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Excess enthalpies of gaseous mixtures of n-alkanes
M-866 J. Chem. Thermo&omics
1979,
11, 1-12
Excess enthalpies n-al kanes C. J. WORMALD,”
of gaseous
mixtures
of
E. J. LEWIS, and D. J. HUTCHINGS
School of Chemistry, lXe University, Bristol BS9 ITS, U.K. (Received 21 December 1977; in revised form I9 April 1978) Plow calorimetric measurements of the excess enthalpy HE of 13 binary mixtures of gaseous n-alkanes from ethane to n-octane are reported. The measurements lie in the temperature range 304.5 to 413.3 K and are at a pressure of 101.325 kPa. These measurements, together with the HE’sfor 7 methane + n-alkane mixtures reported previously, are used to obtain values of the cross term isothermal Joule-Thomson coefficient &, the cross term second virial coefficient Z&, and the geometric-mean-rule parameter e for each mixture. The values of c obtained from the HE’sare in close agreement with those obtained from critical temperatures of the mixtures. The &‘s obtained from the HE’s agree well with those obtained from compression measurements. Comparison with six combining rules is made. It is found that the rule e = 2(Z1Zz)lla(Z1 + Za)-l(V& V&)lla( Vfz)-l fits all the measurements within experimental error.
1. Introduction Measurements of the excess enthalpies HE of 7 mixtures of methane + each of the n-alkanes from ethane to n-octane reported previously”’ were compared with HE’s calculated using a modification (‘1 of the equation of Potter and McGlashan which fits the isothermal Joule-Thomson coefficients of n-hexane, n-heptane, and n-octane, together with the geometric-mean and the Hudson-McCoubrey combining rules. It was found that the experimental HE’s lie approximately midway between values calculated using these rules. To see if this behaviour is unique to methane -t n-alkane, measurements of HE for rz-alkane mixtures which do not include methane have been made. The apparatus, experimental procedure, and purification of materials have been described previously. (‘* 3, The results of our measurements are listed in table 1. Values ofHnatx = 0.5 are in parentheses, and were obtained by fitting the best straight line to HE/4x( 1 -x) against x. The mole fractions x in table 1 are for the n-alkane of lowest molar mass. All experiments were done at atmospheric pressure, and the HE’s were corrected to a pressure of 101.325 kPa. 2. Analysis of the results An equation for the excess enthalpy HE of a binary gaseous mixture at low densities was given previously.(3) We previouslyc3) used a modification’2’ of the correspondingstates equation of McGlashan and Potter together with combining rules for Vi 2, N1 z, a To whom correspondence should be addressed. 0021-9614/79/010001+12 %0.100/O
0 1979 Academic Press Inc. (London)
Ltd.
C. J. WORMALD,
2 TABLE
1. The excess fraction x is that
E. J. LEWIS,
AND
D.
J. HUTCHINGS
enthalpies HE at 101.325 kPa of binary mixtures of n-alkanes. The of the lower molar-mass component. Values of HE at x = 0.5 are in parentheses
T x
X-
HE
304.5 (19.0) 333.2 (13.8) 363.2 (10.2)
0.398 0.434 0.466 0.486 0.377 0.383
17.7 18.8 14.0 14.0 8.9 11.0
0.466 0.522 0.494 0.518 0.411 0.441
18.7 18.5 13.8 13.3 9.2 10.1
372.2 (51.8) 383.2 (46.6) 403.2 (39.5)
0.448 0.480 0.502 0.443 0.475 0.434 0.455
51.3 51.7 51.2 46.1 47.5 38.6 39.2
0.563 0.588 0.594 0.491 0.530 0.468 0.491
51.0 49.3 49.0 45.4 46.2 39.3 39.8
403.2 (120.5) 413.2 (110.3)
0.394 0.458 0.434 0.459
116.6 122.4 110.9 109.3
0.473 0.512 0.480 0.489
118.6 119.6 108.3 110.5
344.2 (47.4) 363.2 (36.5) 396.9 (27.2)
0.521 0.588 0.365 0.443 0.471 0.494
47.1 45.6 34.7 36.1 27.3 27.1
0.632 0.646 0.475 0.548 0.538 0.596
42.4 42.4 36.8 36.0 26.9 26.3
383.2 (62.9)
0.535 0.564 0.586 0.475 0.499 0.366 0.420
63.7 62.0 61.0 49.6 50.4 43.7 48.2
0.643 0.648 0.670 0.530 0.546 0.455 0.486
55.7 56.9 54.9 48.9 48.7 48.6 46.5
403.2 410.2 (94.9)
0.455 cpg3
(89.9) 413.3 (86.5)
0:455 0.363 0.404
96.2 95.7 87.5 87.9 78.7 84.3
0.484 0.515 0.475 0.501 0.437 0.463
93.5 92.4 90.8 92.0 86.8 86.7
Jmol-’
HE X
Jmol-’
H” X
Jmol-l
HE n
Jmol-’
mole
HE X
J moI-l
Cd% + n-C&o 0.524 0.562 0.562 0.619 0.447 0.449
19.0 19.0 14.0 13.5 10.7 9.4
0.595 0.623 0.669
18.2 18.0 11.8
0.661
17.2
0.468 0.506
9.8 9.9
