Effects of Experimental Errors on Thermodynamic Consistency and on

Effects of Experimental Errors on Thermodynamic Consistency and on Representation of Vapor-Liquid ... Note: In lieu of an abstract, this is the articl...
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Effects of Experimental Errors on Thermodynamic Consistency and on Representation of Vaporliquid Equilibrium Data Dean 1. lJlrichson* and F. Dee Stevenson Institute for Atomic Research and Department of Chemical Engineering, Iowa State University, Ames, Iowa 50010

A thermodynamic consistency test based on the “local area” test is developed which permits analysis of vapor-liquid equilibrium data in terms of measurement uncertainties in the independent variables. The effect of random measurement errors on the consistency test and on the uncertainty in the activity coefficients i s illustrated. Uncertainty in the measurement of liquid composition may cause large uncertainty in the activity coefficients but b e difficult to detect b y means of consistency test. Comparing the test results to the null value is shown to b e of less value than comparison with the bounds of a confidence region.

T h e experimental determination of vapor-liquid equilibria is a difficult task. Two basic errors are always present in such experiments: (1) the random errors associated with the analytical procedures; and (2) the systematic errors arising from operational or equipment design error. One of the major problems in determining the accuracy of a set of data is in distinguishing between these random and systematic errors. Many articles in the literature have dealt with this problem. However, the emphasis has generally been on smoothing the data and performing a consistency test. Insufficient emphasis has been placed on an error analysis. The use of error analysis techniques in conjunction with consistency tests provides two independent checks on a set of data. The error analysis indicates the expected uncertainty caused by random measurement errors, whereas the consistency test indicates the total error caused by both random and systematic errors. A comparison of the two indicates the magnitude of the systematic errors and the composition range in which they occur. Finally the error analysis can be used to aid in the selection of a representation equation and can provide weighting factors for fitting the data. Tao (1964) and Chang and Lu (1969) have previously considered using a n error analysis in conjunction with a consistency test. However, their error analyses were developed in approximate form, while the present development is more nearly rigorous. The use of an approximate error analysis can easily lead to incorrect conclusions, particularly with regard to the relative effect of each of the pertinent variables on the consistency test.

The magnitude of f ( x ) depends both on the integration interval, b - a, and the inaccuracy of the data. For perfectly consistent data, f(z) is zero over the complete composition range. However, perfectly consistent data are never obtained and some means of determining acceptable bounds for f(z) is required. These bounds should depend primarily on the Uncertainty introduced by random measurement errors. Such bounds are developed here by evaluating f ( x ) by means of the trapezoid rule and then applying the propagation of error formulas. Using the trapezoid rule, eq 1 is written

Thermodynamic Consistency Test

Error Analysis

The thermodynamic consistency test discussed here is based on the Gibbs-Duhem equation integrated over a n interval from data point a to a n adjacent data point b. The utility of such a test has been illustrated by Stevenson and Sater (1966), who wrote the test equation as f(z) =

J

b

AV

a i = 1

z id In y i

+ S, RANT 2d T ~

lb

AV E d P

The notation f(a,b) is used to emphasize that f is determined for each interval a,b, where a and b refer to adjacent data points, and that the random errors occur only a t these end points. The AH and AV terms are retained as integrals which are respectively assumed to be functions of temperature only and pressure only. Methods of integrating the GibbsDuhem equation for multicomponent syst’ems have been discussed by Li and Lu (1959), 1lcDermott and Ellis (1965), Prausnitz and Snider (1959), and Tao (1962). These methods can be applkd to eq 2 and need not be discussed here. The error introduced by use of t,he trapezoid rule is proportional to the length of the integration interval, b-a, and is generally negligible (Ulrichson, 1970).

