/

LEONARD 1. STlEL

Extensions of the Theorem of Corresponding States Recent advances In the extension

of the slmple theorem permit the

more reliable calculation of

thermodynarnlc and transport

properties of a wlde range of pure

substances and mixtures

n 1873 van der Waals proposed his equation of state I which, when the conditions of the critical point are applied, predicts that the compressibility factor of a fluid is a unique function of the reduced temperature and pressure. This observation led directly to the “theorem of corresponding states” which has formed the bask of many correlations in which a reduced therm- dynamic or transport property is related to the reduced temperature and pressure. In recent years, methods based on extensions of the theorem of corresponding statea have been developed which permit the more accurate calculation of values of thermodynamic and transport properties for certain pure fluids and mixtures. Future work in this area should lead to analogous pro- cedures for a much wider group of fluids and mixtures. To a good approximation, for most fluids, the

functional dependence of the compressibility factor on temperature, pnssure, and molecular properties is

The dimensionless group a / p o represents shape effects, p’/tpo’ the effects of dipole-dipole interactions, and h/ [Pa(mt)lp] quantum effects which are present in certain molecules such as hydrogen and helium at low tempera-

M I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

tures. If only the fist two groups are used, Equation 1 is saictly correct only for simple molecules such as argon, krypton, and xenon. This case corresponds to the use of an intermolecular potential which contains two characteristic parameters, such as the Lennard- Jones potential,

where t is the depth of the potential minimum, u is the collision diameter, and r is the distance between the molecules. For this potential point molecules ’are considered and PO = n.

If the third group is included, Equation 1 is applicable for nonpolar gases and corresponds to the use of an intermolecular potential containing a shape parameter, such as the Kihara potential (s!xictly for a specific shape of core). The Kihara potential can be written as

$(p) = 4€[(”>”-(”)] P P (3)

where p is the shortest distance betwan molecular cores and po is this distance when the potential energy is z m . For a spherical core, p = r - 2 a and po = n - 2 a, where a is the radius of the core. For other shapes of core, a in Equation 1 is a characteristic shape parameter such as the length of a thin rod.

If the fourth group is added, Equation 1 is applicable for polar fluids and a corresponding intermolecular potential is a superposition of the Kihara potential with a term accounting for dipole-dipole interactions,

where g(0) represents the angular dependence of the dipole-dipole interaction. If point molecules are con- sidered, Equation 4 represents the Stockmayer potential which is often used for polar molecules and excludes shape effects. The inclusion of the fifth group in Equa- tion l allows for the quantum corrections to the various intermolecular potentials.

If the conditions of the critical point [z = z,, (bP/ bv),, = 0 , (d2P/dv2)To = 01 are applied to Equation 1, the following relationships result :

(5)

(7)

Hirschfelder, Curtiss, and Bird (22) present plots of z,, KT,/E, and P,u3/e against p2/ea3. The force constants were obtained for the Stockmayer potential from ex- perimental second virial coefficient data. Hirschfelder, Curtiss, and Bird (22) also plot values of KT,/B and P,u3/e against h/u(me)1'2 for the Lennard-Jones potential with quantum corrections.

If Equations 6 and 7 are substituted into Equation 1 the following equation results:

The groups a / p o , p2/epo3, h/po(me)'12 can be represented by empirical macroscopic parameters (J, x , y , which account for shape, polarity, and quantum effects, respec- tively, as follows:

Therefore, Equations 5, 6, and 7 become

zc = f&, x , r) (12)

and Equation 8 becomes

= f l z (TRY ' 2 2 , w Y x> r) ( 1 5) Equation 15 is the resulting farm of the extended

theorem of corresponding states. Similar equations result for other reduced thermodynamic and transport properties. If shape, polarity, and quantum effects are neglected, Equations 12, 13, and 14 predict that z,, K T J B , and P,u3/r are constants, and Equation 15 that the compressibility factor is a function of only the re- duced temperature and pressure, the simple theorem of corresponding states.

Equations 1 and 15 are only approximate in that only one shape, polarity, and quantum parameters are used. Additional shape parameters may be required ; other polarity effects may be important such as dipole-induced dipole, dipole-quadrupole, and quadrupole-quadrupole interactions ; and additional quantum groups can be used ( 7 7). However, Equation 15 is a very substantial improvement over the simple theorem of corresponding states and is probably as complicated a relationship as is presently amenable to analysis.