0.536 0.596
10.6 9.9
0.631 0.644
51.2 47.9
0.666 0.692
47.3 44.0
0.720 0.748
41.2 38.3
0.550 0.587 0.522 0.544
46.4 45.9 39.3 39.3
0.660 0.686 0.595 0.651
41.2 39.9 37.3 36.2
119.7 116.0 110.0 106.9
0.595 0.630 0.595 0.650
114.2 109.1 104.7 100.6
0.695 0.740 0.701 0.792
100.5 88.5 91.1 69.4
40.8 40.8 33.8 31.5 25.2 22.6
0.745 0.748
34.6 34.9
0.788 0.815
31.3 27.5
0.760 0.810
19.2 16.3
0.689 0.705
53.2 52.7
0.749 0.833
46.6 35.2
0.849 0.897
32.2 23.8
0.570 0.607 0.520 0.561
48.0 47.2 48.0 48.8
0.630 0.677 0.575 0.606
44.6 42.5 45.4 44.4
0.724
39.5
0.675 0.794
42.1 29.4
94.9 93.9 88.8 89.4 84.7 87.6
0.625 0.686 0.549 0.590 0.620 0.634
86.7 81.0 87.2 87.6 83.3 77.9
0.742
71.2
0.625 0.677 0.655 0.703
82.5 76.5 77.7 70.7
CaHe + n-C&c
CaHs + n-C&,
n-G&
n-CaHa
403.2 (49.2) 413.2 (47.6)
n-CsHs
0.557 0.585 0.510 0.550
+ n-CBHu 0.674 0.686 0.628 0.683 0.638 0.697 + n-C?Hla
+ 0.534 0.581 0.522 0.540 0.490 0.531
r&J&,
EXCESS
ENTHALPIES
OF
GASEOUS
TABLE
T YE
H” x
Jmol-1
MIXTURES
Jmol-1
363.2 (19.6) 373.2 (17.7) 383.2 (16.4) 393.2 (14.0)
0.331 0.370 0.414 0.450 0.407 0.459 0.383 0.452
16.8 19.6 18.4 17.8 14.6 16.9 11.8 14.7
0.425 0.453 0.477 0.519 0.485 0.513 0.474 0.497
~-GHn 19.7 18.4 16.4 17.9 17.6 15.6 12.7 15.6
403.2 (70.1) 410.5 (62.7) 413.2 (61.9)
0.400 0.449 0.502 0.523 0.441 0.459
69.1 70.3 62.1 63.9 60.1 61.3
0.475 0.520 0.560 0.578 0.484 0.499
68.8 69.0 62.6 59.2 65.5 63.5
343.2 (6.9) 363.2 (4.7) 383.2 (4.0) 403.2
0.417 0.430 0.448 0.480 0.397 0.438 0.481
5.6 5.9 6.0 4.0 4.4 3.1 4.0
0.472 0.485 0.522 0.562 0.485 0.494 0.495
393.2 (18.8) 403.2 (16.2) 413.2 (15.0)
0.347 0.383 0.376 0.421 0.328 0.347
18.4 19.0 14.0 15.1 14.2 13.5
0.430 0.439 0.456 0.480 0.405 0.432
n-G& 18.6 16.9 18.1 16.4 12.5 16.0
403.2 (43.5) 413.2 (39.5)
0.394 0.437 0.421 0.440
43.6 44.4 39.8 37.6
0.445 0.475 0.453 0.462
42.3 42.1 38.4 38.7
403.2 (21.7) 413.2 (19.0)
0.248 0.304 0.280 0.376
16.7 19.8 14.2 19.2
0.376 0.425 0.424 0.450
20.7 21.0 19.1 17.8
403.2 (6.2) 413.2 (4.9)
0.326 0.374 0.370 0.395
6.5 6.0 3.4 4.5
0.385 0.450 0.430 0.481
n-C&L,
n-C&a 9.0 6.0 4.3 6.2 6.3 3.5 6.0
n-C6Hl2
n-G&4
n-G&s i-i 6:4 5.3
3
n-ALKANES
1 --continued
H” x
OF
HE x
Jmol-1
+ n-CsHx, 0.482 18.9 19.9 0.522 18.8 0.560 16.5 0.583 17.0 0.539 0.584 17.9 0.521 11.9 0.596 12.4
H” x
Jmol-’
HE x
J mol-’
0.566 0.595 0.614 0.660 0.611 0.625 0.580 0.622
20.3 17.3 16.6 14.7 14.3 15.6 14.1 14.7
0.625 0.670
19.7 16.8
0.698
12.8
0.670 0.781
13.3 9.6
0.618 0.629 0.638 0.672 0.586 0.648
66.1 64.2 57.9 56.1 60.0 56.4
0.705 0.710
57.8 56.3
0.706 0.755
51.9 44.4
0.556 0.562 0.595
6.2 8.0 2.9
0.597
2.7
0.559
2.9
0.600
4.3
0.522 0.552 0.548 0.577 0.534 0.583
17.9 16.9 13.9 17.6 17.3 13.2
0.622 0.628 0.619
18.8 17.1 15.1
0.596 0.615
13.7 13.5
43.7 44.1 39.9 41.0
0.546 0.592 0.525 0.553
44.5 40.0 40.2 38.5
0.633 0.704 0.574 0.638
40.9 34.3 37.5 35.5
+ n-CeH1s 0.477 20.5 20.7 0.530 0.487 18.2 19.9 0.523
0.550 0.598 0.540 0.581
22.2 21.1 20.4 17.5
0.676 0.753 0.625 0.749
18.1 15.8 16.9 15.0
0.526 0.505 0.580 0.610
6.6 5.0 4.8 3.8
0.598 0.623 0.675 0.744
7.1 4.6 4.4 3.0
+ n-Cd-L3 71.6 69.3 61.7 58.6 60.1 61.7
0.544 0.583 0.594 0.602 0.535 0.558
+ n-Cd-L 0.510 7.5 0.550 5.3 0.574 5.7 0.580 3.9 3.0 0.520 0.554 0.542 ::3’ + n-Cd& 17.8 0.474 0.495 20.6 0.522 17.0 0.531 15.5 0.472 15.1 0.511 14.5 +
0.484 0.511 0.481 0.499
n-C8Hl8
+ n-CsH18 0.481 0.510 0.495 0.553
6.7 7.2 4.5 6.3
C. J. WORMALD,
4
E. J. LEWIS, AND D. J. HUTCHINGS
and T”,, to calculate values of B,, and & and hence obtained values of HE which we compared with experiment. In this analysis we have inverted the procedure so as to find the values of Tf, which give agreement between the experimental and calculated HE’s Equation (6) of reference 3 may be written in the form: Hi =
(2)
~1~2~tW,2--~,,-422),
where HE is the experimental excess enthalpy. The term in @is small and was calculated from the correlation of Chueh and Prausnitzc4) using the combining rules given previously. (3) The terms in B and 4 were calculated from the modified equationc2’ of McGlashan and Potter. This modification is simply the replacement of the exponent 4.5 by (4.18 +O.O4(N- I)}. We used the previous”) arithmetic-mean combining rule N,, = (N1 +N2)/2 and the Lorentz rule for VC,, together with the rule: T;, = c(T;1T;2)“2.
