The enthalpies of sublimation and internal energies of solid argon, krypton, and xenon determined from vapor pressures

A-024 J. Chem. Thermodynamics 1978,10,649-659

The enthalpies of sublimation and internal energies of solid argon, krypton, and xenon determined from vapor pressures H. H. CHEN, C. C. LIM, and R. A. AZIZ Department Canada

of Physics, University of Waterloo,

Waterloo,

(Received 2 May 1977; in revised form 4 November

Ontario

N2L 3G1,

1977)

1. Jieoduction In a previous paper(‘) we reported the enthalpies of vaporization and the internal energies which were derived from the vapor pressures via the Clausius-Clapeyron equation, of liquid argon, krypton, and xenon. Here we present the enthalpies and internal energies of the solids of these substances. We wish to repeat here the statement made in the previous paper that, provided sufficiently precise and copious results are available, the enthalpies and internal energies calculated from vapor pressures are at least as accurate if not more accurate than those obtained from calorimetric measurements. This point of view is further reinforced in the paper by Schwalbe et al (2) who demonstrate that with accurate experimental results one can show the thermodynamic consistency of vapor-pressure and calorimetric results for argon, krypton, and xenon. Accurate values of the internal energy of the solid are of interest because they can be compared with those determined by the Monte-Carlo methodc3# 4, based on newly proposed potential models and assumptions regarding many-body interactions. The true pair potential can be verified by comparing the predictions of the potential with experimental values of dilute gas properties where only two-body forces act. The many-body interactions, which are more complicated and much less known and for which an approximation has been suggested notably by Axilrod and Teller,“’ can be verified by comparing theoretical and experimental values of internal energy. The values of internal energy are presented for only a short range below the triple-point for temperature which range the vapor pressures are of such a magnitude that they can be measured accurately. In most cases, the Monte-Carlo calculations of the internal energy are made only for a single temperature near that of the triple-point or at T = 0 because of the substantial computational time required. It is for this reason that the vapor pressures in the limited temperature range presented here are still of substantial importance. OOZl-9614/78/0701-0649

$02.00/O

0 (1978) Academic Press Inc. (London)

Ltd.

650

H. H. CHEN,

C. C. LIM,

AND R. A. AZIZ

2. Experimental The vapor pressures of solid argon, krypton, and xenon were measured from 74.653 to 83.804 K for argon, from 103.719 to 115.727 K for krypton, and from 142.826 to 161.332 K for xenon. The apparatus and technique used in these measurements were identical to those used for the inert-gas liquids and reference should be made to the previous papers (lp6) for more complete details; however, a few important details bear repeating here. All gases were of research purity and were supplied by Matheson of Canada, Limited. The argon had a purity of 99.999 moles per cent and the massspectrometric analysis provided by the supplier shows the main impurities (IO6 x volume fraction) to be: 02, 1; H,, 1; Nz, 5; Hz0 < 2. The krypton gas was 99.995 moIes per cent pure with the following impurities Iisted: Xe, ~25; Nf, ~25; 02, < 4; Ar, ~4; HZ, ~5; hydrocarbons, ~10. The purity of the xenon gas was given at 99.995 moles per cent and the impurities listed are: Kr, ~50; N,, ~10; 02, ~5; Ar, < 5; Hz, ~5; hydrocarbons, < 10. To preserve the purity of the gases the system was always flushed out several times with the same gas as that being measured after which it was evacuated to high vacuum and then filled with fresh gas which had been subjected to a fresh titanium getter-type purifier made by R. D. Mathis (model GP 100). Mathis claims, in the case of argon, that the gas so treated would have a purity of better than 99.9999 moles per cent. For krypton and xenon, the chief contaminants would be the other inert gases in the untreated gas. The temperatures were measured with a Tinsley platinum resistance thermometer calibrated at the National Physical Laboratory on the basis of the NPL-61 scale. These temperatures were re-expressed in terms of the IPTS-68.(” The resistance of the thermometer was measured with an Automatic Systems Laboratories automatic precision a.c. double bridge, Model A7. A Guildline 10 f2 standard resistor (Type 9330) compared against a Tinsley Wilkins (Type 5684B) 10R standard resistor was used as reference. The precision of measurement of the temperature is If: 1 mK or better. The accuracy of the quoted temperatures relative to the thermodynamic temperature is f 10 mK or better.(*) A Texas Instruments Co.‘s Precision Pressure Gauge (Model 141) was used to measure the pressure. The repeatability of the gauge is +0.008 Torr.? The total uncertainty in the pressure measurements resulting from the combination of (i) the error of interpolation between the calibration points supplied by the manufacturer, (ii) the standard of pressure used by the manufacturer, and (iii) the correction for the pressure in the reference of the bourdon capsule is kO.05 Torr.

