INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1948
C+ulations.
calorimeter, and three rate expressions: heat input, measured temperature rise, and temperature change at no-heat period. A diagrammatic sketch of the apparatus used for the system benzene-methanol is presented in p g u r e 1, and a sample calculation using data obtained with the device is given below.
The basic equation is
% ! d9
=
[WC,
-&I
dT + (C.E.)] [-& - dT,
3 - 60EZ - (60)(3.52)(1.16) = 58.58 tal* per nfin, 4.182
d9
Data. 80% benzene by volume. Desired temperature, 50' C. Weight of flask, stirrer, thermometer, etc., empty C* 9.5042 g r a m Initial weight of flask, etc., plus liquid C 99.7764 grams C 99.6352 grams Final weight of flask, etc., plus liquid 90.2016 grams Average weight of liquid Voltage across coil, 3.52 Amperage, 1.16
++ +
*C=
341
C.E.
=
4.182
11.42 from calibration with pure CeHsand pure CHsOH.
Rates, taken from data plotted in Figure 2. A. Initial radiation, etc., rate at 50" C. = -0.348 1 B. Final radiation, etc., rate at 50" C. = -0.336 J or average = 0.342 C. per min.
counterbalance.
O
0
Temperatures at 15-Second Intervals, C. 50.40 11 21 49.57 50.68 31 50.31 22 49.50 50.83b 32 12 60.23 23 49.44' 50.79 33 13 50.14 24 50.72 34 14 49.58 50.06 25 49.74 50.62 35 15 26 50.52 49.96 36 49.90 16 49.87 27 50.06 50.43 37 17 49.79 28 50.21 50.35 38 18 50.26 49.72 29 50.38 39 19 50.18 40 49.64 30 50.53 20 Current on. b Current off.
C. Average heating rate at 50' C. = 0.630 C. per min.
50.10 50.02 49.93 49.85 49.76 49.68 49.60 49.53 49.44 49.36
O
Therefore 58.58 = [90.2016 C , '
c,
+ 11.421[0.630 + 0.3421
= 48'85 = 0.642 cal. per
90.2016
O
C. at 50" C.
RRICEIVED February 25, 1947.
Thermodynamics of Nonelectrolyte Solutions d
~ 7 -RELATIONS t IN A BINARY SYSTEM OTTO REDLICH AND A. T. KISTER Shell Development Company, Emeryville, Calif. T h e development of specialized processes of distillation and extraction creates a demand for efficient methods of describing the thermodynamic properties of nonelectrolyte solutions. The present paper discusses the examination of experimental data for the composition of a binary solution and its vapor, as functions of the temperature at constant pressure.
T
HE papers of this series attempt to present various conclusions from the thermodynamics of solutions in a form which is most convenient for practical applications. VAPOR-LIQUID EQUILIBRIUM
The design of a distillation column is based, in general, on laboratory data which represent the mole fractions x of the liquid and y of the vapor in equilibrium as functions of the temperature t at a constant pressure P. The functions x and y determine only a single independent function. The known relation between x, y, and t can be used, therefore, for minimizing the influence of experimental errors, which are bound' to be considerable for low concentrations of either component. This problem is usually solved by means of introducing the activity coefficients
van Laar, or Scatchard. This procedure, however, introduces some uncertainty as to whether the deviations of the original values from the smoothed ones are due to experimental errors or t o the insufficientvalidity of the approximation formula. Furthermore, the use of derived function like the activity COefficient instead of the immediate experimental data for the process of smoothing is always disadvantageous, especially if the limits of the experimental error, as in the present case, vary considerably within one set. Any adjustment requires the comparison of the likelihood of different sets of deviations. I t is difficult to exert proper judgment with functions which are somewhat remote from the immediate experimental data. In addition, the problem is complicated by another condition-namely,
yp/xp;; YZ = (1 Y)P/(l X M where p;T and p i are the vapor pressures of the pure components, and by using Duhem's equation
Any set of y1 and yz is inconsistent if it does not satisfy this condition. In general, any reasonable representation of experimental data will satisfy the stability conditions
The considerable labor which would be required in this calculation is reduced t o a fairly satisfactory amount by the use of suitable approximation formulas like the equations of Margyles,
However, in the case of an interpolation or extrapolation over a wide range these conditions may reveal an inconsistency and thus prove useful by restricting the arbitrariness of the representation.