Consider a function H ( z l , z 2 , za, - - - - z k ) There each of the = 1 to k , are independent and contain normally distributed random error with a known standard deviation s2,. The standard deviation for the error in H a t any fixed point is then given approximately by

zit i

(I)

(3) Ind. Eng. Chem. Fundam., Vol. 1 1 ,

No. 3, 1972

287

24

where s is an estimate of u for a finite data set. This equation is generally quite accurate if the coefficient of variation, viz,, s,,/zf, is less than 0.2. Equation 3 may be applied to eq 2 if the independent variables are properly identified. The terms which contain measurement error (the independent variables) are xi, y t , P , and T . However, the x i and v f are normally constrained by material balances so that only N - 1 composition variables are determined analytically. The remaining composition, which can be taken as component N without loss of generality, is calculated from material balance consideration and is taken to be a dependent variable. The definition of the activity coefficient, viz. Yf =

YIPei/xipi

The application of this equation will be illustrated later. Returning to the analysis of the consistency test, we see that eq 3 applies only to a fixed point, while eq 2 contains error contributions from the two points a and b. The standard deviation of f(a,b) is therefore written as

Equation 3 can now be applied to each of the two terms in eq 6. I n evaluating these two terms it is useful to consider x,,, yia, Pa, and T u as a complete set of variables independent of X i b , Y t b , PO, and Tb. The rigorous development of this equation for a multicomponent system is given by Ulrichson (1970). The present discussion is restricted to binary systems with s, sy, s p , and sT independent of composition. t-nder these conditions eq 6 is conveniently written as =

+

+

K Z 2 ~ , 2 KU2su2 K p 2 s p 2

I

I

I

20 16

12 0

4

(4)

is used to write y a in terms of the measured variables. The term ei,defined and used by Black (1958, 1963) as a pressure correction term, is assumed to be free from random error, and the vapor pressure, P i , is a function of temperature only, Equation 4 may be used to evaluate the derivative required when applying eq 3 to eq 2. However, i t is also useful to determine the error in y c which arises from random measurement errors. Applying eq 3 to eq 4 and rearranging terms gives

sf2

I

+

KT2sT2

(7)

The K terms in eq 7 have been evaluated (Ulrichson, 1970) as

K Z 2=

Figure 1. The standard deviation for the percentage of points inside the 68.3% confidence region. Points calculated by assuming normal distribution for f(a,b). Curve i s calculated from a binomial distribution

Equations 7 and 8a-d give the standard deviation estimate for a series of values f(a,b) for a binary system. This means, if f(a,b) is a normally distributed variable, that 68.3% of the f(a,b) values will be smaller in absolute value than sf, while 95.5% will be smaller than 2sP Calculation of f(a,b) and comparison with sf thus determine whether the data contain only random measurement error or both random and systematic error. When only random error is present and a large number of data points are used, the f(a,b) values will be randomly distributed about zero and 68.3y0 of the values will lie within the region defined by +sf (the 68% confidence region). For a finite number of data points the percentage within the confidence region, designated N,, will vary about 68.3 in a manner described by the binomial expansion. The statistical distribution of f(a,b) values was checked by simulating normally distributed random error in x, y, P , and T for a hypothetical binary system and calculating N f (Ulrichson, 1970). The values of f(a,b) were normally distributed and N, varied about 68.3 in a manner well described by the binomial expansion. The uncertainty in ATf, sN,, is plotted in Figure 1 us. the number of intervals between data points. Associating the 1 in 20 probability level with two standard deviations, the expected value of Nf is 68.3 j= 2 s , with 95% probability. Thus for 49 data points (;If = 49), with sN, = 6.7, the value of Nf should lie between 55 and 81.7 with 95% certainty. The comparatively large uncertainty in Nf for a small number of data points illustrates the desirability of taking a large number of data points. Effect of Errors on the Consistency Test

