Redlich Kister Algebraic Representation of T

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February 1948 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 345

y,, 7 2 = activity coefficientsVl, Vz = molal volumes of the liquid components

V , = molal volume in the gaseous stat eB = second virial coefficient

Po, T o = critical pressure and temperatureT = an arbitrarily fixed temperatures = slope factor, Equations 11and 12Y = relative volatility

LITERATURE CITED

1)American Petroleum Institute Research Project 44, NationalBureau of Standards, “Selected Valuesof Properties of Hydro-carbons,” Tables 2k (March 31, 944) and 5k (June 30, 1944).

2)Beatty, H. ,, nd Calingaert, G., IND. NG.CHEH.,26, 04, 041934).

3)Benedict, M.,ohnson, C. A,, Solomon, E., and Rubin, C. C.,

4) Brunjes, A. S., nd Bogart, M. J. P., IND. NG.CHEM., 5, 55

5)Lewis,G. ., and Randall, M., “Thermodynamics,” New York,

(6) Othmer, D . F., and Benenati, R. F., IND. NQ.CHEM.,7,299

7)Scatchard,G., nd Raymond, C. L., J . Am Chem. SOC.,60, 278.

8)Tongberg, C. O., and Johnson, F., IND. NG.CHEM., 5, 733

9)York, R., and Holmes, R. C., Ib i d . , 34, 45 1942).

RECEJVEDovember 26,1946.

Trans. Am Inst. Chem. Engrs. , 41, 71 1945).

1943).

McGraw-Hill Book Co., 1923.

1945).

1938).

1933).

Thermodynamics of Nonelectrolyte Solutions)

ALGEBRAIC REPRESENTATION OF THERMODYNAMIC

PROPERTIES AND THE CLASSIFICATION OF SOLUTIONS

OTTO REDLICH AND A. T. KISTERShell Development Company, Emeryville, Calif.

T h e utilization of aboratory data for the design of distillation columns and other separation equip-

ment requires the efficient representation of extensive experimental data. A flexible, nonarbitrary,

and convenient method is developed for systems of two or more components. This method furnishes

an imm ediate distinction between various types of solutions.

XPERIMENTAL results can be improved or damaged onE heir way from the laboratory to the practical application.in plant design and operation. The treatmen t of experimental

data should eliminate inconsistencies without distorting theresults by imposing arbitrary conditions, it should be flexibleenough to cover all important cases, and it should be pimple inoperation, From this viewpoint the following method was

developed for the representation of thermodynamic propertiesof nonelectrolyte solutions. The present discussion start s from

binary solutions and is later extended to systems of more com-ponents.

SELECTION OF A USEFUL FUNCTION

The use of the activity coefficientsy1 and y 2 is not advisableThe use of two functions necessitates

This

For this reaeonDividing this

1)

since they are redundant.the imposition of a condition-namely, Duhem’s equation.complicates any correct smoothing procedure.

Scatchard’s “excess free energy” is preferable.function by 2.303RT, one obtains

Q = x log YI + (1 - Z og -i2

(z mole fraction of th e first component),which y a y be considereda little more convenient in some numerical calculations.

For various reasons still ’more suitable is the function

dQ/dx = log ( Y I ~ Y ~ ( 2 )

The degree of this function is one unit lower tha n Q as well as thefunctions

log 71 = Q + (1 - ) d Q / d x and log y~ = Q - xdQ/dx (3)

This is a considerable advantage in practical calculations,realized in a special case already by Benedict et al. 1). Another

advantage of the function log y1 /y2) is its simple relation to theexperimental data and to the technically important relativevolatility:

(4)

