GRAPHICAL METHODS FOR DETERMINING A NONLINEAR CONSTANT IN VAPOR
PRESSURE EQUATIONS D0 NA L D G
.
M I L L E
R , Lawrence Radiation Laboratory, University of Calqornia, Lirermore, Calif.
Graphical methods are given for the accurate determination of the nonlinear constant in vapor pressure equations and are applied to the Antoine and Miller-Erpenbeck equations. These schemes, involving derivatives or integrals of deviation functions, are feasible for hand calculation with seven-place logarithms and a desk calculator. Examples of the calculations are given for benzene and hexadecafluoroheptane. The results are especially good with the Antoine equation. In this case, the deviation of C from the best value is about 0.2570 for the integral method and about 0.5% for the more rapid derivative method (an extension of Lu's scheme). An analysis of least square fits as a function of C shows that the best all-round fit is the least square fit to P rather than to log P. hc AN EQUATIOS with a nonlinear constant to experiFzrntal vapor pressure data by least square techniques can be carried out by iterative or hunting procedures on the nonlinear constant. However, such procedures are so tedious by hand calculation that they become practical only with large computing machines. Since large computers are not available to everyone, it is desirable to have a scheme for reducing the hand calculations to manageable proportions. Some graphical methods are given here for obtaining the nonlinear constant in three-constant vapor pressure equations, such as the Antoine equation. The linear constants can then be obtained in the usual least square way. These methods may obviously be used for functions other than vapor pressure equations.
nonlinear constant. For example, consider the vapor pressure equation (Miller-Erpenbeck) : log P
=
.4 -
B T
-
+ log (1 - C T )
(5)
This equation was recently shown ( 7 ) to be a consequence of Dieterici's eqation of state and to give a good fit from the triple point to the boiling point. The derivative of Equation 5 is: m=-B+
LI '
(6)
L(1 - CT)
where L = 2.302585. Lu's method applied to Equation 5 yields: (1
+ a)TrTuC2- (1 +
a)(Tz
+ Tu)C +
LY =
0
(7)
where
Differential Methods
Lu Method. Lu ( 4 ) has recently given a rapid method for determining the nonlinear constant C of the Antoine equation:
where P is the pressure and T the absolute temperature. Bidifferentiating log P w i t h respect to 1 / T or T , A is eliminated, yielding: (2)
where
Combination of tlvo such derivatives at widely separated temperatures T,and Tu results in the elimination of B, yielding a n equation which can be solved for C, namely: (4)
where m l is evaluated a t T iand obtained from the experimental data by graphical or numerical means. With C at hand, A and B may now be obtained by least squares, the method of averages, or graphically. The same technique may be used for arbitrary vapor pressure equations (or any three-constant functions) with one 68
l&EC FUNDAMENTALS
Equation 7 is solved for C by the quadratic formula, and the proper root is the one whose value is about O.64/Tb where T b is the normal boiling temperature. Lu's derivative method gives better values for the C's of Equations 1 and 5 than are obtained from estimation methods ( 7 , 7) which do not use actual data. The C's are determined to about 1 to 5%, resulting in values of the Antoine C, for example, which may be off as much as 2 to 3 units from least square values. (Antoine C values run between 0 and -100, with most between -35 and - 6 5 ; a commonly used "universal" value is -43.) Alternate Method. The disadvantage of Lu's method is that the use of derivatives a t widely separated points neglects all the remaining data. This may be remedied by obtaining derivatives at several other points on the curve, besides T, and Tu,and making use of Equation 2 or 6 . Equation 2 is easily relvritten in the linear forms: T = -C+B'?
(&)
=
-c
I-[
+ hL!?
