THERMODYNAMIC PROPERTIES OF HELIUM AT HIGHPRESSURE
March, 1938
255
Thermodynamic Properties of Helium at High Pressure BY WITOLD JACYNA
It is shown in the preceding papers1 that for helium, neon and hydrogen the equation of state - $ e 9 (1 - e?) vP = R$ ap is valid. v denotes here the volume (m3),p the pressure (kg./sq. cm.) and 4 the absolute temperature (OA.). The above equation for helium fairly covers the experimentally investigated region from 800 to 2’A. and from 0 to 16,000 atm. In the present paper, especially, the region of high pressure ($ 2 100 atm.) is investigated, and it is shown that the quantitative definition of the thermodynamic properties in all the regions of state can be made without any variation of the old2 constants. The equation is based on the so-called “conditions of reality” or on the “theorem of preliminary election"^ which points out that not all the mathematical arbitrary functions accompanying the solution of any differential equation can be used to define the structure of equation of state but only those which fulfil the above-mentioned cofiditions of reality. There are eventually v 2 0, I) 2 0, the “Planck-Hsusen’s criterion” ( & / 3 ~ )6~0, et^.^ Contrary to the molecular-kinetic point of view attacking the knowledge of the individual properties of isolated @article (atom, electron, etc.) the thermodynamic theory of equation of state deals with the matter in bulk regardless of internal structure of the body. The first theoretical explanation of this method is given by Lord Kelvin using the porous-plug experiment for vp-representation of actual gases. The further attempts in this direction are given by Rudolph Plank, Max Jakob, Eichelberg and others to obtain the equation of state for oxygen and steam. A very important opportunity for the convenient resolution of the problem of the thermodynamic equation of state has been given by the recent measurements of the compressibility of fluids and solid bodies accomplished in Cambridge (U. s. A*), Charlottenburg (Germany) and
+
(1) W. Jecyna, 2. Physik. 108, 61, 67 (1936); 106, 267 (1937); see also Phys. Rm.,61, 677 (1937),etc. (2) See, for instance, W. Jacyna, Z . Physik, 96, 246 (1936), ete. (3) W. Jacyna, Bull. Acad. Pel. Sci. Lcllres. A, a t . - N o v . , 376 (1934); also 2. Physik, 28, 133 (1924). See also ibid., 101, 77 (1936). (4) Starting with these conditions Levy’s prbblem [Me COmPt. rend., 87, 449, 554, 649 and 826 (1878)]on the universalityof the generalized form p = pp(u) t ( w ) of vad der Waals’ equation is resolved IW. Jacyna, 2.Physik, 100, 205 (1936)l.
+
Leiden (NetherIands), so #at this problem for helium now can be investigated successfully. Computation of the Exponents K and v At high pressure @I 2 100 atm.) we have e?2 0. Therefore the preceding equation for the ccmsidered region can be rewritten in the simpler form vp = R1/, + a p (aten - +e.) (1) where for helium al = l l ~ oa ,= 0.061085, R = 211.82 and #o = 273.22Oos The product vp ((‘volume-workJJ)cannot be negative, for 2 0. Hence from (I) a t J, = 03 it follows that R - a#ty & 0 (2) The minimum of the difference R-ape’ is defined by the condition
+
(3)
where the expohent v is to the iirst approximation proportional to p”, i. e. Y
From the relations YO
where e 2Z 2.7183 and for helium x = u0 d -0.01245, and we shall set ~p
(41
= vop
(a), (3) and (4)we have s - xe1 (6/R)X -i
l/(.6
-0.0125
(5)
Hence (6)
Now the values of the exponent (K),,ptl. for some temperatwe and pfessure can be computed from (I), i. e.
-”+ 0.0,01875 +e*
(Klexptl.
