INDUSTRIAL AND ENGINEERING CHEMISTRY
802
Vol. 38, No. 8
ACKNOWLEDGMEST
TARLE I v . VALUES O F CY .4ND FOR DIFFERENTDESSITIICS ISO\IETRIC.: CALCULATED FROM D.4Ta ON GESER.AI,IZED 1/4
a
B
0.25 0.50
0,5232 0.5116 0.4925 0.4649
0,1992 0.1881 0.1715 0,1550
1.00 1 .SO
1/Q 2.00 2.50 3.00 3.50
a
P
0.4G23 0,4482 0.4404 0.4492
0.1593 0.1505 0.1341 0.1323
Thc authors wish to express their thanks to Tuan-illou Chang and Shue-rhuan Hu for their assistance in the preparation of this papel hOMEhCLATURE a = 11 11.
The slope and intercept of the bcst straight line d r a n n for a given isometric may be used t o calculate the values of CY an(1 0, R S Equation 4 shows. Table IV presents sricli a calculation. The theoretical values (Y = 0.422 and p = 0.1% are clusc I O those for higher densities. This may be one of tbc explanation> \Thy the results are surprisingly good, for the numeriral valiies n f the const>antsare less import,ant in the rrxyions of low clcnsity. The equation fails a t the critical point as d o r s the origiual vquation of van der Waals, for it would give a critical ratio of 8 3 = 2.67, as compared t o an average critical ratio of 3.7.
= rcduced pressure
0 = T / T c = reduced temperatnie $ = ‘i7/Bc= reduced volume = I-’T’c, = ideal reduced volume Vc, = IZT,, p , = ideal critical volume a , b = \ a n der Kaals constants CY, 3 = griieralizd van der Waal. con-taiitb
+
LITERATURE CITED
(1) Heattie, J. -I.,and Bridgeman, 0 . C . , Pmc. .-lni.A 4 C ~ d.4r& i. Sci., 63, 229 (1928). ( 2 ) Newton, R. H., IKD. Esc;. CHEM.,27, 302 (1935). ( 3 ) Su,Gouq-Jen, thesis, Mass. Inst. Tech., J u n e , 10:19. ( 4 ) Waals, van der, Dissertation, Leiden, 1873.
Generalized Equation of State for Real Gases GOUQ-JEN SU’ AKD CHIEN-HOU CHAPJG .\-ationnl
7 s i n p H i t n I-nicersity,P e i p i n g , Chinn
a generalized i equation of state for real gases is proposed. and the three constants involved are determined. The values of the constants are the same for all gases. The equation is Falid up to about twice the critical density with an average deviation of 2% or less for each gas.
I
r\’THE present study of generalized thermodj-nanlic prupcrt’ies
of real gases, two generalized equations of state have been described (6). One is the generalized form of the Reattie-Bridgeman equation of state, arid the other is that of van der Waals:. Both of these equations are applicable almost up to the critical density. The purpose of this article is to present a generalizeti equation which i d 1 be applicable from lon densities to about twiw the critical density. Thc prol~oseciequatioii is
H
=
H,(1
i i T T
r----
--_____.__ - .
-O B S E R V E O r
--,
+- $)
n-tiere Bo, b, aiid d arc’ gciieralizcd consiniits iiidepciitleiit of t i l ( ’ nature of the gas. Furtlier, since the constants are ilirne11siollless, they h a r e the same valnes no niattvr what consistent set ot’ units is eniployetl. The usual quantities of r d u c e d prc~usurc ( p / p c ) and reduccil temperature ( T / T , ) are denoted in Equation 1 by r and 0, respectively. Instead of employing the usual term, reduced volume V/V., (o is defined as V/V,,, where TTCi = RT,/p,. I’ci is called ideal critical volume, for it would be the critical volumc~of one mole of an ideal gas. T,he term p will be called ideal reduced volum’e because (1)it affords a better correlation of the compressibility data, and (2) i t eliminates the use of t’he term “critical volume”, n-hich is much more difficult to determine than the other two critical quantities, pressure and temperature. I n Inan)cases the critical volume is not given. The ideal critical vol1
unie is su tlcfined that it is esp~essediii terms of the critical pressure and the critical teniperatur?. The thrce constants in Equation 1 were determined from the compressibility data on hydrocarbons presented in the form of generalized isometrics by Su (6): A = 0.472, Bo = 0,160, b = 0.190. 111 determining these constants, emphasis has been laid on the (lata at higher densities, for in that region t,he numerical values of the cunsiants show greatest effects. Function B is first calculated from available data, and constants Boand b are
Present addresq. .Joseph E. Seagram Br Sons. Inc.. Louisville, Icy.
REDUCED TEMPERATURE GENERALIZED lSOMETRlG6
Figure 1
e
~
1,
INDUSTRIAL AND ENGINEERING CHEMISTRY
August, 1946
Maximum Pressure, G35
1%e lie
.1
Hz S Z 0 2
4ir
CO?
CH4 (CzHdLI n-butane Ethane
.Itm.
102 107 114 103 134 103 177 111 243 90 310 351
Maximum Density. 3ioles('L. 10.0 8.5 6.0
10.0 5,5 9.0
?g
6 0 2.7 6.5 10.0
Temp.
