XN EQUATIOX OF STATE; FOR E T H Y L E S E GAS* B Y LOUIS J. GILLESPIE
I n the present paper the equation of state of Beattie and Bridgeman' i + utilized t o smooth and to correlate, as far as is possible, the pressure-volurrrr,temperature data for ethylene gas. The available data are those of Amagat? which extend over a large range of pressure and temperature into the liquid phase; the 24.95' isotherm of Masson and Dolleyj3and about four points o n each of four isotherms by hlathias, Crommelin and Watts.' Data on thf: normal volume of ethylene at oo and I atmosphere are collected and reviewved by Blanchard and P i ~ k e r i n g . ~ KO equation of state has apparently been presented for ethylene, with the exception of virial p v expansions. Aside from the general usefulness of equations of state, especially for gases of considerable chemical activity, an equation for ethylene is especially desirable in view of the work on mixtures of ethylene and argon by Masson and Dolley, which have permitted the calculation of partial molal free energies of the constituents of these binary mixtures.6 The present work was in fact undertaken with a view toward a more complete thermodynamic investigation of these data than has been hitherto possible. Interpolation of the Isothermals For each temperature the pv products of Xmagat were graphically smoothed as a function of the density, using an appropriate deviation function and large scale coordinate paper, and values of the pressure were interpolated for even values of the density in moles per liter (0.5, 1 . 0 ~1.5, etc.). The transfer from Aniagat units of density to moles per liter was effected through the normal density given by Batuccas' and chosen by Blanchard and Pickering,; namely 1.2604 grams per liter. By a similar procedure, the data of hIasson and Dolley and of the Leiden laboratory were interpolated to the same even density values, thus permitting a comparison of data. In the former case the transfer of density units required a knowledge of the ratio of'the volume a t 24.95' to that a t o o , both volunies a t I atmosphere. From Xmagat's data this ratio, V,,/Vo, was calculated to be 1.0936, from >\lasson and Dolley's data, 1.0932. The mean
* Contribution from the Research Laboratory of Physical Chemistry, Massachusetts Institute of Technology, S o . 216. J. A. Beattie and Oscar C . Bridgeman: J. Am. Chem. Soc., 49, 1665 (1927). Ann. Chim. Phys., 16) 29, 68 (1893). Ppoc. Roy. SOC.,103-4, j24 (1923). 4 Cinquihme C0ngrt.s International du Froid, Rome; Premiere commission internationale de 1'Institut Internationale du Froid, Rapports et Communications. Leiden, 1928. 5 Sci. Paper U. S. Bureau of Standards, S o . j29 11926). Gibson and Sosnick: J. .im. Chem. Soc., 49, 2 1 7 2 (1927). 7 J. Chim. phys., 16, 322 (1918).
A S EQU.4TION
OF STATE FOR ETHYLEKE GAS
355
value 1.0934 was chosen. At a subsequent period it was found that the higher ratio was yielded by the equation of state finally derived, but it was found that the effect of the difference, 0.0002, was practically negligible in all relations under consideration. I t was observed in this isothermal interpolation that the pv data a t 24.95' do not approach RT a t low pressures in a perfectly smooth way around I atmosphere. The data are smoother a t all higher pressures. These data were already smoothed once before publication,* but probably without the aid of a value of R T , which in fact could not be obtained without knowledge
FIG. I
Typical plot of an unsmoothed isometric ( 2 . 5 *) showing the isothermally smoothed liter points due to hmagat (circles), Masson and Dolley (square), and Mathias, Crommelin and Watts (crosses). The plot shows the value of the deviation,-p - 34 - 0 . 2 7 j 16t" rn a function of the temperature Centigrade. The line is furnished by the equation of state finally selected. The circles centered on this line are drawn with radii equal to O . j % of the calculated pressures.
of the ratio 1125,/T'0.h like situation obtained with reference to the Leiden isotherms; no way being discovered of smoothing these together with RT values without assuming in this case rather large experimental errors. The interpolated values of the pressures, which are hereafter described as the observed pressures, were now grouped as isometrics for the next step.
