squares fit to Equation 2 for the excess, free energy, Because the relation between free energy and vapor .pressure is nonlinear, the procedure is one of successive approximations. In each iteration the equations are linearized and solved for approximate values of the coefficients, as = g J R T . These are evaluated and revised until new valued differ from those of the preceding step by less than a predetermined amount, e*.
cE(ycv. = g").
In most respects the program follows the general plan of the least squares calculation outlined by Barker (2) Our modifications and generalizations of the original Barker treaiment include: -The equation for the excess free energy includes a "skewing" parameter B (Equation 2) -The program calculates coefficients for various sets of parameters, with a $repent limit of five (m = 4, Eq. 2). EXAMPLE. 'The table shows the results of calculations using CH08B for the system n-pertluorohexane and n-hexane at 25' C. studied by Dunlap and coworkers (6). The system is nearly symmetrical in mole fraction, so we select B = 0.0. We have used their evaluation of the virial coefficient corrections (Wl and 14'2) but have of course not used their vapor composition data in applying the generalized Barker program. C.I(X = 1) and G ~ * ( X= 2) were calculated from the g,,parameters using Equations 7 and 8. The corresponding e.-parameter? give lny,(x = 1) and Iny2(r = 0), since GE, = RTlny. and a. = gJRT. Here the standard deviation c p is defined in the urual way :
-
=
Figwe 2. CurvcJrttiqfor the mpurpm prcsws of the VJfmpnjuorohexane o f 25' C. Tk refmeme line deriucs from the P, for B = 0.0and a i n g l e parmncfrr go = 1337 cal. ptr molt. The circler me the cxpcrimtal pressares, txprrssed os Pah P,, Tk mrm show the improunnmt of the Jrt, keeping B = 0.0 for tluo pnrometns (go and gl), thee pormckrs (go, gl, and gz) and,faur paramctns (go, a,ga undga) hexone bestfit
+
Numbn of Pmomrterr
E.
3
*& *
E.
1317 + 16 -225 38
4
* 8.*
6-
0)
-I
1331 i 32 1337 i 32
1
*
1095 i 46 1545 35
1333 34 1669 zt 20
*
POL..
'P,l.-
I
269.6 313.4 325.7
P2 253.9 294.3 311.3
321.5 324.1 323.3
329.8 330.2 332.9
320.5 324.1 326.6
i
0.6271 0.7200 0.8169
323.0 322.5 319.4
333.1 332.2 318.2
327.3 332.1 331.3
0.9010 0.9488
307.3 269.7
218.1 231.9
303.7 256.4
XI
220.3
PI
261.4 302.9 316.1
0.2857 0.4536 0.6112
0.0000
0.0503 0.1408 0.2095
I
I
5
* *
s.
.E 1308 2 -I34 zt 5 161 i 8 -127 i 12
8.
1301 8 -168 zt 15 200 i 26
E*
G5* (x =
(11)
- No. parameters
Adding additional parameters effects obvious improvement up to the inclusion of 5 3 , but nothing significant is gained by adding gr. The number of iterations required was six for the first stage, but reduced to three for the succeeding stages. Figure 2 illustrates the convergence of the agreement as one uses two, three, and four parameters. We refer all the observed and calculated values to the best values with one parameter (sa = 1337 cal. per mole), and plot Poha- P,. and P2 - P,,etc. (Figure 2).
2
7
SI
e-c=*(x = I )
- Ped
I
i
E" *g. 1331 i 32
EO
W " b ,
No. observations
1207 1729
+
zt 15
+
*
8
I PI
P:, 266.1 302.7 314.0
* .E
1307 zt 2 -134f 6 163 i 13 -129 i 14 -6 i 21 1209 22 1121 12
261.3 303.2 316.5
I
ps 261.3 303.3 316.5
~
319.1 322.7 322.0
322.0 323.9 323.1
321.9 323.8 323.2
322.1 ,323.6 324.3
323.0 322.3 320.5
323.0 322.3 320.5
307.6 261.2
I
306.2 270.3
VOL 55
I
306.1 270.4
NO. 7 J U L Y 1 9 6 3
45
Thermodynamic Functions
Before we can correlate the thermodynamic properties of mixtures we must represent them algebraically. We have selected a function (Equation 1) first suggested by Guggenheim (7) and later popularized by Scatchard ( I 3 ) ,Redlich ( 7 I ) , and Barker (2). T h e excess property, B E , etc.), is expressed as a power series in the mole fraction of the second (or first) component and the coefficients, yn(/zn, u,, etc.), are determined by a least squares procedure.
