The molar volume and expansion coefficient of liquid He4

ANNALS

The

OF

Molar

PHYSICS:

26,

292-306

Volume

and

EUGENE University

(1964)

of California; Los

Expansion

C. KERR Alamos

Coefficient

AND R. DEAN Scientijk

Laboratory,

of Liquid

He4*

TAYLOR Los

Alamos,

New

Mexico

The density of liquid He4under its own vapor pressurehasbeenmeasured over the range0.5-2.8”Kwith a precisionof about 0.006%. A direct pycnometric method was used; the cell was immersed in a liquid Hea bath. Detailed measurements in the neighborhood of the lambda point temperature showed the density maximum to occur O.OOG”K above Z’i . Analysis of these data and other available data has yielded analytical equations from which the coefficient of expansion has been determined. Below 2.14”K the Landau formulations of He II was found to represent the data. In the neighborhood of Ti the density changes logarithmically in ( T - !fh /. Above 2.2OK all available data are fitted as a function of the He4 vapor pressure. The behavior near Tx is analyzed in terms of the theory of Buckingham and Fairbank. I. INTRODUCTIO&

The molar volumes or expansion coefficients of liquid He4 along its saturated vapor pressure curve have been measured with varying degreesof precision over the past several decades.Some of these measurementshave been made by direct observation of the volume-temperature relationship (1-3), some have been obtained from the extrapolation of high pressure PVT data (4, 5), and some have been derived from observations of such related quantities as the dielectric constant and the refractive index (6-8). While the agreement among these various data is generally quite good, the small differences that do exist are still significant in certain thermodynamic relationships. In some regions the data are incomplete or are lacking in detail. In particular, data obtained in the theoretically important region near the X-point have in many casesbeen influenced by the behavior of the surrounding He4 bath at its X-point. The availability of a He3 cryostat, designed originally for measurements on He3 molar volumes (9), presented us with a unique opportunity to investigate this important region around the X-point in a system free of the erratic temperature gradients in the He4bath. In addition, the temperature region of 0.3”-1.2”K, in which few density measurements had been made, was accessible with this * This paper is based on work performed under University the United States Atomic Energy Commission. 292

of California

contract

with

LIQUID

He4:

EXPANSION

COEFFICIENT

393

equipment. The relatively high precision of volume-temperature measurements obtainable from the dilatometer cells used (as indicated by the results of the He3 studies) allowed a detailed study of the lambda region in He* and of the region of a density minimum near 1.17’K. The results obtained from the present investigation as well as comparisons with other studies will be presented in three sections-the low temperature region, the lambda region, and the high temperature region-since the methods of data treatment and interpretation vary considerably from region to region. II.

APPARATUS

AND

MEASUREMEKTS

The He3 cryostat and associatedmeasuring equipment were the same as were used by Kerr and Taylor (9) for measurements of the molar volume and expansion coefficient of He3. This paper may be consulted for detailed information on the cryostat, the dilatometer cells, gas measuring system, temperature measurement, methods, and reduction of the data. Temperatures, in the present case, were based on a He” vapor pressure thermomet,er in terms of the temperature scale, T 62, proposed by Sherman ef al. (10). However, in order to preserve consistency in the comparison of the present results with the data of other investigations, all temperatures are referred to the TSR scale of Brickwedde et al. (12 ). In all, 151 equilibrium volume vs. temperature observations at, saturated vapor pressure were made during the course of four separate series of runs. In addition a large number of measurements of a semiquantitative nature were made at various times in order to study certain aspects of the performance of the equipment). III.

