Review_Cubic Equations of State- Which

7/25/2019 Review_Cubic Equations of State- Which

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Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 8 1

REVIEW

Cubic Equations of

Joseph J. Martin

State-Which?

Departmentof Chemical Engineering, The Universityof Michigan, Ann Arbor, Michigart 48 109

An analysis is made of volume-cubic equations of state , startingwith the most general generating equation fromwhich all specific forms caneasily be derived. One equation is shown to be both the simplest and t h e best inperformance. A new quantity, the t-ch art sum, is presented, which is a useful way to compare all equationsofstate, whether cubic or otherwise,while at the same time permitting excellent predictions ofthe second virial coefficientat the critical temperature fromthe critical compressibility factor. Extensive comparisons of reliable experimentalPVTdata with a number of equ ations of stat e are given for awide range of substa nces from nonpolar to polar,and these demonstrate t h e superiority of t h e simplest two-term cubic equation.

IntroductionSince the time of van der Waals, a century ago, there

has been a steady outpouring of equations of state torep rese nt the pressure-volume-temperature behavior offluids. Th ese have ranged from simple expressions withone or two constants t o complicated forms with upto morethan 50 constants. Th e longer equations have been utilizedfor high-precision work and one can find in the litera tureman y interesting an d useful ones such as the Benedict-Webb-Rubin 3 ) , he Strobridge 3 0 ) , he virial form ofOnnes 2 4 ) to the 17th power in volume, the M artin-Stanford (21), and some recent semitheoretical formula-tions with two or three dozen universal constants an d ahandful of molecular-parameter constants characteristicof the sub stanc e being represented. Although these longand complex equations are desirable for the preciserepresentation of P V T data and calculation of simplethermodynam ic properties, they are not generally preferredfor involved thermodynamic calculations such as vaporpressure a nd lat ent he at of vaporization, mixture behaviorand activity coefficients of mixture components,ormu lticom pone nt vapor-liquid equilibrium ratios becausethe y require tedious manipulation a nd excessive computerstorage in lengthy iterative calculations whichtax even themost modern electronic machines.

T he attractiveness, the n, of the shorter equations l iesin their simplicity of calculation. Th e bulk of these shor tequa tions may be shown to be cubic in volume and theseinclude such well-known forms as van der Waals 3 2 ) ,Clausius (6), Berthelot 4 ) ,Onnes third degree virial 2 4 ) ,Redlich-Kwong 2 7 ) , Wilson 3 3 ) , Barner-Pigford-Schreiner 2) ,Mart in (201, Lee-Edmister 1 5 ) ,Soave (29),Dingran-Thodos (9), Usdin-McAuliffe 3 1 ) ,Redlich 28),Peng-Robinson 2 5 ) ,Fuller (12) ,and Won 34) . It is theseshort cubic equations tha t are to be examined here.

If one is searching for a suita ble equation of state an dhas ruled ou t the e xtended equations as being too cum-bersome and wishes to use a simple cubic, he might thi nkhis problems are over, but m ay be in a quan dary when hefaces the forelisted array of available cubic equations. Howdoes he decide which one to use? A previous stud y of cubicequations by Abbott I ) analyzed some of their charac-teristics and behavior, but did not address the centralques tion, similar to the queen in Snow White who asked,

0019-787417911018-0081 01 0010

Mirror, m irror on t he wall, whos the fairestof them all?Thus, it behooves this study to present a set of argumentswhich will prove beyond any doubt the superiority of oneform over the rest,a formidable objective considering thevast amount of effort that has gone into this subjectpreviously and t he necessityto run down every conceivablealley of appro ach no matt er how blind. In the pu rsuit ofthis objective it is hoped to clear up some of the mythsabout the behavior of cubic equations and the sup posedadvantages of one favorite form over anothe r. Th eprinciples developed here are fundamental and applicableto all equations of state, be they cubicor not.

It has been common practice in recent years to comparethe simpler equations of state by noting how well they

predict saturated liquid volumes and vapor pressuresthrough the equalityof liquid and gas fugacities. Th is hasmerit because of the way such short equations a re used;however, in this work th e comparisons will be made di-rectly to th e PVT behavior in the liquid and gas statesbecause this is more fundamental and through exacttherm odyn ami c relations precludes all equilibrium com-parisons. Also the com parisons will be for compute dpressures rather than computed volumes which is a muchmore severe tes t in the liquid phase . It is only by makingcomparisons with actualPVT data th at the subt let ies inbehavior of the various cubic equa tions can be detecte d.

A. General Source of all Equations of State ThatGive Pressure as a Cubic Function of Volume

Th e analysis will be initiated by setting down t he m ostgeneral form of the volume-cubic equatio n of st ate , withpressure, P , a function of specific volume, V ,and tem-perature , T

which is somewhat different and more inclusive th an onegiven earlier I ) . Here R is the universal gas cons tant,aand 6 are functions of tem perature , andP and y areconstants. Th e latter could also be taken as temp eraturefunctions, but usually the improvement by doingso ismarginal . By specia liz ing the constants and bystraightforward algebraic rearrangement including simpletranslation in volume, all forms of cubic equations can

979 American Chemical Society


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82 Ind. Eng. Chern. Fundarn., Vol. 18, No. 2, 1979

easily be obtai ned from eq1. Following are a nu mb er ofexamples.

I. Let P = y = 0 to get

which is the virial equa tion ( thir d degree) of Onnes24) .11. Rearrange eq 1 o

RT ( V+ p ) aV + 6V ( V + 0)P = V ( V + P)(V + y) V ( V + P)(V +y)

( 3 )

orRT a PRT a y + 6

p = - - (4)

NOW et p = -b = -7, CY PRT = a , a y + 6 = c to giveV + P V ( V + P ) V ( V + P )( V + )

5 )C+

V - b V ( V - b ) V ( V- b ) (V + )RT ap = - -

which is the Lee-Edmister equation 15) , hat as a surpriseto some is just a rearranged form ofP = RT / V - a/ ( V+ P ) + 6/V(V + P)(V - PI .

111. Let 6 = 0 in eq 1 o yield

Now translate by t in volume so t h a t

( 7 )RT ap = - -

V - t ( V - t + P ) ( V - t + y )

Next let t = = b, y = 2t = 2P, and a = a to getR T ap = - -

V - b V ( V + b )

which is the Redlich-Kwong equa tion 27).IV. If in eq 7 we let P = y = t = b , and a = a , we have

which is the van d er Waals equation 3 2 ) .

jus t t as in eq 7

P =

Letting P = b , y = P, and a = a, we have

V. If eq 6 is translate d by + t in volume instead of

(10)a

-Tv - p - t ( V - P - t + P ) ( V - P - t + y)

which is the equation developed by Martin20). I t canalso be obtained by translating the van der Waals eq 9 by1

L .

VI. If in eq 7 wese t t = /3 = b , y = t + c and a = a , weget

RT ap = - -V - b V ( V + C )

(12)

which is the fo rm given by Usdin an d McAuliffe3 1 )andby Fuller 1 2 ) .

VII. Referring again to eq7 , et t = b, P = (2 + &)b,y = ( 2 h ) b , nd a = a to yield

RT ap = - -v - [ V - b + (2 + ) b ] [ V- b + (2 &)b]

(13)

orRT ap = - -

V - b V + 2 b V - b 2

Th e denominator of the last term may be factored t o give

(15)RT ap = - -

V - b V ( V + b ) + b ( V- b )which is the Peng-Robinson equa tion 2 5 ) .

VIII. A form which is not in the li terature but hasintere sting properties is obtaine d from eq 7 by lettingt= b = P + y) /2 , c = - [ p y)/2I2, and a = a , iving

U= - - -TV - b V C

A specialized case of this occurs ify = -P, thenRT ap = - - -v v c (17)

Examples othe r than th e eight presented ca n be given,but they add nothing to the techniques of rearrangementof eq 1.

Clearly, if all cubic equations are merely specialized casesof eq 1, attention should be focused here to determine th echaracteristics of the different forms. Th is may be doneby first solving for the c onsta nts in eq1 by th e classicalmethod of van der Waals 3 2 ) that the first two pres-sure-volume derivatives vanish at th e critical poin t

or by the alternate equivalent technique of using threeequa l volume roots a t the critical point, as explained byMartin and Hou (16). Th e latter is simpler andso will beemployed here. If eq 1 is multiplied out in descendingpowers of volume, it may be written as

v -P y RT a -0 (19)P P

A t the critical point with three equal roots

(V - VC)3 0

v3 3VCV + 3v:v- v,3 = 0

(20)

(21)

Comparing eq 19 and21 for each powerof V a t the criticalpoint, where a and 6 are taken as their critical tempe raturevalues, for v2

which may be expanded to

for V'

for V

It is useful to put these into dimensionless forms withPcV, /RT,= Z,, so that eq 22 becomes

= 1 - 32,P + Y I P,

R TC


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Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 83

eq 23 becomes

CUP P + r)Pc(26)-

RTC- 3 2 2 - p

R2T: R2TC2prp:

and eq 24 becomes

The last three equations are independent, bu t there arefour unknowns, a P, y , and 6 , or their dimensionlessequiv alents , ifP,, V,, and T , are known. Th us, a uniquesolution can be obtained only if one of the unknowns isspecified-usually as zero. T o exploit this approach eq 25and 27 are inserted into eq 26 to get

CUP 6P,2- - - - 2: + 32: 32, + 1 (28)R 2T,2 R3T C 3

which will be examined for its various possibilities.

B. Consideration of Equations for the CriticalIsotherm

Suppose our interest first is only along the critical

temperature l ine and we want to study two-term cubicequations (which th e majority of them are),so 6 is set equalt o 0 and eq 28 becomes

PC - -2: + 32: -32, + 1R 2 T C 2

T he value of2, can be s et as we please,so after somepretrial let us take it to be 0.25. Thi s makes

aPc 27- -R2T: 64

at the crit ical temperature, and

Equa tion 25 then becomes

RTC rpc 1+ - = -64yPc RT, 4

or

+ - = o (32)4RTc 64

Solving gives

YPC 1= -T , 8

Inserting this into eq 31 yields

PPC 1

RT, 8= -

(33)

(34)

Here, the unknowns,a , P , and y , have been solved for asdimensionless quantitie s to be used in th e equ ation ofstatein reduced or dimensionless form which is obtained bywriting eq 1, with 6 = 0, as

p R T V,T , ( aP, R2T:) (R2T,2 VC 2)(35)

or with the definitions of the reducedexper imenta l

_ -pc vpcvcTc PC 2V C2 (VP)(V+ y)

variables PR = PIP,, VR = V/V, , T R = TIT, , and Z, =(PC clRTc) xp

This may be translated byt by replacing V in eq 35 withV - t or V Rn eq 36 or 37 with VR P,/Z&T,. If we thenlet t = P or y so tha t f rom eq 33

C D .

and

(39)

which is a well-known reduced form of the van de r Waalsequation.

It is instructive to note the effect of translation on acalculated or selected 2, (not the experimentalZ,). We firstnote that

which follows from V being replaced with V t ,thus ,giving

Here the superscripts, bt an d at, mean before an d aftertransl ation. For eq 37 transla ted according to eq 38 it isseen from eq 41 th at

2, = 0.25 + 1/8 = 0.375 (42)As emphasized earlier (20,221, eq 39 gives results con-

sisten t with its development onlyif 1, = 0.375 for whichvalue it gives PR = 1 a t the critical point whereV R= 1 a tTR = 1. If, however, the ac tual V ,or Z c is used for a givensubstance, the represen tation along theTR = 1 isothermup to the critical density will be vastly improved eventhough the prediction right atVR= 1 s slightly off. TableI gives a comparison with the National Bureau ofStandards data (13)of the predictions for argon whoseZ,= 0.29121.

Th e earlier discussions(20,21)showed tha t this principlemu st hold for the reduced form of any equatio n of state ;Le., wherever Z, enters , the experimental value should beemployed and not the value used to get the equa tionorpredicted by the equation. Th e reason is simply th at thereduced volume is based on the actua l critical volumesothe value of Z, must also be calculated with the actualcritical volume.

It is of interest n ext to look a t the second virial coef-ficient of tran sla ted equa tions. If eq 7, where6 = 0, isrearranged to solve forz

(43)tP CYRT

z = 1 + - -RT[1 + P / ( V - t ) ] [ V - + y ]

Now in general asP -(44)

RTV - ( T )P


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Table I. Pressures on the Critical Isotherm for Argon Using Values of 2, = 0.375 and 0 . 2 9 1 2 1 in Eq 3 9

Ind. Eng. Chem. Fundam.,Vol. 18, No. 2, 1979

VR P, NBS % dev pR o 2 I % dev

100 0 .033 95 0 .026 462 0 0 . 1 62 0 9 0.128 09

5 0 .540 73 0 . 4 5 1 4 32 .5 0 .834 2 1 0 .750 771.25 0.995 97 0.989 091 .0 1 .000 00 1.000 00

where t is the residual volume. Inserting this into eq43and le t t ing P

-or V

-Differentiating with respect to pressure a t constanttemperature

Th is may be put into reduced form

It has been shown earlier 18) ha t this slope of a constan t

TR line as PR - on the usual compressibility plot of2

vs. P R is just the generalized second virial coefficient,B P J RT, where B is the second virial coefficient in th eexpression, i = 1 +B / V + C / P + .... If we now combineeq 41 and 47 with 2 = Z and Zcat being Z, for any equationwith or without translation

as PR - 0 (48)C - T , ( d z ) ffPCdPR TR R2T,2TR

At th e crit ical tem peratu re this becomes

where BG1is th e generalized secon d virial coefficient a t areduced temperatu reof 1. Th e interesting characteristicof this quantity is thatit is independent of translationbecause t has been eliminated. As an example for eq37

27z ( ) T R = l , p R = O 6 4= 0 . 2 5 + - 0 .671875 (50)

Th us, for van der Waals eq39 it follows automatically th at

P,=o = 0 .671875 (51)

This can be checked by combining eq30, 38, 42, and 47at TR = 1

1 278 6 4

= 0 . 3 7 5 - + - 0.671875 (52)

Th e constancyof eq 50,5 1, and 52 leads one to speculatewhether this quantity, which will be designatedx, hei -char t sum

is th e same for all actual substances. One might guess thisto be so by noting the shape of theTR = 1 ine on the .Zchart for two specific cases as shown in Figure1. If asubstan ce has a small negative slope atPR= 0, it will have

- 2 2 .1 0 .0 3 3 9 0 - 0 . 1-21 .0 0 .163 03 +0.6-16 .5 0 .552 30 t 2.1

10.0 0.862 35 + 3.30 .7 1 .000 04 +0.40 1 . 0 4 1 7 4 +4.2

S u b s t a n c e 12 /

Figure 1. Generalized compressibility or chart.

