Reprinted from THE JOURNAL OF CHEMICAL PHYSICS, Vol. 49, o. 6,2839-2842, IS September 1968 Printed In U. S. A.

PVT Properties of Water. III. Virial Coefficients of D 20 in the Range lS0o-S00°C*

G. S. KELL, G. E. McLAURIN, AND E. WHALLEY Division of Applied Chemist,.y, National Research COlttlCil, Ottawa, Canada

(Received 1 April 1968) MAR 1 7 1969 The equation of state of D 20 has been determined at 25° intervals from 150° to 500°C by the methods

used previously for H20 in which no measurements of the vapor volume are required. The conditions re­produced those used for H 20, and the results were analyzed to give the differences in the second virial coefficient of D20 and H 20 to maximum precision. The enthalpy and energy of dimerization is about 20 cal mole-1 lower for D20 than for H 20, and the entropy of dimeriZation is about 0.03 cal deg-1-mole-1 lower. These values agree qualitatively with other related evidence; a quantitative comparison is not possible at present because the large effects of the quantization of the intermolecular motions, particularly rotational and vibrational, are not known well enough.

1. INTRODUCTION

The difference between the second virial coefficients of D 20 and H 20 measures the difference between their intermolecular forces. Measurements have already been reportedl of the equation of state of H 20 and the determination of virial coefficients. Immediately fol­lowing that work, measurements were made on D 20 in the same apparatus under, as nearly as possible, the same conditions. Hence systematic errors in the two sets of data and the analysis are nearly the same and differences in the virial coefficients can be found with some confidence.

II. METHODS AND RESULTS

The apparatus was that used for steam,! and the techniques were almost exactly the same. Measure­ments were made at the same pressures except that none was made below 1 atm. The oxide layer in the vessel had apparently thickened during the work on H 20 as the production of hydrogen was lower. The mole fractions of hydrogen, determined at the ends of runs were 0.6X 1Q-4 at 450°C and 1.0X 1Q-4 at 500°C. The value at 500°C is not greater than the 1.5 X 10- 4

found for H 20 at 450°C, allowing for the ........ 25 % uncertainty, and is negligible.

The material used was 99.7 % D 20 from Merck, Sharp & Dohme of Canada Ltd. It was thoroughly boiled to degas it before use, and was not otherwise purified. The isotopic composition as measured by infrared absorption by W. H. Stevens of Atomic Energy of Canada Ltd. varied from filling to filling from 98.5-99.7 wt%.

state of the liquid to be determined. Hence the equation of state of liquid D 20 was obtained at 23.8°C and pressures to 100 bar as

Pa-1 (iJp/ iJp h/bar1= 46.8±0.5X 10-6

-3±2X 1O-9(p- pa) / bar,

where Pa is the density at atmospheric pressure pa. The compressibility at 1 atm is therefore 46.8±0.5 X 10-6

barl. The coefficient of p-pa has little significance. The error in the compressibility obtained from the calibration is higher than that of a direct determination because the mass of liquid is less and the number of parameters to be adjusted is larger. A long-term in­stability of the volumometer will, of course, affect the accuracy of the compressibility relative to H 20 but not the accuracy of the volumometer for the vapor measurements.

Eucken and Eigen4 have reported a difference of isothermal compressibility as determined from the sound velocity of D 20 and H20 of 1.7 and 1.4X 10-6

barl at 20° and 40°C, respectively, compared with our value of l.4X 10'-6 barl at 23.8°C. The agreement is satisfactory, but Eucken and Eigen's compressibilities (the sound velocities are not given directly) are nearly 4% too low for H 20. Presumably the same errors oc­curred with D 20.

The experimental conditions for the vapor measure­ments are summarized in Table 1. Measurements were made to saturation or a density of 1.5 mole literI, whichever was smaller. Duplicate measurements were made at 25° intervals between 150° and 500°C. One run at 275°C was discarded because mercury entered the vessel when temperature control was lost overnight, and was not repeated. As with H 20 vapor,! the data for an isotherm were analyzed by the equation

It is, of course, necessary to calibrate the volum­ometer as a function of shaft position and pressure for the particular fluid under investigation. As shown else­where2 the calibration with a particular fluid together Pi with the calibration previously made for H 20 and the known3 compressibility of H 20 allows the equation of Ti

• NRC No. 10359. 1 G. S. Kell, G. E. McLaurin, and E. Whalley, J. Chem. Phys.