Y1 =
-
-
cli
4
INDUSTRIAL AND ENGINEERING CHEMISTRY
342
Vol. 40, No. 2
and the analogous formula for Pz. Here Vl denotes the molal volume of the first component in the liquid state, B1 the second virial coefficient, and El the relative partial molal heat content. An approximation of B1based on Berthelot’s equation for moderate pressures is
where P,, T C = critical pressure and temperature The relative partial molal heat contents are derived from the heat of mixing by the methods discussed by Lewis and Randall ( 5 ) . In principle they can be estimated also from the dat,afor the vapor-liquid equilibrium. The quantity log -02
0.0
-
I
‘
‘
I
I
0.5
I
I
(y;/yL) =
(E1
- z 2 ) ( 7 . - T)/2.3R1’T (7‘4)
can be calculated by means of Equations B and 7 . The second paper of this series (page 345) shows that
1.c
Figure 1. System n-Hexane-Benzene
mde fraotion of n-hexane. Circles indicate t h e experimental data of Tangberg and Johnson (8); broken line indicates caloulation of Beatty a n d Calingaert (2). x
+
log (n/n)
log ( r J r 2 ) d x The relations
What is actually needed for the technical purpose is the relation between z and y. The most diiect correlation method, therefore, will be that one which does not transform the experimental data to any other functions. For the reasons mentioned before this is also the most efficient one. In order to obtain a satisfactory method of deriving consistent final results, one has to express Duhem’s equation in terms of z and y. Condition 3 is then automatically satisfied. This method will be developed in the following sections. ASSUMPTIONS
Equation 1 represents a first approximation for low pressure, and Equation 2 is valid for constant pressure and temperature. They can be used immediately for data of moderate accuracy below atmospheric pressure over a moderate boiling range. If these conditions do not hold, corrections are to be introduced for two reasons. The quality RTln(zyl) represents the difference for the partial molal free energies of the first component in the 8OlUtiOn and in the pure state a t the same pressure P . This quantity is given by RTln(yP/p;) only if the perfect gas laws hold. Furthermore, Equation 1 furnishes the activity coefficients for the equilibrium temperature T . The use of Duhem’s equation requires a reduction of y1and y2 to a fixed temperature T. A second approximation can be based on the following assumptions: (a) The changes of volume accompanying the isothermal mixing of the liquid components and of the gaseous components are negligible. ( 6 ) The equations of state of the gaseous components can be represented in the form
V,
=
RT/P
+B
(3
where V , denotes the molal volume in the gaseous state, and the second virial coefficient B depends only on the temperature. ( c ) The heat of mixing does not appreciably depend on the temperature. The activity coefficients a t the pressure P and the arbitrarily chosen temperature Twill be represented by
The corrections required in the second approximation are thrown into the functions PI and Pa. Their relation with the vapor pressures p i and p i of the components is given by (S, 5, 7 ) :
P
1
log--’ = F j j R T [(Vl P: I
=
0
(7B)
Jo’l
- B d ( P - P 3 - LdT - n/n ( 7 )
I;1
=
-
b(1 - 2)2; Lz = bx2
(7C)
may aln-ays be taken as a first approximation for nonelectrolytes. The coefficient b can be considered to be independent of the Pemperature in the boiling interval. According to Equations 7-4 to 7C, the coefficient b can be estimated from
This calculation, however, is seldom justified. A finite value of the integral on the right-hand side of Equation 7D is usually to be interpreted as due t o esperimental errors. A different interpretation is leqitimate only for very accurate data covering a large boiling interval. DERIVATION OF REQUIRED RELATIOXS
Within the validity range of the chosen approximation, the activity coefficients -1 and y2 do not explicitly depend on t but only through x,so that (9)
Logarithmic differentiation of Equation 6 and substitution of the results into Equation 2 furnish, therefore,
or dt
1:-y
&=SG) \There the slope fact,ors is defined by s = 0.4343/[~ dlog PJdt j-(1 -
5)
dlog Pz/dt]
(12)
For low pressures the quantity RT”s represent’sthe ideal heat of vaporization of one mole of the solution. If condition 11 is satisfied, Duhem’s Equation 2 is also sittialied. The &ability conditions 4, together with 6, furnish only the well known condition that, the quantities 5 - y and dtldz are both either positive or zero or negat,ive. Equation 11 shows that d t l d y has t,hesame sign as x - y. The limiting values of d t / d y for z = 0 and z = 1, which are indeterminat,e in Equation 11, can be expressed by means of the relative volatility
343
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1948
TABLE I. LIMITING VALUES
O1
= y(1
- z)/z(l - y)
(13)
Introducing Equation 13 into Equation 11 one obtains
- t + 01~)(1 -
dtldy = ~ ( l
01)/01
(14)
A similar expression for dt/dx is obtained from Equation 11, and
L
I
\
i I 0.0
I
I
x, Y
Figure 2.
