WA-057

J. Chem. Thermodynamics 1996, 28, 17–27

Second virial coefficients of benzene and

cyclohexane from measurements of the excess

molar enthalpy of (0.5N2+0.5C6H6) and

(0.5N2+0.5C6H12) from 333.2 K to 433.2 K

C. J. Wormald, E. J. Lewis, and A. J. Terry

School of Chemistry, University of Bristol, Bristol BS8 1TS, U.K.

(Received 5 June 1995; in final form 25 July 1995)

Values of the second virial coefficient B of benzene and cyclohexane have been obtainedfrom measurements of the excess molar enthalpy HE

m of {yN2 + (1 − y)C6H6} and{yN2+(1−y)C6H12} measured at standard atmospheric pressure over the temperature range333.2 to 433.2 K. In a hypothetical mixture of an ideal gas with benzene or cyclohexane ameasurement of HE

m is equivalent to a measurement of the isothermal Joule–Thomson coefficientf22 of the hydrocarbon. In a mixture of the nearly ideal gas nitrogen with hydrocarbon, f11

for nitrogen is small. The cross term isothermal Joule–Thomson coefficient f12 is also small andcan be calculated with adequate accuracy using an established combining rule. Potentialparameters which fit B22 and f22 for the hydrocarbon can then be calculated from the HE

m

measurements. It is shown that the HEm measurements yield values of f22 for benzene and

cyclohexane which are in good agreement with values obtained from throttling experiments,and that values of B22 agree well with those obtained from (p, rn, T ). It is suggested that HE

m

measurements on these mixtures might be useful for testing the performance of mixingcalorimeters, and empirical equations which fit the measurements at mole fraction y=0.5are given. 7 1996 Academic Press Limited

1. Introduction

Vapour phase measurements of the excess molar enthalpy HEm for

{yN2+(1−y)C6H6} and {yN2+(1−y)C6H12} at standard atmospheric pressure werefirst reported in 1969.(1) The measurements were made using a differential flow mixingcalorimeter designed to allow the measurement of HE

m for mixtures in which themixing process was either exothermic or endothermic. When constructing the twincalorimeters care was taken to ensure that the pressure drop across each was small,and was the same for both calorimeters. This ensured that any Joule–Thomson effectsin the calorimeters were automatically cancelled by the differential arrangement. The1969 measurements were made over the small temperature range 363.2 K to 383.2 K.

An improved design of flow mixing calorimeter was reported in 1977(2) andagain the differential arrangement was used. This calorimeter was used to

0021–9614/96/010017+11 $12.00/0 7 1996 Academic Press Limited

C. J. Wormald et al.18

make measurements of HEm for the mixtures {yCH4 + (1 − y)C6H6},

{yCH4+(1−y)C6H12}, and {yC6H6+(1−y)C6H12} over the range 363.15 Kto 413.15 K and at a pressure of 101.325 kPa. The calorimeter was later used(3,4)

to measure HEm for 20 mixtures of n-alkane vapours. These measurements were

found to be in excellent agreement with those derived from measurements ofthe excess molar volume VE

m.(5) The programme of standard atmospheric pressureHE

m measurements on mixtures containing benzene or cyclohexane was continuedwith carbon dioxide(6) or water(7) as the second component. High pressure HE

m

measurements on mixtures containing benzene or cyclohexane have been made withcarbon dioxide(8) or water(9) or methanol(10) or ethanol(11) as the second component.These measurements were made with a single mixing calorimeter rather than adifferential arrangement, but at high pressures the speed of flow through thecalorimeter is much reduced and Joule–Thomson effects are negligible.

