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, i t 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, i t 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 \ e r the ideal critical loliinie i* called the ideal reduced loluiiie, to be used i n place of t h e reduced lolunie. It is shown that, for selenteeri gases \+ ithin t h e 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 t h e 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 ratio, = RT,/p,V,, for all gasps: it varies approximately fro111 3 to 5 11-ith an average valur. of almut 3.7. O n account of the
804
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 38, No. 8
August,
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1946
continuity of physical properties, the immediate neighborhood of the critical point (the critical region) has to be excepted also. GRAPHICAL REPRESENTATION
One direct consequence of the modified law is that the compressibility factor, p = p V / R T , is a function of the reduced pressure and reduced temperature: p
=
r ( ~e),
(2)
If we substitute in the defining equation of p = p V / R T the quantities p = r p c and T = OT,,then on rearranging, Tip
=
(31
Sow from Equation 1. 'p
= F(*, 0)
(4)
Combining Equations 3 and 4 gives Equation 2. A graphical rcprcsentation of Equation 2 is often called the p-chart. It may be regarded as a graphical representation of the generalized equation of state. The validity of the p-chart proves the validity at the modified lay. Several authors have made contributions t o the ,u-chart ( 2 , 3, 4, 5 , 8, 9, 16). It n-ould appear to be better t o obtain a p-chart based on the average compressibility behavior of the gases, rather than based on the compressibility behavior of a single substance. Deviations from one gas to another in the generalization are espected. T o reduce the average deviations of several gases and to have a common p-chart applicable to all gases, it would be advisable to present an average w-chart based on the cotnpressibility data of the gases in line Kith the generalization. The p-charts pubhshed by other authors seem to be based on data for only a relatively few gases. A few years ago (IS) the author presented a general or average pi-chart based on the compressibilities of seven hydrocarbons. An extensive comparison was made of compressibility factors derived from experimental data with the generalized curves (13). The over-all average deviation of the seven hydrocarbons from the general p-chart is 1%. The chart applies to carbon dioxide, nitrogen, and steam with about the same degree of accuracy. The figure shows part of the chart with some experimental points for ten gases plotted as illustration. T h e present p-chart agrees fairly well with the results of earlier workers. Other p-charts based on the author's data have appeared elsewhere (a, 6, 1 7 ) . Two other p-charts were made ( I S ) with density-temperature and density-pressure as independent variables. A chart of generalized isometrics was also prepared (IS). T h e data of generalized isometrics were used t o determine the constants of a generalized Beattie-Bridgeman equation of state (16). ANALYTICAL EXPRESSIONS
Several investigators have presented contributions on the reduced equations of state ( 7 , 10, 12). A generalized form of the Beattie-Bridgeman equation of state (1, 15) is given as follows:
.1 = Ad1 - a/p\ B Ba(1 - b / y ) e = e/q@ The constants were determined from the data of generalized isometrics (15): As = 0.4758, a = 0.1127, Bo = 0.18764, b = 0.03833, c = 0.05. These are universal constants, irrespective of the nature of gases. The equation has been shoxn t o hold well for densities nearly up to the critical, and for temperatures as low as the critical. Table I shows the average deviations of seventeen gases from the present modification of the law of corresponding states The
805
first eight gases are directly compared t o the general p-chart (19); the next nine gases are compared with the generalized Beattie-Bridgeman equation of state ( 1 5 ) . T h e over-all average For hydrogen and deviation of the seventeen gases is 1%. helium the pseudo-critical temperature and pressure are employed as defined by Sewton (11).