For several thermodynamic and transport properties such as the second virial coefficient and the viscosity of gases at atmospheric pressure, theoretical equations of the form of Equation 1 in terms of molecular properties can be used directly. Very accurate experimental data are required to establish the molecular constants for each substance. Equations of the form of Equations 3 and 4 can then be used to express the properties in terms of macroscopic quantities. Stiel and Thodos (53) have empirically accounted for shape effects in the Lennard- Jones potential by relating force constants from second virial coefficient and viscosity data to the critical compressibility factor, as follows:

and

V O L 6 0 NO. 5 M A Y 1 9 6 % 51

Tee, Gotoh, and Stewart (55) have presentea slmuar &tionships between the Lennard-Jones force constants and W. Tee, Gotoh, and Stewart (56) also developed the following relationships for the parametera for the Kihara spherical-core potential resulting from second virial coe&ient and viscosity data:

a 2 -

a 1 - 2 -

(18) U

a* = - = 0.1527 + 1.9809 w

a

u = 2.2631 - 0.3278 w (19)

c/KT, = 1.0070 + 2.2450 w (20)

where u = po + 2 a. Stiel and Thodos (50) have found that the group e / g T e and a / ~ ~ * ~ resulting from viscosity data for the Stockmayer potential are approximately independent of the polarity group. For most ranges of temperature and pressure, accurate

theoretical expressions are not available for the thermo- dynamie and transport properties and it is presently more profitable to work k t l y with the macroscopic quation such as Equation 15. Also the experimental data required to obtain the necessary parameters for the maemseopic equations do not have to be of the extreme ~ocuracy required for use with the theoretical equations.

Tkmmdynamk Protawtior of P u n Fluids N u m u s redud state correlations are available for

thc wmpreanibility factor and derived thermodynamic properties accod ig to the simple theorem of correspond- ing states. Lydersen, Greenkorn, and Hougen (30) hve used the critical compressibility factor, q, as a tbird parameter combining both shape and polarity effects w develop tables for +e Compressibility factor and the dcfivrd properties, the enthalpy departure, entropy deparhur, and fugacity coe5cient, for nonpolar and poLar fluids for reduced temperatures up to 15.0, reduced prwurea to 30.0, and values of the critical compressi- Wty factor of 0.29, 0.25, 0.27, and 0.29. Yen and Woods (58) have recently p-ted analytical relation- ships between the reduced density and the reduced temperature, reduced pressure, and critical com- presaibility factor for saturated and compressed liquids. Yen and Alexander (57) have used experimental and derived enthalpy data to improve and wtend the corre- lation of Lydersen, Greenkorn, and Hougen for the reduced enthalpy departure, (H - Ho)/T..

Since the wrrelations of Lydersen, Gteenkorn, and Hougen combine both shape and polarity effects into one third parameter, 1- of accuracy is obtained' for both nonpolar and polar fluids. In Figure 1 values of the reduced density for saturated nonpolar and polar

52 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

0.21 0.72 0.23 6.24 0.25 0.26 0.27 0.28 0.29 0.30 1,

Figun 1. Rclalionship &twm rc&ced dmciry and did mm- prbssibil& factor at T. = 0.9 for nonpolar andp&,?ui&

fluids at TB = 0.9 are plotted against the critical com- pressibility factor. I t can be seen that the line resulting from the correlation of Lydersen, Greenkorn, and Hougen is a good approximation for all the fluids, but that improvement could be obtained by separating shape and polarity effects. For pure substances which do not exhibit excessive

polarity effects, the most accurate generalized method for the prediction of the compressibility factor and other thermodynamic properties is that developed by Pitzcr and co-workers (27). For a nonpolar fluid the reduced vapor pressure can be expressed as

pB = f18(TBr w ) (21)

(22)

Pitzer approximated Equation 21 as

log PB = log PE(0'(T.) + w log PE(l'(TB)

where log PB'& is the simple fluid (w = 0) value. At T. - 0.7, log PEW = -1.000 and log Pa(') was defined to be - 1.000 so that

(23) w = -10 g pE]Ta-o,, - l.Oo0

Equation 23 is thus the definition of the tbird parameter for nonpolar fluids, called the acentric factor. A similar definition of a third parameter for nonpolar fluids, ac, in terms of the reduced vapor pnssure has been proposed by Riedel (42). Guggenheim and Wonnald (13) have also proposed a similar approach in which

the third parameter is the number of carbon atoms in the normal alkane which most closely resembles the given substance. Rowlinson (44) has presented theoretical results for liquids with noncentral forces in terms of a third parameter which is directly related to the acentric factor.