New measurements on (nitrogen+benzene) or cyclohexane have been made forthree reasons. Improvements to the mixing calorimeter and to the flow system haveincreased the accuracy threefold since the 1969 work, and making new and moreprecise measurements seemed worthwhile. Furthermore (nitrogen+cyclohexane) hasbeen the mixture we have used over a number of years to test and develop flow mixingcalorimeters, and a set of accurate measurements on this mixture are useful inthemselves as a test system. Finally in recent papers(12,13) measurements of HE

m formixtures of (nitrogen+methanol or ethanol) made over a range of pressure were usedto obtain isothermal Joule–Thomson coefficients and second virial coefficients ofthese alcohols using an association model. In this work we aim to show that fromHE

m measurements on non-associated fluids second virial coefficients can be obtainedby making measurements at one pressure only, and that to obtain values of B fromthe measurements is a simple and straightforward procedure.

2. Experimental

The measurements were made using nitrogen (0.9998N2), thiophen free analyticalgrade benzene (0.999C6H6), and analytical grade cyclohexane (0.998C6H12). Thehydrocarbons were stored over molecular sieve dessicant, and were used withoutfurther purification.

Measurements of HEm were made using the differential flow mixing calorimeter used

previously.(2) The calorimeter was immersed in a bath of silicone oil and themeasurements were made over the range 363.2 K to 463.2 K at pressures close toatmospheric. As the normal boiling temperature of benzene is 353.25 K and that ofcyclohexane is 353.89 K the HE

m measurements at 353.2 K and temperatures downto 333.2 K were made at progressively lower pressures so as to keep the operatingpressure at around 80 per cent of the saturation pressure of the hydrocarbon. Themeasurements at the lowest temperature 333.2 K were made at a pressure of44.1 kPa, and HE

m was about 45 per cent of the value that it would have been ifmeasurement could have been made at standard atmospheric pressure. At eachtemperature at least five measurements of HE

m were made over the range of mole

B for C6H6 and C6H12 from HEm 19

fraction y=0.4 to 0.6, and the value of HEm at y=0.5 was obtained by interpolation.

Only values of HEm at y=0.5 are reported. The shapes of the HE

m against y curvesdeviate only slightly from parabolic form and are of no interest. The uncertainty onany single measurement of HE

m was usually 0.7 J·mol−1. While the uncertainty on thevalue of HE

m at y=0.5 was usually less than 0.5 J·mol−1 we judge it best to use a globalfigure of 0.7 J·mol−1 for the uncertainty on all the HE

m measurements listed in column3 of tables 1 and 2. The uncertainty on the measurements arises not from errors inthe measured pressure, heater power or flow rates, but from the thermal noise arisingfrom small temperature fluctuations in the silicone oil bath. As this noise was fairlyconstant at all temperatures the uncertainty on a measurement of HE

m is inverselyproportional to the size of HE

m.The measurements of HE

m for (0.5N2+0.5C6H6) are listed in table 1, and those for(0.5N2+0.5C6H12) are listed in table 2. The pressures at which the measurements weremade are listed in column 2. Column 4 lists values of HE

m at p=101.325 kPacalculated from the figures listed in columns 2 and 3. Column 5 is the uncertaintydHE

m on the column 4 values.

3. Strategy for analysis of the measurements

To a first approximation HEm for a binary gas mixture can be written(2)

HEm=y(1−y)p(2f12−f11−f22), (1)

TABLE 1. Value of the excess molar enthalpy HEm of (0.5N2+0.5C6H12). HE

m(p) is the experimental valueat pressure p, HE

m(p+) is at the pressure p+=101.325 kPa, and dHEm(p+) is the uncertainty on this quantity.

f(c) was calculated from equation (8). f is the isothermal Joule–Thomson coefficient of cyclohexane andB is the second virial coefficient

