Second virial cross coefficients for (ammonia + water) derived from gas phase excess enthalpy measurements

doi:10.1006/jcht.2001.0829Available online at http://www.idealibrary.com on

J. Chem. Thermodynamics2001, 33, 1193–1210

Second virial cross coefficients for (ammonia+water) derived from gas phase excess enthalpymeasurements

C. J. Wormalda and B. WurzbergerSchool of Chemistry, University of Bristol, Bristol, BS8 1TS, U.K.

A flow-mixing calorimeter has been used to measure the excess molar enthalpyHEm of

gaseous (ammonia+ water) at the mole fractiony = 0.5, at standard atmospheric pressure,and over the temperature range 383.15 K to 493.15 K. The excess molar enthalpy isnegative, indicating that the strength of the ammonia–water hydrogen bond is greaterthan that of the water–water hydrogen bond. The measurements were analysed first interms of a purely thermodynamic model which assumes that the second virial crosscoefficientB12 and the related isothermal Joule–Thomson coefficientφ12 are adequatelydescribed by an empirical equation of square well form. Information about the strengthof the ammonia–water hydrogen bond was obtained by analysing the measurements onthe basis of a quasi-chemical association model. It was shown that hydrogen bonding inammonia is so slight that to use an association model for the second virial coefficientB11 would be inappropriate. The second virial coefficientB22 of water was writtenB22 =

Bns22−R·T ·K22. The non-specific interactionBns

22 between water molecules was calculatedfrom the Stockmayer potential with parameters appropriate to a water–nonpolar fluidinteraction, and the specific (hydrogen bonding) forces were described by the associationmodel in terms of an equilibrium constantK22 (298.15 K) = 0.36 MPa−1 and an enthalpyof formation of1H22 = −(16.2 ± 2) kJ · mol−1 for the water–water hydrogen bond.The second virial cross coefficient was writtenB12 = Bns

12 − (R · T · K12)/2, and fromthe temperature dependence of lnK12 the enthalpy of formation of the ammonia–waterhydrogen bond was found to be1H12 = −(19.7 ± 3) kJ · mol−1, andK12(298.15 K) =

0.827 MPa−1. Values ofB12 derived from the measurements are given by the equation:

B12/(cm3· mol−1) = 38− 43059· (K/T) − 1.993· exp{1900· (K/T)}.

c© 2001 Academic Press

KEYWORDS: ammonia–water; second virial cross coefficient; excess enthalpy

1. Introduction

In previous publications measurements of the excess molar enthalpyHEm of 22 binary

mixtures containing steam as a component have been reported, and three methods for theaTo whom correspondence should be addressed.

0021–9614/01/091193 + 18 $35.00/0 c© 2001 Academic Press

1194 C. J. Wormald and B. Wurzberger

calculation of thermodynamic properties of steam mixtures have been developed. Thiswork is summarized in a review paper.(1) Excess enthalpy measurements at pressuresin the region of atmospheric and at temperatures in the range 353.15 K to 453.15 Kwere made with a glass flow-mixing calorimeter surrounded by a low-pressure jacket. Adifferential flow-mixing calorimeter was used initially,(2) but more recent measurementswere made with a single stage “plug in” calorimeter(3) which could be very quicklyremoved from the apparatus for modification or repair. Measurements at high pressuresup to 26 MPa and temperatures up to 698 K have been made with two stainless steelflow-mixing calorimeters of different design,(4,5) each contained in a pressure vessel. Thedevelopment of the low-pressure “plug in” calorimeter has made possible a programme ofmeasurements on mixtures of steam with gases which are corrosive, and the first mixturewe have studied is (ammonia+ water)(g) at standard atmospheric pressure. This mixtureis an important working fluid in refrigeration cycles, and, more recently, Kalinaet al.(6)

have suggested that the thermodynamic efficiency of power generating plants might beimproved if (ammonia+ water) were to be used in place of pure water.

The thermodynamic properties of (ammonia+ water) have recently been reviewedby Tilner-Roth and Friend,(7) and these same authors have published a fundamentalequation of state for the Helmholtz free energy of the mixture.(8) Nowarski and Friend(9)

have applied the one-fluid extended corresponding states method to the calculation ofthe thermodynamic surface of (ammonia+ water). According to Thorin(10,11) about30 correlations for the thermodynamic properties of (ammonia+ water) are to be foundin the literature. Weber(12) points out that these are mostly limited to (vapour+ liquid)equilibrium studies and to measurements of the critical locus. No low-pressure(p, V, T)

measurements from which second virial coefficients for the mixture can be extracted havebeen made. Weber(12) has estimated second virial coefficients for (ammonia+ water) onthe basis of a modification of the Tsonopoulos(13) correlation, but the uncertainty involvedin this estimation method only serves to highlight the need for values ofB obtained fromexperiment. We have measured the excess molar enthalpyHE

m for (0.5NH3 + 0.5H2O)(g)

over the temperature range 383.15 K to 493.15 K at standard atmospheric pressure andhave obtained values of the cross term isothermal Joule–Thomson coefficientφ12 andthe second virial cross coefficientB12. These coefficients give information about the pairpotential between the unlike molecules, and of course permit the calculation of residualthermodynamic properties of the gaseous mixture at low densities.