(3)
Starting with 5 = 1.1 an iterative method was used to find the value 5. which balanced equation (1) to within 0.1 J mol-‘. Values of B12, 412, and HE were then computed, and these are listed in table 2. HE allows easy recalculation of 4r2 from equation (2) if a different equation to that used above”) for CprI and 422 is preferred. The uncertainty S4,, in +r2 depends upon the random error o(HE) in HE and upon the systematic error which may be present in the 4’s to which the modified equation of McGlashan and Potter was fitted. c2) The possibility that these 4’s are too positive TABLE 2. Quantities derived from the experimental excess end&pies HE and the standard deviations o(IP). The mixtures are denoted by the number of carbon atoms in the two n-alkanes T E
~
HE
J mol-l
- oWEI
H,”
-ha
J mol-’
J mol-1
cm3 mol-’
0.45 0.45 0.36 0.26 0.39 0.21 0.15 0.35
5.56 5.07 4.63 4.20 4.30 3.40 3.00 3.16
424 396 364 358 348 330 298 285
w1a
cm3 mol-1
-&a
6&a
cm3 mol-1 cm3 mol-1
e
1+2 241.1 250.6 262.4 266.7 269.2 282.2 298.2 303.2 243.2 245.2 250.8 253.2 261.2 263.6 273.2 281.2 290.7
302.2
5.92 5.41 4.88 4.44 4.53 3.62 3.17 3.30 26.5 26.1 24.2 23.6 21.5 20.1 18.1 16.6 15.0 13.6
0.93 0.59 1.12 0.34 0.69 0.74 0.75 0.40 0.76 0.24
24.2 23.8 22.3 21.8 19.9 18.7 16.9 15.6 14.2 13.0
1+3 581 568 547 534 507 514 480 457 433 400
18 17 15 12 15 11 10 13
141 130 118 116 112 105 93 88
6 4 6 4 4 5
37 31
192 187 180 176 166 168 156 148 139 127
13 11 14 9 10 11 10 8 10 6
40
26 :: 29 2: 18
0.992 0.992
0.990 0.998 0.992 1.008 1.008 0.998
0.970 0.966 0.968 0.966 0.968 0.984 0.984 0.986 0.990 0.986
EXCESS
ENTHALPIES
OF
GASEOUS
TABLE
T ii
P J mol-1
wm J mol-’
HE J mol-1
MIXTURES
OF
n-ALKANES
5
2-wnrinued
-ha cm3 mol-1
WlP cm3 mol-’
-&a cm3 mol-1
S&S cma mol-l
e
-
1+4 277.0 284.2 289.0 303.2 306.5 314.2 328.7 373.4 383.2 394.3
48.9 45.5 42.9 36.9 36.0 31.3 30.5 20.9 19.3 17.9
1.7 1.1 1.3 0.9 0.6 2.3 0.7 2.0 1.0 1.4
44.3 41.6 39.4 34.4 33.6 29.3 28.9 20.1 18.6 17.4
318.5 333.2 343.4 353.2 363.2 373.2 383.4 393.2 403.5
67.6 58.0 53.0 48.7 46.0 40.7 37.2 33.7 32.3
1.1 1.1 1.0 0.8 1.1 0.9 1.0 1.3 1.0
61.8 53.7 49.5 45.7 43.5 38.6 35.5 32.2 31.0
343.2 363.2 373.2 383.2 389.2 407.7
97.7 82.2 74.5 67.8 65.3 56.8
389.2 390.2 405.7 408.2 413.3
104.8 104.3 93.4 90.0 86.4
410.2 418.3
611 570 557 506 490 z 332 322 305
63 48 51 40 35 70 33 60 iit
198 184 179 161 155 162 124 97 93 87
21 17 18 14 12 25 12 22 I3 14
0.970 0.960 0.964 0.962 0.956 1.000 0.928 0.948 0.956 0.956
58 52 48 43 45 40 41 47 37
178 163 150 139 121 125 121 120 105
20 18 17 15 16 14 15 17 14
0.950 0.954 0.946 0.942 0.912 0.952 0.962 0.984 0.958
79 69 66 60 89 56
180
27
:z 134 124 108
2”: 21 32 20
0.940 0.904 0.914 0.924 0.908 0.900
74 75 Ii 91
162 157 127 135 131
26 26 26 35 33
0.944 0.936 0.892 0.920 0.918
1+5 567 524 486 458 408 420 407 405 366
I+6 88.5 75.9 69.3 63.5 61.3 53.9
583 483 468 455 426 383
;: 2:6
96.1 95.8 86.8 83.8 80.7
1+7 542 530 447 469 456
137.7 127.5
2.5 1.6
125.4 116.7
525 521
109 90
151 150
38 32
0.910 0.924
304.5 333.2 363.2
19.0 13.8 10.2
0.31 0.40 0.65
17.3 12.8 9.6
2+4 1026 857 728
33 28 28
332 333 233
10 9 9
0.984 0.987 0.992
372.2 383.2 403.2
51.8 46.6 39.5
1.21 0.72 0.35
47.5 43.0 36.9
2+6 1049 987 878
61 51 42
330 310 274
18 15 13
0.980 0.981 0.976
1.1 1.2
I+8
6
C. J. WORMALD,
E. J. LEWIS, TABLE
T ic
-- HE Jmol-1
403.2 413.2
120.5 110.3
344.2 363.2 396.9
47.4 36.5 27.2
383.2 403.2 413.2
o(H=l Jmol-1
HE Jmol-’
AND
D. J. HUTCHINGS
2-contiwed
-ha cmamol-’
Wla ems mol-l
--Baa ana mol-1
6&Z cm amol-l
112 99
354 331
64 51
0.950 0.944
e
2+8
2;
107.9 99.7
1153 1077
0.7 0.5 0.5
41.7 32.6 25.1
1872 1682 1347
68 57 46
574 519 418
19 16 14
0.988 0.997 0.989
62.9 49.2 47.6
1.0 1.0 1.5
56.0 44.5 43.6
3+7 1759 1600 1457
76 67 71
537 490 448
21 20 21
0.989 1.ooo 0.984
403.2 410.2 413.2
94.9 89.9 86.5
1.75 1.5 1.7
84.0 80.1 77.4
1787 1690 1673
101 93 95
541 512 508
28 27 27
0.977 0.971 0.974
363.2 373.2 383.6 393.2
19.6 17.7 16.4 14.0
1.0 i-Y 1:4
17.3 15.7 14.7 12.7
2309 2157 2010 1902
64 61 62 61
694 651 610 579
17 17 17 17
0.996 0.996 0.994 0.997
403.2 410.5 413.2
70.1 62.7 61.9
1.4 1.3 1.2
61.2 55.2 54.5
2362
98 92 89
735 714 700
343.2 363.2 383.2 403.2
6.9 4.7 4.0 4.3
1.3 1.2 1.2 1.1
5.7 4.2 3.7 4.0
3563 3059 2647 2305
84 74 68 57
1023 893 783 689
21 19 18 15
1.ooo 1.000 1.000 0.997
393.2 403.2 413.2
18.8 16.2 15.0
1.3 1.4 1.3
16.5 14.2 13.3
5+7 2991 2812 2630
80 77 72
871 823 774
20 20 19
0.997 0.999 0.998
403.2 413.2
42.6 38.5
1.5 1.0
37.5 34.5
5+8 3320 3096
101 90
953 895
403.2 413.2
21.7 19.0
1.0 1.2
18.6 16.3
4201 3926
101 96
1173 1105
26 25
0.998 0.999
403.2 413.2
6.2 4.9
1.0 1.0
5.6 4.4
5168 4815
100 95
1407 1322
30 28
0.999 1.000
3+6
3+8
