I
JOHN W. TIERNEY' Purdue University, Lafayette, Ind.
Correlation of Vapor-Liquid Equilibrium Data
'
This method depends on accurate appraisal of the experimental method and the source of experimental error. It eliminates the need for individual judgment in smoothing and extrapolating data, but uses human judgment to analyze the sources of error. It can be extended to any problem where there are a number of independent estimates of the same quantity.
Basic Principles
At equilibrium the fugacity of a component in the liquid equals the fugacity of that component in the vapor. fL1
= fVl
and Y1X'(fLl0)%T
=
.YlVl7r
At low pressure the fugacity coefficient, equals 1, and ( f L l o ) T , T equals PI", the vapor pressure of component 1 a t T . Thus VI,
A. Deviations from ideal liquid solutions. B. Deviations from ideal gases. C. Deviations from ideal gas solutions. Deviations of type B are usually not encountered a t moderate pressures. However, it is a simple matter to allow for this kind of deviation by the use of fugacity coefficient charts. Deviations A and C are dependent to a large extent on the interaction between different kinds of molecules and would be expected to be greatest where widely different kinds of molecules are involved. For this reason, effect A would be expected to be much larger ordinarily than C, because the molecules in a liquid are much closer together. This discussion is concerned with the correlation of type A deviations and is limited to binary systems. The principles are general, however, and can be extended to more complicated systems. Present address, Remington Univac, St. Paul, Minn.
Rand
log YI = (1 - x)'[AR
+ BR (4x - l ) ] (7)
log YZ
and T H E correlation of experimental values of equilibrium vapor and liquid compositions has many applications in chemical engineering work. The experimental values can be readily obtained (6). An unequivocal method of presenting and using the data is by plots of y us. x, but it is usually desirable from the standpoint of interpretation of mechanism and extrapolation to more complicated systems to measure deviations from certain standards of behavior called "ideal." These deviations are commonly grouped as follows:
The equations proposed by Redlich and Kister (4)
(X)'[[AR
- B R (3 - 4X)]
are related to the Margules equations by AM = AR
For ideal liquid solutions y1 and yz are equal to 1 over the entire concentration range. For nonideal solutions the activity coefficients are functions of liquid composition, approaching the limiting value of 1 as the mole fraction of the component approaches 1. Experimental 'measurement of pressure, temperature, liquid composition, and vapor composition is sufficient to , PIo calculate both y1 and y ~provided and Pz0 are known as functions of temperature. Conversely, if y1 and yz are known, two of the other four variables may be determined--a, T , x , or y . Normally, equilibrium values of x and y are to be predicted, and thus a knowledge of the activity coefficient as a function of composition is required. Experimentally, the activity coefficients are usually calculated from samples of liquid and vapor in equilibrium. Many ingenious experimental techniques have been devised to obtain such samples (9). These coefficients are then ordinarily correlated with concentration, using some form of the integrated GibbsDuhem equation. The limitations of such integrations are discussed by Ibl and Dodge (3). The two most common forms are the Margules and Van Laar equations, where the activity coefficients are defined as functions of mole fraction by the use of two arbitrary constants. [Carlson and Colburn have discussed some of the advantages and disadvantages of each set of equations (7). Margules. log y, = (1
-
+
x)*[AM ~(BM - A M ~ I( 3 )
BM
=
AR
- BE
+ BR
If more than twg constants are required to correlate the data, Redlich and Kister have proposed three- and four-constant equations. The use of integrated foFms of the Gibbs-Duhem equation has advantages (such as ease of presentation and tabulation of experimental data, correlation of ternary and quaternary data, etc.) which make these equations useful even when the assumptions required in the integration are not satisfied. Whether considered as strictly empirical or as semitheoretical equations, they are often used. Description of Problem
Returning to the experimental problem, a series of vapor and liquid equilibrium compositions is available and is to be used to determine values of A and B in the activity coefficient equations. One experimental measurement is sufficient to determine both constants if a form of equation is assumed, and there will be as many estimates of these constants as there are pairs of experimental points. In a sense, the experimenter has too much information, and it is necessary to disregard or at least discount some of his information. Various methods have been proposed for selecting the best value of the constants. Robinson and Gilliland ( 6 ) recommend that Equations 3 through 6 be converted to straight-line form by plotting either (log rl)/(l x ) 2 us. x for the Margules equation or (log yl)-1/2