1.0
0.5
System n-Hexane-Benzene at 735 Mm. I
Mole fraction of n-hexane in liquid 2 (0)and vapor y ( 0 ) . Short 6traigbt lines reprenent calculated slopes (Equation 11); broken line. indicate calculated limiting slopes (Table I).
The result is
- .)f = 8[1
+41 -
5)
dlnu] dx
(16)
Since OL is positive and finite, and d&
dx
=
(
YIPl)
dx l n q 2
(17)
always finite, dt/dx is also always finite. From Equations 14 and 16 the limiting values listed in Table I are obtained. CORRELATION
The relations derived in the preceding section furnish the means to correlate (z,y,t) data directly, as outlined in the introductory section. The correlation method consists of the following steps: (a) The experimental data are plotted with 1: and y as abscissas and t as ordinate. ( b ) The functions P1 and Pe are calculated according to Equation 7. (c) The slope factors is found according to Equation 12, and the tangents of the (y,t) curve, given by Equation 11, are indicated in the diagram. (d) A smooth curve is drawn through the points representing t. ( a ) The two limiting slopes of dx/dt for z = 0 and t = 1 are taken from this curve, the corresponding values of dy/dt are calculated according to Table I, and the limiting tangents for y indicated in the diagram. (f)A smooth curve is drawn close to the points representing y. This curve must have the slope indicated by the previously drawn tangents. A few remarks will be useful in attaining efficient and quick operation. ( b ) The functions Pl and P, are required only if y1 and yz are to be derived from the final data. Otherwise only dlogPl/dt and dlogPz/dt need be calculated. The values of dlogp;/dt and dlogpildt are more conveniently derived from the usual vapor pressure formulas than from tabulated values of the vapor pres; sure. I n many cases the first approximation P, = p i and Pz = pz will be sufficient. If not, only a crude calculation of the correction term Equation 7 and its derivative is necessary for moderate pressures. ( c ) The slope factor s varies, in general, very little and can be interpolated with respect to t between a few points. If the boiling range is small, it is sufficient to calculate s, and correspondingly the derivatives of log PI and log Pz, for the boiling points of the pure components and, if present, the azeotropic point. (f) Frequently some readjustment of the (y, t) curve and also of the (2, t) curve will be necessary for two reasons. First, the change from the experimental values to those of the smobth curve entails a change in the prescribed slopes (Equation ll), especially through the factor z - y. Wherever this change is appreciable,new slopes should be drawn and the smooth curve changed. Second, the (x,t) curve fixes, according to Table I, certain limiting values
of 01, which are sensitive to experimental errors in x. Better values are obtained by taking smooth values of x and y from the curves, plotting log(yl/yz) = loga - log(P1/Pt) against x, and adjusting the curve in the ranges of low concentrations so that Equation 7B is satisfied. The limiting values of log(yJy8) furnish the limiting values of 01 and, according to Table I, also the limiting slopes. The final (g, t ) curve should strictly satisfy the prescribed slope conditions, even at the sacrifice of close approach to the experimental points. Curves which violate these conditions are inherently inconsistent. Values of y1and yz are calculated from the values of t and y of the smooth curves. The thermodynamic consistency of these values is ensured by the smoothing method. The whole procedure can be carried out quite rapidly after a few examples have been calculated. The method has been found t o be considerably more sensitive and convenient than the usual check of (z,y,t) data by means of Duhem's equation (a). EXAMPLES
Tables 11-IV and Figures 1 and 2 illustrate the correlation for the system n-hexane (I)-benzene ( 9 ) at 735 mm. The molal volumes V1 and Vz of the components are taken from the International Critical Tables, the virial coefficients B1 and Bz are calculated according to Equation 8, and the vapor pressures and their derivatives are found from Antoine formulas (I). The terms log (Pl/p;) and log ( P 2 / p i ) ,which are the corrections for the imperfectionaf the vapor, are given in Table 11.
