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Fluid Phase Equilibria 279 (2009) 4146

Contents lists available at ScienceDirect

Fluid Phase Equilibria

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / f l u i d

Novel correlations between the critical constants of the noble gases

Philip Molyneux

Macrophile Associates, 33 Shaftesbury Avenue, Radcliffe-on-Trent, Nottingham NG12 2NH, UK

a r t i c l e i n f o

Article history:

Received 19 April 2008Received in revised form 18 December 2008

Accepted 20 January 2009

Available online 30 January 2009

Keywords:

Critical constants

Linear correlations between critical

constants

Noble gases

Quantal (quantum mechanical) effects

Residual Volume Effect

Zeronium

a b s t r a c t

For any particular fluid, the set of three critical constants (CC) pressure Pc, temperature Tc and molar

volume Vc has a central importance in defining the physical behaviour of the fluid in the gas andliquid states. However, little attention seems to have been paid in the past to the relations between the

CC of different substances. In the present paper, some simple and apparently novel relations have been

found between the three CC for the set of four noble gases: Ne, Ar, Kr, Xe. Defining the critical quotient

Qc RTc/Pc (where R is the Gas Constant) the correlations may be summarised by the dual equation:

(Vc/cm3 mol1)= 27+ 0.31 (Tc/K)= 3.3+ 0.280 (Qc/cm

3 mol1), which describes the CC data for thequartet

NeXe with an average uncertainty of 0.5%. Regarding the other two noble gases, the two isotopes of

the lightest member, 3He and 4He, show the deviations from these relations that are expected from

quantal effects and their low molar masses; while for the heaviest member, Rn, the correlations enable

a value of 145(5) cm3 mol1 to be estimated for Vc that is not otherwise well defined in the literature. By

contrast, and contrary to the general assumption, the second lightest member, Ne, apparently does not

show appreciable quantaleffects in thearea, so that NeXe may be considered together as a group. These

correlations are compared with the behaviour of a selection of polyatomic fluids; in these comparisons,

theNG dualcorrelation equationprovidesa referenceline defining the presumedsimplestbehaviour.This

and related areas show a Residual Volume Effect, in that extrapolating the equivalent temperature and

energy parameters to zero for the state of zero-mass point particles, referred to here as the hypotheticalelement zeronium (Ze), the system in each case still has a finite intercept; this intercept amounts to

essentially 34%of the averagevolumefor thepresentquartetNeXe,ratherthan the zerovolumeexpected

for this condition. 2009 Elsevier B.V. All rights reserved.

1. Introduction

The set of the three critical constants (CC)for a fluid the critical

pressure Pc, the critical temperature Tc and the critical molar vol-

ume Vc plays a central role in defining the physical properties of

the fluid, particularly its behaviour in the gaseous (vapour) and the

liquid states.1 Primarily, this set defines the critical point for the

substance, that is, the point at which the vapour and liquid states

merge, and the vapourization enthalpy change becomes zero [14].

Furthermore, the corresponding three reduced quantities: Pr, Tr,

Vr, which are obtained by division of these three properties (pres-

sure P, temperature T, and molar volume V) by their respective CC,

are found for simple-molecule fluids to give universal curves defin-

ing the deviations from the ideal gas laws, commonly referred to as

the Law of CorrespondingStates (LCS).Likewise, theories ofequations

of state (EOS) such as the classic van der Waals equation focus on a

Tel.: +44 115 933 4813; fax: +44 115 933 4813.

E-mail address: [email protected] .1 Throughout this paper, the units used for critical constants and derived quanti-

ties are: pressure, MPa; temperature, K; molar volume, cm3 mol1.

derived dimensionless quantity, the critical compressibility factor,

Zc, defined by

Zc PcVcRTc

(1)

where R is the Gas Constant (in the present customary units,

8.3145 cm3 MPaK1 mol1). The experimental values of Zc for

simple-molecule fluids range between about 0.29 and 0.26 [1,2],whereas (for example) the simple van der Waals equation requires

Zc to have a value of 0.375;one criterion of a goodEOS thereforeis

if it yields values ofZc closer to these experimental ones [3,4]. The

parameter has also been used as a criterion of the molecular inter-

actions; for example, the term perfect liquids has been applied

to the four fluids: Ar, Kr, Xe and CH4 for which Zc = 0.292 [5]. In

all of these applications, there is the limitation that the value of Zcdepends upon all three critical parameters, and is subject to their

combined experimental uncertainties.

