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The reduced equation of state, internal energy and entropy of argon and xenon
Levelt,
J. M. H.
Physica
1960
26
361-377
THE REDUCED EQUATION OF STATE, INTERNAL ENERGY AND ENTROPY OF ARGON AND XENON by J.
M. H. LEVELT
Van der \Vaals Laboratoriurn,
Gerneente
*)
Universiteit
Amsterdam
Synopsis Tables
of the equation
at regular
intervals
appropriate have
been
potential.
theoretical
performed
the
for
the
equations
of state.
In the reduction deviations
of state
these
with molecular
deviations
exceed
factors.
parameters factors
of
of argon
and xenon,
in the reduced
form
of state. The reductions the
is discussed.
of the reduction
of argon
are also given,
from corresponding
of the reduction
and entropy are presented
Lennard-Jones The effect
is investigated
6-12
of the un-
and found
to
densities.
equation
as parameters,
molecular
on the results
at liquid
energy
work on the equation
of the reduction
of these factors
constants
that
with
reliability
bc appreciable Tables
internal
and temperature,
for fundamental The
certainty
of state,
of density
and
parameters
the
reduced
with
the
as well as in that with critical
states are established
Possible
xenon,
to be used in the search for general
uncertainties explanations
for argon and xenon.
introduced
by
for the observed
the limited deviations
critical
empirical constants,
It is shown accuracy are offered.
3 1. Introduction. In recent years, much effort has been devoted to the development and improvement of theories on the equation of state of gases and liquids. The cell theory of the liquid state, the cluster expansion of the compressibility factor, the method of the radial distribution function and the Monte Carlo computations all try to give an understanding of experimental behavior in terms of molecular interaction. In the comparison of theory and experiment, the heavy noble gases play an important role. This is due to the fact that in most theories on the equation of state no numerical results can be obtained unless some simplifying assumptions are made, which rule out the application of the theory to all substances not possessing a simple molecular structure. These assumptions are : 1. Classical statistical mechanics may be used for the translational degrees of freedom of the molecules. *) The computations for this paper were performed under Contact DA- 1 l-022-ORD-2526 of the Office of Ordonance Research, at the University of Wisconsin Theoretical Chemistry Laboratory, Madison,
Wisconsin,
U.S.A.
Physica
26
361
-
2.
Internal
degrees
of freedom
are indel)endent
of th(> mutual
1)ositions
of
the molcculcs. 3.
The interaction
potential
of all pair interactions 4.
The
molecular
distances
interaction
function
these them
pressure
is a universal
subject
this
in terms become
description
with critical empirical
of
properties
the
Since
reduced
with
siderations
i-cduc~~l
thcl
if
greatI!,
to results
corresponding
of
constants, and
thrrmosimplifivtl
lcatling
Ia\\
I II thus
is included
tetnl~craturca
measured
in
terms
in
at
the
of
thts
critical
In this
paper
for empirical
experimental
must
‘1‘0 bc useful
as possible. is readily
for these
suitable
experimental
arc available
present
paper selected
and entropy
for argon
to for
obtain
and xenon,
reduced
reduction
as functions
arc
these
should
worker,
of density
be that
and tempcr-
ranges data
thermodynamic the
tht
for cot-r-(‘-
of density
over appreciable
foi
of densit!
data
to the theoretical
small interlrals data
the
con
starting
l)urposes,
the conditions
Furthermore,
accessible
form and at sufficiently
energy
in a form
fulfill
as closely
for
an appropriate
over a large inter\al
accurate
properties
data
\ve vi11 provide
gases,
states.
should
in a way which
in the
point
extend
presented
and temperature
starting
of corresponding
experimental
of state.
the conditions
be accurate,
states
Since
the
fat in its
of these
and the substance
spending
is, in reduced
form
related
states
to fulfill
properties
principles
in addition,
equations
data
and temperature,
often
modified
ant1
l)aram~tcrs
the law of corresponding
gases arc sulq~oscd
we will present
with theory;
of state
usin, u additional
thermodynamic
constants,
on such
l)racticc~ in the clr\~~lol~-
equation
constants,
noble the
is common
of stat<,. Here, one tries to find a general
not follo\ving
states,
based
comparison
internal
1~~. (XX-
the
and densit\-.
form,
critical
classical
critical
the heavy
corresponding
The
(Jf
density
parameters
equations
with
and liquids
true sense.
used
with
constants
experimental
reduced
gases
ature.
o, then
usually,
in reduced
l)ressure,
universal
1). That
parameters.
This reduction
point
sincct
arc
formulation
of reduction
statement,
point
treatments
the older
statcls
this is also true for the other
states.
ment of classical
for
of freedom
through
molecular
those
tcinp~rature
and carried
namely
critical
of reduced
b!-
the substnnccs
dimcnsionl~5s
of I‘ and
formulated
nelvcr
~1 of the
is given
c/’= ~f(v. (T), then
art’ matlc
to a law of corresponding that
interaction
combinations
function
dcgrccs
be noted
molecular
and tcmpratuw
Theoretical
It may states,
the
properties.
by being
function
follo\v a la\\. of corresl~oncling
in apl)ropriatc
of internal
by a universal
of the distanccl,
density
pressing
dynamic
that
four conditions
is, if pr~ssuw,
is obtaiiic~cl by- addition
of the molec~~lcs.
it is assumed
a two-parameter
absence
is described
Y of the centers
If in addition fulfilling
for a group of molecules in the group.
of density have
been
properties.
compressibility and temperature.
factor, The
REDUCED
reductions
EQUATION
have been carried
Lennard- Jones
6-12
potential.
OF STATE
OF ARGON
AND
out with the molecular For
the compressibility
363
XENON
parameters factor
of the
reductions
have also been carried out with the critical parameters. The method of reduction and the choice and accuracy of the reduction factors are described in 9: 2, the results of the reductions are then tabulated. A comparison of these results permits a check of the validity of the law of corresponding states for argon and xenon. This comparison and conclusions drawn from it are contained
in 5 3.
9 2. Method and reduction 9arameter.s. The procedure followed in obtaining reduced thermodynamic properties at regular intervals of reduced density and temperature may be summarized as follows. After a choice of experimental data is made, these data are tabulated, with density and temperature as variables. Then, the property of interest, as well as the entries of the table are brought into reduced form. Finally, since values of the thermodynamic properties are desired at values of the reduced temperature and density which generally do not coincide with the experimental entries of the table, a double interpolation must be performed. In the present research, the experimental information consists of data for the product PV for argon and xenon, as determined by Michels and coworkers 2). The authors converted the experimental data into tables containing PV at regular intervals of density and temperature. From these tabulated values of PV, they derived similar tables for the other thermodynamic properties, for instance, for the internal energy UZ and the internal entropy St. For the noble gases, these functions are the difference in energy and entropy, respectively, between the gas under consideration and an ideal gas at the same density and temperature. Expressed in the critical constants, the density range of the experimental data for argon is from zero up to 2dC, the temperature interval is from T = 0.9T, up to 2.8T,. For xenon the corresponding figures are: d from zero up to 2.5d, and T from l.lTc up to 1.5T,. The accuracy of the PV values is about one part in 10,000, though somewhat less in the regions around the critical point and in the liquid phase. The accuracy of UZ and St is roughly an order of magnitude less. In the reduction with molecular parameters, reduced quantities are obtained as follows: d* = Na3lV,,
T* = kT/e, Ut* = UiINe, Si* = SijNk,
(1)
where N is Avogadro’s number, k is Boltzmann’s constant, E and 0 are the energy and length parameter, respectively, which characterize the intermolecular field, while Vm is the molar volume *). *) The numerical in so-called Amagat
data for argon and xenon were obtained, as usual, by expressing the volume units (of volume). The Amagat unit of volume for a particular gas is equal to
364
J. Al. H. LEVEL’1
When the critical constants are used in obtaining rclduccd tables for the compressibility factor, the rcduccd quantities arc calculatcvl as follo\vs :
where thv intlvs c d(~notc~scritical
constants.
I3g’ means of (1) ant1 (2) the tabulatrd experimental prop-tics and also the entritx of the tables wtw reduced. Then, in the last stall) of thv Compaqtation, a tn.o-dinlensional intcqx)lation was matl~~. ITor the tltGrc,cl rountl valuw of on0 of thcx vnriablcs, a four-1)oint I.agrangian intcrpolatioii \~a5 performed. In the intcrmcdiatc tablt, so obtained, interpolations v.vrt‘ matl~, for the second variabltl. All computations \vtw carrictl out on a I+ndix (;--151) computer, lvitli intcrmvdiatt~ storagt~ on magnetic tap. ‘l‘htl accuracy of the interpolation method was frcclwntly chc~k~tl, for instance by interpolating for the original vspcrimcntal densities from which the 1)V tables had been dcrix~ccl. It turned out that except for the numbers at the cclgcs of thtl tables, tqecially thaw closc~ to the two-l)haw region, that tables obtained by interpolation have: ahnost the samt‘ accuracy as thcb original ant’s, i.c., mostly of thv order of magnitude of thy last digit cluotvtl. Howwr, the reliability of thv tables is dt+cwnined not so much bar thci accuracy of the interpolation procedure, as by the precision of the molecular and critical parameters. This brings us to the problem of sclccting the lwst values of t’ and (r to construct the rctluction factors. In the case of reduction with molecular parameters, thcsv rctluction factors are combinations of t‘ and 0. In the present papc’r, thcw cluantitics were deduced from data for the second virial coefficient as a function of tcmperaturc ; the second virial cwfficicnt in turn was clcrivetl from compwssibility measurements. ‘l‘hc accuraqr of the reduction factors is thcrvfow far less than that of the original comprt~ssibility measurements. ‘This is also trucl for reduction factors from critical constants, since thcsc arc usually not as well known as compressibility data in other regions. Therefore, it is thtl accuracy of the reduction factors rather than that of tht, interpolation process, which determines the reliability of the reduced tables.
REDUCED
EQUATION
OF STATE
OF ARGON
AND XENON
365
As regards the molecular parameters, they may be calculated after a special form for the interaction potential has been chosen. In this paper, the so-called
Lennard- Jones
6- 12 potential
qJ(f-) = 4&+912
-
was selected : W)6},
(3)
since many authors have used this potential, which is not inferior two-parameter potentials, in work on the equation of state.
to other
In the determination of E and a from experimental data for the second virial coefficient B, use was made of the theoretical B* vs. T* relation for the 6-12 potential, as derived by Hirschfelder e.a. 3). The reliability of the molecular parameters is limited, due to the inaccuracies of the measured
Fig. virial
1. Difference coefficient
between
reduced
as a function potential
argon
experimental
of reduced
and theoretical
temperature,
values of the second
if the Lennard-Jones
6-12
is used for argon and xenon.
0 - ref. 4a
xenon
0 - ref. 4c
a - ref. 5
v - ref. 6
0 - ref. 2a, 2e
0 - ref. 2c.
B values, which depend on the location and extension of the density range studied, and on the number and reliability of experimental points in this range. Usually, the reliability of B values is not much better than one percent. Uncertainties are also introduced by the procedure of comparing experimental and theoretical B vs.T curves.
365
J. 31.
H. LEVELT
For xenon, the parameters t‘ and CT as derived 1,~’ LVhalley and Schneider Jd), from compressibility data in the range 0°C up to 7OOY‘, were taken : F,‘k -= 225.3 -~ i l.l”K, cr = 4.070 1 0.013 A. (41 In Fig. 1r, the differences between observed and thaw-ctical plotted as a function of 7’* for this choice of paramettsrs. For argon, the parameters published by Mic hels ~.a.“(‘), +h =
119.8’K,
I-j* valucx arc
cr m=3.405 L\,
(5)
are in agreement with those derived b\. \Vhall~‘~. and SC h nc~itl~~r -In) from 0.33”1<, data in the range - 100°C up to +60”0”(‘, namely, YA = 1 19.49 CT= 3.409 +- 0.007 A. However, as is shoun in I;ig. la, the agreemc~nt between cxlwrimental and theoretical B* vs. 7‘* \,alues is not close at lo\z temperatures. The deviation pattern IZhCX,,.- K*+k,. 1’s. 7‘* of Fig. 1h has been obtained with the parameter pairc:‘k =
119.3”K,
CT2 3.43 .s.