Pitzer tabulated values of log PR(0) and log PR(l) and similar functions for the entropy of vaporization and the saturated liquid and vapor compressibility factor for T , = 0.56 to T , = 1.00. Lyckman, Eckert, and Prausnitz (29) have presented an improved correlation for saturated liquids in terms of the reduced volume, as follows :

u R = U,(O) f OUR(') + 0 2 u R ( 2 ) (24) where u,(O), u,(l), and u , ( ~ ) are tabulated functions of the reduced temperature.

Pitzer expressed the compressibility factor as

(25)

The functions t ( O ) and z ( l ) and similar functions for the reduced enthalpy departure, entropy departure, and fugacity coefficient are tabulated for reduced tempera- tures up to T , = 4.0 and reduced pressures up to P R = 9.0. At the critical point Equation 25 becomes

Z, = 0.291 - 0.08 o (26)

Reid and Valbert (47) defined the derivative com- pressibility factors

% p = % - P,(bz/bPR)TR (27)

ZT Z + T,(bz/bTJpR (28)

and

which were related to reduced temperature, reduced pressure, and acentric factor as

zT = zT(O) + OzT")

z p = Zp(O) + ,zp(l)

(29)

(30)

and

The tables of Pitzer for do) and z(l) were used to obtain values of zT(0), zT( l ) , zp(0), and zP( l ) for reduced tempera- tures from 0.8 to 4.0 and reduced pressures from 0.2 to 9.0. The derivative compressibility factors are useful in calculating partial derivatives such as ( d v / b T ) , =

Pitzer and Curl (35) developed the following equation RzT/P.

for the reduced second virial coefficient:

- BP, R Tc

= (0.1445 + 0.073 W) - (0.330 - 0.46 OJ) T,-l - (0.1385 4- 0.50 O) TR-' -

(0.0121 + 0.097 O) T,-3 - 0.0073 w TR-' (31)

An analytical expression for the second virial coefficient of nonpolar fluids can also be obtained by the combina- tion of the theoretical expression for the Kihara-spherical core potential

B ' / 3 TNV'

~ = [u* , + 3 (21'6)a*2F1 (F) +

3 (2ll3)u*F2 (F) + ( 2 1 / 2 ) F 3 ( ~ T / ~ ) (1 f a*)-3 (32) 1 with Equations 18, 19, and 20 for u, E / K , and a*. The groups FI, Fz, and Fa are tabulated as functions of K T / E by Hirschfelder, Curtiss, and Bird (22). Pitzer and Curl (35) used Equation 31 to obtain relationships for the fugacity coefficient, reduced enthalpy departure, entropy departure, and the heat capacity departure, C, - CPo, in terms of the reduced temperature, reduced pressure, and acentric factor for nonpolar gases at low pressures.

Lydersen, Greenkorn, and Hougen (30) differentiated their tables for the reduced enthalpy difference with respect to reduced temperature at constant reduced pressure to develop a generalized correlation for the heat-capacity departure only for z , = 0.27. Edmister (9) expressed the heat capacity departure as

(C, - C,O) = (C, - C,O)(O) + O(C, - C,0)(1) (33)

Edmister (9) developed plots of (C, - C~)(O) and (C, - CPo)(l) as functions of reduced temperature and reduced pressure by the use of the values of the deriva- tive compressibility factor zT presented by Reid and Valbert (47), through the relationship :

c, - c*o = (34)

For the Joule-Thomson coefficient, 7, Edmister (9) obtained similar plots for the simple and normal fluid contributions to the group (PJRT, ) (7CP), through the use of the relationship :

(3 5 )

For saturated liquids, Bondi (7) discusses theoretical and empirical relationships of the form

C, - Cpo = f14(T,, 01 (36)

Pitzer found that his approach is applicable for nonpolar and slightly polar fluids, called "normal fluids." Normal fluids are defined as those substances whose surface tension constants, yo, can be calculated to within 5% by the following equation :

- - 1.86 + 1.18 o ~ 0 7 ~ 0 ~ ' ~

I', (37)

V O L 6 0 NO. 5 M A Y 1 9 6 8 53

acentric factor of the polar fluid is taken as that of its homomorph. Halm (77) has developed relationships for the reduced second virial d c i e n t of polar fluids as a function of T, o, and x.

