing calorimeter, are fitted by the equation HmE =ðJ  mol1 Þ ¼ :036 þ 0:46635  expf1777  ðK=T Þg:

ð1Þ

This equation is valid over the temperature range (360 to 500 K). The measurements made using the plug-in calorimeter were in agreement with equation (1) to within 1 J  mol1 . When the test measurement was completed the nitrogen supply was disconnected and replaced by sulphur dioxide; again the mixing process was endothermic. As before, the power to the heater was adjusted to restore the pen to the original baseline, and the steady state was monitored for 1.5 h. Flow rate and power measurements were made over the last 0.5 h of this time. Finally the water and sulphur dioxide were switched off, and nitrogen was again passed through the flow calorimeter for (2 to 3) h to check that the original baseline had not shifted. Measurements were made at ambient pressure, and a small adjustment was made to correct them to p0 ¼ 0:101325 MPa. Measurements were made at temperatures from (383.2 to 483.2) K, and at mole fractions close to y ¼ 0:5. Three or four measurements were made at each temperature, and were adjusted to y ¼ 0:5 and averaged. The accuracy was estimated to be no worse than 1:5 J  mol1 . The HmE ðp0 Þ measurements are listed in table 1 and plotted as a function of temperature in figure 1.

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TABLE 1 The excess molar enthalpy HmE ðp0 Þ at p0 ¼ 0:101325 MPa of (0:5H2 O þ 0:5SO2 )(g) measured over a range of temperatures T HmE ðp0 Þ

T

HmE ð0Þ 1

dHmE 1

/12

d/12

383.2 393.2 403.2 413.2 423.2 433.2 443.2 453.2 463.2 473.2 483.2

20.5 17.1 14.2 13.6 11.4 10.5 9.2 8.0 7.8 7.0 6.5

1.5 1.2 1.2 1.2 1.0 1.0 1.2 1.0 1.0 1.2 1.0

)934 )870 )816 )732 )691 )677 )599 )566 )524 )491 )460

3

35 35 30 30 25 25 28 20 20 25 25

1

B12

(J  mol ) (J  mol ) (J  mol ) (cm  mol ) (cm  mol ) (cm  mol ) (cm3  mol1 ) 3

1

B12

(K)

20.0 16.7 13.9 13.3 11.2 10.3 9.0 7.9 7.7 6.9 6.4

1

3

)242 )227 )217 )198 )190 )177 )167 )160 )150 )141 )133

1

16 11 11 11 10 10 9 7 7 9 7

The quantity HmE ð0Þ is yð1  yÞp0 ð2/12  /11  /22 Þ, and the quantity dHmE is the uncertainty on both and HmE ð0Þ. /12 is the cross term isothermal Joule–Thomson coefficient, B12 is the cross term second virial coefficient, and the uncertainties on these quantities are d/12 and dB12 , respectively. HmE ðp0 Þ

FIGURE 1. Experimental measurements of HmE ðp0 Þ at p0 ¼ 101:325 kPa for the mixture (0:5H2 O þ 0:5SO2 )(g). s, table 1. The curve through the measurements was calculated from the association model described in Section 6 with the parameters K12 ð298 KÞ ¼ 0:18 MPa1 and DH12 ¼ 14:9 kJ  mol1 . The upper curve was calculated assuming no association between water and sulphur dioxide.

3. Thermodynamic analysis of the measurements The excess molar enthalpy of a binary gaseous mixture can be written [10] as a sum of terms which are a function of the second virial coefficient B and the third virial coefficient C, etc. HmE ¼ HmE ðBÞ þ HmE ðCÞ þ   

ð2Þ

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HmE ðBÞ can be written as: HmE ðBÞ ¼ yð1  yÞpð2/12  /11  /22 Þ  ðp2 =RT ÞðB/  yB11 /11  ð1  yÞB22 /22 Þ;