16)
Although there is some improvemr~nt, the esperimwtal cur\.<’ again docx not show a close fit to the theoretical one at IOLVtt~mlxeraturw. Hoa-c~vrr-, with the parameters (4) and (6), the deviation patterns for q+Jll (1;ig. 171) and xenon (Fig. lr) arc almost identical. Keduced thermodynamic properties have bc~ln tabulatc~d for xgon with the set (5). They are shown in th(a ‘I’ablcs 1~ III. A surve!. of the diffcrenws introduced by using the second set, (6), is given in Table IS. I-or xenon, the calculated with the lw-amctc’rs (4), reduced thermodynamic prolwrtirs, are shown in the Tables IV-\‘I. In the reduction with critical +zvanzetcvs, the following L-alues ha\~l bc~w used for the critical constants. For argon d, 1 300.4 Amagat
units,
7‘, -: 150.86”K
‘g),
with estimated accuracies of 3 parts in 1000 for the clensity anti 0.02%: the temperature. For xenon. d, =
186.3 Amagat
with an accuraq- of two parts temperature. The results of the reduction Tables \‘I1 and \,‘III.
units,
7‘, = 289.74”K
in 1000 in thcx density Lvith critical
parameters
-lb),
(7) in (8)
ancl 0.001’ (‘ in thtl art’ shoLvn in thcb
4 3. Conqbarison of tabulated @q!wties, and concl~~~io~s. The comparisons that may be made fall into three classes. First, the influence of a small uncertaint?; in the molecular reduction factors may be invcstigatetl b!computing the reduced properties for the two sets of parameters (5) and (6)
REDUCED
EQUATION
OF STATE
TABLE
-
ld* T*l -__
Reduction 0.050
1.15 1.20 1.25 1.30 1.35
0.7976 0.8138 0.8281 0.8412 0.8529
1.40 1.45 1.50 1.60 1.70
AND
XENON
367
I
The equation of state PV/KI‘ of argon. with the molecular uarameters elk = 119.8”K and o = 3.405 is
7 _
OF ARGON
I 0.200
0.250
O.‘OO I 0.150
0.300
0.325
0.350
0.400
-
0.6491 0.6771 0.7021 0.7244
0.5482 0.5840 0.6158
0.4872 0.5270
0.4112 0.4570
0.3538 0.4045
0.3313 0.3844
0.3125 0.3683
0.2870 0.3493
0.8632 0.8727 0.8815 0.8970 0.9103
0.7445 0.7629 0.7796 0.8092 0.8346
0.6444 0.6704 0.6943 0.7366 0.7732
0.5629 0.5956 0.6258 0.6794 0.7260
0.4990 0.5375 0.5733 0.6374 0.6935
0.4517 0.4957 0.5368 0.6111 0.6766
0.4342 0.4809 0.5248 0.6044 0.6747
0.4210 0.4706 0.5173 0.6024 0.6776
0.4087 0.4649 0.5178 0.6146 0.7002
1.80 1.90 2.00 2.10 2.20
0.9218 0.93 I7 0.9406 0.9484 0.9554
0.8567 0.8759 0.8930 0.908 1 0.9216
0.8048 0.8326 0.8574 0.8794 0.8992
0.7667 0.8025 0.8346 0.8633 0.8890
0.7429 0.7866 0.8258 0.8608 0.8925
0.7347 0.7862 0.8324 0.8739 0.9114
3.7371 0.7926 0.8424 0.8870 0.9274
0.7444 0.8039 0.8573 0.9053 0.9486
0.7762 0.8438 0.9047 0.9593 1.0086
2.30 2.40 2.50 2.70 2.90
0.9617 0.9674 0.9726 0.9815 0.9891
0.9338 0.9449 0.9550 0.9724 0.9873
0.9170 0.9332 0.9480 0.9738 0.9956
0.9 123 0.9334 0.9527 0.9866 1.0152
0.9210 0.9470 0.9706 1.0125 I .0478
0.9452 0.9760 1.0040 I .0537 1.0957
0.9640 0.9972 1.0274 1.0811 1.1263
0.9878 1.0234 1.0560 1.1135 1.1619
1.0531 1.0936 1.1306 1.1957 1.2503
3.10 3.30 3.50
0.9955 1.0011 1.0061
1.oooo 1.0109 1.0206
1.0143
1.0304 1.0445
1.0397 1.0609 1.0794
1.0780 I. 1042 1.1271
1.1316 1.1627 1.1898
1.1650 1.1985 1.2276
I .2034 1.2392 1.2703
1.2972 1.3373 1.3722
0.450
0.500
0.550
0.575
0.600
0.625
0.650
0.675
0.700
0.2162 0.3312 0.4378 0.5368
0.1512 0.2827 0.4049 0.5176 0.6218
0.2343 0.3738 0.5024 0.6207 0.7296
0.3465 0.4931 0.6272 0.7503 0.8633
0.4919 0.6440 0.7831 0.9099 1.0260
0.8303 0.9233 1.0095 1.1633 1.2967
0.9673 1.0633 1.1520 1.3100 1.4464
d* ‘-+ 1.15 1.20 1.25 1.30 1.35
1 0.2813 0.3527
0.306 I 0.3882
0.3779 0.4714
1.40 1.45 1.50 1.60 1.70
0.4202 0.4840 0.5440 0.6535 0.7502
0.4652 0.5375 0.6053 0.7284 0.8366
0.5585 0.6396 0.7153 0.8519 0.9715
0.7936
0.9367 1.0614
0.7183 0.3078 0.8908 1.0397 1.1691
1.80 1.90 2.00 2.10 2.20
0.8359 0.9122 0.9805 1.0417 1.0969
0.9323 1.0171 1.0928 1.1603 1.2210
1.0765 1.1690 1.2514 I .3246 1.390 1
1.1706 1.2665 1.3518 1.4274 1.4949
1.2821 1.3810 1.4688 I .5463 1.6153
1.4130 1.5145 I .6042 1.6831 1.7530
1.5654 I .6692 1.7599 1.8394 1.9095
2.30 2.40 2.50 2.70 2.90
1.1467 1.1919 1.233 1 I .3054 1.3659
1.2756 1.325 I 1.3701 1.4489 1.5145
I .4488 1.5019 I .5500 1.6339 1.7033
1.5553 1.6097 1.6589 1.7447 1.8154
1.6769 1.7323 I .7823 1.8695 1.9408
1.8152 1.8712 1.9216 2.0096 2.0808
1.9717 2.0274 2.0784 2.1664 2.2366
2.1487 2.2041 2.2546 2.3412 2.4097
2.3478 2.4020 2.4514 2.5360 2.6321
3.10 3.30 3.50
1.4175 1.4616 1.4998
1.5701 1.6174 1.6580
1.7617 1.81 10 1.8530
1.8747 1.9243 1.9664
2.0006 2.0500 2.09 17
2.1406 2.1892 2.2300
2.2956 2.3429 2.3822
2.4676
2.6579
-
-L
0.6287 0.7141
-
-
-
J. M. H. LEVELT
368
TABLE
i-
Reduction
\
a*
-
0.100
0.050
-
L
1.20 1.25 1.30 1.35
-0.443 -0.43c -0.419 -0.409
-0.879 -0.850 -0.825 -0.804
1.40 1.45 1.50 1.60 1.70
-0.4oc -0.392 -0.385 -0.372 -0.361
-0.785 -0.77c -0.755 -0.73c -0.709
1.80 1.90 2.00 2.10 2.20
-0.352 -0.345 - 0.338 -0.333 -0.327
2.30 2.40 2.50 2.70 2.90 3.10 3.30 3.50
1.20 1.25 1.30 1.35
7Jg/N& of argon. = 119.8”k and 0 = 3.405 A
i 0.250
0.200
0.150
T*
s
II
The reduced internal energy with the molecular oarametersaik
0.300
0.325
0.350
I
,
- 1.256 - 1.214 -1.180
- 1.581 - 1.535
- 1.922 - 1.866
-2.233 -2.174
- 2.378 - 2.320
-2.517 -2.463
- 1.152 -1.129 - 1.108 - 1.0% - 1.044
-
1.499 1.469 1.442 1.399 1.364
- 1.824 - 1.790 - 1.760 -1.711 _ 1.672
-2.129 -2.094 -2.063 -2.013 - 1.972
-2.276 -2.242 -2.211 -2.161 -2.119
-2.421 -2.387 -2.357 - 2.307 -2.266
-0.694 -0.677 -0.665 -0.654 -0.644
- 1.019 -0.998 -0.981 -0.965 -0.951
-
1.334 1.309 1.287 1.268 1.251
-
1.639 1.611 1.586 1.564 1.545
-
1.937 1.906 1.879 1.855 1.834
-2.084 - 2.052 - 2.024 - 1.999 - 1.977
-2.230 -2.198 -2.169 -2.143 -2.120
- 0.322 -0.317 -0.312 -0.305 -0.299
- 0.635 -0.625 -0.617 -0.602 - 0.589
-0.938 -0.926 -0.914 -0.892 -0.874
-
1.235 1.220 1.206 1.177 1.153
-
1.527 1.509 1.492 1.458 1.429
~ -
1.814 1.794 1.774 1.736 1.702
-
1.956 1.935 1.914 1.873 1.837
- 2.098 - 2.076 -2.054 -2.010 - 1.971
-0.294 -0.291 -0.288
-0.580 - 0.574 -0.568
-0.860 -0.850 -0.842
- 1.135 -1.122 -1.111
- 1.675 - 1.654 - 1.636
- 1.808 - 1.785 - 1.765
- 1.940 - 1.915 - 1.893
0.500
0.550
0.600
0.625
0.650
0.675
-3.986
-3.934 -3.903
-4.140 -4.112 -4.087 - 4.055
- 4.295 - 4.266 -4.241
- 4.450 -4.420 - 4.393
- 4.027 -4.001 -3.975 -3.923 -3.872
0.400
, ,
-
0.450
-
-
- 1.407 - 1.390 - 1.375
-
0.575 -
-3.836 -3.806 -3.782 -3.751
- 2.785 -2.742
-3.10 -3.057 -3.021
-3.37 -3.337 -3.306
-3.66 -3.628 -3.600
1.40 1.45 1.50 1.60 1.70
-2.705 - 2.674 -2.647 -2.599 - 2.557
-2.989 - 2.96 1 -2.936 -2.890 -2.848
-3.278 - 3.253 -3.229 -3.183 -3.141
-3.574 -3.550 -3.526 -3.480 - 3.435
-3.724 -3.700 -3.628 -3.581
-3.876 -3.851 -3.826 - 3.776 -3.727
1.80 1.90 2.00 2.10 2.20
-2.520 -2.487 - 2.457 - 2.429 - 2.404
-2.811 -2.776 -2.744 -2.714 -2.687
-3.102 -3.065 - 3.030 - 2.998 -2.967
-3.393 - 3.353 -3.314 -3.278 -3.244
-3.538 -3.496 -3.455 -3.416 -3.380
-3.681 -3.636 -3.593 -3.552 -3.513
-3.823 - 3.775 -3.729 -3.685 -3.643
2.30 2.40 2.50 2.70 2.90
-2.380 -2.355 -2.331 -2.282 -2.238
-2.660 - 2.633 -2.606 -2.551 -2.501
-2.938 -2.907 -2.877 -2.815 -2.758
-3.211 -3.177 -3.142 -3.072 -3.006
-3.344 -3.309 -3.272 -3.196 -3.125
-3.475 -3.437 -3.398 -3.318 -3.241
-3.603 -3.563 -3.521 -3.435 -3.352
-3.727 -3.684 - 3.639 -3.547 -3.459
-3.848 -3.801 -3.752 -3.654 -3.559
3.10 3.30 3.50
-2.202 -2.172 -2.145
-2.450 -2.424 -2.391
-2.710 -2.667 -2.628
-2.950 -2.898 -2.851
-3.065 -3.008 -2.957
-3.175 -3.114 -3.058
-3.281 -3.214 -3.153
-3.381 - 3.309 -3.241
-3.475 -3.396 -3.321
-
-
-
-3.958
-3.676
-
-
-
-
REDUCED
EQUATION
OF STATE
TABLE
o.050
AND
369
XENON
III
The reduced internal entropy &/Nk of argon. with the molecular parameters c/k = 119.8’K and u = 3.405 A.