P(a",
is approximately that of a skewed parabola; in othcr cases (such as S-shaped curves), it has no virtue, and oiir may as well use Equation 1. There are many advantages in fitting experimental data to Equations 1 or 2. Appropriate statistical rules can be applied in eliminating spurious values. ,41though uncertainties in P may vary over the range of Y and will differ mith worker and experimental method, proper weighting allows all available data for a given
m
P = x(l
-
y n ( l - 2x)"
X)
(1)
30
n=O
T h e behavior of the first three terms in this widely used equation is shown in Rowlinson's book, "Liquids and Liquid Mixtures" (72). When us. x is sharply skewed, the series of Equation 1 converges only slowly and is rather inefficient. This suggested a modification which proved useful in minimizing the number of parameters needed to fit a set of data. \2'e introduce a pre-selected skewinq constant B: where - 1 < B < 1 :
20 0
0 x
10
O
-101
I
I
I
I
I
I
I
I
0.5
0.0
I
1
1 .o
Concentration, X
B is not evaluated by least squares and, once selected, is used for all properties and all temperatures for a given system in order to keep the various functions easily differentiable. It is for this reason that we do not use a general power series in the denominator. T h e behavior of the first four terms in this skewed function is shown in Figure 1 ( B = 0 . 6 ) . Equation 2 can serve the purpose of introducing other functional forms, such as the volume fraction form preferred by Hildebrand and coworkers (8) or the van Laar equation used extensively by chemical engineers. For an excess function in the simple volume fraction form:
where p1 and cpz are volume fractions and pl and p2 are the molar volumes of the two pure liquids. I t is then easy to rewrite this expression in the form :
Figure 7 .
The jurictioni F7br=
4 7 - s ) ( 7 - 2r)tz jor B 7 B 1 7 - 2u)
-
=
0.C
and n = 0, 7; 2, 3
property of a system to be used in determining
J?~.
We have assumed x to be the independent variable and assigned all uncertainty to the programs are possible.
I' values,
but alternate
Calculational Sequences
Our ultimate goal is to d m d o p a general program which will accept all the thermodynamic data a \ ailable for a particular system (vapor pressures, solubility data, heats of mixing) together with appropriate 1% eightin% factors, perform a statistical analysis and synthesis, and yield values ofg, in Equation 1 or 2 for the excess free energy, their temperature and pressure deril ati! es, and standard deviations for all of these. CH08A. The obvious place to begin is to fit directl) measured excess properties (RE, etc.) by the method of least squares, obtaining parameters (hn, u p , etc.) which can later be correlated with values of g,, and their derivatives. T h e matrix in\ ersion method of least squaring is used. Output includes coefficients >,,, calculated values of f.
vE,
in which it is obvious that B = i o = 2 k 1 7 ' 1 8 z / ( P 14- Vz),andyl
-(PI
-
=)z
=
.
pz)/(p1+ pz), . = 0.
A reasonably good choice for B can h e made b y selecting that value which causes the maximum in the first term of the series to occur at the same mole fraction as the maximum in a n appropriate property (for example, the heat of mixing). If we define this mole fraction as xlnnx,this leads to a n equation for B:
I t will normally suffice to round B o f f to a n eiren tenth except, of course, for extreme skewness ( B > 0.9). Equation 2 is useful only when the shape of the excess function 44
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
and standard deviations for all of these. T h e use of CH08A to process thermodynamic data for alcohol and fluorocarbon systems studied in our laboratories is illustrated in recent publications ( I , 1 1 , 75). Further details and copies of the Fortran source program are abailable from the authors. CHOIB. This program takes total vapor pre\.ureliquid composition data and determines the beit least
d.
Put the COOLBITE on touah
\ \
THE NEW
\
STR
1
'Exclusive ventilated saw tooth rotor distributes and circulatesgreater amunt of cooling air than any conventional unit
1m-PULVERILm w...m.mu
produces granular grinds without excessive fines!