T
(9)
Thus -C, the intercept of the T us. (l/nl'*) or Tvs. T/(-rn)'* curves may readily be obtained graphically or by least squares. Equation 6 cannot be put into a linear form. I t may, however, be linearized as follows: Assume a reasonable value of
C, such as 0.64/Tb, denote this estimate by C1, replace C by ( C , f 4C,), factor (1 - C1T) out of the denominator, expand
29 -
the other factor [I - T4C1/(1 - C , T ) ] into a series i n the numerator, and collect terms, neglecting all which are of order ACl2or higher. This process yields: Ci T 3
2.8 2.7
-
2.6 -
+
2.524
-
a s 0 a 2.3
Thus, AC1 is the slope of the best straight line through a plot of the left hand side against the bracket on the right hand side. If 4C1 is large. repeat with C P = C1 ACl and calculate 4C2. Usually, two such iterations give a good enough result. Because Equation 5 does not give as good a fit as the Antoine equation, one often finds a curve in the shape of a shallow bowl. Hence, it is more objective to determine AC by the method of averages or 1 east squares rather than graphically. Much better results are obtained by this "extended derivative" method than b\ the simple Lu scheme, C being determined to about 0.57,. Estimation of m. I,u obtains the derivatives m, graphically from a plot of log P us. I / T , using straight line segments. Because C depends on small differences between derivatives, their determination from a plot with such a small curvature (see Figure 1) results in a large uncertainty. However, it is possible to improve the estimate of m considerably by means of deviation functions. The simplest one is a Kirchhoff equation, which is equivalent to passing a straight line through the two end points of the data plotted as log P us. 1/T (see Figure 2). The deviations 6, are calculated by subtracting the Kirchhoff equation from the experimental data:
+
6K =
lOg P -
(OK
-
bK/T)
22
-
2.1 -
1.8 1.7 -
20
1.9
t
I
I
I
I
28
2.9
30
3.1
1
I
I
3.2
33
3.4
35
1 0 3 n
lo3
Figure 1. Plot of log P vs. (I/T) showing actual very slight curvature
for benzene
log P
(1la)
where bK
aK =
= -
(log P? - 10s Pi)
log P $. h K u ,
ua
-
Figure 2. Schematic diagram of Figure 1 with curvature of vapor pressure curve greatly exaggerated
u1
i
=
1 or2
(1lc)
Here u = 1/T, a, and b , are the intercept and slope of the Kirchhoff line, respectively. and 1 and 2 refer to the data points a t the highest and lowest temperatures. Because the Kirchhoff line contains the major portion of temperature dependence, the deviations from linearity, which are to be accounted for by the nonlinear C in Equations 1 or 5, are greatly magnified (see Figure 3). The slope m is given by:
__ ---
Vapor pressure curve Straight line (Kirchhoff) deviation function Antoine deviation function using an estimated Co
141
where j refers to arbitrary temperatures. The derivatives dg/du or d6/dT are obtained a t T, from a plot of 6 us. u or 6 us. T by graphical differentiation or by numerical differentiation formulas such as Rutledge's (5). Because b , is the major portion of the derivative, a n error in the determination of the 6 derivatives is not as important as it would be in the direct determination of d log P/d (1/ T ) from Figure 1. Even greater magnifications can be obtained using deviation functions such as Equation 1 to give 8A [with a Co = -43 or calculated from Thomson's rules (7)] or Equation 5 to give 6.,{ (with CO= 0.64/Tb). The appropriate equations are:
I
I
280
290
I
300
310
320
1 330
1
1
340
350
36C
T
Figure 3. benzene
Plot of deviations
EK
of Equation 1 l a vs. T for
Solid line is best "by eye" curve through points. T I and Tz do not coincide exactly with actual d a t a points. Points r e a d off solid line a t equal intervals can b e used in numerical differentiation formulas to get derivatives. They may also b e used in Simpson's rule to get the a r e a J K ' under the solid line
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Integral Methods
and
The better the experimental data, the better these more complex deviation functions work for determining derivatives. Discrepancies and experimental error even in the best data become very prominent, as shown for benzene (2) in Figures 4 and 5; this technique is thus useful for a critical examination of data.