Reduction ;*:I
OF ARGON
1
0.100
’ o.150 1 1
0.200
1
0.250
~
~
0.300
0.325
~
0.350
i
0.400
1.20 1.25 1.30 1.35
-0.178 -0.167 -0.159 -0.151
-0.360 -0.336 -0.317 - 0.30 1
-0.472 -0.447
-0.621 -0.586
-0.761 -0.718
~ 0.887 -0.842
-0.946 -0.902
- 1.003 -0.962
- 1.1 16 - 1.082
1.40 1.45 1.50 1.60 1.70
-0.145 -0.139 -0.134 -0.126 -0.119
-0.288 -0.276 -0.266 -0.250 -0.238
-0.426 -0.410 -0.396 -0.373 -0.355
-0.560 -0.539 -0.521 -0.493 -0.471
-0.687 -0.663 -0.643 -0.612 -0.588
-0.810 -0.785 -0.764 -0.731 - 0.706
-0.870 -0.846 -0.825 -0.792 -0.767
-0.931 -0.907 -0.887 -0.855 -0.829
~ 1.056 ~ 1.034 -1.015 -0.984 -0.959
1.80 1.90 2.00 2.10 2.20
-0.114 -0.110 -0.106 -0.104 -0.101
-0.228 -0.220 -0.213 -0.208 -0.203
-0.341 -0.330 -0.321 -0.313 -0.306
-0.454 -0.441 -0.429 -0.420 -0.412
-0.569 -0.554 -0.541 -0.530 -0.521
- 0.686 -0.670 -0.656 -0.644 -0.634
-0.747 -0.730 -0.715 -0.703 -0.693
-0.809 -0.792 -0.777 -0.764 -0.753
-0.938 -0.920 - 0.904 -0.891 -0.879
2.30 2.40 2.50 2.70 2.90
-0.099 - 0.097 - 0.095 -0.092 - 0.090
-0.199 -0.196 -0.192 -0.186 -0.182
-0.301 -0.296 -0.291 -0.283 -0.277
-0.405 -0.399 -0.393 -0.383 -0.374
-0.513 -0.505 - 0.498 -0.486 -0.476
-0.625 -0.617 - 0.609 -0.594 -0.583
-0.683 -0.674 -0.666 -0.650 -0.638
-0.743 -0.734 - 0.725 -0.708 -0.695
-0.868 -0.858 -0.848 -0.829 -0.815
3.10 3.30 3.50
-0.089 -0.087 ~ 0.086
-0.179 -0.176 -0.174
-0.272 -0.269 -0.265
-0.368 -0.364 -0.360
-0.468 -0.463 -0.458
-0.574 -0.567 - 0.56
-0.629 -0.621 -0.614
-0.685 -0.676 -0.669
-0.802 -~-0.792 -0.781
,“iI-* -1.20 1.25 1.30 1.35
0.450
i
0.500
~
/
0.550
~
1
0.575
1
0.600
I
’ ~
~
-1.769 ~ 1.747 - I.728 ~ 1.704
- 1.847 ~ 1.828 - 1.803
IjpII,
0.650
1
0.675
- 1.976 ~ 1.953 - 1.933 - 1.91 1
~ 2.089 - 2.065 - 2.044 -2.021
~ 1.240 - 1.212
- 1.380 - 1.357
- 1.541 - 1.520
~ 1.652 ~ 1.633 - 1.609
1.40 1.45 1.50 1.60 1.70
- 1.189 -1.170 - 1.152 - 1.123 - 1.098
-
1.336 1.318 1.302 1.273 1.248
- I.501 ~ 1.483 - 1.467 - 1.438 -1.411
~ -
1.590 1.573 1.556 1.526 1.497
~
1.684 1.666 1.649 1.618 1.588
-
1.783 1.764 1.747 1.714 1.682
1.80 1.90 2.00 2.10 2.20
~ ~ -
1.076 1.057 1.041 1.027 1.013
- 1.225 - 1.205 -1.188 - 1.172 - 1.157
- 1.387 - 1.365 - 1.345 ~ 1.328 -1.311
-
1.472 1.450 1.429 1.410 1.393
~ ~
1.561 1.538 1.516 1.495 1.477
~ ~ -
1.654 1.629 1.605 1.584 1.564
2.30 2.40 2.50 2.70 2.90
- 1.001 -0.990 ~ 0.979 -0.958 -0.941
-1.144 - 1.131 - 1.119 - 1.095 - 1.075
~ -
1.297 1.283 I .269 1.241 1.218
- 1.377 - 1.362 ~ 1.348 - 1.317 -1.293
~ 1.460 - 1.444 - 1.429 - 1.396 -1.370
- 1.546 ~ 1.529 - 1.512 ~ 1.478 -1.450
- 1.635 - 1.617 - 1.599 - 1.562 -1.532
- 1.728 - 1.708 - 1.688 ~ 1.649 -1.617
3.10 3.30 3.50
-0.927 -0.915 -0.904
- 1.059 -1.045 - 1.034
~ 1.199 -1.182 -1.168
- I .273 -1.254 -1.238
- 1.348 -1.328 -1.311
- 1.426 -1.404 -1.385
- 1.506 -1.483 - 1.462
~ 1.589 -1.563 1 -1.541
1
370
J. .?I. 11. LEVELT ‘I’ \ l’,l.I:
I\
REDUCED
EQUATION
OF STATE
TARLE
Reduct
OF ARGON
371
XENON
\‘I
The reduced internal entropy St,‘iVk of xenon. 7 with the molecular parameters e/k = 225.3”K and B = 4.070 A.
cl* I‘\. T*
0.100
0.150
0.200
0.250
0.300
I .35 1.40 1.45 1.50 1.60
-0.297 -0.283 -0.272 -0.261 -0.243
-0.438 -0.417 -0.400 -0.385 -0.359
-0.574 -0.546 - 0.524 -0.504 -0.472
-0.701 - 0.667 -0.642 -0.619 -0.582
-0.819 -0.783 -0.755 -0.732 -0.692
1.70 1.80 1.90
-0.229 -0.218 -0.210
-0.339 -0.324 -0.314
-0.447 -0.430 -0.418
- 0.555 - 0.536 -0.523
0.500
0.550
0.600
0.650
\
AND
0.350
0.400
-0.874 -0.838 -0.81 1 -0.787 -0.747
- 0.928 -0.894 -0.867 -0.843 -0.803
- 1.034 - 1.003 -0.978 -0.956 -0.917
~ -
-0.663 - 0.643 -0.629
-0.717 -0.697 -0.684
-0.773 - 0.753 - 0.739
-0.887 -0.867 -0.854
- 1.008 -0.988 -0.975
0.700
0.750
0.800
0.850
0.325
d
0.450 1.143 1.117 1.095 1.074 1.037
I
?‘* 1.35 1.40 1.45 1.50 1.60
- 1.261 _ 1.240 - 1.221 ~ 1.202 ~ 1.166
~ ~ -
1.393 1.375 1.359 1.341 1.305
~ 1.542 ~ 1.526 -1.511 - 1.493 -1.456
~ -
1.70 1.80 I .90
-1.137 -1.117 - 1.105
~ 1.276 ~ 1.256 - 1.243
~ 1.426 ~ 1.406 - 1.392
- 1.590 - 1.567 - 1.552
-
1.709 1.694 1.678 1.660 1.621
-
1.895 1.879 1.862 1.842 1.801
- 2.099 -2.080 -2.062 -2.040 ~ 1.994
- 1.766 - 1.955 - 1.741 - I .928 - 1.724 - 1.907
I
I
~ -
2.320 2.299 2.277 2.253 2.202
-2.158 -2.127 -2.103
,
-
2.558 -2.533 - 2.508 - 2.480 - 2.423 - 2.374 - 2.338 -2.310
for argon (Table IX). Then, the validity of the law of corresponding states may be checked by comparison of reduced isotherms of argon and xenon in the case that molecular parameters are used for reduction. (Table X). Finally, a similar check may be made for properties obtained by reduction with the critical parameters of argon and xenon (Table XI). Table IX shows that values of PVIRT, Ui* and St*, obtained with the molecular parameters (5) and (6) for argon, show a satisfactory agreement below the critical density. However, in the dense gas and at low temperatures discrepancies become apparent for the compressibility factor. They must be attributed mainly to the influence of the difference in G, since the density reduction factors differ by 2% whereas the temperature factors differ by 4 parts in 1000 only. The percentage deviations in Ug* are almost constant since Ui* varies roughly proportional to the density. In general, the internal functions Ui* and Se* are less sensitive to changes in reduction parameters than the compressibility factor. The conclusion must be that reduced experimental compressibility data at high densities, and a fortiori in the liquid state, should be considered with due reticence, since small uncertainties in the reduction parameters have a large effect on the factor PVjRT in this range. Turning to the comparison of reduced argon and xenon isotherms obtained with molecular parameters, Table X shows deviations from corresponding states for the thermodynamic properties especially at high densities. The
_
0.95 1.00 1.05 1.10 l.lS
1 O.d7dO O.hY97
1.20 1.25 1.30 I AS 1.40
~ 0.9236 ~ 0.9331 0.9359 ~ 0.941 I 0.9459
1.45 1.;10
0.764h 0.7875
0.6939 0.7230 0.7484 0.7710
O.Sh4c 0.~~09 I 9.6469 0.6797 0.7037
O.dS40 0.846s 0.8777 0.8daO 0.8973
0.7911 O.t?092 0.8254 O.d406 0.8542
0.734:i 0.7Sd2 0.7795 0.7988 O.alhh
1.70 1.>10
0.9,?‘0.1 0.9.?‘43 0.961,5 0.9676 0.97.1 I
0.9059 0.9 I .\” 0.927n 0.9390 0.9503
0.3667 0.87&% 0.8989 0.9145 0.9.320
I.90 2.00 2.10 2.20 2.30
0.977:; 0.9620 0.9856 0.9390 0.9920
0.9.59b 0.9676 0.9751 0.9dlh 0.9675
0.94s7 0.9577 0.948s 0.97d I 0.9hS9
1.60
O.h601
0.5199 ~ O.-ihii:: O.S?Sh ~ O.Sl9ii 0.5Ohl ~ O.ShS2
0.71.16 0.7395 0.7630 0.7H46
O.iL129 O.rt479 0.h747 0.6979 0.9la2 0.9&l 0.9520 0.964 0.9738 0.9903
o.s309
OA:,Y4 0.4490 0.5033
O..ln30 0.4237 0.4d24
0.442,5 1 0.6062 O.b754 i 0.6435 0.70.55 0.6776 0.7AE 0.70?9 0.7.511.\ 0.7.17a
0.5704 O.h17,j, 0.6.5.59 0.5909 0.7233
I).S.S.)O 0.,59Fh 0.6406 0.5794 0.71i.3
0.5.154 0.,5%2 0.6322 0.5747 Cl.7142
OAO45 0.9228 o.as57 0.~842 !I.9091
0.7816 0.8032 O.d419 0.87.55 0.9050
.7044 .71192 .;336 .a723 .9oi3
0.7,533 0.7812 O.d3 12 0.07so 0.9134
0.74if> 0.7796 o.u.53 O.c;840 0.9269
0.7soa 0.78SO O.d444 0.9002 0.947,<
0.93 I I 0.9so7 0.94d2 0.9839 0.99.10
0.9310 0.9542 0.9751 0.9937 1.010s
,936,) .9631 .9872 .OOd7 .02jI
0.9474 0.9777 1.ooso 1.0294 l.O,514
0.9hk 0.99X6 I .029 1 1.osv I.061 I
0.9894 1.024R 1.0404 1.0907 1.1178
I .07 l!i ~ 1.0897 1 l.lOh4 ~ I.1217 I 135;
1.lrj.P. 1.1239 1.142.1 1.1.i95 1.17S.i
1.1425 1.1~51 i.l%a 1.2044 1.2217
/.I’
2.1
3.2937 7.4490 I.5892 II.7162 >.;i.315
0.4230 o.sc79 0.7343 O.r(h99 0.9937
0.6013 0.7745 0.9290
O.hi!,5.3
I
2.40 2.50 2.hO 2.70 2.80
1 .045b ~ 1,0619 ~ 1.0764 I .0900 1.I025
cl, I.? 1-r 0.95
1.oo
1.0s 1.10 I.15
0.3371 0.3208, ~ 0.40.52 ~ 0.3940 ’ 0.4485 ~ 0.442s
0.2246 i 0.3259 0.4200 0.5072,
0.2439 0.35X 0.4575 0..%519
0.2637 0.4050 0.5160 0.6176
0.206 0.3499 0.4809 0.4003 0.709
I
0.3121 O..J914 0.4654
0.3128 O.j992 0.4790
0.5260 o.sd49 0.6393 0.6898 0.7346
0.5344 0.S9dl 0.4572 0.7118 0.7626
~ 0.5541 0.6231 O.htji>9 ~ 0.7460 ~ O.dO07
0.5879 0.4424 0.731 I 0.7944 0.053.5
0.6339 0.7la9 0.7927 O.bh07 0.9236
0.7109 0.7945 0.8752 0.9474 1.0144
0.0034 0.8993 0.9826 l.OS91 1.1294
I.9345 1.0321 I.1 195 1.1994 1.2723
1, 1:J02 ~ 1.1991. 1.2903, I A730 1.4491
0.7801
; 0.9S73 1.0142
0.8096 0.8536 0.9327 1.0019 1.0~20
0.8SlS 0.8989 0.9d40 ~ I.0584 1.1237
0.90~1 0.9588 1.0499 1.129.3 1.19?!?