Three examples applying both a consistency test and a n error analysis to a set of binary vapor-liquid equilibrium data are given. The first example is for a hypothetical system with positive deviations from Raoult's law and symmetrical activity coefficient curves. It illustrates the effect of measurement errors on the consistency test and on the scatter in the activity coefficient data. The second example is for an approximate Llargules equation representation of the ethanol-chloroform system. Having established the effect of various measurement errors on this system, the test procedure is applied to the actual ethanol-chloroform data in the third example. The modified hlargules equations In

71 =

x22[A

+ 2 zl(B - A ) ]

(9)

+ 2 ~ ( --4B)1

(10)

and In 288

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7 2

=

n2[B

2.21

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6.0

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-

-

1.8 I.6

71

-

1.4

I.2 I

-

0.0004 0.0002

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LIQUID

I I I I I I 0.40 0.60 0.00 MOLE FRACTION, XI

1.0

Figure 4. The error produced in and the 68.3% confidence region from example 1 . x = 1 OOAyl/yl (imposed random error) 0.005I

I

0.20

il

LIQUID MOLE FRACTION, X I

Figure 2. The activity coefficients from example 1 . = Margules equation ( A = 8 = 0.693);0 = data with imposed random error (s, = 0.001) 0.0006

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0.00

I,O

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sr

-0.0004

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-0.0006o.oo I3

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0.20

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0.60

LIQUID MOLE FRACTION, X I

Figure 3. The consistency test results and the fidence region from example 1

0.8

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\-I I.o

LIQUID MOLE FRACTION, X,

1.0

Figure 5. The effect of uncertainties in x , y, P, and T on the standard deviation predicted for the consistency test. Example 1

68.3y0 con-

were used to specify a thermodynamically consistent set of activity coefficients for a hypothetical system by selecting values for A and B. -4set of ,lf interior points (xl = 0 and x1 = 1 are not included) for $1 were chosen so that xI0 xl, = l/(X 1). Then, with constant temperature and fixed vapor pressures, values of P and yl were calculated from P = xlylPl X Z ~and ~ Py1Z= xlylP1/P for each of the Ji values of xl. Normally distributed random errors of zero mean were next generated for each of the four variables xl, yl, P , and T. The random errors were calculated from IBM subroutine GAUSS by specifying the standard deviation estimates s,, sy, s p , and sT as input to the subroutine. The new variables xl', ylr, P', and T' were calculated by adding the random error to the original variables. New activity coefficients were then calculated from the variables containing random error. Equations 2 , 7 , and 8:t-d, with AH = AV = 0, were applied to the data set yE' to obtain values of f(a,b) and sf for each of the Jf - 1 intervals between adjacent data points. The effects of various types of measurement errors were illustrated by comparison of the f(a,b) and S f values. An indication of the meaning of the consistency test results was obtained by considering the resulting error in the activity coefficients with respect to the confidence region is,. The activity coefficient error mas expressed, in percent, as

+ +

E(rr) = 100AYJYt = ( - I C r - Yz)/Yz' The error was expressed in this manner because if the 7%' values represented actual data containing only random error, then the 7%values would be the result of perfect smoothing.

The values of E(?$) are related to 100syr/yi in the same manner as f(a,b) is related to sp Examples are used to illustrate the effect of random measurement errors and verify the propagation of error formulas. Example 1. A test case was arbitrarily chosen with A = B = 0.693, P1 = 300 mm, Pz = 700 mm, d P l / d T = d P z / d T = 15 mm/'d, and T = 300'K = constant. The effect of random error in the liquid mole fraction is first illustrated with 49 interior data points ( X = 49). The consistent activity coefficient curves, y r , and the values of yz7calculated with s, = 0.001 and sy = s p = sT = 0 are shown in Figure 2. The results of the consistency test are shown in Figure 3 \There f(a,b), as calculated from eq 2, and the 68.374 confidence region formed by the *sf curves (from eq 7 ) are both plotted. The shape of the confidence region indicates that random error in the liquid mole fraction may cause considerably larger f(a,b) values near the extremes of composition than in the midrange. I n this particular case 37 of the 48 f(a,b) values lie within the bounds of +sp This gives Nf = 77%, which is well within the 55 t o 88% range expected for a set of 49 data points. Figure 4 shows how random error in zl propagates to the activity coefficient for component 1. The percent error imposed on the activity coefficient, E ( y l ) , and the 68.3% confidence region, +lOOsy,/yl, are plotted against xl. The growth in both the error and the error bounds is obvious as z1 approaches zero. Letting N,, be the percent of E(y1) values which is within the 68% confidence region, we find Xy, = 67.3y0. This value is well within the confidence region defined by Figure 1. Figure 5 shows the effect on sf,as given by eq 7 , of random measurement errors in each of the independent variables Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972