2. Experimental

A “plug in” flow-mixing calorimeter suitable for measurements of the excess enthalpyHEm

of low pressure gases had been previously described.(3) The advantage of the design isthat, in about 10 min, the calorimeter can be removed from the apparatus and dismantledfor modification or repair, or another calorimeter can be plugged into the flow system. Asthe original glass calorimeter contained some copper components, it was unsuitable formeasurements on ammonia mixtures, so a new calorimeter was constructed in which theonly metals used were stainless steel and (nickel+ chromium) alloy. As it was sometimesnecessary to purge the calorimeter with air, it was essential to avoid the possibility of the

Second virial cross coefficients for (ammonia+ water) 1195

1 2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22 23

24 25

26

27

28

FIGURE 1. The flow calorimetric apparatus.

catalytic oxidation of ammonia, and platinum was not a suitable material of construction.The reverse flow labyrinth design was the same as that used previously.(3) The ancilliaryapparatus was however different, and as it has a number of novel features it is worthdescribing.

The apparatus for supplying a controlled flow of the gaseous components is shown infigure1. Water from burette 1 or reservoir 2 was pumped by metering pump 3 into a flashboiler 4 which consisted of a stainless-steel tube (1 m long, 2.5 mm i.d.) coiled round acylindrical heater rated at 540 W. A thermocouple on the tube exit was used to control thepower supplied to the flash boiler heater. The outlet tube 5, and some other tubes, werewound with heating tapes, and in figure1 these are indicated by broken lines.

The nitrogen used for the measurements was mass fraction 0.9995 and the ammoniawas mass fraction 0.999. Steam was generated from ordinary distilled water which wasboiled under an atmosphere of nitrogen before use. A cylinder of nitrogen or ammonia wasattached to valve 6 and the flow rate was adjusted so that a slow stream of the gas vented toa fume cupboard via flow indicator 7. This consisted of a capillary tube (10 cm long, 2 mmi.d.) with a silicone-oil-filled manometer fixed across it, and a tap to drain out the oil whenrequired. The flow of gas to the calorimeter was generated by diaphragm compressor 8which was fitted with a by-pass valve 9, an outlet pressure gauge 10, and a 5 dm3 ballastvolume 11. Gas at atmospheric pressure was admitted to the compressor via valve 12.Valve 13 controlled the flow of compressed gas into the flow meter 14, along heatedpipe 15, and into the calorimeter. If the compressor was switched on with valve 6 closed,atmospheric air was drawn into the apparatus though flow indicator 7 and was pumpedthrough the calorimeter. This facility was useful for purging ammonia from the apparatus,and for obtaining a stable base-line on the chart recorder linked to the thermometer in the


calorimeter, by running overnight. Thermocouples fitted to heated tubes 5 and 15 allowedthe power to the heating tapes to be adjusted until the temperature of the gas entering theheat exchange coils in the silicone oil bath 16 was about 5 K hotter than the temperatureof the oil.

The “plug in” calorimeter 17 was mounted in a glass tube 18 (5 cm i.d. and 45 cmlong). The heat exchange coils 19 (3.2 m long, 10 mm i.d.) were made of glass tubeing.The tubes at the bottom of the heat exchange coils were connected to two concentric tubeswhich terminated in a hemispherical socket joint, and this mated with a ball joint on thecalorimeter. Details of this concentric tube arrangement were given previously.(3) The twogases flowed over a heater and through a reverse flow labyrinth containing stainless steelmesh to promoted turbulent flow and good mixing.(3) Gas mixture left the calorimeterthrough ball and socket joint 20. If the clip was removed from joint 20, and springs (notshown) which press the calorimeter against the lower hemispherical socket were removed,the calorimeter could be lifted out of the apparatus. The silicone oil bath 16 was capableof operation at temperatures to(523± 0.01) K.

The pressure drop across the calorimeter was measured on mercury manometer 21. Gasmixture leaving the calorimeter was cooled by condensers 22 and 23. Condensate wascollected in bulb 24. Non-condensible gas, nitrogen, flowed through condenser 23 and thenthrough bubble flowmeter 25. Here the nitrogen flow rate was measured with an accuracy of±0.5 per cent by timing the rise of a soap bubble between two reference marks. The bubbleflow meter was used, with air flowing through the apparatus, to check the calibration of therotary flow meter 14. To check the flow rate when measurements on (ammonia+ water)(g)were made, a slow stream of tap water was admitted to the apparatus along tube 26 and wasinjected into the flowing gas stream at the top of condenser 22. A dilute aqueous solution ofammonia was allowed to drain out of the apparatus via tap 27. The solution was collectedin a 1 dm3 volumetric flask, and the time to fill the flask to the fiducial mark was measured.The molarity of the ammonia solution was determined by titration with hydrochloric acid,and the flow rate was calculated from the total amount of ammonia collected in the flask.This method was accurate to±0.5 per cent. Tap 28 was used to drain the ammonia solutionwhen the flow rate was not being measured.