3. Results The results shown in tables 1, 2, and 3 were fitted to the three-parameter log,,(p/Torr)

= a - b(K/T)-

c log,,(T/K),

equation: (1)

by minimizing X(pcXpt-pcarc)‘. All points were given equal statistical weight; table 4 summarizes the results of the fit. Equation (1) was used to evaluate values of t Throughout

this paper Torr = (101.325/760) kPa.

VAPOR

PRESSURES

TABLE

TIK 83.804 83.763 83.732 83.701 83.453 83.195 82.815 82.560 82.306 82.178

p/Torr 516.85 513.96 511.79 509.60 492.67 475.49 451.08 435.34 420.11 412.54

T/K 81.936 81.697 81.459 81.202 80.916 80.599 80.292 80.039 79.793 79.528

TABLE

TIK I 15.727 115.631 115.532 115.376 115.273 115.142 114.822 114.475 114.139 113.782 113.482

p/Tom

545.16 539.99 534.69 526.36 521.03 514.26 497.90 480.79 464.56 447.93 434.33

TIK

113.157 112.832 112.522 112.193 111.745 111.421 111.099 110.731 110.381 110.037 109.695

TABLE

TJK 161.332 161.279 161.245 161.162 161.067 161.043 160.963 160.857 160.842 160.727 160.659 160.623 160.484 160.337 160.321 160.163

609.86 607.56 606.08 602.50 598.41 597.39 593.95 589.45 588.80 583.95 581.14 579.63 573.86 567.82 567.15 560.69

OF AI-(S), Kr(s), AND Xc(s)

651

1. Vapor pressure of solid argon [Torr = (101.325/760) kPa] p/Tom

398.59 384.29 372.36 358.76 344.18 328.55 314.01 302.42 291.51 280.14

T/K

79.261 79.017 78.772 78.504 78.199 77.940 77.657 77.396 77.151 76.889

269.02 259.20 249.63 239.50 228.45 219.36 209.80 201.27 193.51 185.53

76.625 76.383 76.150 75.885 75.631 75.387 75.124 74.915 74.653

p/Torr 177.76 170.88 164.46 157.41 150.89 144.82 138.54 133.69 127.85

2. Vapor pressure of solid krypton [Torr = (101.325/760) kPa] p/Torr 419.94 406.00 393.01 379.65 361.97 349.66 337.75 324.57 312.42 300.90 289.74

TIK 109.365 109.033 108.691 108.574 108.356 108.337 108.007 107.661 107.339 107.006 106.661

p/Torr 279.31 269.13 259.02 255.68 249.40 248.82 239.68 230.36 222.00 213.60 205.19

T/K

p/Torr

106.325 105.993 105.659 105.328 104.999 104.679 104.380 103.719

197.23 189.63 182.26 175.18 168.36 161.97 156.17 143.96

147.978 147.679 147.256 146.800 146.553 145.989 145.542 145.029 144.527 144.066 143.631 143.181 142.826

p/Torr -216.21 210.81 203.34 195.53 191.41 182.30 175.34 167.60 160.31 153.87 147.99 142.10 137.59