289

Figure 6. The effect of uncertainties in x , y, P, and T on the standard deviation predicted for the activity coefficient of component 1 . Example 1

0.012

-

0.008

-

0.006

-

LIQUID MOLE FRACTION, X ,

Figure 7. The effect of uncertainties in x , y, P, and T on the standard deviation predicted for the consistency test. Example 2

z, y, P , and T . The levels of error were arbitrarily chosen for illustrative purposes as s, = 0.004, sy = 0.001, s p = 0.5, and sT = 0.05. Five cases are shown illustrating the effect of error in each variable alone and error in all four variables simultaneously. Note t h a t the contribution of s, to s, is insignificant except as x1 approaches zero or unity. Thus the consistency test does not effectively detect random errors in liquid composition. Figure 6 shows the s,,/yl curves for the same levels of error as shown in Figure 5. The curves for sy2/y2(not shown) are similar if plotted against XZ. For the level of error shown in Figure 5 the errors in temperature and pressure have the dominant effect on sI. However, for the same levels of error, Figure 6 shows that the dominant'effect on s,,/yl is from errors in z and y. Thus, comparison of Figures 5 and 6 indicates that errors in measurement of liquid composition may cause large errors in the activity coefficients while yielding small values of f(a,b) from a consistency test. It is, therefore, extremely important that the error in each of the independent variables be estimated before attempting to assess the meaning of the results of a consistency test. T h a t is, large measurement uncertainties in x may cause large errors in the data but the small values of f(a,b) would tend to misleadingly indicate accurate data. Example 2. The effect of random measurement errors on the local area consistency test and on the accuracy of the activity coefficients is illustrated here by using the ethanol290 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972

