HEATSOF MIXIKGOF XON-ELECTROLYTE SOLUTIOKS
Feb., 1961
275
HEATS OF &IIXlNG OF SON-ELECTROLYTE SOLUTIONS. 11. PERFLUORO-n-HEPTANE + ISOOCTANE AND PERFLUORO-n-HEXANE n-HEXANE')2
+
B Y A. G. WILLIAMSON3AND R. L. SCOTT Department of Chemistry, University qf California, Los Angeles, California Received June 20, 1960
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Heats of mixing have been measured for the systems perfluoro-n-hegtane isooctane (2,2,4-trimethylpentane) a t 30 and 50" and for the system perfluoro-n-hexane n-hexane at 25 and 35 . The results are correlated with the excess Gibbs' free energies of mixing measured by other workere.
+
Introduction Almost the only directly measured heats of mixing for fluorocarbon hydrocarbon systems are three experiments by Mueller and Lewis4 and two very recent' measurements by Dyke, Rowlinson and T h a ~ k e r . ~ Most of the values quoted in the literature for hea,ts of mixing and for entropies of mixing of these systems are derived from the temperature variation of the Gibbs' free energies of mixing, a notoriously inaccurate method. Since the measured heats of mixing of the system perisooct'ane are not in agreement fluoro-n-heptane with the values (obtained from the free energy of mixinge,' we have begun our study with this system.
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rest; this is probably due to a small amount of phase separation induced by the instantaneous cooling of the mixture at the moment of mixing. The heat of mixing is sufficient to cool the mixture from 25' to below the critical solution temperature (22.65') X C ~ H=~ 0.63).9 ~ Unfortunately the stirring in the calorimeter was not sufficiently vigorous to cause rapid mixing of the two layers once they were formed so that it became very difficult to obtain good measurements in this region.
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TABLE I HEATSOF MIXINGOF n-C,H16 Exper.
X*-C8H18
Experimental Apparatus.-The design and testing of the calorimeter and auxiliary apparatus are described in a previous paper! Materials.--A sample of perfluoro-n-heptane purified in this department, by Dr. A. P. Watson was used. This had a refractive index 1.2588) slightly higher than the values previously reported6 for highly purified perfluoro-nheptane (n% 1.2582). However, only one peak was obtained from a, gas chromatographic analysis a t 26" using a 2.5 m. column with C21F44 as the stationary phase. The column had been shown t o resolve readily mixtures of fully and partially fluorinated compounds. The sample of CTF16 may contain small quantities of branched chain perfluoroheptanes which cannot be resolved by our present gas chromatographic equipment. This should not seriously affect the values of the heats of mixing. Eastman "'spectra"-grade isooctane (2,2,4-trimethylpentane) was used from a freshly opened bottle. The perfluoro-n-hexane and n-hexane were supplied by Professor R. D. Dunlap. Their purification and properties are described elsewhere.g*10
+ i-CsH18
A8/cal.
T
Ca1cd.a
Calod. Q
0.1515 281 278 279 .2483 393 397 396 .3393 466 467 466 ,4327 507 500 502 .5113 501 504 508 ,5356 498 501 506 ,5865 493 49 1 496 .7458 412 412 415 .8066 357 357 358 T = 50" 0 2447 391 392 393 .3529 476 475 473 .5065 512 514 508 .6235 492 49 1 484 .7360 425 425 422 Four coefficients were used in equation 1; h,, h1, h z , ha: b 14 obsn. assumed indep. a 9 obsn. a t 30°, 5 obsn. a t 50". of temperature. =
30"
Results The experimental results have been represented The experimental results are shown in Tables I by equations of the type and 11. It can be seen from the data that for both A R ~ (-lz)[ho hi(1 - 22) h,(l - 2 ~ ) ~. I systems the variation of the heat of mixing with (1) temperature is very small. The values in Table I1 where x is the mole fraction of the hydrocarbon with a superscript are lower in comparison with the (second component). All the equations were ob(1) This work is supported by the U. S. Atomic Energy Commission tained by the method of least squares on the under Project 13 of Con1,ract AT(ll-1)-34 with the University of CaliWestern Data Processing Center IBM 709 comfornia. puter using a program devised by Mr. D. B. Myers (2) Presented a t the 135th National Meeting of the American of this department. The number of constants used Chemical Society, Boeton, Mass., April 9, 1959. (3) Department of CEemist,ry, University of Otago, Duncdin, New (four) is the smallest number for which the deviaZealand. tions between the experimental results and those (1) C. R. Mueller and J. E. Lewis, J. Chem. Phys., 85, 1166 (1956). given by the equation are of about the same (5) D. E. L. Dyke, J. 9. Rowlinson and R. Thacker, Trans. Faraday magnitude as the estimated experimental uncerSOC.,56, 903 (1959). (6) C. R. hluellerand J. E. Lewis, J . Chem. Phys.. 26, 278 (1957). tainties. The constants for equation 1 are shown (7) A. G. Williamson, R. L. Scott and R. D. Dunlap, ibid., 80, 325 in Table I11 along with values of rhnand values of (1959). U A H given by the relation ( 8 ) A. G . Williamson and R. L. Scott, THISJOGRNAL, 64,440(1960).