4+6
4+8
E
0.985 0.991 0.989
5-i-6
0.995 0.994
6+8
7+8
EXCESS ENTHALPIES
7
OF GASEOUS htIXTURI?S OFn-ALKANES
because of heat leaks cannot be discounted. reference 3 we obtain
From the first term of equation (6) in
(4) 6412 = P.ewB11x(l -xl)” +@41 II2 + @~22)2~I”2. The uncertainties 6+,, and 6+22 were taken to be 2 per cent of q511 and c$~~, and a(H’) is the standard deviation of the measurements listed in table 1 obtained from HE/4x(l -x) against x. The uncertainty 6B,, in B12 was calculated from the equation : &2
(5)
= S~,,(dB,,/dT)(d~iz/dT)-'.
The temperature derivatives were calculated from the motied equation of McGlashan and Potter. The &,‘s listed in table 2 show no trend with temperature. The mean value of <,, for each mixture is listed in table 3. The average uncertainty on the ca’s is +0.007. It is TABLE 3. Comparison of < obtained from experiment with equation (7). c. is obtained from our HE
measurements, eb from virial-coefficient measurements, COfrom critical-temperature measurements, and & is calculated from equation (7) -.-1+2 l-i-3 1+4 1+5 1+6 1+7
I+8
2+3 2+4 2+5 2+6 2+7 2+8 3+4
e. 0.997 0.997 0.960
0.951 0.917 0.922 0.917 0.987 0.979
1.012
0.991
0.995 0.969 0.949 0.918
1.002
0.992 0.978 0.959
0.947 1.001
0.976 0.958
0.941 0.925 1.009
0.990 0.988 0.982
l.ooo
:E
O:!B6 0.987 0.976 0.965 0.953 0.942 0.997
3+5 3+6 3+7 3+8 4+5 4+6 4+7 4+8 5+6 5+7 5+8 6+7 648 7+8
0.991 0.991 0.974 0.996 0.988 0.999 0.998 0.994 0.998 0.999
0.994 0.983 0.965
0.938 l.ooo
0.994 0.999
0.982 0.988 0.968
0.963 0.999
1.002 1.002 1.003
l.ooo
0.997 0.986
1.001 1.002 1.001
0.992 0.984 0.976
0.967
0.998
0.994 0.989 0.983 0.999 0.996
0.992 0.999 0.997 0.999
interesting to compare these 5,‘s with e’s obtained from other experiments. In a recent compilation of measurements of the second virial coefficients of n-alkane mixtures made by Pompe and Spurling c5) the most extensive set of measurements is that of Dantzler, Knobler, and Windsor (‘) who measured the quantity E = B, 2- +(B, 1+ Bz2) for all 15 binary mixtures of the nalkanes from methane to n-hexane. Using a similar iterative procedure to the above, and using the modified equation of McGlashan and Potter to obtain B, 1 and Bz2,we calculated values of g which we refer to as &,. As the measurements on each mixture were made at 4 temperatures the value of &, included in table 3 is the mean. The only measurements on mixtures containing higher n-alkanes are those of BM made by McGlashan and Potter(‘) on propane + n-heptane and propane + n-octane, and &,‘s obtained from these measurements are also included in table 3. Values of t obtained from measurements of the critical temperature Ts of binary mixtures of n-alkanes including the mixtures from ethane to n-octane are listed by Hicks and Young. (‘I Their value 0.892 for ethane + n-heptane should be
8
C. J. WORMALD,
E. J. LEWIS,
AND D. J. HUTCHINOS
0.982.t’) Values of < obtained from Tb measurements will be referred to as <,, and these are listed in table 3. Inspection of the three sets of 5’s shows the overall agreement to be remarkably good. While a few of the es obtained from both E and TM measurements are greater than unity this is probably experimental error. As the uncertainty in the critical temperatures of these mixtures is often about 1 K, and as the measurements are usually all close to one end of the composition range, the cc’s are probably of no better accuracy than our &,‘s. Agreement between the three sets of r’s is to within the combined experimental error. Indeed the &,‘s obtained from our HE’s deviate on average by less than 1 per cent from the corresponding cc’s. 3. Comparison of 5 with combining rules We have investigated 6 combining rules for T”,, from which we have obtained values of < by dividing through by (T;1T&)“2. The equations for c are t, = (K,~o,2>1’2(~~2>-‘, t2 = 2~1(~1~2)1’2(~, +I,)l, +zJ1, 53 = 2~:(z~z,)1’2(z, C4 = 2(T;,T;,)1’2(T;,+T;,)-1, & = 2(T;,T;2)1’6((T;,)“3+(T~2)1’3}-1,
(6)
(7) (8)
(9) w-3 56 = 2(TflT”,2)“25~~f(V”,2)2~la2~Tfl(~C,~a2)2+T”,2(~‘,2al)2)-‘. (11) Here c1is the polarisability, Z the ionization potential, and VC,, is calculated from the Lorentz rule. c3) Equation (8) i s the combining rule of Hudson and McCoubrey.(lo) Equation (7) is a modification of (8) made following our observation(‘) that HE for methane + n-alkane lies midway between values calculated using c = 1 and equation (8). Equation (6) may be used instead of (7) when the ionization potentials Zr and I2 are similar. Equation (9) is the harmonic-mean rule of Fender and Halsey,“‘) equation (10) is a similar rule suggested byHicks and Young, (*) and equation (11) was suggested Munn.