TABLE 11. C.4LCULATION O F Pl/py AND Pz/p," (EQUATIONS 7 AND 8) FOR SYSTEM n-HEXANE-BEKZENE AT 735 MIX. 67.7 140.2 1225 735 0 94.4 1039 509.9 0,0052 0.1588
t
v1
-
BA
PI log ( P I / P 3 V2
-
B1
I)*
io'p (P*/&) log ( P ; / P ; )
74.0 141.7 -1177 894 -0.0042 95.3 -999 627.4 0,0024 0,1537
79.0 142.9 -1141 1038 -0.0077 95.9 -969 735 0 0,1499
TABLE 111. CALCULATION OF s (EQUATION 12) FOR SYSTEM n-HEXANE-BENZENE 0 79.0
5
t
d d d d 8
loa I):/& log P y / d t log pg/dt log Pz/dt
..... .....
0.01347 0.01297 33.48
0.5 69.2 0.01359 0.01291 0.01440 0.01394 32.36
0.763 67.7 0.01373 0.01308 0.01459 0.01417 32.56
1
67.7 0.01373 0.01308
..... .....
33.20
INDUSTRIAL AND ENGINEERING CHEMISTRY
344
TABLEIV. CALCULATION O F d t l d y t
78.4 0.004 0.009 0,561
,2,
- u(1 - Y) 2--Y
33.4 18.7 0.354 0.212
8
-dt/dg log a
log
(YI/Yd
77.7 0,025 0.000 0.621 33.3 20.7 0.396 0.253
AND y ~ / " / s(EQUATION 11) FROH BENZENE AT
DATAO F TOXGBERG AND 735 Mni. .
77.3 0,043 0,083 0.526 33.3 17.5 0.304
69.8 0.430 0,545 0.464 32.4 15.1 0,201 +0.049
0.160
76.0 0.075 0,153 0.602 33.1 20.0 0.34s 0.203
73.8 0.215 0.333 0.536 32.8 17.6 0.261 0,112
67.8 0.713 0,753 0.215 32.5 7.0 0,089 -0,065
Vol. 40, No. 2
JOHXSON
67.7 0.763 0.790 0,199 32.6 6.5 0.067 -0,086
(8) FOR SYSTEM TL-HEXAXE67.5 0.889 0.893 0.047 32.9 1.5 0.018 -0,136
67.4 0.965 0.966 0.031 33 . O 1.0 0.013 -0,141
67.3 0.99 0.99 0.000
.....
0.0 0.000 -0.154
curve and the y curve leads to the boiling temperatures 78.75 arid 67.4' C. for benzene and hexane, respectively, compared with the values 79.02" and 67.68' calculated for 736 mm. from t'he literature (1). Apparently all temperature measurements are about 0.26" C. too low. The effect of this discrepancy on thc present calculation is entirely negligible, although it would affcct, the values of y1 and yZ by about 170. The curve in Figure 1 represents the equat,ion
I:
log ( y i / y ~ )= 0.185(1 - 22)
0.6
XI
I
Y
! 0.8
0.7
Figure 3.
I 0.9
1.0
System Acetone-Water
Mole fraction of acetone i n liquid x (0 Brunjes and Bogart, 0 Othmer and 0 ) . Straight linea indicate and Benenati) and vapor calculated slopes (Equation 11).