T p HEm(p) HE

m(p+) dHEm(p+) f(c) f B

K kPa J·mol−1 J·mol−1 J·mol−1 J·mol−1 cm3·mol−1 cm3·mol−1

333.2 44.1 40.3 99.0 3.1 1.9 −4043 −1178338.2 63.0 56.1 94.2 1.8 1.7 −3883 −1137343.2 63.9 53.5 88.2 1.6 1.5 −3733 −1097348.2 85.6 68.9 82.9 1.2 1.3 −3592 −1060353.2 94.4 74.5 80.0 0.9 1.1 −3459 −1024358.2 97.2 75.1 78.3 0.8 1.0 −3335 −991363.2 101.0 72.9 72.9 0.8 0.8 −3217 −959368.2 101.8 71.3 71.0 0.7 0.7 −3106 −928373.2 100.9 67.7 68.0 0.7 0.6 −3002 −900383.2 102.0 64.1 63.7 0.6 0.5 −2809 −846393.2 101.2 58.9 59.0 0.5 0.3 −2636 −797403.2 100.9 54.3 54.5 0.5 0.2 −2480 −752413.2 101.6 50.8 50.7 0.5 0.1 −2338 −711423.2 101.8 47.8 47.6 0.5 0.1 −2209 −673433.2 100.9 45.3 45.5 0.5 0.0 −2092 −638

C. J. Wormald et al.20

TABLE 2. Values of the excess molar enthalpy HEm of (0.5N2+0.5C6H6). HE

m(p) is the experimental valueat pressure p, HE

m(p+) is at the pressure p+=101.325 kPa, and dHEm(p+) is the uncertainty on the quantity.

f(c) was calculated from equation (8). f is the isothermal Joule–Thomson coefficient of benzene, and Bis the second virial coefficient

T p HEm(p) HE

m(p+) dHEm(p+) f(c) f B

K kPa J·mol−1 J·mol−1 J·mol−1 J·mol−1 cm3·mol−1 cm3·mol−1

333.2 46.7 41.4 94.8 2.7 1.3 −3901 −1117338.2 61.2 51.7 88.7 1.7 1.2 −3739 −1077343.2 66.7 53.5 83.9 1.5 1.0 −3589 −1039348.2 81.3 62.7 79.4 1.0 0.9 −3448 −1003353.2 96.0 71.7 75.9 0.8 0.8 −3315 −968358.2 98.6 71.0 72.9 0.8 0.7 −3191 −936363.2 101.0 69.6 69.8 0.7 0.6 −3074 −905368.2 101.3 66.1 66.1 0.7 0.5 −2965 −876373.2 102.0 64.7 64.2 0.6 0.5 −2861 −849383.2 101.8 60.1 59.8 0.8 0.4 −2671 −797393.2 101.8 54.8 54.5 0.5 0.3 −2500 −751403.2 101.1 51.5 51.5 0.5 0.2 −2347 −708413.2 100.9 46.9 47.5 0.5 0.1 −2209 −669423.2 100.6 45.2 45.5 0.5 0.1 −2083 −633433.2 100.6 41.0 41.5 0.5 0.0 −1969 −601

where

f=B−T dB/dT. (2)

This simplified equation will serve to illustrate the principle of the method if weanticipate for a moment some values of the isothermal Joule–Thomson coefficientf that come out of the analysis. For the mixture (nitrogen+cyclohexane) at 373.2 Kf11 for nitrogen is −36 cm3·mol−1, f22 for cyclohexane is −2863 cm3·mol−1, and thecross term f12 calculated from potential parameters for the pure components asdescribed below is −264 cm3·mol−1, rather less than 10 per cent of f22 forcyclohexane. In the analysis to follow we shall use the Kihara(14) potential togetherwith the combining rules

o12=(1−k12) (o11o22)1/2 (3)

a12=(a11+a22)/2 (4)

s12=(s11+s22)/2. (5)

If, instead of nitrogen we mix the cyclohexane with an ideal gas with molecules ofzero volume and for which o11=0, then because of equation (3) o12 will be zero andboth f11 and f12 will be zero. The mixing experiment will therefore be a way ofmeasuring the isothermal Joule–Thomson coefficient f22 of cyclohexane. Indeed themixing process can be regarded as an expansion of the cyclohexane into the spaceprovided by the ideal gas, from an initial pressure p1 to a final pressure p2 which is

B for C6H6 and C6H12 from HEm 21

the partial pressure of cyclohexane in the mixture, and which is almost equal to p1/2.Once f22 is known, B22 can be calculated.