The equations of state of Hirschfelder et al. (20) wntain a. as a third parameter and z. as a fourth parameter and therefore should be applicable to a wide group of substances, including polar fluids. Hirsch- fclder e l al. (27) also present relationships for the reduced enfhalpy departure, entropy departure, fugacity co- e%rcient, heat capacity departure, and the derivatives @PJbp.), and (bP,/bT,)p, resulting from their equations of state.

For quantum fluids, Gum, Chueh, and Prausnitz (14) have presnted relationships for effective critical temperatures and critical pressures in terms of the clasi- cal critical constants, the molecular mass, and the tem- perature which permit Pitzer's tables for thermodynamic properties to be used directly. The acentric factors of the quantum fluids are assumed to be zero. A similar approach has been used by Gum, Chueh, and Prausnitz (15) for the Joule-Thomson inversion curves of quantum fluids.

The approach used by Pitzer and co-workers (27) for nonpolar fluids can also be used to establish rela- tionships for the thermodynamic properties of quantum fluids. For quantum fluids, shape effects are usually not important because the molecules are small in size and w can be taken to be zero. With this simplification the parameter y can be calculated from its molecular defini- tion h / [ u ( v ~ e ) ~ ~ ] , since values of the force constants u and e am available for the quantum correction to the Len- nard-Jones potential (22). Hirschfelder, Curtis, and Binl (22) present plots of the reduced vapor pressure,

0.08

0.06

0.04

0.02

x 3

np I

2 I -0.M

-0.04 ri

nw1noaol

-0.08 -O.O6 w -0.06 -0.04-0.02 0 0.02 0.04 0.06

P 8 / c , the reduced molar liquid volume, u/u', and re- duced surface tension, rd/e, as functions of K T / ~ and h/u(rne)l'* for a number of quantum fluids.

Transport Propdes of P u n Fluids Several generalized comlations are available for the

reduced viscosity p / p e or p / p * as a function of reduced temperature and p m u r e according to the simple theorem of corresponding states. Thodos and co- workers (54) have presented plots of the reduced vis- cosity against reduced temperature and reduced p m u r e for a wide range of individual substances which can be used to calculate viscosities for substances having a similar value of the critical compressibility factor or acentric factor.

Isotherms of the viscosity of a gas against density are parallel, indicating that a unique relationship should exist between p - p* and the density. A theoretical justification of this form of relationship for moderately dense gases has been presented (26). Jossi, Sticl, and Thodos (24) have obtained expressions for the viscosity of pure nonpolar substances in the dense gaseous region by the use of a relationship between p - p* and density, along with dimensional analysis considerations. Dean and Stiel (8) have modified the relationship of J&, Stiel, and Thodos on the basis of more recent e x p i - mental data. The resulting relationship is

(,i - p*)f = 10.8 X 106 (&a) pa - 8-1.11 h''? (42)

where f = Te*8/M'1'P:1a. A relationship for 01 - p * ) t as a function of reduced density for polar fluids has also been developed by Stiel and Thodos (57).

The theoretical relationship for the viscosity of a non- polar gas at atmospheric p m u r e is

266.93 X lo-' ~ F T UW+)*(KT/~)

where the colliion integral Q@*)* is a tabulated function of KT/C for the Lennard-Jones potential (22). Bromley and Wilke (5) obtained a nlationship for the viscosity of nonpolar gases by combining Equation 43 with the constant values of IT,/* and u/ve*' for the Lennard- Jones potential. Stid and Thodos (47) found that the group p*E for nonpolar gass a t a constant reduced temperature is essentially independent of the acentric factor, as shown in Figure 4 for TE = 1.0. This o h a - tion indicates that the viscosity of nonpolar gases at atmospheric pressure can be accurately characterized by a two-parameter intermolecular potential such as the Lennard-Jones potential, a result confirmed by Tee, Gotoh, and Stewart (55, 56). Stiel and Thodos (47) obtained the following relationships for the viscosity of nonpolar gases:

p* = (43)

p*E = 34.0 X 10-5 TEo.M, Tn < 1.5 (44)

VOL 6 0 NO. 5 Y A Y 1 9 6 0 55

x 10-5

32 0 0.040.080.120.160.200.240.280.32 0.360.40 w

F i e 4. R&imuinp bumm r r h e d v*coriry and accnbic facm fa gaw at nbMrpharic pressure and T p = 7.00

and

&*E 17.78 X 10-’[4.58 TR - 1.67]”*, T p 1 1.5

(45)