Table 11 5Chart Sum, E , or a Number of Substances-

substance Z C a E

N e 0.3090.3050.2950 . 2 9 40 . 2 9 10 . 2 9 10 .2900.2890.2880.2880.28 30 . 2 8 10.279

0 .2790.2780.2780.2780.2740 .2740 .2740.2740 . 2 7 10 .2700.2700.2680.2670.2640 . 2 6 10.2600.259

0 . 2 5 90.2480.2480 .2420.2380 . 2 3 4

0.3250.3250.3300 .3240 .3300.3320.3330 . 3 4 10.3390 .3330 . 3 4 10.3370.340

0.3400.3450 .3360.3390.3450 . 3 5 00 .3410.3480.3450.3520.3630.3650.3460.3680.3680.3600.355

0.3520.3560.3580.37 00.3650.340

0.6340.6300.6250.6180.6210.6230 .6230 .6300 . 6 2 10 . 6 2 10.6240.6180.619

0.6190 .6230 . 6 1 40.6170.6190.6240.6150.6220 .6160.6220 .6330 .6330 . 6 130 . 6 2 40.6290 . 6 2 00 .61 4

0 .6110 .6040.6060.6120.6030.574

average (exclusive of H,O) = 0.62

- dz/dPR )TR = ,, = = BPc 1R Tc,

a high t,, nd vice versa. T he verification of thi s idea isshown by Ta ble 11, where i t will be seen that the i- cha rtsum is remarkably constant. Th e only notable exceptionis water, which is highly polar, butso also is ammoniawhich behaves like the other substances. In fact, whatvariability there is in may be due to lackof precision


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Ind. Eng. Chem. Fund am., Vol. 18, No. 2, 1979 85

Now three arbitrary values of 2, (0.23, 0.25, 0.27)areinserted into eq 65 and each equation is translated byt P, / RTc to give a good f i t at about twice the crit icaldensity. As shown in the pioneer paper(20) on volumetranslation, a cubic equation cannot be adjusted to fit thecritical isotherm a t reduced densities of both1.5 and 2.0.Th e translation selected in m ost cases in this pap er willgive deviations t ha t are slightly negative a t twice th ecritical density a nd of the ord er of25% positive a t abo ut

1.5 time s the critical density. If interes t is in reduceddensities only to 1.5, the translation can be reduced an da good f i t obtained a t 1.5, but large negative deviationswould occur a t twice the critical density and would be farworse if the equation were to be used at even higherdensities. Table I11 gives the comparison withNBS da tafor argon for the three values of2,. For all densities upto th e critical the deviations are well below1% for 2, =0.25. Th e results are not so good for 2, = 0.23, whererelatively large negative deviations occur, and for2, = 0.27,where large positive deviations occur for d ensitie s belowthe critical. Above the critical the th ree equa tions givesimilarly large deviations. Th e best equa tion overall is for2 = 0.25, translated by 0.082, and can be written

in the experimental determina tion of2 , a n d B P J RT, , or(dZ/dPR)TR=l o, though it is possible tha t better da ta willshow ten&g to be slightly lower for some compoundssuch as alcohols and ketones. Th e average value of =0.62 allows one to predict with considerable confidence thesecond virial coefficient a t the critical tempe ratu re fromthe critical compressibility factor for any substance

( 5 4 )

Returning to eq 37, it may be translated in VRby (0 +t )P, /Z T, instea d of just tP,/@T,. Taking @ P, / RT, =118 as before

27 /64( 5 5 )RpR= -

z,VR P c / RTc 1/8 z , V R - tP,/RT,) '

which is the reduced form of eq11.The development leading to eq 37 was on the as-

sumption of a value for2, of 0.25, but the procedure couldhave followed Exa mple V of the several cubic equation sby setting p = y. Thi s would have made eq25 become

and eq 27 become (6 = 0)

- - - 2:'P,'

R2T( 5 7 )

Combining eq 56 and 57 to eliminate P P J RT, yields

4 Z C 3 2, + 6 2 , 1 = 0 ( 5 8 )Th is may be solved by Cardan's me thod for the roots,2,= 114, 1, a n d 1. Since 2, = 1 implies ideal gas at thecritical point, only the first root is meaningful in thisapplication and it is identical with the assumption whichgave eq 30, 33, and 34.

Because eq 55 differs from eq 39 only by translatio n, itfollows th at = 0.671875. As pointed out earlier 20),however, the translation characteristic makes eq55 muchmore powerful tha n eq 39 because 2, is not constrainedto 0.375 and can be made more realistic.

If in eq 29 2, is taken to be less than0.25, the followingresults

2, = 0.24; aP,/R2T,2 = 0.4390; = 0.6790 (59)

2, = 0.23; aP, /R2T: = 0.4565; = 0.6865 (60)

Since the desired value of2 is 0.62, it is clear that to make2, < 0.25 is going in th e wrong direction. For valuesgreater than 0.25, we get

2, = 0.26; aP, /R2T: = 0.4052; E= 0.6652 (61)

2, = 0.27; aP, /R2TC2= 0.3890; X = 0.6590 (62)2, = 0 .3 ; aP, /R2Tc2= 0.343; = 0.643 (63)

2, = 1/3; aP, /R2T, ' = 0.2963; =0.6296 (6 4)

Here th e trend is toward0.62, but it must be noted thatis indicative of the behavior only up to the critical

density, so the behavior above the critical density orvolumes less than the critical volume) must be studied.Therefore, eq 36 will first be combined with eq25 , 26 , 27to obtain after translation

PR = TR/(Z,VR - t P, / RT, ) -ZC3+ 32,' - 3 2 , +l ) / [ ( Z , V R- tP,/RT,) ' + ( 1 - 3 2 , ) ( Z C V ~ t P, / RT,) +

Z23 (65)

or in its reduced form before a specific translation is made

( 6 7 )2 7 / 6 4

T RPR = -Z,VR - tP, /RTc B,VR + 1 / 8 t P, / RT J 2which is the equivalent of eq55 or 37 with translation.

The problem of large deviations forp R above 1.2 ischaracteristic of all cubic equations of sta te, for the actua lda ta tend to follow a fourth-degree equation as discussedpreviously (21). The major difference between the manydifferent cubic equations that have been proposed iswhether they fit better at ab out two-thirdsor one-half the

critical volume. One of the major ad vanta ges of eq67 istha t a small change in translation will permit fitting exactlywhere one chooses. In obtaining eq65 from eq 25,26,and27, it was no t necessary to solve explicitly for/3 and y , butif one wishes to do this, he would find th atp and y are realnumbers for 2,s 0.25, but imaginary or complex numbersfor 2, > 0.25.

We can now show how someof the well-known cubicequa tions fit into the general pattern . As shown in ex-ample 111, if we let y = 2P and t = p , we get the Red-lich-Kwong equation. Using dimensionless quantities, eq25 becomes

1 - 3 2 ,PPc

RTC

and eq 27 with 6 = 0 gives2p2pc2- - - 2:R 2 T C 2

Eliminating PP, /RT, between eq 68 and 69 yields anexpression for 2, (similar to eq 58)

92: - 182,' + 122, - 2 = 0 ( 7 0 )Again this may be solved by Cardan's method to give tworoots that are complex and the third which is real anduseful

2, = 0.246693 (71)

Th e translation for eq67 is, theref ore, from eq7 1


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86 Ind. Eng. Chem. Fundam., Vol. 18, No. 2 , 1979

" m 3 m C . l m t - m m w o3 o o * t - w m o m w o o m o 2 35 0 0 0 0 0 0 0 0 A i i ~ ~ e i m

I + + + + + + + + A * + W

+ + I c o a ,

tP, PP, bP, 1 - 32,- - - - - = - - - 0.08664035 (72)RT, RT, RT, 3

and using eq 71 in eq 65 a t the crit ical temperature

0.4274802-

RP -- Z,VR 0.08664035 Z,VR(Z,VR+ 0.08664035)

(73)

which is the reduced form of the Redlich-Kwong equatio nbefore tem perature s other th an th e critical are considered.Table I11 gives the results predicted by this equ ation an dit will be seen tha t thefit is similiar to th at of eq 66. Thi sis expected because the two equations have a similarmathematical form as discussed before(20) and since theRedlich-Kwong equa tion gives Zcat= 0.246693 +0.08664035 = 113 according to eq 41, while th e exa mpleof eq 66 gives2cat = 0.25 + 0.082 = 0.332, which valuesarealmost identical. T he Redlich-Kwong, however, mu st beslightly inferior because its Z-chart sumis = 0.246693+ 0.42748 = 0.674173, which is to be compared with0.671875 of eq 66 and the desired value of about 0.62.Furth ermo re, the Redlich-Kwong will not fit oth er sub-stances nearly as well as for argon whereas eq 66 can easilybe made to doso by translation . Natura lly,if one wishesto fix th e translation for many substan ces and never allowit to vary (thu s, fixed2cat), his can be done by using anaverage value th at is best for all the substances involved.Of course, the Redlich-Kwong equation itself could betranslatedto give a better fit of the experimental data thangiven by eq 73 bu t it would be no better t ha n eq 67 andwould be more complex in form.

From example VI1if we let

2 - d 2P = (2 + f i ) b and y = (2 &)b or y =

we obtain the Peng-Robinson equation. Now eq 25 be-comes

= 1 - 32, (74)PPc

and eq 27 becomes

Eliminating pPC2/R2Tc2ields the cubic equation in2,

As with eq 70 two roots are complex and th e third an d

useful root is2, = 0.229605235 (77)

This may be inserted into eq 75 to give according toExample VI1

= 0.077796074 (78)P, bP, PPC- -

RT, RT,- (2 + &)RT,Using eq 77 an d 78 in e q 65 givesP R = TR/(Z,VR - 0.077796074) -

0.457235529/[z,vR(zcvR +0.077796074) +0.077796O74(Zc V~ 0.077796074)l (79)

which is the reduced form of the Peng-Robinson equatio n


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the critical if the n umerator of the second term is takenas a tempe rature function (Le.,a ( T ) which is 0.195112 a tT,). For tempe rature s below the critical and densities upto the critical, the equation does an excellent job.

For the three -term cubicit is instructive to s tar t withthe conditions of example11, backtrack to I, and thengeneralize. In eq 1 let P = -7, as done to get eq 5. Fromeq 25 it follows th at 2, = 113. Combining eq 26 and 27the n gives a relation betweency and 6

for the critical isotherm . Ta ble I11 gives the predictionsof thi s equation, and it is seen tha t these are somewhatlower than those of the equation based on2, = 0.23 andtranslated by 0.09. If the latter had been translated by0.0774 so that i t s Zcat = 0.3074 which is almost the sameas 2, = 0.229605 + 0.077796 = 0.307401 in the Peng-Robinson, the two would predict almost the same. Lookinga t the Z-chart sum for the Peng-Robinson equa tion, wefind 0.229605+ 0.457236= 0.68684. Recalling againthat the desired value is 0.62, it is seen that eq 79 issomewh at worse tha n eq 55or 67 or even the Redlich-Kwong equation.

From example VI if we lety = P + c and t = 0, we getth e Usdin-McAuliffe or Fuller equation(12). We note tha tif c = 0, we obtain 2, = 0.25 from eq 56, 57, and 58, whichwith the indicated translation gives the van der Waalsequation (39). If c = P, eq 68 ,60, and 70 give2, = 0.246693which with translation gives the Redlich-Kwong equation(73). If we take y = m@,which is equiv alent toc = (m -1)P or P + y = (m + 1)P, m may be set at will. In thisprocedure eq 25 becomes

PP, 1 - 32,(80)- -RT, m + l

and eq 27 with 6 = 0 becomes

P2PC2 2:(81)

R2T,2 m

Once m is set,P can be eliminated between eq80 and 81to give a cubic in2, whose roots can be det ermin ed as ineq 58 or 70. Alterna tively, 2, may be set and the twoequations solved form. This is easier because m occursin a quadratic rather than a cubic. As an example, take2, = 0.245 which determinescyPc/R2T?by eq 29. Theneq 80 and 81 give m = 2.34962058, and eithe r eq80 or 81gives PPJRT, = 0.079113438. N oting th atP = t , eq 65yields

TR 0.430368875PR =Z,VR 0.079113438- ZcVR(ZcV~0.106773124)

(82)

Th e results of this are tabulated in Table111. Had eq 67been translated by 0.075 instead of0.082, it would havegiven almost the sam e results. Th is is because of similarityof form andZ,st. For eq 82 it isZFt = 0.245 + 0.079113438= 0.324113438 and for eq 67it would be 0.25 + 0.075 =0.325. Th e two equa tions are of abou t equivalent sim-plicity, but it is easier to fix the transla tion in eq 67 thanto assume 2, and calculate m,P, and cy in eq 65 to get eq82.

From example VI11 it is seen through eq 25 thatif 2,< 113, b turns out to be a positive number and th e resultsare similar t o those from eq 65 for 0.25C 2, < 113. If 2,> 113, however, th e value of the 2-chart sum c an be mad eclose to 0.62 and the predictions are fantastically goodbelow the critical density. Th is is shown in Tab leI11 for2, = 0.42, which gives the eq uatio n

(83)0.195112

( Z , V R ) ~(0.239140)2-

RPR = ZCVR t 0.13

whose 2-chart sum is = 0.42 + 0.195112 = 0.615112.It might be expected that this would be a good equation

to use for tempe ratures away from the critical and densitiesup to the critical. Unfortunately,it does not have theability to inflect properly a t the critical as can be seen bythe low pressures above the crit ical density. Th is effectis carried back t o lower densitiesfor temperatures above

Now let cyPc/R2T,2= 113, which makes6P:/R3T: = 1/27.Put ting th is into eq 27 makes = y = 0, which gives thesimple virial eq 2 that in reduced form is

This equation can be obtained directly as in example I byletting = y = 0. Th e predictions of eq 85 are given inTable 111. A t low densities the results are good, but athigher densities it gives significant deviatio nsso tha t theaverage overall densities to twice the critical are notsogood as for th e two-term eq 55 or 66. Equa tion85 can betranslated to improve the predictions a t high density, butthis will worsen tho se below the critical density a nd th eaverage predictions will be no bet ter t ha n eq 66.If interestis only in densities less than the critical, eq85 can betranslated by just th e right amou nt to do a slightly bette rjob tha n any of the other cubics. Th is is because its 2-chartsum, E, of 113+ 113 = 213, is less th an and closer to 0.62than any of the others.

Th e Lee-Edmister equation 5 permits settingb a tsomething other tha n0 which is the u niqu e value givingthe virial equation. An equivalent way to do this in eq 84is to set aP,/R2T,2 at a value othe r tha n 113, since eq 5has been shown to be a form of eq1 with P = -7. IfcyP,/RT, i s taken as 8/27, then 6PC2/R3Tc3 0 a n dPyPc2/R2Tc21/2 7 from eq 27 so tha t in reduced formeq 1 becomes

m I n

Th is is just a reduced form of eq 17 previously consideredand it predicts high for densities below the critical and lowfor densities well above the critical.

By trial it is found tha t the best three-term equation forargon, with P = - y is obtained by lettingaP,/R2T,2 = 0.34which makes 6P,2/R3T2= 0.437037037 a nd /3-yP,2/R2T,2= -11150. Th is gives

pR= --- R 0.34 0.0437037037~ C V R ( Z , V R )~1/150 Z,VR[(Z,VR)~1/150]

(87)

which is the reduced form of eq1 ha t is equivalent to eq5. Tab le I11 gives the results for this eq uatio n andit isseen that i t is about as good, bu t no be tter th an eq 66.Varying cyP,/R2T,2does not improve the situation. Itsthir d term , therefore, does not giveit any advantage a t thecritical tem per atu re over two-term cubic equations. Fortemperatures away from the critical,it does offer thepossibility of two temperature functions (i.e.,a(? ) ndG(T))compared to just one for two-term equations, bu t thisdoes no t giveit any better overall behavior because of theoverpowering defect of all cubic equations at reduceddensities in the range of 1 .5 to 2.0.