5 )

where g is the mole fraction in the vapor and p i and p i are the

vapor pressures of the pure components, the vapor being 118-

sumed to be perfect.Furthermore, the function log -y~/r~)rovides an efficient

tool for eliminating inconsistencies in the experimental data.Since , according to Equation 1, assumes the value zero forx = 0 and x = 1,we derive from Equation 2

a = y 1 - z ) / x ( l - y)

log a = log Y l l r n ) -I- log PYP,”)

for

Figure 1 shows a simple example for the application of this condi-

tion. The only curve for log (rl/r3which represents the da ta in

accordance with Equation 6 is the iero line-that is, the system

is perfect within the limits of experimental error. The deviationsfor low concentrations of either component are safely recognized

t o be due to experimental errors. One could hardly arrive so

quickly and cogently at the same result by another method.The relation

7)

following from Equations 5 and 6 is sometimes useful. If thevalues of (Y r:fer to a constant temperature, th e integral is equal

to log ( p l / p z ) . If Y is derived from equilibrium measurementsover a moderate temperature interval, the integral can be easily

estimated since usually varies only little with the tempera-ture. Equation 7 s based only on the assumptions that thevapor is perfect and that the dependence of the activity coeffi-cients on the temperature may be neglected.



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 voi. 40, No.

Figure 1. 2,2,4-Trimethylpentan~Methylcyclohex-ane a t 741 Mm.

Measurements of Harrison and Berg (3). Zero line represent.perfect solution.

S E R I E S DEVELOPMENT

The development of functions like Q, log y, etc., into power

aeries with respect to x has been suggested already by hlargules.Por nonelectrolytes it is always permissible and at least in nu-merous cases useful. The usefulness, however, depends greatlyon the form of the series. Kohl ( 7 ) , n reviewing the variousmethods of representing activity coefficients, pointed out that theseries should be developed in such a way that the higher. terms areoorrections of the terms of lower order. In this way he avoids

the inconvenience of having the result appear as a small differenceof large terms. Also important is the fact that the coefficientsof a properly chosen series furnish a natural classification of

various systems, as will be s h o m later.Since Q = 0 for x = 0 and x = 1,each term must contain the

factor x ( 1 - ). It is desirable to develop the series with respect

to a variable which is somehow symmetric with respect to the two

components. The simplest variable of this kind is 22 - 1, which

merely changes it s sign on exchange of the components. Thusthe most useful development appears to be

Q = ~ ( ls ) [ B + C ( 2 2 - 1) + D ( 2 2 1) + 1 ( 8 )

The coefficients are determined either by plotting

or, preferably, from a diagram of I

+ D(l - 2.r)ll - 8r l z ] + , . . 10)

TYPES OF S O L U T I O N S

TYPE1.

TYPE.

The simplest case is the perfect solution for which

The next type, characterized by B 0; C = D = . . ,

log(r,/rz) = 0.

= 0, is represented by a straight line in a diagram of log(yl/r2)

against x This straight line must, according to Equat,ions 6 o r10, pass through zero at x = 0.5.

TYPE . The type B 0; C 0 ; D = . . . = 0 corresponds

to what is frequently called t'he equation of ?vlargules. If a largenumber of experimental points of extremely high accuracy is to berepresented, the method of least squares with properly chosenweights will be adopted for the determination of the constants B

and C. In general, however, the following method furnishesin a few minutes the best results.

The method consists of plotting the experimental data for

log(yl/yz) against 2 drawing a preliminary curve, and reading thevalues of log(yl/yz) for a number of characteristic points listed in

Tab leI . For type 3-that is, D = &only points 1, 3, 5 , 7, and9 need be considered.

In points 3 and 7 the value of log(yl/yJ is independent of C.In addition, the values of t he function in these two point,s are

equal if three terms are sufficient in Equation 10. This conditionmay be assumed to be satisfied in all cases except for extreme

deviations from the perfect solution-that is, especially for in-

completely miscible systems. The equa lity of the values oflog(yl/rt) for p0int.s3 and 7 constitutes, therefore, in many cases,a check of the experimental data. The value of B - D / 3 , or, if

D = 0, of B, is calculated from the best value of log(yl/y2) for

point,s 3 and 7.The value of C is derived from point 5 . If D = 0, the qui tnt i -

ties B- C) and ( - B - C) must represent reasonable values forpoints 1 and 9. ew curve is drawn through the points 1, 3, 5 ,

7, and 9 calculated according t o Table I with the assumed values

of B and C.The deviations of the experimental points from this curve are

to be judged with regard to Equation 6-that is, only such varia-tions of the representative curve are permissible which d o notchange the total area under the curve. With a little practico onesees immediately whether a slight adjustment of B and C will im-

prove the agreement.The type B 0, D 0, and c = 0 actual1,y exists.