LOW

TEMPERATURE

REGIOP;

In the temperature region 0.4’-214°K some 75 volume-temperature observations were made. A special effort was made to obtain data below 1.0% since very few observations have previously been reported in this region. Although over twenty volume-temperature measurements were obtained at these low temperatures, the results were disappointing. At temperatures below about O.VK, oscillations of the liquid meniscus of varying frequencies and amplitudes prevented precise determinations of the meniscus height. From 0.6’ t’o O.S’K, these oscillations were so intense that no menisci could be observed at all. Accompanying the oscillations was a vastly increased heat influx to t’he cell system which occasionally prevented pumping of the He” bath to lower temperatures. On the few occasions when special pumping bechniques permitted access to the region below 0.6”K, the observed volumes were erratic and were apparently too low by 0.02-0.03 %I, indicating that some helium was being held in the upper capillary either as liquid or as a thick refluxing film. The uncertainty in these

294

KERR

AND

TAYLOR

measurements was of the same order of magnitude as the total volume change below l.O”K, and consequently the measurements were considered to be of dubious value for studies of the shape of the volume-temperature curve. Values for the molar volume and expansion coefficient below l.O”K can be obtained, however, by use of the data above l.O”K for normalization of a theoretical functional dependence of the expansion coefficient on temperature. Landau (12,lS) has shown that the entropy S of liquid helium, at sufficiently low temperatures, may be written as the sum of the contributions of the phonon and roton excitations and has derived expressions for the separate entropies. Thus,

s = Sph-I-ST s

ph

(1)

= 2s’ ii4 T3 45fi3 c3 p

ST =

2p1’2 Icl”T’P” &)3/z

(2) * (1 + 3kT/2A)

h3

exp (-~/lcT)

where k, fi, p, and T have their usual meanings, c is the phonon velocity (equal to the velocity of first sound at T = O”K), and p, p, , and A are parameters of the energy vs. momentum curve in the region of the energy minimum. In principle, the constant pressure expansion coefficient at various temperatures could be evaluated from these equations by differentiating them with respect to pressure and summing the roton and phonon contributions. However, the temperature dependence of several of the parameters and, in particular, of their pressure coefficients are known with insufficient accuracy for such a procedure. Atkins and Edwards (2) have described a method for normalizing the resulting aP equations to experimental entropy and expansion coefficient data in regions where accurate data are available, without destroying the essential characteristics of the Landau model. We employ their procedure, utilizing the data of this research and other more recent data, in evaluating the normalization parameters of the Landau formulation. The calculations of the phonon contributions to the entropy and expansion coefficient are straightforward, and we evaluate these with the same data as used by Atkins and Edwards. They express the roton contributions in the form:

in which a and b are functions of the pressure derivatives of p, p, , and A. It is assumed that the variations in a and b are sufficiently small that they can be treated approximately as constants. If this assumption is justified, a plot of the left side of Eq. (4) against the temperature function on the right-hand side

LIQUID

He4:

EXPANSION

COEFFICIENT

295

should yield a straight line over a significant temperature range. In order to make such a plot, cyr values were obtained by subtracting the CQ,~contribution from the constant pressure expansion coefficients derived from the volumetemperature measurements of this research. The velocity of first sound, u1 , at various temperatures was selected from a smooth curve representing the experimental results of Van Itterbeek et al. (1 4, 15) and Laquer et al. (16). The parameters P, p, , and A were initially taken from the neutron scattering data of Yarnell et al. (17) who obtained (at l.l”K) P = 0.16 mHe, pO,/fi = 1.92 A-‘, and A/k = 8.65”K. However, in order to effect agreement with the experimental entropy at 1.1% it was necessary to assign p = 0.19 mHe The parameter A/Ii at temperatures other than l.l”K, was calculated from the relation A/k = 8.68 - 0.00842”“K also given by Yarnell et al. (17). Bendt ef al. (18) give several other functions for the temperature dependence of A/k which we also investigated in an attempt to improve the over-all fit to the experimental entropy at temperatures above about 1.8”K. The improvements obtained were not significant within the framework of the empirical method being employed here. For the experimental entropy, htkins and Edwards used values derived from Hercus and Wilks (19) which appear to be about 10 % too high in comparison with the entropy derived from the specific heat measurements of Kramers et al. (i?~), Hill and Lounasmaa (21 ), and Lounasmaa and Kojo (22). Accordingly, we used entropies derived from the latter sets of measurements. The resulting plot of Eq. (4) is shown in E’ig. 1 along with the similar results obtained by Atkins and Edwards (2). The vertical shift between the two curves results primarily from the different entropy values used in the two calculations. The change in slope and the better fit at higher temperatures for our curve is mostly a consequence of including the temperature dependence of A/k in Eqs. (3) and (4), although some additional effect results from the differences between the CY,‘s used in the two cases. The coefficients, a and b, of Eq. (4) resulting from the straight line of Fig. 1 are -0.835 and +0.383, respectively and can be compared to the corresponding values of - 1.95 and +0.57 of Atkins and Edwards (2). Once these two coefficients have been determined, the above process can be reversed and o(~and aph calculated and combined to give the total expansion coefficient, an , over the entire range of interest. It should be noted that, of necessity, all calculations have been performed with reference to the constant pressure expansion coefficient, although it is the expansion coefficient at the saturated vapor pressure, (Y,~, which is directly related to our experimental data. The difference between these two expansion coefficients is often neglected or is calcu lated in an approximate manner. Goldst’ein (23) has given thermodynamically rigorous equations for obtaining C, , C,. , and CY~from the properties observed along the saturated vapor pressure curve such as p, (Y* , C,s , 7(1, and (dp/dT),;,,. .

296

KERR

AND

TAYLOR

X

FIG. 1. Plot of Y vs. X according from Atkins and Edwards.

to Eq. 4. Open circles from this research. Solid circles

These relationships have been used throughout this paper whenever such interconversions have been necessary. Integration of the calculated CY.vs. T curve was used to obtain the excess molar volume with respect to the reference volume of 27.5864 cm3 mole-’ at 1.172”K. The resulting V, vs. T curve from 0°K ( V0 = 27.5793 cm3 mole-‘) to the X-point is shown by the solid line of Fig. 2. For comparison, the experimental molar volumes from this research as well as some directly observed and derived volumes from other sources are included. At temperatures above about 1.8”K, a gradual deviation between the volumes obtained from the Landau model and those obtained experimentally is observable in Fig. 2. This difference may be ascribed partially to the increasing effects of phonon-roton interacti0n.s and partially to approximations made in the normalization of the Landau model to the experimental data. The “smoothed” values of the molar volumes and expansion coefficients as given in Table I for the region below the X-point use the results of the Landau model calculation where applicable, but in the higher temperature region they are obtained by graphical and analytica. smoothing, through the experimental

LIQUID

He4 :

EXPANSION

297

COEFFICIEXT

18

19

eo

FIG. 2. The molar volume of He4 below the lambda-point. The solid line is based on t.he Landau model normalized as discussed in Section III. The lightly dashed line (- - -) is based on a graphical integration of the expansion coefficient data of Atkins and Edwards. The heavier dashed line (- - -) near the lambda-point shows the best fit, of the present, results.

data, to match the analytical expression which applies very close to the h-point (see Section IV). The average deviation of the results of this research from the smoothed values of Table I is 0.006 % for the temperature range l.0”-2.14°1i, with a maximum deviation of 0.026 5%. The extrapolated isopycnal data of Edeskuty and Sherman (5) are, on the average, 0.09 !Z, lower than the present smoothed data in this region. Jlolar volumes obtained by normalization at 1.2”K and graphical integration of the expansion coefficient data of Atkins and Edwards (2) are higher than the present results by less than 0.01% up to 1.75”K but deviate up to 0.03 5, at l.YK and by nearly 0.06 70 near the X-point. The results of this graphical integration are shown by the dashed line in Fig. 2. The over-all agreement could, of course, be improved by normalization at an intermediate temperature. The refractive index

298

KERR

AND

TAYLOR

TABLE SMOOTHED

VALUES

OF THE MOLAR

T(“K)