8/17

88 Ind. Eng. Chem. Fundam.,Vol. 18, No. 2, 1979

P I

1 2 14 16 -R 8 2 0 2 2 2 4O

F i g u r e 3. Generalized vapor pressures with critical-point slopes.

Figure 2. Typical pressure-temperature graph.

C. Consideration for Temperatures Other Thanthe Critical

As mentioned for eq1, he volume constants,/3 and ycould be dependen t upon tem perature ; however, here theywill be assumed to be temperature independent as iscustomarily do ne, and also any translation in volume willbe con stant, whilea and 6 will be considered to be tem-perature d ependent. Since eq 66 is as good or better thanall of the other two-term cubic equations and at the sametime is the simplest in form,it will be examined first. T omake its a a function of tempera ture, the least complicatedexpression that can be employed isTR , and it may beintroduced into eq 66 as

~ - 1 . 0+ I

F i g u r e 4. Behavior of the second derivativ eof pressure with respectto temperature a t lower temperatures.

the critical pressure is 48.34 atm , so1/TR = 1.7285 andPR = 0.020687. Placing th is point in Figure 3 shows th atthe slope of the critical isometric is about 6.1. (Becauseof the inhe rent lack of precision in a derivative comparedto a primitive, this should no t be taken to be necessarilybe t te r than f5% .) T o utilize this, differentiate eq88 toget

Th is is jus t a slight modification of what Berthelot(4 )did80 years ago when he put T in the denominator of thesecond term of the van der Waals equation.

T o determin e the value ofn, use is made of the slopeof the critical volume line on a pressure-temperaturediagram or th e slope at the high end of a vapor-pressureplot. As developed and emphasized in earlier papers(16,19,20,21), he critical isometric is essentially, thou ghnot exactly, straight for all substances an d its slope is theslope of the v apor-pressure curve just a s it approaches thecritical point. Figure2 shows this characteristic an d alsoshows tha t for rathe r high tem peratures, whereT R- ,the critical isometric (an d all the othe r isometrics) tend sto curve down very slightly; however, this effect is minorcompared to others and is outside the range of temper-

atures encountered most often and considered in thisdiscussion.Figure 3 shows the generalized vapor-pressure plot of

log PR s. l/T R, presented several time s in the literature(16,19,21). Th e param eter on each curve is the slopeM= dPR/dTR as T R - . All that is needed to establishwhich curve a substance follows, and t hu s what will be theslope of th e critical isometric, is the critical point a nd onevapor-pressure point suchas he normal boiling point. Th ecurves for different substances never cross each other onthis plot so tha t a single vapor-pressure point determinesa curve all the way to th e critical point with a unique slopea t the critical point. (Th is is often useful for gettinginterme diate vapor pressures.) In the case of argon whichis th e example used in thi s discussion, the no rmal boiling

point is 87.28 K, the critical tem per atu re is 180.86 K, and

One quickly notes that for all isometrics ( dP R/ dT R) Vsgiven by eq 89 decreases with tem peratureif n is a positiveconstant. This is in accordance with the experimental factsfor volumes greater than the critical volume, as shown inFigure 2, but it is contrary to the facts for volumes lessth an the critical volume, an d of course, the critical volumeline has little appreciable curvature as mentioned above.Also, as the volume decreases to about half the criticalvolume, (dPR/dTR)Vends to become constant again. Inano ther way (dz pR/dT R2)Vs negative for large volumes,zero at a bout the critical volume, positive for volumes lesstha n the critical, and negative again for volumes less th anhalf the critical volume. Th is is shown in Figure 4 wherethe solid line is the experimental data.

Despite the inability of eq88 or any similar two-termcubic, to give the correct curva ture to the isometrics bymaking CY a function of temperature,it is common practiceto use such equations andf i t the dat a as best as possible.Th e dashed line in Figure4 hows how this approxim ationworks. Th e results, although not really precise, are goodenough to make useful calculations. T he principal reasonthe approximation works is that the curvature of theisometrics is small and other d eviations are of muc h greatersignificance.

To app roxim ate the critical isometric with eq 88, onecan use eq 89 to calculate its slope a tT R = 1 and T R = 1.5and note the dow nward curvature. Th e reason for selectingthis tempera ture range is not only tha t most situationsinvolve tempe ratures belowT R = 1.5, but th at the crit ical

isometric is linear between1 and 1.5 and only s tar ts to


9/17


Table IV. Comparison of Six Equations of State with Data ( I 3 ) for ArgonMa r t i n - wi t h Ya r t i n- wl t h

TC = i o 0 0 . 5

1 . 21 . 5

150. 86X

F C = 2 0 0 .7i

1 . 21 . 5

0 0. 8

4 8. 3 4a t m

C

1 . 23 . 4 1

1 . 53 . 87

y m o l e ,

R = 1, 082j j j j l at r r . 2

q r r o l e . K 1. 52 . 3 . 9 5

c . 29121 1. 2z = 1

1. 51 U 3 . 6 . 9 9

11 . 21 . 5

1 . 21 . 5

1 . 21 . 5

i . 21 . 3

1 . 21. 5

1 . 21. 5

1. / 1. 4 . 991i . 21 . 5

1 / 1 . 6 . 9 6

. / 0. 8 1

d 3 . 9 1

1, l . l 1

j.; 1. 2 1

1 . 21 . 5

l Vi . 8 . 9 2

1 . 21 . 5

; / 2 . 8 6

1 . 21 . 5

p RD a t a

, 0 2 00 1. 0 3 39 5, 0 4 08 8. 0 5 12 7, 1 07 0 4. 1 6 20 9, 1 9 80 8, 2 5 15 9, 2 2 81 3. 30565, 38079, 4 9 18 3, 4 30 0 8, 5 4 0 73, 7 0 43 2, 9 4 21 2. 7 3 59 1. 834211 . 2 1 341 . 7 4 30. 9 2 68 2* . 9 63 2I . 58612. 4836. 995601 . 9 C 023 . 2 2 17. 999462 . 0 34 -3 . 6 1 67. 9 9 99 92 . 2 1 744 . 0 4 711 . 00062. 39834 . 52521 . 0 0 582 . 6 1 055 . 0 7 38, 9 6 55 1

3. 19666 . 4 7 5 9. 790031. 34264 . 1 9 938. 5241. 6 4 322 . 1 5 345 . 9 7 6311 . 6 1 9, 4 5 8 9 44 . 03209 . 0 6 8 21 6, 318

1. 3635

R e d l i c h - S o a v eKwong

( A ) B l

R D e v . R oi l . ., 0 2 00 2 1. 01 , 0 2 00 5 + . 1 7. a 3 39 4 - . 03 , 0 3 33 4 - . 03, 0 4 08 7 - . 03 , 0 4 08 7 - . 03, 0 5 12 5 - . 03 . 0 5 12 6 - . 01, 1 0 71 6 +. 11 , 1 0 75 6 +. 49. 1 6 1 8 7 - . 1 4 , 1 6 1 8 7 - . 14, 1 9 78 1 - .14 . 1 9 78 4 - . 12, 2 51 30 - . 11 , 2 51 55 - . 0 2, 2 2 84 1 t . 1 2 , 2 29 10 t . 1 3, 3 0 49 7 t . 2 2 , 3 0 49 7 - . 2 2, 3 80 02 - . 2 0 , 3 80 14 - . 1 7, 4 9 09 2 - . 1 9 , 4 91 93 7 . 0 2. 4 3 12 8 +. 2 8 , 4 3 25 1 - . 5 7, 5 3 99 3 - . 15 , 5 3 9 9 3 - . 13, 7 02 56 - . 2 5 , 7 0 30 2 - . 18, 9 3 9 9 8 - . 2 3 , 9 4 39 0 1 . 1 9, 7 4 16 9 1 . 1 9 , 7 4 27 8 t . 9 3, 8 3 82 7 t . 50 , 8 3 83 7 +. 501 . 2 1 30 +. 2 l 1. 2147 +. 361 . 7 50 2 c . 4 1 1 . 7 6 51 1 1. 3, 9 35 42 r . 9 3 , 9 3 57 6 +. 96, 9 6 82 6 +. 53 , 9 6 82 6 +. 5 31 . 6 0 40 +1. 1 1 . 6 0 77 t l . . ,2 . 5 0 49 t . 5 6 2 . 3 3 66 +2. 1, 9 9 93 1 t . 38 , 9 9 93 7 - . 3 d1 . 9 4 7 5 +2 . 5 1 . 9537 1 2. 83 . 2 8 02 +l . 8 3 . 3 3 3 9 +3. 51 . 0 0 00 t . 0 5 1 . 0 00 0 - . 3 52 . 1 2 48 t 3 . 4 2 . 1 3 25 +3. 83. 7314 t 2 . 3 3 . 7 6 78 - 4 . 21 . 0 0 33 t . 33 1 . 0 0 33 + . 3 32 . 3 19 5 1 4 . 6 2 . 3 2 88 1 5. 04 . 1 6 04 +2 . 8 4 . 2 4 05 - 4 . 81. 0195 +.9 1 . 3 19 5 t ? ?2. 5429 - 6. 0 2 . 5 5 39 +6 . 34 . 6 7 02 - 3 . 2 4. 7650 +5. 31 . 0 5 98 +5. 4 1 . 0598 - 5 . 42 . 8 0 76 + 7 . 5 2 . 8 2 04 a , ?3 . 2 4 57 7 3 . 4 5 . 3 5 60 t 5 . 61 . 1 4 71 t 1 9 . 1 . 1 48 6 +1 9.1 . 2 6 46 +18. 1 . 2 6 16 +; 8.3. 5202 l o . 3 . 5 3 70 +l l .6 . 6 6 43 1 2. 9 6 . 8 0 81 +5 . li . 1 43 5 - 4 5. 1 . 1 5 33 +46.1 . 7 4 32 +3 0 . 1 . 7 4 32 - 3 0.4 . 6 0 21 +l o . 4 . 6 23 1 - 1 0 ,5 . 5 9 0 5 + 78 8 . 7 7 07 - 2 . 91 . 1 5 42 +78. 1. : 860 - 8 3.2 . 6 7 17 - 24. 1. 6717 - 2 4 .6 . 2 6 00 +4.8 6 . 2 8 55 - 5 . 211. 2 77 - 2 . 9 11 . 497 -1. :. 9 5 35 1 +l o8 1. 0468 +1284 . 3 1 80 +7. 1 4 . 3 1 80

- 7. 13 . 8 11 3 - 2 . 8 8 . 8 41 6 - 2 . 51 5 . 11 7 - 7. 4 1 5 . 3 5 8 - 5 . 8

3. 42748

P e n g -Robi nson

, 0 2 00 3 + 07, 0 3 39 0 - . 16, 0 4 08 2 - . 1 5, 0 51 20 - . i 4, 10703 . O O, 1 6 08 9 - . 7 4, 1 9 56 2 - . 7 4. 2 50 02 - . 6 2, 2 2 69 0 - . 5 4, 3 01 60 - 1 . 3, 3 7 5 7 7 - >. 3, 4 8 62 3 - 1 . 1, 4 2 59 0 - . 9 7, 5 30 23 - 1 . 9, 6 89 05 - 2 . 2. 9 2 4 5 2 - 1 . 9, 7 23 81 - . 8 3, 8 2 04 0 - 1 . 71 . 1 78 4 - 2 . 11 . 7 0 52 - 2 . 2, 9 24 50 - . 2 5

1 . 3 50 8 - 2 . 22 . 4 2 35 - 2 . 4. 99654 t . 09; . 8709 - 1 . 53 . 1 46 7 - 2 . 3, 9 9 9 8 8 + . a 42 . 0 2 91 - 1 . 23 . 1 2 9 8 - 2 . 41 . 3 0 0 2 +. 0 22 . 1 9 64 - . 953 . 9 3 8 1 - 2 . 7L. 0039 +. 332 . 3 7 93 - . 7 94 . 3 80 2 - 3 . 21 . 0 18 2 +? . 22 . 5863 - . 9 44 . 8 6 55 - 4 . 11 . 0 0 69 + 4 . 31 . 1 0 84 +3 . 73 . 1 06 3 - 2 . 96 . C 0 9 3 - 7 . 2, 8 3 21 7 - 5 . 31. 3416 - . 073 . 8 4 11 - 8. 57 . 4 72 3 - 1 2., 5 4 76 4 - 15.1 . 8 117 - 1 5.4 . 9 0 27 - 1 8.9 . 3 9 49 - 1 9.- . @ E 0 2 - 118

2 . 6 52 3 - 3 4.6 . 4 52 1 - 2 9.11. 9 79 - 2 7.

, 9 5 46 3 - . a 9

M a r t i n

( D )

2 D e v .

. 0 2 00 1 - . 0 4, 0 3 39 4 - . 03, 0 40 88 - . 0 2, 0 51 26 - . 0 2. l o 69 9 - . 05, 1 61 89 - . 1 2, 1 97 89 - . l o, 2 5 14 2 - . 0 7, 2 28 06 - . 0 3, 3 05 03 - . 2 0, 3 80 31 - . 1 3, 4 9 13 6 - . l o, 4 3 0 48 + . a 9. 59004 - . l 3, 7 0 35 1 - . 12. 9 4i 4 5 - . 07, 7 4 05 7 + . 6 3. 8 3 32 7 - . 111. 2156 +. 431 . 7541 + . 6 4, 93480 +. 86, 9 6 7 9 7 +. 501. 6080 t l . 92 . 5 0 9 7 +1 . 1, 9 9 93 3 + . 3 71. 9515 +2 . 73 . 2 8 13 C 1 . 91. 0300 . Os2 . 1 2 75 +3 . 61 . 6 9 74 + 2 . 21 . 0 0 2 9 +. 2 92 . 3 1 93 + 4 . 64 . 1 4 78 12 . 51 . 0 1 74 +1. 72 . 3 3 7 1 +5 . 84 . 6 4 4 2 +2 . 71 , 0 5 4 4 +4 . 82 . 7 9 23 +7 . 05 . 1 99 5 - 2 . 5: . 1235 +1 6 .1 . 2 40 1 - 1 6.3 . 4 6 62 8 . 46 . 5 4 66 +1 . 1i . 0 77 3 +36.1 . 6 6 55 t 2 4 .4 . 4 6 14 7 6. 2

, 9 9 92 2 5 4 .2. 4740 +15.

1 8 . 7 5 2 - 7 . 5, 6 3 5 39 +3 8.

3 . 8 62 2 - 4 . 28 . 1 38 2 - 1 0.1 4 . 0 9 4 - 1 4 .

a . 3 31 8 - 2 . 3

5 . 9407 - . 60

~ ...

s m a l l e r L i n e a rt r a n s l a t i o n I s o m e t r i c s

( E ) g ( F ) ~

R D e v . R D e v .