(12) We compare these rules with the T’s listed in table 3. As the ionization energies(13) for the n-alkanes from ethane (1.87 x lo-‘s J) to n-octane (1.64 x IO-‘s J) are much the same, whereas that for methane is 2.11 x 10-r* J, it is informative to examine the mixtures containing methane separately from the rest. For each mixture 5’s were calculated from equations (6) to (ll), the standard percentage deviations from the experimentally determined r’s were obtained, and these are listed for the two groups of mixtures in table 4. Comparison with the &,‘s obtained from compression experiments was restricted to the measurements on the 15 mixtures reported in reference 6. The measurements of Z?, for propane + n-heptane(‘) give a value of &, which is a little low though still in agreement with that calculated from PM but &, for propane + n-octane(‘) is in poor agreement with other values. Table 4 shows that for each mixture both <, and c2 agree equally well with experiment, which suggests that the ionization potentials in equation (7) are unimportant. This term is largest for methane + n-octane for which t1 = 0.901 and r2 = 0.894, a difference of only 0.8 per cent. As the comparison between the r’s obtained from
EXCESS ENTHALPIES
OF GASEOUS MIXTURES
OF n-ALKAIWS
9
TABLE 4. Standard percentage deviations of values of e calculated from equations (6) to (11) from values obtained experimentally Mixtures containing methane e, Cb f: ii 4
Mixtures without methane t-e Cb c
0.95 1.23 6.93 3.97
1.45 1.65 4.89 4.22
0.63 0.68 0.46 2.79
0.37 0.46 1.44
0.89 0.87 0.85 2.28
5.10 10.6
3.92 7.63
1.72 4.10
1.48 2.30
1.60 3.32
experiment and equation (7) is worth making in detail, values of & are included in table 3. While equations (8) to (11) all appear to give better agreement with the results containing no methane, this is deceptive. Most of the mixtures in this group contain n-alkanes of not very different chain length so that the mean value of < is 0.985 whereas for the methane mixtures it is 0.825. For similar molecules even poor combining rules give reasonable values of r. Of equations (8) to (11) the harmonic-mean rule, equation (9), is the best. The Hudson-McCoubrey rule is not as good as earlier work suggested.(14) A good test of equation (7) is the comparison shown in figure 1 of the experimental HE’s for methane + n-alkane with HE calculated using the combining rule as described previously (‘) This figure should be compared with figure 2 of reference 1. In view of the wide difference in pair potentials which we might expect for methane + ethane, methane + n-octane, and n-heptane + n-octane, it is surprising to find a combining rule which fits the measurements on all mixtures to within experimental error. The possibility that our measurements (2) of 4 for n-hexane, n-heptane, and n-octane may be in error must be considered. We therefore repeated the analysis of our HE’s using the original equation of McGlashan and Potter(‘) rather than the modified equation. (2) Agreement between the &,‘s and tc’s is not now so good. The ca’s for the 7 methane mixtures are increased by an average of 3.0 per cent, and the 5;s for the 13 remaining mixtures are increased by an average of 0.45 per cent. For the mixtures of methane with the n-alkanes from ethane to n-octane the ga’s calculated using the original equation are 1.001, 0.999, 0.987,0.986,0.959,0.961, and 0.945. These figures should be compared with the /&‘s and &,‘s listed in table 3. When the HE and E@) results are analysed using the modified rather than the original equation of McGlashan and Potter, closer agreement between the ra’s and &,‘s is found. This suggests that the modification made to the equation so as to fit the 4’s is an improvement, and that the 4’s are not seriously in error. 4.
Cross-term second virial coetllcients
As cross-term second virial coefficients are a primary source of information about pair potentials between unlike molecules, it is important to compare the B,,‘s listed in table 2 with those obtained by other methods. It is convenient to do this by plotting the measurements on a reduced scale using the parameters T; 2 = (T,“, Ti2)1’2 and V; 2
10
C. J. WORMALD,
E. J. LEWIS, AND
D. J. HUTCHINGS
125
I \ \ 100
\ \ \ \ \ \ - \\ \ \
\\
50
\
‘\
\ \ \ \ \
\ \ \ \ \ \ \ \ \
h \ \
\
\\
\
\
\ Q
\ a\ \ b\
\\
\
‘s \
25
250
5 T/K
FIGURE 1. Excess enthalpies of methane + each of the n-alkanes from ethane to n-octane. The dashed Iines are calculated by the method described previously@’ using the modified correspondingstates equation’“) of McGlashau and Potter together with the combining rule T,Oa= &(Z’i’lT&)l/a, where & is given by equation (7). 0, reference 1.
