I
MARY G. MYLES and R. E. WINGARD Chemical Engineering Department, Auburn University, Auburn, Ala.
Calculating Activity Coefficients How accurate must your data be? b Experimentally reproducible?
b Thermodynamically consistent?
B,
m I s G the chi-sauare statistic. the engineer can quickly compare the theoretical equations with the experimental data to see if they are significantly different from each other; by means of the coefficient of correlation, the engineer can easily indicate the degree of relationship between the experimental data and the theoretical equations. Insofar as these four binary systems are concerned: the experimental, thermodynamically inconsistent data are successfully predicted by several types of Gibbs-Duhem theoretical equations. The theoretical data fulfill the criterion that log y 1 / y 2 dx = 0. Therefore, in a separation problem, it is recommended that a type of Gibbs-Duhem equation should be used to describe these four binary systems. Deviations from ideal behavior of chemical solutions are important in the design and operation of separation processes and are calculated from experimental measurements of composition, temperature, and pressure. I t would be advantageous to have theoretical equations which would exactly fit experimental data so that the vapor-liquid equilibrium curve of a solution could be extrapolated or even predicted. Many theoretical equations have been derived. The purpose of this investigation was: first, to compare the experimental equilibrium composition-temperature data of four binary solutions with several of these theoretical equations; and second, to show that by two simple statistical tests one can compare the experimental data with the theoretical equations and indicate if they are significantly different from each other, and can indicate if the degree of correlation between them is significant. The experimental and theoretical data were analyzed further for thermodynamic consistency using the
so1
This article shows that, by means of the chi-square statistic and the coefficient of correlation, one has a quickly and easily applied pair of tools to test the consistency of experimental data. The chemical engineer uses such data to determine certain design factors and those data under much the same conditions as they were obtained. It matters not whether they are thermodynamically consistent because the system, if it deviates from ideal behavior, is not going to behave any differently in scaled-up equipment. What is important is that the data be experimentally consistent and reproducible and consistent within itself. Then the chemical engineer can place some reliance on it.
criterionh' log. ?2 dx = 0, as proposed 7 2
bv Herincton ( 4 ) and Redlich and Kister
for comparison with the revised van Laar equations ( 2 ) were
Working Equations
Four theoretical equations were used in the analysis. All were derived from the Gibbs-Duhem partial differential equation
y2 =
AB,'T (A
+
:)2
(3)
where
A , B=constants.
6 In
Second, these van Laar rquatiocs. as revised by.Carlson and Colbuin ( 2 ) :were used :
where XI.
In
xz = mole fractions of comoonents
1 and 2 in the liquld, 71, y2 = activity coefficients, T : P = absolute temperature
log
A
=
(1
and
+%)'
(4)
Dressure.
First, original van Laar equations used in the analysis of one b i n a q solution
log y 2 =
VOL. 53, NO. 3
+z)z B
(1
MARCH 1961
219
Third, a new equation proposed by Saphtali (6) was also used. He has suggested that a failure of the conventional manner of plotting the van Laar revised equations does not rule out the data in every case. By his combining Equations 4 and 5, he obtained
K
of the molar latent heats of vaporization of the loiver boiling component to that of the higher boiling component, at the respective boiling points, -21 = slope of the line, Z = . \ { X I C, =In = the ratio
z
[L$] )?
The intercepts are 4 2 when log yz=O and .\/B when log 71’0. If dGl is plotted against Equation 6 tvill fit data which are not thermodynamically consistent and will represent adjusted curves which are thermodynamically consistent. The constants in Equations 2 through 6 were determined from the experimental activity coefficients which were calculated from
G2,
^(
i”
(7)
Px
Chi-square indicates the probability that the differences between the experimental data and the theoretical data are due to chance. The usuany accepted “level of significance” of differences is a probability of not more than one in 100 that the differences are due to chance. Fisher’s chi-square table ( 3 ) was used to obtain the significance levels of the calculated chi-squares. The significance of the variance between the theoretical and experimental data was double checked by the “product-moment’’ coefficient of correlation (7). This statistic was used to indicate the degree of relationship between the experimental data and the theoretical data
where
+
xi
In calculating constants from the slopes and intercepts of straight lines, the least squares method \$as used to determine the locations of the lines. Theoretical vapor compositions were calculated a t varying temperatures along the experimental boiling curves. !\-hen based on 71,
where
when based on yr,
dz = experimental vapor composition minus the arithmetic mean experimental vapor composition, d, = theoretical vapor composition minus the arithmetic mean theoretical vapor composition, I = coefficient of correlation.