=
k P(VL@O)Istm.0.076631
(7)
where (v&)latrn. = 57,904 kgm./kg. is the “Amagat unit” for helium in the meter-kilogramsecond system and the ( ~ p ) , , ~values ~ , . are defined by Wiebe, Oaddy and HeinsJ7and also Brid&man’s8experimental data. Figures 1 and 2 represent a synthesis of several experimental data (6) &e W. J e y n d , Acta Phys. Pol., I, 410 (1934); 4, 243 (1935); also 2. Physik, 96, 262 (1936). and 97, 107 (1936),etc. (6) W. Jacyna. ibid., 103, 61 (1936). (7) R. Wiebe, V. L. Gaddy and Conrad Heins, Tms JOURNAL, 68, 1721 (1931). (8) P. W. Bridgman, Ptoc. A m . Acad. Sci.. 69, 173 (1924); 70, 1, 286 (1835); 71, 387 (1836); 74, 46 (1937); I. of Gcol., 44, 653 (1936); J . Chem. Phys., 3, 697 (1936); Phys. Rev., 48, 826, 893 . (19%); J . Applied Phys. (U.S. S. k.1, 8, 328 (1937); P Y ~Not. &ad. Sci., 33, 202 (1937),etc.
556
WITOLD JACYNA
VOl. 60 YO’
=
-0.01251
YO
(9) and K O = -0.00000145 The selection of values (9) has been made in order to approach more nearly Bridgman’s experimental data at pressure p 2 10,000 atm. and to simplify the use of the formulas. The results of calculation for some isotherms are given in Table I
1
0’ 0
5000 10,000 15,000 p, kg./sq. cm. Fig. l.-Graphical representation of the synthesis of several experimental data for helium at 65” (pressure in kg./sq. cm.).
In order to make the above-mentioned synthesis of the several data a t high pressure, i. e., to join the results of the measurement due to Wiebe, Gaddy and Heins (p 6 1000 atm.)-which I consider here as being correct-with Bridgman’s measurement and calculation (3000 4 p 6 15,000 atm.), I take the two following fundamental series of computations: (1) calculation of the thermodynamic properties under the assumption that Bridgman’s experimental data for p < 10,000 atm. are exact and (2) analogous calculation under the assumption that Bridgman’s experimental data in the years 1924 for helium are subjected to some systematic error, namely, in the lower range of the pressure values. The computations show that in the latter case the PlanckHausen’s criterion‘ and also the conditions C, 5 0 and C, 2 0 are satisfied in all the region of pressure to 30,000 atm., while in the first case the specific heat at constant volume already has become negative at 8500 atm. Thus the first assumption appears to be unwarranted; the computations show also that the exponent K is almost independent of temperature over the whole range of Wiebe, Gaddy and Heins’ experiments, so that it can be put down as K = KOp evo’p= l (8) where KO = -0.000001920, yo1 = -0.01655 and x = l/,. Practically without appreciable systematic deviation from the experimental data one can write
10,000 15,000 20,000 Kg./sq. cm. Fig. 2.-(up)~~-Data for helium in Amagat units; pressure in kg./sq. cm.
0
5000
In many cases the true value of volume, v, is necessary. From the equation of state (I) it follows that this value is composed additionally from the ideal gas volume Rd. = R$/# (10) and from the “volume-deviation” (total) Au = v
- Vid.
= a(llfi@
- $e.)
(11)
Hence, the volume can be represented by the sum V
=
Vid.
f AV
All these values diminish with the increase of pressure a t constant temperature, but the isobaric variations are positive for volume and negative for volume-deviation Av at all values of pressure. It is, however, remarkable that the volume-deviation at 100 atrn. and 0’ which is equal to nearly 5% of “ideal gas volume,” q d . becomes a t 15,000 atrn. and the same temperature 85% of the corresponding value of volume v and 500% of the corresponding ideal gas volume, vid. (see Table 11).
This fact gives an explanation of the final form of the equation without the ideal gas volume R $ / p for the condensed state. Gay-Lussac’s Coefficient and Specific Heat C, and C, and their Relations.-The simple relation can be found for the true Gay-Lussac coefficient where, in accordance with (I)
($),
=
- aey
THERMODYNAMIC PROPERTIES OF HELIUM AT HIGHPRESSURE
March, 1930:
557
TABLE I “VOLUME-WORK” v p OF HELIUM IN AMAGAT UNITS Calculated from equation (1) (calcd.) in comparison to the experimental data (exptl.) of Wiebe, Gaddy and Heins (from 103.3 to 1033.3 kg./sq. cm.) and of Bridgman (from 3000 to 15,000 kg./sq. cm.). The vp data in parentheses ( ) are doubtful; *Bridgman’s extrapolation of Keyes’ equation. r
9 , kg./sq. cm.