Now f11 for nitrogen is small, and B11 and f11 for this substance are well fittedby the Kihara spherical core potential using the parameters(15) o11/k=139.5 K,a11=0.03526 nm, and s11=0.3526 nm. Furthermore it has been established(4) that formixtures of n-alkanes and similar fluids the term (1−k12) can be calculated from theequation

(1−k12)=2(s311s

322)1/2(s−3

12 ) (I1I2)1/2(I1+I2)−1 (6)

where I is the ionisation energy. If we now guess a set of values of o22/k, a22, ands22 for cyclohexane, we can calculate a value of f22, use equations (3)–(6) to calculatef12, and hence compare the right hand side of equation (1) with the experimentalvalue of HE

m. The set of potential parameters for cyclohexane which causes equation(1) to balance will therefore yield both B22 and f22 for cyclohexane.

4. Analysis of the measurements

To analyse the measurements we make use of the full form of equation (1) whichhas been published previously.(2) It is convenient to write the equation with the termwhich is a function of third virial coefficients on the left hand side.

HEm−f(c)=y(1−y)p(2f12−f11−f22)−(p2/RT)(Bf−yB11f11−(1−y)B22f22),

(7)

where

f(c)=(p2/RT)(c−yc111−(1−y)c222), (8)

and

c=C−(T/2)(dC/dT ). (9)

Terms B, f and c without subscripts refer to properties of the mixture. B and f arequadratic function of the mole fraction y, and c is a cubic function.

The term f(c) on the left hand side of equation (7) is small, about 1 per cent ofHE

m and can be estimated with adequate accuracy using the corresponding statescorrelation of Chueh and Prausnitz.(16) The appropriate combining rules for thecalculation of cross terms have been given previously.(2)

The first step of the analysis was to calculate the term f(c) and subtract it fromHE

m. The second step was to use a grid search method to find the best set of potentialparameters for cyclohexane. This was done be setting up a 3-dimensional grid of 10values of a22 between 0.0 and 0.05 nm, 10 values of s22 between 0.2 and 0.5 nm and10 values of o22/k between 500 and 1000 K. At each grid point the local values of a22,s22, and o22/k were used to calculate values of B22, f22, B12, and f12 at each temperatureat which HE

m measurements had been made. The difference between each calculatedand experimental value of HE

m was computed, and from the sum of the squares of

C. J. Wormald et al.22

the differences a value of the standard deviation s was obtained. This value was storedat the appropriate grid point. By searching the whole of the a, s, and o/k spacecontained within the boundaries of the grid, the point of minimum standarddeviation was obtained. While this procedure sounds to be time consuming it wasgreatly speeded up by using a grid-hop algorithm which started from a point at thecentre of the grid and located the minimum in about 20 steps. This was done bycomputing the standard deviation at the six nearest neighbour positions surroundingthe central grid point. The values a22, s22, and o22/k at the position of the smallestvalue of s were then used as the next guess, and again s for the six nearest neighbourswas calculated and the procedure was repeated. When the minimum value of s hadbeen located the spacing between the grid points was reduced by a factor of 10, theprocedure was repeated, and a set of potential parameters which gave an even lowervalue of s was obtained. When further repetition of the grid search gave nofurther reduction in the value of s, the set of parameters corresponding to thispoint were used to compute B22 and f22 for cyclohexane. The HE

m measurements for(nitrogen+benzene) were analysed the same way. Using this technique theparameters of the Kihara potential for cyclohexane were found to be a=0.112 nm,s=0.5464, and o/k=810 K. The parameters for benzene were found to bea=0.102 nm, s=0.5076 nm, and o/k=860 K. These parameters are similar to thoseobtained by other workers, and a compilation of the different sets is made in table 3.Values of B and f for cyclohexane and benzene calculated from these Kiharapotential parameters are listed in tables 1 and 2.