-
VOL. 50, NO. 4
APRIL 1958
707
us. x / ( l - x) for the Van Laar equation. By fitting the best straight line to the points, the constants can be determined. The difficulty is in determining the best straight line. The experimeter knows he should not give equal weight to all points, but he has no criteria for determining just how much weight to give each point. The selection of the best line thus becomes a matter of individual judgment. Rose and others (7) describe a method which avoids the arbior activity trary smoothing of x coefficient data by using T - x data and then by a trial and error process determining the values of the constants that best fit this curve. This method suffers from the same fundamental disadvantage as the previous one, in that the experimenter must use his judgment in smoothing the T - x data. Redlich, Kister, and Turnquist (5) describe a similar method. A method is needed to determine the relative reliability of each experimental measurement. T o do this, it is first necessary to analyze the sources of experimental error. The experimental quantities x , y, T, and T a r e required for each determination, and, in addition, PI' and P2' must be known as functions of temperature. Each of these quantities is a possible source of error. Measurement of Temperature and Pressure. The instruments used to measure temperature and pressure are usually accurate. as they can be compared with known standards. I t is not, however, at all certain that the desired temperature is being measured, Such items as superheating, partial reflux, radiation, and conduction losses may lead to substantial error. These errors are caused by poor still design. If at all possible, the still should be checked using a test mixture. Errors of temperature measurement are not likely to be random errors-but rather systematic errors due to faulty design or operation of the unit. Ordinarily less difficulty is encountered with pressure measurement-the motion of the generated vapor usually giving enough flushing action to remove noncondensable gases. Once again, however, errors in pressure measurement are more likely to be consistent or systematic errors rather than random errors. Measurement of Vapor Pressure of Pure Components. These should be measured in the apparatus being used and should be consistent with pressure and temperature measurements. Measurements of Composition. The least accurate and most difficult experimental measurement is ordinarily the composition of the phases in equilibrium. Systematic errors are encountered in sampling techniques and both random and systematic errors in analysis. Random errors are very likely in the analysis
708
-for example. the reading of a buret or the weighing of a sample. Often the absolute error is approximately constant -for example, the contents of one drop of solution. I n some spectroscopic analyses the percentage error is constant.
The problem is thus simplified to obtaining the standard deviation of each measurement. This could be done by repeating each measurement a number of times and then calculating the standard deviation. However, if A is not a directly measured quantity but is calculated from other measurements using the relation A = q5 (x, y ) , if s, and s, are the standard deviations of x and 4: and if x and y are assumed to be normally distributed,
Derivation of Equations
The method described corrects only for random errors. As these are most likely to occur in the determination of phase composition, the foIIowing treatment is based on the assumption that the only source of random error is in composition measurements. I t is further assumed that the standard deviation of both liquid and vapor measurements is the same. This assumption is normally justified, because both liquid and vapor samples are usually removed as liquids and analyzed in the same way. If a series of independent measurements of a quantity is available, they can be averaged, provided a relative importance or "weight" be assigned to each measurement (8, 70). The best estimate of the measurement is then given by
Thus, if the standard deviations of a series of measurements are known, the standard deviation of a quantity calculated from them can be determined. Further, a series of independent estimates of this same quantity can be averaged if the standard deviation of each estimate is known. Applying these principles to the Margules equation
and It can further be shown that and if the absolute error is constant,
W, - SZ2 WZ s,2
(9)
s, = su =
The weights to be assigned to each measurement are thus inversely proportional to the squares of the standard deviations of the measurements. The weight of a measurement is a relative term only. The weight of a single measurement is meaningless; if, however, two or more measurements are available, their relative reliability can be calculated.
A similar expression can be written for constant B.