Vm

0.00 0.05 0.10 0.15 0.20 0.25 0.30

27.5793 27.5793 27.5793 27.5793 27.5793 27.5793 27.5794

0.000 0.000 0.001 0.004 0.009 0.017 0.028

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40

27.5794 27.5795 27.5796 27.5798 27.5800 27.5803 27.5807 27.5811 27.5816 27.5822 27.5829 27.5836 27.5843 27.5850 27.5856 27.5861 27.5863 27.5863 27.5859 27.5851 27.5838 27.5820

1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80

27.5795 27.5761 27.5719 27.5667 27.5603 27.5536 27.5450 27.5352

a x

103

VOLUMES W’K)

I

AND EXPANSION Vm

01 x

103

COEFFICIENTS

OF LIQUID

T(OK)

V,,,

He4

01 X 103

1.85 1.90 1.95 2.00 2.05 2.10 2.15

27.5237 27.5106 27.4953 27.4787 27.4587 27.4352 27.4069

-8.71 -9.98 -11.45 -13.2 -15.5 -18.4 -25.4

2.60 2.65 2.70 2.75 2.80 2.85 2.90

27.7341 27.7942 27.8581 27.9259 27.9978 28.0738 28.1538

42.0 44.6 47.3 50.0 52.8 55.6 58.6

0.046 0.068 0.098 0.134 0.180 0.227 0.280 0.343 0.408 0.461 0.496 0.514 0.505 0.465 0.386 0.261 0.094 -0.137 -0.416 -0.754 -1.14 -1.61

2.16 2.162 2.164 2.166 2.168 2.170 2.171 2.1715 2.1719 2.1720 2.1721 2.1725 2.173 2.174 2.176 2.178 2.180 2.182 2.184 2.186 2.188 2.190

27.3993 27.3976 27.3958 27.3940 27.3920 27.3897 27.3886 27.3879 27.3872 27.38704 27.3869 27.3867 27.3865 27.3862 27.3860 27.3859 27.3860 27.3862 27.3864 27.3868 27.3872 27.3876

-30.0 -31.2 -32.8 -34.9 -37.9 -43.0 -48.1 -53.1 -64.9 --m -29.6 -17.8 -12.7 -7.63 -2.55 0.42 2.53 4.17 5.50 6.63 7.61 8.48

2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

28.2382 28.3266 28.4192 28.5156 28.6232 28.7312 28.8441 28.9620 29.0847 29.2125 29.3450 29.4829 29.6252 29.7729 29.9257 30.0844 30.2467 30.4151 30.5889 30.7682 30.9533 31.1443

61.6 64.6 67.6 70.7 73.8 76.9 79.7 83.1 86.1 89.1 92.1 95.1 98.0 100.9 103.8 106.8 109.6 112.7 115.4 118.5 121.5 124.6

-2.13 -2.72 -3.38 -4.03 -4.80 -5.71 -6.57 -7.59

2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55

27.3904 27.4136 27.4471 27.4867 27.5296 27.5757 27.6250 27.6777

11.71 21.5 27.0 30.1 32.3 34.6 36.9 39.4

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40

31.3413 31.5476 31.7542 31.9704 32.1932 32.4231 32.6402 32.9049

127.7 131.1 134.1 137.3 140.6 144.0 147.6 151.0

data of Edwards (7) are also subject to normalization since, within limits, the selection of the molar polarisability is somewhat arbitrary. With a molar polarizability of 0.12435 selected, his data agree very well with the present results, as shown in Fig. 2, except at the lowest temperatures where the deviation reaches a little over 0.02 %.

LIQUID He4 :

EXPANSION

289

COEFFICIENT

The earlier data of Kerr (3) in this region are omitted since his data in this region appear to be consistently low by about 0.3 %, although his data at higher temperatures (see Section V) do not show this discrepancy. After Edeskuty and Sherman (5) pointed out the existence of an apparent discontinuity in Kerr’s data, a careful examination was made of the raw data and computations, but no significant clue as to the origin of the discrepancy was found. IV.