0.9819 I.0360 I.1328 1.2169 1.2903
I .0760 1.1322 I .2X54 I .,12.38 l.400d
1.1941 1.2541 1..!‘610 1.4532 l.S331
1..1403 I.4026 I.5133 l.hO’13 1.0903
1.51nc I .5dJO 1.694jJ 1.792.: 1._7SC~
I
1.20 1.25 1.30 1.35 1.40
0.5271 o.sal3 0.63l.i 0.6779 0.7209
1.45 1.50 1.60 1.70 1.30
0.7610 0.7983 0.8655 0.9244 0.9742
1.90 2.00 2.10 2.20 2.30
1.0221 1.0630 1.0999 1.1329 I, 1626
1.0641 1.1085 3.1485 1.1843 I.2165
l.llhh 1.1644 1.2074 1.2461 1.2804
I.1014 1.2327 3.2736 1.3197 1.3564
I.2631 1.35Sl I.3136 1.4124 1.3633 1.46Jo 1.4067 1.5091 I .445&> I .5457
1.4ha.S I.,5282 1.5816 I.6288 1 .h709
I .h03 1 1.6648 1.7197 I .7683 1.a113
1.7OlC 1X246 1.8808 1.9301 I .973s
1.9470 2.0112 2.0677 2.1169 2.1599
2.40 2.50 2.60 2.70 2.60
I.1896 1.2142 1.2366 1.2571 1.2759
I .24S7 1.2723 1.295s 1.3185 1.3388
1.3120 1.3404 1.3663 1.389d 1.4115,
1.319s 1.4200 1.447.3 1.4723 1.4952
I.4307 I.5124 I..541 I I .Sh72 1.591 I
I .7oct7 1.7427 I .7733 l.dOlO 1.8261
l.aSOO Id345 I .9 15,s 1.9433 1.9636
2.0125 2.0473 2.0779 2.1054 2.1304
2.19.;9 2.233~) 2.2429 2.2894 2.3136 --
’ 0.8207 , 0.8938
-
1 .Suh.i 1.6193 I .h492 1.6763 1.7008
1542 2276 2 2835 2. 3317 i 2. 3736
14;
2.41 I I 2.44nl
REDUCED
EQUATION
OF STATE
TABLE
Reduction --
1 0.2
( 0.3
1
0.4
1
0.5
1
0.6
AND
373
XENON
VIII
The equation of state the critical parameters
with
OF ARGON
PV/KI‘ of xenon. T, = 289.74”K and
~ 0.7
186.3 am. 1 1.1
~
0.3340 0.4010 0.4633 0.5210
0.3163 0.3882 0.4554 0.5178
0.3061 0.3836 0.4562 0.5236
0.6404 0.6142 0.5942 0.6742 0.6518 0.6355 0.7051 0.6864 0.6737 0.7337 0.7183 0.7091 0.7602~0.7480~0.7420
0.5807 0.5742 0.5754 0.6258 0.6234 0.6287 0.6677~0.6689 0.6782 0.7065’0.7112 0.7241 0.742610.7507,0.7670
0.5860 0.6436 0.6971 0.7468 0.7932
~
~ 2.2
/
i 0.8
1 0.9
/
de = 1.0
1.2
i
1.3
?‘r
1.15 1.20
‘0.8391 0.8531
0.7692 0.7895
0.6448 0.6777 0.7069 0.7330
1.25 1.30 1.35 1.40 1.45
0.8655
0.8077 0.8241 0.8390 0.8526 0.8652
0.7564 0.7114 0.6728 0.7775 0.737010.7026 0.7968,0.7604 0.7299 0.8145’0.7819 0.7550 0.8308 0.8017 0.7782
--_. 1 ,:_
0.5773 0.6168 0.6518 0.6832
1
1.53.6
1.7
0.5181 0.5634 0.6038 0.6401
1
1.8
0.4670 0.5174 0.5626 0.6034
1.9
7. 1.05 1.10 1.15 1.20
0.3037 0.3878 0.4665 0.5394
0.3129 0.4043 0.4895 0.5683
0.3354 0.4348 0.5269 0.6119
0.3766 0.4843 0.5834 0.6746
1.25 1.30 1.35 1.40 1.45
0.6068 0.6690 0.7268 0.7805 0.8305
0.6409 0.7079 0.7700 0.8276 0.8813
0.6900 0.7619 0.8284 0.8901 0.9474
0.7582 0.8484 0.9655 0.8349 0.9299 1.0512 0.9058 1.0049 1.1300 0.9714 1.0741 1.2024 1.0321 1.1382 1.2694
0.4402 0.5561 0.6623 0.7596
0.5321 0.6559 0.7688 0.8718
0.4236 0.4785 0.5281 0.5731
)
2.0
0.3873 0.4462 0.5000 0.5491
~
2.1
0.3575 0.4203 0.4782 0.5316
1
2.3
1
2.4
2.5
0.6580 0.8245 0.7889 0.9613 0.9078 1.0850 1.0157 1.1970 1.1138 1.2031 1.2850 1.3602 1.4295
1.2984 1.3905 1.4747 1.5518 1.6228
1.7819 2.0554 2.3780 1.8535 2.1266;2.4473
2.7550 2.8210
reduced data for xenon deviate about three times more from the argon values, than the difference between the two argon sets (Table IX) amounts to. Finally, as regards the reduction by critical parameters, Table XI shows that a somewhat closer agreement is obtained between the two gases in this case. However, the compressibility factor of xenon is lower again than that of argon at high densities. Before drawing any conclusions from the observed deviations from corresponding states, we must make sure that these deviations are real, and that they cannot be explained by uncertainties of the reduction factors. This may be done as follows. The Amagat density and Kelvin temperature of the PV/RT isotherms of argon may be multiplied by scale factors such that these isotherms are transformed into the compressibility isotherms of xenon, in the temperature range from 1.05 T, to 1.45 T, and for densities from zero up to 2d,. By requiring that the minima in the PV/RT vs. d isotherms coincide, one finds that the scale factor for the temperature is equal to 1.931 f 0.002. Then, the scale factor for the density is found by requiring the coincidence of the high density range of the isotherms. This factor is 0.633 f 0.001. With this choice of scale factors, the experimental PV!RT values of argon and xenon agree to about one percent in the range of overlap. Therefore, in comparing tables of reduced properties of argon and xenon, we can only expect to find corresponding states, if the ratio of
374
J.
hl.
II.
LEVELT
the temperature reduction factors used in a particular comparison is equal to 1.931, and the ratio of the density reduction factors is equal to 0.633. R summary of the actual ratios of these reduction factors, for reduction with molecular as well as with critical parameters, is given in Table XII. It is seen that they deviate considerably from the “ideal” ratios 1.931 and 0.633. We must check whether these differences may be due to uncertainties of the molecular or critical parameters.
REDUCED
EQUATION
OF STATE
OF ARGON
TABLE
r
Cornmrison
of reduced
T*
properties _ _
Reduction
with molecular
d* = 0.3
I
375
XENON
X
of amon and xenon.
d’ = 0.2
AND
parameters.
I
1
-I
d* = 0.4 Xe
225.3 4.070 1.4 1.7 1.9
0.5629 0.7263 0.8025
1.4 1.7 1.9
- 1.499 - 1.364 -1.309
1.4 1.7 1.9
-0.560 -0.471 -0.441
0.5596’ - 0.0033 0.7184 -0.0076 0.7925 -0.0100 -1.474 -- 1.328 --1.275 -0.546 -0.447 l-O.418
0.3749 0.6484 0.7843 - 2.091 _ 1.912 - 1.851
-2.129 - 1.972 - 1.906
A
0.718: 3; 1.169 I 1.381( 1
0.50311-0.21521 0.9267 -0.2424 1.1309 -0.2501
1.4 -3.278 1.7 -3.141 1.9 -3.065
-3.167 -3.013 -- 2.955
to.111 j-0.128 to.110
- 3.876 -3.727 -3.636
-3.716 -3.565 -3.504
1.4 _ 1.336 1.7 ~ 1.248 1.9 -1.205 -
- 1.240 - 1.137 -1.105
to.096 to.111 to.100
- 1.684 ~ 1.588 ~ 1.538
- 1.526 - 1.426 _ 1.392
TABLE
XI
I
d, = 0.6 Ar
1 Xe
1
+0.065 to.091 +0.081
~ 1.003 -0.887 -0.854
+0.053 +0.072 +0.066
0.65
0.96731 1.4464 1.6692
0.642( 1.103 I.3211
+0.160 +0.162 +0.132
-3.999 -3.841 - 3.774
+0.158 i0.162 1+0.146
- 1.694 - 1.590 - 1.552
-
- 0.324: ~ 0.343: - 0.347‘
of reduced compressibility factors of argon and xenon. Reduction with critical parameters. 1
A
- 2.640 -2.466 -2.406
d” =
I
--0.035 -0.051< - 0.059:
Ar 119.8 3.405
-0.0869 -0.1144 -0.1249
Comparison
- 2.705 -2.557 -2.487
d” = 0.6
I
i
+0.038 +0.060 +0.055
-0.810 -0.706 I-O.670
0.722; 0.892;
1.4 0.4652 0.8366 1.7 1.0171 1.9
/
+0.014 1+0.024 1+0.023
Xe 225.3 4.070
Ar 119.8 3.405
Tr
+0.025 +0.036 +0.034
d* = 0.5
T*
0.45 1: 0.676t 0.786: !I
A
1 Ar
d, =
1.2
1 Xe
/
A
1 Ar
dr =
1.8
/ Xe
/
d, = 2.0 A
1 Ar
!
Xe
1
n
I0.7363 1.5188 I0.6580 1.1996 1.4295 I -0.0783 1.1138 -0.0893 -0.0858 As for the deviations from corresponding states found in the reduction with critical fmmneters (Table XI), it is clear from Table XII that one of the critical temperatures should be in error by +yo (i.e., 1.5% for xenon, 0.8”C for argon) and a critical density by 2.1 y. (i.e., 3.7 Amagat units for xenon, 6 Amagat units for argon) to account for the difference. In view of the claimed experimental accuracy (5 2), this is not very probable. Since the use of critical parameters for reduction thus leads to deviations from corresponding states for these heavy noble gases, it is clear that the construction of semi-empirical equations of state in terms of a modified principle of corresponding states (9 1) may meet difficulties in the dense gas and the liquid state.
376
Table XI I also shows that in the reduction Lvith wzolectr/av~n~n~~z~ic~s,that paramctcr pair (6) for argon is clearly the better \vhtw corrc,spontling statcas with xc‘non arc conwrncd. Ncvcrthrlcss, one of the t‘, k \.alucs shoultl by> modified nith 2.3”~~ (i.tx., 2.3^(‘ for argon, 5.2”C for xcwon) ant1 ant’ of thr (T values by 2.1 ‘I;, (i.v., 0.067 I$ for argon, 0.085 ipi for scwon) to achitl\,cs coincidence of rcductd argon and xenon isotherms. These chang-(1s ~xc~c~l the. cxpectcd uncertainties of thcb molecular parameters (4 2). ‘1‘11~uw of rtlasowble values for the molecular param&ers of argon and xt’non thus results in deviations from corresponding states. Tlic reason maJ- bc that the I.c~nnnrtl*Jones potcmtial is not a proper description of the intt~rmolccular ficlltl of these gases. Studies of crystal properties and transport phenomclna ha\~ rcvr~aled deficiencies of this potential Jds’i). An inspection of Fig. 1, showing the difference of observed and theoretical H” as a function of 7‘* adds to suspicion concerning the, applicability of the Iznnartl-Jonc% 6-l 2 l)ottwtial for argon and xenon. Suppose that though the I,t>nnard-Jones 6 -12 potential fails, some othc,r two-parameter function holds for these gases. Then, tht, rcduccd 1;” 1’s. I‘* graphs of these gas’s must coincide after proper scaling, although thq may deviate from the corresponding cur\Tc for tht, 6-12 potential. l;rom the deviation pattern K* - IS*th of Figs. II’, and lr, it is clear that this coincidence is achieved with the parameter pair (6) for argon and (4) for xenon. Thus, with this choice of parameters a law of corresponding states should hold for argon and xenon. Since this is not the cast, the assumption of additivity or the limitation to a universal two-parameter potential function must be in error. Unfortunately, it is not possible to sq)aratc thtl two effects using gas P-I’-T
data only.
Acknowledgement. The author is indebted to Professor _J. 0. Hirschfclder for his suggestions and helpful aclviw concerning this research, and to Professor J. de Boer for his kind interest. Without the stimulating criticism of Dr. E. G. I>. Cohen the paper cvoul~l never have reached its present state. Mrs. Marion Taylor has deserved the gratitude of the author for developing an efficient method for carrying out the two-dimensional interpolations on the Bendix G-l 511. She was very helpful in performing the
REDUCED
EQUATION
OF STATE
OF ARGON
AND
377
XENON
tedious numerical work in those parts of the tables where information of most interest but hardest to obtain. Received
is
13-3-60 REFERENCES
14 b) C) 24 b) C) d) e) f) 8)
De Boer,
3)
Hirschfelder,
Pitzer,
J. and Michels, K. S., J. them.
Guggenheim,
A., Physica Phys.
E. A., J. Chem.
583.
Phys. 13 (1945) 253.
Michels,
A., Wijker,
Michels,
A., Lunbeck,
Michels,
A., Wassenaar,
T. and Louwerse,
Michels,
A., Wassenaar,
T., Walkers,
Michels,
A., Levelt,
J. M. H. and De Graaff,
A., Levelt,
J. M. H. and Walkers,
Michels, Levelt, Liquids.
Hub.
5 (1938) 945.
7 (1939)
and Wijker,
J. M. H., Thesis,
Amsterdam
J. O., Curtiss,
John Wiley
W., Physica G. J., Physica
New York
Whalley, Habgood,
c)
Whalley,
E., Lupien,
d)
Whalley,
E. and Schneider,
5)
Holborn,
L. and Otto,
6)
Beattie,
J. A., Barriault,
7)
Kihara,
T., Rev. Mod. Phys. 23 (1953) 831.