If the percentage error is constant, then
y
x(1
Table I. Equations Margules Constants B,w(2 - 6 ~ ) A.cr(4 - 62) + 1 - x x)' 1 - x
z
y(l -
XI2
c C(1 - 2 x ) -c
xy1 - x ) +
9
2c x(l
-y) - 62)
Av(4 - 6 ~ ) Bnr(2 x
(z)v+i-;)[P(I =
D
(1
X
Van Laar Constants
+
r-x> A'X
29AvN
-
(1 - x)' (1
+ i..;,?]
Nx XY(1 y)$(
INDUSTRIAL AND ENGINEERING CHEMISTRY
=
- (%)($)($)
- Y)
VAPOR-LIQUID EQUILIBRIUM DATA
and
T o evaluate the partial derivatives given above it is necessary to know A and B as functions of x and y . These are complex expressions, and are best evaluated by an indirect method. By rearranging Equations 3, 4, 5, and 6,
Figure 1. Relative weights for Margules constants as a function of mole fraction for the system n-heptane-toluene
x - MOLE F R A C T I O N
Av = logy1 [l
+ (kx log Y1
In the above procedure, it was assumed that the weights for the experimental points were determined at the experimental liquid composition. An equally valid procedure would be to calculate weights a t the experimental vapor compositions. The trial and error required in the solution given above makes the method rather long and tedious. Fortunately, good results can be obtained by a much simpler method, in which experimental values of A , B, x, and y are used in Equations 14, 15, 16, and 17. This reduces the computation to a straightforward set of calculations at each experimental point. All calculations made to date have yielded essentially the same result by the simpler method. 4
Thus, as AM and BM are functions of
y1,72, and x , and y1 and y2 in turn are functions of x and y, the following relations can be derived :
Application of Equations
Similar equations can be derived for constant BMand for constants AV and BY in the Van Laar equation. Equations 18 and 19 can be evaluated' by using Equations 1, 2, 14, 15, 16, and 17. The results are given in Table I. The quantities A and B, which are being determined, have been substituted into these equations for simplification. A rigorous method for determining the best value of A and B would proceed as follows : 1. Calculate an A and a B for each pair of experimental x - y values. 2. Assume a value for each constant (2 and and then determine for each experimental liquid composition the weights W , and W,, using the equations in Table I plus Equations 11, 12, and 13. If any vapor compositions are required, they should be calculated (from A , B, Po,T , and x ) . 3. Using the calculated weights and the experimental values of A and B, determine and from Equation 8. 4. If these values check-the assumed values from step 2, the calculation is complete. Otherwise, it is necessary to return to step 2.
z),
The equilibrium data in Table I1 are from a recent article ( 2 ) . The authors of that article obtained Margules constants of lM = 0.155 and BM = 0.088 by a graphical procedure. Using the technique described in this paper, it was assumed that a constant absolute error existed (Equations 12 and 13)) and values of AIM = 0.155 and BM = 0.0913 were calculated. Some of the significant intermediate values are listed in Table 11. (The calculations in Table I1 were made using the simpler method described above. Use of the more rigorous procedure gave values of AM = 0.156 and BM = 0.0913.) The check between the graphical and weighting method is good and is probably due to the fact that the Margules equation fits the data very well over most of the range. An examination of point values of AM in Table I1 shows a range of 0.130 to 0.238. The majority of the values, however, are between 0.14 and 0.17, and it is not surprising that the average is 0.155. However, an arithmetic average of the point AM values is 0.177. I t is obviously necessary to give more weight to the more reliable values. The variation of weight with mole
n
-
HEPTANE
fraction is shown in Table I1 and Figure 1. The decrease in reliability as the concentration extremes are approached is pronounced. Values of A M for liquid concentrations greater than 0.8 mole fraction heptane or less than 0.1 are virtually useless. This is borne out experimentally by comparing the four estimates of AM above 0.8 mole fraction heptane-0.238, 0.270, 0.214, and 0.283. This variation is much greater than that observed in the remainder of the data, and is due to lack of precision in measuringy. The fact that all four are higher than the correct average value is discussed below. The best estimate of Afif (0.35 mole fraction heptane) is obtained a t a different concentration from B M (0.65 mole fraction heptane). The curves appear to be symmetrical about 0.5 mole fraction, but this is due to the low relative volatility of the system and the fact that AM and BM are similar in magnitude. If an analysis of experimental procedure indicated that a constant per cent error was being made in the analysis rather than a constant absolute error, Equations 12a and 13a would be used giving the weights shown in the last two columns of Table 11. As might be expected, there is now a very noticeable shift in reliability to lower values of x . I n fact, the best value for AM is now the most dilute. BMis not affected as much. The average values obtained are AM = 0.159 and BM = 0.0846, not greatly different than before. If the experimental data were less consistent, greater discrepancies would result. The analysis given above for the data in Table I1 is based on the assumption that the Margules equation correlates the data, and that the only sources of error are random variations in liquid and vapor composition. As this illustration is intended to demonstrate a technique only, it is not important whether these conditions are met in the example. Actually, the first step in the VOL. 50,
NO. 4
APRIL 1958
709
Table II.