48, 3805 (1968). 2 G. S. Kell and G. E. McLaurin, "Virial Coefficients of Meth­

anol from 150° to 300°C" (unpublished). 3 G. S. Kell and E. Whalley, Phil. Trans. Roy. Soc. London

A258,565 (1965) .

where mi and mo are the masses in the screw injector at

• A. Eucken and M. Eigen, Z. Elektrochem. 55, 343 (1951). 2839 RC-10359

2840 KELL, McLAURIN, AND WHALLEY

TABLE I. Virial coefficients of DiO.

(1) (2) (3) (4) (5) (6) (7) (8)

Conditions of experiment

Pressu~", 'Greatest Mean temp No. of , range mass lO-aV.oM -B 1O-3C tT

(0C) points (bar) (g) (g-cm3 mole-I) (emS mole-1.) (Cm8 mole-i) (mg)

150.00 8 1.0-4.0 3.2 27.26 343 0.8 ±0.06 ±13

150.00 8 1.0-4.0 3.2 27.29 338 0.3 0.02 4

174.99 10 l.0-7.6 5.9 27.35 269 0.8 0.02 2

174.99 10 1.0-7.6 5.9 27.35 269 0.7 0.02 2

200.02 7 1,,0-14.1 10 .7 27.45 206 -35 0.6 0.02 4 7

200.01 7 1.0-14~ 1 10.7 27.47 202 -40 0.7 0.02 4 8

224.99 9 1.1-23.7 17.9 27.48 177.8 -13.4 1.0 0.01 1.6 1.7

224.99 9 1.1-23.7 17.9 27.49 174.6 -18.0 0.8 0.01 1.2 1.3

249.99 9 1.1-3l.0 22.5 27.514 153.1 -6.6 1.1 0.012 1.0 0.9

250.00 10 1.0-34.7 25 .8 27.524 152.1 -7.4 0.8 0.007 0.6 0.4

274.99 9 1.1-56.8 43.9 27.526 135.6 -1.0 3.7 0.017 0.8 0.4

299.99 9 1.1-56.8 39.6 27.600 117.1 -0.4 1.8 0.010 0.5 0.3

299.99 9 1.0-64.2 46.3 27.579 118.3 +0.3 1.5 0.007 0.3 0.1

324.94 10 1.0-64.2 42.5 27.631 103.8 0.7 2.0 0.010 0.5 0.2

324.94 11 1.0-71.5 48.5 27.633 104.1 0.9 3.1 0.013 0.5 0.2

350.06 11 1.0-71.5 44.8 27.675 92.0 1.0 1.2 0.006 0.3 0.1

350.06 11 1.0-71.5 44.8 27.668 92.3 1.1 1.1 0.005 0.2 0.1

374.99 12 1.0-78.9 47.0 27 .728 81.5 0.9 1.0 0.004 0.2 0.1

374.99 12 1.0-78.9 47.0 27.713 82.4 1.2 0.7 0.003 0.1 0.1

400.01 13 1.0-86.3 49.0 27.752 73 .9 1.3 1.2 0.005 0.2 0.1

400.01 13 1.0-86.3 49.0 27.760 73.5 1.1 1.4 0.006 0.3 0.1

425.08 13 1.0-86.3 46.3 27 .801 66.3 1.2 0.9 0.004 0.2 0.1

425.08 13 1.0-86.3 46.3 27.801 66.3 1.1 0.9 0.004 0.2 0.1

450.04 14 1.0-93.7 48.2 27.843 59.9 1.1 0.9 0.004 0.2 0.1

450.05 14 1.0-93.7 48.2 27.846 ' 59 .8 1.0 0.6 0.003 0.1 0.1

475.02 14 1.0-93.6 46.0 27 .891 54.2 1.0 0.8 0.003 0.2 0.1

475.02 14 1.0-93.6 46.0 27.890 54.2 1.0 1.0 0.004 0.2 0.1

500.02 15 1.0-10l.0 47.8 27 .939 49.0 0.9 1.0 0.004 0.2 0.1

500.02 14 5.3-101.0 47.8 27.939 49 .1 0.9 1.0 0.005 0.2 0.1

PVT PROPERTIES OF WATER. III 2841

pressures Pi and zero, vespectively, Ti the absolute temperature at which the ith point was measured, W=V~oM/R, where Vvo is the volume of the pressure vessel at zero pressure and temperature T, M the mo­lecular weight, R the gas constant, " the compressibility of the pressure vessel, and B, C, etc., are the second, third, etc., virial coefficients. In the present work the molecular weigh t of the vapor varied from run to run because of isotopic variations, so it was not possible to smooth W as a function of temperature as was done for H20 vapor.1 Consequently the data were fitted by adjusting~, W, B, and C for each run. The results are given in Table I along with the standard error IT of the mass as found from the equation. They do not differ significantly from those previously found for H20.1 In a small region near 325°C the fourth virial coefficient could be fitted significantly, but this fit is not reported because irregularities would then be introduced into Band C as functions of temperature. Although the analysis just described does not yield significant differences between the second virial coefficients of H20 and D20, a modified analysis that uses the fact