Representation of the Isometrics For each density, the pressure was plotted as a function of the absolute temperature. Fig. I shows a typical plot, that for 2 . 5 moles per liter. The line, shown for comparison only, was calculated from the final equation. It 8 See Fig. 2 , ref. So. 3, where irregularities in the original data are obviously not entirely removed in the smoothing.
356
LOUIS J. GILLESPIE
is clear that the results of Masson and Dolley, of the Leiden Laboratory, and of Amagat differ too much to permit utilization of all data in the final smoothing. Only the Amagat data are extensive enough to determine an equation of state; such an equation must be based on Aniagat's data alone. It is also clear from the line drawn that the equation does not include a certain t r m d in the Amagat data. This trend, evidenced by an inflection of the isometric., suggests a t first a discordance between the high temperature (137', 198.;") and the low temperature (0-100') data. Exclusion of the high temperature data would however leave d2p/dt2positive, whereas this is as a rule negative (above the critical volume)Q.It seemed best therefore to include all temperatures and to assume such a curvature of the isometrics as would keep them within about 0.5 percent of the measured pressures. Following the procedure of Beattie and Bridgemanlo a value of c was found which would secure this, a t least from a density of zero to about 7 moles per liter, with some difficulty as regards the highest temperature-pressure corner of the field. The other constants of their equation were then determined as described by them. The constants so determined are given in Table I. TABLE 1 Constants in Beattie-Bridgeman Equation of State for Ethylen?.
R
o
08206
A0 6 152
Bo
a
o 04964
o
b
12156
o 03597
C
22
68 IO^
Mol n-t 2 8 031
+
The equation is p = RT(I - e).(v B)/v2 - A/v2 Where A = A, ( I - a/v) B = Bo ( I - b/v) E = c/vT3 v = volume in liters of a mole p = pressure in international atmospheres T = 273.13 t"C.
+
The agreement of the equation with the observed pressures of Amagat is exhibited in Table 11, which lists the observed pressures and the deviation A p (observed minus calculated pressure) in atmospheres. The average deviation (taken without regard to sign) over the entire range considered is 0 . 4 j per cent. This range goes to a density 8 mols per liter, slightly higher than the critical density. The agreement is fair. The equation holds very well indeed up to 7 mols per liter, the average deviation being 0.36per cent. Table I11 shows the magnitude of the disagreement, already noted by Masson and Dolley, between their results and those of Amagat. There is agreement only at low densities. It is however precisely at such low densi9 Onnes and Keesom: Encyklopadie der mathematischen Wissenschaften, Art. V IO, page 756 (1912);also Communications from the Phys. Lab. U n i v . of Leiden, 11, Supplement 23, p. 142,Leiden (1g12). 'OProc. Am. Acad. Arts Sei., 63,229 (1928).
AN EQCATIOS O F STATE FOR ETHYLENE GAS
3 57
3 58
LOCIS J. GILLESPIE
ties (1.1 to 2 3 ) that the Leiden data differ, and a t all temperatures. from the equation, as is shown in Table Is’, for which the unsmoothed Leiden data were utilized. The data deviate more strongly than those of Masson and Dolley and in the opposite direction. If the percentage deviation is plotted as a function of the density, it is found not to approach zero asymptotically a t zero pressure. This is not attributable to a defect in the equation. Even though the purity of Amagat’s ethylene was not so great as it is now possible t o obtain, the fact is important that the equation furnishes numbers, smooth in two dimensions, which must be considered to approach correctness, in smooth fashion, as the pressure approaches zero. There exist relations in which the disagreement between Masson and Dolley and Amagat, which reaches 4 6 5 up to a density of 8 > is not so important as might appear from Table I11 alone. Compare the discrepancy of 3 7 c a t a density of 4.5 with the error of 8 jTCjwhich is the percentage deviation of the perfect gas law from the observed value. At a density of 6, the perfect gas law is in error by 1 2 2 % . TABLE 111 Comparison of Calculated and Observed (Masson and Dolley) Isotherms at 24 95O
Ap, Density
Ah
p calc.