Calculation of Margules Constants for System Heptane-Toluene Constant Absolute
Mole % Heptane 5
Y 0.053 0.124 0.191 0.194
0.030 0.074 0.122 0.123
0.271 0.282 0.323 0.337 0.395 0.430 0.445 0.492 0.505 0.555 0.602 0.650 0.703 0.774 0.833 0.882 0.917 0.958
0.184 0.193 0.228 0.240 0.294 0.329 0.345 0.399 0.411 0.470 0.527 0.588 0.655 0.742 0.813 0.868 0.906 0.952
AM
0.171 0.159 0.177 0.153 0.162 0.157 0.174 0.130 0.163 0.141 0.158 0.143 0.142 0.147 0.167 0.167 0.186 0.198 0.238 0.270 0.214 0.283
BM -0.712 -0.125 -0.087 -0.0095 0.0151 0.0360 0.0438 0.0923 0.0800 0.0941 0.0952 0.0922 0.101 0.0864 0.0980 0.0892 0.0991 0.100
0.0919 0.0847 0,0956 0.0013
13.3 6.04 3.84 4.41 3.00 2.96 2.64 2.76 2.48 2.52 2.45 2.57 2.60 2.82 3.16 3.84 4.97 8.20 14.99 29.10 54.0 207
-
22.4 9.92 6.57 6.56 4.96 4.82 4.41 4.29 3.97 - 3.96 - 3.84 - 3.79 - 3.84 - 3.99 - 4.31 - 4.90 - 6.07 - 9.26 - 15.71 - 29.28 - 55.1 - 200
- 443
-
-
-
-
-
78.5 27.7 31.5 13.9 12.3 8.8 8.74 5.95 5.12 4.67 3.86 3.73 3.19 2.84 2.68 2.62 2.88 3.45 4.43 5.59 10.50
497 84.8 32.4 32.0 14.7 13.8 10.20 9.29 6.53 5.43 5.02 4.02 3.84 3.21 2.85 2.61 2.53 2.66 3.11 4.07 4.92 4.95
0.000 0.001 0.007 0.007 0.032 0.039 0.073 0.081 0.170 0.238 0.282 0.427 0.463 0.647 0.820 0.949 1.000 0.863 0.616 0.367 0.239 0.098
0.031
0.154 0.359 0.332 0.618 0.649 0.785 0.798 0.948 0.943 1,000 0.990 0.975 0.870 0.728 0.536 0.337 0.136 0.043 0.012 0.003 0.000
xw
0.155 FM = 0.0913
analysis should be a careful evaluation of sources of error, followed by application of the pertinent equations. This was not done in the example because firsthand knowledge of experimental techniques was not available. Caution must be used in applying these methods. This can be illustrated by considering the example. If the assumptions stated above were valid, the experimental A,,{ and B M values should be randomly distributed. Even a casual examination of the point values of A.,f and B.M shows that this is not the case. There is a definite trend toward higher AM and lower a t the concentration extremes. This implies that one of the assumptions above is incorrect. If a new equation, such as the Van Laar, is used, a new analysis is required. However, the possibility that some systematic error rather than a random error is present should not be overlooked. In Table 11, for example, the variation of (bA,) / ( b x ) y The is remarkably similar to thatof A,. same is true of (bA,3f)/(dy)x,except that the signs are reversed. The derivatives of BJr with respect to x and J are similarly consistent. This suggests that one possible cause for the nonrandom variations of the An* and B.w is that the x’s are consistently high or the j ’ s consistently low. A plausible explanation, therefore, for the nonrandom variation in AM and BAwis that small systematic errors in composition are occurring. These errors will have the least effect on the most reliable values, and the experimenter may be justified in neglecting them. Any results based on this assumption should be properly qualified.