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(9) R. D. Dunlap, C. 6. Murphy and R. G. Bedford, J . Am. Chem. SOC.,80, 83 (1958). (10) R. G. Bedford and R. D. Dunlap, ibid., 80, 282 (1958).
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A. G. WILLIAMSON AND R. L. SCOTT
276
TABLE I1 HEATSOF hfIXING OF n-CsFla
+ n-C&
AH/cal. 2C6H14
E
~ Calcd. ~ Ir ~ Ca1cd.c ~ Ca1od.d .
5" = 25"
0.2897 436 437 441 439 .4867 519 517 513 516 ,5975" 483 489 492 .. 463 .. 455 .6080(2S")o 462 460 .. 452 .G729" 453 ,7129" 426 437 .. 429 372 377 368 .io55 367 278 281 T = 35" 0.1518 280 282 365 365 366 .2149 370 437 434 434 430 .2841 493 491 490 ,3753 496 513 516 515 .4856 516 508 514 514 .5359 508 483 493 494 ,6227 494 ,7538 424 408 416 419 314 315 318 .a479 317 .8976 239 241 240 237 0 Suspected phase separation. Four coefficients used in equation 1; ho, hl, h 2 , . h . * 7 obsn. at 25O, 10 obsn. a t 35". 17 obsn., assumed indep. of temperature. 13 obsn. (omitting four data where phase separation suspected j, assumed indeD. of temDerature.
where the di's are the individual deviations and m and n are the number of experimental measurements a i d the number of terms in the equation, respectively. Also shown in Table I11 are the values of u A r i for the equation using one less term and one more than the number listed. The apparently very small temperature coefficient of AH suggests that me might reasonably regard AH as independent of temperature. The third set of constants for each system in Table I11 was obtained by combining the results for the two temperatures into a-single curve. Tables I and I1 show the values of AH calculated from the individual equations and from the equations for the combined results. The column labelled calcd. (a) was obtained from the equation for the particular temperature and that labelled calcd. (b) was obtained from the equation for the combined results for the two tempera tures.