For polar gases at atmospheric pressure, Stiel and lAodos (48) have developed relationships for #*E as a fuaetton of the reduced temperature with the critical wmpresibility factor as a third parameter. If shape and polarity effects are considered for polar gases at atmospheric pressure, the functional form is

(46) p*E = .fir(Tm 0, x)

Equation 46 was expanded into a Taylor series truncated afar quadratic terms, and the coefficients were deter- mined at Ta = 0.9 by fitting the equation with data for both normal and polar fluids with values of x obtained by Halm and Stiel (78). The effect of the terms con- taining w was found to be significant because of the large values of the acentric factors of polar fluids. A plot of &*E - O.*& - Awx against x is shown in Figure 5.

Since shape effects on the viscosity are negligible for quantum fluids, the functional form for the viscosity of gases at atmospheric pressure is

pi€ = f d T m Y) (47)

w h y = h/[c(me)v*/’l for the Lennard-Jones potential. In Figure 6 values of p*E at T p = 1.0 are plotted against y fur aeveral quantum fluids. It can be seen that p*E for quantum fluids is strongly dependent on y.

For saturated liquids, Reid and Sherwood (40) com- bined the relationship of Jossi, Stiel, and Thodos (24) f q (r - p*)E as a function of the reduced density with 4uati0m 44 and 45 and values of the reduced density obtained h m the correlation of Lydersen, Greenkorn, and aougen (30) to obtain plots of pE as a function of TB and 1.. In Figure 7 a plot of pE against w for saturated liquids at T. = 0.7 is presented for simple fluids and nor- mal p a r e s . It can be seen that a smooth relationship for the normal paraffins is obtained and that there is only a ivak dependence of rE on w for the higher meFbers of

56 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

.. 12.0

x 10.0 2 8.0

1 6.0 E, 4.0

I O - - 0.02

-0.06 -0.08

*a 2.0

a -0.04

-0.08-0.06 -0.04-0.02 0 0.07. 0.04 0.06 0.08 X

Figure 5. C w e l d k n for rediced oinosiry of polmgms at Ta = 0.9

the series. There are large deviations for simple fluid., possibly caused by quantum effects. Similar deviations of p.‘/[(me)’”] from a unique relationship with K T / ~ for simple fluids have been noted (7). Boon, Legros, and Thomaes (2) found that the monatomic molecules argon, krypton, and xenon followed the theorem of corresponding states exactly, while methane deviates considerably, indicating that the viscosity of liquids is very sensitive to the exact shape of the molecules. There are also deviations of the reduced viscosity from the normal paraffin relationship for isomers and slightly polar substances, similar to the behavior of the surface tension. These deviations increase at lower tempera- tures. For quantum liquids, plots of p.“/[(rne)’”] against K T / ~ and h/[lr(me)’”] are presented by Kerrisk, Rogers, and Hammel (25).

For the thermal conductivity Gamson (72) developed a correlation of k/k. against T, and Pw Thodos and co-workers (54) have presented plots of the reduced thermal conductivity against reduced temperature and pressure for a wide range of substances. For dense gases, Stiel and Thodos (49) have developed relation- ships between (k - k*)xI : and reduced density, where

The theoretical equation for the thermal conductivity T:/fiM1Ia/P:/a.

of a monatomic gas at atmospheric pressure is

1989.1 X lo-’ d / n ~ (48) ~ W ) * ( K T/e) k,” =

where the collision integral is tabulated as a func- tion of K T / ~ for the Lennard-Jones potential (22). The thermal conductivity of polyatomic gases does not follow the theorem of corresponding states because of internal degrees of freedom. For polyatomic gases the correction factor proposed by Stiel and Thodos (49) can be used

k,*/k,* = 0.307 C./R 4- 0.539 (49)

Theoretical relationships for the thermal conductivity of polyatomic and polar gases a t atmospheric pressure have been proposed by Mason and Monchick (37).

Riedel (43) obtained the following relationship for

10-3

the thermal conductivity of liquids a t atmospheric pressure :

(50) Cini-Castagnoli, Pizzella, and Ricci (7) show that the group k m 1 / 2 u 2 / t d 2 is a unique function of K T / B for simple liquids. Kerrisk, Rogers, and Hammel (25) present plots of km*’2u2/Ke1i2 against h/ me)^'^] and K T / B for quantum liquids.