26
P., Physica
(1949) 627. 15 (1949) 689. 20 (1954) 99. J., Physica
22 (1956)
17.
24 (1958) 659. 24 (1958) 769.
(1958).
b)
Y. and Schneider,
H. W. andschneider,
l:,
G. J. and Dawson,
4a)
Physica
Physica
G. J., Physica
C. F. and Bird,
and Sons, Inc.,
E., Lupien,
Hk.,
R. J. and Walkers,
R. B., The molecular
Theory
of Gases and
(1954).
W. G., Can. J. Chem. 31 (1953) 722.
W. G., Can. J. Chem. 32 (1954) 98, 164.
Y. and Schneider,
W. G., Can. J. Chem. 33 (1955) 633.
W. G., J. them.
J., Handbuch
Phys. 23 (1955)
der Experimentalphysik
R. J. and Brierly,
1644.
8, 2.Tei1, p. 153. Leipzig,
J. S., J. them.
Phys. 19 (1951)
1222.
(1929).
J. M. H.
Physica
1960
26
361-377
THE REDUCED EQUATION OF STATE, INTERNAL ENERGY AND ENTROPY OF ARGON AND XENON by J.
M. H. LEVELT
Van der \Vaals Laboratoriurn,
Gerneente
*)
Universiteit
Amsterdam
Synopsis Tables
of the equation
at regular
intervals
appropriate have
been
potential.
theoretical
performed
the
for
the
equations
of state.
In the reduction deviations
of state
these
with molecular
deviations
exceed
factors.
parameters factors
of
of argon
and xenon,
in the reduced
form
of state. The reductions the
is discussed.
of the reduction
of argon
are also given,
from corresponding
of the reduction
and entropy are presented
Lennard-Jones The effect
is investigated
6-12
of the un-
and found
to
densities.
equation
as parameters,
molecular
on the results
at liquid
energy
work on the equation
of the reduction
of these factors
constants
that
with
reliability
bc appreciable Tables
internal
and temperature,
for fundamental The
certainty
of state,
of density
and
parameters
the
reduced
with
the
as well as in that with critical
states are established
Possible
xenon,
to be used in the search for general
uncertainties explanations
for argon and xenon.
introduced
by
for the observed
the limited deviations
critical
empirical constants,
It is shown accuracy are offered.
3 1. Introduction. In recent years, much effort has been devoted to the development and improvement of theories on the equation of state of gases and liquids. The cell theory of the liquid state, the cluster expansion of the compressibility factor, the method of the radial distribution function and the Monte Carlo computations all try to give an understanding of experimental behavior in terms of molecular interaction. In the comparison of theory and experiment, the heavy noble gases play an important role. This is due to the fact that in most theories on the equation of state no numerical results can be obtained unless some simplifying assumptions are made, which rule out the application of the theory to all substances not possessing a simple molecular structure. These assumptions are : 1. Classical statistical mechanics may be used for the translational degrees of freedom of the molecules. *) The computations for this paper were performed under Contact DA- 1 l-022-ORD-2526 of the Office of Ordonance Research, at the University of Wisconsin Theoretical Chemistry Laboratory, Madison,
Wisconsin,
U.S.A.
Physica
26
361
-
2.
Internal
degrees
of freedom
are indel)endent
of th(> mutual
1)ositions
of
the molcculcs. 3.
The interaction
potential
of all pair interactions 4.
The
molecular
distances
interaction
function
these them
pressure
is a universal
subject
this
in terms become
description
with critical empirical
of
properties
the
Since
reduced
with
siderations
i-cduc~~l
thcl
if
greatI!,
to results
corresponding
of
constants, and
thrrmosimplifivtl
lcatling
Ia\\
I II thus
is included
tetnl~craturca
measured
in
terms
in
at
the
of
thts
critical
In this
paper
for empirical
experimental
must
‘1‘0 bc useful
as possible. is readily
for these
suitable
experimental
arc available
present
paper selected
and entropy
for argon
to for
obtain
and xenon,
reduced
reduction
as functions
arc
these
should
worker,
of density
be that
and tempcr-
ranges data
thermodynamic the
tht
for cot-r-(‘-
of density
over appreciable
foi
of densit!
data
to the theoretical
small interlrals data
the
con
starting
l)urposes,
the conditions
Furthermore,
accessible
form and at sufficiently
energy
in a form
fulfill
as closely
for
an appropriate
over a large inter\al
accurate
properties
data
\ve vi11 provide
gases,
states.
should
in a way which
in the
point
extend
presented
and temperature
starting
of corresponding
experimental
of state.
the conditions
be accurate,
states
Since
the
fat in its
of these
and the substance
spending
is, in reduced
form
related
states
to fulfill
properties
principles
in addition,
equations
data
and temperature,
often
modified
ant1
l)aram~tcrs
the law of corresponding
gases arc sulq~oscd
we will present
with theory;
of state
usin, u additional
thermodynamic
constants,
on such
l)racticc~ in the clr\~~lol~-
equation
constants,
noble the
is common
of stat<,. Here, one tries to find a general
not follo\ving
states,
based
comparison
internal
1~~. (XX-
the
and densit\-.
form,
critical
classical
critical
the heavy
corresponding
The
(Jf
density
parameters
equations
with
and liquids
true sense.
used
with
constants
experimental
reduced
gases
ature.
o, then
usually,
in reduced
l)ressure,
universal
1). That
parameters.
This reduction
point
sincct
arc
formulation
of reduction
statement,
point
treatments
the older
statcls
this is also true for the other
states.
ment of classical
for
of freedom
through
molecular
those
tcinp~rature
and carried
namely
critical
of reduced
b!-
the substnnccs
dimcnsionl~5s
of I‘ and
formulated
nelvcr
~1 of the
is given
c/’= ~f(v. (T), then
art’ matlc
to a law of corresponding that
interaction
combinations
function
dcgrccs
be noted
molecular
and tcmpratuw
Theoretical
It may states,
the
properties.
by being
function
follo\v a la\\. of corresl~oncling
in apl)ropriatc
of internal
by a universal
of the distanccl,
density
pressing
dynamic
that
four conditions
is, if pr~ssuw,
is obtaiiic~cl by- addition
of the molec~~lcs.
it is assumed
a two-parameter
absence
is described
Y of the centers
If in addition fulfilling
for a group of molecules in the group.
of density have
been
properties.
compressibility and temperature.
factor, The
REDUCED
reductions
EQUATION
have been carried
Lennard- Jones
6-12
potential.
OF STATE
OF ARGON
AND
out with the molecular For
the compressibility
363
XENON
parameters factor
of the
reductions
have also been carried out with the critical parameters. The method of reduction and the choice and accuracy of the reduction factors are described in 9: 2, the results of the reductions are then tabulated. A comparison of these results permits a check of the validity of the law of corresponding states for argon and xenon. This comparison and conclusions drawn from it are contained
in 5 3.
9 2. Method and reduction 9arameter.s. The procedure followed in obtaining reduced thermodynamic properties at regular intervals of reduced density and temperature may be summarized as follows. After a choice of experimental data is made, these data are tabulated, with density and temperature as variables. Then, the property of interest, as well as the entries of the table are brought into reduced form. Finally, since values of the thermodynamic properties are desired at values of the reduced temperature and density which generally do not coincide with the experimental entries of the table, a double interpolation must be performed. In the present research, the experimental information consists of data for the product PV for argon and xenon, as determined by Michels and coworkers 2). The authors converted the experimental data into tables containing PV at regular intervals of density and temperature. From these tabulated values of PV, they derived similar tables for the other thermodynamic properties, for instance, for the internal energy UZ and the internal entropy St. For the noble gases, these functions are the difference in energy and entropy, respectively, between the gas under consideration and an ideal gas at the same density and temperature. Expressed in the critical constants, the density range of the experimental data for argon is from zero up to 2dC, the temperature interval is from T = 0.9T, up to 2.8T,. For xenon the corresponding figures are: d from zero up to 2.5d, and T from l.lTc up to 1.5T,. The accuracy of the PV values is about one part in 10,000, though somewhat less in the regions around the critical point and in the liquid phase. The accuracy of UZ and St is roughly an order of magnitude less. In the reduction with molecular parameters, reduced quantities are obtained as follows: d* = Na3lV,,
T* = kT/e, Ut* = UiINe, Si* = SijNk,
(1)
where N is Avogadro’s number, k is Boltzmann’s constant, E and 0 are the energy and length parameter, respectively, which characterize the intermolecular field, while Vm is the molar volume *). *) The numerical in so-called Amagat
data for argon and xenon were obtained, as usual, by expressing the volume units (of volume). The Amagat unit of volume for a particular gas is equal to
364
J. Al. H. LEVEL’1
When the critical constants are used in obtaining rclduccd tables for the compressibility factor, the rcduccd quantities arc calculatcvl as follo\vs :
where thv intlvs c d(~notc~scritical
constants.
I3g’ means of (1) ant1 (2) the tabulatrd experimental prop-tics and also the entritx of the tables wtw reduced. Then, in the last stall) of thv Compaqtation, a tn.o-dinlensional intcqx)lation was matl~~. ITor the tltGrc,cl rountl valuw of on0 of thcx vnriablcs, a four-1)oint I.agrangian intcrpolatioii \~a5 performed. In the intcrmcdiatc tablt, so obtained, interpolations v.vrt‘ matl~, for the second variabltl. All computations \vtw carrictl out on a I+ndix (;--151) computer, lvitli intcrmvdiatt~ storagt~ on magnetic tap. ‘l‘htl accuracy of the interpolation method was frcclwntly chc~k~tl, for instance by interpolating for the original vspcrimcntal densities from which the 1)V tables had been dcrix~ccl. It turned out that except for the numbers at the cclgcs of thtl tables, tqecially thaw closc~ to the two-l)haw region, that tables obtained by interpolation have: ahnost the samt‘ accuracy as thcb original ant’s, i.c., mostly of thv order of magnitude of thy last digit cluotvtl. Howwr, the reliability of thv tables is dt+cwnined not so much bar thci accuracy of the interpolation procedure, as by the precision of the molecular and critical parameters. This brings us to the problem of sclccting the lwst values of t’ and (r to construct the rctluction factors. In the case of reduction with molecular parameters, thcsv rctluction factors are combinations of t‘ and 0. In the present papc’r, thcw cluantitics were deduced from data for the second virial coefficient as a function of tcmperaturc ; the second virial cwfficicnt in turn was clcrivetl from compwssibility measurements. ‘l‘hc accuraqr of the reduction factors is thcrvfow far less than that of the original comprt~ssibility measurements. ‘This is also trucl for reduction factors from critical constants, since thcsc arc usually not as well known as compressibility data in other regions. Therefore, it is thtl accuracy of the reduction factors rather than that of tht, interpolation process, which determines the reliability of the reduced tables.
REDUCED
EQUATION
OF STATE
OF ARGON
AND XENON
365
As regards the molecular parameters, they may be calculated after a special form for the interaction potential has been chosen. In this paper, the so-called
Lennard- Jones
6- 12 potential
qJ(f-) = 4&+912
-
was selected : W)6},
(3)
since many authors have used this potential, which is not inferior two-parameter potentials, in work on the equation of state.
to other
In the determination of E and a from experimental data for the second virial coefficient B, use was made of the theoretical B* vs. T* relation for the 6-12 potential, as derived by Hirschfelder e.a. 3). The reliability of the molecular parameters is limited, due to the inaccuracies of the measured
Fig. virial
1. Difference coefficient
between
reduced
as a function potential
argon
experimental
of reduced
and theoretical
temperature,
values of the second
if the Lennard-Jones
6-12
is used for argon and xenon.
0 - ref. 4a
xenon
0 - ref. 4c
a - ref. 5
v - ref. 6
0 - ref. 2a, 2e
0 - ref. 2c.
B values, which depend on the location and extension of the density range studied, and on the number and reliability of experimental points in this range. Usually, the reliability of B values is not much better than one percent. Uncertainties are also introduced by the procedure of comparing experimental and theoretical B vs.T curves.
365
J. 31.
H. LEVELT
For xenon, the parameters t‘ and CT as derived 1,~’ LVhalley and Schneider Jd), from compressibility data in the range 0°C up to 7OOY‘, were taken : F,‘k -= 225.3 -~ i l.l”K, cr = 4.070 1 0.013 A. (41 In Fig. 1r, the differences between observed and thaw-ctical plotted as a function of 7’* for this choice of paramettsrs. For argon, the parameters published by Mic hels ~.a.“(‘), +h =
119.8’K,
I-j* valucx arc
cr m=3.405 L\,
(5)
are in agreement with those derived b\. \Vhall~‘~. and SC h nc~itl~~r -In) from 0.33”1<, data in the range - 100°C up to +60”0”(‘, namely, YA = 1 19.49 CT= 3.409 +- 0.007 A. However, as is shoun in I;ig. la, the agreemc~nt between cxlwrimental and theoretical B* vs. 7‘* \,alues is not close at lo\z temperatures. The deviation pattern IZhCX,,.- K*+k,. 1’s. 7‘* of Fig. 1h has been obtained with the parameter pairc:‘k =
119.3”K,
CT2 3.43 .s.