L
A sesious disadvantage of the derivative techniques is that differentiation magnifies the effects of experimental errors. However, if instead of differentiating 6, it is integrated over the whole range of data, not only are all the data used but the effects of experimental errors are smoothed out. This idea has been used previously in the analysis of diffusion data ( 3 ) . To fix these ideas, the analysis is carried through for the Antoine equation. The plan is to integrate the analytical expression for gA, assuming the data are fit by an Antoine equation. The correct value of C is the one which when substituted into this integral gives the area determined from the actual experimental data by means of a plot of Equation 13a. Suppose Equation 13a is used as the deviation function with a n estimated value of Co. Suppose further that the data are fitted by the Antoine equation. Then the following equation should fit the experimental 6,’s:
where A , a d , B, C, b,, and COare constants. The constants b, and COare known by Equation 13b and assumption, respectively, whereas [ A - a,], B, and C are not. However, a t the end points of the data, 6 = 0-Le., at T 1and Tz. (If the data are not exact, T I and Tz are the most widely separated points where the best curve through the data goes through and aA2 are set equal zero. See Figure 4.) Therefore, if to zero a t T1 and T z ,it is possible to solve for [A--a,] and B in terms of the known TI, TZ, and Co and the unknown C. One obtains:
\
a
[ A - 0.41 = (Ti
b.i(C
- Co)
+ Co)(Tz + Co)
280
290
300
320
310
330
340
350
T
Figure 4. Plot of deviations 6~ from Equation 1 3 0 vs. T for benzene using Co = -51 This deviation function magnifies experimental errors. A derivative obtained from this graph contributes less than 1 part in 2000 to the value of rn or n. JA’ is the algebraic sum of the positive and negative areas between the solid curve and the line 6~ = 0
Substitution of Equations 16a and 16b into l 5 b gives an implicit equation for C in terms of 8A, T , and known constants. Integration of BA between any two temperatures will eliminate T . Going from T I to TBis the simplest and makes best use of the data. If the analytical integral is denoted by J , one obtains:
Substitution of Equations 16a and 16b and rearrangement yields :
1
,
280
I
I
I
290
300
310
1
I
I
I
320
330
340
350
J
360
T
Figure 5. Plot of deviations 6M from Equation 14a vs. T for benzene using Co = 1.83 X 10-3 Note the S shape. JM’ is the algebraic sum of the positive and negative areas between the solid curve and the line 6~ = 0
70
l&EC F U N D A M E N T A L S
Thus, the value of J between T1 and TZ is a function of C and the known constants b,, CO,TI, and Tz. However, the actual numerical value of this integral (denoted by J’) can be obtained from the experimental data by plotting the experimental bA’s Z.J. T and using Simpson’s rule or other numerical or graphical integration schemes. Putting this “experimental” value J’ in Equation 18 permits solving in principle for the only unknown quantity, namely, C. I n
practice it is advisable to calculate values of J for three or four values of C, bracketing the expected one; plot J as a function of C; and read off the value of C at J = J ’ . (Simple iteration diverges.) Obviously the same technique will work with the integration:
I :=
S
\
L
(19)
BAd(l/T)
giving a different analytical expression. I n Table I are collected equations for both types of integral, as well as a similar set for the Miller-Erpenbeck equation (Equation 5) using the 6Jf of Equation 14a. T h e expressions are denoted by three letters: the first is the equation to be fit (A for Equation 1 or M for Equation 5); the second is the deviation function [A (13a), M (14a), or K ( l l a ) ] , and the third is the type of integral (J or I). The subscript on Z or J refers to the deviation function used to obtain the experimental plot. I n certain expressions (AKI, MKJ), the Kirchhoff expressions could not be obtained by simply setting Co = 0 ; a separate integration (or a limiting process) was required. Before discussing the practicality of these equations for calculation and any comparisons with least square results, let us consider certain properties of the least square fits of the Antoine and Miller-Er penbeck equations.