FIGURE 1. The excess molar enthalpy HEm of (0.5N2+0.5C6H12) (upper curve) and (0.5N2+0.5C6H6)

(lower curve). w, tables 1 and 2. r, Reference 1. ——, calculated using parameters for the Kiharaspherical core potential listed in table 3.

B for C6H6 and C6H12 from HEm 23

TABLE 3. Parameters of the Kihara spherical core potential for benzene and cyclohexane

Benzene Cyclohexane Referenceo/k a s o/k a s

K nm nm K nm nm

832 0.1143 0.5335 18, 1964975 0.2411 0.4938 15, 1966850 0.1020 0.5102 783.5 0.1124 0.5666 19, 1977856 0.1020 0.5074 783 0.1124 0.5578 20, 1988810.1 0.1020 0.5282 751.5 0.1124 0.5664 6, 1992860 0.1020 0.5076 810.0 0.1120 0.5464 this work

The contribution of third virial coefficients to HEm is small, but not negligible. To

show the size of this contribution we have listed f(c) calculated from theChueh–Prausnitz(16) correlation in column 6 of tables 1 and 2. For both mixturesstudied and at temperatures below 400 K f(c) is slightly smaller than the uncertaintydHE

m on HEm. Although f(c) is small, and is at the most 1.5 per cent of HE

m, to neglectit would introduce a systematic error which would make f22 about 1 per cent toonegative. However, it can be seen from tables 1 and 2 that at temperatures above400 K, f(c) is about half dHE

m and rapidly becomes negligible with increasingtemperature, and it is clear that in this region no serious error would arise if thirdvirial coefficients were neglected altogether. For mixtures in which association of themolecules occurs third virial coefficients cannot be calculated adequately using acorresponding states correlation, and fourth virial coefficient terms may beimportant. For mixtures of (nitrogen+methanol(12) or ethanol(13)) it was necessary tomake HE

m measurements over a range of pressure, and hence to measure thecontribution from higher virial coefficients. The above procedure used for(nitrogen+benzene or cyclohexane) in which measurements are made at a singlepressure will only work for non-associated fluids for which an adequate third virialcorrelation is available, or at temperatures at which third and higher virial coefficientsare negligible.

5. Sensitivity to the choice of combining rule

For (nitrogen+cyclohexane) equation (6) gives (1−k12) = 0.91 and for(nitrogen+benzene) (1−k12)=0.92. Equation (6) has been tested well for mixturesof non-polar fluids and found to be superior to five other combining rules.(4) Howeverit is of interest to investigate the sensitivity of the value of f obtained for cyclohexaneor benzene on the choice of combining rule. The results for (nitrogen+cyclohexane)at 373.3 K serve to illustrate the point. The experimental value of HE

m (table 1) is68.0 J·mol−1, and HE

m calculated from equation (6) is 63.3 J·mol−1. f11 for nitrogenis −37 cm3·mol−1, and f22 for cyclohexane obtained from Kihara parameters whichbest fit the whole of the HE

m measurements is −3002 cm3·mol−1. Equation (6) gives(1−k12)=0.91, and equations (3)–(5) lead to f12=−270 cm3·mol−1.

C. J. Wormald et al.24

If instead of equation (6) we use the combining rule of Hudson and McCoubrey(17)

we obtain (1−k12)=0.85, and the corresponding value of f12 is −233 cm3·mol−1. Ifwe set (1−k12)=1 then f12=−323 cm3·mol−1. Using these two values in equation(1) leads to f22=−2928 cm3·mol−1 and −3108 cm3·mol−1 respectively. Comparisonof these figures with −3002 cm3·mol−1 obtained using (1−k12)=0.91 shows that thetwo alternative combining rules change f22 (and also B22) by about 3 per cent.