3. Vapor pressure of solid xenon

T/K

p/Torr

T/K

160.007 159.844 159.691 159.680 159.495 159.428 159.316 159.312 158.522 158.382 157.810 157.596 157.113 157.056 156.642 156.123

554.38 547.84 541.82 541.38 534.17 531.49 527.22 527.02 497.30 492.21 471.77 464.33 447.80 445.93 432.22 415.55

155.548 154.994 154.616 154.192 153.701 153.192 152.645 152.143 151.670 151.202 150.764 1SO.296 149.827 149.440 148.983 148.464

P/TO=

397.71 381.16 370.17 358.17 344.66 331.12 317.07 304.60 293.27 282.39 272.47 262.24 252.32 244.36 235.25 225.25

652

H. H. CHEN,

TABLE

4. Summary of vapor-pressure

C. C. LIM,

AND

R. A. AZ12

results: number N of points; standard errors a(p) of estimate of pressure

a

b

C

Solid Ar

Kr

Xe

14.653 8:&l

39

103.719 to 115.727

41

142.826 to 161.332

61

7.661180

414.6464

7.823114

417.0581

7.729900

577.8645

8.333203

589.5989

1.775552

805.1050

7.344499

794.0826

0.0692412

0.2432569

-0.1642996

0.020

(la) a

0.018

(1)

0.039

(la)

0.022

(1)

0.042

(la)

0.033

(1)

Variances and covariances in coefficients of equation (1)

Ar Kr Xe

1O%,

10%7*

1030,

1O%&

10%0,

lo%,,

3.0433 4.3315 4.0764

4.6244 8.5952 10.707

1.3021 1.7489 1.5541

1.4073 3.6713 4.2655

6.0216 14.696 16.100

3.9628 7.5808 6.3293

a Equation (la) is the traditional log&/Torr) = a - b(K/T).

two parameter equation used to present the vapor pressure:

(dp/dT),,, which were required in the Clausius-Clapeyron relation. Such values of (dp/dT),,, are valid since the further smoothing action of the already smooth results is such that A(dp/dT),,, is a monotonically increasing function of T and that A2(dp/dT),,, is a constant. 4. Analysis The enthalpy of sublimation AH was determined Clausius-Clapeyron equation :

at various temperatures

The internal energies U, were calculated using the thermodynamic U, = -AH+p,,,(V,-I/,)-RT’B’(T)/T/‘,,

(R = 8.31441 J K-’

from the

relation: mol-‘),

(3)

where B’(T) is the derivative with respect to temperature of the second virial coefficient and subscript “sat” refers to saturated vapor conditions. The results are presented in tables 5, 6, 7. For the molar volumes, V, of the solid phases, recent data of Peterson et ~1.“) were used for argon, those of Losee and Simmons(“) were used for krypton, and those of Swenson and Anderson(“) were used for xenon. The molar volume VB of vapor was calculated from second virial coefficients B(T). The second virial coefficients were calculated from recently proposed potentials: The Barker-Fisher-Watts(3) potential for argon, the K2 potential of Barker et al.“‘) for krypton, and the X4

VAPOR

TIK 74.0 74.5 75.0 75.5 76.0 76.5 77.0 77.5 78.0 78.5 79.0 79.5 80.0 80.5 81.0 81.5 82.0 82.5 83.0 83.5 83.8050

V&m3

TABLE

5. Enthalpy

mol - 1

V./cm3

40071 36972 34151 31578 29229 27082 25118 23319 21670 20155 18764 17484 16305 15219 14217 13292 12437 11646 10914 10235 9850.2 TABLE

UK 102.0 102.5 103.0 103.5 104.0 104.5 105.0 105.5 106.0 106.5 107.0 107.5 108.0 108.5 109.0 109.5 110.0 110.5 ill.0 111.5 112.0 112.5 113.0 113.5 114.0 114.5 115.0 115.5 115.7768