I-C
Figure 6. Plot for benzene of sum of the squares of the residuals of log P ( S L ~and ) of P ( S p ) as a function of the Antoine C, obtained from a least squaring of A and B of Equation 1 The t w o curves have been superimposed on a n arbitrary logarithmic scale. Note how much sharper the minimum is of S p than of SLP. The minima C P and C L P a r e - 5 1 . 5 6 and - 5 0 . 8 6 , respectively
least Square Fits
T o determine how well the nonlinear constants obtained. graphically compared with least square values, least square programs for the Ani:oine and Miller-Erpenbeck equations were prepared for a n IBM 650. Instead of iteration, the following “hunting” procedure was used to obtain C: An estimated value of C is assumed, the best A and B are determined by the usual linear least squares, and the residuals of log P and the sum of their squares are calculated. A larger value of C: say C E, is put in, the process repeated, and the sums of squares of the residuals are compared. If the new sum is smaller than the previous one, E is added to (C E), and the procedure continued. However, whenever a sum is found to be larger than the prwious one, the next increment is taken of the previous one. Thus a smaller C is put in as minus next, and the process continues with smaller C’s and the smaller increment until again a sum is larger than the previous one, and so on. Thus to start, E is added to C until the first larger sum of squares is found, whereupon there occurs the first reversal of the increment added. At this first reversal, - ~ / 4 is added; at the second, +e/l6; a t the third, - ~ / 6 4 . I n this lvay, the minimurn is bracketed by continually shrinking bounds until it is located to the accuracy desired. The size of the original e and the factor by which it was reduced o n reversal could be altered a t will. Moreover, the residuals of P and the sum of their squares were also computed, so that the code could control this sum to find its minimum as well. I n this way the best least square fit for either P or log P could be found, a convenience not readily available in a n ordinary iterative code. Considerable difficulty was experienced in the preparation of these codes owing to cancellation and round-off; it became necessary to use 10-digit extended precision routines. This code was also used to determine systematically the variation of the sum of the squares of the residuals of P (denoted by s), and of log P (denoted by XI,,) as a function of C. The results are of some interest. Examples of S, and S,, for
\ I
Table 1.
Deviation Function Integrals I and J for Antoine and Miller-Erpenbeck Equations
+
+
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71
Table II.
Comparison of Graphical Methods with least Squares for Hexadecafluoroheptane
Antoine Equation 7’2 -Co
.___-
‘Method
Least squares: P Least squares: log P Lua
Extended derivativeh
a
-C 63.13 62.34 60 33 62.67 62 40 62 80 63 41 62 61 62 80 62 95 63 01 63 01
Deriuatiues from I K curoe.
TI ,..
..
,
. . . , . ,
271 271 271 270 271 270 271 271
...
.,.
...
. ..
...
. .. ...
, . .
0 2 5 7 2 7 2 5
363 363 363 363 363 363 363 363
3 3 3 1 1 1 1 1
0 0 0 51 51 60 33 60 33 60 33
-
SP 0.34 1.03
8.67 0.58 0 93 0 47 0 42 0 64 0 47 0 38 0 36 0 36
Method
Least squares: P Least squares: log P Lua Extended derivativeb
103c 1.966 2.018 1.99,; 2.018 2.015 2.019 2.021 2.022 2.025 2.020 2.021 2.020
LMiller-Erpenbeck Equation T, T~ 103 co ,
.. ...
. .. , . .
, ,
...
...
, , ,
...
...
270.7 271.2 271.5 270.7 271.2 270.7 271.2 271.5
363.3 363.3 363.2 362.7 362.7 363.2 363.2 363.1
0 0 0 1.83 1.83 1.9925 1.9925 1.99?s
. . , .