Range, C. 400 t o -217 400 to -217 400 t o -150 "0 t o -244 400 to -149 100 to -117 200 to -145 100 t o 0 200 to 0 325 t o 150 300 to 150 27.5 to 50
Sumber
Total of Av. Point8 Deviatgn 160 0,556 170 0.374 209 0.692 297 1.24 154 0.922 1.25 114 152 0,865 1.99 127 60 0.638 64 1.98 82 1.79 82 l.i6 Grand average 1,07
803
Tlie comprcsibility data lor the fir,t ten g:iscs are those presented by Beattie and Bridgeman (I) : the &ita for ethane and n-butane were taken from the W O I . ~of Beattie rl n d eo-workers (17:. For helium arid liydropen Seivton's pseudo-ciitical temperaturi. T ' , and peutlo-witical pi 'UI'C p ' c arc used (4):
Y'A
1',
- 8;
71;
= 11.
+ 5,
\\here Tc i, expressed in degrecxs Kelvin ant1 pc in a1llltJspllt?I'e.~. The avcragt: critical density for gases is a t 1 'q? = 3 . 7 , awiming iyc. critioal ratio, rc, eqiials 3.7. DISCUSSION
This equation falls into thr general form of the T~~rcxntz c.(lli:ition of s h t e (3): 2
determined according to the method of determining the B-func1 ion i n Beattie-Bridgeman's equation of state ( 1 ) . Then the values for -4 are calculated from the equation; Bo and b are known, anti the observed values of T , p, and 0 are substituted. The value> for A do not vary much in the higher density region. Therefore, instead of expressing 9 as a density function, it is given a constaiit \ d u e Of course, linear isonieYrics are assnnieti in tlie presvnt n o r k . Tlie ahility of t h e proposed equatioii t o reproduce tlie conipressibility tlnta is slio\vn in Figure 1. The average deviation is 1% helon- die critical density and 2 c 2 above t,lie critical density. To test the validity of the equation further, detailed calculations \yere made for the conipressibility behavior of twelve gases. Table I compares calculated \Tit11 observed values over various ranges of pressure and temperature. The over-all average deviation for the twelve ga
.llthough R is a constant. i:i the Lorcntz equation, it is expressed a volunio function in the present n o r k , and anoilier constant. h. is atlded; or the proposed equation may be regarded a? :I .iniplified, generalized form of Beattie-Bridge:iian's equation I ) ( -tatc in Jyhich constnnts c and a are omitted. :E
LITERATURE CITED
Beartie. J . .1.,and Bridgeinnil. 0. C., Pioc. A m . A c i ~ r E . - l i , f u S c ; . . 6 3 , 229 (1928). Benttie, .J. Ai1.. Su,G . J., and Pi:iini,d, G . L.. J . ;im.P h o n . Soc.. 61 2(j, 926 (1939). Loreiitz. W i c d . Arm., 12, 11'7. RGO (18811. S e n c o i l , R.H., ISD. E s u . CHEX..27, 302 (1935). Pu,G . J., thesis. Mass. Inst. T e c h . , J u n e , 19:*i. Y u , G. J.. and Chang. C. H., J . -4ni. f ' h m z . , Y r i c , . 68, 1080 ( I H A I j , . ISD. ICSG, C H E N . . 38,600 ClOdR).
Modified Law of Corresponding States for Real Gases -4 riiodificatioti
of the la\+ of corresponding states is propohed. A term called the ideal critical lolunie is defined, and the raLio of lolunie o\er the ideal critical loliinie i* called the ideal reduced loluiiie, to be used in place of the reduced lolunie. It is shown that, for selenteeri gases \+ ithin the temperature arid pressure ranges studied, the oler-all alerage deliation is 1%. The lalue of the critical ratio is riot a restriction or a criterion for the applicabilit? of the modified l a w .
I
S 1881 van der JVaals proposed the well known law of corre-
sponding states. The original k i of~ van der Waals has been s. and its practical applications slionn to be inexact in many t are often regarded F i t h reserve. The present paper aims to present a niodification of the law of corresponding states 7r.hich Till be applicable to all gases in gencral, n-ith a deviation from a fraction of a per cent to a few per cent which i.3 allo\wl)le in most practical problems. Vci = R T c j p cand is called ideal critical volume, since it \vould be the crit,ical volume for one mole of ideal gas. A fen- years ago this term was called the pseudo-critical volume ( I S ) . T 7 'Vci 1
Present address, Joseph E . Seagrain Br Sons, Inc., Louisville, K y
is txlled the ideal reductd volrinie and designated by io. i l t d r i c i ~ ~ ~ temperature (7'; 7'2 is a, and reduced pressure ( p , ; p c ~i.j T . .Iccorcling to the proposed inodified h\v of corrcy)onding a t a t c - . a generalized rehtioii exists such t h a t ;\a,os 5 ' =
0
',
1
In other noid-, the itlenl rctliicul volnnie is 3 iini of the reduced pressure ant1 the reduced tcmperiturc. irrespectivc. of tile nature of the gas. One immediate advantage oi the nioilificaiioii is that the use ut the term "critical volume" is avoided. The critical voliinic i d m u c h more difficult to determine than critical pressure and crit ital temperature, and in many cases is lackitig. =\riy uncertaiiit! involved in the critical volume term is thus avoided. The idual critical volume is defined in terms of the critical pressure a i d critical temperature. It, is believed that the ni'idification affords a better correlation of compressibility behavior both in accuracy and in scope of application. One point is an exception-tlic critical point. The original law as well as the present modified form requires the same critical = RT,/p,V,, for all gasps: it varies approximately fro111 ratio, 3 to 5 11-ith an average valur. of almut 3.7. O n account of the