The apparatus was tested by making measurements ofHEm for (nitrogen+ water)(g),

and these were found to agree with previous work(14) to within ±1 per cent. Experimentswere begun by flowing pure steam through the apparatus at a flow rate of approximately1.2 mmol· s−1 for 3 h to establish a baseline on the chart recorder linked to the glass-coated platinum resistance thermometer in the calorimeter. Nitrogen was then admitted tothe apparatus and the flow rate was adjusted until it was the same(1.2 mmol· s−1) as forwater. The mixing of (nitrogen+ water)(g) is an endothermic process, and the deflectionof the chart recorder pen, which was to the left, was observed for about 1.5 h. Ammoniawas then substituted for nitrogen, and the flow rate was adjusted to 1.2 mmol· s−1. For(ammonia+ water)(g) the mixing process is exothermic and the chart recorder deflection,which was to the right, was observed for 1.5 h. Again the reading on the chart recorder wasnoted. Next, with (ammonia+ water)(g) still flowing, the calorimeter heater was switchedon and the power was adjusted to give a further increase of the chart recorder deflectionuntil it was about twice that produced by the exothermic mixing process. This was observed


for 1.5 h. Finally, the ammonia was switched off, and pure steam was again allowed to flowthrough the apparatus for 2 h to recover the baseline obtained at the beginning of the run.

The excess enthalpy of (ammonia+ water)(g) was calculated two ways. The first wayrelies on the fact that accurate measurements ofHE

m for (nitrogen+ water)(g) at standardatmospheric pressure are available,(14) and these are fitted to within experimental error(±1 J· mol−1) by the equation:

HEm/(J · mol−1) = −0.36+ 0.46635· exp{1777· (K/T)}. (1)

This equation is valid over the temperature range 360 K to 500 K. To calculateHEm

for (ammonia+ water)(g) the ratio of the chart recorder diflections obtained from themeasurements on (nitrogen+ water)(g) and (ammonia+ water)(g) were multiplied bythe value ofHE

m for (nitrogen+ water)(g). To use this scaling method, it was necessaryto ensure that the molar flow rates of the two gas mixtures were the same to within±1 per cent, but with the pumping system shown in figure1 this was not difficult toachieve. A second way of measuringHE

m for (water + ammonia)(g) was to comparethe size of the deflection produced by switching on the calorimeter heater with the sizeof the deflection produced by the mixing process. Agreement between the two methodswas always to within±2 per cent, and there was no systematic difference between them.Several measurements were made at each temperature, and these were averaged. Onlyvalues ofHE

m at mole fractiony = 0.5 are reported; the shapes of theHEm against mole

fraction y curves deviate only slightly from parabolic form and are of no interest. Thevalues ofHE

m(y = 0.5) are listed in table1 and plotted in figure2. The uncertaintyδHEm

on theHEm measurements is±1.5 J· mol−1, which is a little greater than the uncertainty

±1 J· mol−1 on the measurements(14) for (nitrogen+ water)(g).

3. Thermodynamic analysis of the measurements

The excess molar enthalpy of a binary gaseous mixture can be written(15) as a sum of termswhich are a function of the second virial coefficientB and the third virial coefficientC, etc.:

HEm = HE

m(B) + HEm(C) + . . . . (2)

Here,HEm(B) can be written as:

HEm(B) = y·(1−y)·p(2·φ12−φ11−φ22)−(p2/R·T){B·φ−y·B11·φ11−(1−y)·B22·φ22},

(3)where y is the mole fraction of component 1,p is the pressure, and the isothermalJoule–Thomson coefficientφ is given by:

φ = B − T(dB/dT). (4)

Subscripts 11 and 22 refer to pure components ammonia 1 and water 2, and subscript 12refers to the cross interaction, whileB without a subscript refers to the mixture and is givenby:

B = y2· B11 + 2y(1 − y) · B12 + (1 − y)2

· B22. (5)

For the mixture,φ is given by a similar equation.


TABLE 1. The excess molar enthalpyHEm(po) at the pressurepo

= 0.101325 MPa of (0.5NH3+ 0.5H2O)(g) measured over a range of temperatureT . The quantityHE

m(0) is y(1 − y) · po·

(2φ12−φ11−φ22), and the quantity±δHEm is the uncertainity on bothHE

m (po) andHEm(0). Here,

φ12 is the cross-term isothermal Joule–Thomson coefficient,±δφ12 is the uncertainty onφ12, B12is the cross term second virial coefficient, and±δB12 is the uncertainity onB12

T

K

HEm(po)

J · mol−1

HEm(0)

J · mol−1

±δHEm

J · mol−1

φ12

cm3 · mol−1

±δφ12

cm3 · mol−1

B12

cm3 · mol−1

±δB12

cm3 · mol−1

383.15 −35.5 −35.3 1 −1894 20 −361 7

388.15 −33 −32.8 2 −1783 39 −344 15

393.15 −28 −27.9 2 −1628 40 −319 15

403.15 −24 −23.9 1 −1448 20 −291 7

413.15 −22 −21.9 1 −1321 20 −272 7

423.15 −19 −18.9 2 −1186 39 −250 15

433.15 −15 −15 1 −1040 20 −225 7

443.15 −12.5 −12.5 2 −932 39 −206 15

453.15 −10 −10 1 −831 20 −187 7

463.15 −9 −9 1 −766 19 −176 7

473.15 −8 −8 1 −705 19 −165 7

483.15 −7 −7 1 −649 19 −154 7

493.15 −6.5 −6.5 1 −606 18 −146 7

When third virial coefficients are available, values ofHEm(C) can be calculated. Hill

et al.(16) proposed an equation for the third virial coefficient of water, but there is no similarequation for ammonia. AtT = 373 K and standard atmospheric pressure the contributionto the residual enthalpy of water from second virial coefficients is 217 J· mol−1, andthe contribution arising from third and higher virial coefficients is 2 J· mol−1, which isnegligible compared with the second virial term. We shall assume that in the mixturethe largest contribution toHE

m(C) arises from interactions between groups of three watermolecules, and as these are negligible compared with the contribution toHE

m(B), othercontributions toHE

m(C) can also be assumed to be negligible.Accurate values ofB22 and φ22 for water can be calculated from the equations of