, 0 19 94 - . 3 5 , 02008 +. 32, 0 3 39 2 - . OX , 0 33 93 - . 0 5. 04087 - . 0 4 . 0 40 86 - . 0 6, 0 5 12 6 - . 0 2 , 5 12 51 - . 0 3, 1 05 94 - 1 . 0 , 1 0 79 5 +. 88, 1 6 15 0 - . 3 5 , 1 61 71 - . 2 3, 1 97 70 - . 1 9 , 1 97 53 - . 2 8, 2 5 13 6 - . 0 9 , 2 51 26 - . 1 3, 2 2 52 8 - 1 . 3 , 2 2 98 4 + . 7 5, 3 0 36 6 - . 65 , 3 0 43 9 - . 4 1. 37968 - . 2 9 , 3 78 93 - . 4 8, 4 9 12 2 - . 1 2 , 4 90 78 - . 21, 4 23 45 - 1 . 5 , 43375 +. 85. 3 35 88 - . 9 0 , 5 3 80 9 - . 4 9, 7 0 19 1 - . 3 4 , 69861 - . X i, 9 4 13 1 - . 0 9 , 93940 - . 2 9, 7 2 93 7 - . 8 9 , 7 43 11 t . 9 8, 8 29 97 - . 0 5 , 8 3 43 9 +. 0 21 . 2 1 43 - . 3 2 1. 1995 - . 9 3L . 7 5 51 + . 6 9 1. 7472 i . 2 4, 9 2 73 8 1 . 0 6 . 93436 +. 8 l, 9 61 37 - . 1 9 , 9 6 49 4 +. 181. 6107 - 1 . 6 1. 5766 - . 5 32 . 5 0 96 - 1 . 1 2. 4942 + . 4 3, 9 98 04 - . 2 5 . 9 9 88 0 t . 321 . 9 5 52 + 2 . 9 1. 8983 - . O B3 . 2 5 68 +i . 4 3 . 2484 + . d 3. 9 9 99 9 +. 0 5 1. 0000 +. 05

3 . 6648 t 1 . 3 3. 6497 +. 911. 0006 +. 06 1 . 0 0l 5 + . l 52 . 3 0 72 - 4 . : 2 . 2 3 26 +. 684 . 0855 +. 95 4. 0791 +. 791. 0069 7 . 6 3 1 . 0 116 +1 . 12. 5037 + 4 . 4 2. 4254 t l . 14 . 5 3 1 4 C . 1 4 4.3461: + . 4 61. 0255 +2 . 0 1 . 0388 +.32 . 7 2 34 + 4 . 3 2. 5473 - 1 . 45 . 0 2 51 - . 9 6 5 , 0 60 5 - . i o1. 0167 +5. 3 1 . 0 78 9 t i 2.1 . 1 2 98 t 5 . 7 1 . 1 8 09 +11.3. 2625 +2 . 0 3 . 2 i 9 7 - . 6 66 . 1 4 89 - 5 . 1 6 . 2 7 ~3 3 . 1, 8 11 23 t 2 . 7 1 . 0 0 14 +27.1 . 3 7 1 3 +2 . l 1 . 5 0 76 +1 2.3 . 9 8 79 - 5 . 0 4 . 0 3 a i - 3 . d7 . 5 28 7 - 1 2. 7 . 8 3 43 - 8 . 1, 44780 - 3 : . , 87751 +36.1 . 8 1 66 - 1 6. 2 . 113 1 - 1 . 74 . 9 7 33 - 17. 5 . 2 1 59 - 1 3 .9 . 2 5 00 - 2 0 . 9 . 8 6 42 - 1 5.- . 3 69 1 - 1 80 , 5 0 25 2 +9 . 3

2 . 5 4 51 - 3 7. 3 . 1 37 2 - 2 2.6 . 3 0 88 - 3 0. 6 . 9 0 39 - 2 4 .11. 421 - 3 0. 12. 5 . 7 - 2 3.

2 . 1 2 67 1 3 . 3 2. 0599 +. 2a

? 7 6 4T no j 5( D ) P n

? _ V R - 0 . 0 8 66 4 TR 0 5 f V ? c VRf 0. 08564; R - cVR- O. 082 l ~cV, +0. 043)c R

T

R 0 . 42748 [ 1 +0 . 4 8 1 - ? R 0 5 ) TR - __ 7 / ( 6 4~, ~ )( E ) PR =

Z c VR- 0 . 0 67I Bj Pa = = C v R - 3 . 0 8 5 5 4 Z c V R ( Z c V R + O . 08664

0 . 4 57 23 6 [ 1 +0 . 3 75 ( 1 - T Ro j l 2 0 . 5 - 0 . 1 7 81 2 5T Ra T R -

i c R - 0 . 0 7 5 (;cvR+0.0j)25 V - 0 . 0 7 7 8 F V ~ Z c V R + 0 . 0 7 7 8 ) + 0 . 0 7 7 8 ( T c v R - o o 7 7 8 ) I F ) R =

C R( c ) 2 2 = -___- -c R

droop a li t t le as TR - . In th e case of eq 88 for argonwhose 2 = 0.29121 one quickly find s tha t forn = 0.55 atVR = 1, eq 89 gives

() = 6.857 when TR = 1dTR vwhich is high, and

() = 5.888 when T R = 1.5dTR v

which is low. Th e slope a t an average tem per atu re ofTR= 1.25 is 6.25 which compa res favorably with th e 6.1 fromthe reduced vapor-pressure charts,so

27 / 64TR055

( i c V ~ 0.043)2(90)

Rp R = -

ZCVR- 0.082

should be a good equation for argon. Th is is shown to betrue in Table IV where eq 90 is prese nted a s eq(D) andcompared with theNBS dat a on argon. (In making thesecomparisons one can emphasize either the absolute de-

viations or percentage deviations depending on the ulti-mate use of the equation of state. Here a little more weightis given to th e percentage deviation.)

Th e selection of the am ount of translation depen ds inpart upon the critical compressibility factor and the twoare simply related in a linear fashion. To fit data a t abouttwice th e critical den sity, bu t have fairly high positivedeviations (20-2570) at about two-thirds the crit icalvolume, along the critical isotherm , use

(91)

To fit dat a well around two-thirds the critical volume, buthave high negative dev iations a t twice th e critical andhigher densities, use

tP, /RT, = 0.8572, - 0.1674


10/17

90 Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979

tP, /RT, = 0.7522, 0.152 (92)

Th ese e quatio ns give for argon th e values of 0.082 and0.067, respectively. Clearly anyth ing in between is ac-ceptable d epending upon where the fit isto be maximizedabove the critical density.

A variation of (D) is the temperature function. In (E )it has been made linear with the slope of the criticalisometric taken as 6.155, and the translation made a little

less than (D) and (E). Th e performance is exactly whatwould be expected. Below th e critical density it predictstoo high at the lowest temperatures because it does notallow the isometrics to curve down as they really do. Inth e region of the critical densityit is superb because it doesnot force downward curva ture on isometrics tha t are es-sentially straigh t, as does eq(D). At th e higher densitiesthe effect of curvature of th e isometrics is completelydom inated by the large deviations in the critical isotherma t reduced densities around 1.5 to 2.0, and its smallertranslation makesit bette r at 1.5 and worse at 2.0. On anoverall basis (E) performs as well as any othe r equatio n,so one must conclude t ha t in the two-term cubic linearisometrics are as good as isometrics that always curvedown.

A number of years ago there were claims that theRedlich-Kwong equa tion was the best of the two-pa-rameter equations of stateso it has been selected forcomparison as eq (A) in Table IV. For densities up to 1 /4th e critical it is not as good as(D). For densities aroundthe critical the two a re similar.At 2/3 the critical volume(A) is not as good as (D), bu t a t twice the critical densityit is a little better though a slight change in translationwould make (D) just as good. Overall the two equation sare ab out a standoff for argon, but oth er substances willreveal appreciable differences. Also th e Redlich-Kwongcan never be quiteas good as eq (D)or (E) or eq 55 becauseits 2-chart sum is = 0.33333 + 0.42748 0.08664 =0.67417, which is slightly greater than that of(D).

There have been several attempts to improve theRedlich-Kwong equa tion either by changing the tem-perature function or by changing the volume-functionconst ants. Soave chose th e former procedure, replacingthe simple l/ 'Td.5with [ l + m(1 TRo.5 ) ]2 .e made m afunction of the acentric factor(26) which is just an em -pirical parameter somewhat equivalent to the fundamentalquant i ty, M ,bu t which does not con stitute a basic fluidcharacteristic as doesM which is the slope of th e criticalvolume line onP-T coordinates. Soave's eq (B) is shownin Table IV and the results indicate no improvement overthe Redlich-Kwong as the two predic t almost the samesince they are identical a t the crit ical temperature andsince it is for substances with lower experimental2, tha ta variable te mpera ture function, such as Soave's orl / T R n ,

is needed in place of 1/TRo,5 .Th e Peng-Robinson eq ( C ) has been shown to be aspecific case of the all-inclusive eq1 presented in thisstudy, but i t represented a new development becauseitwas not based on previous well-known equations.It usesthe same temperature functionas Soave, [ l + ~ 1 - R' ,~)] ' ,with a function of the acentric factor. Its principaldifference from the Soave equation is tha t it predicts thecritical compressibility factor t o be 0.3074 insteadof 1/3.This is not beneficial for argon, butit does help forsubstances whosez, is around 0.26 to 0.27. Table IV showsth at below the critical density it is inferior to (A),(B),or(D) because its .%chart sum,2 = 0.3074+0.4572 - 0.0778= 0.6868, is higher th an for (A ),(B),or (D). The mainreason for the large negative deviations in this range is tha t

the generalized second virial coefficient at the critical

Pable V.

substance Z C M a

Ar 0.291 6.1 0.332c, H4 0.281 6.5 0.337C4F8 0.278 7.8 0.336i-C 0.270 7.2 0.352CHF, 0.259 7.4 0.352

0.242 7.4 0.37 0

-(dZ/dP&R=l, p R = o = -BG,.

temperature for (C) s -0.379 whereas th e tru e value forargon is -0.332. A t 213 the critical volume the Peng-Robinson is better than (D) while at twice the criticaldensity, it is not so good.

Th e conclusion from the d ata on argon is th at n o oneequation stand s clearly above the others, even though n oother equation is better than(D),and (D) is as simple asany other in form. In order to test the versatility of anyof the equations, i t is necessary to look at other substancescarefully chosen over a range of parameters t ha t determinethe PVT behavior. The se have been shown to bez,, M ,and (dz/dPR),,=,, as PR- . Sub stance s have been se-lected that have dependable wide-rangePVT d a t aavailable an d are listed in Tab le V.

Th e last two substances in Table V are polar, but th atdoes not affect the p rinciples involved unless there is a highdegree of chemical association of vap or-phase m olecules.It would have been more desirable to have selected anonpolar sub stance with a lowz,, such as octane or nonane,but th e experimentalPVT dat a for these areso extremelylimited that no reliable comparisons can be made.

For ethylene the temp eratu re function to be used in eq67 is TR0.@hile th e translation is 0.074 and th e resultingeq (D) up t o the critical density is clearly superior to theRedlich-Kwong, the Soave, the Peng-Robinson, and th eChueh-Prausnitz ( 5 ) . (See Ta ble VI.) As expected, thelatter equation is better th an t he Redlich-Kwong, Soave,and Peng-Robinson below the critical because the con-stan ts of the Redlich-Kwong were arbitrarily adjusted toget a better fit for saturated vapor volumes which, ofcourse, are densities below the critical. Even in this range,however, the C hueh-Prausnitz is inferior to eq(D),partlybecause it has the same temperature function as theRedlich-Kwong. Above th e critical density its deviationsare so very large it mus t be dismissed as a serious con-tende r for the thron e of cubic equations. At reduceddensities of abou t 1.5 the Peng-Robinson is bet ter tha n(D),but this is simply because th e translation for(D) hasbeen selected by eq 91so tha t i t will do a better job neartwice the critical an d higher densities. As with argon, th etranslation could have been set by eq 92 and the fit a tp R= 1.5 would have been equalto th at of the Peng-Robinson,

while still being far better th an th e Peng-Robinson belowthe critical density. Equation(F) has been introduced hereto show how another simple tempe rature function,ek(l-Td,behaves. It gives abou t the same fit as the tempe raturepower function, TRn. The more complex temperaturefunction of Soave,or the linear temperature function 20)could just as easily have been chosen and t he results wouldhave been as indicated for argon. Th e overall conclusionfor ethylene is th at eq(D) is both the best an d the simplest.

Table VI1 gives the comparisons for perfluorocyclo-butan e. Up to a reduced density of0.8 eq (D) s betterthan a l l the others, particularly if reliance is placed on t hehigher precision data . From the critical density to abou t1.8 times the critical the Peng-Robinson is very slightlybetter. In general, the comparisons are abo ut the same

as for argon.


11/17

Ind. Eng. Chem. Fundam.,Vol. 18, No. 2, 1979 91

Table VI. Comparison of Six Equations of State with Data ( I 0 ) fo r Ethylene

C 2 H 4ETHYLENE

Tc =2 8 2 . 3 5 K

Pc =5 0 . 4 1 9 7 bar

- C = 7 . 6 3 5gmo 1 / d r n 3

R = , 0 8 3 1 4 3 3b a r . d n 3 / q r n o l e . F

Z = 0 . 2 8 1 3 0 4

0 . 8

1 . 5

3 . 5

5 . 0

6 . 5

7 . 5

8 . 0

1 0 . 0

1 1 . 0

1 2 . 0

1 3 . 0

1 4 . 0

1 5 . 0

1 5 . 5

T u

- 2 51 07 5

1 5 0

- 2 5107 5

1 5 00

7 51 5 0

107 5

1 5 0107 5

150107 5

1 5 01 07 5

1 5 01 01 5

1 5 0107 5

1501 07 5

1 2 50

5 01 0 0

07 5

05 0

- 2 50

3 0

P , bard at a

1 3 . 9 0 31 6 . 5 8 22 1 . 4 0 12 6 . 8 4 3

2 2 . 0 5 42 7 . 6 8 43 7 . 5 0 24 8 . 4 3 44 0 . 9 8 47 3 . 3 2 31 0 3 . 4 95 0 . 1 4 99 3 . 9 3 21 4 2 . 0 05 1 . 1 9 31 1 2 . 7 71 8 2 . 3 05 1 . 3 1 71 2 6 . 0 52 1 2 . 4 75 1 . 3 5 91 3 3 . 3 32 2 9 . 1 65 2 . 5 7 91 7 1 . 8 73 1 3 . 5 75 6 . 1 0 22 0 1 . 1 23 7 2 . 1 46 5 . 2 0 22 4 1 . 9 73 7 9 . 1 45 2 . 5 9 72 1 6 . 3 93 8 2 . 1 38 1 . 0 7 53 8 0 . 9 11 3 3 . 5 73 7 5 . 4 93 9 . 0 0 21 7 2 . . 3 13 3 1 . 6 8

Red 1 c h-Kwonc

( A )

P D e v .