ð3Þ

where y is the mole fraction of component 1, p is the pressure, and the isothermal Joule–Thomson coefficient / is given by: / ¼ B  T ðdB=dT Þ:

ð4Þ

Subscripts 11 and 22 refer to pure components, water 1 and sulphur dioxide 2, and subscript 12 refers to the cross interaction. B without a subscript refers to the mixture and is given by: 2

B ¼ y 2 B11 þ 2yð1  yÞB12 þ ð1  yÞ B22 :

ð5Þ

/ for the mixture is given by a similar equation. When third virial coefficients are available values of HmE ðCÞ can be calculated. Hill et al. [11] have proposed equations for both the second and third virial coefficients of water but as there is no similar equation for the third virial coefficient of sulphur dioxide we must be content for the moment to omit any consideration of the HmE ðCÞ term, and analyse the measurements at the level of second virial coefficients only. In Section 6 the size of HmE ðCÞ and its effect on the analysis is estimated. 4. Analysis using an association model Accurate values of B22 and /22 for water can be calculated from the equations of Hill et al. [11] or LeFevre [12]. These equations are in such a close agreement that there is little to choose between them. A table giving values of B22 calculated from both correlations has been published by Eubank et al. [13]. It was shown previously [14] that the Stockmayer potential parameters for water in its interaction with hydrocarbons or slightly polar fluids with which it does not form hydrogen bonds, are e=k ¼ 233 K, r ¼ 0:312 nm. Water has a dipole moment of 6:171  1030 C  m, and the corresponding reduced dipole moment is t ¼ 1:238. Second virial coefficients for sulphur dioxide were examined recently [15] and are fitted by the Stockmayer parameters e=k ¼ 443 K, r ¼ 0:294 nm. The dipole moment l of sulphur dioxide is 5:44  1030 C  m, and the corresponding reduced dipole moment is t ¼ 0:604. Cross terms B12 and /12 were calculated from the usual combining rules e12 =k ¼ ð1  k12 Þðe11 =k  e22 =kÞ1=2 ;

ð6Þ

r12 ¼ ðr11 þ r22 Þ=2;

ð7Þ





¼ ðt11  t22 Þ t12

1=2

:

ð8Þ

The rules assume that the molecules are spherical, that the dipole moment is located at the centre of the sphere, and that induction forces are negligible. It was previously

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found that the best combining rule for the calculation of (1  k12 ) for mixtures of nalkanes [16] and mixtures of (water + n-alkane) [17] is 1=2 ðI1 þ I2 Þ1 ; ð1  k12 Þ ¼ 2ðr311 r322 Þ1=2 ðr3 12 ÞðI1 I2 Þ

ð9Þ

where I is the Ionisation energy. For water I1 ¼ 2:018  1018 J and for sulphur dioxide I2 ¼ 1:97  1018 J. These combining rules give ð1  k12 Þ ¼ 0:998, e12 =k ¼

320:6 K, r12 ¼ 0:303 nm and t12 ¼ 0:865. E Measurements of Hm for gaseous mixtures of water vapour mixed with a fluid with which it does not hydrogen bond were analysed using a quasi-chemical association model [18] which is easily extended. As water itself is hydrogen bonded, B11 and /11 can be written as the sum of a non-specific contribution and a term involving an equilibrium constant K11 , and an enthalpy of association DH11 . B11 ¼ Bns 11  RTK11 ;

/11 ¼ /ns 11 þ K11 DH11 ;

ð10Þ

where superscript ns refers to a non-specific contribution. Likewise the second virial cross coefficient B12 and isothermal Joule–Thomson coefficient /12 can be written B12 ¼ Bns 12  ðRTK12 Þ=2;

/12 ¼ /ns 12 þ ðK12 DH12 Þ=2;