Uncertainties on the values of B22 and f22 arising from experimental errors areproportional to dHE

m/HEm and as can be seen from tables 1 and 2 are about 1 per cent.

If we allow that the uncertainty on f(c) could be 25 per cent, and that there couldbe an uncertainty on the value of (1−k12) of 3 per cent, we conclude that the overalluncertainty on the values of B22 and f22 listed in tables 1 and 2 is no greater than2 per cent.

6. Fitting equations

As the mixture (nitrogen+cyclohexane) is recommended as a test system for flowmixing calorimeters, it is convenient to give an empirical equation which fits themeasurements at y=0.5 and p=101.325 kPa.

HEm/J·mol−1=−37.5+6.9270×104(K/T )−3.5416×107(K/T )2+

9.0586×109(K/T )3. (10)

Similarly for (nitrogen+benzene),

HEm/J·mol−1=−26.1+5.1626×104(K/T )−2.7470×107(K/T )2+

7.7561×109(K/T )3. (11)

The second virial coefficients listed in tables 1 and 2 are conveniently fitted byequations of square well form. For cyclohexane

B/cm3·mol−1=401.6−257.6 exp(604.9 K/T ), (12)

and for benzene

B/cm3·mol−1=361.1−230.9 exp(618.1 K/T ). (13)

Differentiation of equations (12) and (13) gives values of f which differ by less than0.5 per cent from the values listed in tables 1 and 2. The equations are valid overthe range 300 to 450 K.

7. Comparison with other work

The previous measurements of HEm for these mixtures made at the temperatures

363.2, 373.2 and 383.3 K were 70, 65 and 59 J·mol−1 for (nitrogen+benzene) and 78,72, and 67 J·mol−1 for (nitrogen+cyclohexane). These early measurements are

B for C6H6 and C6H12 from HEm 25

FIGURE 2. The zero pressure isothermal Joule–Thomson coefficient f of benzene (upper curve) andof cyclohexane (lower curve). w, Reference 19. r, Reference 21. ——, calculated from equations (12)and (13).

approximately 3 per cent bigger than those reported in this work, but were madeusing the very first design of flow mixing calorimeter which was not of the reverseflow design and which was subject to heat leaks.

Isothermal Joule–Thomson coefficients for cyclohexane and benzene have beenmeasured by Francis et al.(21) and Wormald et al.(19) Values of the zero pressureisothermal Joule–Thomson coefficient are plotted in figure 2 where the measurementson the two fluids are displaced by 500 cm3·mol−1 for clarity. The two sets(19,21) ofmeasurements are in good agreement except for the measurements on cyclohexanemade by Wormald et al.(19) at 383.2 K and 393.3 K, which seem to be about 3 per centtoo low. The solid curves drawn on the figure were calculated from equations (12)and (13) which best fit the values of B and f listed in table 1 and 2. Agreement withthe Joule–Thomson measurements is to within the combined experimental error.Whereas the curve generated from equation (13) is an excellent fit to themeasurements of f for benzene at all temperatures, the curve generated fromequation (12) is about 1 per cent more negative than the measurements of f forcyclohexane at temperatures below 350 K.

As the aim of this work was to show that second virial coefficients can be obtainedfrom HE

m measurements, values of B calculated from equations (12) and (13) arecompared with a selection of literature values in figure 3. The two curves aredisplaced by 500 cm3·mol−1 for clarity. Agreement with these measurements, which

C. J. Wormald et al.26

FIGURE 3. Second virial coefficients B of benzene (upper curve) and cyclohexane (lower curve). Thepoints on the upper curve are as follows. r, References 22 and 23. r, Reference 24. q, Reference 25.t, Reference 26. w, Reference 27. The points on the lower curve are as follows. t, Reference 26. r,Reference 28. q, Reference 29. w, Reference 30. r, Reference 31. ——, calculated from equations (12)and (13).

were all obtained by (p,rn,T ) techniques, is to within the combined experimentalerror.

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