calculated from the Lorentz rule. With this choice the B12’s for different mixtures lie almost on a single curve, in marked contrast to graphs in which either the geometric mean or Hudson-McCoubrey rules for TE, are used. In figure 2 we compare our B12’sfor methane + ethane to methane + n-hexane with reference 6. The solid lines in figure 2 were calculated from the modified equation@) of McGlashan and Potter, and to be consistent we recalculated the B,,‘s reported in reference 6 using this equation. Recalculation makes a significant difference only for methane + n-pentane and methane + n-hexane for which the new B,2’~ are on average 7 per cent less negative. The upper part of figure 2 shows that our B,z’s agree as well with the calculated curves as do the reference 7 measurements in the lower part. A similar comparison for mixtures containing no methane is made in figure 3 where BIZ’s for mixtures of propane + higher n-alkanes are plotted. Both the BIZ’s from reference 6 for mixtures of propane with n-butane, n-pentane, and n-hexane, and our B, i’s for mixtures with n-hexane, n-heptane, and n-octane are in very close agreement
U.i
1.?5
09
1.0
1.1
1.2
1.3
1.4
1.5
16
T/T&
FIGURE 2. Cross-term second virial coefficients for methane + each of the n-alkanes from ethane to n-hexane. The solid lines are calculated as described in the text for the mixtures methane + ethane, methane f n-butane,andmethane + n-hexane.The reducingparametersare T& = &(IIJ&)‘la and VfSgiven by the Lorentz rule. The Bla’s listed in table 2 are plotted on the upper curves and those of reference 6 are plotted on the lower curves. I:), methane + ethane; A, methane + propane; V, methane + n-butane; 0. methane + n-pentane; +, methane -t n-hexane.
b" 1 .A'--L' 3 ,o L0.7
iwe--I_0.8
..- I--L... 0.9 T/T,]
i
FIGURE 3. Cross-term second virial coetlicients for propane + n-alkane from n-butane to n-octane. The solid lines arc calculated as described in the text for propane + n-butane, propane + n-hexane, and propane + n-octane. The reducing parameters are the same as for figure 2. 0, propane + n-hexane (this work); q , propane + n-heptane (this work); 0, propane + n-octane (this work); A, propane + n-butane (reference 6); V, propane + n-pentane (reference 6); 8, propane + n-hexane (reference 6); x, propane + n-heptane (reference 7); +, propane + n-octane (reference 7).
12
C. J. WORMALD,
E. J. LEWIS, AND
D. J. HUTCHINGS
with the calculated lines. The reference 7 results are too positive, and conclusions”4) about combining rules made on the basis of these measurements must be revised. For the remaining 11 mixtures on which we have made HE measurements the B12’s are, on average, in as close agreement with the calculated HE’s as are the B12’s for the propane mixtures. These comparisons suggest that the B,,‘s obtained from our HE measurements are no less accurate than those obtained from compression measurements, and that the limits of error 6B,, in table 2 are probably too wide. REFERENCES 1. Hutchings, D. J.; Lewis, E. J.; Wormald, C. J. J. Chem. Thermodynamics 1978, 10, 559. 2. Al-Bizreh, N. ; Wormald, C. J. J. Chem. Thermodynamics 1978, 10,231. 3. Wormald, C. J. J. Chem. Thermodynamics 1977, 9, 901. 4. Chueh, P. L.; Prausnitz, J. M. AZChEJ 1967, 13, 896. 5. Pompe, A.; Spurling, T. H. C.S.I.R.O. Aust. Div. Appl. Organic Gem. Tech. Pap. No. 3 1976, 1. 6. Dantzler, E. M. ; Knobler, C. M.; Windsor, M. L. J. Phys. Chem. 1968, 72, 676. 7. McGlashan, M. L.; Potter, D. J. B. Proc. Roy. Sot. A 1%2,267,478. 8. Hicks, C. P.; Young, C. L. Chem. Rev. 1975,75, 119. 9. Hicks, C. P. Personal communication. 10. Hudson, G. H.; McCoubrey, J. C. Trans. Faraday Sot. 1960,56,761. 11. Fender, B. E. F.; Halsey, G. D. J. Chem. Phys. 1962, 36, 1881. 12. Munn, R. J. Trans. Faraday Sot. 1961,57, 187. 13. Reed, R. J. Zon Production by Electron Impact. Academic Press: London. 1962. 14. Cruickshank, A. J. B.; Windsor, M. L.; Young, C. L. Trans. Faraday Sot. 1966,62,2341.
1979,
11, 1-12
Excess enthalpies n-al kanes C. J. WORMALD,”
of gaseous
mixtures
of
E. J. LEWIS, and D. J. HUTCHINGS
School of Chemistry, lXe University, Bristol BS9 ITS, U.K. (Received 21 December 1977; in revised form I9 April 1978) Plow calorimetric measurements of the excess enthalpy HE of 13 binary mixtures of gaseous n-alkanes from ethane to n-octane are reported. The measurements lie in the temperature range 304.5 to 413.3 K and are at a pressure of 101.325 kPa. These measurements, together with the HE’sfor 7 methane + n-alkane mixtures reported previously, are used to obtain values of the cross term isothermal Joule-Thomson coefficient &, the cross term second virial coefficient Z&, and the geometric-mean-rule parameter e for each mixture. The values of c obtained from the HE’sare in close agreement with those obtained from critical temperatures of the mixtures. The &‘s obtained from the HE’s agree well with those obtained from compression measurements. Comparison with six combining rules is made. It is found that the rule e = 2(Z1Zz)lla(Z1 + Za)-l(V& V&)lla( Vfz)-l fits all the measurements within experimental error.