Discussion Correlation of Heats and Gibbs Free Energies of Mixing.-The simplest test of consistency between the free energy and heat of mixing over a range of temperatures is to compare the measured heat of mixing with the value derived from the temperature coefficient of the free energy of mixing. This widely used approach is, however, not a very good one; an uncertainty of 2 cal. mole-' in the values of GE over a 20" range would lead to an unceJtainty of about 40 cal. mole-' in the estimate of A Z . A better method of correlating GE and A B is via the relation (3)
which on integration gives (4)
Thus, if the values of ARare first fitted by an analy-
ltol. 65
tical expression such as equation 1 we may use this expression and the values of GE/T a t any temperature T to calculate a set of values of GE'/T' at some arbitrary reference temperature T'. If the expression used to represenkthe values of AI? is correct and all the values of GE and AH are consistent then the values of oE'/T' will lie on a single smooth curve. For the two systems considered here this procedure can be further simplified. If the heat of mixing is independent of temperature, so also is the entropy of mixing. Copequently we can calculate the excess entropy S*(T) for each of the temperatures from the equation dE(T) = [ A g
If
- GE(T)]/T
(5)
oEis expressed as a power series with coefficients
gn analogous to the coefficients hn in equation 1, we have for each temperature T @ = z(1 - x)[go Ol(1 - 2,) 82(1 - 2,)2 ., .I
+
+
+
,
/\ "&/ I
Equation 6 be With equation to yield coefficients Sn of the allalogoUS excess entropy series sn(T) = ih,
- gn(T)I/Y'
(7)
The coefficients sn should be independent of temperature if the hn's are, so we have a test for consistency. If these are reasonably constant we may average them (weighting differently if appropriate) and substitute back into equation 7 to obtain improved values of the gnls for each temperature. This treatment has the virtue of representing all the experimental data in a thermodynamically consistent manner. As a further test of the quality of the data we have used the free energy equation to predict the consolute temperatures for the two systems by solving the equations b2AG -a22 -
baAd = 0 - 0) -
ax 3
and comparing the values of Tc and xc obtained with the observed results. n-C7FI6 i-CsHls.-In analyzing the rcsults for this system we have recalculated the excess Gibbs free energies of mixing from the experimental vapor pressure data of hlueller and Lewis6 a t 30, 50 and 70". Since these authors give reasons for distrusting the vapor compositions we have used only the total pressure and liquid composition from which equations for the excess Gibbs' free energy are obtained using the method described by Barker.l' We have also used rather different estimates of the second virial coefficients of the vapors from those used by Mueller and Lewis.6 The second virial coefficients shown in Table I V were obtained in the following way. The available data for perfl~oro-n-alkanes'"-'~ were plotted yielding second virial coefficients as a function of temperature. Assuming that these com-
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(11) J. 4. Barker, Austral. J . Chern., 6 , 207 (18i3). (12) E. 1,. Pare and J. G. Aston, J . Am. ChPm. Soc., 7 0 , 5663 (1948). (13) IC. E. RlacCormack and W. G. Schneider, J . Chern. P h y s . , 19,
845 (1951). (14) C . Booth, this department, unpublished value of -3110 for n - C I F n at 5 ~ ~ .
c111.a
HEATSOF MIXINGOF NON-ELECTROLYTE SOLUTIOXS
Feb., 1961
277
TABLE I11 FROM LEAST S Q U A R E S FITS
hn f Uhn V A L U E S
(ABin calories)
System: n-CiHls 30' (a)
n=O 1 2 3
2019.6 69.3 536.9 -480.8
oh113 4 5 System: n-CaH14
25O (c)
n=O 1 2 3 uAR3 4 5
9 48 50 168 6.7 4.3 4.6
+ i-CsHls50' 2055.4 -14.5 424.3 -302.0
+ n-C&r 350 (e)
(a)
30'
9 48 64 236 3.4 2.9
+ 50'
(b)
2031.9 21 5 507.1 -368.3
8 42 51 151 6.0 5.1 5.3
..
25'
+ 35'
25O
(d)
+ 35O-suspected
phase
sep. data (e)
2063.5 261.5 245.1 -1285.0
19 2062.9 10 2049.2 14 2065.1 10 79 3 0 46 138.6 54 19.8 49 146 553.1 47 507.3 71 510.5 51 401 -410.7 121 -694.6 170 -421.1 139 9.9 7.5 11.5 7.5 5.5 4.8 7.9 5 5 6.1 4.6 8.1 5.9 u A 8 a = std. dev. of A R using coefficients given here; uAB3 = std. dev. using only three coefficients (ho, hl, h ~in) equation l ) ; uAH6 = std. dev. using h,. h,, . . ., ha. In none of these systems ie the uae of 5 coefficients warranted. Three are probably sufficient in eome; the use of four in this paper is based on a preference for consistency.