!\.here
y
= mole fraction of one component
7~
= total pressure, = vapor pressure of one component
in the vapor,
P
a t the boiling point.
Fourth, these Norrish and TwiSg single-constant, linear, equations ( 7 ) , using the molar latent heats of vaporization and the boiling points of the pure components, were used In yi =
( K - 1)* ~
[Kln
-
I?;
K 1 - (R- l ) x 2 \
( I ; - l)x?]
The binary solutions I\ hich u ere analyzed were carbon tetrachloridefurfural ( 7 7 ) , acetone-furfural (5), ethyl acetate-furfural (8), and furfural-furfury1 alcohol (70).
high coefficient of correlation indicates that the relationship is not due to chance. The cut-off level is usually one chance in 100 that the relationship is not a true one, or that it is due to chance, and is called the 1% level of significance. This is sometimes called “practical certainty.” The thermodynamic consistency of the experimental and theoretical data was tested by plotting log y1/y2 LIS. XI. The positive and negative areas bounded by the curve and the x-axis should be equal in magnitude. The log. y /
Testing Theoretical Data The statistic, chi-square (7), \vas used to indicate significant differences be-
tween the experimental and theoretical vapor compositions :
(8) ‘t
where ye = j’t =
experimental vapor composition, theoretical vapor composition.
A’
MOLE FRACTION CARBON TETRACHLORIDE IN L i a w
MOLE FRACTION ACETONE
I N LIQUID
The logarithm of the ratio of the activity coefficients as a function of composition for the binary solution Left.
220
C a r b o n tetrachloride-furfural
a t 760 m m .
INDUSTRIAL AND ENGINEERING CHEMISTRY
Right.
Acetone-furfural a t 760 mm.
CALCULATING A C T I V I T Y COEFFICIENTS y ? ds = dQ = 0, since the change in
free energy is equal to zero at constant temperature ( 4 , 9 ) .
Table I.
Experimental Equilibrium Composition-Temperature Data
Theoretical V a p o r Compositions, Chi-square, and Coefficients o f Correlation. Furfural a t 760 Mm.
Discussion of Results
Carbon Tetrachloride-Furfural. The chi-square test indicates that the equilibrium data predicted by the van Laar original and/or revised equations are not significantly different from the experimental data (Table I). The ver)high coefficients of correlation indicate almost perfect relationship between experimental data and these two theoretical equations. The chi-square tests of the differences between experimental data and the Naphtaliequation and theSorrish and Txvigg equationsare 39and 52, respectively (lvhen the calculations were based on carbon tetrachloride), and both are significant at the 1y0probability level. This indicates that the differences betLveen the experimental data and the equations of Saphtali, and Sorrish and Tivigg are too large to be ascribed to chance. The coefficients of correlation between the experimental data and these t\vo theoretical equations are 0.564 and 0.403, both of xvhich indicate a low relationship. Hoxvever, Lvhen the calculations u w e based on furfural, the Saphtali equation and the Sorrish and Txvigg equations predict the data exactly, as indicated by the low chi-square values and high coefficients of correlation. Acetone-Furfural. Table I1 shoivs the experimental and theoretical vapor composirions. calculated from the activity
0
Cxperimen tal Mole fraction CCIJ in Liquid, Vapor, xi ti1
1.0 0.906 0.793 0.699 0.573 0.481 0.373 0.260 0.156 0.065
0.0
1.0 0.989 0.984 0.977 0.971 0.966 0.957 0.938 0.892 0.703 0.0
Temp.
Theoi etiral l l o l e frartion CClr iri vapor 2h 3c Based on CClr (?I)
la
e.