Hill et al.(16) or LeFevre.(17) These equations are in such close agreement that there is littleto choose between them. A table giving values ofB22 calculated from both correlations hasbeen published by Eubanket al.(18) A simpler equation, which, over the temperature range300 K to 800 K is as good as either, is:

B/(cm3· mol−1) = 46.5 − 23848· (K/T) − 10.858· exp{1383· (K/T)}. (6)

Virial coefficientsB11 for ammonia are listed in the compilation of Dymond and Smith.(19)

The best of these data(20–22) and some more recent measurements(23,24) are well fitted by


30

20

10

0

– 10

– 20

– 30

– 40

373 413

ξ = 0.994

ξ = 1.16

ξ = 1.20

ξ = 1.18

HE m

/(J.

mol

–1 )

453

T/K

493

FIGURE 2. Experimental measurements ofHEm(po) at the standard pressurepo

= 101.325 kPa forthe mixture(0.5NH3 + 0.5H2O)(g).◦, table1. The curve through the measurements was calculatedfrom the association model described in Section6. The other curves were calculated as described inSection3.

the equation:

B/(cm3· mol−1) = 27.2 − 13650· (K/T) − 9.430· exp{960· (K/T)}. (7)

Two methods for obtaining second virial cross coefficientsB12 from HEm measurements are

possible, and the simplest of these will be described first.

4. Analysis using an equation of square-well form

Previous work has shown that the(p2/RT) term in equation (3) is small, 5 per cent or lessof the first term which is proportional to the pressure. If(p2/R · T) = 0, then puttingφ11for ammonia andφ22 for water into equation (3) would yield values ofφ12 from whichvalues ofB12 can be obtained. The(p2/R · T) term contains the quantityB · φ whereBis given by equation (5) andφ is given by a similar equation. AlthoughB12 is not initiallyknown, a simple iterative procedure allows it to be calculated. The procedure makes use ofan equation based on the square-well potential which relatesB12 andφ12. If values ofφ12are fitted to the equation:

φ12 = a + b · {1 − (c/T)} · exp(c/T), (8)

then values ofB are given by:

B12 = a + b · exp(c/T). (9)

The procedure was first to set the(p2/RT) term in equation (3) to zero and for each valueof HE

m to calculate a value ofφ12. These values were fitted by the method of least squares


to equation (8) and the adjustable parametersa, b, andc were determined. Values ofB12andφ12 from equations (8) and (9) were then used to calculate the(p2/RT) term in (3),this term was subtracted fromHE

m, and new values ofφ12 were calculated. These wereagain fitted to equation (8), new values ofB12 were obtained from (9), and the calculationcycle was repeated. After 4 iterations the values ofB12 became constant, and are given bythe equation

B12/(cm3· mol−1) = −51− 1.6236· exp{2022· (K/T)}. (10)

The above method assumes no intermolecular potentials for ammonia or water, and nocombining rules. It is a purely thermodynamic analysis. Any equation which accuratelyfits the values ofB or φ for ammonia or water can be used, and these equations do nothave to be of the same form. While equation (10) is satisfactory over the temperaturerange, here (110 K) of the measurements, it is not the best. This simple functional form isnot capable of representing the second virial coefficientB22 of steam over the full 500 Ktemperature range of the measurements, and a consequence of this inadequacy is that if theequation is used to fitB22 values over, for example, one third of the range, a fit to withinexperimental error can be achieved, but on differentiation the equation may generate valuesof φ22 which are in error by 5 per cent or more. If this is true forB22, it is also true forthe ammonia–water cross coefficientB12. Despite these limitations, the above method isquite satisfactory for the calculation of the(p2/RT) term in equation (3) which is small,and for obtaining values ofφ12 from the HE

m measurements. At this point we note thatat the three experimental temperatures (383.15, 433.15, and 483.15) K the values ofφ12given by equation (10) are (−2047,−1032, and−605) cm3

· mol−1, and the values ofB12 are (−369,−224, and−158) cm3

· mol−1, respectively. TheseB12 values are in goodagreement with the values listed in table1 which were obtained by the method described inSection7. Equation (10) is valid only over the experimental temperature range, and shouldnot be used to extrapolate to other temperatures.

5. The calculation of H Em from pair potentials

Accurate measurements ofB for both water and ammonia are available and are plottedagainst temperature in the lower part of figures3 and4. Potential parameters for nitrogen,ammonia, and water are listed in table2. The values ofB for ammonia, which has a dipolemoment of 4.903· 10−30 C · m, can be fitted to within the accuracy of the measurementsby the Stockmayer potential with the parametersε/k = 310 K,σ = 0.265 nm, and reduceddipole moment t∗ = 0.959. The curve through the measurements shown in figure4 wascalculated with these values. The values ofB for water are of greater accuracy than thosefor ammonia, and it is not possible to fit them over the full temperature range, and to withinthis accuracy, with the Stockmayer potential. However, over the temperature range 350 Kto 550 K a fit which is as good as the fit to the ammonia measurements can by achieved.Water has a dipole moment of 6.171· 10−30 C · m, and the parameters of the Stockmayerpotential areε/k = 372 K,σ = 0.267 nm, and t∗ = 1.238. The curve through the secondvirial coefficients of water shown in the lower part of figure4 was calculated on the basisof these parameters.