pounds form a family similar to similar to that observed by McGlashan and Potter for the n-alkanes15 the curve for perfluoro-n-heptane was estimated and hence values a t 30 and 70" were obtained. For isooctane, values of the second Tirial coefficients were obtained a t 25 and 99.24' from the calorimetric latent heat of vapoyization and the vapor pressure using the Clapeyron-Clausius equation in the form
The required data, were obtained from the tables of API research project 44.16 The two values of the second virial coefficient thus obtained were used t o calculate values of V , and T, in the equation" B / v 0 = 0.447[3.375 - 2.375 exp (0.936TC/T)] (10)
which was then used as an interpolation formula to give the values a t the required temperatures. The estimated uncertainty of the values shown in Table I V is about 3-4 X 102 cm3. mole-'. TABLE IV SECOND VIRIAL COEFFICIENTS (B) OF n-C,FI6 AND i-CsHls 1, "C. 30 50 70 i-C8H18Blcm.3 mole -1 -2350 -1470 -3180 n-C;F,, B / ~ r n mole . ~ -1 -3900 -3100 -2500
It was assumed that BIZ
=
+ BZZ)
(11)
where B12is the contribution to the second virial coefficient from interactions between unlike molecules. This assumption, while almost certainly incorrect, will not cause an error-of more than 2-3 cal. mole-' in the cbalculation of GE. (15) M. L. McGlashan and D. J. B. Potter, "Proc. of Conference on Thermodynamic and Transport Propertiea of Fluids," Institution of Mechanical Engineers and International Union of Pure and Applied Chemistry, London. 1957. (16) American Petroleum Institute Research Proiect 44, Selected Values of Physical and Thermodynamic Properties of Hydrocarbona and Related Compounds, Tables 3K and 3M, (Pittsburgh, Carnegie Press, 1953). (17) E. A. Guggenheim, "Mixtures." Oxford University Preaa, 1952.
The actual calculations of the gn's were made on the IBM-709 of the Western Data Processing Center at U.C.L.A. using a program developed by one of us (R.L.S.) as part of the Petroleum Research Fund project (PRF-366A). This program, using essentially the least squares procedure first proposed by Barker" but with modifications and generalizations, processes total vapor pressure data and yields sets of gn (for varying numbers of coefficients) together with standard deviations. The computer results show clearly that the vapor pressure data are not precise enough to justify more than three coefficients, so in what follows we restrict ourselves to three. (A consideration of column b of Table I11 shows that the heat of mixing data barely justify four.) Table V shows the raw coefficients g n for the three temperature and the entropy coefficients Sn derived from these and the three parameter hn'S (ho = 2033 cal., hl = -70 cal., hz = 492 cal.). The standard deviations for the g,'s are approximately 6, 10 and 20 cal. for go, g,, and g2, respectively, at all three temperatures. TABLE V THERMODYNAMIC C O E F F I C I E N T S FOR CeH, t, "C. 30 50 gdcal. 1302 (1308) g h l . --112(-116) gzlcal. 173 (171) s,/cal. deg.-l 2.41 sl/cal. deg.-l 0.14 sl/cal. deg. -1 1.05
T H E SYS'IEM
C7Fit1
+
70 1204 (1806) -123(-121) 111 (128) 2 41 0.15
1250 (1254) -116(-118) 164 (149) 2.42 0.14 1.01
1.11
The entropy coefficients sn are certainly constant within the experimental uncertainties, so we average these and write for the excess free energy GE
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GE/(cal.) = r(1 - z)[(2033 - 2.41T/(deg.)) (-70 - O.lBT/(deg))(l - 2 2 ) (492 l.OGT/(deg.))(l
-
-
(12)
The smoothed values of gn for the three experi-
A. G. WILLIAMSON AND R. L. SCOTT
278
T'ol. 65
551 cal.) to obtain the entropy coefficients shown in Table VI. TABLE VI THERMODYNAMIC COEFFICIENTS FOR THE SYSTEM n-C6FI( n-C& (yn's from Dunlap, et d . 9 1, "C. gdcal. gdcal.
deal. so/cal.deg.-l sl/cal. deg. -l se/cal. deg. -l
0
0.2
0.4
Fig. 1.-Excess
+
functions for C,FI~ CsH,, at 50'.