TABLE II. Difference of second virial coefficients of H.O and DtO. a

Temp -Bulo -BD2o BU2o-BD2o (0C) (em' mole-I) (cm3 mole-I) (cml mole-I)

150 342 343 342 338

175 273 269 259 269

200 222 226 3 226 225 220

225 188.2 190.1 2.8 187.4 191.1

250b 159.5 160.7 1.1 159.7 160.7

275 136.98 137.72 0.77 136.93

300 117.11 117.83 0.70 117.32 118.01

325 101.96 102.38 0.38 101.97 102.30

350 89.45 89.83 0.22 89.78 89.84

375 79.78 79.70 ~O

79.46 79.77

400 71.02 71.11 0.08 71.02 71.08

425 63.59 63.61 0.03 63.57 63.61

450 57.20 57.23 0.05 57 .16 57.23

• The values of B here contain systematic errors and are to be used only to obtain differences.

b The values for H.O and 0.0 are not fully comparable at this tem­perature.

i E u 2

'-. o ~

CD , 'l.1

CD'"

o~ 0 _____

o~----------------------------~~~~~ .1

200 300 tOC

FIG. 1. Difference between the second virial coefficients of H.O and D.O. B is more negative for D.O. The curve is.obtained with AAHo=20 cal mole-I and AASo= -0.03 cal deg-I·mole-I.

that the measurements were made in the same appa­ratus under the same conditions can do so.

Before analysis of the experimental data to obtain these differences, simulated experimental data were generated and investigated to find the best method of analysis. Ten runs were made, adding experimental error from a random number generator to data gener­ated with values of ~, W, B, C, and D that roughly represent water in the upper part of our temperature range. The best method of analysis was taken to be the one in which B varied least from run to run. Such variation was about one-tenth as large when only ~, W, and B were adjusted as when mo, W, B, and C were, and was less when the pressure range was large-it is particularly important that the highest pressures be common to the runs compared. It seems clear that differences in the values of B for H20 and D20 obtained in this way will be more nearly correct than those ob­tained by other methods.

The values of BaJa, BD •o, and the differences between them obtained in this way using values to the highest common pressure are shown in Table II, and the differences are plotted in Fig. 1. The differences would be in error if the third and higher coefficients of H 20 and D20 differed. There is no evidence on this point, but it seems likely, by analogy with the small differences in the second coefficient in Table II, that any difference is small and probably unimportant for present purposes. There are no previous determinations of the isotope effect on the second virial coefficient of water.

III. DISCUSSION

The main interest in the present data is the difference in the intermolecular potentials of H20 and D20. It is probably unfruitful to look for differences in the parameters of, say, Stockmayer's potential function, as there are too many parameters for the accuracy of the data and they do not fit the values for water well.1

A better approach is to attribute the gas imperfection to

2842 K ELL, MeL A URI N " AND W HAL' LE Y

dimerization. In its simplest form this model yields the equation6

(1)

where bo is the "hard-sphere excluded volume" and Kd is the equilibrium constant for dimerization in partial pressures. Hence

In[ (B- bO)H20/ (B-bo) D20]= In[KdH20/ K dD20]

=-~~Go/RT

= -~~Ho/RT+~~so/R,

(2)

where ~~Go, ~~Ho, and ~~SO are the differences be­tween the standard Gibbs energy, enthalpy, and entropy of dimerization of H20 and D20.