Several systems of methanol and hydrocarbons can bo approxi-mately represented by a corresponding function. The diagram of

1og(y1/y2) indicates this type immediately by the S-shape of the

representative curve. The absolute values of the function at,z =

0 and x = 1 , are equal, and the curve passes through zero a,f,x =

0.5. The values of B and D are easily derived from points 2, 4,6,

and 8 (Table I).TYPE , If none of the constants B , C, and D equals zero, C is

calculated from point 5 (Table I), B from points 2 and 8, and Dfrom point,s 4 and 6 in which the value of t'he 11 term reaches a

maximum.

TYPE.

DISCUSSION OF TYPES

A principal advantage of a classification like that discussed in

the preceding section lies in its close connection with the nature ofthe solutions and of the components.

It is well known from the work of Hildebrand, Scatchard,Guggenheim, and others that type 2 very well approximates

systems the components of which are not associated, inte ractonly moderately, and have approximately equal molal volumes.

For systems which do not satisfy the last condition, Scatchard( 5 )suggested the relation (Equation 11).

TABLE. CHARACTERISTICOINTSh-0. 1 2 3 5

0 0 .1464 0.2113 0,2969 0 . 5f - ; D 0.7071B C/4 0,5773 B - D/3) 0.4082 B - 1)/3) + C 4 c/2N O . 8 7 6 . . /

1 0.8536 0.7887 0.7041 . . .fog y I / y I ) - - C - D -0.7071B - C 4 -0.5773 B- D / a ) -0.4082 B- 0 / 3 ) + C 4 . . .



February 1948 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 341

2.303RTQ = AViVzX(1 - z)/[xVI + (1 - x VZ] 11)

where Vi and V2 are the molal volumes of t he components. If

this equation is developed into a series similar to Equation 10,

th e comparison of the coefficients furnishes

B = 2AViV2/2.303RT(Vl f Vz

C

D = B(Vi - VdZ/(V1+

Vz)

- B Vi - Vz)/(Vi + Vz)

(12)

The D term contributes less than 0.011B to log(yl/yz) if thelarger molal volume exceeds the smaller by less than 50 .

Scatchard's equation therefore can be replaced in this case fordata of moderate accuracy by a formula of type 3 with thevalue of C given in Equat ion 12, This formula is usually more

convenient than the original equation.Figure 2 illustrates ?n application of this formula. Scatchard's

equation has been found to approximate well all hydrocarboneystems except those containing benzene.

It will be shown in another paper of this series th at associationof one of the components produces a contribution to log(-yi/rz)

which has approximately the character of the ll term in Equation10. Types 4 and 5 therefore will be discussed in connection withassociation. A few results, however, should be antic ipated here.

Interassociation between the two components tends to diminishthe influence of the term characteristic of association. This ex-

plains why systems of highly associating substances often belongto type 3.

The association term is maintained if both components associatebut not with each other, Apparently the system benaene-

cyclohexane belongs to this type. According to Hildebrand (4 )both substances are slightly associated. The measurements of

Scatchard, Wood, and Mochel(6) can be represented by Equat ion

10 with a finite value of D.It should be pointed o ut, however, thab the existqnce of this

small D term could be ascertained only by measurements of highaccuracy. Within the limits of error of the usual technique thesystem would belong to type 3 rather than type 5. Thus the

classification presented here furnishes approximations of various

degrees. This flexibility may be considered to be an advantage ofa power series.

MULTICOMPONENT SYSTEMS

The series used for binary systems can be extended withoutThe definition of theifficulty to multicomponent systems.

function Q by Equation 1 s now replaced by

Q = X Z ~og ~k (13)

The relation of this function with the free energy furnishes for the

activity coefficient

The differentiation is to be performed at constant pressure andtemperature. Also, all mole fractions are to be kept constant

except the one indicated in the differential quotient.It is convenient for some calculations t o represent Q as a sum

of terms Q h which are homogeneous of the degree h in the molefractions xk Equation 14 can then be written as

(15)og y r = Z B Q h / a X r - ( h - ) Q h '

The functions

log (rr/y*) = a Q / a ~ r- aQ/axa (16)

offer the same advantages as the function defined in Equation 2.Their degree is one unit lower than that of 6? or logy?,and they areclosely connected with the relative volatility.