16)
Although there is some improvemr~nt, the esperimwtal cur\.<’ again docx not show a close fit to the theoretical one at IOLVtt~mlxeraturw. Hoa-c~vrr-, with the parameters (4) and (6), the deviation patterns for q+Jll (1;ig. 171) and xenon (Fig. lr) arc almost identical. Keduced thermodynamic properties have bc~ln tabulatc~d for xgon with the set (5). They are shown in th(a ‘I’ablcs 1~ III. A surve!. of the diffcrenws introduced by using the second set, (6), is given in Table IS. I-or xenon, the calculated with the lw-amctc’rs (4), reduced thermodynamic prolwrtirs, are shown in the Tables IV-\‘I. In the reduction with critical +zvanzetcvs, the following L-alues ha\~l bc~w used for the critical constants. For argon d, 1 300.4 Amagat
units,
7‘, -: 150.86”K
‘g),
with estimated accuracies of 3 parts in 1000 for the clensity anti 0.02%: the temperature. For xenon. d, =
186.3 Amagat
with an accuraq- of two parts temperature. The results of the reduction Tables \‘I1 and \,‘III.
units,
7‘, = 289.74”K
in 1000 in thcx density Lvith critical
parameters
-lb),
(7) in (8)
ancl 0.001’ (‘ in thtl art’ shoLvn in thcb
4 3. Conqbarison of tabulated @q!wties, and concl~~~io~s. The comparisons that may be made fall into three classes. First, the influence of a small uncertaint?; in the molecular reduction factors may be invcstigatetl b!computing the reduced properties for the two sets of parameters (5) and (6)
REDUCED
EQUATION
OF STATE
TABLE
-
ld* T*l -__
Reduction 0.050
1.15 1.20 1.25 1.30 1.35
0.7976 0.8138 0.8281 0.8412 0.8529
1.40 1.45 1.50 1.60 1.70
AND
XENON
367
I
The equation of state PV/KI‘ of argon. with the molecular uarameters elk = 119.8”K and o = 3.405 is
7 _
OF ARGON
I 0.200
0.250
O.‘OO I 0.150
0.300
0.325
0.350
0.400
-
0.6491 0.6771 0.7021 0.7244
0.5482 0.5840 0.6158
0.4872 0.5270
0.4112 0.4570
0.3538 0.4045
0.3313 0.3844
0.3125 0.3683
0.2870 0.3493
0.8632 0.8727 0.8815 0.8970 0.9103
0.7445 0.7629 0.7796 0.8092 0.8346
0.6444 0.6704 0.6943 0.7366 0.7732
0.5629 0.5956 0.6258 0.6794 0.7260
0.4990 0.5375 0.5733 0.6374 0.6935
0.4517 0.4957 0.5368 0.6111 0.6766
0.4342 0.4809 0.5248 0.6044 0.6747
0.4210 0.4706 0.5173 0.6024 0.6776
0.4087 0.4649 0.5178 0.6146 0.7002
1.80 1.90 2.00 2.10 2.20
0.9218 0.93 I7 0.9406 0.9484 0.9554
0.8567 0.8759 0.8930 0.908 1 0.9216
0.8048 0.8326 0.8574 0.8794 0.8992
0.7667 0.8025 0.8346 0.8633 0.8890
0.7429 0.7866 0.8258 0.8608 0.8925
0.7347 0.7862 0.8324 0.8739 0.9114
3.7371 0.7926 0.8424 0.8870 0.9274
0.7444 0.8039 0.8573 0.9053 0.9486
0.7762 0.8438 0.9047 0.9593 1.0086
2.30 2.40 2.50 2.70 2.90
0.9617 0.9674 0.9726 0.9815 0.9891
0.9338 0.9449 0.9550 0.9724 0.9873
0.9170 0.9332 0.9480 0.9738 0.9956
0.9 123 0.9334 0.9527 0.9866 1.0152
0.9210 0.9470 0.9706 1.0125 I .0478
0.9452 0.9760 1.0040 I .0537 1.0957
0.9640 0.9972 1.0274 1.0811 1.1263
0.9878 1.0234 1.0560 1.1135 1.1619
1.0531 1.0936 1.1306 1.1957 1.2503
3.10 3.30 3.50
0.9955 1.0011 1.0061
1.oooo 1.0109 1.0206
1.0143
1.0304 1.0445
1.0397 1.0609 1.0794
1.0780 I. 1042 1.1271
1.1316 1.1627 1.1898
1.1650 1.1985 1.2276
I .2034 1.2392 1.2703
1.2972 1.3373 1.3722
0.450
0.500
0.550
0.575
0.600
0.625
0.650
0.675
0.700
0.2162 0.3312 0.4378 0.5368
0.1512 0.2827 0.4049 0.5176 0.6218
0.2343 0.3738 0.5024 0.6207 0.7296
0.3465 0.4931 0.6272 0.7503 0.8633
0.4919 0.6440 0.7831 0.9099 1.0260
0.8303 0.9233 1.0095 1.1633 1.2967
0.9673 1.0633 1.1520 1.3100 1.4464
d* ‘-+ 1.15 1.20 1.25 1.30 1.35
1 0.2813 0.3527
0.306 I 0.3882
0.3779 0.4714
1.40 1.45 1.50 1.60 1.70
0.4202 0.4840 0.5440 0.6535 0.7502
0.4652 0.5375 0.6053 0.7284 0.8366
0.5585 0.6396 0.7153 0.8519 0.9715
0.7936
0.9367 1.0614
0.7183 0.3078 0.8908 1.0397 1.1691
1.80 1.90 2.00 2.10 2.20
0.8359 0.9122 0.9805 1.0417 1.0969
0.9323 1.0171 1.0928 1.1603 1.2210
1.0765 1.1690 1.2514 I .3246 1.390 1
1.1706 1.2665 1.3518 1.4274 1.4949
1.2821 1.3810 1.4688 I .5463 1.6153
1.4130 1.5145 I .6042 1.6831 1.7530
1.5654 I .6692 1.7599 1.8394 1.9095
2.30 2.40 2.50 2.70 2.90
1.1467 1.1919 1.233 1 I .3054 1.3659
1.2756 1.325 I 1.3701 1.4489 1.5145
I .4488 1.5019 I .5500 1.6339 1.7033
1.5553 1.6097 1.6589 1.7447 1.8154
1.6769 1.7323 I .7823 1.8695 1.9408
1.8152 1.8712 1.9216 2.0096 2.0808
1.9717 2.0274 2.0784 2.1664 2.2366
2.1487 2.2041 2.2546 2.3412 2.4097
2.3478 2.4020 2.4514 2.5360 2.6321
3.10 3.30 3.50
1.4175 1.4616 1.4998
1.5701 1.6174 1.6580
1.7617 1.81 10 1.8530
1.8747 1.9243 1.9664
2.0006 2.0500 2.09 17
2.1406 2.1892 2.2300
2.2956 2.3429 2.3822
2.4676
2.6579
-
-L
0.6287 0.7141
-
-
-
J. M. H. LEVELT
368
TABLE
i-
Reduction
\
a*
-
0.100
0.050
-
L
1.20 1.25 1.30 1.35
-0.443 -0.43c -0.419 -0.409
-0.879 -0.850 -0.825 -0.804
1.40 1.45 1.50 1.60 1.70
-0.4oc -0.392 -0.385 -0.372 -0.361
-0.785 -0.77c -0.755 -0.73c -0.709
1.80 1.90 2.00 2.10 2.20
-0.352 -0.345 - 0.338 -0.333 -0.327
2.30 2.40 2.50 2.70 2.90 3.10 3.30 3.50
1.20 1.25 1.30 1.35
7Jg/N& of argon. = 119.8”k and 0 = 3.405 A
i 0.250
0.200
0.150
T*
s
II
The reduced internal energy with the molecular oarametersaik
0.300
0.325
0.350
I
,
- 1.256 - 1.214 -1.180
- 1.581 - 1.535
- 1.922 - 1.866
-2.233 -2.174
- 2.378 - 2.320
-2.517 -2.463
- 1.152 -1.129 - 1.108 - 1.0% - 1.044
-
1.499 1.469 1.442 1.399 1.364
- 1.824 - 1.790 - 1.760 -1.711 _ 1.672
-2.129 -2.094 -2.063 -2.013 - 1.972
-2.276 -2.242 -2.211 -2.161 -2.119
-2.421 -2.387 -2.357 - 2.307 -2.266
-0.694 -0.677 -0.665 -0.654 -0.644
- 1.019 -0.998 -0.981 -0.965 -0.951
-
1.334 1.309 1.287 1.268 1.251
-
1.639 1.611 1.586 1.564 1.545
-
1.937 1.906 1.879 1.855 1.834
-2.084 - 2.052 - 2.024 - 1.999 - 1.977
-2.230 -2.198 -2.169 -2.143 -2.120
- 0.322 -0.317 -0.312 -0.305 -0.299
- 0.635 -0.625 -0.617 -0.602 - 0.589
-0.938 -0.926 -0.914 -0.892 -0.874
-
1.235 1.220 1.206 1.177 1.153
-
1.527 1.509 1.492 1.458 1.429
~ -
1.814 1.794 1.774 1.736 1.702
-
1.956 1.935 1.914 1.873 1.837
- 2.098 - 2.076 -2.054 -2.010 - 1.971
-0.294 -0.291 -0.288
-0.580 - 0.574 -0.568
-0.860 -0.850 -0.842
- 1.135 -1.122 -1.111
- 1.675 - 1.654 - 1.636
- 1.808 - 1.785 - 1.765
- 1.940 - 1.915 - 1.893
0.500
0.550
0.600
0.625
0.650
0.675
-3.986
-3.934 -3.903
-4.140 -4.112 -4.087 - 4.055
- 4.295 - 4.266 -4.241
- 4.450 -4.420 - 4.393
- 4.027 -4.001 -3.975 -3.923 -3.872
0.400
, ,
-
0.450
-
-
- 1.407 - 1.390 - 1.375
-
0.575 -
-3.836 -3.806 -3.782 -3.751
- 2.785 -2.742
-3.10 -3.057 -3.021
-3.37 -3.337 -3.306
-3.66 -3.628 -3.600
1.40 1.45 1.50 1.60 1.70
-2.705 - 2.674 -2.647 -2.599 - 2.557
-2.989 - 2.96 1 -2.936 -2.890 -2.848
-3.278 - 3.253 -3.229 -3.183 -3.141
-3.574 -3.550 -3.526 -3.480 - 3.435
-3.724 -3.700 -3.628 -3.581
-3.876 -3.851 -3.826 - 3.776 -3.727
1.80 1.90 2.00 2.10 2.20
-2.520 -2.487 - 2.457 - 2.429 - 2.404
-2.811 -2.776 -2.744 -2.714 -2.687
-3.102 -3.065 - 3.030 - 2.998 -2.967
-3.393 - 3.353 -3.314 -3.278 -3.244
-3.538 -3.496 -3.455 -3.416 -3.380
-3.681 -3.636 -3.593 -3.552 -3.513
-3.823 - 3.775 -3.729 -3.685 -3.643
2.30 2.40 2.50 2.70 2.90
-2.380 -2.355 -2.331 -2.282 -2.238
-2.660 - 2.633 -2.606 -2.551 -2.501
-2.938 -2.907 -2.877 -2.815 -2.758
-3.211 -3.177 -3.142 -3.072 -3.006
-3.344 -3.309 -3.272 -3.196 -3.125
-3.475 -3.437 -3.398 -3.318 -3.241
-3.603 -3.563 -3.521 -3.435 -3.352
-3.727 -3.684 - 3.639 -3.547 -3.459
-3.848 -3.801 -3.752 -3.654 -3.559
3.10 3.30 3.50
-2.202 -2.172 -2.145
-2.450 -2.424 -2.391
-2.710 -2.667 -2.628
-2.950 -2.898 -2.851
-3.065 -3.008 -2.957
-3.175 -3.114 -3.058
-3.281 -3.214 -3.153
-3.381 - 3.309 -3.241
-3.475 -3.396 -3.321
-
-
-
-3.958
-3.676
-
-
-
-
REDUCED
EQUATION
OF STATE
TABLE
o.050
AND
369
XENON
III
The reduced internal entropy &/Nk of argon. with the molecular parameters c/k = 119.8’K and u = 3.405 A.