1 3 . 9 6 6 t . 4 61 6 . 5 6 9 - . 082 1 . 3 2 1 - . 3 72 6 . 7 1 8 - . 4 7

2 2 . 3 6 7 t 1 . 42 7 . 6 8 8 t . 0 13 7 . 2 8 9 + . 5 74 8 . 0 7 4 - . 7 44 1 . 5 6 1 t 1 . 47 2 . 8 0 0 - . 711 0 2 . 4 4 - 1 . 15 0 . 3 8 3 + . 4 79 3 . 7 6 1 - . 181 4 0 . 9 5 - . 7 45 1 . 2 2 7 t . 0 71 1 4 . 1 7 t 1 . 21 8 2 . 2 0 - . o s5 1 . 6 6 5 C . 681 2 9 . 5 6 t 2 . 82 1 3 . 5 0 t . 4 94 5 2 . 3 2 +1. 91 3 8 . 3 2 + 3 . 72 3 0 . 9 0 t . 7 66 2 . 1 3 4 t L 8 .1 8 5 . 3 6 +7. 83 1 7 . 7 5 t 1 . 37 4 . 4 9 5 + 3 3 .2 1 9 . 7 5 t 9 . 33 7 5 . 8 4 t . 9 99 4 . 8 4 5 + 4 5 .2 6 4 . 9 9 t 9 . 53 8 8 . 1 8 t 2 . 49 4 . 1 4 5 + 7 9 .2 5 0 . 2 4 t 1 6 .3 9 7 . 4 8 + 4 . 01 3 5 . 4 7 +67.4 0 4 . 0 2 + 6 . 11 9 6 . 7 3 t 4 7 .4 0 8 . 3 2 + 8 . 71 1 6 . 6 5 +1992 3 7 . 1 5 t 3 8 .3 7 5 . 9 7 +13.

0 . 4 2 7 4 8~~ -Z c V R - 0 . 0 8 6 6 4 T R 0 '5 ? c V R( Z c V R+ 0 . 0 8 6 6 4 )

Soave( a )

P Dev.

1 3 . 9 3 2 + . 2 11 6 , 5 7 0 - . 0 72 1 . 4 1 3 t . 0 62 6 . 9 3 1 + . 3 3

2 2 . 2 5 1 t . 8 92 7 . 6 9 2 +. 033 7 . 6 0 4 s . 2 74 8 . 8 0 4 t . 7 74 1 . 3 8 0 +. 977 4 . 3 9 3 + 1 . 51 0 6 . 1 0 t 2 . 55 0 . 4 1 5 t . 5 39 6 . 8 4 9 +3. 11 4 8 . 1 1 t 4 . 35 1 . 2 7 8 t . 1 71 1 9 . 1 4 +5. 71 9 3 . 7 1 +6. 35 1 . 7 3 1 +. E11 3 5 . 9 7 t 7 . 92 2 8 . 3 6 +7. 55 2 . 3 9 3 t 2 . 01 4 5 . 5 0 t 9 . 12 4 7 . 5 4 t 8 . 06 2 . 2 4 3 +18.1 9 5 . 9 3 + 1 4 .3 4 2 . 2 5 + 9 .27 4 . 6 2 3 3 3.2 3 2 . 1 9 +15.4 0 4 . 6 7 + 8 . 79 4 . 9 9 4 + 4 6 .2 7 9 . 3 9 +15.4 1 5 . 1 2 +11.9 2 . 2 7 7 t 7 5 .2 5 9 . 9 6 + 2 0 .4 2 1 . 0 0 t 1 0 .1 3 3 . 3 6 + 6 4 .4 2 2 . 6 2 t11 .1 9 4 . 3 7 t 4 6 .4 2 0 . 6 1 + 1 2 .1 0 8. 5 8 t 1 7 82 3 4 . 6 6 + 3 6 .3 8 2 . 2 5 + 1 5 .

0 . 4 2 7 4 6 1 1 + 0 . 6 15 ( 1 - T q 0 ' 5 ) 1 2

Z v ( Z v t 0 - . 9 8 6 6 4 )T R

TR -

C R C RZ c V R - i l . 0 8 6 6 4

0 . 4 5 7 2 3 6 [ 1+0. 504 ( 1- T R0' 5)

f V - 0 . 0 77 8 Z C V R ( i V t O . 0 7 7 8 ) + 0 . 0 7 7 8 ( 5 c V R - 0 . 0 7 7 8 )C R C R

P e n g -Ro b i n s o n

( Ci

P De v.

1 3 . 7 7 9 - . E 91 6 . 3 7 4 - 1 . 32 1 . 1 5 0 - 1 . 22 6 . 6 0 5 - .E 9

2 1 . 9 0 5 - . 6 72 7 . 1 8 7 - 1 . 83 6 . 8 4 5 - 1 . 84 7 . 8 0 4 - 1 . 34 0 . 7 3 2 - . 6 27 1 . 9 0 2 - 1 . 91 0 2 . 0 9 - 1 . 44 9 . 8 9 5 - . 519 2 . 9 2 9 - 1 . 11 4 0 . 8 7 - . E O5 1 . 1 8 2 - . 0 21 1 3 . 0 6 + . 2 51 8 1 . 7 4 - . 315 1 . 3 9 6 + . 1 51 2 7 . 4 8 t l . l2 1 1 . 7 8 - . 3 25 1 . 6 1 9 +. 501 3 5 . 3 3 > 1 . 52 2 8 . 0 3 - . 4 95 5 . 8 9 0 t 6 . 31 7 4 . 1 2 t l . 33 0 4 . 8 4 - 2 . 86 1 . 8 5 9 +l o .2 0 0 . 1 0 - . 503 5 2 . 8 9 - 5 . 27 2 . 0 4 8 +11.2 3 2 . 5 0 - 3 . 93 5 1 . 5 1 - 7 . 35 8 . 7 7 9 1 1 2 .2 0 3 . 0 3 - 6 . 23 4 2 . 4 8 - 1 0 .7 8 . 3 0 4 - 3 . 43 2 5 . 0 2 - 1 5.1 0 7 . 8 3 - 1 9 .2 9 8 . 2 : - 2 1 .2 2 . 4 5 6 - 4 2 .1 2 7 . 3 9 - 2 6 .2 5 0 . 6 0 - 2 4 .

Ma r t i n3 )

P Dev.

1 3 . 8 9 4 - . 0 61 6 . 5 3 0 - . 312 1 . 3 1 7 - . 3 92 6 . 7 3 2 - . 4 2

2 2 . 1 5 5 t . 4 62 7 . 5 8 1 - . 3 73 7 . 2 8 0 - . 5 54 8 . 1 2 2 - . 6 44 1 . 2 3 7 t . 6 27 2 . 9 1 9 - . 5 51 0 2 . 5 5 - . 9 15 0 . 2 6 1 t . 2 29 3 . 9 6 4 t . 0 31 4 0 . 8 5 t . 8 15 1 . 2 3 3 t . 0 61 1 4 . 0 5 t l . l1 8 0 . 9 1 - . 7 65 1 . 5 3 1 t . 4 21 2 8 . 6 2 + 2 . 02 1 0 . 3 9 - . 9 85 1 . 9 5 5 t 1 . 21 3 6 . 6 6 + 2 . 52 2 6 . 3 8 - 1 . 25 8 . 5 6 9 +11.1 7 7 . 1 7 +3. 13 0 2 . 4 4 - 3 . 66 6 . 8 4 8 t 1 9 .2 0 4 . 6 9 t 1 . 83 5 0 . 2 4 - 5 . 98 0 . 1 8 0 t 2 3 .2 3 9 . 0 3 - 1 . 23 5 2 . 2 7 - 7 . 17 0 . 3 0 8 + 3 4 .2 1 4 . 1 6 - 1 . 03 4 7 . 7 6 - 9 . 09 4 . 6 0 1 t 1 7 .3 3 5 . 8 2 - 1 2 .1 2 9 . 2 2 - 3 . 33 1 5 . 2 7 - 1 6 .4 5 . 0 6 6 t 1 6 .1 5 1 . 2 6 - 1 2 .2 7 2 . 2 6 - 1 8 .

T"

C h ue h -P r a us n i t z

( E l3

P Dev.

1 3 . 9 3 9 + . 2 61 6 . 5 4 4 - . 2 32 1 . 3 0 1 - . 4 72 6 . 7 0 3 - . 5 2

2 2 . 2 7 8 + l . O2 7 . 6 1 0 - . 2 73 7 . 2 2 7 - . 7 34 8 . 0 2 8 - .E 44 1 . 2 2 6 1 . 5 97 2 . 5 6 8 - 1 . 01 0 2 . 2 7 - 1 . 24 9 . 8 8 5 - . 539 3 . 4 4 7 - . 5 21 4 0 . 8 3 - . E 35 0 . 6 4 8 - 1 . 11 1 3 . 9 2 t l . 01 8 2 . 2 8 - . 015 1 . 1 3 8 - . 3 51 2 9 . 4 9 t 2 . 72 1 3 . 9 0 + . 6 85 1 . 8 6 4 + . 9 81 3 8 . 3 9 t 3 . 82 3 1 . 5 2 + 1 . 0

Ma r t i n - wi t hE x po ne nt i a lTempera t ur e

F un ct i o n( F i

P D e v.

1 3 . 9 4 3 t . 2 91 6 . 5 3 0 - . 3 22 1 . 3 0 2 - . 4 62 6 . 7 6 4 - . 30

2 2 . 3 1 9 t 1 . 22 7 . 5 7 8 - . 3 83 7 . 2 3 6 - . 7 14 8 . 2 2 2 - . 4 44 1 . 3 9 7 +1. 07 2 . 6 4 9 - . 9 21 0 3 . 0 6 - . 4 25 0 . 2 3 8 + . 1 89 3 . 4 4 7 - . 5 21 4 1 . 8 1 - . 145 1 . 1 9 6 + 011 1 3 . 2 3 t . 4 11 8 2 . 4 3 + . 0 75 1 . 4 8 4 + . 3 31 2 7 . 5 7 + 1 . 22 1 2 . 3 3 - . 065 1 . 9 0 2 t l . l1 3 5 . 4 9 +1. 62 2 8 . 5 5 - . 2 7

6 2 . 4 3 0 t 1 9 . 5 8 . 4 9 3 + 1 1 .1 8 6 . 6 0 1 8 . 6 1 7 5 . 4 8 + 2 . 13 2 0 . 0 0 t 2 . 1 3 0 5 . 5 7 - 2 . 67 5 . 5 7 8 t 3 5 .2 2 2 . 0 9 + 1 1 .3 7 9 . 5 3 t 2 . 09 7 . 1 4 0 + 4 9 .2 6 8 . 9 4 +11.3 9 3 . 3 5 t 3 . 89 7 . 9 0 4 +86.2 5 5 . 7 2 t 1 8 .4 0 4 . 6 1 +5. 91 4 1 . 8 1 t 7 5 .4 1 3 . 7 8 + 8 . 62 0 6 . 9 1 t 5 5 .4 2 1 . 6 1 t 1 2 .1 2 7 . 5 7 t 2 2 72 4 9 . 9 4 + 4 5 .3 9 0 . 9 5 t 1 8 .

6 6 . 7 5 9 + 1 9 .2 0 2 . 7 2 1 . 9 03 5 3 . 8 9 - 4 . 980. 073 t 2 3 .2 3 6 . 7 7 - 2 . 23 5 3 . 6 6 - 6 . 77 1 . 8 2 4 t 3 7 .2 1 1 . 1 8 - 2 . 43 4 6 . 8 3 - 9 . 29 6 . 2 9 7 t 1 9 .3 3 2 . 9 7 - 1 3 .1 3 1 . 1 0 - 1 . 93 1 1 . 5 8 - 1 7 .5 5 . 1 5 2 + 4 1 .1 5 3 . 2 3 -11.2 6 9 . 1 8 - 1 9.

Table VI11 gives the comparisons for isopentane. Upto th e critical temperature eq(D) is clearly th e best. Abovethe crit ical temperat ure the Peng-Robinson and eq(D)give abo ut the same average deviation, both of which ar ebet ter t han the Redl ich-Kwong, the Soave, and theChueh -Prausnitz. At the higher densities the Peng-Robinson and eq (D) are similar. Th e conclusion is th ateq (D) is the sim plest an d be st for an overall fit for iso-pentane.

Tab le IX gives the comparisons for trifluoromethane.

At all densities eq(D) is superior to th e other equationswhile retaining the simplest form.Tab le X gives the re sults for ammonia. Again for all

densities eq (D) is superior to all the others.Considering the performance for the wide range of

substances the inescapable conclusion is that eq(D),oreq 67 with the temperature function,TRn,(or A + BTR)is no t only the simplest but the best of the two-term cubics.T h e thre e-te rm virial does have a slightly lower value ofE, but it is one term more complicated. Certainly, thereis no reason to go to o ther more complex two-term cubics,such as the Peng-Robinson, for they do not do as well as(D).

One of the reasons for the su perio rity of(D) is foundin Table V, which lists the PVT behavior parameters,e,,M ,and B G ~ ,ne of which is no t indepen dent because it has

been shown that = f , B G ~s practically constant at0.62. Using a single empirical parameter, such as theacentric factor, in a correlating e quation of sta te can notpossibly do as well as (D) unless there is a consistentrelation betweenM and 2, or BG1,which the re is not forthe six substances chosen. T he Soave an d Peng-Robinsonequations, for example, are constrained t o fixed values of2, (1/ 3 and 0.3074, respectively) with the temp eraturefunction allowed to vary with a single param eter, but noallowance is made for different values ofe,. Equation (D)does not have this pitfall asit is not locked int o a fixed2, for all substan ces. Since in(D) 2, = 0.25 + tP, /RT,,2, varies from 0.25+ 0.082 = 0.332 for argon down t o 0.25+ 0.04 = 0.29 for ammonia. At the same time the tem-perature function,TRn,s varying simultaneously withMand 2, as shown by eq 89 whereM and the t ransla t ion,tP,lRT,, enter into the slope of the critical isometric.Thus, accounting for differences ine,, having the smallestvalue of E, better performance, an d inh erent simplicityare the compelling features that make(D) superior to allothers.

Clearly, other temperature functions besidesTRncan beused in eq 67 an d they will perform a s described for argonand ethylene, these being the linear an d exponen tialfunctions. T he temp erature function of Soave could easilybe used b ut n othin g is gained by its more complex form.


12/17

92 Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979

Table VII. Comparison of Four Equations of Sta te with Data ( I 7 ) for Perfluorocyclobutane

TC =

699. 27R

Pc =

F c =

4 03 . 6 gs i .

3 8. 7 1b / f t 3 .

MW=

2 00 . 0 4 4 .

R = 10.131F

1bnole .R

1 . 3 7 5 1 8i . 3 7 7 5 8

1 9 1. 7 8545. 24617.246 8 2. 9 25 2 7. 6 9618. 14688. 25

1 7 . R 320. 1123.3925.7336.924 5 . 0 5

1 8 . 3 6 323.47323.297

+ 3 . 2i l .