1. Introduction Measurements of the excess enthalpies HE of 7 mixtures of methane + each of the n-alkanes from ethane to n-octane reported previously”’ were compared with HE’s calculated using a modification (‘1 of the equation of Potter and McGlashan which fits the isothermal Joule-Thomson coefficients of n-hexane, n-heptane, and n-octane, together with the geometric-mean and the Hudson-McCoubrey combining rules. It was found that the experimental HE’s lie approximately midway between values calculated using these rules. To see if this behaviour is unique to methane -t n-alkane, measurements of HE for rz-alkane mixtures which do not include methane have been made. The apparatus, experimental procedure, and purification of materials have been described previously. (‘* 3, The results of our measurements are listed in table 1. Values ofHnatx = 0.5 are in parentheses, and were obtained by fitting the best straight line to HE/4x( 1 -x) against x. The mole fractions x in table 1 are for the n-alkane of lowest molar mass. All experiments were done at atmospheric pressure, and the HE’s were corrected to a pressure of 101.325 kPa. 2. Analysis of the results An equation for the excess enthalpy HE of a binary gaseous mixture at low densities was given previously.(3) We previouslyc3) used a modification’2’ of the correspondingstates equation of McGlashan and Potter together with combining rules for Vi 2, N1 z, a To whom correspondence should be addressed. 0021-9614/79/010001+12 %0.100/O
0 1979 Academic Press Inc. (London)
Ltd.
C. J. WORMALD,
2 TABLE
1. The excess fraction x is that
E. J. LEWIS,
AND
D.
J. HUTCHINGS
enthalpies HE at 101.325 kPa of binary mixtures of n-alkanes. The of the lower molar-mass component. Values of HE at x = 0.5 are in parentheses
T x
X-
HE
304.5 (19.0) 333.2 (13.8) 363.2 (10.2)
0.398 0.434 0.466 0.486 0.377 0.383
17.7 18.8 14.0 14.0 8.9 11.0
0.466 0.522 0.494 0.518 0.411 0.441
18.7 18.5 13.8 13.3 9.2 10.1
372.2 (51.8) 383.2 (46.6) 403.2 (39.5)
0.448 0.480 0.502 0.443 0.475 0.434 0.455
51.3 51.7 51.2 46.1 47.5 38.6 39.2
0.563 0.588 0.594 0.491 0.530 0.468 0.491
51.0 49.3 49.0 45.4 46.2 39.3 39.8
403.2 (120.5) 413.2 (110.3)
0.394 0.458 0.434 0.459
116.6 122.4 110.9 109.3
0.473 0.512 0.480 0.489
118.6 119.6 108.3 110.5
344.2 (47.4) 363.2 (36.5) 396.9 (27.2)
0.521 0.588 0.365 0.443 0.471 0.494
47.1 45.6 34.7 36.1 27.3 27.1
0.632 0.646 0.475 0.548 0.538 0.596
42.4 42.4 36.8 36.0 26.9 26.3
383.2 (62.9)
0.535 0.564 0.586 0.475 0.499 0.366 0.420
63.7 62.0 61.0 49.6 50.4 43.7 48.2
0.643 0.648 0.670 0.530 0.546 0.455 0.486
55.7 56.9 54.9 48.9 48.7 48.6 46.5
403.2 410.2 (94.9)
0.455 cpg3
(89.9) 413.3 (86.5)
0:455 0.363 0.404
96.2 95.7 87.5 87.9 78.7 84.3
0.484 0.515 0.475 0.501 0.437 0.463
93.5 92.4 90.8 92.0 86.8 86.7
Jmol-’
HE X
Jmol-’
H” X
Jmol-l
HE n
Jmol-’
mole
HE X
J moI-l
Cd% + n-C&o 0.524 0.562 0.562 0.619 0.447 0.449
19.0 19.0 14.0 13.5 10.7 9.4
0.595 0.623 0.669
18.2 18.0 11.8
0.661
17.2
0.468 0.506
9.8 9.9
0.536 0.596
10.6 9.9
0.631 0.644
51.2 47.9
0.666 0.692
47.3 44.0
0.720 0.748
41.2 38.3
0.550 0.587 0.522 0.544
46.4 45.9 39.3 39.3
0.660 0.686 0.595 0.651
41.2 39.9 37.3 36.2
119.7 116.0 110.0 106.9
0.595 0.630 0.595 0.650
114.2 109.1 104.7 100.6
0.695 0.740 0.701 0.792
100.5 88.5 91.1 69.4
40.8 40.8 33.8 31.5 25.2 22.6
0.745 0.748
34.6 34.9
0.788 0.815
31.3 27.5
0.760 0.810
19.2 16.3
0.689 0.705
53.2 52.7
0.749 0.833
46.6 35.2
0.849 0.897
32.2 23.8
0.570 0.607 0.520 0.561
48.0 47.2 48.0 48.8
0.630 0.677 0.575 0.606
44.6 42.5 45.4 44.4
0.724
39.5
0.675 0.794
42.1 29.4
94.9 93.9 88.8 89.4 84.7 87.6
0.625 0.686 0.549 0.590 0.620 0.634
86.7 81.0 87.2 87.6 83.3 77.9
0.742
71.2
0.625 0.677 0.655 0.703
82.5 76.5 77.7 70.7
CaHe + n-C&c
CaHs + n-C&,
n-G&
n-CaHa
403.2 (49.2) 413.2 (47.6)
n-CsHs
0.557 0.585 0.510 0.550
+ n-CBHu 0.674 0.686 0.628 0.683 0.638 0.697 + n-C?Hla
+ 0.534 0.581 0.522 0.540 0.490 0.531
r&J&,
EXCESS
ENTHALPIES
OF
GASEOUS
TABLE
T YE
H” x
Jmol-1
MIXTURES
Jmol-1
363.2 (19.6) 373.2 (17.7) 383.2 (16.4) 393.2 (14.0)
0.331 0.370 0.414 0.450 0.407 0.459 0.383 0.452
16.8 19.6 18.4 17.8 14.6 16.9 11.8 14.7
0.425 0.453 0.477 0.519 0.485 0.513 0.474 0.497