.
sp 19.5 57.7 28.9 57.7 53.2 59.3 62.6 64.3 69.6 60.9 62.6 60.9
Deriuativu from J K curoe.
the Antoine equation applied to benzene are given in Figure 6 and are drawn on a logarithmic scale to emphasize the dramatic improvement with an appropriate C. Similar graphs are obtained using the Miller-Erpenbeck equation and also with hexadecafluoroheptane. As might be expected, the C which makes S, a minimum (denoted by C,) does not coincide with the one which makes S,, a minimum (denoted by C,,), although they are close. The minimum for S, as a function of C is very much sharper and deeper than the one for SLp. Moreover, because of the relative shallowness of S,,, the value of C which gives a minimum for S, also gives a value of S,, which is near its minimum value. The converse is obviously not true. Consequently it seems that the best allround fit is a least squares fit to P rather than to log P. This is also reasonable from the standpoint that absolute errors of measurement of vapor pressures are more constant than percentage errors. Comparisons of Equations 1 and 5 for goodness of fit with benzene (2) and hexadecafluoroheptane JHDFH) ( 6 ) showed the Antoine equation to be superior to the Miller-Erpenbeck equation by a small but noticeable margin (although in the H D F H case, Equation 5 is being used above its limit of validity -i.e., the normal boiling point). Examples of Calculation
The integration techniques were carried out on experimental data for benzene (2) and H D F H ( 6 ) . To investigate their suitability for hand calculation in terms of such questions as loss of significant figures and the sensitivity of the results to minor changes in TI,Tz,or the Simpson’s rule integral, all calculations were carried out on an IBM 650. The following conclusions were obtained.
Antoine Equation.
S o n e of the I integrals (Equations
20, 21) are suitable for hand calculation, as more than seven
significant figures are required to obtain the graphical C to the nearest 0.05 to 0.1. Ho\vever, the J integrals (Equation 18) are all feasible with seven-place logarithms and a desk calculator, since only six or seven significant figures are needed, including one place for round-off. Further ana1)sis of the J calculations shows that when the Kirchhoff deviation function is used (Co = O), the value of C is fairly sensitive to small errors in TI, T z ,or the Simpson’s rule integral J’ (see Table 11, AKJ rows). “hen Co is Mithin about 10 units of the proper C, the effects of minor variations of 71 and Tl or of drawing the best curve become much less. The results become more reproducible (see Table 11, AAJ rows), and the graphical values are within 0.1 to 0.4 units of C p and usually between it and CLp. If Co is within 3 or 4 units of C, then the graphical values are within 0.1 or 0.2 units of C,. The resulting C’s give an excellent fit to the data. Miller-Erpenbeck Equation. Both the I and J equations are feasible for hand calculations with seven-place logarithins and a desk calculator, the fewest number of significant figures being required with the J integrals (Equations 22 and 23). The value of C is much less sensitive to errors in T I , T P ,or J’ when COis close to the proper value. Gaod preliminary estimates of CO are l / T c or 0.64/T0. The graphical C’s are all in good agreement with one another (to about 0.005 X and with the ones from the derivative method (to about 0.02 X but are not in as good agreement with the least square C’s, as was found in the Antoine case. From the two cases analyzed here, it seems that subtracting 0.04 X l o p 3from the graphical C will give a value within 0.01 X of C,. The poorer agreement of graphical and least square values for
Table 111.
__ Method
Least squares: P Least squares: log P Lua Extended derivativeb
a
72
-C 51.56 50.86 50.79 51.34 50.85 51.36 51.63
Derivatives from IK curue.
I&EC
Comparison of Graphical Methods with least Squares for Benzene Anloine Equation !lliller-Erpenbeck Equation Ti T? -Co Sp Method 103C TI T? 103 c0 .. . ... .. 0 . 0 0 7 Least squares: P 1.795 .., ... ... .,. , . . ., 0.178 Least squares: log P 1.825 , . . ... ... ... ... .. 0 . 1 9 4 Lua 1.842 ... ... ... Extended derivativeb 1 811 ... ... ,.. ... ... .. 0.105 0 283.8 354.1 0 0.180 1 . 8 3 4 2 8 3 . 8 354.1 1.76 1 . 8 3 8 2 8 2 . 7 354.0 284.0 3 5 4 . 0 43 0.103 1.83 1 . 8 4 1 280.0 354.3 292 3 5 3 . 5 51 0 , 0 9 8 MMJ
Deriuatiues from J K curue.