103’3
206.7 620.0
1033.3
{ { { {
Calc. Ca1cd.-exptl.
- 70
0
50
65
100
200
1.2345
1.289
....
1.417 05
1.781 05
1.2855
1.340
1.467
0
....
1
1.830 (2) 2.018 (3)
0.797 05
0.925 0
1.052 0
0.849
0.976
1.1035
0
0
0
1.048
1.174
0
0
1.300 0
1.234
1.358
Calcd. -exptl. ::;:::-expt1.
...
Calcd. -exptl. p , kg./sq. cm. Calc. t = 65”
t, o c .
-35
{
1.479 1 1.661
1.4835 0
0
3000 2.49 2.31 (2.88)*
0
5000 3.20 2.99 (3.72)*
and
1.5335
....
1.058 15
1.7155
1.840
,...
1 10000
7000 3.86 3.66
4.77 4.60 (5.73)*
... .,
2.193 (4) 30000 9.79
(4)
15000 6.16 6.11 (7.61)*
......
......
the preceding paperlofor helium at 100 kg./sq. cm. ?so
$O =R -
P
+ a$o (Ilex - ev)
c(t) =
depends in the temperature range from -70 to ZOO0 on the pressure only; hence 1 1 laper = a, =
-11
m
= 1.251
- 0.1194
2’ (3 - 5 3 (A)-‘
Hence, generally, a t high pressure it can be put C, = 1.251
-
R + ap(lleK - e v ) q0 where am = (v-oo)/mt denotes the mean value of the Gay-Lussac coefficient, and t the temperature measured from 0’. Therefore the Gay -Lussac coefficient for helium diminishes uniformly a t room temperature with the increase of pressure, but at sufficiently low temperature and pressure p < 50 kg./sq. cm. an increase of a and a,,,can be seen.9 cy
Cp
- 0.1194 (3 - 5$)(&)-‘
(12)
The specific heat a t constant volume, C,, on the contrary, varies with the pressure on the isothermal line most appreciably. The values of C, C,,, C, and C,/C, = K calculated with the equation
where, as passim, A = 1/42’7, #o = 273.22 and for
TABLEI1 helium R = 211.82-are assembled in Table 111. COMPARISON OF TEE VALUES OF SPECIFIC VOLUME v AND OF “VOLUME-DEVIATION” Au IN CC./RG. COMPUTED The difference of the two specific heats at high pressure increases with the increase of temperaFROM EQUATIONS (1) AND (11) c I , OC. ture a t p = const. but uniformly diminishes with p, kg/sq. cm. - 70 0 100 200 103.3
1 :::lv 1
446 30.1
104v 1033‘3 IO‘ Av
30’000
i
104 D IO‘ Av
587.5 29.6
69.1 27.5
82.9 27.15
18.25 16.8
18.6 16.6
794 28.9
998 28.1
103 26.6
123 26.05
19.1 16.45
19.6 16.25
The above discussion concerning the GayLussac coefficient shows that a t high pressure in the region of temperature considered the specific heat a t constant pressure, C,, depends on the temperature alone. Hence, C, at high pressure is equal to C,(t) a t the lower limit of the considered interval of pressure. I have calculated in (9) W . Jacyna, 2. Physik, 95, 692 (1935).
increasing pressure at t = const. This shows that the specific heat at constant volume, C,, increases with the increase of pressure and a t high pressure, as follows from Table 111, is nearly equal to the specific heat, C,, so that the ratio C,/C’ = K approaches unity at the lower temperature limit. JouleKelvin and Joule-Gay-Lussac EfTects.Both effects for the pressures 100 kg./sq. cm. are discussed in the preceding papers“ in con(10) W.Jacyna, ibid., 92, 661 (1934), where the formula (21)must be expressed as C(1)
- (3
- &).
-
1.2610 C 0.1194 p
(11) W.Jacyna, refs. 9 and 10.