TABLE I. AVERAGE DEVIATIOXS OF GASESFROM THE ~ I ~ D I F I I C D LAW
Gas n-Butane Ethane Ethylene n-Heptane Methane Isopentane Propane Water
Critical R,bv. Ratio Deviation 0 71 3.61 0 63 3.51 0 82 3.58 2 47 3.99 1.44 3.46 0.95 3.77 0 66 3.71 1.44 4.30
Critical Ratio Deviation % bv. GaP
Air 3.52 Argon 3.44 Carbon dioxide 3.58 Ethyl ether 3.82 Helium 3.28 Hydrogen 3.27 h-eon 3.37 Kitrogen 3.43 Oxygen 3.42 Grand average
0.28 0.67 0.86 1.61
0.73 1.01 0.51
0.32 0.22 0.90
dnother geneializccl equation of state of t h e b a n drr TTaals t v p ~ has also been presented ( 1 4 ):
e
lT=---
@ - - $
tX
6)
c"2
where a and 9, are universal constants. The numerical values of the constants are deduced by imposing on Equation 6 thc two conditions of a critical point-namely, *
\\-e obtain CY = 27/64 = 0.422, p = 1/8 = 0.123. Equation 6 with the values of a and j3 so determined, independent of actual compressibility data, has been shown to be applicable to seventeen gases with a n average deviation for each gas of 570 or less; the temperature and density regions studied Fere the same as in those of Equation 5. Such a gratifying result is not obtainable, it is believed, with the original van der Waals reduced equation of state:
i 7) where $ = V / V ,
=
reduced volume DISCUSSION
The question arises as t o whether the variation of the critical ratio, ro = RTc/p,Vc,would restrict the applicability of the law of corresponding states. The answer is negative if the law is modified as described above. T h e critical ratio enters only a t the critical point, a t m-hich the modified law as v,-ell as the original law fails. Other than the critical point, the critical ratio plays no role in the present treatment. T h e criterion of constancy of the critical ratio is not a criterion of the applicability of the modified law. I n fact the modified law applies t o all gases irrespective of the value of the critical ratio, as evidenced by Table I. The original law of corresponding states would require the follon-ing to be true instead of Equation 3: T$
pro
(8)
= pr,e
In other words, the original law would require to be a universal function of K and 6: = i-(T, 0)
pro
instead of P 9)
For substances which have the same or approximately the same critical ratio, p and pre have the same function. For substances which have appreciably different values of the critical ratio, the crucial test ~ o u l dbe vihether p or p r c is a function of K and 6.
INDUSTRIAL AND ENGINEERING CHEMISTRY
806
Investigation so far points to the conclusion that p, not we,is a universal function of 7 and e. For instance, steam has a critical ratio of -1.3 as contrasted to the average value of 3.7, a difference of 16%. It conforms t o the p-chart with a n average deviation of even less t.han 2YO. If we had correlated pro instead of p as a function of T and 8, the result would not have been so good as shown by Table I or the chart. An indirect proof o f the validity of the present modification is the tyork of Xcirton (2 1) on the fugacity chart. H e showed that for twenty-four substances, deviating within 4% from the stanclard curvw, with some exceptions the i"ollo\ring relation holds:
f/P = S ( T , 8)
Vol. 38, No. 8
The use of the ideal reduced volume enables one to c o ~ ~ r c l a1lI ti r~. thermodynamic properties in terms of 'p and 0 or ip antl ?r \\.it11 the same success as in terms of IT and 8. It is hoped that, t l l c . present discussion may help in paving the way for more cxteiisivt. and confident use of the law of corresponding states in p r w t ical as well as in theoretical treatment of the thermodynamic, propi+ tim of real gases ACKNOWLEDGMENT
T h e author is grateful to Jamrs A. Ikattie antl I,zv& for advice and encouragement
I\
IT'aritbii
(101 '11)