FUNDAMENTALS
sp 0.90 1.91 3.55 1.16 2.67 3.08 3.42
this equation apparently arises because the 6 plots are not fit as \vel1 by the Miller-El-penbeck equation as by the Antoine. Sumerical comparisons of all the graphical methods applied to hexadecafluoroheptane and benzene are given in Tables I1 and 111, respectively. Table I1 also shows the effect of different TI and ?'? on the value of C for different choices of Co. Over-All Assessment
Antoine Equation. T h e graphical integration method using J integrals gives excellent results. When Co is within 3 to 5 units of C: the graphical C is about 0.25y0 (0.1 to 0.2 units) from C,. T h e calculations are readily carried out with a desk calculator and seven place logarithms. T h e "extended derivative'' method is simpler and more rapid, but gives a slightly less accurate result, about 0.5% (0.1 to 0.4 units) from C,. T h e derivatives should be obtained from deviation functions. T h e Lu technique, which gives C to 1 to 5y0,is a little faster than the "extend'ed derivative" method, but is not always too accurate. Miller-Erpenbeck Equation. Because this equation doesn't fit the 6 curvcs well enough, the integral techniques
appear less useful. T h e .'extended derivative" scheme seems the best, everything considered, giving C within about 0.5% from C,, (not Cp). Acknowledgment
T h e author thanks Joseph Brady, Jr., and Bradley Johnston of the Computation Department for their aid \vith the calculations. Literature Cited
(1) Erpenbeck, J. J., Miller, D. G.. Ind. En,?. Cheni. 51, 329 (1959). (2) Forziati, A. F.: Camin, D. L.: Rossini, F. D.: J . Resrarch -Yatl. Bur. Standards 45, 406 (1950). (3) Fujita, H., Gosting, L.? J . Phys. Chim. 64, 1256 (1960). (4) Lu, B. C. Y . :Can. J . Chem. Eng. 38, 33 (1960). (5) Margenau, H., Murphy. G.. "Mathematics of Physics and Chemistry," 2nd ed., p. 473: Van Nostrand, Princeton, N. J.: 1956. (6) Oliver, G., Grissard. J.: J . Am. Chem. Soc. 73, 1688 (1951). (7) Thomson, G. \V., in "Techniques of Organic Chemistry," .4.Weissberger. ed.: 3rd ed., Vol. I, Pt. 1>p. 473. Interscience, New York, 1959. RECEIVED for review April 30. 1962 ACCEPTED November 15, 1962 Work performed under the auspices of the U . S. Atomic E n e r g Commission.
REDUCED STATE CORRELATION FOR T H E ENSKOG MODULUS OF SUBSTANCES OF SIMPLE MOLECULAR STRUCTURE G E D l M l N A S D A M A S I U S A N D G E O R G E T H O D O S The Technological Institute, .Yorthwestern Cniuersity. Evanston, Ill.
A reduced state correlation has been developed between the Enskog modulus, bpx, and reduced pressure and temperature for substances having critical compressibility factors approximately equal to z, = 0.291 from PVT data available in the literature for argon. From the value of the Enskog modulus at the critical point for argon, a generalized correlation has also been developed which can be used to estimate bpx values for substances having different compressibility factors.
c
interest has recently been exhibited in methods for the calculation of the transport properties of substances in their dense gaseous and liquid states. Several attempts have been made to calculate the viscosity, thermal conductivity, and self-diffusivity of pure substances at high pressures from the Enskog relationships ( 6 ) . These relationships. developed from considerations of kinetic theory, can be expressed as follows: ONSIDERABLE
rection factor accounting for the probability of collisions. These relationships require for the calculation of the transport properties accurate values of the Enskog modulus, b p x , over a complete range of temperatures and pressures. Therefore. in the present study a n attempt has been made to develop a reduced state correlation for the Enskog modulus of substances whose critical compressibility factor is approximately 0.291.
Development of Reduced State Correlation
T h e Enskog modulus is also present in the Enskog equation of state where h =
2
3
x
u3
; for rigid spherical molecules and
x
is a cor-
P
T + up2 Rxp (1 + b p x )
(4)
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