-
I
= IO0
5
11.41
-0.2
I
21.2;
+O
1.5
29.66 36.77 42.71 47.63 51.7 54.9 j7.6 59.8 61.7 63.3 65.0
+0.3 0.6
3 3.5 4 4.5 5
5.5 6 6.5
Density
1.0
A P, 6 4.6
p cnlc
7 7.5 8 8.j 9 9.5
-0.02
0.5
2 2.5
obs - calc, calc. 66.7 68.7
4.2
74.1
77.8
- 2.5 - 4’7 - 6.9 - 8.7
82.4 87.9 94.7
1.5 1.9
IO
2 . j
I1
102.8
3.‘
11.;
3.7 4.0
I2
112.3 123 5 136.6 151.6
10.5
-10.0
-IO. j
’
12.j I3
4.6 4.6
3.1 I .6 - 0.3
71.2
-10.5 - 9.0
TABLE Is’ Percentage deviations betu-een Leiden experimental values and the values calculated. The calculated pressure is always greater. Temp./Serial order
3
2
1.4
$0. IO0
I
.o
1.2
1.5
170
1.1
I
.o
1.3
I
.o I . I414
I
1 . 2
1.4
1.6474
2.2644
IO.
20.18~ Density at 2 0 . 1 8 ~
I
.o 1.3356
I
.6
5
4
I
1.3 0.9
- I . 36°C
2
.o .6
__ 1.8 ~
__ ~
AN EQCATION O F STATE FOR ETHYLESE GAS
359
Calculation of the Normal Density The equation does not necessarily furnish upon calculation the same value of the density at oo and I atmosphere as that which was used to obtain i t ) but may give a better value. Table V shows the summary of Blanchard and Pickering (5) with the addition of the value calculated from the equation.
TABLE V Normal Density of Ethylene, in grams per liter Source
Density
Source
Density
Leduc
I . 2605
Batuecas
I . 2604
Stahrfoss
I .2 6 1 0
Equation
1.2599
Blanchard and Pickering selected the value of Batuecas. This selection is supported by the value here found.
Calculation of Critical Data In the isothermal smoothing of the pu values under the critical temperature it was noticed that the calculated pressures became almost a zero function of the density and indeed, without great care in the smoothing, t'he pressures actually decreased with increasing density. K h e n the equation was derived it was thought interesting to see whether it would exhibit this trend. It was found that the equation gave a 7. j" isotherm with an inflection at about 7 moles per liter. By a succession of trials it was found that a n inflection occurred at as high a temperature as 8.5", but not at 8. j 5 ' , when the pressures were calculated with a precision of 0.0027c. The critical temperature is therefore given by the equation as 8.5', to the nearest 0.1'. Pickering" selects the value 9. i o ,a difference of 0 . 4 7 ~on the absolute temperature, which is of course the quantity calculated by the equation. Since this selection, Masson and Dolley (3) obtained 9.35', the lowest recent experimental value. The critical pressure calculated is 49.19 atmospheres against 50.9 selected by Pickering, a difference of 3.4yc. The critical density calculated is 6.4 to 6 . 5 , selected value 7 . 9 in moles per liter. Here the errors seem to accumulate, making a positive disagreement. Equations Tvhich represent the measured pressures are not generally expected to furnish correct critical constants (nor do those which are derived from critical constants represent the measured pressures), so that the success of the equation for ethylene in furnishing at least the critical temperature is surprising. The agreement may of course be accidental. l1
Sci. Papers, U. S. Bureau of Standards, ?To. 541 (1926).
360
LOUIS J. GILLESPIE
summary
The constants in the equation of state of Beattie and Bridgeman have been determined for ethylene gas from the data of hmagat. Using atmost " C , R = 0.08206, they are A, = 6.152, pheres, liters per mole, T = 273.13 a = 0.04964, Bo = 0 . 1 2 1 5 6 , b = o.o3j97$ and c = 22.68 IO?. The representation of Amagat's data is good up to a density of 7 molcs per liter and fair t o 8, slightly above the critical density; the average deviations being o 36 and o 45 per cent respectively. The equation does not represent closely the isotherm of Masson and Dolley, except at low pressures, in accordance with their statement that this isotherm does not agree with interpolations from Amagat's data The critical temperature calculated from the equation agrees t o o 4y0 with that observed, the difference being 1 . 2 " ; the critical pressure calculated is 3.4Yc in error. The normal density calculated from the equation, 1.2 599, supports the value of Batuecas as against that of Stahrfoss, being lower than either.
+