The development of Q nto a power series can be appropriateIygeneralized without difficulty. It is useful to rewrite Equation 8

in the more symmetric form

Qlz = ZIXZ[BIZ Clz(~1- 2,) + DIZ(ZI- Z ) ~+ . . I 17)

The series for a ternary system is then conveniently represented

by

Q Q23 + Q3i + Q i z

+ 2 1 2 2 2 3 [ C + D i(~z- 3 ) D2(23 - 21) + . . I (181

The number of coefficients required for a series running to acertain degree has been discussed by Benedict et al. ( I ) . Thepresent representation offers again the advantage of a natu ral

classification by means of terms of decreasing importance.

Equations 16, 17, nd 18furnish for a ternary system

log ( 7 z / 7 i ) = Blz(21- 2) + Cm[3(21- 2 z ) * - 11/2+ DlZ(Z1- XZ)[(Xl - 2 ) 2 - 4x12221 + . . .+ 23[B23 - B81 f Cz3(222 - 3 + c31(221 - 3 )

+ Dz8 32s2- 4x228 + 3 ) - D31(32l2- 42123 + 2 3 )+ c ( Z l - 2 ) + D1(-2321 2x122 + 22x3 - ZZ)+ 02 +23zl + 22122 - 22x3 - i z ) f s . I 19)

The actual calculation is simple, since the higher te rms are alwayssmall and often negligible. Cyclic permutat ion of the subscripts

furnishes log( 3 / 7 2 ) and log(y~/y ,~).For a system of four components Q is represented by the sum of

the six binary functions of Equation 17, the four terms containing

21x223, etc., and a term containing X I X Z X ~ X P The further gen-eralization is obvious.

EXAMPLE:HEPTANE-METHANOL-TOLUENE.he measure-

ments of Benedict et al. I ) of vapor-liquid equilibria in thesystem n-heptane-methanol-toluene furnish an opportunity toshow that even accurate measurements in a ternary system can be

sufficiently well represented by the functions Q for the binary

systems alone, provided that adequate expressions for thesesystems have been found.

Systems with an associating component like methanol are

properly represented by means of the association function

A ( K , x ) = 22 log (1 + K ) -210g (1 - 22 + 11 + 4 K ~ ( l ~) ] )2 ) / 2 (1- 2) (20)

0.1

t o

0.05

t 0

m--

0.05

0.110 x 0 . 5

Figure 2. Toluene-2,2,4-Trimethylpentane at760 M m .

Menmaremento of Driakamar, Brown, and White 3). Curverepremntm Santahard a equation.



348 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 Vol. 40, No. 2

coefficients-namely, eight from the binary jystems and threeTABLE1. RELATIVE OLATILITYF ~ - H E PT A N E - ~ ~ E T H A N O Lerived from the ternary system. The practical importance of

the restriction of the number of empirical coefficients need not bestrrssed

A T 760 AIM.

RIethanol/Heptane

1 oc. x Obsvd. Benediet This paper

CONCLUSIBSS60.60 0 .138 16.1 16.45 16. 559.47 0.178 12.6 12.65 12.5558.93 0,390 4.4 3 4.45 4.36

58.82 0.668 1.46 1.49 1.47 The use of the funct'ion log( rl/rz ) for the representation of0.810 0.696 0.691 0.702

free energy da ta of binary solutions is advisable. A suita-8.81

59.01 0 ,885 0,423 0,407 0.419

59.90 0.946 0,242 0,243 0.246 bly chosen power series represents this function in a flexible and

unbiased ~iay. The coefficients can be derived quickly from

TABLE11.

t , oc.70.2566.4465.5864.4764.1063.7963.6763.5863.6263.94

RELATIVEOLATILITYF >\IETHBSOL-TOLCENF:

AT 760 MM.

RIethanol/Toluene

22 Obsvd. Benedict This paper

0.1300.2660,4070.5930.6920.7790 ,8430,8520.9270,969

19.09.905 , 9 43.102.161 .541 .231 .010 . 810 .61

15.079.755.723 072.171 . 6 51.190 .990.800.62

1 7 , s9 . 8 35.933.132.201 .581 . 2 11 . 0 10.800.81

rxperimental data.The nature of various classes of solutions is expressed in the

magnitude of these coefficients. The behavior of log(y , / y 2 )

therefore can be predicted to a certain degree. Scatchard'scquation furnishes a good approximation for systems of hydro-carbons except those containing benzene.