Reduction ;*:I
OF ARGON
1
0.100
’ o.150 1 1
0.200
1
0.250
~
~
0.300
0.325
~
0.350
i
0.400
1.20 1.25 1.30 1.35
-0.178 -0.167 -0.159 -0.151
-0.360 -0.336 -0.317 - 0.30 1
-0.472 -0.447
-0.621 -0.586
-0.761 -0.718
~ 0.887 -0.842
-0.946 -0.902
- 1.003 -0.962
- 1.1 16 - 1.082
1.40 1.45 1.50 1.60 1.70
-0.145 -0.139 -0.134 -0.126 -0.119
-0.288 -0.276 -0.266 -0.250 -0.238
-0.426 -0.410 -0.396 -0.373 -0.355
-0.560 -0.539 -0.521 -0.493 -0.471
-0.687 -0.663 -0.643 -0.612 -0.588
-0.810 -0.785 -0.764 -0.731 - 0.706
-0.870 -0.846 -0.825 -0.792 -0.767
-0.931 -0.907 -0.887 -0.855 -0.829
~ 1.056 ~ 1.034 -1.015 -0.984 -0.959
1.80 1.90 2.00 2.10 2.20
-0.114 -0.110 -0.106 -0.104 -0.101
-0.228 -0.220 -0.213 -0.208 -0.203
-0.341 -0.330 -0.321 -0.313 -0.306
-0.454 -0.441 -0.429 -0.420 -0.412
-0.569 -0.554 -0.541 -0.530 -0.521
- 0.686 -0.670 -0.656 -0.644 -0.634
-0.747 -0.730 -0.715 -0.703 -0.693
-0.809 -0.792 -0.777 -0.764 -0.753
-0.938 -0.920 - 0.904 -0.891 -0.879
2.30 2.40 2.50 2.70 2.90
-0.099 - 0.097 - 0.095 -0.092 - 0.090
-0.199 -0.196 -0.192 -0.186 -0.182
-0.301 -0.296 -0.291 -0.283 -0.277
-0.405 -0.399 -0.393 -0.383 -0.374
-0.513 -0.505 - 0.498 -0.486 -0.476
-0.625 -0.617 - 0.609 -0.594 -0.583
-0.683 -0.674 -0.666 -0.650 -0.638
-0.743 -0.734 - 0.725 -0.708 -0.695
-0.868 -0.858 -0.848 -0.829 -0.815
3.10 3.30 3.50
-0.089 -0.087 ~ 0.086
-0.179 -0.176 -0.174
-0.272 -0.269 -0.265
-0.368 -0.364 -0.360
-0.468 -0.463 -0.458
-0.574 -0.567 - 0.56
-0.629 -0.621 -0.614
-0.685 -0.676 -0.669
-0.802 -~-0.792 -0.781
,“iI-* -1.20 1.25 1.30 1.35
0.450
i
0.500
~
/
0.550
~
1
0.575
1
0.600
I
’ ~
~
-1.769 ~ 1.747 - I.728 ~ 1.704
- 1.847 ~ 1.828 - 1.803
IjpII,
0.650
1
0.675
- 1.976 ~ 1.953 - 1.933 - 1.91 1
~ 2.089 - 2.065 - 2.044 -2.021
~ 1.240 - 1.212
- 1.380 - 1.357
- 1.541 - 1.520
~ 1.652 ~ 1.633 - 1.609
1.40 1.45 1.50 1.60 1.70
- 1.189 -1.170 - 1.152 - 1.123 - 1.098
-
1.336 1.318 1.302 1.273 1.248
- I.501 ~ 1.483 - 1.467 - 1.438 -1.411
~ -
1.590 1.573 1.556 1.526 1.497
~
1.684 1.666 1.649 1.618 1.588
-
1.783 1.764 1.747 1.714 1.682
1.80 1.90 2.00 2.10 2.20
~ ~ -
1.076 1.057 1.041 1.027 1.013
- 1.225 - 1.205 -1.188 - 1.172 - 1.157
- 1.387 - 1.365 - 1.345 ~ 1.328 -1.311
-
1.472 1.450 1.429 1.410 1.393
~ ~
1.561 1.538 1.516 1.495 1.477
~ ~ -
1.654 1.629 1.605 1.584 1.564
2.30 2.40 2.50 2.70 2.90
- 1.001 -0.990 ~ 0.979 -0.958 -0.941
-1.144 - 1.131 - 1.119 - 1.095 - 1.075
~ -
1.297 1.283 I .269 1.241 1.218
- 1.377 - 1.362 ~ 1.348 - 1.317 -1.293
~ 1.460 - 1.444 - 1.429 - 1.396 -1.370
- 1.546 ~ 1.529 - 1.512 ~ 1.478 -1.450
- 1.635 - 1.617 - 1.599 - 1.562 -1.532
- 1.728 - 1.708 - 1.688 ~ 1.649 -1.617
3.10 3.30 3.50
-0.927 -0.915 -0.904
- 1.059 -1.045 - 1.034
~ 1.199 -1.182 -1.168
- I .273 -1.254 -1.238
- 1.348 -1.328 -1.311
- 1.426 -1.404 -1.385
- 1.506 -1.483 - 1.462
~ 1.589 -1.563 1 -1.541
1
370
J. .?I. 11. LEVELT ‘I’ \ l’,l.I:
I\
REDUCED
EQUATION
OF STATE
TARLE
Reduct
OF ARGON
371
XENON
\‘I
The reduced internal entropy St,‘iVk of xenon. 7 with the molecular parameters e/k = 225.3”K and B = 4.070 A.
cl* I‘\. T*
0.100
0.150
0.200
0.250
0.300
I .35 1.40 1.45 1.50 1.60
-0.297 -0.283 -0.272 -0.261 -0.243
-0.438 -0.417 -0.400 -0.385 -0.359
-0.574 -0.546 - 0.524 -0.504 -0.472
-0.701 - 0.667 -0.642 -0.619 -0.582
-0.819 -0.783 -0.755 -0.732 -0.692
1.70 1.80 1.90
-0.229 -0.218 -0.210
-0.339 -0.324 -0.314
-0.447 -0.430 -0.418
- 0.555 - 0.536 -0.523
0.500
0.550
0.600
0.650
\
AND
0.350
0.400
-0.874 -0.838 -0.81 1 -0.787 -0.747
- 0.928 -0.894 -0.867 -0.843 -0.803
- 1.034 - 1.003 -0.978 -0.956 -0.917
~ -
-0.663 - 0.643 -0.629
-0.717 -0.697 -0.684
-0.773 - 0.753 - 0.739
-0.887 -0.867 -0.854
- 1.008 -0.988 -0.975
0.700
0.750
0.800
0.850
0.325
d
0.450 1.143 1.117 1.095 1.074 1.037
I
?‘* 1.35 1.40 1.45 1.50 1.60
- 1.261 _ 1.240 - 1.221 ~ 1.202 ~ 1.166
~ ~ -
1.393 1.375 1.359 1.341 1.305
~ 1.542 ~ 1.526 -1.511 - 1.493 -1.456
~ -
1.70 1.80 I .90
-1.137 -1.117 - 1.105
~ 1.276 ~ 1.256 - 1.243
~ 1.426 ~ 1.406 - 1.392
- 1.590 - 1.567 - 1.552
-
1.709 1.694 1.678 1.660 1.621
-
1.895 1.879 1.862 1.842 1.801
- 2.099 -2.080 -2.062 -2.040 ~ 1.994
- 1.766 - 1.955 - 1.741 - I .928 - 1.724 - 1.907
I
I
~ -
2.320 2.299 2.277 2.253 2.202
-2.158 -2.127 -2.103
,
-
2.558 -2.533 - 2.508 - 2.480 - 2.423 - 2.374 - 2.338 -2.310
for argon (Table IX). Then, the validity of the law of corresponding states may be checked by comparison of reduced isotherms of argon and xenon in the case that molecular parameters are used for reduction. (Table X). Finally, a similar check may be made for properties obtained by reduction with the critical parameters of argon and xenon (Table XI). Table IX shows that values of PVIRT, Ui* and St*, obtained with the molecular parameters (5) and (6) for argon, show a satisfactory agreement below the critical density. However, in the dense gas and at low temperatures discrepancies become apparent for the compressibility factor. They must be attributed mainly to the influence of the difference in G, since the density reduction factors differ by 2% whereas the temperature factors differ by 4 parts in 1000 only. The percentage deviations in Ug* are almost constant since Ui* varies roughly proportional to the density. In general, the internal functions Ui* and Se* are less sensitive to changes in reduction parameters than the compressibility factor. The conclusion must be that reduced experimental compressibility data at high densities, and a fortiori in the liquid state, should be considered with due reticence, since small uncertainties in the reduction parameters have a large effect on the factor PVjRT in this range. Turning to the comparison of reduced argon and xenon isotherms obtained with molecular parameters, Table X shows deviations from corresponding states for the thermodynamic properties especially at high densities. The
_
0.95 1.00 1.05 1.10 l.lS
1 O.d7dO O.hY97
1.20 1.25 1.30 I AS 1.40
~ 0.9236 ~ 0.9331 0.9359 ~ 0.941 I 0.9459
1.45 1.;10
0.764h 0.7875
0.6939 0.7230 0.7484 0.7710
O.Sh4c 0.~~09 I 9.6469 0.6797 0.7037
O.dS40 0.846s 0.8777 0.8daO 0.8973
0.7911 O.t?092 0.8254 O.d406 0.8542
0.734:i 0.7Sd2 0.7795 0.7988 O.alhh
1.70 1.>10
0.9,?‘0.1 0.9.?‘43 0.961,5 0.9676 0.97.1 I
0.9059 0.9 I .\” 0.927n 0.9390 0.9503
0.3667 0.87&% 0.8989 0.9145 0.9.320
I.90 2.00 2.10 2.20 2.30
0.977:; 0.9620 0.9856 0.9390 0.9920
0.9.59b 0.9676 0.9751 0.9dlh 0.9675
0.94s7 0.9577 0.948s 0.97d I 0.9hS9
1.60
O.h601
0.5199 ~ O.-ihii:: O.S?Sh ~ O.Sl9ii 0.5Ohl ~ O.ShS2
0.71.16 0.7395 0.7630 0.7H46
O.iL129 O.rt479 0.h747 0.6979 0.9la2 0.9&l 0.9520 0.964 0.9738 0.9903
o.s309
OA:,Y4 0.4490 0.5033
O..ln30 0.4237 0.4d24
0.442,5 1 0.6062 O.b754 i 0.6435 0.70.55 0.6776 0.7AE 0.70?9 0.7.511.\ 0.7.17a
0.5704 O.h17,j, 0.6.5.59 0.5909 0.7233
I).S.S.)O 0.,59Fh 0.6406 0.5794 0.71i.3
0.5.154 0.,5%2 0.6322 0.5747 Cl.7142
OAO45 0.9228 o.as57 0.~842 !I.9091
0.7816 0.8032 O.d419 0.87.55 0.9050
.7044 .71192 .;336 .a723 .9oi3
0.7,533 0.7812 O.d3 12 0.07so 0.9134
0.74if> 0.7796 o.u.53 O.c;840 0.9269
0.7soa 0.78SO O.d444 0.9002 0.947,<
0.93 I I 0.9so7 0.94d2 0.9839 0.99.10
0.9310 0.9542 0.9751 0.9937 1.010s
,936,) .9631 .9872 .OOd7 .02jI
0.9474 0.9777 1.ooso 1.0294 l.O,514
0.9hk 0.99X6 I .029 1 1.osv I.061 I
0.9894 1.024R 1.0404 1.0907 1.1178
I .07 l!i ~ 1.0897 1 l.lOh4 ~ I.1217 I 135;
1.lrj.P. 1.1239 1.142.1 1.1.i95 1.17S.i
1.1425 1.1~51 i.l%a 1.2044 1.2217
/.I’
2.1
3.2937 7.4490 I.5892 II.7162 >.;i.315
0.4230 o.sc79 0.7343 O.r(h99 0.9937
0.6013 0.7745 0.9290
O.hi!,5.3
I
2.40 2.50 2.hO 2.70 2.80
1 .045b ~ 1,0619 ~ 1.0764 I .0900 1.I025
cl, I.? 1-r 0.95
1.oo
1.0s 1.10 I.15
0.3371 0.3208, ~ 0.40.52 ~ 0.3940 ’ 0.4485 ~ 0.442s
0.2246 i 0.3259 0.4200 0.5072,
0.2439 0.35X 0.4575 0..%519
0.2637 0.4050 0.5160 0.6176
0.206 0.3499 0.4809 0.4003 0.709
I
0.3121 O..J914 0.4654
0.3128 O.j992 0.4790
0.5260 o.sd49 0.6393 0.6898 0.7346
0.5344 0.S9dl 0.4572 0.7118 0.7626
~ 0.5541 0.6231 O.htji>9 ~ 0.7460 ~ O.dO07
0.5879 0.4424 0.731 I 0.7944 0.053.5
0.6339 0.7la9 0.7927 O.bh07 0.9236
0.7109 0.7945 0.8752 0.9474 1.0144
0.0034 0.8993 0.9826 l.OS91 1.1294
I.9345 1.0321 I.1 195 1.1994 1.2723
1, 1:J02 ~ 1.1991. 1.2903, I A730 1.4491
0.7801
; 0.9S73 1.0142
0.8096 0.8536 0.9327 1.0019 1.0~20
0.8SlS 0.8989 0.9d40 ~ I.0584 1.1237
0.90~1 0.9588 1.0499 1.129.3 1.19?!?