1 . 9 0

1 8 . 2 4 020.383

- 2 . 51 1. 41 . 6 91 . 4 41 3 . 31 1. 21 . 5 7i 1 . l

1 6 . 1 9 52C.331

i 2. 211. 1+ . 4 4r . 1 9- 2 . 5* . l o1 . 9 81 . 4 3

- 2 . 6- 1 . 3- 1 . 2- 1 . 6- . 4 6- 3 . 1- 2 . 1- 2 . 11 2. 31 . 9 5

18. 1402 0 . 3 2 3

1 1. 9- 1 . 1+. 56- . 37- 2 . 2* . 9 41 . 4 31 . 4 4

- 2 . 2- . 77- 1 . 2- 1 . 2- . 4 4- 2 . 5- 1 . 5- 1 . 61 2. 2+ 1 . 3r 70

i . 3 80 621 . 3 8 35 1. 6 6 C 7 8 :, 682863, 6 8 4 3 3 5

23.25025.8433 8 . 3 1 24 5 . 5 9 25 1 . 4 0 2

23.19125.7783 7 . 8 2 745.36751.150

2 3 . 2 2 025.82437.7264 5 . 4 7 551. 330

25.85238.4214 5 . 7 8 15 1 . 4 2 65 9 . 6 7 6

1 . 4 7l 4 . l11 . 6+. 62+2. 2

51.115 8 . 4 0

6 9 . 2 07 7 . 2 08 6. 5 09 6 . 7 0

, 465549 572. 63

641. 527 i 1 . 1 87 8 5. 0 7862.42

5 9 . 0 4 0

6 7 . 8 8 27 6 . 7 4 786.2789 5 . 7 7 9

5 8 . 6 3 2

6 7 . 4 0 67 6 . 2 1 585.4939 5 . 1 4 5

5 s . 6 5 a

6 7 . 6 8 37 5 . 6 3 86 5 . 5 3 29 5 . 5 3 2

6 8 . 1 6 77 6 . 6 8 98 5 . 6 7 79 5 , 0 4 06 6 . 3 8 98 5 . 8 2 9

- 1 . 5- . 66- . 9 5- 1 . 711. 6- 2 . 4- 1 . 8- 2 . 1+ 4 . 61 2. 5i l . 0i 4 . 8- 2 . 9- 1 . 4i 4. 21 2. 91 2. 3i 2. 4- . a 6- 2 . 2i . 1 5- 1 . 6- 2 . 8- 3 . 5- 3 . 91 . 3 8- 2 . 0- 3 . 7- 4 . 5- 5 . 0i 2. 2- 1 . 9- 4 . 0- 4 . 3- . 7 6- 3 . 1- 5 . 2- 5 . 9- 6 . 711. 7- 4 . c- 6 . 3+ . 7 1- 4 . 0- 6 . 5- 7 . 9- 8 . 9c . 2c- 4 . 4- 7 . 2- 9 . 0- 1 0 .

- 1 . 9- . 5 8- . 4 9- . 9 5i . 3 0- 2 . 3- 1 . 3- 1 . 3* 3 . 1- 1 . 7- . 9 3+ 2 . 8+1. 8r 1 . 3+2. 0t 2 . O

, 4 116 9 2 5 7 1. 3 27 1 0 . 1 97 6 1 . 5 2851.835 5 9. 6 26 1 6 . 1 0

6 5 . 3 78 7 . 9 59 7 . 4 7

1 0 7. 7 06 2 . 6 2

5 5 . 5 6 9

9 6 . 2 3 1- 0 6. 2 86 4 . 5 7 5

8 5 . 8 9 665.0668 5 . 2 3 29 5 . 4 6 9105.596 4 . 0 7 7

6 5 . 0 7 98 5 . 7 2 29 6 . 0 0 01 0 5 . 9 96 4 . 0 1 0

9 5 . 7 0 91 0 5 , 4 06 5 . 4 9 77 3 . 7 0 08 3 . 5 6 5

, 406226. 407?67 71 . 92

8 2 . 7 311 3. 7 91 2 7. 31 4 3. 0 91 8 7. 7 5208. 372 1 9. 3 31 9 9. 52 6 6. 73 1 2. 0257. 913 0 2. 1 5362.814 2 1. 5 34 7 9. 113 0 8. 2 93 7 2. 5 54 5 9. 7 7543.76625.93 1 3. 4 0419.00495. 505 58 . 9 0

73.1746 3 . 5 0 :i 1 6. 9 61 2 9. 6 31 4 4. 9 2

7 2 . 6 0 182.84d11 5. 5 01 2 7. 9 71 4 3. 0 4188. 15208. 76219. 731 9 7. 6 22 6 3. 4 3

7 2 . 8 5 96 3 . 3 1 2115.731 2 8. 8 2144. 361 8 9 . 4 02 11. 1 22 2 2. 4 91 9 9. 4 52 6 7. 0 8

, 4 3 8 1 4 8, 218651, 218911, 219291, 126033

6 8 9 . 1 5596. 026 3 6 . 2 86 8 9 . 9 66 3 4. 0 36 7 0. 1 9

i . 1 4+ I .- . 5 2- . c 4

11 9 . 2 6130.961 4 5 . 1 2195.672 1 4 . 4 7

1 1. 21 . 9 61. 88t . 8 8

z c =

C. 27801 1 9 1. 4 72 1 2. 6 32 2 3 . 4 32 0 1. 1 9268. 64

r . 2 21 . 1 81 . 1 7- . 9 4- 1 . 2- 1 . 3- 1 . 9- 2 . 0- 1 . 6- 1 . 2- . a 4- 1 . 8- 2 . 0- 1 . 6- i . l- . a 0- 1 . 0- 1 . 4- 1 . 8- 1 . 0-1 .1-1 . 6- 1 . 1

- . a 7- . 5 6-. 06

, 125177, 126257, 126582

i 1 . 36 8 9 . 6 06 5 1 . 4 47 6 8 . 5 58 4 9 . 8 4

2 2 4 . 4 6204.232 6 4 . 4 23 0 5 . 3 0258.31297.433 5 2 . 6 74 0 6 . 9 74 6 0 . 5 6309. 46364. 964 4 2 . 9 95 1 9. 4 0594. 553 2 0. 2 24 1 0 . 9 24 7 5 . 7 8534. 68312. 794 6 5 . 9 6

1 1 . 8

, E 51 . 7 3

i l .- . 0 3+ . I 4- . l a14.18

258. 29302. 733 6 5 . 1 44 2 6. 0 5

+ . 7 01.151 . 1 9t . 6 411. 1i l . 4- , 371 .441 1. 111 . 61 2. 01 . 9 1i l . 21 1. 1+2. 0+, 7 5+1. 1- 2 . 91 2. 61 3. 11 1. 21 1 . 7

3 0 8. 0 12 5 3. 0 7296. 273 5 7. 1 0416. 65

311 .44* . l a 6 7 7 1 699.08

7 6 1 . 6 98 5 1 . 6 9941.691 0 3 1 . 7

2 5 6. 6 5300.78361.364 1 9. 4 7

- . 4 9- . 4 5- . l o- . 4 8

4 8 5, 6 6 4 7 5. 819 475. 77 - . 70- . 3 51 . 3 6- . 3 2- . 4 9- . 8 21. 641.29- . 3 31.061 . 0 7+ . 0 5C.36- . 3 9- . 6 81 . 6 81 . 4 0

*. 08C078 699.06761.698 5 1 . 6 9941. 69i G3 1 . 76 8 5 . 8 9

309.44374. 174 6 4. 6 5552. 53

302.71365.044 5 2. 4 65 3 7. 6 7

307. 20371. 214 5 8. 2 9541. 106 2 0. 8 011 3 . 6 04 2 0. 2 34 9 3 . 8 85 5 9. 2 53 7 0. 7 2

6 3 8. 1 6316. 254 2 4. 0 3500.69569.953 7 2. 7 34 8 5 . 7 76 4 2. 7 47 1 9 . 0 78 6 7. 9 1380. 48

620.99310. 134 1 3 . 0 54 8 6. 5 5553. 14

, 0665347 6 1 . 6 78 2 0 . 8 68 7 5 . 4 9699.08761.69851.69896. 699 8 6 . 6 9694. 68769.118 6 4 . 3 36 9 9 . 0 87 6 1 . 6 98 5 1 . 6 99 4 1 . 6 91031. 76 9 9 . 0 87 6 1 . 6 98 5 1 . 6 9941. 69

* . 0 5 3 38 5 369.974 8 0. 6 26 2 9. 0 4700. 830 4 1. 8 43 7 6. 0 0544. 007 4 2. 0

3 6 5. 8 54 7 2. 7 86 2 1. 9 0694. 688 3 7 . 1 0

480.856 2 9 . 4 27 0 0 . 1 48 3 6. 0 93 7 8 . 5 4546. 187 4 4. 0 7396. 765 6 2. 4 67 8 3. 0 89 8 8. 4 811 82 . 74 0 2. 7 6631.339 3 3. 9 31 2 14 . 01 4 77 . 6402.56564.357 1 2. 4 29 8 9. 2 34 8 9. 7 9

903. 771 2 81 . 31 6 32 . 74 1 3. 1 37 8 9. 6 51 2 8 5. 11 7 4 0. 62167. 5

596. 116 5 9. 8 3785.C)382. 555 2 2. 4 5

, 044823

* . 0 4 0 33 9

375. 775 3 7. 5 3737. 163 9 4 . 4 05 5 4. 0 2

553. 037 6 4 . 8 6398. 20569. 56896.461 0 33 . 81252. 94 0 2. 8 26 4 2. 4 39 7 2. 6 91288. 61 5 92 . 24 0 4. 6 9571.71

- 1 . 2- . 6 5- . 2 7- . 7 3

6 9 5. 3 7398. 31535. 817 2 7 . 0 4912.34

13. 11 . 6 91 2. ;

1 . 2 81 . 3 2+ . 7 R

395. 48558. 117 7 7. 5 49 9 0. 2 611 99 . 31 0 2 . 1 86 1 9. 0 89 1 9. 7 21 2 15 . 5

1 3 . 714.4+4. 5

7 7 5. 8 09 8 9. 8 11197. 14 0 2. 4 56 2 2. 7 3928. 02

- . 2 2- . 0 5- . l aC.07+ . 5 91.901 . 5 1- . I 81 . 0 31 2. 2+I.

1. 71- . l a- 1 . 4i . 1 4i 2. 0+1. 5- . 1 2- 2 . 0- . 3 311.6+2. 5i l .

1 0 93 . 04 0 3 . 0 0591.818 5 3 . 5 81106. 5

* . 032031 + . 1 71 3. 8i 5 . 81 6. 0- 5 . 6* . 2 0i 2. 9i 5 . 0

1 2 21 . 81 5 05 . 6404. 03542. 98702. 969 8 3. 9 6485.148 9 4. 0 11 2 85 . 9

1 0 3 1 . 76 9 9 . 5 27 3 8 . 7 97 7 6 . 9 48 5 2 . 5 9

1 5 08 . 34 0 3. 9 0555. 1695.49 7 3. 4479.26873.641 2 69 . 81 6 65 . 5403. 00741.701 2 53 . 81 7 73 . 32295. 6453. 00a73.301 3 - 3 . 44 3 6 . 9 ;8 3 5. 3 9

1352. 4404. 465 3 5. 4 26 6 0. 0 69 0 0. 8 5

. 030030 + . I 4- 3 . 6- 5 . 0- 7 . 5

7 3 0. 3 41035. 34 9 5. 1 2

943. 841 3 71 . 0

1 6 . 4+3. 318. 0i 8. 0

i l . 1A1. 2

i 2. 3+1. 3

' . 026593 7 1 6 . 6 98 0 6 . 6 96 9 6 . 6 99 8 6 . 6 96 1 9 . 0 8761.69851. 699 4 1 . 6 91 0 31 . 7

4 7 5 . 2 38 2 5 . 4 211 6 1. 51486. 84 2 3 . 0 1738.0511 7 3. 31 5 9 2. 41 9 99 . 0

- . 8 5- 5 . 5- 8 . 5-11.

1 5. 0- . 4 9

+2. 2

t 3 . 5+ . 9 11 7 79 . 84 2 2. 8 08 2 9. 8 71 3 89 . 41 9 22 . 92 4 33 . 9

16. 1 1662. 9 - . l 5+1. 3

.~- 2 . 0+2. 5. 022879 14. 9

t l 2 .- 1 1 .

- 5 . 0+ a . 4

408.407 7 5. 0 51 2 81 . 81767. 82 2 36 . 0

1 4. 5i 2. 2- . 3 1- 2 . 6

1 6. 5c 2 . 5- 1 . 8- 5 . 6

- 6 . 4- 1 0.- 1 3.

, 022497

* . 0 2 00 2 0

707. 28782. 898 5 7 . 5 26 9 9. 0 87 6 1. 6 9

470. 50 1 3. 9 4 8 2. 8 8 16 . 69 8 3. 6 8 11 3 .11 56 . 9 r l l .481.08 +l a .989.10 t l 8 .

460.10 c 1 . 69 1i . 1 8 + 4 . 31 3 39 . 7 C 2. 04 33 . 1 4 t 6 . 48 8 6. 8 7 C6 . 1L 5 1 3 . 5 + 1 . 12 11 4. 2 - 2 . 72 6 92 . 5 - 5 , l4 3 7. 0 1 t 8 . 78 3 5. 9 1 1 7 . 0

4 6 8 . 0 7 C3. 38 5 8 . 0 8 - 1 . 81 22 6 . 5 - 6 . 64 8 1. 4 5 - 1 8 .8 7 3 . 4 8 1 4 . 5

9 2 6 . 0 1 1 6. 01341. 4 C2. 1447.51 + o .9 0 9 . 1 0 C8.8

8 5 1. 6 99 4 1 . 6 91 3 31 . 76 9 8. 87 5 2. 1 4

1 4 9 7 . C2 1 71 . 82 8 54 . 8c 0 2 . 0 0781 .40

1415. 0 - 5 . 51 9 3 6. 1 - 11 .2441. 6 - 1 5 ,4 92 . 7 5 1 2 3 .837.81 1 7 . 2

1 6 87 . 1 1 1 3 .2 3 52 . 3989. 3 16 .34 .1

4 91 . 8 3 i 2 2 .9 39 . 1 2 i 2 0.

1 5 1 5. 8 c 1 . 32 07 3. 1 - 4 . 52 59 4. 4 - 9 . 1453.11 113.659.75 110.1180. 1 16. 2i 4 67 . 6 1 2. 61621. 1 1 . 5 44 8 7. 3 2 t 2 0 .1081. 3 t 10 .1 3 9 1. 7 i 4 . 0

522. 74 +20.1 0 7 7. 3 +8. 61 8 0 5. 9 - 2 . 32 4 75 . 1 - 9 . 03 1 0 0. 9 - 1 4 .4 0 3 . 8 4 +53.9 1 7. 8 4 11 5 .1 7 0 1. 3 c l . 82 7 3 6. 1 - 6 . 63 68 8. 6 - 1 2.2 4 2. 7 8 13 0 .1 3 6 4 . 0 - 6 . 32304. 1 - 1 2 .3548. 9 - 1 8.

1 9 0 3. 2 c .41

, 019670

796. 81838. 868 6 2. 0 46 9 5. 9 57 6 3. 7 6

1111 .41430. 416L2.44 0 5. 9 09 8 7. 4 0

1119. 0 + 69 1302.7 117. 1161.2 i 4 . 51 3 7 8. 2 - 3 . 7 1636. 5 - 1 4 . 1460.6 +2. ;i 5 19 . 1 - 5 . 8 1817. 3 + 1 3 . 1623. 4 + 68572. 90 i l l . 5 65 . 2 9 - 3 9. 4 6 0. 1 6 1 i 3 .1 0 6 6. 8 110. 1 2 33 . 8 1 25 .