0

– 200

– 400

– 600

300 400

T/K

B/(

cm3. m

ol–

1 )

1

2

5

4

3

500

FIGURE 3. The second virial coefficient of water and related quantities. 1:4, second virialcoefficients measured by Keyeset al.(29) fitted by the Stockmayer potential with the parametersε/k = 372 K, σ = 0.267 andt∗ = 1.238. 2: Bns

22 for water equated toB for CH3F. 3: Values of

B23 for water–nitrogen.�, Rigbyet al.(30); ◦, Richardset al.(14). 4: Bns22 for water calculated from

B23 for water–nitrogen as described in the text. 5:Bns22 for water calculated from the parameters

ε/k = 233 K,σ = 0.312 nm, andt∗ = 1.238, as described in the text.

TABLE 2. Parameters of the Lennard–Jones potential fornitrogen and the Stockmayer potential for ammonia andwater.ε/k is the well depth,σ is the collision diameter,µ is the dipole moment,t∗ is the reduced dipole moment,

and I is the ionization energy

Substanceε/k

K

σ

nm

1030· µ

C · mt∗

1018· I

J

Nitrogen 95 0.3698 0 0 2.50

Ammonia 310 0.265 4.903 0.959 1.634

Watera 372 0.267 6.171 1.238 2.018

Waterb 233 0.312 6.171 1.238 2.018

a These Stockmayer parameters for the water–water interactionfit measurements ofB for water over the temperature range 350 Kto 550 K. b These Stockmayer parameters for water are for thecalculation of B12 for an interaction of a water molecule withanother molecule, polar or non-polar, with which no hydrogenbond is formed.(31) Equations (11) and (14) must be used for thecalculaion ofε12/k.


0

– 50

– 100

– 150

300 400

T/K

B/(

cm3. m

ol–

1 )

1

2

3

500

FIGURE 4. The second virial coefficient of ammonia and related quantities. 1: Values ofB11for ammonia calculated from the Stockmaker potential with the parametersε/k = 310 K, σ =

0.265 nm, andt∗ = 0.959. ×, Glowska,(23); ◦, Beattieet al.(20); �, Hirschfelderet al.(21) 2:B13 measurements for ammonia–nitrogen fitted by the Lennard–Jones potential with the parametersε/k = 161.1 K andσ = 0.3174 nm.4, Glowska.(33) 3: B11 for ammonia calculated as described inthe text fromB13 for ammonia–nitrogen.

Cross termsB12 andφ12 can be calculated with the usual combining rules:

ε12/k = ξ(ε11/k · ε22/k)1/2, (11)

σ12 = (σ11 + σ22)/2, (12)

t∗12 = (t∗11 · t∗22)1/2. (13)

The rules assume that the molecules are spherical, that the dipole moment is located at thecentre of the sphere, and that induction forces are negligible. It was previously found(25)

that the best combining rule for the calculation ofξ for mixtures ofn-alkanes and (water+ an alkane)(26) is:

ξ = 2(σ 311 · σ 3

22)1/2

· (σ−312 )(I1 · I2)

1/2· (I1 + I2)

−1 (14)

where I is the ionization energy. For ammonia,I1 = 1.634 · 10−18 J and for waterI2 =

2.018· 10−18 J. These combining rules giveξ = 0.994,ε12/k = 343 K,σ12 = 0.2635 nm,and t∗12 = 1.104. WhenB12 andφ12 are obtained from these parameters, values ofHE

m canbe calculated from equation (3). The calculatedHE

m curve, which is shown in the upperpart of figure2, is positive not negative, and is clearly wrong.

It is of interest to see if theHEm measurements can be fitted by treatingξ as a purely

empirical parameter. In the lower part of figure2, curves generatedξ = 1.16, 1.18, and


1.20 are shown. While these are clearly better than the curve generated usingξ = 0.994 itis at once apparant that non of these curves reproduces the shape of the experimentalHE

magainst temperature graph. The choiceξ = 1.18 most nearly fits the measurements, but is9 J· mol−1 too small atT = 373 K and 3 J· mol−1 too large atT = 473 K.

That the fit to the measurements is improved by increasingξ from 0.994 to 1.18 indicatesthat the potential for the ammonia–water interaction is deeper than expected on the basisof equation (14), and suggests that there is a specific interaction, hydrogen bond formation,between the molecules. A model which takes this into account is required, but such a modelmust also take into account any specific pairwise interaction between ammonia molecules,and between water molecules. Before such a model can be used, a method of calculatingthe non-specific and specific contributions to these pairwise interaction energies must beestablished.

