2254
W. H. MEARS,E. ROSENTHAL, AND J. V. SINBA
Physical Properties and Virial Coefficients of Sulfur Hexafluoride
by W. H. Mears, E. Rosenthal, and J. V. Sinka Specialty Chemicals Division, Allied Chemical Corporation, Morristown, N e w Jersey
(Received Soztember 4, 1968)
On the basis of new experimental PVT data and of selected literature values, this paper derives a Martin-Hou equation of state which correlates, within i=O.18% standard deviation, data in the experimental ranges of 25130°, 10-75 bars, and 1.05-13 cc/g. A vapor pressure equation with six constants describes the data within =t0.18% standard deviation between -45 and +45.64", the critical temperature. A density equation with five constants correlates data between -41 and +42" with a per cent standard deviation of =tO.lO'%. The Martin-Hou equation serves to compute virial coefficients and a variance-covariance matrix treatment indicates their probable accuracy.
Introduction Sulfur hexafluoride is a gaseous dielectric of great importance in the electrical and communications industries. It alone or in a mixture with other gases has some possibilities as a refrigerant. Its symmetric shape and the presence of the six fluorine atoms make it of theoretical interest in the study of intermolecular forces. Inspection of the available information on S F P made it apparent that a comparison of available information, supplemented by additional data recently generated a t Allied Chemical C o p , would be needed prior to producing thermodynamic tables or to computing virial coefficients. This article reports our additional SF6 data together with the results of the comparison with available data. Based on this work, we have computed correlation equations for the vapor pressure and for the vapor and liquid densities. Finally, the Martin-Hou equation herein derived serves to develop virial coefficients, and the probable accuracy of these is evaluated by covariance matrix treatment. Experimental Details and Results
Material for Measurements. The SFe used in our measurements was produced commercially by Allied Chemical Corp. Its purity determined by vapor phase chromatography was 99,96%. Prior to conducting the measurement, it was degassed by repeated pumping and cooling in liquid nitrogen.8 Vapor Pressure, The methods for determining these data and details of the accuracy of the input variables are published elsewhere. Temperatures were measured to 3t0.01" on a platinum resistance thermometer calibrated by the U. S. National Bureau of Standards (degrees Centigrade Int. 1948). The claimed accuracy of the pressure measurements is better than 3t0.2Oj,. There also exists work by Otto and Thomas,8 Clegg, Rowlinson, and Sutton,' and other^.^^'^ Based on a preliminary correlation, the data of Otto and Thomas were in excellent agreement with Allied's. Accordingly, four- and six-constant1'*'2equations were derived, by a T h e Journal of Physical Chemistry
least-squares technique, for t,hese data, eq 1 and 2. (Xote: in this paper discard italicized digits for hand calculations.) log P b a r s = 4.41780 - 889'185 __- - 0.56334 X T°K
+
i o - 3 ~ ~ 1o.120040 ~
log P b a r s =
x
10-57701~ (1)
- 1.12347406 - 816.48995 + T"K
0.02928734dT01i - 0.40107649 X lO-*T"I 1.0 cc/g had a standard per cent deviation of *0.25Oj,. Based on the
Acknowledgments. The authors wish to thank Professor C. M. Knobler of the Department of Chemistry of The University of California, Los Angeles, for comments and helpful suggestions, and also Miss Pauline Lissy for editorial and typing assistance.
- 0
RWU
virials is probably within 2% or at least, within 6% of the true values. Any additional check of the reliability of the MartinHou equation for calculating virial coefficients will depend on the availability of high quality data of known accuracy. Assuming the Martin-Hou equation is a good physical representation, the standard error calculations have shown that the fourth virials can be calculated to within 10% or better and the fifth virials to within 20% or better, depending on the temperature.
