A Generalized van der Waals Equation o f State for Real Gases GOUQ-JEN SU' &ND C H l E 3 HOU CHANG \atinno1 Tsirip H c r t r L nit.erait.v, P e i p i n p , Chino
0
N E of the classicai equatiori,< of state for flujds and the best knon-n of its type is due t o van der Waals ( 4 ) ; for one mole of gaf this equation is:
In procesb industriea which inrol\e the uat. of moderate or high pressures, ii h o w ledge of the compressibilitie. of the gases o r Iapors in question is essential f o r the design and operation of the equipnient. In the follow i n p three paper- sorn(* generalized cyuations of *late will be presented and a rnodified l a w of correspoiiciinp states for g a s e s w i l l be proposed. 'I'he first paper in the seriei deals with the pcnrralised \ a n der U aals equation of s t a t e . I t is shown that the propowd equation i- good for seienteen gases alrnost 111' to the critical den-itj with an a\erage de\iation of 3% or less.
Tiit original reduced equatiorl 01 van lier W a a k i.:
w h i d i states that the reducwf vriluriir. i,. a
iiii11,tioti i)f
tlic. t ~ c ~ ~ i r i c c ~ l
pressure arid thc reducwl t ~ I n ~ J I L ~ ~ t Il tl ris l ~ kiiowii . :I- t h theorcm of corresponding states. -1modification of this t h c o r m I J ~c,l,i.rt..piiritliiig ' states rvill I J ~ . . propoaed here. 6 is defined as T','(RT, 'p.), The dcnominatoi RT,/p, is the critical volume which would ho occupied 11y ani, molf. of perfect gas a t the critiral tempfdrature and the critical i.5 substituted for R T , ' p , arid is called tlir ideal prcssurt.. lrcL critical volume. -1fen- ycws ago (3)tliii same quantity was dcfined and called pseudo-critical volume. The ratio 6 = T7,,'Trc, iL< called the ideal reduced volumc. The propoyd modification of the theorem of rorrcqx)nding atntw i i th:rt N fiirictinn existi surh that ftT,@ *
HI = 0
18:
I n other words, the ideal reduced voluniij i > :I t'uitctiun Ut t 1 t 1 ~w dured pressure and tho reducrd teinperaturr. This papc.r employ,. an equation of stat(' of the van der Waals type t o reprexcrit t l i c . functional relation emhodied in Equation :3, .\ detailed discnbsioii of the modified tlic~~rrrn nil1 IIP given i n I III,third paper o f thiseries (page 803 ). The proposril pi~ri~~i~alizt~d t:quatiori ib: ti 3 - +
U
i r = - - -
w-httru u arid p will tw uriivrrnal
r#9
+
~ ' ~ J l l ~ t : l l l t , 3ri~giirtllcss , ot
1 tit,
nature of the substances. To obtain the numerical values o f tlic i.otistants. we impoEquation 4 t h e conditions for t h r critical pfoint:
oil
8CO
7r
=
v 4
- 0.125
-
0.422 62
~~
~
I~~quatiori 4.4 may be intcxrpri:itdc as that effect due to the
voliiirir
of the nioleculen, expresseil 1): thc: uriiversal constant p = 1 'I. in retliic.cd units. The effect 'cluc t o t h e intcrniolccular attrnctioil \vi1 hc O C / ~ ? where , a is a uiiiviwa, constant equal to 27/64 in rerliiw3i3 units. Constants a and d :in tiimerisionless and will IJI,I t i c same tvhatever consistelit (if