0.9819 I.0360 I.1328 1.2169 1.2903
I .0760 1.1322 I .2X54 I .,12.38 l.400d
1.1941 1.2541 1..!‘610 1.4532 l.S331
1..1403 I.4026 I.5133 l.hO’13 1.0903
1.51nc I .5dJO 1.694jJ 1.792.: 1._7SC~
I
1.20 1.25 1.30 1.35 1.40
0.5271 o.sal3 0.63l.i 0.6779 0.7209
1.45 1.50 1.60 1.70 1.30
0.7610 0.7983 0.8655 0.9244 0.9742
1.90 2.00 2.10 2.20 2.30
1.0221 1.0630 1.0999 1.1329 I, 1626
1.0641 1.1085 3.1485 1.1843 I.2165
l.llhh 1.1644 1.2074 1.2461 1.2804
I.1014 1.2327 3.2736 1.3197 1.3564
I.2631 1.35Sl I.3136 1.4124 1.3633 1.46Jo 1.4067 1.5091 I .445&> I .5457
1.4ha.S I.,5282 1.5816 I.6288 1 .h709
I .h03 1 1.6648 1.7197 I .7683 1.a113
1.7OlC 1X246 1.8808 1.9301 I .973s
1.9470 2.0112 2.0677 2.1169 2.1599
2.40 2.50 2.60 2.70 2.60
I.1896 1.2142 1.2366 1.2571 1.2759
I .24S7 1.2723 1.295s 1.3185 1.3388
1.3120 1.3404 1.3663 1.389d 1.4115,
1.319s 1.4200 1.447.3 1.4723 1.4952
I.4307 I.5124 I..541 I I .Sh72 1.591 I
I .7oct7 1.7427 I .7733 l.dOlO 1.8261
l.aSOO Id345 I .9 15,s 1.9433 1.9636
2.0125 2.0473 2.0779 2.1054 2.1304
2.19.;9 2.233~) 2.2429 2.2894 2.3136 --
’ 0.8207 , 0.8938
-
1 .Suh.i 1.6193 I .h492 1.6763 1.7008
1542 2276 2 2835 2. 3317 i 2. 3736
14;
2.41 I I 2.44nl
REDUCED
EQUATION
OF STATE
TABLE
Reduction --
1 0.2
( 0.3
1
0.4
1
0.5
1
0.6
AND
373
XENON
VIII
The equation of state the critical parameters
with
OF ARGON
PV/KI‘ of xenon. T, = 289.74”K and
~ 0.7
186.3 am. 1 1.1
~
0.3340 0.4010 0.4633 0.5210
0.3163 0.3882 0.4554 0.5178
0.3061 0.3836 0.4562 0.5236
0.6404 0.6142 0.5942 0.6742 0.6518 0.6355 0.7051 0.6864 0.6737 0.7337 0.7183 0.7091 0.7602~0.7480~0.7420
0.5807 0.5742 0.5754 0.6258 0.6234 0.6287 0.6677~0.6689 0.6782 0.7065’0.7112 0.7241 0.742610.7507,0.7670
0.5860 0.6436 0.6971 0.7468 0.7932
~
~ 2.2
/
i 0.8
1 0.9
/
de = 1.0
1.2
i
1.3
?‘r
1.15 1.20
‘0.8391 0.8531
0.7692 0.7895
0.6448 0.6777 0.7069 0.7330
1.25 1.30 1.35 1.40 1.45
0.8655
0.8077 0.8241 0.8390 0.8526 0.8652
0.7564 0.7114 0.6728 0.7775 0.737010.7026 0.7968,0.7604 0.7299 0.8145’0.7819 0.7550 0.8308 0.8017 0.7782
--_. 1 ,:_
0.5773 0.6168 0.6518 0.6832
1
1.53.6
1.7
0.5181 0.5634 0.6038 0.6401
1
1.8
0.4670 0.5174 0.5626 0.6034
1.9
7. 1.05 1.10 1.15 1.20
0.3037 0.3878 0.4665 0.5394
0.3129 0.4043 0.4895 0.5683
0.3354 0.4348 0.5269 0.6119
0.3766 0.4843 0.5834 0.6746
1.25 1.30 1.35 1.40 1.45
0.6068 0.6690 0.7268 0.7805 0.8305
0.6409 0.7079 0.7700 0.8276 0.8813
0.6900 0.7619 0.8284 0.8901 0.9474
0.7582 0.8484 0.9655 0.8349 0.9299 1.0512 0.9058 1.0049 1.1300 0.9714 1.0741 1.2024 1.0321 1.1382 1.2694
0.4402 0.5561 0.6623 0.7596
0.5321 0.6559 0.7688 0.8718
0.4236 0.4785 0.5281 0.5731
)
2.0
0.3873 0.4462 0.5000 0.5491
~
2.1
0.3575 0.4203 0.4782 0.5316
1
2.3
1
2.4
2.5
0.6580 0.8245 0.7889 0.9613 0.9078 1.0850 1.0157 1.1970 1.1138 1.2031 1.2850 1.3602 1.4295
1.2984 1.3905 1.4747 1.5518 1.6228
1.7819 2.0554 2.3780 1.8535 2.1266;2.4473
2.7550 2.8210
reduced data for xenon deviate about three times more from the argon values, than the difference between the two argon sets (Table IX) amounts to. Finally, as regards the reduction by critical parameters, Table XI shows that a somewhat closer agreement is obtained between the two gases in this case. However, the compressibility factor of xenon is lower again than that of argon at high densities. Before drawing any conclusions from the observed deviations from corresponding states, we must make sure that these deviations are real, and that they cannot be explained by uncertainties of the reduction factors. This may be done as follows. The Amagat density and Kelvin temperature of the PV/RT isotherms of argon may be multiplied by scale factors such that these isotherms are transformed into the compressibility isotherms of xenon, in the temperature range from 1.05 T, to 1.45 T, and for densities from zero up to 2d,. By requiring that the minima in the PV/RT vs. d isotherms coincide, one finds that the scale factor for the temperature is equal to 1.931 f 0.002. Then, the scale factor for the density is found by requiring the coincidence of the high density range of the isotherms. This factor is 0.633 f 0.001. With this choice of scale factors, the experimental PV!RT values of argon and xenon agree to about one percent in the range of overlap. Therefore, in comparing tables of reduced properties of argon and xenon, we can only expect to find corresponding states, if the ratio of
374
J.
hl.
II.
LEVELT
the temperature reduction factors used in a particular comparison is equal to 1.931, and the ratio of the density reduction factors is equal to 0.633. R summary of the actual ratios of these reduction factors, for reduction with molecular as well as with critical parameters, is given in Table XII. It is seen that they deviate considerably from the “ideal” ratios 1.931 and 0.633. We must check whether these differences may be due to uncertainties of the molecular or critical parameters.
REDUCED
EQUATION
OF STATE
OF ARGON
TABLE
r
Cornmrison
of reduced
T*
properties _ _
Reduction
with molecular
d* = 0.3
I
375
XENON
X
of amon and xenon.
d’ = 0.2
AND
parameters.
I
1
-I
d* = 0.4 Xe
225.3 4.070 1.4 1.7 1.9
0.5629 0.7263 0.8025
1.4 1.7 1.9
- 1.499 - 1.364 -1.309
1.4 1.7 1.9
-0.560 -0.471 -0.441
0.5596’ - 0.0033 0.7184 -0.0076 0.7925 -0.0100 -1.474 -- 1.328 --1.275 -0.546 -0.447 l-O.418
0.3749 0.6484 0.7843 - 2.091 _ 1.912 - 1.851
-2.129 - 1.972 - 1.906
A
0.718: 3; 1.169 I 1.381( 1
0.50311-0.21521 0.9267 -0.2424 1.1309 -0.2501
1.4 -3.278 1.7 -3.141 1.9 -3.065
-3.167 -3.013 -- 2.955
to.111 j-0.128 to.110
- 3.876 -3.727 -3.636
-3.716 -3.565 -3.504
1.4 _ 1.336 1.7 ~ 1.248 1.9 -1.205 -
- 1.240 - 1.137 -1.105
to.096 to.111 to.100
- 1.684 ~ 1.588 ~ 1.538
- 1.526 - 1.426 _ 1.392
TABLE
XI
I
d, = 0.6 Ar
1 Xe
1
+0.065 to.091 +0.081
~ 1.003 -0.887 -0.854
+0.053 +0.072 +0.066
0.65
0.96731 1.4464 1.6692
0.642( 1.103 I.3211
+0.160 +0.162 +0.132
-3.999 -3.841 - 3.774
+0.158 i0.162 1+0.146
- 1.694 - 1.590 - 1.552
-
- 0.324: ~ 0.343: - 0.347‘
of reduced compressibility factors of argon and xenon. Reduction with critical parameters. 1
A
- 2.640 -2.466 -2.406
d” =
I
--0.035 -0.051< - 0.059:
Ar 119.8 3.405
-0.0869 -0.1144 -0.1249
Comparison
- 2.705 -2.557 -2.487
d” = 0.6
I
i
+0.038 +0.060 +0.055
-0.810 -0.706 I-O.670
0.722; 0.892;
1.4 0.4652 0.8366 1.7 1.0171 1.9
/
+0.014 1+0.024 1+0.023
Xe 225.3 4.070
Ar 119.8 3.405
Tr
+0.025 +0.036 +0.034
d* = 0.5
T*
0.45 1: 0.676t 0.786: !I
A
1 Ar
d, =
1.2
1 Xe
/
A
1 Ar
dr =
1.8
/ Xe
/
d, = 2.0 A
1 Ar
!
Xe
1
n
I0.7363 1.5188 I0.6580 1.1996 1.4295 I -0.0783 1.1138 -0.0893 -0.0858 As for the deviations from corresponding states found in the reduction with critical fmmneters (Table XI), it is clear from Table XII that one of the critical temperatures should be in error by +yo (i.e., 1.5% for xenon, 0.8”C for argon) and a critical density by 2.1 y. (i.e., 3.7 Amagat units for xenon, 6 Amagat units for argon) to account for the difference. In view of the claimed experimental accuracy (5 2), this is not very probable. Since the use of critical parameters for reduction thus leads to deviations from corresponding states for these heavy noble gases, it is clear that the construction of semi-empirical equations of state in terms of a modified principle of corresponding states (9 1) may meet difficulties in the dense gas and the liquid state.
376
Table XI I also shows that in the reduction Lvith wzolectr/av~n~n~~z~ic~s,that paramctcr pair (6) for argon is clearly the better \vhtw corrc,spontling statcas with xc‘non arc conwrncd. Ncvcrthrlcss, one of the t‘, k \.alucs shoultl by> modified nith 2.3”~~ (i.tx., 2.3^(‘ for argon, 5.2”C for xcwon) ant1 ant’ of thr (T values by 2.1 ‘I;, (i.v., 0.067 I$ for argon, 0.085 ipi for scwon) to achitl\,cs coincidence of rcductd argon and xenon isotherms. These chang-(1s ~xc~c~l the. cxpectcd uncertainties of thcb molecular parameters (4 2). ‘1‘11~uw of rtlasowble values for the molecular param&ers of argon and xt’non thus results in deviations from corresponding states. Tlic reason maJ- bc that the I.c~nnnrtl*Jones potcmtial is not a proper description of the intt~rmolccular ficlltl of these gases. Studies of crystal properties and transport phenomclna ha\~ rcvr~aled deficiencies of this potential Jds’i). An inspection of Fig. 1, showing the difference of observed and theoretical H” as a function of 7‘* adds to suspicion concerning the, applicability of the Iznnartl-Jonc% 6-l 2 l)ottwtial for argon and xenon. Suppose that though the I,t>nnard-Jones 6 -12 potential fails, some othc,r two-parameter function holds for these gases. Then, tht, rcduccd 1;” 1’s. I‘* graphs of these gas’s must coincide after proper scaling, although thq may deviate from the corresponding cur\Tc for tht, 6-12 potential. l;rom the deviation pattern K* - IS*th of Figs. II’, and lr, it is clear that this coincidence is achieved with the parameter pair (6) for argon and (4) for xenon. Thus, with this choice of parameters a law of corresponding states should hold for argon and xenon. Since this is not the cast, the assumption of additivity or the limitation to a universal two-parameter potential function must be in error. Unfortunately, it is not possible to sq)aratc thtl two effects using gas P-I’-T
data only.
Acknowledgement. The author is indebted to Professor _J. 0. Hirschfclder for his suggestions and helpful aclviw concerning this research, and to Professor J. de Boer for his kind interest. Without the stimulating criticism of Dr. E. G. I>. Cohen the paper cvoul~l never have reached its present state. Mrs. Marion Taylor has deserved the gratitude of the author for developing an efficient method for carrying out the two-dimensional interpolations on the Bendix G-l 511. She was very helpful in performing the
REDUCED
EQUATION
OF STATE
OF ARGON
AND
377
XENON
tedious numerical work in those parts of the tables where information of most interest but hardest to obtain. Received
is
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