mental temperatures are shown in Table V (the italicized values in parentheses.) The agreement is excellent for go and g1 and good even for g2 considering that the standard deviation of the experimental g2 is approximately 20 cal. In Fig. 1 are shown the smooth curves for BE, and TgEat the middle temperature, 50". From equation 12 one calculates a critical solution temperature T, = 304°K. (31") and a critical composition x, = 0.67. These are only in fair agreement with the observed values,'* T , = 296.9 (23.7') and xc = 0.62. It should be noted that equation 12 predicts partial miscibility at 30°, while the experimental fact is that the two liquids are miscible in all proportions. Better agreement is hardly to be expected for several reasons: the experimental data are subject to considerable uncertainty, and the equations for the critical constants are very sensitive to small uncertainties in the coefficients gn. Moreover while the correct theoretical form for the thermodynamic functions in the critical region is unknown, the flatness of the coexistence curve and the flatness of the chemical potential p as a function of 2 in the vicinity of T, require a more complex expression than equation 12. Probably a requirement that the excess free energy equation fit the coexistence curve (or a t least the point xc, Tc)would be a valuable addition to a data processing program, and we hope to develop such a program ultimately. n-C6F14 n-Ca14.-The partial vapor pressures hexane system were of the perfluorohexane measured by Dunlap, et u Z . , ~ ~at 25, 35 and 45". From the vapor pressure data and measured vapor compositions, they calculated excess free energies (using the appropriate corrections for gas imperfection) checked consistency with the Gibbs-Duhem relation, and fitted the resulting values to an excess free energy power series by the method of least squares. They concluded that only three parameters were justified, so we have combined their gn's with our three parameter temperature-independent hn's (ho = 2064 cal., hl = - 114 cal., hz =
eE
+
25 35 45 1310 (1301) 1271 (1275) 1245 (1250) -200 (-197) -200 (-200) -199 (-203) 15A (143) 172 (1a.9) 50 ( 1 1 5 ) 2.53 2.57 2.57 0.29 0.28 0.27 I .32 1.23 1.57
0.8
0.6
XC~HIP.
+
+
(18) J. H. Hildebrand, B. B. Fisher and H. A. Benesi, J . Am. Chem. Soc., 72, 4348 (1950).
(19) R. D. Dunlap. R. G . Bedford, J. C. Woodbrey and S. D. Furrow, i b i d . , 81, 2927 (1959).
If, as before we average the s,'s for the three temperatures, we may write for the excess free energy
+ +
GE(cal.) = 1 ( 1 - r)/(2064 - 2.56T/(deg )) (-114 - 0.28T/(dep.j)(l - 2z) (551 - 1.37T/(deg.))(l - 2zjz] (13)
The smoothed free energy coefficients gn calculated from equation 13 are shown in italics in Table VI; the agreement is good except for the coefficients g2 and s2 which vary erratically. It should be noted that the separation of heat and entropy obtained above with the aid of our calorimetric measurements differs markedly from that which Dunlap and his co-workers calculated from the temperature dependence of the vapor pressure. Our coefficients h, differ from theirs by several times their estimated error. We have also processed the total vapor pressure data of Dunlap, et al., using our Barker method computer program. In this, the data on vapor composition ape not used, but for consistency me used the same reasonable estimates of virial coefficients which Dunlap used. Surprisingly the results of these computations indicate that the total vapor pressure data clearly justify four parameters. (The standard deviation of the vapor pressure is reduced from about 3 t o 1 mm. in going from three to four parameters; no significant further improvement is obtained by adding a fifth). In Table VII, we show these new coefficients and the entropy coefficients obtained by combining these with the kn's of Table 111, column e. From Table VI1 the agreement is significantly better, and as before we obtain an equation for the excess free energy
+
-
WE(cal.) = s(l - r)[(2065 2.59T/(deg )) (20 - 0.55T/(deg.))(l - 22) (511 - 1.12T/(deg.))(l 2 ~ )(-4.21 ~ 1.08T/(deg.))(l - 2 ~ ) ~(14) ]
-
+
+
+