These CC are also importanttechnicallyin chemical engineering,

both directly in relation to thebehaviour of fluids,and indirectly for

the correlation and prediction of other physical properties [6,7].

As a preliminaryto considering thecorrelations that are thespe-

cific topic of this paper, it is useful to introduce a new quantity, the

0378-3812/$ see front matter 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.fluid.2009.01.008

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42 P. Molyneux / Fluid Phase Equilibria 279 (2009) 4146

critical quotient Qc defined by

Qc RTcPc

(2)

This is a volume-dimensioned quantity that is found to give linear

correlations when used in place of Pc alone.

2. Correlations between the critical constants

2.1. Critical constants for the noble gases

The above discussion of CC is of course part of the common

currency of physicalchemistry. However, it is evident fromthe liter-

ature cited that theCC values forany specific fluid are considered by

the physicochemical communitysimplyas a given set of parameters

that are characteristic (but otherwise unexplained or unrelated) for

that fluid. In particular, little if any interest seems to have been

shown in the literature as to whether or how these parameters for

different fluids might be related to one another.

The present paper points out some simple correlations that are

apparent between these parameters, specifically as applied to thefour noble gases (NG): Ne, Ar, Kr, Xe; the outer two NG, He and

Rn, are considered somewhat separately. This set of elements (still

commonly referred to eitheras therare gases or theinertgases)

is ideal fortesting such correlations, since they are monatomic with

spherical molecules (atoms) and with any interactions between the

atoms only of the van der Waals/London dispersion type. For these

reasons, any theory of molecular behaviour must first be shown to

be applicable to the NG before it is worthwhile testing with other

less simple fluids.

The literature values for the CC for the noble gases are sum-

marised in Table 1. To get the best consensus on these data, those

listed are the averagesfrom six published data-sets[813]. The data

for Ne and Rn require special note.

In thecase of Ne,it hasbeen suggestedthat therelativelygreater

uncertainty in the literature values ofPc may reflect contamination

by He in some of the samples used; on the other hand, the appar-

ently low uncertainty in Vc seems to reflect the use of a single early

value that has been copied from one listing to another. Neverthe-

less, the data for this NG seem to be sufficiently well defined to

behave consistently in the correlations observed.

InthecaseofRn,thevalueslistedfor Pc and Tc arefromtherecent

re-evaluation by Ferreira and Lobo [14] of the early data (and still

apparently the only precise ones) that were obtained by Gray and

Ramsay [15]. From its format, the value for Vc of 140.00 cm3 mol1

listedby Polinget al.[13] is evidently an estimate; thevalue given in

Table 1 is an extrapolationfrom the present Case C linearcorrelation

for the quartet NeXe (Section 2.4). This Vc value is important for

example in connection with the diffusion of Rn from radiogenic

rocks into the environment.In considering the correlations for the NG considered below in

Sections 2.22.4, the data are compared with those for two other

groupsof fluids,both to putthis behaviour in the wider context and

to show how it relates to that of fluids in general.

The first comparison group is a trio of related molecules: H2,

O2, and H2O. Here, H2 is a light gas that is expected, like the iso-

topes of He, to show quantal (quantum mechanical) effects (QE), as

discussed in Section 4; in this case, since the three isotopic forms:

H2, HD, D2, have very similar CC values (average range 14%) [13],

the latter two have been omitted in the comparisons. For O2, its

CC values are coincidentally close to those for Ar, differing by only

3% on the average; this suggests that one may act as substitute for

the other in considering physical behaviour in particular applica-

tions. Lastly, with H2O, at normal temperatures and pressures (forexample, 100 C, 0.1MPa) the vapour (steam) shows no sign of the

association by hydrogen bonding that is prominent in the liquid

state [16]; however, such association would be expected to be sig-

nificant at the critical point because of the much higher density

(Vc =56cm3 mol1compare molar volume Vm =18cm

3 mol1 for

the liquid at ambient temperatures).

This second comparison group comprises fluids with molecules

of the type MY4 (tetrahedral) and MY6 (octahedral, where Y = F in

all the examples used), which may be referred to as quasispheri-cal molecules (QSM) since they would become effectively spherical

by rotation in these fluid states. For example, the tetrahalides MX4have been studied by Hildebrand and co-workers as proposed heav-

ier analogues of the NG [17]. The intermolecular forces throughout

will again be of thevan derWaals/Londondispersiontype,although

(unlike the situation with the NG) these forces will no longer be

central.