PRESSURES

Vg/cz3 54461 51329 48405 45674 43121 40734 38500 36409 34449 32612 30889 29272 27753 26327 24986 23724 22537 21418 20365 19371 18435 17551 16717 15929 15185 14481 13816 13186 12856

mol-’

24.15 24.17 24.20 24.22 24.24 24.26 24.29 24.31 24.33 24.36 24.38 24.41 24.43 24.46 24.48 24.51 24.53 24.56 24.59 24.61 24.63

V&m3 29.32 29.34 29.36 29.38 29.40 29.42 29.44 29.46 29.48 29.50 29.52 29.55 29.57 29.59 29.61 29.63 29.65 29.68 29.70 29.72 29.74 29.77 29.79 29.81 29.84 29.86 29.88 29.91 29.92

Ar(s),

Kr(s),

AND

of sublimation and internal energy fTorr = (101.325/760) kPa1

mol - 1

6. Enthalpy

OF

(dp/dT)/Torr

K -1

19.925 21.436 23.038 24.733 26.526 28.421 30.420 32.529 34.752 37.092 39.553 42.140 44.858 47.709 50.699 53.832 57.113 60.545 64.134 67.883 70.258

AH/kJ 7.8721 7.8669 7.8614 7.8556 7.8496 7.8432 7.8366 7.8297 7.8224 7.8147 7.8066 7.7982 7.7894 7.7802 7.7707 7.7608 7.7504 7.7395 7.7282 7.7163 7.7087

of sublimation and internal energy [Torr = (101.325/760) kPa] mol - 1

(dp/dT)/Torr 14.852 15.674 16.532 17.426 18.360 19.332 20.345 21.399 22.496 23.638 24.824 26.057 27.337 28.666 30.045 31.476 32.959 34.496 36.088 37.737 39.443 41.209 43.035 44.923 46.873 48.889 50.970 53.118 54.336

K - 1

AH/U 10.993 10.988 10.982 10.976 10.970 10.963 10.957 10.950 10.943 10.936 10.928 10.920 10.913 10.905 10.896 10.888 10.879 10.870 10.860 10.850 10.840 10.830 10.819 10.808 10.797 10.785 10.773 10.761 10.754

Xc(s)

653

of solid argon U./kJ

mol-’ z!z t zt i zt f + f + rt f zt 31 i * 3~ zt f f jz f

0.0126 0.0121 0.0117 0.0113 0.0110 0.0107 0.0105 0.0103 0.0101 0.0100 0.0099 0.0099 0.0098 0.0099 0.0099 0.0100 0.0101 0.0102 0.0104 0.0106 0.0107

of solid

-7.2719 -7.2638 -7.2554 -7.2468 -7.2380 -7.2290 -7.2199 -7.2104 -1.2007 -7.1908 -7.1805 -7.1699 -7.1591 -7.1480 -7.1367 -7.1250 -7.1130 -7.1006 -7.0878 -7.0746 -7.0663

zt 0.0141 F 0.0137 f 0.0133 x!r 0.0130 zk 0.0127 A= 0.0125 i 0.0124 f 0.0123 i: 0.0123 zk 0.0123 rJ, 0.0124 It 0.0125 41 0.0127 z!c 0.0129 + 0.0132 jr 0.0135 i 0.0139 zk 0.0143 ztO.0148 f 0.0153 zt 0.0156

krypton

mol - 1 * 0.017 zt 0.016 3~ 0.016 i 0.016 i 0.015 Z!C 0.015 zt 0.015 zk 0.015 i 0.014 zk 0.014 rir 0.014 31 0.014 i 0.014 i 0.014 i 0.014 zlz 0.014 I 0.014 ZJI 0.014 It 0.015 + 0.015 zk 0.015 zk 0.015 31 0.016 5~ 0.016 i 0.016 + 0.016 rfr: 0.017 f 0.017 f 0.017

mol-’