77.0 78.0 78.0 78.5 79.0 80.0 83.5 89.0 100.0 124.5 161.7
Carbon Tetrachloride-
1.0 0.995 0.957 0.948 0.946 0.928 0.945 0.905 0.840 0.667 0.0
1 .O
1.0 0.995 0.975 0.979 0.978 0.959 0.955 0.916 0.859 0.654 0.0
0.973 0.881 0.822 0.758 0.716 0.714 0.705 0.718 0.670 0.0
Chi-square based on y, 0.57 39.21
1.04 0.987'
0.997
0.564
1.0 0.996 0.905 0.935 0.775 0.722 0.722 0.586 0.650 0.573 0.0
57.03 0.403
Based on furfural (yz) 1 .o 0.906 0.793 0.699 0.573 0.481 0.373 0.260 0.156 0.065 0.0
1.0 0.989 0.984 0.977 0.971 0.966 0.957 0.938 0.892 0.703 0.0
77.0 78.0 78.0 78.5 79.0 80.0 83.5 89.0 100.0 124.5 161.7
1.0 0.975 0.968 0.966 0.964 0.961 0.953 0.935 0.890 0.701 0.0
0.070
1.0 1.0 0.973 0.988 0.968 0.978 0.967 0.971 0.965 0.965 0.962 0.960 0.954 0.951 0.935 0.933 0.890 0.888 0.701 0.700 0.0 0.0 Chi-square based on yz 0.070 0.025
1.0 0.987 0.978 0.972 0.967 0.962 0.952 0.935 0.890 0.702 0.0 0.014
T
1.00 a Van Laar's original equations. Sorrish and Twigg's equations.
01 02 03 0 4 0 5 0 6 0 7 0 0 0 9 MOLE FRACTION ETHYL ACETATE I N LIQUID
'>
1 * 000
Vaii Laar's revised equatioiij.
0.997
0.996
Kaphtali'a equation.
10
)
MOLE FRACTION FURFURAL IN LIQUID
The logarithm of the ratio of the activity coefficients as a function of composition for the binary solution Left.
Ethyl acetate-furfural a t
760 mm.
Right.
Furfural-furfuryl alcohol a t 25 mm.
VOL. 53,
NO 3
MARCH 1961
221
Table II.
Experimental Equilibrium Composition-Temperature Data
Theoretical V a p o r Compositions Calculated from the Activity Coefficients Based on Furfural ( 7 2 ) . Square and Coefficients o f Correlation. Acetone-Furfural at 760 Mm.
Experimental Mole fraction acetone in Liquid, x1 Vapor, YI 1.0 0.938 0.873 0.795 0.732 0.663 0.630 0.599 0.515 0.446 0.375 0.258 0.185 0.0
Temp.,
1.0 0.992 0.987 0.984 0.982 0.980 0.978 0.977 0.969 0.966 0.951 0.923 0.890 0.0
Theoretical Mole fraction acetone in vapor
c.
Zb
3c
4d
56.1
1.0 0.992 0.988 0.984 0.981 0.978 0.976 0.975 0.969 0.965 0.955 0.929 0.890 0.0
1.0 0.997 0.995 0.992 0.989 0.987 0.985 0.983 0.976 0.971 0.960 0.934 0.890 0.0
1.0 0.996 0.993 0.990 0.986 0.983 0.981 0.979 0.972 0.967 0.957 0.930 0.895
57.9 59.4 61.8 64.8 67.3 68.8 70.1 74.3 76.3 81.7 90.8 100.0 161.7
Chi-
0.0
C hi-Square
0.007
0.059 r 0.999
0.998
Van Laar’s revised equations.
Table 111.
Saphtali’s equation.
0.007 0.998
Xorrish and T w g g ’ - equation-.
Experimental Equilibrium Composition-Temperature Data
Theoretical V a p o r Compositions Calculated from the Activity Coefficients Based on Furfural ( y ? ) . ChiSquare and Coefficients of Correlation. Ethyl Acetate-Furfural at 760 Mm.
Experimental Mole fraction ethyl acetate in Liquid, xI Vapor, vi 1.0 0.885 0.758 0.711 0.625 0.528 0.422 0.298 0.218 0.216 0.140 0.0
Theoretical T ~ ~ ~ . ,Nole fraction ethyl acetate in vapor O
1.0 0.986 0.967 0.961 0.950 0.931 0.898 0.856 0.792 0.784 0.681 0.0
c.
77.2 80.5 85.5 87.1 90.0 93.9 100.8 109.5 117.0 117.5 127.5 161.7
’ 25
1.0 0.968 0.947 0.941 0.932 0.919 0.892 0.847 0.791 0.789 0.686 0.0
3c
4*
1.0 0.990 0.977 0.973 0.962 0.948 0.919 0.868 0.806 0.804 0.696 0.0
1.0 0.991 0.978 0.973 0.962 0.947 0.917 0.866 0.803 0.805 0.696 0.0
Chi-square
0 189
0 243
0 225
r 0.999
Van Laar’s revised equations.
Table IV.
e
Saphtali’s equation.