1 5 90 . 5 +19.2 1 43 . 2 1 1 3 .6 0 8. 2 0 + l o .1 23 0. 7 1 2 4 .

, 0 1 7 9 31

* . 0 1 77 9 5

i 0 50 . 2 1 6. 48 0 1. 6 0861. 366 9 9. 0 87 6 1. 6 9851.69

9 4 1. 6 91 0 31 . 76 6 1. 0 76 9 9. 2 77 6 1. 6 98 5 1. 6 99 4 1. 6 96 3i . 2 96 9 9. 2 7

1 3 38 . 41 8 95 . 4434.159 9 1. 7 61 8 4 7 . 62:20.a3607. 02 6 4. 6 8798. 61671. 02929. 34186. 91 8 6. 2 11 4 55 . 62621. 24 3 01 . 7

1 36 5 . 5 1 2 . 01795. 5 - 5 . 36 0 8 . 6 4 - 4 0 .1089.5 19 . 91753. 7 - 5 . 1

2 39 3. 1 - 1 2.3 0 1 3. 2 - 1 7.8 6 1 . 4 4 - 2 261 31 8 . 6 l 6 5 .2 0 4 3. 8 1 22 .3 0 5 1. 0 t 4 . 24 0 2 2. 7 - 3 . 91169. 0 +5282206. 9 +52.3125. 1 119.4 45 3 . 3 r 2 . 4

1 3 7 C . l 1 2. 41 8 63 . 3 - 1 . 74 9 3. 5 4 i l 4 .1 04 5. 2 i 5 . 91 8 06 . 8 - 2 . 22 53 6. 7 - 6 . 63 2 39 . 3 - 1 0 .3 46 . 1 6 i 3 1.6 53 . 9 6 - 6 . 9

2 0 86 . 2 113 .2901. 5 16. 63 68 2. 4 i 2 . 17 3 3. 2 0 1 1 7 71318. 6 C 6 5 .

+ * . 0 1 4 3 5 6

2 2 45 . 8 + 3 4 .3526.4 +20.4 7 49 . 7 +13 .895. 51 13812206. 9 +52.

1 66 0. 3 - . E 42 77 7. 7 - 5 . 23849. 3 - 8 . 11 85 . 2 1 - . 5 4* . 0 1 29 2 01305. 9 - 1 0.2 29 7. 6 - 1 2.3673. 4 - 1 5 .

7 6 1. 6 98 5 1. 6 9

3364. 9 C28.4 9 67 . 7 115 .

* Highe r p r e c ~ s ~ o n a t a* * Extens ion o f h i g h e r p r e c i s i o n d a t a t o tw i ce t he c r i t i c a l dens i t y

A ) I C )

PR ~ R 0. 42748 p = L . 4 57 23 6 l r 0. 8 9 l - T R0' j i 1T v - 0 .08664 I v - 0 . ~7 7 8 z v ( t v + ~ . o ~ ~ ~ ~ ~ ~ . o ~ ~ ~ I z ~ v ~ - o . o ~ ~C R C R c H C R

181 I D )

i =TR 0 . 4 2 7 46 Iltl 0 l l - T Ro' , 1 TR 27/64TR1"

p x = - -TCVR-0. O8664 TcvR(IcvR+O. 8664) Z C R- 0 . 071 ( TcVR+2. 05412


13/17


Table VIII. Comparison of Five Equations of State with Data ( 8 ) or Isopentane

I SOPENTANE

T, K

TC = 310

33060. 39K

p C = 3 3 . 3 7 a t m3 5 0

V = .306c m3 gmol

370

R = 82. 05606 390c m a t mgrnole . K-z = 4100. 270297

430

450

455

470

5 0 0

600

( A ) -

Z

, 9 44 5, 9 39 5, 9149. 9671, 8771. E832. 9729, 8 30 1. 9045. 9775, 7 7 26, 8 2 86. 9207. 9811. 7014, 7 71 5, 8589, 9335. 9840, 6087, 6 29 0. 7325. E132. E822, 9437, 9864, 4649. 4875, 6219. 7104

, 7822. a445, 9006, 9 52 1. 9883, 4204. 4547. 5 39 1. 6460, 7256, 7923, 8512, 9046, 9 5 3 9, 9 88 7, 6046. 4386, 3 6 8 0. 2949. 2585. 2295. 5 89 1. 7641, 8926

, 9 89 9. 7651. 6106. 4532. 3932, 3 6 71. 4363. 5771. 7097. a200. 9152. 9919

1. 0606. 9309. a059, 6996. 6625, 6 93 7, 7539. 7919. E327. a747, 9 17 0. 9588. 9959

P, at m

1. 351

2. 471

4 . 1 941

6 . 6 641

841

14. 6412

841

20. 6320161 2.

841

28. 41282420

1612841

30.713 02824201 612

841

1 5 0100

806 05 040302010

1200150100

80605 040302010

1300250200150100

806 05 040302010

1

10.08

0. 42748

Re dl i c h-Kwong

( A i

P Dev.

1 . 3 67 3 t 1 . 31 . 0 0 92 t . 9 22 . 5 13 8 t l . 81. 0065 . 6 54. 2855 t 2 . 34 . 0 8 66 t 2 . 21 . 0 0 47 t . 4 76 . 8 48 4 t 2 . 84 . 0598 t 1 . 51 .0033 t . 3310. 416 t 3 . 38 . 1 9 32 t 2 . 44. 0419 +1. 11 . 0 0 24 t . 2 41 5. 1 96 t 3 . 817. 332 +2. 88 . 1 28 5 t 1 . 64 . 0 2 08 t . 7 21 . 0 0 17 t . 1 721. 466 t 4 . 120. 765 +3. 81 6. 4 20 t 2 . 41 2. 2 09 t l . 78. 0839 +1. 14 . 0 1 94 t . 4 91. 0011 t . l l29. 377 t 3 . 42 8. 9 57 t 3 . 424. 679 t 2 . 820. 427 +2. 1

16. 248 +1.612. 128 +1. 18 . 0521 +. 654 . 0 119 t . 3 01. 0007 +. 0731. 448 +2. 430. 865 t 2 . 928. 829 t 3 . 024. 580 +2. 420. 369 +1. 91 6. 2 17 t 1 . 412. 112 +. 938 . 0 4 56 t . 5 74 . 0 1 06 t . 2 71. 0007 +. 07213. 19 +42.142. 38 +42.115. 98 145.8 9. 7 62 t 5 0.7 5. 6 30 t 5 1.57. 007 +43.3 0. 5 70 t 1 . 920. 227 t l . l1 0. 0 47 t . 4 7

1. 0004 +. 042 58 . 6 8 t 2 9.199. 00 +33.134. 81 +35.105. 90 +32.6 8. 0 07 t 1 3.5 0. 4 07 t . 8 140. 127 t . 3230. 110 t . 3720. 042 + 2110. 007 + 071. 0000 003 09 . 9 5 t 3 . 32 67 . 1 5 t 6 . 9218. 57 +9. 31 61 . 4 6 f 7 . 6100. 12 +. 1278. 270 -2 . 25 8. 4 62 - 2 . 64 8. 8 28 - 2 . 339. 206 - 2 .029. 538 - 1 . 519. 789 - 1 .19. 9470 - . 53, 99948 - . 05

IP =R - cVR- 0. 08664 TR0' 5ZcVR(ZcVR+0. 08664)

18) T 0.42748[1+0.82(l - T:' 5) 1P = ZcVR- 0. 08664 - ZcVR(ZcVR+0. 08664)

(CI T 0.457236[1+0.71(1-T-0 51 l 2

Soave( B l

3P Dev.

1 . 3 6 17 t . 8 71 . 0 0 62 t . 6 22 . 4 9 83 t 1 . 2

Peng-Rob n so n

(Cl

P Dev.

1 . 3 5 87 t . 6 41. 0046 +. 462. 4888 +. 761 . 0 0 27 t . 2 7. 0043 t . 43

4 . 2486 t 1 . 4 4 . 2224 +. 774 . 0 5 34 t l . 3 4 . 0 2 95 c . 7 41 . 0030 t . 30 1 . 0015 t . 156 . 7719 t 1 . 7 6 .7098 t . 754 . 0363 t . 91 4 . 0136 +. 341 . 0 0 21 t . 2 1 1. 0006 t . 0610 .274 +1 .9 10. 144 t . 638 . 1151 11. 4 8 . 0306 t . 384 . 0258 1. 651. 0015 f . 1 514.965 +2. 21 2. 2 02 t 1 . 78. 0811 t 1 . 04 . 0 1 86 t . 4 71. 0011 +. 1121. 141 +2. 52 0. 4 77 t 2 . 41 6. 2 81 t 1 . 81 2. 1 45 t 1 . 28. 0591 +. 744 . 0 13 9 t . 3 51. 0008 t . 0829.066 +2. 32 8. 6 80 t 2 . 424. 548 t 2 . 320. 356 t 1 .8

1 6. 2 10 t 1 . 312. 109 t . 918 . 0447 1. 564 . 0102 t . 261. 0006 t . 0 63 1. 2 10 t 1 . 630. 683 +2. 32 8. 7 12 t 2 . 524 .518 t 2 .22 0. 3 34 t 1 . 71 6. 1 98 t 1 . 212. 102 . e 58 . 0 41 9 t . 5 24 . 0097 t . 241. 0006 t . 06216. 29 t 44.1 45 . 0 8 t 4 5 .118. 47 +48.92. 002 t53 .77. 689 155 .

30. 759 +2. 520. 280 +1. 41 0. 0 57 t . 5 7

1. 0005 t . 05271. 23 +36.210. 34 +40.144 . 32 144 .114 . 2 1 t 4 3 .73. 808 +23.53. 483 t 7 . 04 1. 3 57 t 3 . 430. 594 +2. 020. 210 +1. 110 .042 t . 421. 0003 t . 03346. 50 t 16.300. 71 +20.248. 00 +24.184 . 47 123.112 . 6 4 t 1 3.85. 995 +7. 56 2. 3 67 t 4 . 051. 358 t 2 .74 0. 7 11 t l . 830. 325 +1. 120. 115 t . 5810 .023 t . 231 . 0 00 2 t . 0 2

58. 735 +47.

4 . 0041 1 . 101. 0001 t . 0114. 720 +. 5512. 029 +. 238 . 0 0 04 t . 0 13 . 9 9 80 - . 0 5, 9 99 77 - . 0 220. 749 t . 5820. 093 t . 4716.000 0011. 977 +. 197 . 9 8 23 t . 2 23. 9942 +. 15. 9 99 52 - . 0 528.690 +. 992 8. 2 51 t . 9 02 4. 0 41 t . 1 719. 955 - . 23

1 5. 9 31 - . 4 011. 9 49 - . 4 37 . 9715 - . 363 . 9916 - . 21. 99944 - . 0631. 016 t 1 . 03 0. 2 95 t . 9 828. 167 1 . 6023. 998 - . 011 9. 9 34 - . 3 315. 027 - . 4611. 9 44 - . 4 77 . 9 6 96 - . 3 83 . 9 91 3 - . 2 2. 99945 - . 061 32 . 3 9 - 1 2.9 5. 2 56 - 4 . 781. 169 t1 . 567.056 +12.5 9. 3 84 t l 9 .4 9. 2 32 t 2 3 .30. 069 +. 231 9. 8 86 - . 5 79. 9496 - . 50

, 99938 - . 06182. 32 - 8 . 8148.38 - 1 . 0110 . 2 9 t 1 0.9 2. 2 86 t 1 5 .6 6. 7 79 t 11 .51. 691 +3. 440. 248 + 622 9. 8 40 - . 5 31 9. 8 36 - . 8 29 . 9 41 6 - . 5 8. 9 99 29 - . 0 7275. 19 - 8 . 3244. 77 - 2 .1208 .56 t4 . 3162. 59 +8. 4105. 42 t5 . 481. 939 t 2 . 460. 103 t . 1749. 730 - . 543 9. 6 19 - . 9 529. 678 - 1 . 11 9. 8 12 - . 9 49 . 9 4 33 - . 5 7. 99937 - . 06

Ma r t i n( D i

P Dev.

1. 3478 - . 16. 9 98 78 - . 1 27 . 4 6 42 - . 2 3. 99914 - . 094 . 1 7 57 - . 3 43 . 9 8 77 - . 3 1. 99944 - . 066. 6342 - . 393 . 9 9 23 - . 1 9. 9 99 59 - . 0 410. 039 - . 417 . 9 7 91 - . 2 63 . 9 9 60 - . l o. 99975 - . 031 4. 6 01 - . 2 711. 9 79 - . 1 77. 9923 - . l o3 . 9 9 81 - . 0 5. 99986 - . 012 0. 6 45 t . 0 72 0. 0 19 t . 0 916. 013 . OB12. 004 +. 038. 0000 003. 9998 00, 99991 - . 0128. 629 +. 752 8. 2 30 t . 8 224. 173 t . 7220. 093 t . 47

16. 044 +. 271 2. 0 17 t . 1 48 . 0 0 45 t . 0 64 . 0 00 4 t . 0 11. 0000 0030. 939 +. 7530. 329 t l . l28. 317 t l . l24. 193 t . 8020.102 + . 5 11 6. 0 49 t . 3 012. 019 +. 168. 0053 +. 074 . 0007 + 021. 0001 + 01114 . 7 6 - 2 3.88. 008 - 1 2.77. 113 - 3 . 66 5. 6 29 t 9 . 459. 081 t l 8 .4 9. 9 07 t 2 5.

Chueh-P r a us n i t z

( El

P De v.

1. 3648 t l . l1. 0078 +. 782 . 5 06 5 t 1 . 51 .0054 t . 544. 2662 t 1. 84 . 0693 t 1 .71. 0038 1. 386. 8032 +2. 24. 0456 +1. 11. 0025 +. 2510. 317 +2. 48. 1375 +1. 74. 0300 +. 751 . 0 01 7 t . 1 71 4 . 9 9 3 t 2 . 412. 213 +1. 88 . 0 83 3 t 1 . 04. 0187 +. 471 . 0011 +. 1121. 056 t 2 . 120. 396 t 2 . 016. 228 t1 . 412. 115 + 968. 0464 +. 584 .0108 t . 271. 0006 t . 0628. 530 +. 4228. 172 t . 6124 .234 1. 982 0. 1 63 t . 8 1

1 6. 0 98 t . 6 112. 051 +. 438. 0205 +. 264. 0045 t . l l1 . 0 00 3 t . 0 33 0. 4 36 - . E 929.973 - . 0928. 160 t . 5724. 174 +. 7220. 122 +. 6116. 074 t . 4612. 038 +. 328 . 0152 t . 194. 0034 . OS1. 0002 +. 02259.67 t73 .169 .39 t69 .1 36 . 1 0 t 7 0 .1 03 . 1 9 t 7 2.8 5. 4 67 t 7 1 .6 2. 0 25 t 5 5 .