chloroform data reported by Scatchard and Raymond (1938). This system differs from the previous one in that it has unsymmetric activity coefficients and exhibits an azeotrope. These data have been tested by successive approximation of the vapor pressure surface (Mixon, et al. (1965)) and found to be quite accurately measured. (The reported data taken a t 45°C are plotted in Figure 13). Carlson and Colburn (1942) indicate that the Margules equations satisfactorily represent this type of data. By smoothing and extrapolating the data to z1 = 0 and $1 = 1, the Margules constants A = 1.5740 and B = 0.5019 were obtained. Although these constants may not be the best representation of the data, they are used to simulate the system and illustrate the effects of measurement errors 011 the accuracy of the data. The reported vapor pressure data (Scatchard and Raymond, 1938) were plotted to obtain the slopes of the vapor pressure curves a t 45°C. Taking component 1 as ethanol, these were estimated as dPl/dT = 8.709 mm/"C and dPz/dT = 15.89 mm/"C. The vapor pressures were P I = 172.76 mm and PZ = 433. 54 mm. Nonidealities in the vapor phase and the volume change of mixing were neglected. Csing eq 7 and 8a-d, with the Margules constants and vapor pressure data given above, the effect on s, of uncertainties in z, y, P , and T was determined for various levels of error. Figure 7 shows the sl curves for specific values of s,, sy, s p , and ST taken individually. The most dramatic difference between the results for this system and the previous one is in the effect of sy. Here the presenceof a n azeotrope near 21= 0.15 causes sy to approach zero. Equation 8d shows that K , approaches zero as x1 approaches yl. Thus the error in measuring the vapor phase composition near an aqeotrope, although it may be very important for purposes of applications, has little effect on the consistency test. The effect of s, on the local area test is again significant only a t the extremes of composition. The effect of measurement errors in z, y, P , and T on sy1/yl and sy2/y2 is similar to that shown in example 1 except that the unsymmetric activity coefficient curves cause a slightly different effect for each component. The results of imposing random error on this simulation of the ethanol-chloroform system and applying the consistency test and error analysis are shown in Figures 8 to 10 for J!f = 49. The standard deviation estimates s, = 0.001, sy = 0,001, s p = 0.02, and sT = 0.03 were used to characterize the uncertainties in z, y, P , and T . This particular choice of error levels was dictated by the results shown later in example 3. Figure 8 shows the results of the consistency test where N , = 56.2%. The effect of sy on sf is apparent in the shape of the s, curve, even though the error in temperature is the dominant variable. Figures 9 and 10 present E ( y J and the confidence regions for components 1 and 2, respectively, where N , , = 59.2y0 and ivy, = 71.4%. These values for N,and N , are within the expected range. Example 3. Ethanol-Chloroform Data. Scatchard and Raymond (1938) reported 25 data points for the ethanolchloroform system a t 45OC. At each point the values of zl, yl, and P were reported a t constant temperature. Equation 4, with et = 1, was then used to calculate the activity COefficients. This example illustrates the application of the consistency test in assessing the accuracy of the llargules equations as candidate representation equations for the activity coefficients. Equation 2 was used to calculate f (a$) for each of 24 intervals. The results are plotted against the mole fraction of ethanol (zl) in Figure 11. The first step in this procedure

6.0 0.008

t

1

X

k 4.0

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t\

1

0.004 fb,b)

0.000 -0.004

0.20

0.40

LIQUID MOLE

0.80

0.60 FRACTION,

1.0

XI

Figure 8. The consistency test results and the 68.3y0 confidence region from example 2. s, = sy = 0.001, sp = 0.02, and ST = 0.03

should always be a check on the randomness of the distribution of the f(a,b) about zero. For a very large number of data points containing only random error, half of the f(a,b) values should be positive and half negative. The xsquare test conveniently measures the significance of the deviation from a 50: 50 split. I n the present case, 7 of the 24 values are greater than 0 and the value of x-square is 4.2. The X-square test thus indicates the probability is less than 1 in 20 that f(a,b) is randomly distributed about zero. The most likely cause of this nonrandom distribution is the neglect of the pressure correction term 8, (Mixon, et al., 1965). The results here suggest that the X-square test, by itself, may quickly indicate errors in the data before the confidence regions are calculated. Such tests may also be applied to restricted ranges, e.g., 51 = 0.3 to 0.8 in Figure 11, to check maldistribution in f(a,b). The next step in the test procedure is to estimate the magnitude of the random errors i n X I , y l , P , and T which would be required to produce the distribution observed for f(a,b). Such error estimates are developed here without applying pressure corrections to the data. The points in Figure 11 do not diverge rapidly from zero, either to the left of z1 = 0.1 or to the right of z1 = 0.9. The errors in x1 are therefore assumed to be small because sf is significantly increased by s, only in this region. A few calculations of sf and Xf for selected values of s,, sy, s p , and sT quickly indicate that reasonable estimates of the level of random measurement error associated with these data are s, = 0.001, sY = 0.001, s p = 0.02, and ST = 0.03. With these estimates of error the dominant terms influencing sf are sy and sT. This level of error is quite low for vapor-liquid eqilibrium data and reflects careful measurements. The 68.3% coiifidence region determined by these error estimates is also plotted in Figure 11. The value of N , obtained from Figure 11 is 75%. Other slightly different estimates of the measurement error associated with these data could obviously be used, but the general nature of the confidence regions would not be significantly changed. Alternatively, values of et could be estimated to determine whether this would reduce the spread in f(a,b) and allow smaller error estimates. However, assuming that 8, = 1, t h a t only random errors are present, and that AV = 0, the data are observed to satisfy the Gibbs-Duhem equation within the estimated measurement uncertainties. d frequent reason for testing the thermodynamic consistency of a set of published data is to determine whether the data are sufficiently reliable to use in equipment design. The activity coefficients are usually represented by some type of equation if the data are deemed acceptable. The consistency test presented here provides estimates of the magnitude