LAMBDA-POINT

REGION

Three series of measurements were made, comprising a total of approximately 60 observations, during which the helium meniscus was observed continuously while the temperature was varied in small increments from about 2.14’K to about 2.21”K. About one-third of these observations were made within 2 mdeg of TX . The results are shown in Fig. 3. A similar curve which included the results of the first of these series of experimental observations has been published previously (24, 25). Small differences between the earlier curve and Fig. 3 are the result of the application of minor calibration corrections and of a resmoothing of our temperature scale in this region. The precise value chosen for the lambda temperature corresponds to the point of intersection of the slightly extrapolated volume-temperature curves below and above the transition temperature. Tk is defined with reasonable precision because the He II curve tends to infinite slope. The resulting X-point temperature coin I

I

I

0 SERIES . SERIES v SERIES

-

I 5 FIG.

represent

I

2.16

I

2.17

I

2.18

2.19

TEMPERATURE

t-K)

I

I

II P l!Z

u

3. The molar volume of He4 in the vicinity of the lambda-point. the best least squares fit of the data according to Eq. (5).

:2

The

solid

lines

300

KERR

AND

TAYLOR

tides, to within the experimental accuracy, with that given by the Tss temperature scale. In the subsequent analysis TX is taken to be 2.1720”K. The most striking feature of Fig. 3 is that the volume minimum, at a temperature a little less than 6 millidegrees above the X-point, is clearly resolved in spite of the fact that the total volume change is only 0.0011 cm3 mole-’ or 0.004 % between the X-point and the minimum. The sensitivity of the dilatometer was 0.008 volume per cent per millimeter of height (9) and the meniscus height was observed to 0.02 mm. The temperature of the volume minimum corresponds, within experimental accuracy, to that obtained by Chase et al. (8) from dielectric constant measurements but is higher than the l-2 mdeg shift reported by Edwards (7) from refractive index measurements. The experimental data within about 0.030”K of the X-point can best be expressed by the equation: log I’, = -4 + B 1 T -

Tx 1 + C 1T - TX 1log 1T - TX 1

(5)

Least squares solutions of the experimental volume-temperature data, treating the regions above (I) and below (II) the X-point independently, gave the following values for the coefficients: AI = AIi = log I’& = 1.4375411, B,, = 0.002102,

2% = 0.013287,

CI = 0.007331

C*r = -0.007313.

The resulting expansion coefficients along the saturated vapor pressure curve are 103a,r = 37.92 + 16.88 log (T - TX) 103as~~ = 2.47 + 16.84 log (Tx -

T)

(6) (7)

Figure 4 shows the expansion coefficients of He4 near the X-point as derived from the data of this research as well as from the data of other sources (2, 7, 8). The points shown on this graph represent expansion coefficients obtained by differentiating the original volume, dielectric constant, or refractive index data at selected temperatures. Each set of data showsthe samegeneral characteristics, i.e., an asymptotic approach to a logarithmic function near the X-point and a step function displacement between the values of the expansion coefficients above and below the X-point. However, the slopesand displacements resulting from the several experimental approaches can be seen to differ by significant amounts. Buckingham and Fairbank (25) have derived several relationships between observable thermodynamic quantities which are valid in the region of a transition characterized by the absenceof a latent heat, but at which the constant pressure heat capacity becomes infinite. The particular relationship of interest here is that, between the specific heat and the expansion coefficient (25),

(8)

LIQUID

FIG.