30. 485 t1 . 6 29 .977 - . OS2 0. 115 t . 5 8 2 0. 0 21 - . 0 110. 014 +. 14 10. 004 t . 04

1. 0001 + 01 1. 0000 001 62 . 9 8 - 1 9. 3 06 . 0 4 t 5 3.1 38 . 2 3 - 7 . 8108. 13 +8. 19 2. 7 22 t 1 6.68. 934 +15.53. 337 +6. 741. 267 +3. 230. 437 +1.520. 115 +. 5810. 015 + 151. 0000 00256.45 -15 .2 32 . 3 9 - 7 . 0202. 59 +1. 2162. 12 18. 1107. 74 +7. 78 4. 0 87 t 5 . 161. 634 +2. 750. 894 +1. 840. 425 t l . l30.164 +. 5520. 042 t . 211 0 . 0 0 5 t . 051. 0000 00

231. 02 +54.151. 68 +52.1 1 6 . 3 8 +45.7 0. 615 t l 8 .4 9. 8 62 - . 2 839.416 - 1.52 9. 7 33 - . E 91 9. 8 91 - . 5 59 . 9 7 26 - . 2 7. 9997 - .03342. 43 +14.291. 56 +17.234. 59 117.1 68 . 9 4 t 1 3 .100. 97 t . 9778 .041 -2 . 55 8. 0 37 - 3 . 348. 471 - 3 .138. 952 - 2 .629. 386 - 2 . 11 9. 7 15 - 1 . 49 . 9 29 1 - . 7 1, 9 99 30 - . 0 7

TR - 27/ 64TR1'

ZcVR-0. 064 ( ?cVR+0. 06112

0. 4450

c R

T R

TcVR- 0. 0906 T R O ' % V ( ?cVRt 0. 0906)

m - K -P = zcVR- 0. 0778 zcVR( z V +0 .07781 t0 .0778(ZcVR-0 .07781C R

The differences of all temperature functions are notsignificant at high densities because of the inhe rent defectof all cubics for reduced densities of1.5 to 2.0.

the simplest and i t can be written either as eq67 with atempe rature function such asT R " in the second term oralternatively as

D. The Best Two-Term Cubic Equation of StateIn summary, the best two-term cubic equation is also


14/17

94 Ind Eng. Chem. Fundam., Vol. 18, No. 2, 1979

Table IX. Comparison of Four Equations of State with Data (18) for T rifluoromethane

C H F 3

TRI FL UOROMETHANEv

f t 3 / l b

TC = , 99819

P c =

538. 33 R

. 70008701. 42 p s i

F, = 32. 78 l bi f t

R = 10. 7315 . 31066psi' f t 3 / l t mol e . RMW = 70. 019

z = . 259344 . 20158

. 12836

, 082677

, 055465

. 047352

, 033604

, 028644

, 023589

, 023646

, 021531

, 020316

. 016045

T, R

400. 1466. 4517. 66602. 60687. 91412. 49463. 35504. 59590. 22691. 63455. 44493. 01546. 04614. 02679. 54419. 89505. 55536. 95538. 33502. 52642. 41103. 64510. 78548. 42591. 05632. 09670. 58705. 91519. 08539. 42576. 68613. 11650. 58100. 06534. 86552. 38587. 93643. 00694. 8531. 685 6 3 . 0 6606. 44545. 6574. 22602. 73628. 97657. 22683. 05545. 69568. 67581. 21613. 06634. 82656. 98

5 3 8 . 3 3561. 005 7 6 . 8 6596. 89624. 23536. 11538. 33558. 08517. 12591. 56617. 85536. 54538. 33559. 85575. 60593. 64610. 16523. 915 3 8 . 3 3548. 51561. 74512. 13598. 68500. 02505. 25514. 58522. 155 3 8 . 3 3544. 27

p , P S 1data

55. 616 1. 2 016. 3090.25

104. 1779. 4592. 46

103. 13123. 92148. 08179. 78203. 82236. 26276. 15313. 09270. 8298. 0330. 7332. 2315. 2430. 34 84 . 3409. 0416. 9548. 06 12 . 2672. 5725. 5525. 8589. 9699. 2802. 0906. 1

1036. 5654. 3742. 79 10 . 0

1162. 61392. 5

684. 38 34 . 1

1082. 2764. 1

1014. 71268. 91496. 41148. 61913. 1

715. 01019. 01224. 41504. 11749. 91995. 4

705. 81024. 31254. 51555. 71974. 3

693. 2115. 7

1014. 31313. 61649. 11978. 2

106. 67 3 6 . 6

1097. 51314. 61698. 11991. 0

680. 8781. 1974. 8

1229. 11435. 81971. 5

461. 8645. 3965. 4

1226. 61192.41967. 9

R e d l i c h -Kwong

( A 1

P D e v

5 7. 2 53 t 3 . 06 7. 8 84 t 1 . 07 6. 0 56 - . 3 289 .532 - . E O1 03 . 0 1 - 1 . 18 7. 1 01 t 3 . 39 3. 9 44 t 1 . 61 03 . 4 8 t . 3 41 23 . 1 6 - . 6 2146. 29 - 1 . 2187 . 06 t4 .1208 . 02 +2 . l237. 20 +. 402 14 . 2 3 - . 6 9209.58 -1 . 1281. 36 +3. 9304. 56 t 2 . 2332. 68 +. 603 33 . 9 1 t . 5 2373. 06 - . 57425. 53 - 1. 14 18 . 4 1 - 1 . 2422. 99 +3. 44 80 . 1 8 t . 6 9543. 96 - . 74604. 54 - 1 . 36 6 0 . 7 3 - 1 . 8111 . 8 5 - 1 . 9541. 28 +4. 1600. 11 +1 . 86 91 . 2 8 - . 2 11 90 . 0 6 - 1 . 58 84 . 0 8 - 2 . 41006. 4 - 3 . 06 67 . 11 t 2 . 1145. 29 . 35911. 11 - 1 . 1113 4. 3 - 2 . 41 34 9. 3 - 3 . 16 94 . 0 8 t 1 . 4833. 49 -.071066. 6 - 1 . 47 72 . 0 8 t 1 . 01029. 5 +1. 51 28 1. 4 t . 9 91509. 4 . E 71151. 0 +. 141968. 8 - . 22828.09 t 6 . 91 09 4. 6 t l . 41306. 3 t 6 . 71 59 7. 1 t 6 . 21838. 3 +5. 12 08 0. 8 i 4 . 3

9 51 . 8 5 t 3 5.1 31 7. 5 t 2 9 .1569. 5 +25.1883. 8 +21.2306. 0 117.1 01 7. 2 t 4 7 .1 04 5. 6 t 4 6 .1 3 8 8 . 2 t 3 7.111 3. 1 t 3 0.2058. 2 t 25.2395. 5 +21.116 7. 5 t 6 5 .1 20 1. 8 t 6 3.1 61 0. 6 t 4 1.1 90 5. 5 t 3 9.2 23 9. 3 t 3 2 .2541. 4 +28.1 33 9. 3 t 9 7 .1455. 3 t 86 .1611 . 8 +72 .1950. 7 159.2167. 7 t 51 .2 71 5. 2 t 3 8.2855 .1 t 5123056. 1 +3733393. 0 t2513 6 6 4 . 5 +1994239. 7 t1364449. 2 t126

S o a v e

I B )

P D e v.

56. 818 t2 . 36 1. 6 86 t . 1 275. 999 - . 3989 . 707 - . 6 0103. 30 - . 7481. 410 +2. 593. 525 +1. 2103. 29 +. 161 2 3. 4 4 - . 39141. 11 - .65184. 15 t 2 . 8206. 75 11. 4237. 42 +. 49276. 31 t . 06313. 38 +. 092 17 . 5 5 t 2 . 53 02 . 4 2 t l . 5332 . 59 t . 57333. 91 t . 52315. 91 t . 194 32 . 1 5 t . 4 34 88 . 1 7 t . 9 2418. 68 t2 . 4481 . 15 t 1 . 0552. 11 + 75618. 88 t l . l6 80 . 7 2 t l . 2736. 88 +1. 65 70 . 3 1 t 2 . 86 01 . 1 6 t 1 . 9111 . 0 5 t l . 18 16 . 6 8 t 1 . 8923. 63 +1. 91 06 2. 5 t 2 . 56 6 5 . 0 6 t 1 . 7755 . 98 t 1 . 8931. 63 +3. 01212. 3 t 4.31463. 9 t5 . 1693. 42 +1.3858. 51 +2. 91135. 1 +4. 9785. 90 +2. 91096. 7 +8 . l1401. 4 t 10 .1 67 1. 0 t 1 2 .1 96 9. 1 t 1 3.2232. 1 + 1 3 .846. 43 +9. 21169. 9 t15 .1421. 2 +11.1 78 0. 8 t l 8 .2 01 4. 0 t 1 9 .2368. 7 t 19.

9 51 . 8 5 t 3 5.1396. 1 + 3 6 .1102. 8 t 36.2085. 5 +34.2599. 8 +32.1011. 2 t46 .1045. 6 +46.1461. 7 +44.1 8 5 7 . 5 +41.2276. 8 t 38.2681. 5 t 36.116 0. 2 t 6 4.1 20 1. 8 t 6 3 .1697. 7 5 5 .2055. 9 t50 .2461. 7 t45 .2829. 1 +42.1315. 0 + 9 3 .1455. 3 +86.1711. 3 t16 .2055. 1 +61.3218. 2 +61.2982. 4 +51.2612. 8 +4582831. 4 t 3393236. 1 t 2353551. 7 +19C4239. 7 +1364488. 3 +128

Penq-Robi nson

I C )

P Dev.

56. 645 +1. 967. 398 +. 291 5. 6 73 - . E 289. 328 - 1. 0102. 98 - 1. 18 0. 9 48 t 1 . 99 2. 9 15 t . 5 61 02 . 6 8 - . 4 41 22 . 7 1 - . 9 7146. 27 - 1 . 21 82 . 6 7 t 1 . 62 04 . 3 4 t . 2 52 34 . 5 7 - . 7 12 12 . 9 8 - 1 . 2309 . 65 - 1 . 1293 .34 t . 942 91 . 6 5 - . 1 2321. 11 - 1 . 13 28 . 4 6 - 1 . 13 69 . 6 2 - 1 . 5424 . 83 - 1 . 34 80 . 5 3 - . 7 8410. 63 +4 . 04 71 . 6 2 - 1 . 1539. 81 - 1. 56 04 . 6 5 - 1 . 26 64 . 8 2 - 1 . 1119 . 5 4 - . E 2530. 31 t . 89588. 32 - . 27693. 00 - . E9193. 81 - 1 .0896. 24 - 1. 11029. 5 - . 686 51 . 3 8 t . 4 7142. 15 - . 07911. 93 +. 21116 9. 5 t . 6 01 40 6. 5 t 1 . 0588. 97 +. 68841. 58 +. 911098. 1 +1. 5774.88 t1 . 41 05 6. 5 t 4 . 11 3 3 3 . 3 +5. 11584. 3 +5. 91 85 0 . 9 t 5 . 92091. 5 t6 . 0807 . 27 +4 . 21 09 1. 3 t l . 71328. 4 +8. 51646. 1 t9 .51911. 2 19 . 22117. 5 t9. 1

808. 11 +15.1 20 0. 9 t 1 7 .1 47 2. 6 t l 7 .1812. 0 1-16.2269. 2 115.823. 18 +19.853. 57 +19.1220. 2 +20.1 56 9. 7 t 1 9.1 94 0. 4 t 1 8 .2304 . 3 +16.8 94 . 11 t 2 1 .930 .59 t26 .1366. 0 t 2 4.1681. 0 t 2 2.2 03 8. 3 t 2 0 .2362. 3 t 19 .9 35 . 4 9 t 3 7 .1 05 7. 9 t 3 5.1286. 1 +32.1 58 1. 9 t 2 9.1 81 2. 1 t 2 6 .2394. 0 +21.1098. 2 t 1341287. 9 t1001 62 4. 7 t 6 8.1 89 6. 5 t 5 4 .2413 . 1 +38.2683. 3 +36.

Ma r t i nI D1

P D e v.

56.364 + 1 . 461. 316 t . 261 5. 1 55 - . 7 189. 466 - . E7103. 09 - 1. 08 0. 4 96 t 1 . 39 2. 9 04 t . 4 8102. 19 -.33122. 98 - . 76146. 48 - 1. 11 81 . 8 8 t 1 . 2204. 47 +. 32235. 43 - . 3 5214. 07 -.75310. 49 - . E 3272.63 t . 68298. 09 1. 03328. 54 - . 6 532 9. 81 - . l o311 . 64 - . 95426. 75 -.E3481. 57 - . 56410.97 t . 48414. 18 - . 57543. 32 - . E56 01 . 9 0 - . 7 0667. 02 - . E1120.26 - . 72530. 18 . E 3590. 46 +. 09697. 34 - . 0 37 98 . 0 8 - . 4 9898.50 - . E41021. 0 - . 926 51 . 4 0 t . 4 91 44 . 5 2 t . 2 4915. 19 +.571166. 6 +. 341391. 6 - . 06688. 99 +. 69843. 94 +1. 21097. 2 +1. 4114. 52 +1. 41 05 0. 5 t 3 . 51315. 3 t 3 . 11550. 7 t 3 . 61196. 3 t 2 . 72014. 6 12. 18 01 . 7 0 t 3 . 41080. 6 + 6 . 01 29 8. 7 t 6 . 11593. 7 + 6 . 01834. 7 +4. 92074. 0 +3. 9

17 5. 48 t l 0 .1142. 8 +12.1391. 5 t 11.1697. 4 t 9 . 12101. 9 +6. 5118. 43 +12.806. 74 + 1 3 .1145. 8 t 13 .1462. 9 t11 .1193. 5 +8. 82112. 8 +6. 8822. 98 +16.8 56 . 6 2 t 1 6 .1253. 3 +14.1535. 1 t12 .1850. 0 t 9 . 02131. 6 +7. 18 24 . 5 1 t 2 1 .935. 59 +20.1144. 5 +17.1 40 4. 1 t 1 4 .1606. 5 112.2 11 0. 0 t 2 . 0525. 33 +12.

686. 45 t 6 .4969. 97 +. 471196. 5 -2 . 51670. 7 - 6 . 81 84 1. 6 - 6 . 4

0. 42748p = i R

ZcVR-0. 08664 TR0 5EcVR( ZcVRt 0. 08664)

( C ) 0 . 4 512 36 11 +0 . 1 5 ( 1 - T ~ 5 ) P = TR -

ZcVR-O. 0778 Z V ( f cVRt O. 077 8) o. 01 8 ( ;cVR-O. 077 8C R

0. 42748 i l t 0. 86 u - T ~ ~ ) I D 21/ 64TR( B )

TRP R = -

zcVR- 0 . 08664 ~cVRl ~cVR+0. 08664) zcVR- 0. 055 l z c V R t 0 . 0 7) 2

where A TR) 2 7 / 6 4 T ~ ~ ,= t P c / RTc ,he translation,and C = 1/8 - B. Th e translation is set to fit best at areduced density of 1.5 or 2.0, depending upon w hat be-havior is sought, and n is selected to give a reasonable

average slope M of the criticcl isometric basedon t h egeneralized vapor-pre ssure diagram. It has been shownthat no other two-term cubic can be developed to besuperior to eq 93 beca

Review_Cubic Equations of State- Which

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