From equation 14 we estimate a critical solution temperature T, = 297°K. (24") in very good agreement with the observed value of 295.8'K. (22.7"). This critical point however is at a critical composition xc = 0.73 which agrees less well with the observed value of 0.63. If we use equation 13 instead we derive a very unsatisfactory critical solution temperature T, = 317 (44") at x, = 0.67. If we use the free energy equations of Dunlap and coworkers directly (Le., the unsmoothed gn's of Table
Feh., 1961
HEATS O F LfIXISG
O F ?;OX-ELECTROLY TE
SOLUTIONS
279
VI) we find that these predict partial miscibility at all three temperatures, including 45". TABLE VI1 THERMODYNAMIC COEFFICIENTS FOR THE SYSTEM n-CZ1, n-CsHl, (g,'s from Barker analysis of data of Dunlap, et al.l9) t , "C. 25 35 45 dcal. 1298 (1293) 1265 (1267) 1234 (1241) deal. -136(--144) -151 (-149) -162 (-266) 92 /cal . 162 (177) I75 (166) 159 (166) gdcal. -125 (-99) -72 (-88) -67 (-77) so/cal. deg.-1 2.57 2.60 2.61 st/cal. deg.0.52 0.56 0.57 sn/caI. deg.-l 1.17 1.09 1.11 s3/cal. deg.-l - 0.99 -1.13 -1.11
+
Direct comparison of the coefficients of equations 13 and 14 is difhult because gl and g3 in equation 14 do the work of gl alone in equation 13. The Barker three coefficient treatment yields still another set of gn's and we have obtained therefrom the following equation for the excess free energy
0
0.2
Fig. 2.-Excess
0.4 0.6 0.8 zcfl14. functions or C ~ F I ~C8Hll at 35".
+
+
methylcyclohexane methylcyclohexane it is 17O or more depending on the choice of slope for the temperature variation of go. These discrepancies possibly arise from the inaccuracies in the experimental data as the concave dE/(cal.) = s(l - 2)[2064 - 2.60T/(deg.) + sections of the plots of AG vs. x are only 2-3 cal. (-114 - 0.19T/(deg.))(l - 22) mole-' "deep." They may also result from the (551 - l.IGT/(deg.))(l - 2 ~ ) ~(15) ] form of the function used to fit the data. In particEquation 15 differs from equation 13 principally ular, if the experimental data of Dunlap, et al., n-Ci"r at 25" are plotted as AG us. in the smaller values of s1 and sz. Equation 15 is well as &E VS. x it can be seen that although the only slightly better than equation 13 for evaluating J:forasn-C6F14 critical constants, yielding T, = 309°K. and xc = empirical equation does in fact fit the experimental points with u = f 2 cal. mole-', the plot of AG 0.71. suggests that one measurement around X C ~ H ,=~ 0.9 Figure 2 shows the excess functions for C6F14 C6HI4at 35" obtained from equations 13 and 14. which appears to be less accurate than the other It should be noted that neither of these equations points could be the cause of a spurious concavity in gives the minimum in the excess entropy reported the least squares curve. This apparent discrepancy by Dunlap, et at. Their reported s2 = 5.3 is much is not obvious from the plot of GE us. x. In this too large (a minimum at 2 = 1/2 occurs when s2> case it appears that the false pr_ediction of T, from the analytical expressions for GE arises out of the sa). The better fit of total pressure and critical solu- procedure used in fitting these expressions without tion temperature (but not critical composition) for first taking into consideration all the available data equation 14 suggests that it is preferable, but this is for the system. Now that computer methods are coming into by no means certain. Clearly a careful re-examination of the whole problem of deriving free energies more common use for fitting analytical expressions from vapor pressure data is called for, with partic- to experimental results the danger of errors of this ular reference to whether the extra effort required type is increased. We suggest that careful considto measure vapor compositions is worthwhile. eration of the criteria for thermodynamic consistAgain. if the equations of Dyke, Rowlinson and ency is necessary to ensure that the final represenThacker5 are simplified by assuming that GE varies tation of the data is as meaningful as possible. linearly with temperature, it is found that the critiAcknowledgments.-We wish to thank Mr . cal solution temperatures predicted from the ther- D. B. Myers of this department for the computer modynamic data are higher than the observed program used in the least squares treatment of the values. For perfluorocyclohexane cyclohexane results and Dr. G. N. Malcolm of the University of the discrepancy is about 8" and for perfluoro- Otago for helpful discussions.
+
+
+