4. The

expansion

coeffcient,

He4:

EXPANSION

01~ , near

the

301

COEFFICIENT

X-point

as a function

of log 1 I” -

2’~ 1

where t = T - Tx(P). Since this equation applies with increasing rigor as the transition temperature is approached, a plot of C,TJT vs. a,px/p should give a curve which asymptotically approaches a straight line near the transition temperature. Such a plot is shown in Fig. 5. Here, the C, values were calculated from the C, data given by Buckingham and Fairbank (25). The expansion coefficients 0~~were obtained by applying appropriate corrections to our deduced cyBvalues. It can be seen that the data from the two sides of the transition temperature each approach and follow a straight line for an appreciable distance away from the transition temperature. In the linear region the displacement of the two straight lines reflects the discontinuity at the X-point of the specific heat (25) as well as the discontinuity in the expansion coefficient [Eqs. (6) and (7)]. Both lines in the present case have slopes from which (dP/dT), = PA’ , the limiting slope of the lambda-line, is equal to - 118.4 atm “K-l. The intercepts at a= = 0 give TA( dS/G’T) t , equivalent to C, in the Pippard relations (dG), equal to 6.38 joules gm-’ OK-’ above TX and 5.34 joules gm-’ ‘Ii-l below Tx . Chase et al. (8) analyze their data above and below Th in terms of a single straight line, implying not only the same slope but the same intercept C, . However, when their data are plotted in the same manner as above, two definite parallel lines of slope -95.4 and Co’s of 6.62 and 6.26 (in the same units as above) are obtained. The data of Atkins and Edwards (2) are consistent with the param-

302

KERR

AND

TAYLOR

t 01 -0.06

-0.06

-a04

-a02

0

a02

0.04

(fAwp(*K-‘)

Fig. 5. Parametric plot of the specific heat of He4 as a function near the X-point. Values of 1 T - 2’~ 1 are indicated.

of the expansion

eters proposed by Buckingham and Fairbank, viz., PA' =

-130

coeffjcient

f

10 and

C, = 5.0 f 0.6. Very few direct determinations of the X-point as a function of pressure have been made. Lounasmaa and Kaunisto (27) and Swenson (28) have measured the X-point for a number of pressuresabove 7.8 atm; extrapolation of these data to the saturated liquid indicate an initial slope of -98 atm OK-l. Keller and Hammel (29) have observed several PA , Tx points at pressuresbelow 800 mm Hg in connection with another experiment and have obtained values which tend to support a limiting X-line slope of about - 120 to - 130 atm ‘K-l. The wide range of values for PA' obtained by these various approaches indicates the need of better direct measurements of this quantity. The existence of the two parallel lines as shown in Fig. 5 may be real, or it may be a spurious effect resulting either from small errors in the specific heat and/or expansion coefficient data or from methods of treating the original data to obtain these derivative quantities. The slopes, intercepts, and separation (if any) of the asymptotic straight lines such as shown in Fig. 5 are related directly to the coefficients of the equations which expressthe specific heat and expansion coefficients in terms of log ( T - TX I. These coefficients in turn are very sensitive to small variations in the original data, to the selection and weighting of the original data points, and to the particular analytical methods used to obtain the derivatives of the observed quantities.

LIQUID V.

He4:

EXPANSIOS

HIGH-TEMPERATURE

COEFFICIENT

303

REGION

In this series of measurements, fifteen volume-temperature measurements were made in the region 2.2’-2.8’K. Observations were not extended to higher temperatures because the uncertainties in the gas space corrections were heginning to exceed the general precision of the experimental method. Also, the He” cryostat was not designed for carefully controlled operation at much higher temperatures. Several types of analytical formulations were investigated not only to find the best representation for the present results but also to incorporate the results of other measurements at higher temperatures into a consistent equation. The best, and also the simplest, representation found for the available data in the range 2.2”4.4’K gives the molar volume as a linear function of the He4 vapor pressure ( 19.58 scale) (11) with a small exponential correction term to correct systematic deviations at the higher temperatures. Thus: l’,,(cm”

mole-‘)

= 27.1040 + 0.0067224P

(P in mm Hg at 0°C and standard

-

1.692 X 10-‘4””

(9)

gravity)