The present examination of the CC values for the NG group

NeXe has revealed three linked linear correlations, listed here as

CasesA, B and C and discussedin Sections 2.22.4 respectively. It is

useful to present the equations for these at this initial stage before

they are discussed individually.

Case A : Vc = 27(2)+ 0.31(1)Tc (3)

Case B : Qc = 86(2)+ 1.12(2)Tc (4)

Case C : Vc = 3.3(4)+ 0.280(1)Qc (5)

It will be noted that these are not three independent equations,

since they involve only three independent variables; however, it is

convenientto consider themall in turn,becauseeach has distinctive

features.

In the above Eqs. (3)(5), since all the cases refer specifically to

the quartet NeXe, the coefficients are put only as their numerical

values, rather than being given symbols. In each case, thenumerical

value given is the mean, with in parenthesis the estimated stan-

dard deviation in the last quoted decimal place. This usage will be

followed throughout this paper.

2.2. Case A: critical volume versus critical temperature

TheplotofVc against Tc for the noblegases andother comparison

molecules is shown in Fig. 1. In the case of Rn, the Vc value plotted

(filled circle) is that obtained by extrapolation from Case C below

(Table 1). Excepting the two helium isotopes, the plot is essentially

linear according to Eq. (3) above; the average deviation for the four

NGs in Fig. 1 is about 2%. The strong deviations for the two helium

isotopes may be attributed to the QE resulting from the low molar

mass, and the different character of the liquid/vapour equilibrium

in these cases [18]. With Ne, however, there is no marked deviation

for this fluid from the straight line in Fig. 1; this is surprising, since

similar although lesser QE are to be expected because of its low

molar mass (Section 4).

The finite intercept of Vc (27 cm3 mol1) for Tc0 presumablyrelates to the limiting condition of zero-mass point particles. The

occurrenceof this finite intercept is reminiscentof the covolumes,

with values in the range 1340cm3 mol1, that arise for the limit

of zero molar mass in the additive group-contribution schemes for

molar volumes of pure substances [6], and for partial molar vol-

umes of solutes [19,20]. From the molecular viewpoint, Stillinger

has suggested this parameter may arise fromthe zero-point motion

for these limiting particles [19]. For concreteness of discussion, and

in view of the importance of this limiting state in the present and

the related correlations (Section 2.6), it is convenient to name this

hypothetical limiting situation (zero-mass point particles) as the

element zeronium (Ze), characterised from the present Cases by

the data in Table 1; this is accordingly so-plotted (asterisk) in Fig. 1.

It is convenient to quantify this intercept by reference to the NGquartet, as 34(2)% of the average Vc value for the four gases. As

discussed below (Section 2.6), parallel behaviour is shown at the

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P. Molyneux / Fluid Phase Equilibria 279 (2009) 4146 43

Table 1

Critical constants and derived quantities for the noble gases.a

Gas Atomic number, Z Relative atomic mass, Mr Pc (MPa) Tc (K) Vc (cm3 mol1 ) Qc (cm

3 mol1)b Zc

3He 2 3.017 0.116(1) 3.32(1) 72.23(1) 239(2) 0.302(4)4He 2 4.003 0.228(1) 5.24(6) 57.5(3) 191(1) 0.301(1)

Ne 10 20.180 2.71(6) 44.5(2) 41.7(1) 137(3) 0.305(7)

Ar 18 39.950 4.89(2) 150.9(1) 74.9(3) 257(1) 0.292(2)

Kr 36 83.800 5.50(1) 209.40(1) 91.7(6) 317(1) 0.289(2)

Xe 54 131.300 5.84(1) 289.75(5) 119.0(8) 413(1) 0.288(2)

Rn 86 222.00 6.2(2)c 377.1(1)c 145(5)c 506c 0.287c

Zed 0 0 0 0 27(2)d 86(2)d 0.31(2)d

a Tabulated values of the CC (except for Rnsee below) are averages from listings in the literature [813], except for the underlined values, which are either assigned zero

values, or estimates (see below and text). Numbers in brackets are estimated standard deviations in the quoted last decimal place.b Critical quotientsee Eq. (2).c Rn:the values ofPc and Tc are fromthe re-evaluation by Ferreiraand Lobo [14] of theoriginal data of Grayand Ramsay [15]. The value ofVc has thenbeen estimated from