a SCHNEIDER
- 50
Appendix
-
100
-150
. 2 W
I
$
-200
m
-
250
-300
- 350
0
I50
50
T
zoo
250
OC
Figure 3. SFP, virial coefficient B: 0, calculated eq 3;'4 0, MacCormack and Schneider;aj4 A, CIegg, et aZ.;I and 0 , Harnana2
error analysis of the Appendix, B values from Allied data agreed with those computed from eq 3 of this paper up to 100" within one standard deviation. This agreement worsened to two standard deviations between 100 and 150" and t o three a t 150 and 175". One concludes that agreement is excellent below 100" and good in the rest of the temperature range. The agreement of Allied's virials with the literature was similar to that for eq 3 virials shown in Figure 3. The Clegg' data were 2% more negative than the Allied data while those of Hamann2and of MacCormack and Schneider3J were within 6% maximum. From these results it would seem that the accuracy of B The Journal of Physical Chemistry
This section describes statistical methods for obtaining the coefficients of the Martin-Hou equation and for calculating the standard errors of the derived virial coefficients. The development of the equations used in the calculation of standard errors may be found in the literature.20 Referring to Martin's papers on his equation of state," he gives a set of primary physical parameters (R, T,, P,, V,, m, P, Tb, T ' , K , y, and n) from which the coefficients of his equation (R, b, T,, A z , B,, Cz, Aa, B I , C3, Ad, A5,B6, and c6) are calculated. In his papers he shows how the parameters (0, TI,, T ' , K , y, and n) may be estimated or assigned reasonable values if R, To,P,, V,, and m are known. A computer program has been written which will start from an initial set of values for the primary physical parameters, and will vary them in a systematic manner to obtain a "best fit" for a set of PVT data. I n this process, some of the parameters may be assigned fixed values, some may be varied independently, and some may be calculated from the estimation equations. The "best fit" is defined as the point at which 21 [P,(measd - P1(calcd)12,the sum of the squares of the differences between the measured and the calculated pressures, is a minimum. Assume there are m PVT points and that n parameters (xt; i = 1, n) are varied, Then for a small variation of the parameters
where the Apt represent the changes in calculated pressures and the Ax, represent the changes in the parameters. For small changes in the XI, the partial derivatives, bPt/bxj,may be assumed t o be constant. Thus, for "points" very near it, finding the best fit becomes a simple least-squares type calculation in which equations A1 are solved for the Ax,. The sum of squares, 8, where
PHYSICAL PROPERTIES AND VIRIAL COEFFICIENTS OF SF6 S = c[Pt i
+ A p t - Pt(measd)12
2261
Ayj =
C dY5 -.AX, ax,
(J' =
1 , . . ., p; i = 1 , . . .,n) (A8)
I
is the quantity to be minimized. Equations A1 may be represented in matrix form by ax =
p
eq A8 in matrix form
Or
Y
(-42)
where
=
BX
(-49)
where
... a =
... X =
(""');8-(bp,) Axn
APn
The covariance matrix, V, for the yt, is given by
(-43)
Letting A = h a, (where & represents the transpose of a),the least-squares solution to eq A1 or A2 is given by
X
=
A-'&P
where A -l is the inverse of A. The standard error for the measured pressure, u, can then be obtained from g2
S
= -
m - n
(444)
where S represents the minimum value of the sum of the squares. The covariance matrix, U,of the xl, is given by
u = A-102
(A5)
uza* = A{i-'u2
(A6)
V = SUB
(All)
= vtt
(AW
also UYt2
Equation A l l may be reapplied for each new set of parameters which are functions of a preceding set, Thus, for the calculation of the standard errors for the virial coefficients, the covariance matrix for the set of primary physical parameters which were varied was calculated using eq A5. The covariance matrix for the equation coefficients was then calculated using eq A l l . Finally, the covariance matrix for the derived virial coefficients was calculated with eq A l l and their standard errors with (A12). With regard to the calculation of the partial derivatives used, the following method was used to calculate them. Fordyl/dxj, values of y&), yt(xj -I- 61, yr(xj 26), yr(xj - 6), and yt(xj - 26) were calculated, where 6 is a small increment in xj. These values were then substituted in a formula from Savitzky and Golay21to get the value of by,/bxj. This method was used as opposed to calculating Ay/Ax, so that larger values of 6 could be used to avoid round-off errors, while also compensating for the nonlinearity of the functions.
+
and also is the where uzl is the standard error for xa, and i t h diagonal element of A-l. If a set of (p) parameters, yr, is considered which are functions of the xa,such that
yj = Y,(x~) ( j = 1 , . . .p; i = 1 , . . ., n) (A7) then for small changes in the xt
(21) A. Savitsky and M. J. E. Golay, Anal. Chem., 36, 1627 (1964).
Volume 73, Number 7 July 1060