August, 1946
INDUSTRIAL AND ENGINEERING CHEMISTRY
10.0
400 t o -217 -217 -150 -244 -146 -117 -14.7 0 0 72.5 t o 150
6.0 10.0
5.5 9.0
g
6.G 2, I
400 3Oii
'(K lill I(11i ill
21 I I)
.. 2 3 . 6 2
- 50 -81.10
- 100 -121.19
2 i . 94 0.12 23.76 0.12 !9,56 0.10 17.46 0.10 15,36 0.09 13.26 0.08 11.99
0.oc
11.15 0.0ti 10.15 0.05 9.04 0.04 7.74 0.04 6.92 0.02 6.03 0.01
- 130
5.65
0 5.16 -0.01 "' 144.4H 5.03 -0.01 - 146.32 4.94 -0.03 Av, c/o deviationb 0.44
- 141.53
0.2;1
0.229
-0.003 0 5
1 u 1 5
2.tr
x
(1
9 5 Av. !/c deviationb a
b
compressibility behavior of nitrogen, carbon dioxide, ai111>teal[ n i t t i approxiinatcsly t h c ~same degree of accuracy. Tlw ~ Y ~ I C I J lated values are compared with these gcncdized isometric valuw in Table 111. The highest ideal critical density is 3.5, d i i c ~ l li+ close to the average critical density 1/4 = 3.7. T h e total a v a age deviation is about 3Yc. Thus for the stiventeen gawh (including steam) studied, the equation holds, with a deviation of .5% or less, with few exceptions u p to nearly t,he crit,ical dmisity This result is considered good, i r r a s m i i r h as (only two tiiiivtma 1 rmstants we user1 in thp equation.
400 t o 400 to 200 t o 400 t o 100 to " 0 to 100 t o 200 t o
8.5
56.59 0.46 48.03 0 4G 39. 4.: 0 41 ,3.5 16 0.3P
$i.!f'J
:30,84
40.48 0.76 39,87 0.68 3 3 . 86 0.59 33.22 0.56 30.05
1.07 7 2 , 86 1.01 j',.)7 1 11.92 33.11
0.86
0.34 26.54
0.30 23.93 0.26 22.22 0,25 20.17 0.22 17.87 0.17 15.19 0.14 13.50 0.07 11.65 0.03 10.83 -0.02 9.84 -0.05 9.56 -0.08 9.38 -0.10 0.87
0.48 26.50 0.38 22.3i 0.31 19.76 0.17 16.89 0.Oi 15.63 -0.04 14.05 -0.11 13.62 -0,16 13.34 -0.20 1.29
DISCUS SIOY
C onstants 01 and 3 are really functions of density, ewii 1 1 M I 'i-sunie linear isometrics. The actual values for a and fl 111a\ b t r~al(~iilatet1 from the knoir n data of generallaid isometril- ( J
0.84 98.29 1,75 80 37
1 . ti2 71.36 1.50 62.31 1.34 33.28 1.20 4 7 . 80 1.OG 44.18 0.99 39.83 0.85 z34.96 0.67 29.29 0.53 25.71 0.31 21.75 0.12 20.01 -0.06 17.81 -0.20 17.23 -0.26 16.86 -0.30 1.69
.., I..
1 0 1 -hi
89.Y; 2.2.3 i8.3b 2.00 fi6,ii' I .84 39.78 1.64
5.5 , 1:J 1 .33 49.54 1.31 43.28 1.04 :3.j.96 0.80 31.38
0.55 26.26 0.18 24.02 -0.06 21.16 -0.30 20.42 -0.38
. .
...
... '14.23
1.13 36.82
0.75 30.46 0.2b 17.69 -0.04 24.14 -0.37 23.21 -0.48
19.95 -0.42 2.08
22.64 -0.54 2.37
0.270 -0.000
0.283 -0.000
0.296 -0.000
0.31U -0.001
0.448 -0.007
0.477 - 0 004
0.535 0.000
fn.002
0.563
C.593 +o.o05
0.764 -0.014 0.978 -0.011 1.121
0.832 -0.003
0.507 -0.001 0.901 i.n.009
1.023
1.088
1.216 to.035 1.279 +0.029 1.334 -0,030
2 866
1,687
i0.008
I 092 -0.010 1.260
1.201 1-0.030 1 442 ~ 0 . 0 1 5 C0.063 1.645 1.425 4-0 062 c o . 1 0 0 1.546 1.829 t0.058 tO.098 1.667 2.013 - 0 008 +O.OZR 1.583
2.59'3
0.959
I ,iO?