(1) Beattie, J . A , , and Bridgeinan, 0. C . , Pioc., A m . A c u d . I ' I * . \ r ~ . , 6 3 , 229 (1928). 1 2 ) Beattie, J. .1.,and Stockmeyer, W. H., P h y s . Soc. f < c ' / j l . / ' I O O , ,\,. Physics, 7, 195 (1940). ( % Brown, G . G., Souders, 1I.,Jr., and Sniith, It. I,,, I ~ I I . C H E Y . , 24, 515 (1932). i - i i ('ope, J. D., Lewis. TV. K . , and Weber, H. C., Iln'd.. 27. $v7 (1931). (5) Dodge, B. F., Ibid.. 24, 1353 (1332). (6) Keenan, J. H., "Thermodynamics", p. 360, New York. JOIIII Tiley 8: Sons, Inc., 1941. ( 7 ) Keyes, F. G., J . Am. Chenz. Soc., 60,1761 (1935). ( 8 ) Lewis, W. K., IXD.ESG. CHEN.,28, 257 (1936). (9) Lewis, W.K., and Luke. C. D., I b i d . , 25, 725 (1938); Oil ./. 32, KO.40, 114 (1934). (IO) .\faron, S. H., and Turnbull, D., IKD.l k ~ CHBX., . 33, 40s (1941); 34, 544 (1942); J . -4m. C'hem. ,%IC., 64, 44, '210.5 * (1342). (11) Sewton, R. H., IND. E ~ GCHEM.. . 27, 302 (1935). (12) Onnes. H. K., and Keesom, TI-. H., Encyc. del, l l a ~ l i .\\'I--., Band V , Teil I. 615 (1911). (13) d u , G. J., thesis. Mass.. Inst. Tech., June, 1937. (143 S u , G . J., and Chang, C . H., ISD. Esc. C H E h i . , 38, 800 ( I94kij (15) Su,G. J., and Chang, C. H., J . Am. C'hem. SOC.,68, 1080 ( 1 9 N (16) \Tatson. K. >I., and Smith, 11. L., iVatl. Petiolriim .\'cws. 28 S o . 97 (19361, (17) 'Teher, H . C., T h e r n i o d y ~ ~ a n i i cf usr Cheiiiical 1,:iigiIii Yew Yo1,k. .John \Viley & Sons, Ini~.,l9:JY. I
Tht~i~efui~c~ t'imrii Kcwton's results we may conclude that t h t . pchart, p = ( ( 7 , e), or the modified law, 'p = F ( T , e), will be app1icd)le t o the tn-enty-four gases studied with about the sanit: degrw of accuracy. The question of constancy of the critical rat,io docls not enter the picture of the fugacity studies. This samc qiicstiori is not involved in the present generalization. Thc. 1)r(wnt inotlification of t,he lair of corresponding states, whic*ilslions that p and not p r c is a function of T and 8, may be consiilcrcd t o serve as a rat,ional basis for the p-rhart and othc,r relatoti correlations. Th?t this is a valid moclification is seeu from the rather wide divergencies of the known values of r0 fi,oni constancy. h number of other investigators used the law essentially in t>heniotlified form without explicitly pointing out that any modification was involved. Probably some of these investigators assumed that the validity of their correlation was limiteri by the lack of constancy of the critical ratio T ~ . With the removal of the constancy of the critical ratio as a necessary condition for the validity of the law of corresponding states, 0111' may feel that the validity of the p-chart and other related ror.rclations is not affected by the variation of critical ratio.
l ' ' r i m
~~~~
I
ACTION OF ANTIFOULING PAINTS Maintenance of the Leaching Rate of Antifouling Paints Formulated with Insoluble, Impermeable Matrices JOHN D. FERRY' AND BOSTWICK H. KETCHUR.1 Woods Hole Oceanographic Institution, V'obds Hole, Mass.
A
-A EFFECTIVE: antifouling paint must release toxic contiiiuously over a prolonged period. Selection of a toxic wit11 ii moderately lorn solubility, as described in the first two papers of this series (f, g ) , can facilitate attainment of this result. H o x -
ever, the paint must be formulated t o provide a mechanism for the eventual dissolution of toxic particles which are originally buried deep below the surface. Orie possible mechanism for the prolonged leaching of toxic from the intcrior of a paint is based on the use of a matrix ivhich is sufhciently permeable so that water can enter the paint film and dissolve the toxic, and the toxic ions can subsequently diffuse to the surface and be released there. Experiments with certain matrices which possess substantial permeabilities to water vapor have been described by Young and Schneider ( 5 ) . Successful paints can also be made with binders which have very low permeability t o water and ions and are insoluble and inerodible. 1
Present address, University of Wisconsin, Madison, Wis.
The niechnnisni 0 1 pruluiiged 1t:acfiing in such prtirit., \rliic.ll rt'quire a high loading of the toxic pigment, to he effectirc:, is tii+ cussed in the present paper. Still another mechanism, ~hic.11 p('rmits paints to be formulated n-ith considerably lower tusii. i o : i t l b ings, will be described in a later paper of this series (.{). PAIBTS WITH HIGH TOXIC LOADING
The leaching behavior of paints with very high loadiiiga of t o 1 1 1 can be explained on the basis of continuous contact, of toxic p a r ticles throughout the paint structure so that, as soon as one p31ticle is dissolved, another beneath it is exposed to solvent ac.tiorr. These have been colloquially t,ermed "cannon ball" paints, sinc~, the particles are pictured as arranged roughly like tmhefamiliar structure of a pile of cannon balls, with the binder filling the interstices. T h e high loading of toxic which is required to provide coritinuous contact demands a strong, tough binder to ensure that ttria