ð11Þ

where DH12 is the molar enthalpy of water–sulphur dioxide association. Here K11 and K12 are equilibrium constants at the temperature T for the association of two component 1 molecules, and one component 1 + one component 2 molecule. The slope of the graph of the logarithm of (K11  p0 ) or of the logarithm of (K12  p0 ) against reciprocal temperature gives a value of DH11 or DH12 , the enthalpies of specific association between the like and unlike molecules. Here the equilibrium constants are multiplied by p0 to make the product dimensionless. Values of K11 and K12 are normally reported at T ¼ 298:15 K. For water Bns was calculated from the parameters [14] e=k ¼ 233 K, r ¼ 0:312 nm and t ¼ 1:238. The difference between Bns and B for water was calculated over the temperature range (350 to 500) K from the correlation of LeFevre et al. [12]. The slope of a graph of ln (K11  p0 ) against reciprocal temperature yielded an enthalpy of association of DH11 ¼ ð16:2  2Þ kJ  mol1 . This enthalpy was used to calculate the value K11 ¼ 0:36 MPa1 at the temperature 298.15 K. The value DH11 ¼ ð16:2  2Þ kJ  mol1 is close to the value calculated using ab-initio methods for the water–water hydrogen bond by Hideaki and Morukuma [19].

5. Water–sulphur dioxide association Where there is association between water (1) and sulphur dioxide (2) the cross term coefficients are given by equation (11). The non-specific terms were calculated from the above Stockmayer parameters for water and sulphur dioxide using the combining rules given by equations (6)–(9). The interaction parameter calculated from equation (9) has the value ð1  k12 Þ ¼ 0:998. With the exception of K12 and DH12 all quantities in equation (3) are known. As described previously [4] these parameters can be obtained iteratively. The quantities B12 and /12 in the (p2 =RT ) term

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in equation (3) were first set equal to the arithmetic mean of B11 and B22 , and /11 and /22 . The (p2 =RT ) term was subtracted from HmE , and equated to the term in p. From the term in p values of the quantity ðK12  DH12 Þ=2 were obtained at each experimental temperature. To separate K12 and DH12 advantage was taken of the fact that K12 is a function of temperature but DH12 is independent of temperature. From the slope of a graph of lnðK12  DH12 =2Þ against reciprocal temperature an initial value of DH12 was obtained, thus allowing first values of K12 to be calculated at any temperature. The (p2 =RT ) term in equation (3) was again calculated and subtracted from HmE , and new values of K12 and DH12 were obtained. After four iterations the value of the enthalpy of association DH12 became constant and the value of K12 at T ¼ 298:15 K was calculated. The final values are K12 ð298:15 KÞ ¼ 0:18 MPa1 and DH12 ¼ ð14:9  2Þ kJ  mol1 . The curve through the experimental points shown in figure 1 was calculated using these values. The upper curve was calculated assuming no association between water and sulphur dioxide. Comparison with the values for water, K11 ð298:15 KÞ ¼ 0:36 MPa1 and DH11 ¼ ð16:2  2Þ kJ  mol1 shows that the water–sulphur dioxide association energy is about 90 per cent of the water–water association energy. Finally values of B12 and /12 obtained from this analysis are listed in table 1 and the values of B12 are plotted against temperature in figure 2. Curve a through the measurements was calculated from the association model described above. Curve b is B11 for water calculated from the correlation of LeFevre et al. [12], and curve

FIGURE 2. Values of the second virial cross coefficient B12 for water–sulphur dioxide plotted against temperature. s, table 1. Curve (a) through the measurements was calculated from the association model described in the text with the parameters K12 ð298 KÞ ¼ 0:18 MPa1 and DH12 ¼ 14:9 kJ  mol1 . Curve (b) is B11 for water, curve (c) is B22 for sulphur dioxide. Curve (d) is Bns 11 , the second virial coefficient of water in the absence of hydrogen bonding. The broken curve (e) is B12 for H2 O–SO2 in the absence of association calculated from the parameters which generate curves (c) and (d).