U,/kJ - 10.163 -10.155 -10.146 -10.137 -10.128 -10.118 - 10.109 - 10.099 - 10.090 - 10.080 - 10.070 - 10.060 - 10.049 - 10.039 - 10.028 -10.018 - 10.007 -9.995 -9.984 -9.972 -9.960 -9.948 -9.936 -9.923 -9.910 -9.897 -9.884 -9.871 -9.863

mol-

*

I 0.024 $1: 0.024 IO.024 jL 0.024 i 0.024 f 0.024 f 0.025 * 0.025 i 0.026 I 0.026 j; 0.027 f 0.028 I+Z 0.028 i- 0.029 4 0.030 zt 0.031 f 0.032 f 0.034 It 0.035 i 0.036 zt 0.038 -c 0.040 zk 0.041 zk 0.043 zt 0.045 i- 0.047 f 0.049 f 0.051 41 0.052

654

H. TABLE

T/K 143.0 144.0 145.0 146.0 147.0 148.0 149.0 150.0 151.0 152.0 153.0 154.0 155.0 156.0 157.0 158.0 159.0 160.0 161.0 161.395e

Fe/cm3

mol - 1

63250 58178 53576 49393 45586 42119 38956 36067 33426 31008 28794 26763 24898 23183 21605 20152 18813 17577 16435 16017

H.

CHEN,

7. Enthalpy

V&m3

C. C. LIM,

AND

R. A. AZIZ

of sublimation and internal energy [Torr = (101.325/760) kPa] mol-r

37.77 37.80 37.84 37.87 37.91 37.94 37.98 38.02 38.05 38.09 38.13 38.16 38.20 38.24 38.28 38.31 38.35 38.39 38.43 38.45

(dp/dT)/Torr

K -1

12.663 13.665 14.728 15.857 17.053 18.321 19.662 21.079 22.577 24.156 25.822 27.576 29.423 31.364 33.404 35.546 37.792 40.148 42.614 43.622

of solid

xenon

m/kJ

mol-’

15.261 15.253 15.243 15.234 15.223 15.212 15.200 15.188 15.175 15.161 15.146 15.131 15.115 15.098 15.080 15.061 15.041 15.020 14.999 14.990

f 0.026 zt 0.026 f 0.026 i 0.027 z!z 0.027 f 0.028 z!z 0.028 f 0.029 ZJi 0.030 i 0.031 zt 0.032 rE 0.034 f 0.035 i 0.037 z!z 0.039 f 0.041 * 0.043 f 0.045 f 0.048 f 0.049

U,/kJ - 14.102 - 14.088 - 14.073 - 14.057 - 14.042 - 14.025 - 14.009 -13.991 -13.974 -13.955 - 13.936 -13.917 -13.897 -13.876 -13.855 -13.832 -13.810 - 13.786 -13.762 -13.752

mol-1 f f f f f f f f f zt f f rt f f f f f f f

0.029 0.029 0.029 0.029 0.030 0.031 0.032 0.033 0.034 0.035 0.037 0.039 0.040 0.043 0.045 0.047 0.050 0.053 0.056 0.057

T/K FIGURE 1. Deviation curves for enthalpy of sublimation. Ziegler and co-workers and ‘p’ to present values.

Subscript

‘Z’

refers

to values

of

VAPOR

PRESSURES

OF Ads), K&J), AND Xc(s)

655

potential of Barker et al. t1 3, for xenon. There is evidence (14) to indicate that the virials derived from accurate intermolecular potentials are more reliable than those from low-temperature experimental data. Values of B’(T) required in equation (3) were obtained by differentiating the quadratic relation used to express the second virial coefficients, determined as described above, as a function of temperature. The values of (dpldT),,, were evaluated from the three-parameter vapor pressure equation (1). Ziegler and co-workers c’-“) have calculated the enthalpy of sublimation of Ar, Kr, and Xe by using results currently available to them. The comparison between their results and ours is presented in figure 1 over the limited temperature ranges in which we made our measurements. Agreement is best for krypton and worst for xenon. As mentioned in the introduction, we feel that the precise and copious results presented here allow one to calculate more precisely the enthalpy of sublimation. Since they did not present the values of internal energy, we have calculated these using equation (3) with their values of AH and psat.Other quantities appearing in the equation are those we have used in calculating our values. Deviation plots of internal energy are shown in figure 2. As expected, the agreement is similar to that for the enthalpy of sublimation.