(&)
-4
where p
-
558
Voi. 60
WITOLD JACYNA
TABLE I11 SPECIFICHEATAT CONSTANT V O L (C,) ~ AND RATIO C,,/C, = K, AND TIE DIFFERENCE C, C,,CQYPUTED BY THE HELPOF EQUATION5 (12) AND (13) ,oc-.
-
p , kg./ sq. cm.
103.3
1033.3
1
1
-
c, Cp;
c,
co
c.
CP;
CO
15,000
14;
cv
30.000
/ c p ;
c,
70
0.786 ,491 1.625 0.828 .4495 1.54 1.138 0.139 1.12 1.206 0.071 1.06
0
200
0.764 .491 1.64 0.800 .455 1.57 1.091 0.164 1.15 1.168 0.0875 1.075
0.760 .491 1.65 0.789 .462 1.585 1.040 0.211 1.20 1.127 0.124 1.11
nection with the investigations of Roebuck and Osterberg.I2 The recent consideration18 on the thermodynamic theory of these effects shows that for helium above 50 kg./sq. cm. at any temperature the Joule-Kelvin effect pj must be negative; the negative values of Joule effect pu can be observed, however, a t the temperature $ > 200'A. But in the region of state considered both effects are negative. This conclusion is confirmed by the new calculation (Table IV) accomplished with the help of the equations
TABLE IV J O U L E - ~ L V I NEFFECT (pj) AND JOULE-GAY-LUSSAC EFFECT (p,) FOR HELIUM AT HIGHPRESSURESCOMPUTED FROM EQUATIONS (14) AND (15) IN "C./(RG./SQ. CM.) 9 . kg./aq. cm.
103.3 1033.3
i i
-1O'pi -1O'p, -1O'pj -1O'p.
15,000
{ -108pj
30,000
i
-1O'p. -1O'pj
-1O8pCc,
P
-70
58 6 52.5 9 36.5 11.5 31.5 10
i
, 0c.0
59 7.5 53.5 10.5 37
12 32 10.5
200
59 12 53.5 13 37.5 13.5 32.5 11
upon temperature at high pressure generally cannot be neglected. Perhaps the variability in some cases (coefficient of expansion, Joule-Kelvin effect, etc., see for instance the formulas 13, 14 and 15) is joined to the variability of the negative volume-deviation with the temperature. Nevertheless it seems to me that the exact estimation of this doubtful effect is not yet possible in the present state of experimental knowledge, and that the above thermodynamic synthesis gives, therefore, a result which has not been, so far as I know, attained with such a great quantitative adequacy, as is given here.
Summary Starting with the thermodynamic equation of state for helium-which covers the whole range and of experimental data of Holborn and Otto, Kamerlingh Onnes and Boks, Keesom and Kraak, Bridgman, Wiebe, Gaddy and Heins, and others where p j = 14.% = (st/@),; u denotes the from 2 to 800'A. and from 15,000 kg./sq. cm. to internal energy and j is the enthalpy. zero-are given (1)the tabular and graphical comThe absolute values of the Joule-Kelvin effect and synthesis of several experimental wpI p j decrease uniformly with the increase of pres- parison data at high pressure, (2) the thermodynamic sure but these values of the Joule-Gay-Lussac efproperties of helium: the "volume-work" vp, the fect I pu 1 attain a maximum approximately at specific volume v and volume-deviation Av, co7000 kg./sq. cm. (Table IV). A small increase of ejicient of expansion a,specific heat C,,C,, C,both effects with the increase of temperature a t C, and the ratio C,/C, = k , Joule efect pu and high pressure can almost be disregarded. But Joule-Thomson efect pj, which all are computed from the other extrapolations it results that the in the region from -70 to 200'C. and from 103.3 dependence of some thermodynamic properties to 30,000 kg./sq. cm. and respectively tabulated.
I
(12) J. R. Roebuck and H.Osterberg, Phys. Rev., 48, 60 (1833); ibid., 4S, 333 (1934). (13) W. facyna, loc. cit.; also 2. Phyrik, 96, 119 (1936).
INSTITUTE OF METROLOGY RECEIVED FEBRUARY 3, 1937 LENINGRAD, U. S. S. R.