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0.20

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0.40

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0.60

I.oo

0.80

LIQUID MOLE FRACTION, X I

Figure 9.The error produced in y l and the 68.3% confidence region from example 2. x = 100Ayl/yl (imposed random error)

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(u

2.

1

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4.0

a. -4.0 -6.0

I I I I I I I I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 LIQUID MOLE FRACTION OF ETHANOL, XI

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0

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0.9

Figure 10. The error produced in y2 and the 68.3y0 confidence region from example 2. x = 100Ay2/y2(imposed random error) 0.006

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1

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0.004 0.002 f(o,b)

I-Lx-

0.000 +

-0.002

x

- 0.004

xx X X

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I

X

Figure 1 1. The consistency test results and the 68.3% confidence region for the ethanol-chloroform data. s, = sy = 0 . 0 0 1 , ~=~0.02, and+ = 0.03

of the error in the independent variables which can, in turn, be used to estimate the uncertainty in the activity coefficients. The two-constant Margules equations are used to illustrate such a procedure. Equations 9 and 10 are combined to give In

y1/y2 =

z d l - 3xJA

+ z1(2 - 3x1)B

(11)

Equation 11 was fit to the ethanol-chloroform data by a least-squares procedure. No weighting was used although as a weighting factor would using l/[S,,2/y~2 Sy,z/yz2]1'* undoubtedly improve the reliability of the results. Somewhat

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sent these data. The results suggest also that the rather common technique of using a representation equation to fit the data and force thermodynamic consistency is of questionable value unless the equation represents the original data within the uncertainty of the experimental measurements. This again emphasizes the need for experimental estimates of the uncertainty of the measurements. Conclusions

I 0.20 LIQUID MOLE

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FRACTION OF ETHANOL,

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0.80

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I.oo

XI

Figure 12. Analysis of the accuracy of a least-squares fit of the Margules equations used to represent the activity coefficient of ethanol 5.6 5.0

4.4 3.8 3.2

Ti 2.6

0.6 , ! j

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0.1

0'?2

I I I I I I 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9

LIQUID MOLE

1.0

FRACTION OF ETHANOL, X I

Figure 1 3. Activity coefficients for ethanol-chloroform. - - - graphical smoothing and = least-squares fit of the Margules equations

---

similar weighting techniques have been suggested by Ross (1970) and Tassios (1967). The constants A = 1.601 and B = 0.470 result from fitting eq 11 via least squares to the activity coefficient data. These constants are used in eq 9 and 10 to calculate 7%.Letting :y refer to the original data, the error in the activity coefficients is then A y I / y I = ( y t r - Y J ~ ~ ' Equation . 5 is used to calculate s r l / y I with sz = 0.001, sy = 0.001, sp = 0.02, and ST = 0.03, as estimated from the consistency test. The 68.3% confidence region and E ( y i ) are then plotted as in Figure 12. This figure indicates that the Margules equations are not exact models of the liquid phase behavior because of the nonrandom nature of the deviations. The results for component 2 are similar. The indicated error of 5y0 in y1 is much greater than the uncertainty indicated by the consistency test. The value of 16% obtained for N,, is essentially meaningless because of this nonrandom behavior. The Margules equations with A and B as determined above should be rejected as representation equations if i t is desired to fit the data with the accuracy justified by the consistency test. Another representation equation, a different fitting procedure, or perhaps an interpolation formula should be sought. The three-constant Margules equations, incidentally, do not significantly reduce the lack of fit. The activity coefficients and the curves obtained from the least-squares fit are shown in Figure 13. This figure misleadingly indicates a rather good fit of the data. The rigor or accuracy of the consistency test is obviously immaterial if the Margules equations, with a 5y0error, are used to repre292 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972