(6
The differences between the values of log ( P l / p ; ) for the three temperatures in Table I1 furnish a sufficiently accurate estimate of dlog(Pl/p;)dt. This quantity added to dlogp;/dl furnishes dlogPl/dt in Table 111,in which z and y represent the mole fraction of n-hexane. The analogous calculation is carried out for the second component. The slope factor s is compdted according t o Equation 12. Table IV contains in the first three lines the data of Tongberg and Johnson (8). The slope factor s is graphically interpolated from Table 111,and &/dy is calculated according to Equation 11. Equation 13 furnishes log CY. The values of log (PllPZ)are interpolated from the figures of Table I1 and subtracted from log CY to give log (mlrd Figure 1 shows the values of log ( n / y J against x. The values extrapolated for x = 0 and x = 1 are used for the calculation of the limiting slopes d t / d x and d t l d y in Table V.
TABLE v. LIMITINGS L O P E S (TABLE I) FOR SYSTEM n-HEXANE-BENZENE 2
0
log ( - / l l Y l ) log (Pl/PZ)
0.203 0.143 0,346 33.5 -40.8 -18.4
log a 8
df/dz Wdzl
1 -0,167 0,154 -0.013 33.2 1.0 1.0
The experimental (2, y, t ) values are plotted in Figure 2. The short straight lines indicate the slopes found in Table IV. The limiting slopes (broken lines) are taken from Table F'. Figure 2 shows that a curve can be d r m n very close t o the y points which satisfies the slope condition. The calculated slopes for y = 0.545, 0.753, and 0.790 are slightly steeper than that of a smooth curve; but, according to Equation 11, a very small decrease of the values of y, well within reasonable experimental errors, will produce perfect agreement. The extrapolation of the
- 0.018[ - 1 + 6 ~ ( 1 - ~ ) ]
Condition (7B) indicates that the data cannot be represented without a small systemat,ic deviation, JThich, however, is of the order of magnihde of the accidental errors. The representation of the same data by Beat,ty and Calingacrt (1;broken line in Figure 1) is consistent but does not sufficiently well eliminate the influence of accidenial errors by properly smoothing t,heresulk. As a second example, Figure 3 s h o w part of the (s,y,t) diagram of acetone (l)-water (2), according to the data of Brunjes and Bogart (4)and of Othmer and Berienati ( 6 ) . The data of York and Holmes (9) are not included because of considerable scattering. A preliminary curve is drawn through the y points, and the slope, calculated by means of Equation 11, is indicated for y = 0.80, 0.85, and 0.90. The concordant, results of the two sets of measurements closely define a y curve, but this curve does not satisfy the calculated slopes. The differences, although not great, appreciably exceed the magnit,ude of the accidental errors. The conclusion is inevitable that both sets, in spite of the close agrcement, are affected by a significant systematic error. INCOMPLETE DATA
Equation 11 can be used for calculating z if only the relation bet'ween y and t is known. This equation furnishes
Since s varies only little, it can be approximated by linear interpolation with respect to t between the boiling temperatures of the pure components. If the system is azeotropic, s is interpolated between t,he boiling points and the azeotropic point, for which x = y. With the results for z obtained by means of Equation 18, the values of s are checked, and the Calculation, if necessary, is repeated. In order to obt'ain y from (2, 1 ) data, one calculates CY by means of Equation 16. For z = 0 and z = 1, the values of CY are determined from ds/dt according t,o Table I. For an azeotropic point, one has LY = 1. For a first approximation, dlna/dx is linearly interpolated between these points. The results for CY are used for a second approximation of dlnm/dx, and so on. -4 procedure like this is sensitive t o experimenta,l errors and will he u s d only as a subaidiary method if complete data are lacking. NOMESCLATURE
mole Elactions of the first component in the liquid x, 1~ and vapor, respectively P = total pressure p ; , p i = vapor pressuies of the pure components P I , Pp = functions defincd in Equations 6 and 7 =
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
February 1948
(3) Benedict,
y,, 7 2 = activity coefficients
345
M.,Johnson, C. A,, Solomon, E., and Rubin, C. C.,
Trans. Am. Inst. Chem. Engrs., 41,371 (1945).