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c is B22 for sulphur dioxide. Curve d is Bns 11 for water calculated from the parameters e=k ¼ 233 K, r ¼ 0:312 nm and t ¼ 1:238. The broken curve e is B12 for water–sulphur dioxide in the absence of association calculated from the Stockmayer potential parameters for SO2 and the Bns 11 parameters for water. The difference between the broken curve e and the curve a through the B12 values is an indicator of the extent of association between H2 O and SO2 . The extent of self association in water is clearly much greater, as can be seen from the much bigger gap between curve d which is Bns 11 , and the lower curve b which is B11 for water. The equilibrium constant K11 (298 K) for water is 0:36 MPa1 , twice the value (0:18 MPa1 ) of K12 (298 K) for H2 O–SO2 . The association energy obtained from the gas mixing experiment is ð14:9  2Þ kJ  mol1 but this quantity is a measure of the specific (hydrogen bonding) contribution to the intermolecular interaction energy, and does not include dispersion or dipole-dipole forces. As shown previously [3] a rough estimate of the non-specific interaction energy DH ns can be obtained from the equation DH ns ¼ ð1  k12 ÞNkðe11 =k  e22 =kÞ

1=2

:

ð12Þ

For sulphur dioxide e22 =k ¼ 443 K, for water e11 =k ¼ 233 K, ð1  k12 Þ ¼ 0:998, and equation (12) gives DH ns ¼ 2:6 kJ  mol1 . The sum of the specific and non-specific energies is ð14:9 þ 2:6Þ ¼ ð17:5  3Þ kJ  mol1 . This energy is of course not for the complex in the minimum energy configuration but for a Boltzmann-weighted average over all orientations and all separations from zero to infinity. It should therefore be less than the energy (14:6 kJ  mol1 ) at the minimum energy configuration calculated by Li and McKee [1]. These authors give no estimate of the uncertainty on their calculated energy, but it is well known that different ab-initio methods give different binding energies, and it would not be surprising if the uncertainty was 3 kJ  mol1 .

6. Discussion The value of DH12 obtained from the quasi-chemical analysis may be in error due to the neglect of third virial coefficients in the analysis, but from the information gained on the pairwise H2 O–SO2 interaction it is now possible to make an estimate of how big this error might be. For H2 O–SO2 the value DH12 ¼ 14:9 kJ  mol1 is slightly smaller than DH12 ¼ 16:2 kJ  mol1 for H2 O–H2 O, and it is therefore not unreasonable to assume that the third virial cross coefficients C112 and C122 will be no larger than the third virial coefficient C111 for water. As SO2 is a non-hydrogen bonded fluid, it is to be expected that C222 will be considerably less than C111 for water. The contribution of third virial coefficients HmE ðCÞ to the heat of mixing is given by HmE ðCÞ ¼ ðp2 =RT Þ  fw  yw111  ð1  yÞw222 g;

ð13Þ

where w ¼ C  ðT =2Þ  ðdC=dT Þ:

ð14Þ

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99

In equation (13) w without a subscript is for the mixture and is given by; 2

3

w ¼ y 3 w111 þ 3y 2 ð1  yÞw112 þ 3yð1  yÞ w122 þ ð1  yÞ w222 :