6. Self association in ammonia and in water vapour

The analysis presented in Section4 is purely thermodynamical, it is a good method forobtaining B12 and φ12 from HE

m, but it provides no information about the strength ofthe ammonia–water interaction, nor does it allow comparison with the strength of theammonia–ammonia and water–water interactions. The analysis presented in Section5is clearly inadequate, although it suggests than an ammonia–water hydrogen bond isformed on mixing the component gases. Previous measurements ofHE

m for gaseous (argon+ water) were analysed using a quasi-chemical association model(27) which is easilyextended. If both pure components are self associated thenB11, B22, φ11, andφ22 canbe written as the sum of a non-specific contribution and a term involving an equilibriumconstantK11 or K22:

B11 = Bns11 − R · T · K11; B22 = Bns

22 − R · T · K22 (15)

and

φ11 = φns11 + K11 · 1H11; φ22 = φns

22 + K22 · 1H22 (16)

where superscript ns refers to a non-specific contribution. HereK11 andK22 are equilib-rium constants at temperatureT for the association of two component 1 molecules and twocomponent 2 molecules. The slopes of graphs of lnK11 or ln K22 againstT−1 give valuesof 1H11 and1H22, the enthalpies of specific association between the like molecules. Val-ues ofK11 andK22 are normally reported at the temperatureT = 298.15 K.

A criticism of the association model is that the separation ofB into non-specific andspecific terms is somewhat arbitrary. The non-specific contributionBns is sometimesobtained by assuming that it is the same as the value ofB for a homomorph molecule.In the case of steam,(28) for example, it has been assumed thatBns is the same asBfor CH3F. This is not a bad choice, as the two molecules have the same dipole moment,6.171· 10−30 C · m. It is not always possible to find an appropriate homomorph, and in thecase of ammonia, it is particularly difficult.

We have developed a method for applying the association model to pure polar fluidswhich avoids the above problems. Accurate second virial cross coefficients for a mixture


of the polar fluid with another fluid with which there is no specific interaction must beavailable, and combining rules must be assumed. The procedure is best illustrated withwater as the example.

Second virial coefficients for water are shown in figure3, curve (1), where themeasurements of Keyeset al.(29) are plotted. Several other sets of virial coefficients areavailable, but as the Keyes measurements lie close to the best line through all the datait is sufficient to plot just the one set. The curve through the points was generated fromthe Stockmayer potential with the parametersε22/k = 372 K, σ22 = 0.267 nm, andt∗22 = 1.238. These parameters are appropriate to the water–water interaction in whichthere is a hydrogen bond, but they are not appropriate to an interaction with water in whichthere is no hydrogen bond. For the latter interaction the well depth will be less than thatfor water–water, dipolar and quadrupolar forces will still be present, but hydrogen bondingwill be absent. The non-specific termBns can be estimated(28) by equating it toB forCH3F, and the Stockmayer potential parameters areε/k = 204.8, σ = 0.3448 nm, andt∗ = 1.044. These parameters generate curve (2) in figure3.

A good example of a water–non polar fluid mixture is (water+ nitrogen). The interactionof nitrogen with water is dominated by London dispersion forces; dipole-induced–dipoleand dipole–quadrupole forces are small by comparison, and can be neglected. In whatfollows, it is convenient to denote quantities associated with ammonia by subscript 1, forwater by subscript 2 and for the non-polar fluid, in this case nitrogen, by subscript 3. Thesecond virial cross coefficientB23 for water–nitrogen is largely a function of the dispersionforces between these unlike molecules, and a simple analysis ofB23 data gives informationabout the water–non-polar fluid pair potential, in the absence of hydrogen bonding.

Values of the second virial cross coefficientB23 for water–nitrogen are shown ascurve (3) in figure3. The measurements of Rigby and Prausnitz(30) were obtained frommeasurements of the solubility of water in compressed nitrogen, and the measurementsof Richardset al.(14) were derived from gas-phase excess enthalpy measurements. TheLennard–Jones parameters which best fit theB23 measurements areε23/k = 143 Kandσ23 = 0.3762 nm. Lennard–Jones parameters for nitrogen areε33/k = 95 K andσ33 = 0.3698 nm. Equation (12), written in the form:

σ22 = 2 · σ23 − σ33, (17)

givesσ22 = 0.3826 nm. The ionisation energyI3 of nitrogen is 2.5 · 10−18 J, that of wateris I2 = 2.018· 10−18 J, and equation (14) givesξ = 0.9938. Equation (11), written in theform:

ε22/k = {(ε23/k)/ξ}2(ε33/k)−1, (18)

givesε22/k = 218 K. The water molecule has a dipole moment of 6.171· 10−30 C · m, andthis is present although it contributes nothing to the water–nitrogen interaction energy.The parametersε22/k = 218 K, σ22 = 0.3826, andµ = 6.171 · 10−30 C · m givea reduced dipole momentt∗22 = 0.718. These are the Stockmayer potential parametersappropriate to water in its interaction with nitrogen, and they make possible the calculationof the contribution of dispersion forces to the second virial coefficient of water, in theabsence of a hydrogen bond. Curve (4) in figure3, which is just above the curve for


CH3F, was calculated from these parameters. At high temperatures, curves (2) and (4)become indistinguishable. As the interaction between ammonia and water has a term whicharises from the dipole moments of the two molecules, it is necessary to find some way ofestimating the potential parameters for water in its interaction with a polar molecule withwhich it does not form a hydrogen bond. Chloromethane, which has a dipole momentof 6.237 · 10−30 C · m, is such a molecule, andHE

m measurements on (chloromethane+ water)(g) and (chloroethane+ water)(g) have been reported.(31) Analysis of these andother HE

m measurements of mixtures containing water(31) vapour yielded the parametersε/k = 233 K, σ = 0.312 nm, andt∗ = 1.238 for Bns

22 for water. These parameters,together with values ofξ calculated from equation (14), make possible the calculation ofthe second virial cross coefficient for water and any polar or non polar molecule with whichno hydrogen bond is formed. Values ofBns

22 for water calculated using these parameters areshown as curve (e) in figure3. It is a little surprising that curve (2) for CH3F lies closer tocurve (4) than curve (5), but this may be due to the strong repulsive force associated withthe fluorine atom.