A gni,ph of In[(B-bo)H20! (B-bo)D20], obtained from the data in the present paper and the virial coefficients of steam reportedearlier,1 against l / T,or alternatively of T In[(B-bo)H2o/ (B-bo)D20] against T, gives values of ~~Ho and ~~SO. The assumed value of bo makes little difference to the values of ~~Ho and ~~SO. The value assumed was 40 cm3 mole-l for both H20 and D20. In the approximation that the molecular motions of the monomer and dimer that contribute to the heat capacity are adequately described by classical mechanics, the difference of heat . capacity between monomer and dimer is isotopically invariant, and conse­quently ~~Ho and , ~~SO should be independent of temperature. This approximation has been made. The values obtained are ,

="",20 cal mole-I,

[SO(Mz) -2S0(M)]H20-[SO(M2) -2S0(M)]D20

="",-0.03 cal deg-I·mole-l , (3)

at 600 oK, where M refers to the monomer molecule. The difference of the entropies is of course independent of the standard pressure, arid the difference in enthalpy is equal to the difference in internal energy.

,The enthalpy difference is about 0.5% of the enthalpy of dimerization of water (about -3.8 kcal mole-I) found by Eq. (1). The curve of the calculated values of BH20 - BD20 is plotted in Fig. 1.

Before discussing these values it is worth considering briefly what is implied by an isotopically variant inter­molecular potential. To an accuracy far exceeding that

6 J. O. Hirschfelder, F. T. McClure, and 1. F. Weeks, J. Chern. Phys. 10, 201 (1942). But see D. E. Stogryn and J. O. Hirsch­felder, J . Chern. Phys. 31, 1531 (1959); 33, 942 (1960).

of the present ' me,aStirements, the potential energy of two molecules in terms of all the interatomic coordinates is isotopically invariant. An isotopically variant poten­tial energy of two molecules in terms of the ' inter­molecular coordinates implies therefore an isotopically variant coupling between the intramolecular and inter-

, molecular energies; , The present measurements are consistent with pre­vious evidence for a difference in intermolecular poten­tials for H20 and D20. The heat of sublimation of the liquid6 at 25°C is 331 cal mole-l greater for D20 than for H20, and the heat of vaporization of the solid7 at OaK is 609 cal mole-l greater for D20 than for H20 compared with a heat of sublimation of H20 of 11.31 kcal mole~l. These figures are, however, not directly comparable to the heat of dilnerization in the vapor because the intermolecular rotational and translational zero"point energy of liquid water and ice is high and contributes a large proportion of the heat of sublimation from the static lattice, and this contribution ,differs considerably for H20 and D20. When allowance is made for its the heats of sublimation from the static crystalline lattice at OaK differ much less than the heats of sublimation of the' actual crystal. The intermolecular translational lattice vibration frequencies of corresponding vibrations of H20 and D20 ice are in the ratio 1.036±0.010 according tb Leadbetter's9 arialysis 'of the heat capac­ities, and at least some of them are in the ratio 1.034± 0.002 from direct observation by infrared absorption,Io The deviation from the ideal ratio of 1.054, calculated assuming that only the differences in mass 'of H20 and D20 need to be taken account of, is not due to the anharmonicity of the lattice vibrations, but on an intermolecular model of the vibrations must be ascribed to a slightly higher force constant for D20 than for H20. Again, however, this result is not quantitatively com­parable with the difference of virial coefficients. Never­theless it is clear that H20 and D20 molecules do differ­in a sense in their intermolecular energies. '

ACKNOWLEDGMENTS

• We are greatly indebted to A. Lavergne for help with the apparatus, and to Dr. W. H. Stevens, of Atomic Energy of Canada Limited, Chalk River, for isotopic analysis of the samples.

6 F. D. Rossini, J. W. Knowlton, and H. L. Johnston, J. Res. Nati. Bur. Std. 24,369 (1940) .

7 E. Whalley, Trans. Faraday Soc. 53, 1578 (1957) . 8 :rhe information on th~ intermolecular vibration frequencies

avaIlable when the calculatIOns of Ref. 7 were performed has been superseded. Calculations based on recent better data do not however, change the conclusions significantly. '

9 A. J. Leadbetter, Proc. Roy. Soc. (London) A287,403 (1965) . 10 J . E. Bertie and E. Whalley, J. Chern. Phys. 46, 1271 (1967).