Vl, Vz = molal volumes of the liquid components V , = molal volume in the gaseous state B = second virial coefficient Po, To = critical pressure and temperature
T
= an arbitrarily fixed temperature s = slope factor, Equations 11 and 12 (Y = relative volatility LITERATURE CITED
(1) American Petroleum Institute Research Project 44, National Bureau of Standards, “Selected Values of Properties of Hydrocarbons,” Tables 2k (March 31,1944) and 5k (June 30, 1944). (2) Beatty, H.A,,and Calingaert, G., IND. ENG.CHEH.,26,504,904 (1934).
(4) Brunjes, A. S.,and Bogart, M. J. P., IND. ENG.CHEM.,35,255 (1943). (5) Lewis, G.N., and Randall, M., “Thermodynamics,” New York, McGraw-Hill Book Co., 1923. (6) Othmer, D. F., and Benenati, R. F., IND.ENQ.CHEM.,37, 299 (1945). (7) Scatchard, G.,and Raymond, C. L., J . Am. Chem. SOC.,60,1278. (1938). (8) Tongberg, C. O., and Johnson, F., IND.ENG.CHEM.,25, 733 (1933). (9) York, R., and Holmes, R. C., Ibid., 34,345 (1942). RECEJVED November 26,1946.
(Thermodynamics of Nonelectrolyte Solutions)
ALGEBRAIC REPRESENTATION OF THERMODYNAMIC PROPERTIES AND THE CLASSIFICATION OF SOLUTIONS OTTO REDLICH
AND
A. T. KISTER
Shell Development Company, Emeryville, Calif. T h e utilization of laboratory data for the design of distillation columns and other separation equipment requires the efficient representation of extensive experimental data. A flexible, nonarbitrary, and convenient method is developedfor systems of two or more components. This method furnishes an immediate distinction between various types of solutions.
E
XPERIMENTAL results can be improved or damaged on their way from the laboratory to the practical application. in plant design and operation. The treatment of experimental data should eliminate inconsistencies without distorting the results by imposing arbitrary conditions, it should be flexible enough to cover all important cases, and it should be pimple in operation, From this viewpoint the following method was developed for the representation of thermodynamic properties of nonelectrolyte solutions. The present discussion starts from binary solutions and is later extended to systems of more components.
a = y(1
- z ) / x ( l - y)
(4)
for log a = log ( Y l l r n ) -I- log (PYP,”)
(5)
where g is the mole fraction in the vapor and p i and p i are the vapor pressures of the pure components, the vapor being 118sumed to be perfect. Furthermore, the function log (-y~/r~) provides an efficient tool for eliminating inconsistencies in the experimental data. Since &, according to Equation 1, assumes the value zero for x = 0 and x = 1, we derive from Equation 2
SELECTION OF A USEFUL FUNCTION
The use of the activity coefficients y1 and y 2 is not advisable since they are redundant. The use of two functions necessitates the imposition of a condition-namely, Duhem’s equation. This complicates any correct smoothing procedure. For this reaeon Scatchard’s “excess free energy” is preferable. Dividing this function by 2.303RT, one obtains
Q
= x log YI
+ (1 -
Z) log
-i2
(1)
(z mole fraction of the first component) , which y a y be considered a little more convenient in some numerical calculations. * For various reasons still ’more suitable is the function dQ/dx = log
(2)
( Y I ~ Y ~
The degree of this function is one unit lower than Q as well as the functions log 71 = Q
+ (1 - s)dQ/dx
and log y~ = Q
- xdQ/dx
(3)
This is a considerable advantage in practical calculations, realized in a special case already by Benedict et al. (1). Another advantage of the function log (y1/y2) is its simple relation to the experimental data and to the technically important relative volatility :
Figure 1 shows a simple example for the application of this condition. The only curve for log (rl/r3which represents the data in accordance with Equation 6 is the iero line-that is, the system is perfect within the limits of experimental error. The deviations for low concentrations of either component are safely recognized t o be due to experimental errors. One could hardly arrive so quickly and cogently at the same result by another method. The relation
(7) following from Equations 5 and 6 is sometimes useful. If the values of .(Y r:fer to a constant temperature, the integral is equal to log ( p l / p z ) . If (Y is derived from equilibrium measurements over a moderate temperature interval, the integral can be easily estimated since usually varies only little with the temperature. Equation 7 is based only on the assumptions that the vapor is perfect and that the dependence of the activity coefficients on the temperature may be neglected.