FIGURE 2. Deviation curves for internal energies. Subscript ‘Z’ refers to values of Ziegler and coworkers and ‘p’ to present values

5. Discussion THE INTERMOLECULAR

POTENTIAL

FOR SPHERICAL

ATOMS

Because the inert-gas atoms have a closed-shell structure, they are spherically symmetric and thus the forces between them are central. In the dilute gas, only two-body interactions are involved and hence, the forces are pairwise additive. This means that the total intermolecular potential energy Q, of an assembly of N molecules can be written as a sum of terms, one for each pair of molecules :

656

H. H. CHEN,

C. C. LIM,

AND R. A. AZIZ

Here ri and rj are the position vectors of the ith and jth molecules respectively. A number of true pair potentials have been proposed, (e.g. Bobetic and Barker,(“) Barker et al.,(3,129 13) elc.), applicable for the dilute gas phase. For the condensed phase, however, the forces are no longer pairwise additive and many-body forces must be taken into account in the calculation of equilibrium condensed-phase properties. The potential energy @ for a given configuration of N molecules can be written as @(ri,

43(ri,

. . ., i
rj,

rJ+

. . .,

(5)

where ri, . . . , rN are their position vectors. Here 42 and $3 are the pair and triplet potential functions respectively. Experimental evidence”‘) indicates that the series is rapidly convergent for inert gas systems. Axilrod and Teller(‘) showed that third-order quantum-mechanical perturbation theory implies the presence of three-body triple-dipole interactions (DDD),. Beyond this term there are higher multipole terms: (DDQ),, (DQQ)3, (QQQ)3, and also higher-order perturbation terms, for example, (DDD),. Bobetic and Barker,(l*) and Barker et ~1.‘~’ “,r3) have accounted for these many-body effects in argon and krypton to a very good approximation by including only third-order triple-dipole interactions. That this approximation is satisfactory appears to be due to the fact that third-order dipole-quadrupole terms are nearly cancelled by the fourth-order triple-dipole term. An alternate approach is to use an effective central pairwise-additive potential whose parameters adjust themselves as best they can to the properties of the condensed phase. In other words, the effective potential, in an indirect way, takes into account the effect of various many-body forces. An example of such an effective potential is the Lennard-Jones (12,6) potential. This potential which has been found adequate in predicting equilibrium properties, is a central pairwise-additive potential containing only two adjustable parameters. THE INTERNAL

ENERGY

The internal energy is normally calculated on the basis of a potential model using Monte-Carlo techniques. If the potential chosen is a true pair potential then the internal energy U can be approximated by u

=

u,+u,+u,,

(6)

where U, and Uo are classical and quantum contributions due to interactions between pairs of atoms and U, is the contribution due to three-body forces. If the potential is an effective one, e.g. Lennard-Jones (12,6), then u = u,,+u,. (7) Unfortunately only a few calculations of internal energy near the triple-point temperature have appeared in the literature. In table 8 our derived values of internal energy are compared with those calculated by Monte-Carlo techniques. We see that our value of internal energy of solid argon at 80 K is in excellent agreement with that

VAPOR

PRESSURES

OF A+),

Kr(s), AND Xc(s)

657

TABLE 8. Comparison of the internal energy

T/K

Ar

80

Kr

110 115

present -7.1591

U,/kJ mole1

L.-J. 12,6

zt 0.0127

-7.22 al)

- 10.007 rt 0.032 -9.884 zk 0.049

Monte-Carlo calculations B.B. -7.19 (21) B.B.