ð15Þ

The approximation that all the third virial cross coefficients for (0:5H2 O þ 0:5SO2 ) and their temperature derivatives are the same as the third virial coefficient of water and its temperature derivative is likely to yield an overestimate of the size of the third virial coefficient of the mixture, but it is a quantity easy to calculate. Gallagher [20] has proposed a polynomial equation for the second and third virial coefficients of steam which are consistent with the NBS/NRC steam tables [21]. Values of these virial coefficients are listed in reference (22) At T ¼ 383:2 K the value of HmE ðCÞ is 1:52 J  mol1 and at T ¼ 483:2 K HmE ðCÞ is 0:14 J  mol1 . If the best curve through the HmE measurements is lowered by 1:5 J  mol1 at 383.2 K, the effect is to increase DH12 from ð14:9 to  16:8Þ kJ  mol1 , and make the total energy ð16:8 þ 2:6Þ ¼ 19:4 kJ  mol1 . This is still smaller than the total energy 22 kJ  mol1 for the H2 O–H2 O interaction [23], but is in worse agreement with the Li and McKee [1] value of 14:6 kJ  mol1 , and reinforces the view that their value is too low. The effect of a 1:5 J  mol1 decrease in HmE on the value of B12 at T ¼ 383:2 K is small, it makes B12 only 7 cm3  mol1 less negative, and this is well within the uncertainty 16 cm3  mol1 listed in table 1. The values of B12 calculated from the association model with DH12 ¼ 14:9 kJ  mol1 , which best fits the measurements listed in table 1, are fitted by the equation B12 =ðcm3  mol1 Þ ¼ 29:6  39290  ðK=T Þ  3:0973  ½expf1493  ðK=T Þg : ð16Þ

References [1] W.-K. Li, M.L. McKee, J. Phys. Chem. A 101 (1997) 9778–9782. [2] C.J. Wormald, in: P.R. Tremaine, P.G. Hill, D.E. Irish, P.V. Balakrishnan (Eds.), Proc. 13th Int. Conf. Properties of Water and Steam (IAPWS), NRC Research Press, Ottawa, 2000, pp. 355–364. [3] C.J. Wormald, B. Wurzberger, Phys. Chem. Chem. Phys. 2 (2000) 5133–5137. [4] C.J. Wormald, B. Wurzberger, J. Chem. Thermodyn. 33 (2001) 1193–1210. [5] C.J. Wormald, J. Chem. Thermodyn. 9 (1977) 901–910. [6] C.J. Wormald, J. Chem. Thermodyn. 29 (1997) 701–714. [7] C.J. Wormald, C.N. Colling, J. Chem. Thermodyn. 15 (1983) 725–737. [8] C.J. Wormald, M.J. Lloyd, J. Chem. Thermodyn. 26 (1994) 101–110. [9] P. Richards, C.J. Wormald, T.K. Yerlett, J. Chem. Thermodyn. 13 (1981) 623–628. [10] J. Doyle, D.J. Hutchings, N.M. Lancaster, C.J. Wormald, J. Chem. Thermodyn. 29 (1997) 677–685. [11] P.G. Hill, R.D.C. MacMillan, Ind. Eng. Chem. Res. 27 (1988) 874–882. [12] E.J. LeFevre, M.R. Nightingale, J.W. Rose, J. Mech. Eng. Sci. 17 (1975) 243–251. [13] P.T. Eubank, L.L. Joffrion, M.R. Patel, W. Warowny, J. Chem. Thermodyn. 20 (1988) 1009–1034. [14] C.J. Wormald, N.M. Lancaster, J. Chem. Soc. Faraday Trans. 84 (1988) 3159–3168. [15] C.J. Wormald, Phys. Chem. Chem. Phys. 4 (2002) 4008–4013. [16] C.J. Wormald, E.J. Lewis, D.J. Hutchings, J. Chem. Thermodyn. 11 (1979) 1–12. [17] G.R. Smith, M. Fahy, C.J. Wormald, J. Chem. Thermodyn. 16 (1984) 825–831. [18] P. Richards, C.J. Wormald, Zeit. Phys. Chem. N.F. 128 (1981) 35–42. [19] H. Hideaki, J. Morukuma, J. Am. Chem. Soc. 99 (1977) 1316–1332.

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[20] Gallagher, personal communication. [21] L. Haar, J.S. Gallagher, G.S. Kell, NBS/NRC Steam Tables, Hemisphere Publishing Corp., Washington, DC, 1984. [22] C.J. Wormald, N.M. Lancaster, J. Chem. Soc. Faraday Trans. 1 84 (1988) 3141–3158. [23] A.J. Stone, The Theory of Intermolecular Forces, Clarendon Press, Oxford, 1996.

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