Eq. (5), as shown in Fig. 4see also Section 2.1.d Hypothetical element zeronium as the limiting state of zero-mass point particlessee Eqs. (4) and (5), and Section 3.

macroscopic (molar) level for the crystal state, and at the molecu-

lar level with the LennardJones (LJ) parameters; it may therefore

be referred to collectively as the Residual Volume Effect (RVE) as

discussed specifically in Section 3.With thefirst comparison group,the point forH2 deviates above

the correlation line because of the QE for this light gas, as seen

with the He isotopes. The point for H2O lies far to the right (off

scale) because of the high Tc value, evidently related to the strong

hydrogen bonding as already discussed. For O2, as already noted,

here and in other plots the point lies (coincidentally) close to Ar.

2.3. Case B: critical quotient Qc versus critical temperature Tc

It is useful to preface this Case by considering firstly the simple

plot of critical pressure Pc (MPa) against critical temperature Tc (K)

for the noble gases as shown in Fig. 2. The curve for NeXe may be

fitted by a rectangular hyperbola

Pc =7.4(1)Tc

[77(3) + Tc](6)

Fig. 1. Critical volume Vc

versus critical temperature Tc

. Symbols: () noble gases

as labeled the continuous trend line and its broken extrapolation relates only to

the NG quartet NeXe (Eq. (3); () (estimated value ofVc Table 1); () H2, O2 and

H2O; () sym-MY4 as labelled; () Zesee text and Table 1.

Although the data for Rn are available (Table 1), they have been

omitted in deriving the above coefficients for conformity with the

other Cases; however, it is notable that its plotted point lies very

close to thecorrelationcurve. The average deviation forthe four NGin Fig. 2 is 0.6%. The form of the curve of Eq. (6) indicates that the

origin, that is, for the hypothetical limiting element Ze (asterisk

in Fig. 2) lies on the curve, so that this corresponds both to Tc0

and to Pc0.

The points for 3He and 4He lie close to the curve, but evidently,

this is only because both parameters are small and the points are

therefore close tothe origin. The point for Nelies close tothe curve,

indicating again that QE do not have an important effect on the CC

for this gas, which is contrary to a common expectation (Section 4).

The form of Eq. (6) suggests that for these NG, at the limit

of indefinitely large Tc (presumably, indefinitely large molar

mass), the critical pressure should approach a limiting value of

Pc =7.4MPa.

Onemethod of testing thehyperbolicform of Eq.(6) isto plotthe

ratio Tc/Pc against Tc, which should be linear. However, it is more

Fig. 2. Critical pressure Pc versus critical temperature Tc . Symbols as Fig. 1point

for Rn from experimental values (Table 1). The continuous trend curve and its bro-

ken extrapolation (rectangular hyperbola) relate only to the data for the NG quartet

NeXe (Eq. (6)); the horizontal broken line is the extrapolatedlimiting critical pres-

sure Pc = 7.4MPa.

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Fig. 3. Critical quotient Qc versus critical temperature Tc . Symbols as Figs. 1 and 2.

The continuous trend line and its broken extrapolation (Eq. (4)) relate only to the

data for the NG quartet NeXe.

fruitful to use the quantity RTc/Pc, that is, the volume-dimensioned

critical quotient Qc as already defined by Eq. (2). This plot of Qcagainst Tc is shown in Fig. 3. For the NGsetNeXetheplot is essen-

tiallylinear according to Eq.(4) above;the average deviation forthe

fourNG is about 0.5%. AswithFig.2, the datafor Rn havebeenomit-

ted from fitting the line, but its point nevertheless lies close to this

line. The marked positive deviations for the two He isotopes may

again be attributed to the QE resulting from the low molar mass,

thelower mass isotope as expected showing the greater deviations.

However, with Ne, as with Case A, there is again no marked devi-

ation from the straight line than seen with the other NG in Fig. 3,

despite the QE commonly expected for this light gas (Section 4).

The finite intercept (Qc =86cm3 mol1) for Tc0 in Fig. 3 again

presumably relates to the limiting conditionof pointmass particles,

i.e., zeronium (Ze) as introduced in Section 2.2, and defines the Qcvalue for this state (Table 1). This is discussed further in Section 3.

For the first comparison group, H2 is strongly deviant upward

like the He isotopes presumably again because of QE (Section 4).