Finally, at the temperature 383.15 K, the difference between curve (5) and curve (1)for water was obtained, and equated toR · T · K22, thus yielding a value ofK22. Thisprocess was repeated atT = 493.15 K where the difference gave a value ofK22at this higher temperature. The quantity(K22 · po), where po is standard atmosphericpressure, is dimensionless, and the slope of a graph of ln(K22 · po) againstT−1 yieldedan enthalpy of association of1H22 = −(16.2 ± 2) kJ · mol−1. This enthalpy was usedto calculate the valueK22 = 0.36 MPa−1 at the temperature 298.15 K. The value1H22 = −(16.2±2) kJ · mol−1 is very close to the value calculated using ab-inito methodsfor the water–water hydrogen bond by Hideaki and Morukuma.(32) It should be pointedout that, although this simple association model provides a good fit to the second virialcoefficient of water over the range 350 to 500 K, a fit at lower temperatures can only beachieved by using slightly different parameters. The values ofK22(298.15 K) and1H22obtained from the two point fit described above vary according to the temperatures chosen,and by selecting different temperatures, values of1H22 between−16 and−18 kJ· mol−1

can be obtained. It was for this reason that we chose the temperatures 383.15 K and493.15 K as these lie at the boundaries of our experimental measurements, thus ensuringthe best possible fit over this range.

Second virial coefficientsB11 for ammonia are plotted against temperature in figure4.The curve drawn through the results was calculated from the Stockmayer potential withthe parametersε/k = 310 K,σ = 0.265 nm andt∗ = 0.959. These parameters give valuesof B11 very close to those obtained from equation (7). To estimate the non-specific secondvirial coefficient Bns

11 for ammonia we followed the procedure used for water. Accuratemeasurements of the second virial coefficient of (ammonia+ nitrogen) mixtures havebeen measured by Glowska.(33) Cross coefficientsB13 have been obtained and these areshown as the upper curve in figure4. The Lennard–Jones parameters which best fit themeasurements areε13/k = 161.1 K andσ13 = 0.3174 nm. Combining these with theparametersε33/k = 95 K andσ33 = 0.3698 nm for nitrogen on the basis of an equationsimilar to equation (17) yields σ11 = 0.265 nm. The ionization energy of ammonia isI1 = 1.634 · 10−18 J and that of nitrogen isI3 = 2.5 · 10−18 J, so equation (15) yields


ξ = 0.952. Finally, an equation similar to equation (18) givesε11/k = 301 K. As nitrogenhas no dipole moment there is no dipole–dipole coupling with ammonia andt∗13 = 0.However, ammonia has a dipole moment of 4.903· 10−30 C · m and this corresponds to avalue oft∗11 = 1.074. The parametersε11/k = 301 K,σ11 = 0.265 nm, andt∗11 = 1.074can now be used to calculate the non-specific contribution ofBns

11 to the second virialcoefficient of ammonia, and this is shown by the broken curve in figure4. The curve isonly 10 cm3

· mol−1 less than the curve drawn through the values ofB11 for ammonia, andlies on the upper bound of the possible experimental error of theB11 measurements. On thebasis of this evidence, we conclude that hydrogen bonding in ammonia gas at temperaturesabove 300 K is so slight that it can be neglected, and gaseous ammonia can be regarded asa non-associated fluid.

7. Ammonia–water association

Where there is association between ammonia (1) and water (2), the cross term coefficientsare given by:

B12 = Bns12 − (R · T · K12)/2, (19)

and:

φ12 = φns12 + (K12 · 1H12)/2. (20)

The non-specific terms were calculated from the above Stockmayer parameters forammonia and water with the combining rules given by equations (11) to (14). Theinteraction parameter calculated from equation (14) has the valueξ = 0.984. With theexception ofK12 and 1H12 all quantities in equation (3) are known. As described inSection4 these parameters can be obtained iteratively. The quantitiesB12 andφ12 in the(p2/R · T) term in equation (3) were first set equal to the arithmetic means ofB11 andB22, andφ11 andφ22. The(p2/R · T) term was subtracted fromHE

m, and equated to theterm in p. From the term inp values of the quantity(K12 ·1H12)/2 were obtained at eachexperimental temperature. To separateK12 and1H12, advantage was taken of the fact thatK12 is a function of temperature but1H12 is independent of temperature. From the slopeof a graph of ln(K12 · 1H12/2) againstT−1 an initial value of1H12 was obtained thusallowing first values ofK12 to be calculated at any temperature. The(p2/R · T) term inequation (3) was again calculated and subtracted fromHE

m, and new values ofK12 and1H12 were obtained. After four iterations the value of the enthalpy of association1H12became constant and the value ofK12 atT = 298.15 K was calculated. The final values areK12(298.15 K) = 0.827 MPa−1 and1H12 = −(19.7 ± 3) kJ · mol−1. Comparison withthe values for steam,K22(298.15 K) = 0.36 MPa−1 and1H22 = −(16.2± 2) kJ · mol−1,shows that the water–ammonia hydrogen-bond energy is about 3 kJ·mol−1, or about 20 percent, stronger than that for the water–water hydrogen bond.