The series used for the free energy can be advantageously usedalso for other propertie., such as heat content and entropy.

ACKNOWLCDGIIENT

The authors cxpress their

gratitude to It. W. Millar andG. Muller for helpful discus-sions on t,he subject of this and

the preceding paper. L. Korba

assisted the authors in the

- ~ .

TABLE1. RELATIVE70L.ITILITICY O F n - I l c P . r . ~ ~ ~ - - 1 ~ ~ T H , 4 ~ O L - - T n L U ~ S E4T 760 lIA>I,Heptanc/Toluene IIcthanol/Toluene

t , ae. El I2 O h s v d . Bencdict This paper O b s v d . Benedict Thiq paper

90 .77 0.0684 0,8486 2.75 2.78 2 .72 1 . 15 1 .19 1.2469 .96 o.1.51.0 0.7733 2.11 2.26 2.32 1.58 1 .64 1 .80 calculations.61.59 0 .0 7 8 7 0,7494 2 .58 2 . 6 3 2 .48 1 .91 1 .77 1 . 8 859.97 0.3412 0.5409 1 .65 1 .8 1 .83 3 . 8 3 . 9 4 .20

m61.35 0.1 717 0.603 8 2 . 0 7 2.17 2 .08 3 .22 3 .25 3 .26 LITERATURE CITED82.01 0.0911 0,6114 2.43 2.42 2.25 3.13 3 . 0 6 3 . 0 864.34 0.0962 0,3711 2.24 2 . 1 3 2.04 7.14 6 . 8 6 .9262.96 0.2533 0,3172 1 , 8 5 1.92 1 .82 8 . 3 8 . 5 8 . 6 (1) Benedict, M . , Johnson, C.

A., Solomon, E., and

Rubin, L. C., Trans. Am.Inst. Chem. Engrs., 41 ,371 (1945).

( 2 ) Drickarner, H. G . , B r o w n , G. G., and IVhite, R . R., Ibid.,

4) Hildehraild, J. H., J . them. p h y s . , 7, 234 (1939).

In tkis equation the mole fract,ion z refers to the associating

The three binary systems have been found to be represented b v

41, 55 (1945).component, and is a Onstant characteristic Of this component'

(3) Rarrison, J . M , , andBerg, L,,IND, st CHEM., 8, 117 (1946).

means of the coefficients (1 = heptane, . 2 = methanol, 3 = (5) Scatchard, G., Chem. Revs. , 8, 321 (1931).

toluene)(6) Scatchard, G . , JVood. S. E., and M o o h e l , J. M., . P h ~ s . hem.,

7) W o h l , K . , T,.ans. Am,. h t . C hem. E i i g i s . , 42, 215 (1946).

43, 119 (1930).

B23 = 0.95; Bai = 0.117; Biz = 1.178

Kz = 3.8; c3 = 0.018; D 2 = 0.155 (21) R E C E I V E D November 2 5 , 1946

ail o ther coefficients being zero. Accordingly,

the followingequations were used for the ternary

system:

log (?1/? 3) =

0 2 2 8 ~ ~O . 1 1 7 x 3 xi) + 0 O l 8 [ z , ( ~ ~2s3)

(23)

The corrections for the imperfection of the vaporcalculated by Benedlct et at . ue ie used.

The results are compared n ith the experinlentaland calculated values of BenediLt e l al in Tables11,111, and IV. These tables contain the tem-perature, t h e mole fractions in the Irqurd, and therelative volatilities observed and calculated by

Bcnerhct et al. and by means of Equations 22 and23.

The volatilities calculated by mean3 01 theseequations agree with the observed values ap-proximately a5 well as the values calculated byBenedict et al.; but Equations 22 and 23 con-tain only six coefficients, all derived from the

binary systems. Bonedict et d ntroduced eleven

k Za Z8 - 2Z1)] + 0.155~2(3~: Z l X z + 2:)

Furfural Solvent Extraction Unit for Lube Oil

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Redlich Kister Algebraic Representation of T