0.997
0,995
Xorrish and Twigg’s equations.
Experimental Equilibrium Composition-Temperature
Data
Theoretical V a p o r Compositions Calculated from the Activity Coefficients Based on Furfural ( y l ) and Furfuryl Alcohol ( ~ 2 ) . Chi-square and Coefficients o f Correlation. Furfural-Furfural Alcohol at 2 5 Mm.
Theoretical -__ Mole fraction furfural in vaDor T ~ ~ ~ . 3,c , based on 4 , d based on
Experiment a1 Mole fraction furfural in Liquid, SI Vapor, ~1 1.0 0.953 0.858 0.714 0.590 0.521 0.405 0.322 0.201 0.079 0.0
Naphthali’s equation.
222
O
1.0 0.973 0.912 0.821 0.727 0.664 0.560 0.477 0.330 0.147 0.0
C.
68.5 68.8 69.5 70.9 72.3 73.2 76.0 76.5 78.4 81.1 85.0
d
Yl 1.0
72
Yl
5.84
1.0 1.0 0.988 0.973 0.915 0.920 0.829 0.840 0.735 0.766 0.696 0.722 0.619 0.612 0.501 0.558 0.358 0.436 0.161 0.258 0.0 0.0 Chi..Square 9.69 1.21
0.995
0.983
0.975 0.915 0.816 0.761 0.720 0.665 0.554 0.407 0.192 0.0
Xorrish and Twigg’s equations.
INDUSTRIAL AND ENGINEERING CHEMISTRY
0:997
Y2
1 .o 0.977 0.931 0.855 0.783 0.736 0.624 0.568 0.443 0.260 0.0 11.17 0.997
coefficients based on furfural. The theoretical vapor compositions. calculated from the activity coefficients based on acetone, were erratic (more than 1.0 mole fraction) and are not included in this table. The chi-square test indicates that the small differences between experimental data and theoretical equations are due to chance. The coefficients of correlation are very high (0.983 and above) and indicate almost perfect relationship between experimental data and theoretical equations. Ethyl Acetate-Furfural. This binary system was predicted sarisfactorilv by three theoretical equations (Table 111). The chi-square test indicates that the differences between the experimental data and theoretical equations are due to chance, and the coefficients of correlation indicate almost perfect relationship. Furfural-Furfuryl Alcohol. T h e van Laar original and revised equations. and the Margules equations did not apply to this binarv system. T h e chi-square values in Table IV do not indicate a significant difference between experimental data and the Kaphtali equation and the Norrish and Twigg equations. This is substantiated by the coefficients of correlation which indicate a high relationship between experimental data and the theoretical equations. Thermodynamic Consistency. The figures (pages 220 and 221) show the logarithm ofthe ratiooftheactivitvcoefficients for both experimental and theoreticaldata. I t can be seen that the experimental data are thermodynamicallv inconsistent, whereas the theoretical data are thermodynamically consistent.
literature Cited (1). Arkin, Herber, Colton, K. K., “Statistical Methods,” p. 109, Barnes and Noble, Inc., New York, 1957. (2) Carlson, H. C., Colburn, A. P., IND. EPG. CHEM.34, 581 (1942). ( 3 ) Fisher, R. A , , “Statistical Methods for Research bvorkers,” 6th ed., Oliver and Boyd, Edinburgh, 1936. (4) Herington, E. F. G., .Vuture 160, 610 (1947). ( 5 ) Lewis, P. R., ‘Vapor-Liquid Equilibrium Relationships for the System .4cetone-Furfural at 760 mm. Pressure.” The Alabama Polytechnic Institute. Auburn, Ala., unpublished M.S. thesis, 1956. (6) Naphtali, L. M., private communica‘tion,-1956. (7) Norrish, R. S., Twigg. G. H.. IND. ENG.CHEM.46, 201 (1954). (8) Piazza, M., Engineering Bull. No. 32, Engineering Experiment Station, Alabama Polytechnic Institute, Auburn, .\la. (July 1958). (9) Redlich, O., Kister, A. T.: IND.ENG. CHEM.40, 345 (1948). (10) Wingard, R. E., Durant, W. S., J . Alabama Acad. Sci.27, I1 (1955). (11) Wingard, R. E., Durant, W. S.: Tubbs, H. E., Brown, W. O., IND. ENG.CHEM.47, 1757 (1955).
RECEIVED for review December 9, 1959 ACCEPTED November 8, 1960