The point for H2O deviates in the other direction, again presum-

ably because the hydrogen bonding gives a high Tc value, as already

discussed.

2.4. Case C: critical volume Vc versus critical quotient Qc

TheplotofVc versus Qc isshownin Fig.4.InthisCase,thestraight

line fit for NeXe is given for Eq. (5) above. The average deviation

here for the four NG is only about 0.4%. The closeness of the fit

is in this case remarkable in view of the fact that all three CC are

involvedin this type of plot.In particular,it gives more confidencein

extrapolating toobtain anestimate ofthe valueof 145(5) cm3 mol1

for Vc of Rn as given in Table 1 (see also Section 2.1).

The close fit of the helium isotopes to the trend line in Fig. 4 is

quite remarkable because of the QE which show up in the previous

plots (Figs. 13); even the small deviation seen with 3He could be

ascribed to uncertainties in the small values ofTc and Pc along with

the propagation of errors in taking their ratio.

Forthefirstcomparisongroup,H 2 alsoliesclosetotheline,while

H2O is relatively closer than with the other plots.Likewise,for thesecond comparison group,the data forthe QSM

also lie on or remarkably close to the line of Eq. (6). Taking this a

Fig. 4. Critical volume Vc versus the critical quotient Qc. Symbols as Figs. 1 and 2.

The continuous trend line and its broken extrapolation relate only to the data for

the NG quartet NeXe (Eq. (5)). The dotted line indicatesthe critical compressibility

factor Zc = 0.292 for perfect fluids (see Section 1).

stage further, Fig. 5 shows the same plot with extended axes to

include the heavier QSM; the 13 compounds shown nowfit the line

with an average deviation of about 3%, although there seems to be

some downward deviation from the line at the upper end. The plot

is particularly interesting as it contains a number of compounds of

technical importance, such as SF6 and UF6.

To obviate further crowding at the upper end of this plot, the

data available [8] for the Group 4 tetrachlorides TiCl4

, ZrCl4

, HfCl4 have been omitted; in fact these data are anomalous, showing a

decrease in the values plotted with increase in molar mass, which

requires further investigation.

Fig. 5. Critical volume Vc

versus thecritical quotient Qc

extended ranges. Symbols:

() sym-MF6 (octahedral); other symbols as Figs. 1 and 2. The continuous trend line

and its broken extrapolation relate only to the data for the NG quartet NeXe (Eq.

(5)).

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P. Molyneux / Fluid Phase Equilibria 279 (2009) 4146 45

It is evident that this Case C plot promises to be useful for fluids

in general, involving as it does all three values of the CC, where it

may be appliedfor thepurposesof display,discussion andinterpre-

tation; the NG linear correlation then provides a reference line for

this purpose. This should reveal aspects of the CC data not evident

from the conventional listings in tabular form [813].

Taking this a stage further, the use of the three parameters Tc,Vc and Qc in the three linear plots in Figs. 1, 3 and 4 indicates the

possibility of using 3D plots to display all three parameters. Going

beyond the 2D representation by stereo-optic pairs, this requires

the use of a computer display with software to allow the rotation

of the 3D plot.

The correlation line in Figs. 4 and 5 also explains the fact that

the values of the critical compressibility factorZc in most cases drift

downwards with increasing molar mass, as seen from the data in

Table 1 for thenoble gasesfrom 0.305 forNe down to 0.288 for Xe.

This arises because the correlation plot has a small but finite posi-

tive intercept(3.3 cm3 mol1Eq. (5)), andsincelinesof constantZcradiate from the origin in this type of plot, then these sweep across

the plots with lower slope in traversing from lower to higher molar

mass. Thedrift inZ

c values is thereforean artefact of this finite ordi-nate intercept, leading to Ne having a markedly different value ofZcfrom ArXe, which has customarily ascribed to QE with this light

NG (Section 4). It seems that these types of effects have not been

noted previously because such data have usually been treated on a

piecemeal basis and presented at most in tabular form, rather than

using plots of the present type.

It should be noted that although the line in Figs. 4 and 5 is

extended back to meet the ordinate axis, the actual extreme lower

limit is the Ze (zeronium) point as plotted (asterisk), which also lies

close to the correlation line.