151.5 156

- 13.964 It 0.034 -13.876 f 0.043

-7.154 caoJ K2 -9.991 f 0.029 W)

-9.95 (a) L.-J. 12, 6

Xe

B.F.W.

x4 - 13.944 f 0.042 w?)

- 14.14 @I’

predicted”” on the basis of the Barker-Fisher-Watts potential and the inclusion of only the (DDD), term for the many-body interaction. For the internal energy of krypton, our value agrees with that of Klein and Murphy”‘) based on Bobetic-Barker potential to better than 0.7 per cent. Even better agreement is found for the recent Monte-Carlo value of Jacobs and Card (22) who used the K2 potential of Barker et a1.(r2) Both groups included only the Axilrod-Teller triple-dipole term. The B.F.W. potential for argon and the K2 potential for krypton give excellent predictions for both macroscopic and microscopic properties involving only pair interactions. It is evident, from the excellent agreement we find here between the theoretical and experimental values of internal energy, that the potential together with the (DDD), term gives an excellent description of the condensed phase for these systems. In other words the Axilrod-Teller term adequately accounts for the many-body effects. The internal energy for xenon due to Jacobs and Card(22’ using the X4 of Barker et ~1.~‘~) potential with terms (DDD),, (DDQ)3, (DQQ)3, and (DDD), gave a value of -(13.944 + 0.042) kJ mol-’ at 151.5 K which compares favorably with our value of -(13.964 f 0.034) kJ mol-‘. APPENDIX The estimated errors in AH arise mainly from a combination of the errors in (dp/dT),,,, T, and V,. In calculating the error AZ in z = (dpldT),,,, we included the error arising from the fit of the results to equation (1) and the uncertainties in measuring p and T, i.e. AZ = a, + I@z/Q)Ap( +I(az/aT)ATj, where 0, = ((a2jab)%,2 +(a2/a#g

+2(a2fab)(aZ/ac)a~~)*”

;

0: is the variance of parameter b, 6: is the variance of parameter c, and u& the covariance of parameters b and c of equation (1). The variances and covariances are included in table 4.

658

H. H. CHEN,

C. C. LIM,

AND R. A. AZIZ

To obtain the error in AH we combined the error in (dp/dT),,, and the uncertainties in T and V,. Since V, was obtained from the second virial coefficients through the equation pv, = RT(1 +B/V,), we included the error in the second virial coefficient in addition to the uncertainties in the measurement of pressure and temperature. The error in U, included not only errors affecting AH but also that due to B’(T). We took 10 mK to be the uncertainty in Tea, as suggested in Table 7 of reference (8). Our triple-point temperatures are slightly higher than some of the published values but the temperature differences fall within the above chosen uncertainty. We have examined the effects of a downward shift in temperatures on our values of the enthalpy of sublimation (AH) and the internal energy (Us) and have found them to be small. For example, in the case of argon we reduced all our temperatures by 6 mK so as to bring the triple-point temperatures in closer agreement with those of Furukawa(23) and Ancsin.(24) Our values of AH and U, in the worst cases were changed by no more than 0.2 J mol-‘. The uncertainty in V, depends on the temperature and for the worst cases again they are no more than 30 cm3 mol-’ for argon, 42 cm3 mol-’ for krypton, and 71 cm3 mol-’ for xenon. Table 9 shows the various contributions to the error in AH and U,. In the table, all contributions to the uncertainty in AH and U, arising from temperature errors either directly (e.g. through T) or indirectly, e.g. through Az or B’(T), were combined to indicate the effect of temperature errors on AH and U,. Similarly all contributions, direct and indirect, due to pressure uncertainties were lumped together. TABLE

9. Contributions

to the uncertainty 6AH in LUS in worst cases, and contributions uncertainty SU. in U, in worst cases from