For the self-diffusivity of nonpolar gases at atmospheric pressure, Stiel and Thodos (52) have obtained the follow- ing relationships :

( p a ) * E = 0.464 X l op6 [1.391 TR - 0.381I2l3 for TR _< 1.5 (51)

and

( p D ) * E = 0.488 X T,0.777 for TR > 1.5 (52)

Similar to the behavior of the viscosity, the group ( p D ) * E at a constant reduced temperature is essentially independent of the acentric factor for nonpolar gases. Values of the self-diffusivity of polar gases can be calcu- lated from the theoretical relationship with collision integrals obtained for the Stockmayer potential by Monchick and Mason (32). Stiel and Thodos (52)

“U

0 0.5 1.0 1.5 2.0 2.5 3.0

Y

Figure 6. Juids

Relationsh+ between f i*F at TR = 7.0 and y for quantum

UJ

4

” 0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40

0

Figure 7. f o r saturated liquids at T R = 0.7

Relationsh+ between reduced viscosity and acentric factor

have shown that the group ( p D ) E is independent of reduced density for p R 5 1.0, so that Equations 51 and 52 can also be used for this region. Slattery and Bird (45) have presented a generalized chart of the ratio PD/(PD)* as a function of reduced temperature and reduced pressure, based on Enskog’s theoretical relation- ship. However, this correlation was shown not to predict the behavior indicated by the experimental data for p R _< 1.0 (52). Naghizadeh and Rice (33) have found that the group Dm1iz/&2u is not a unique function of K T / E for simple liquids. Preston, Chapman, and Prausnitz (39) have recently presented relationships for the viscosity, thermal conductivity, and self-diffusivity of saturated simple liquids. The force constants used to reduce the transport properties and temperature were determined from experimental liquid viscosity and ther- mal conductivity data.

Thermodynamic and Transport Properties of Mixtures

For the calculation of the thermodynamic and trans- port properties of mixtures from correlations based on the simple theorem of corresponding states, Kay’s rules for the pseudocritical temperature and pressure have been widely utilized :

Pitzer and Hultgren (36) used experimental volumetric data for 12 binary gas mixtures at several compositions to establish values of the pseudocritical temperature, pseudocritical pressure, and acentric factor which permit Pitzer’s tables for the compressibility factor to be used for mixtures. They found that quadratic terms in the mole fractions were sufficient to relate the constants to composition for the binary mixtures considered. The Pitzer and Hultgren pseudocritical rules can be general- ized to multicomponent mixtures as follows (23) :

i j

(55)

(57) i j

VOL. 60 NO. 5 M A Y 1 9 6 8 57

the thermal conductivity of liquids at atmospheric presure:

x10-3

(50) Cini-Castagnoli, Pizzella, and Ricci (7) show that the group km*W/Xc'l' is a unique function of K T / E for simple liquids. KulisL, Rogers, and Hammel (25) present pie@ of km*W/& against h/[u(m~)~'*] and K T / ~ for quantum liquids. For the self-diffusivity of nonpolar gases at atmospheric

pressure, Stiel and Thodas (52) have obtained the follow- ing relationships:

(pS)*[ = 0.464 X 10-5 [1.391 TB - 0.381]"* for TB 5 1.5 (51)

and

(pD)*E = 0.488 X TRo.777 for T. > 1.5 (52)

Similar to the behavior of the viscosity, the group (,as)*€ at a constant reduced temperature is essentially independent of the acentric factor for nonpolar gases. Vdues of the self-diffusivity of polar gases can be calcu- lated &om the theoretical relationship with collision integral8 obtained for the Stockmayer potential by Mondtick and Mason (32). Stiel and Thodos (52)

w 0 0.5 1.0 1.5 2.0 2.5 3.0

Y

0

Figurr 7. Rel&'rnlu~ bdween reduced msm' ly and ace& fadm for safurated Iiprddr at Ts = 0.7

have shown that the group (p& is independent of reduced density for p B 5 1.0, so that Equations 51 and 52 can also be used for thii region. Slattery and Bird (45) have presented a generalid chart of the ratia ~ / ( P S ) * as a function of reduced temperature and reduced pressure, based on Endog's theoretical relation- ship. However, this cornlation was Shawn not to predict the behavior indicated by the experimental data for pm 5 1.0 (52). Naghizadeh and Rice (33) have found that the group Sml'a/~*au is not a unique function of KT/E for simple liquids. Preston, Chapman, and Prausnia (39) have recently presented relationships for the viswsity, thermal conductivity, and self-diffusivity of saturated simple liquids. The force constants used to reduce the transport properties and temperature were determined from experimental liquid viscosity and ther- mal conductivity data.