Finally values of B12 and φ12 obtained from this analysis at each experimentaltemperature are listed in table1. The values ofφ12 are plotted against temperature infigure5 in which the upper curve isφ11 for ammonia and the lower curve isφ22 for water.The curve through the values ofφ12 was calculated from the association model describedabove. The figure shows that at temperatures below(385± 5) K, φ12 for ammonia–water


0

– 1000

–2000

400 450 500

T/K

φ/(

cm3. m

ol–

1 ) 1

2

3

FIGURE 5. Values of the cross term isothermal Joule–Thomson coefficientφ12 for ammonia–water(curve 1) plotted against temperature.◦, table1. Curve 2 isφ11 for ammonia, curve 3 isφ22 forwater. BelowT = (385± 5) K, φ12 is more negative thanφ22.

T/K

B/(

cm3. m

ol–

1 )

1

3

2

350

0

– 200

– 400

– 600

– 800

400 450 500

FIGURE 6. Values of the second virial cross coefficientB12 for ammonia–water (curve 1) plottedagainst temperature. Curve 2 isB11 for ammonia, curve 3 isB22 for water.◦, table 1. BelowT = (325± 20) K, B12 is more negative thanB22.


is more negative thanφ22 for water. The values ofB12 listed in table1 are plotted againsttemperature in figure6 where the upper curve isB11 for ammonia and the lower curve isB22 for water. The curve through the values ofB12 was calculated from the associationmodel described above. At temperatures below(325± 20) K, B12 for ammonia–water ismore negative thanB22 for water. The values ofB12 and, after differentiation, the valuesof φ12, are well fitted by the equation:

B12/(cm3· mol−1) = 38− 43059· (K/T) − 1.993· exp{1900· (K/T)}. (21)

The maximum uncertainty±δB12 is given as a function of the temperatureT by theequation:

± δB12/(cm3mol−1) = 5 + 0.02 · exp{2500· (K/T)}. (22)

Equation (22) was obtained by repeating the analysis of the excess enthalpy measurementswith values of(HE

m ± δHEm) where the uncertaintyδHE

m was taken to be±2 J · mol−1 atT = 383.15 K and±1 J· mol−1 at T = 493.15 K. Values of±δB12 were calculated fromthe difference between the values ofB12 obtained from the analysis made with values of(HE

m + δHEm) and(HE

m − δHEm).

The water–ammonia complex has been examined by microwave and tunable far-i.r.laser spectroscopy(34,35) and molecular beam electric resonance spectroscopy.(36) Themolecular beam results show that the complex is almost linearly hydrogen bonded withthe nitrogen atom of ammonia linked to a hydrogen atom of the water molecule. TheO . . . N separation is 0.2983 nm. Several groups have performed ab-initio calculations onthe linear complex, and on other possible geometries. For the linear complex, Yeoet al.(37) calculated the dimerization energy to be−26.9 kJ· mol−1. For comparison, the sameproperty, calculated at the same level of theory, of the linear water dimer was found tobe−19.6 kJ · mol−1. More recently, Sadlejet al.(38) calculated structures and energeticsfrom second-order Møller–Plesset perturbation theory. The nature of the intermolecularinteraction in the complex was investigated by symmetry-adapted perturbation theory.The best estimates of the binding and association energies of the linear complex are−26.8 kJ · mol−1 and−17.4 kJ · mol−1. The above calculations suggest that the bindingenergy of the linear water–ammonia complex is about 30 per cent greater than that of thelinear water–water dimer.

The association energy obtained from the gas mixing experiment is−(19.7 ± 3) kJ ·

mol−1 but this quantity is a measure of the specific (hydrogen bonding) contribution tothe intermolecular interaction energy and it does not include dispersion or dipole–dipoleforces. As shown previously,(39) a rough estimate of the non-specific interaction energy1Hns

m can be obtained from the equation:

1Hnsm = ξ · N · k(ε11/k · ε22/k)1/2. (23)

For ammonia,ε11/k = 310 K; for water, ε22/k = 233 K; and while ξ can becalculated from equation (14) it is adequate to set it to unity. Equation (23) gives1Hns

m = −2.2 kJ· mol−1, so that the sum of the specific and non-specific energies is−(19.7 + 2.2) = −(21.9 ± 3) kJ · mol−1. This energy is of course not for the linearcomplex in the minimum energy configuration but for a Boltzmann weighted average over


all orientations and all separations from the collision diameter to infinity. It should thereforebe less than the energy at the minimum energy configuration. If the lower bound of theuncertainty is applied, the value−(21.9) kJ· mol−1 becomes−(18.9) kJ· mol−1, which is1.5 kJ· mol−1 greater that the value−17.4 kJ· mol−1 obtained by Sadlejet al.(38) Bearingin mind the difficulties implicit in making ab-initio calculations and the approximationsmade in the analysis of the gas mixing experiment this level of agreement is good.

Our finding, that from gas mixing experiments, the enthalpy of association of the water–ammonia dimer is about 20 per cent greater than that of water–water dimer seems to be inaccord with the results of ab-initio calculations.

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(Received 17 October 2000; in final form 24 January 2001)

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Second virial cross coefficients for (ammonia + water) derived from gas phase excess enthalpy measurements