Ar ti Xe

(T=74K) (T= 102K) (T = 143 K)

from Ar Kr Xe

(T=74K) (T= 102K) (T = 143 K)

&AH/J mol-’ Sp 0.5

6T

6B

0.1 0.4 0.5

4.3 4.3 4.3

1.3 2.6 10.5

12.6 16.9 26.3

ST

6B

total

4.4 4.4 4.4

1.4 2.8 11.3

14.1 23.8 28.8

6.9 9.6 11.0

6&/J mol-l fit” Sp 0.8

6.2 1.3

7.5 10.4 11.8

a The error in U, due to the ‘fit’ includes the contributions

to the

total

from uz and the quadratic fit of B(T).

REFERENCES 1. Chen, H. H.; Lim, C. C.; Aziz, R. A. J. Chem. Thermodynamics 1975, 7, 191. 2. Schwalbe, L. A.; Crawford, R. K.; Chen, H. H.; Aziz, R. A. J. Chem. Phys. 1977,66,4493. 3. Barker, J. A.; Fisher, R. A.; Watts, R. 0. Mol. Phys. 1971, 21, 657. 4. McDonald, I.; Singer, K. J. Chem. Phys. 196!& 50, 2308. 5. Axilrod, E. M. ; Teller, E. J. Chem. Phys. 1943, 11, 299.

VAPOR 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

PRESSURES

OF Ar(s), Kr(s), AND Xc(s)

659

Chen, H. H. ; Aziz, R. A. ; Lim, C. C. Con. J. P/tys. 1971,49, 1569. Bedford, R. E.; Durieux, M.; Muijlwijk, R.; Barber, C. R. ht. J. Sci. Metr. 1969, 5, 47. Metro&ia 1%9, 5, 35. Table 7. Peterson, 0. G. ; Batchelder, D. N.; Simmons, R. 0. P&r. Rev. 1966, 150, 703. Losee, D. L. ; Simmons, R. 0. Pltys. Rev. 1968, 172,944. Swenson, C. A.; Anderson, M. S. T. E. Symposium, Corning, N. Y. 1971, Barker, J. A.; Watts, R. 0.; Lee, J. K.; Schafer, T. P.; Lee, Y. T. J. Chem. Phys. 1974,61,308. Barker, J. A. ; Klein, M. L.; Bobetic, M. V. IBM J. Res. Devel. 1976, 20, 222. Barker, J. A. J. Chem Phys. 1975,63,2767. Ziegler, W. T.; Mullins, J. C. ; Kirk, B. S. Tech. Rep. No 2, June 15, 1962, NutlBur. Stand. U.S. Contract No. CST-7238. Ziegler, W. T.; Yarbrough, D. W.; Mullins, J. C. Tech. Rep. No. 1, July 15,1964, Nutl Bur. Stand. U.S. Contract No. CST-1154. Ziegler, W. T. ; Mullins, J. C.; Berquist, A. R. Tech. Rep. No. 3, April 29,1966, N&But-. Stand. U.S. Contract No. CST-1154. Bobctic, M. V.; Barker, J. A. Phys. Rev. 1970, B2, 4169. Barker, J. A. In Rare Gas Solidr Vol. 1. Klein, M. L.; Venables, J. A.: editors. Academic Press: New York. 1976. Fisher, R. A.; Watts, R. 0. Mol. Phys. 1972,23, 1057. Klein, M. L.; Murphy, R. D. Phys. Rev. 1972, B6, 2433. Jacobs, P. W. M.; Card, D. N. Personal communication. 1976. Furukawa, G. T.; Bigge, W. R.; Riddle, J. I. Temperature, Its Measurement and Control in Science and Industry Vol. 4, Part 1. Instrument Society of America: Pittsburgh. 1972. p. 231. Ancsin, J.; Phillips, M. J. Metrologia 1969, 5, 77.