Thormodynamic and Tmnrpolt Propeltios af Mixhrnr

For the calculation of the thermodynamic and trans- port properties of mixtures from correlations based on the simple theorim of corresponding states, Kay's r ~ l w for the pseudocritical temperature and p m w e have been widely utilized:

T a = C wtTu (53)

pen = c wcpn (54) i

i

Pitzer and Hultgren (3) used experimental volumetric data for 12 binary gas mixtures at several compositions to establish values of the pseudocritical temperature, pseudocritical pressure, and acentric factor which permit Pitzer's tables for the compressibility factor to be used for mixtures. They found that quadratic te rm in the mole fractions were sufficient to relate the constants to composition for the binary mixtures considered. The P i e r and Hultgren pseudocritical rules can be general- ized to multicomponent mixtures as follows (23) :

Tm = C C W W T U ~

0, = C c WdWPij

(55)

(56)

(57)

' I

p a = c c WPPM : I

i i

V O L 6 0 NO. 5 M A Y 1 9 6 % 57

Halm (77) found that the m r s in second virial w- efficient8 calculated in this manner from Equation 31 for a number of nonpolar mixtures were comparable to thosc resulting from this equation for pure nonpolar &lids.

An alternate approach is to use the macroscopic relationship, Equation 31, to calculate B, directly with pseudocritical constants. This latter approach is simi- lar to that used by Dean and Stiel (8) for the viscosity of gaeeous mixtures, since a theoretical relationship in terms ot the v i d t i e s of the pure compoaents is avail- &le for this property (22). pi- and Hultgren (36) &lated second virial coefficients from Equation 31 lor a 40% n-butane, 60% methane mixture in the range 378" to 510' K by the use of the pseudocritical con- #ants established from the high prewure volumetric data for this system. The maximum deviation betwken experimental and calculated virial coefficients was 6 cm'/mol.

O'+~ell and Prausnitz (34) have presented combin- ing RlEes for the parameters of their pure fluid relation- diip which enable the calculation of the second virial -ts of gas mixturea Containing polar components. Fw a$mciatiug mixtures a binary interaction parameter is required. A similar approach using w and x has been developed by Halm (77). In the latter study con-

bly better results were obtained with Equa- tions 62 and 63 and relationships for the force constants in terms of w and x than from the direct use of the macrosmpic equation and the Prausnia and Gunn

wlxl). Prausnitz and Myers (38)

barc,presmted procedures for the calculation of the h virial coefficients of mixtures containing quantum @as. 4f thetPitzer and Hultgren peeudocritid rules, Equa-

tioos 55-57, are used the following relationship can be d&vcd for the fugacity coefficient of the rth component in a nonpolar mixture (23) :

fvith x, = W n ~ 1

%.quantities f./P, z,, and (H - Ho), are calculated fFom the cordations for the pure .components by the ltlc $..&e .. pseudocritical dw. Similar &tionships an obtained for other pseudoctkical rules (40). Joffe. and-Zudkevitch (23) compared experimental fuga+ aocfficientr and those calculated from Equation 64 and &&ate that good agreement is,obraincd in most Cases.

on of Equation' 64 to mixtuns contaiding U a n h l k a $ k u e n t s is- atraigh&ward once

on' & are estalhhcd for the pure wmponent ' M y n a + c . ppperties, aqd pseud . k d a a m i n l d for tlk.&&

200 x 10-5

0.1 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 10 1,

Rigurt 8. nixhum at abnos&ricprmsure

Ralarionhip batwen @*E and T p fa &gen-#thy*tU

For vapor-liquid equilibria calculations, Equation 64 can be used directly to determine fugacities for the components of both the vapor and liquid mixtures. Pitzer and Hultgren (36) obtained good results in this manner for equilibrium compositions in the propane isopentane system. An alternate procedure is to use Equation 64 for the vapor mixture and to use activity coefficients for the liquid components through the rela- tionship

4 = 7' P

where ( f l /P ) ' is the fugacity of the pure liquid component at the temperature and pressure of the solution. Chao and Seader (6) have presented a relatiodp'for (fl/P)' as a function of reduced tempera- p~essure, and acentric factor for nonpolar liquids. Lyckman, Eckert, and F'rausnitz (28) developed a correlation for (fl /P)' at zero pressure as a function of reduced temperature apd W . Lyckman, Eckert, and Rauanitz (29) have also presented a correlation of the reduced solubility param- eter 8/P;la against reduced temperatwe and acentric factor which can be used to caloulpte activity coefficients for components forming regular solutions, as outlined by Chao and hde r (6).