710
Conclusions
The method presented for determining the constants in the Margules and Van Laar equations depends on an accurate appraisal of the experimental method and the sources of experimental error. Once this is done, it is possible to make quantitative allowance for the effect of these errors on the reliability of each experimental measurement. The principal advantage of this method over other methods is that it eliminates the necessity for individual judgment in smoothing and extrapolating data. Instead, the element of human judgment is used to analyze the sources of error. The improvement which can be gained is illustrated by the different values of constants 1 , and Z?2,fobtained from the data in Table 11, depending on the type of error assumed. Any other methods for determining the constants would give the same value in either case. Although this method has been developed only for the vapor-liquid equilibrium problem, it can be readily extended to any problem where there are a number of independent estimates of the same quantity.
W x, y y Y P
constants in activity coefficient equations 2,B = calculated best values of A and B = (log, l O ) - l = 0.4343 C D = a constant f = fugacity N = AV/BV Po = vapor pressure of pure component s, S = standard deviation T = temperature
INDUSTRIAL AND ENOINEERING CHEMISTRY
-
AM -B.ir
WSM
0.006 0.036 0.105 0.097 0.232 0.260 0.377 0.367 0.536 0.631 0.689 0.828 0.854 0.961 1.000 0.974 0.849 0.591 0.359 0.188 0.114 0.042 =
=
0.159 0.0846
relative reliability or weight of a quantity = liquid and vapor compositions, mole fraction = activity coefficient = fugacity coefficient = total pressure =
Redlich and Kister equations Literature Cited
(7) (8)
=
Wa, 1.000 0.918 0.882 0.826 0.748 0.727 0.660 0.618 0.523 0.438 0.432 0.349 0.320 0.237 0.165 0.103 0.054 0.018 0.005 0.001 0.000 0.000
SUBSCRIPTS 1, 2 = components 1 and 2 V , L = vapor or liquid M , V , R = Margules, Van Laar, or
Nomenclature
A, B
=
Constant % Error
(9)
(10)
Carlson, H. G., Colburn, A. P., IND.END.CHEM. 34, 581 (1942). Hipkin, H., Myers, H. S., Zbid., 46, 2524 119541. Ibl, N. V., Dodge, B. F., Chem. Eng. Scz. 2, 120 (1953). Redlich, O., Kister, A. T., IND. ENG.CHEM. 40, 345 (1948). Redlich, O., Kister, A . T., Turnquist, C . E., Chem. Eng. Progr., Symp. Ser., No. 2,48,49 (1932). Robinson, C. S., Gilliland, E. R., “Elements of Fractional Distillation,” McGraw-Hill, New York, 1950. Rose, A,, Williams, E. T., Sanders, W. W.,Henry, R. L., Ryan, J. F., IND. ENG.CHEM.45,1568 (1953). Sh:rwood, T. K., Reed, C. E., Applied Mathematics in Chemical Engineering,” p. 357, McGrawHill. New York, 1939. Tierney, J. W.,Smith, J. M., Ann. Rev. Phys. Chem., 7 , 21 (1956). Worthing, 4.G., Geffner, J., Treatment of Experimental Data, pp. 189202, Wiley, New York, 1943. RECEIVED for review March 20, 1957 ACCEPTEDSeptember 16, 1957
Division of Industrial and Engineering Chemistry, 131st Meeting, ACS, Miami, Fla., April 1957.