2.5. The Law of Corresponding States

Experience during the preparation of this paper has shown that

the present correlations may be confused with, or may be taken

to be a special form of, the Law of Corresponding States (LCS). It is

therefore necessary to point out the distinctions between the two

forms of behaviour.

The LCS relates to the three reduced parameters Pr, Tr, and Vr, as

obtained by dividing the actual value of the specified quantity by

the corresponding critical value. The Law then specifies that for a

range of simple fluids, if two of these parameters are the same for

the different fluids, then the third parameter will also be the same;

in addition, plots for fixed values ofTr of the reduced compressibil-

ity factor Zr against the reduced pressure Pr are the same for the

different fluids. This applies more or less exactly to molecules of

complexity at least up to pentane [1,2].

However, by contrast, the present correlations are concerned

directly with the critical values, whereas in the LCS these havealready been absorbed into the reduced values. In addition, in the

present case the correlation plots arestraight lines (Figs. 1, 3 and 4),

whereas in thecase of theLCS thespecified plots of plots of reduced

compressibility factor Zr versus Pr (fixed Tr) arenonlinear. A further

point of distinction from the latter plots is that the third parame-

ter is not kept fixed in the present plotsfor example, in Fig. 1, the

value ofPc varies along the linear plot ofVc against Tc.

It should be evident, therefore, that the present correlations

represent a quite distinct form of behaviour from the LCS.

2.6. Links with other areas

These three correlations: Cases A, B, andC, aresurprising both in

theirsimplicityand inthe fact donot seem tohavebeennoted previ-ously in the literature, either in theoretical treatments [15,10,11],

or in the prediction methods that are currently used in chemical

engineering [6,7]. They are evidently important from the practical

viewpoint for the noble gases in three ways: (a) they confirm the

internal consistency of the CC data-set (Table 1), (b) they show that

with Ne any QE are not significant in this area, so that this element

can be considered on a par with Ar, Kr, and Xe (Section 4); and (c)

they enable a rational estimate tobe made ofVc for Rn which isnot

otherwise well defined in the literature (Table 1).Although these correlations between experimental data are

therefore evidently both significant and useful, experience during

the preparation of this paper showedthat they tendedto be viewed

as purely empirical and liable to be dismissed as such by the

scientific community. In fact, the correlations may be viewed as

a quantitative example of the Periodic Law, which initially was of

course only empirical and without any theoretical background. It

can be shown [21] that similar correlations for the NG apply to the

comparable datafor the solid(crystal) state [22,23], and the param-

eters controlling the LennardJones (612) potential [24], but this

is outside the scope of the present paper.

3. Residual Volume Effect

This anomaly hasalready been discussedfor considered CC (Sec-

tion 2), where it is seen that the joint limits Pc0, Tc0, referred

to as zeronium (Ze), there is still a finite value for Vc. As already

noted, such an effect also applies [21] to the crystal data [22,23]

and to the LennardJones parameters [24]. It then amounts to an

overall average of 34% of the mean volume for the NG set NeXe.

From the molecular viewpoint, the volume parameter Vc (relat-

ing to repulsions between the molecules in close proximity) and

the temperature parameter Tc (which may be taken to be an

energy parameter relating to attractions between the molecules)

are dependent on the number of electrons in the molecule, increas-

ing as this number increases; in the case of the volume parameter

this is mitigated by the contraction effect because of increasing

nuclear charge. Thusin eachcase, the limiting situationcorresponds

to zero-mass point particles, where with zero number of electronsthere ought to be zero energy and zero volume; thus the zeronium

state does have a finite volume, and indeed without this anomaly

the state would not be noteworthy.

It is perhaps surprising that this RVE has apparently not been

noted before, or notbeen commentedon before. For its seems even

from a cursory examination of the data for the noble gases that

the volume values for the lighter elements are much higher than

they should be, judged from the values for the heavier members. A

similar general effect may be readily seen with the sizes of atoms

and monatomic ions in the solid state [25].

4. Quantal (quantum mechanical) effects (QE)

As already noted, the present phenomena are influenced byquantal (quantum mechanical effects (QE)), which are expected to

occur most markedlywith thelightestgasessuch as theHe isotopes

and to a much lesser extent Ne; they would also be expected to

occur with dihydrogen andits isotopic forms (H2,HD,D2) [5,10,26].

Such effects could be the reason for the deviations from normal

behaviour seen for the CC data in Figs. 13, although they appar-

ently disappearin the plot ofVc versus Qc in Fig. 4. In general terms,

the QE add a repulsive contribution to the EOS, leading the high

values ofVc observed for3He and 4He (Figs. 1 and 3).