Conclusions The extension of the theorem of corresponding states

proposed by Pitzer has found wide utiEty lor the calcula- tion of the thermodynamic and transport ppoperties of pure substances and mixtures. Future work in this area will enable the calculation of thw properties for a wider group of substances by the inclusion of additional parameters. Accurate exp to enable the functional re1 parameters to be established. When these relationships are available, the only input data rquired for thc generation of a complete range d thermodynamic and transport properties of a pure fluid an one vapor pnssure point for a nonpolar substance and two vapor pressure p e t s for a polar substance to establish the ~~cce~ssly

parameters. C o m b i g d a for thc c c i t i d tempera-

ture, critical pressure, and shape, polarity, and quantum parameters will enable the pure component relationships to be used for mixtures. For mixtures containing dis- similar components, experimental volumetric data for the binary systems are presently required for highly accurate calculations of most properties.

Nomenclature characteristic shape parameter, such as the radius of a

dimensionless shape parameter for Kihara spherical core

coefficient of w x term in quadratic expansion for p*E second virial coefficient, cm3/g-mol heat capacity a t constant pressure, cal/g-mol O K heat capacity a t zero pressure, cal/g-mol O K heat capacity a t constant volume, cal/g-mol O K fugacity coefficient fugacity of the pure component a t the temperature and

angle dependent term of dipole-dipole interaction Planck’s constant enthalpy, cal/g-mol enthalpy at zero pressure, cal/g-mol thermal conductivity, cal/sec cm O K thermal conductivity a t atmospheric pressure, cal/sec

thermal conductivity a t critical point, cal/sec cm O K thermal conductivity of a monatomic gas at atmospheric

thermal conductivity of a polyatomic gas a t atmospheric

molecular mass molecular weight pressure, atm critical pressure, atm reduced pressure, PlP, distance between molecules gas constant, 1.987 cal/g-mol O K entropy of vaporization, cal/g-mol temperature, O K

spherical core

potential, 2 a / u - 2 a

pressure of the solution

cm O K

pressure, cal/sec cm O K

pressure, cal/sec cm K

= critical temperature, O K = reduced temperature, T/T, = volume, cm3/g-mol = critical volume, cms/g-mol = hypothetical volume of liquid at absolute zero, cma/g-

= reduced volume, v / v , = mole fraction = polarity factor = quantum mechanical factor = compressibility factor = critical compressibility factor = pressure derivative compressibility factor = temperature derivative compressibility factor

mol

Greek Letters

a0

Y Y i YO 6 a)* a,

II K x P

P

P P P c

PO

E

P*

5

60

= Riedel parameter = surface tension, dynes/cm = activity coefficient of component i = hypothetical surface tension a t absolute zero, dynes/cm = solubility parameter = self-diffusivity of gas at atmospheric pressure, cm2/sec = self-diffusivity, cm2/sec = maximum energy of attraction, erg = Joule-Thomson coefficient = Boltzmann constant, 1.3805 X 10-6 erg/’ K = thermal conductivity parameter, M1/zT ,1 /~ /Po*~3 = dipole moment of molecule, Debyes = viscosity of gas a t atmospheric pressure, g/cm sec = viscosity, g/cm sec = viscosity parameter, Tol~~ /M1/2P02 /3 = shortest distance between molecular cores = density, g/cm3 = critical density, g/cm3 = shortest distance between molecular cores when the

potential energy is zero

I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

p~ = reduced density, p / p c U = collision diameter, A $ ( r ) = intermolecular potential function W = acentric factor W,2)* = reduced collision integral

Subscripts and Superscripts

C = critical value i, j , r m n = normal fluid (0) = simple fluid function (1 ) (2)

= components of a mixture = pseudocritical property of a mixture

= normal fluid correction term = polar fluid correction term; also normal fluid quadratic

term in Equation 24

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