'Presenting a completely new grihding technique, the STR MIKRO-PULVERIZER handles thermoplastics and other tough, resilient. heabsensitive materials more efficientlv at i n e r e a d capacities . . . will produce 20 to 60 mesh grinds without refrigerants! Equally important. it has proven ideal ~. for production of granu& grinds without ohjectibnable fines. Key to this performance is a special ventilated saw tooth rotor with integral radial fans which provides a shearingpulverizing action while circulating a greater amount of cooling air than any conventional unit. Details in Bulletin STR1. Send Coupon lor Actual Sample Qrlndf
PULW€RIZINO
, !
.,_",
Summit, N.J.
i.~ 1
~ ~~~~~~~~~~~~~~~~~~~~~ { ~ ~ ~ f i p,'$!
Bulletin STR.1. in addition. N~~~
I I Address
F,rm Nlme
II-c8Y --------------,-,,,,
MACHINERY
Zone
s
t
a
t
e
MIKRO-PRODUCTS
40 Chatham Road * Summit, New Jersey DIVISION OF SLICK INDUSTRIAL COMPANY
Sales, service and mannfacturing facilities throughout the world
PRINMNQ CONVnlNQ UKLECTINE
VOL 5 5
NO. 7 JULY 1 9 6 5
47
-One can weight all experimental points as desired -The matrix inversion method allows standard deviations of all the coefficients and the pressure as well as covariances to be computed -Once in each iteration the sums Z ( b P / b d are recomputed. These sums change but slightly, so Barker computed them only once. For high precision, however, the IBM 7090 can easily recompute them occasionally -Vapor compositions are calculated at each experimental point Recently, vapor pressure data for several fluorocarbon systems have been processed by C H 0 8 B (7, 15). A detailed example is given on page 45. Copies of this program and further details are available from the authors. Since not all workers choose to express their results in the same analytical form, it seems desirable to report certain quantities whose general significance does not depend upon the precise equation used. Three such quantities are the values of P a t equal mole fractions and the limiting partial molar quantities at infinite dilution :
.All our programs include calculations of these quantities and their standard deviations.
using vapor pressures and liquid or vapor compositions to calculate free energies. I n fact, it is desirable to use T,, x , data to restrict the values of g, (that is, require that the excess free energy equation fit the coexistence curve). We are developing a program (called CHOSC) to d o this. Bellemans and coworkers in Belgium (3, 4 ) have recently proposed a method for obtaining GE from total pressure-vapor composition data (dew point curves) that is very similar to the Barker method. W e are now working on a program to perform these calculations (CH08D). NOMENCLATURE g,, = coefficients y n when h,, = coefficients y,, when
Y Y
G” HE k = constant in Equations 3 and 4 m = maximum number of coefficients y,, in Equation 1 or 2 n = summation index in Equation 1 or 2 s, = Coefficients yY2when Y = SE u , ~ = coefficients y.. when Y = V” ZL’ = virial coefficient correction in Barker method x = mole fraction of component 2 in liquid phase y = mole fraction of component 1 in vapor phase I3 = skewing parameter in Equation 2 F,‘ = tlic function x(1 - x ) ( l - 2n)”i[l - B(1 - 2x11 Mihich is multiplied by the coefficient y r ,in Equation 2 C E = excess free energy of mixing per mole of mixture = partial molar excess free energy of mixing G E H” = excess heat per mole of mixture P = pressure R = gas constant T = absolute temperature F E = excess volume per mole of mixturc p = molar volume of pure liquid i; = any excess thermodynamic property ai, = ,cqdivided by It?’ *, = activity coefficient = =
e
=
ar> -
Preliminary Correlations and Future Programs
G
=
1
=
ll’illiamson and Scott (15) have used C H 0 8 A and C‘H08B to correlate heats of mixing and rxcess free n-hexane energies for the systems perfluoro-n-hexane and perfluoro-n-heptane isooctane. T h e thermodynamic equation :
0
=
standard deviation (see Equation 1 1 ) 2nd virial coefficient corrrction in Barker m c t h d volume fraction
+
+
01% - 1
Subscripts 1 = component 1 2 = component 2 12 = 1-2 interaction i = ith component = nth coefficient or term in Equation 1 or 2 n
REFERENCES
requires that for each n:
(1) Andersen: D. L.: Smith, I,,Ibrrf. 6 5 , 275 (1961).