These QE may be quantified in the present context in terms of

the thermal de Broglie wavelength at the critical temperature, c[27]

c h/(2mkBTc)1/2 (7)

where h is the Planck constant, kB is the Boltzmann constant, and

m is the molecular (atomic) mass. Putting this on a volume basis,

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the cube of this wavelength must be compared with the critical

molecularvolume Vc/NA (where NA is the Avogadro constant), with

theirratio giving a dimensionless quantal effectparameter defined

by:

NAh3/(2mkBTc)

3/2Vc (8)

For QE to be negligible, 1 (27). Using the values ofTc and Vcfrom Table 1 gives thevalues of forthe NG: 3He, 1.3; 4He,0.56; Ne,

0.03; Ar, 0.0001, with correspondingly smaller values for the higher

NG. Thevaluesfor thetwoHe isotopes are in line with thedeviations

seen in the CC plots (Figs. 13). However, with Ne the small value

of indicates [27] that the QE are not expected to be appreciable,which accords with theabsence of any markeddeviation forthis NG

in the correlation plots (Figs. 14). In this respect therefore, QE do

not haveas great aninfluenceon the CCbehaviour ofNe ashasoften

been assumed [5,10,26], and this supports the present observation

that the quartet NeXe behaves as a single group.

5. Conclusions

From the personal viewpoint, the present investigations startedout simply to try to clear up some anomalies in the literature

values for the sizes of gas molecules. This then became focussed

on the NG as the simplest fluids for which the molecular (i.e.,

atomic) size might be expected to be best established. This in

turn led to the linear correlations between the critical constants

for the NG presented here. For the CC area, the NG quartet: Ne, Ar, Kr, Xe, shows good lin-

ear correlations between the three critical parameters Tc, Vc and

Qc. These correlations are surprising in their simplicity and their

apparent novelty. The linearcorrelation between Vc and Qc has the immediateprac-

tical application of enabling the Vc of Rn to be estimated by

extrapolation.

The positive intercept on this same plot is the cause of theobserved downward drift in the critical compressibility factor Zcfor the group NeXe.

The helium isotopes 4He and 3He show the deviations in that

order from these correlations that may be ascribed to the

expected quantal effects (QE) from such light gases. Theabsence ofsuch deviations with thenextlightestmember, Ne,

indicates that the QE are not important in these CC correlations

andthatthe group NeXe may be considered as coherent a group;

this is contrary to a common perception. The correlation equations for NeXe provide a rational basis for

the display, comparison and correlation of the three CC values for

fluids in general. An important limiting case in these correlations is the extrap-

olation to limiting state of zero-mass point particles; this state

is made concrete by naming it as the element zeronium (Ze).

This is expectedly the zero limit for the temperature and volume

parameters. However, in this extrapolationthe volume parameter

showsa finiteintercept;thisis referred toas theResidualVolume

Effect. It leads to the paradox that the limiting state zeronium,

although it lacks any electrons, is still able to exert a repulsive

force at sufficiently small distances. This remains to be clarified

from the theoretical viewpoint.

List of symbols

CC critical constant(s)

h Planck constant (6.6261034Js)

kB Boltzmann constant (1.3811023J K1)

LCS Law of Corresponding States

Me methyl group (CH3)

NA Avogadro constant (6.0221023 mol1)

NG noble (rare, inert) gas(es)

Pc critical pressure (Pa)

Qc critical quotientEq. (2) (cm3

mol1

)QE quantal (quantum mechanical) effects (Section 4)

QSM quasispherical molecule(s)

R gas constant (8.3145 cm3 MPaK1 mol1)

RVE Residual Volume Effect (Section 3)

Tc critical temperature (K)

Vc critical molar volume (cm3 mol1)

Zc critical compressibility ratioEq. (1)

Ze zeroniumhypothetical limiting state of zero-mass point

particles

Greek letters

thermal de Broglie wavelengthEq. (7) (m) quantal effect parameterEq. (8)

References

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[2] P.W. Atkins, Physical Chemistry, Oxford Univ. Press, Oxford, 1978,pp. 3747.

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McGraw-Hill, New York, 2001, pp. 2.23 and 4.8.[27] Z. Yan, Eur. J. Phys. 21 (2000) 625631.

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