The linear portion of this equation was derived solely from the data of this research, while the exponential portion was obtained by evaluation of deviations from this straight line of the available data from other direct and derived density or volume measurements. As was shown in Section IV, the data immediately above the X-point is best expressed as a function of log (T - TX). The short region (2.22”-232°K) between these two analytical representations has been smoothed by graphical and analytical methods to complete the tabulation of “smoothed” values given in Table I. Deviations from the linear portion of Eq. (9) of the data of this research as well as the data from various other sources are shown in Fig. 6. The contribution of the exponential term is shown by the heavy broken line. The parallel dashed lines represent a region of 3~0.2 % in the molar volume and contain over 85 5% of the available data points from all sources. It may be noted that the contribution of the exponential term to the molar volume is less than 0.01% at temperatures below about 3.4”K. The data points of this research show an average deviation of 0.011% from the smoothed curve. The eighteen points obtained by Kerr (,3) in the region of 2.2”-4.4”K have an average deviation of 0.06 %. Equation (9) gives a better representation of his data than the equation given in his original paper. The molar volumes obtained by Edeskuty and Sherman (5) from the extrapolation if isopycnal measurements are on the average about 0.25 % lower than those given by Eq. (9). Similar extrapolations from isopycnal data by Keesom and Keesom (4) are about 0.27 % too high in the same respect and in addition show some systematic trend with temperature. Onnes and Boks ( l), who made t,he

304

KERR

AND

TAYLOR

0 0 $ $ 0

THIS RESEARCH KERR ONNES (L SOKS EDWARDS CHASE, MAXWELL. 8 MILLET1 l GRESENKEMPER 8 HAGEN P EDESKUTY a SHERMAN . KEESOM B KEESOM

-0.1

e2

’ 2.4

2.6

I 2.6

I 3.4 SO 3.2 TEMPERATURE

I 3.6 (‘Kl

16

I 4.0

4.2

I 4.4

FIG. 6. Deviation of the experimental molar volumes of He4 from an equation of the form V, = a + bP4 , where P4 is the vapor pressure of He4. The heavy dashed line shows the contribution of the exponential term in Eq. (9). The envelope shown by the lightly dashed lines represents a region of ~kO.2% from Ey. (9).

earliest measurements of the He4 density, obtained results which differ only 0.09 % on the average from Eq. (9)) and most of this deviation is contributed by the two points above 3.8”K. Densities or molar volumes obtained indirectly by calculation from refractive index or dielectric constant measurements cannot be compared in the same manner as above since these require the assumption of a constant molar polarizability, which must be chosen either by normalization to a density at some selected temperature or by assuming the best average value with respect to a selected density curve. The dielectric constant measurements of Grebenkemper and Hagen (6), when converted to molar volumes by assumingan average molar polarizability of 0.1230 cm3 mole-‘, show an average scatter of 0.18 % about Eq. (9). Similar measurements of Chase et al. (8), calculated using an average molar polarizability of 0.12312, show an average deviation of 0.09 % and exhibit a definite trend with temperature. Molar volumes derived from the refractive index data of Edwards (?‘), using his original normalization at 3.7”K, are up to 0.35 % higher than the present results and again show a systematic trend within this range. However, if his results are recalculated, using a polarizability of

LIQUID

He4:

EXPANSION

COEFFICIENT

305

0.12427, the average deviation is reduced to 0.11 %, although the systematic deviation is still apparent (see Fig. 6). There thus appears to be a real, though small, systematic difference between the molar volumes measured directly and those derived from optical or dielectric measurements. This may result from small systematic errors in any of the original observations, from lack of true constancy of the poalrizability of He4, or from the inapplicability of the Clausius-Mosotti or Lorenz-Lorentz formulas to within the degree of precision considered here. At the present time, the available data do not permit an unequivocal decision as to the source of the discrepancy. RECEIVED:

*June 17, 1963

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