90.4Y
",!I 71.81 2.33 W.10 2.18 59.21 1.84 il.46 1 . ,54 42.47
+0,018 t 0 . 0 2 4
1.303 1.410 1.50'9 t o . 0 3 7 t 0 . 0 5 1 -0.0.58 1.591 1,94(J 1.882 -0.079 4-0 093 -0 103 1.839 2.041 2.236 r o . ~ 1 2 + o 182 - 0 146
.(
I .3'&
, .
I 87
3.44
.. 96.13 3.90 881! 3.71
108.50 4 79. 99.3:3
121.08 5.i4
1.02 i34.38 0.43 31.04 0.01 26.78 -0.42 25.66 -0.57 24.96
-0.65
2.61
..
2 I!.
133.91 0.73
3 . O:(
, .
2.6h
4.51,
.. ~.~
42,01
0.336 +O.OOI 0.648
.~
.,.
53.91 3.08 77.11 2.9% ij8.87 2.52 39.58 1.99 48.80 1.54
..
)..
2.11
67.71 2.55 54.97 1.98 17.01 1.36
75,59 3,lH 61.00 2.14 51.82 1,73
YB..5j
3.79
!11.54 4.46
2.81i
,..?
..
2.20
l38.05
0.62 34.09 0.08 29.12 -0.41 27.78 -0.62
41.48 0.86 36.88 0.19 31.18 -0.37 29,60 -0.62
44.66 1.09
39.44 0.34 32.91 -0.32 31 15 -0.59
26.84 -0.73 2.74
28.62 -0.77 2 8.i
30.00 -0.79 "7'3
..
ti1.00 2.56 $7.60 1.29
j6.47 2.13
2.12
1. I!.
11.83 0.57 34.41 -0.94 32.42 -0.55 31.11 -0.79 2.89
0.3k 1.04
I .62 2.Of
+0.002
0.363
0.389 to.003
0.441 +0.003
0.493 co.003
+0.004
to 005
0.545
0.59s
-0.008
+a
0.705 010
0.75'3 +o.o11
0.866 +0.011
U.975 -0.013
1.084 +o.oi6
+o.ois
1,212 -0.034
I 334 tn.042
1.450 c0.043
1.688 t0.048
1.920 -0.056
2.152 +o.060
+o.otx
1.712
1.913 c0.093
2.850 3.246 +0.113 f 0 . 1 3 4 3.820 4.396 t0.175 t0.217 4.860 ,. -0 224 i.
3.636 +o.l53 4.981 C0.27C.
-0.078
2.066 f0.09.5 2.310 C0.012
2.340 +0.128 2.6313 f0.017
2 580 -0 134 2.067 -0.047
2,172 t0.127 2.652 -0.171 3.108 4-0.174 3.626 40.084
2.515
3.113
3.2413
3.461
T h e first fiwure for each value of 1 ' 6 is the observrd reduced gres-ure: Total averEge deviation, 3%.
1 I.i>
. "
,.
..
94.68 2.93
0.256 -0.002
co.010
..
' ,
0 243 -0.002
0.417 -0.011 0.693 -0.028 0.863 -0.034 0.950 -0.029 0.990 -0.009 0.999 -0,011 0,994 --0.0.59
80 i
2.100 +0.095 2.4G-1 2.732 t O . 1 5 2 4-0.153 3.01.5 3.388 +O.IW f0.204 3.600 4.089 +O.l8ti +0,195 4.281 4.900 +0,lli t0.118 3,823
3.54:
2.414
+o.ion
3.254 +0.14? 4.0'90 +0.181 5 043 4 0 187
2.39U
..
.. :3 04!1
1.193
_
I
.3.196
the second is the differenre, t,hSPryrd .- c:rlcul;ita~l,
I
.
.
I
,
,
2 810
.,
2 !gib
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,