The use of a n error analysis in conjunction with the local area consistency test in judging the accuracy of vapor-liquid equilibrium data has been illustrated. The error analysis provides valuable additional information: 1. It provides a quantitative meaning for the consistency test rather than a qualitative comparison with the null value. 2. It provides a means for determining whether the data are adequately represented by an equation for the liquid phase behavior and may avoid both misrepresentation of good data and excessively complex representation of poor data. 3. It provides a means for assessing the importance of nonideal vapor phase behavior and of the heat and/or volume change of mixing, 4. It illustrates that the consistency test does not effectively detect random measurement errors in the liquid composition nor in the vapor composition near a n azeotrope. This procedure should prove useful in analyzing the effect of given uncertainties in the analytical techniques to be used before a set of experimental vapor-liquid equilibrium data are taken. I t should be helpful in detecting systematic errors as the data are being taken. And it will indicate the accuracy of a representation equation after the data have been taken, even if uncertainties in the experimental measurements are not known. The authors have for some time privately questioned the value of thermodynamic consistency tests because clear-cut decisions about the consistency of the data could rarely be made. The error analysis and thermodynamic consistency test illustrated here do not change that fact. However, the results of the consistency test are now properly interpreted in the contest of probabilities, and some indication of the importance of e,, A H , AV, and the systematic errors will usually be obtained by considering the distribution of f(a,b) within a given confidence region. Finally, the authors would like to propose that the words consistent and inconsistent no longer be applied to experimental data. Obviously experimental data which contain any measurement error are inconsistent. Only mathematical relationships may be consistent, since only such relationships will be exact solutions of the Gibbs-Duhem equation. Experimental data should then be said to satisfy or not satisfy the Gibbs-Duhem equation within the given experimental uncertainty. If the data do not satisfy the Gibbs-Duhem equation it is because of either excessive random error or the presence of systematic error, or both. Nomenclature

A, B a, b

= =

E(yJ =

Margules equation constants, eq 9 and 10 indicate specific, adjacent data points in a set of vapor-liquid equilibria data the percent error in the activity coefficient, 100. (71'

f (a$)

=

H

= =

AH

-

YI)/YlT

the trapezoid rule approximation of eq 1 defined by eq 2 hypothetical function used in eq 3 enthalpy change of mixing a t constant T and P per mole of solution formed

M N hTk

P pk R S S‘Vk

T

Av X

Ax Y AY Zk

Y

e U

= number of data points generated in random error

test cases = number of components in a multicomponent system = percent of points greater than sk in absolute value , y2 where k may be f, y ~ or = total pressure = vapor pressure of component k = gas constant = estimate of standard deviation u = standard deviation on N k where k may be f o r y = absolute temperature = volume change of mixing a t constant T and P per mole of solution formed = liquid phase mole fraction = random error generat,ed in x a t a given datum point = vapor phase mole fraction = random error generated in y a t a given datum point = set of independent variables in eq 3 = activity coefficient defined by eq 4 = pressure imperfection term used in eq 4 = standard deviation

SUBSCRIPTS a, b = data point identification i, j = component identification f , x, y , P , T , N = hr, s, or u applies t o variables so indicated

SUPERSCRIPTS r = indicates variable containing random error

Literature Cited

Black, C., Ind. Eng. Chem. 50,391 (1958). Black, C., Derr, E. L., Papadopoulos, M. N., Ind. Eng. Chem. 55(9). 38 (1963). CarlsoG H. C., Colburn, A. P., Ind. Eng. Chem. 34,581 (1942). Chang, S.-D., Lu, B. C.-Y., “International Symposium on Distillation,” Brighton, England, Part 3, p 22, Sept 1969. Li, J. C. M., Lu, B. C.-Y., Can. J. Chem. Eng. 37,117 (1959). McDermott, G., Ellis, S. R. M., Chem. Eng. Sci. 20,293 (1965). Mixon, F. O., Gumowski, B., Carpenter, B. H., IND.ENG. CHEM.,FUNDAM. 4,455 (1965). Prausnitz, J. M., Snider, G. D., A.I.Ch.E. J. 5 , 75 (1959). Ross, J. F., Paper presented a t Third Joint Meeting AIChEIMIQ, Denver, Colo, Sept 1970. Scatchard, G., Raymond, C. L., J. Amer. Chem. SOC.60, 1278 (1 0.18 ). \----,.

Stevenson, F. D., Sater, V. E., A.I.Ch.E. J. 12,586 (1966). ENG.CHEM.,FUNDAM. 2, 119 (1962). Tao, L. C., IND. Tao, L. C., Ind. Eng. Chem. 56 (2), 36 (1964). Tassios, D., “Prediction of Binary Vapor-Liquid Equilibria; Members of a Homologous Series in a Common Solvent,” unpublished Ph.D. thesis, University of Texas, Austin, Texas, 1967

Ulrichson, D. L., “Effect of Experimental Error in Vapor-Liquid Equilibrium Data on Thermodynamic Consistency,” unpublished Ph.D. thesis, Iowa State University, Ames, Iowa, 1970. RECEIVED for review March 15, 1971 ACCEPTED February 24, 1972 Work was performed in the Ames Laboratory of the U. S. Atomic Energy Commission. Contribution No. 2797.

Statistical Thermodynamics of Group Interactions in Pure n-Alkane and n-Alkanol-1 Liquids Tsung-Wen lee, Robert A. Greenkorn, and Kwang-Chu Chao* School of Chemical Engineering, Purdue U n i v u d y , Lafayette, Ind. 47907

A partition function i s developed for the description of nonpolar and polar chain molecule liquids b y combining the cell theory and the quasichemical lattice theory. On the basis of this partition function, the interaction properties of methylene and methyl groups are determined from analysis of literature data on n-alkanes. An explicit procedure i s devised and followed for the stepwise decomposition of the molecular properties into group properties, including the interaction energy parameters, the core volumes, and the external degrees of freedom of the methylene and the methyl groups. The interaction properties of the hydroxyl groups are then determined from analysis of literature data on n-olkanol-1 liquids. The hydrogen-bonding energy thus determined agrees with accepted values. The core volume, degrees of freedom, and energy parameters of the hydroxyl groups are evaluated. The densities and heats of vaporization calculated from the theory for the n-alkanol-1 liquids are in good agreement with data.

A group is an identifiable structural unit of a molecule, such

as a methyl group or a hydroxyl group. A few kinds of groups make up a large number of the molecules of interest in chemical processes. This fact has provided the motivation for numerous investigations of pure substance and solution properties from the viewpoint of group contributions. The success of these efforts confirms that the intermolecular forces, being short-ranged, can be validly considered, for a large class of molecules, to be of a local nature between groups that come into close proximity due to molecular motion.

Langmuir (1925) suggested the premise that the force field of a group is independent of the nature of the rest of the molecule. The electronic structure of molecules suggests that Langmuir’s principle of independent action cannot be strictly valid, but is a good approximation for many molecules. The degree of approximation can be improved by refined classification to reflect the nature of the rest of the molecule. Notable developments of group contribution to solution properties have been made by Pierotti, et al. (1956, 1959), Deal, et al. (1962), Wilson and Deal (1962), Hermsen and Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972

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