~-GHn 19.7 18.4 16.4 17.9 17.6 15.6 12.7 15.6
403.2 (70.1) 410.5 (62.7) 413.2 (61.9)
0.400 0.449 0.502 0.523 0.441 0.459
69.1 70.3 62.1 63.9 60.1 61.3
0.475 0.520 0.560 0.578 0.484 0.499
68.8 69.0 62.6 59.2 65.5 63.5
343.2 (6.9) 363.2 (4.7) 383.2 (4.0) 403.2
0.417 0.430 0.448 0.480 0.397 0.438 0.481
5.6 5.9 6.0 4.0 4.4 3.1 4.0
0.472 0.485 0.522 0.562 0.485 0.494 0.495
393.2 (18.8) 403.2 (16.2) 413.2 (15.0)
0.347 0.383 0.376 0.421 0.328 0.347
18.4 19.0 14.0 15.1 14.2 13.5
0.430 0.439 0.456 0.480 0.405 0.432
n-G& 18.6 16.9 18.1 16.4 12.5 16.0
403.2 (43.5) 413.2 (39.5)
0.394 0.437 0.421 0.440
43.6 44.4 39.8 37.6
0.445 0.475 0.453 0.462
42.3 42.1 38.4 38.7
403.2 (21.7) 413.2 (19.0)
0.248 0.304 0.280 0.376
16.7 19.8 14.2 19.2
0.376 0.425 0.424 0.450
20.7 21.0 19.1 17.8
403.2 (6.2) 413.2 (4.9)
0.326 0.374 0.370 0.395
6.5 6.0 3.4 4.5
0.385 0.450 0.430 0.481
n-C&L,
n-C&a 9.0 6.0 4.3 6.2 6.3 3.5 6.0
n-C6Hl2
n-G&4
n-G&s i-i 6:4 5.3
3
n-ALKANES
1 --continued
H” x
OF
HE x
Jmol-1
+ n-CsHx, 0.482 18.9 19.9 0.522 18.8 0.560 16.5 0.583 17.0 0.539 0.584 17.9 0.521 11.9 0.596 12.4
H” x
Jmol-’
HE x
J mol-’
0.566 0.595 0.614 0.660 0.611 0.625 0.580 0.622
20.3 17.3 16.6 14.7 14.3 15.6 14.1 14.7
0.625 0.670
19.7 16.8
0.698
12.8
0.670 0.781
13.3 9.6
0.618 0.629 0.638 0.672 0.586 0.648
66.1 64.2 57.9 56.1 60.0 56.4
0.705 0.710
57.8 56.3
0.706 0.755
51.9 44.4
0.556 0.562 0.595
6.2 8.0 2.9
0.597
2.7
0.559
2.9
0.600
4.3
0.522 0.552 0.548 0.577 0.534 0.583
17.9 16.9 13.9 17.6 17.3 13.2
0.622 0.628 0.619
18.8 17.1 15.1
0.596 0.615
13.7 13.5
43.7 44.1 39.9 41.0
0.546 0.592 0.525 0.553
44.5 40.0 40.2 38.5
0.633 0.704 0.574 0.638
40.9 34.3 37.5 35.5
+ n-CeH1s 0.477 20.5 20.7 0.530 0.487 18.2 19.9 0.523
0.550 0.598 0.540 0.581
22.2 21.1 20.4 17.5
0.676 0.753 0.625 0.749
18.1 15.8 16.9 15.0
0.526 0.505 0.580 0.610
6.6 5.0 4.8 3.8
0.598 0.623 0.675 0.744
7.1 4.6 4.4 3.0
+ n-Cd-L3 71.6 69.3 61.7 58.6 60.1 61.7
0.544 0.583 0.594 0.602 0.535 0.558
+ n-Cd-L 0.510 7.5 0.550 5.3 0.574 5.7 0.580 3.9 3.0 0.520 0.554 0.542 ::3’ + n-Cd& 17.8 0.474 0.495 20.6 0.522 17.0 0.531 15.5 0.472 15.1 0.511 14.5 +
0.484 0.511 0.481 0.499
n-C8Hl8
+ n-CsH18 0.481 0.510 0.495 0.553
6.7 7.2 4.5 6.3
C. J. WORMALD,
4
E. J. LEWIS, AND D. J. HUTCHINGS
and T”,, to calculate values of B,, and & and hence obtained values of HE which we compared with experiment. In this analysis we have inverted the procedure so as to find the values of Tf, which give agreement between the experimental and calculated HE’s Equation (6) of reference 3 may be written in the form: Hi =
(2)
~1~2~tW,2--~,,-422),
where HE is the experimental excess enthalpy. The term in @is small and was calculated from the correlation of Chueh and Prausnitzc4) using the combining rules given previously. (3) The terms in B and 4 were calculated from the modified equationc2’ of McGlashan and Potter. This modification is simply the replacement of the exponent 4.5 by (4.18 +O.O4(N- I)}. We used the previous”) arithmetic-mean combining rule N,, = (N1 +N2)/2 and the Lorentz rule for VC,, together with the rule: T;, = c(T;1T;2)“2.
(3)
Starting with 5 = 1.1 an iterative method was used to find the value 5. which balanced equation (1) to within 0.1 J mol-‘. Values of B12, 412, and HE were then computed, and these are listed in table 2. HE allows easy recalculation of 4r2 from equation (2) if a different equation to that used above”) for CprI and 422 is preferred. The uncertainty S4,, in +r2 depends upon the random error o(HE) in HE and upon the systematic error which may be present in the 4’s to which the modified equation of McGlashan and Potter was fitted. c2) The possibility that these 4’s are too positive TABLE 2. Quantities derived from the experimental excess end&pies HE and the standard deviations o(IP). The mixtures are denoted by the number of carbon atoms in the two n-alkanes T E
~
HE
J mol-l
- oWEI
H,”
-ha
J mol-’
J mol-1
cm3 mol-’
0.45 0.45 0.36 0.26 0.39 0.21 0.15 0.35
5.56 5.07 4.63 4.20 4.30 3.40 3.00 3.16
424 396 364 358 348 330 298 285
w1a
cm3 mol-1
-&a
6&a
cm3 mol-1 cm3 mol-1
e
1+2 241.1 250.6 262.4 266.7 269.2 282.2 298.2 303.2 243.2 245.2 250.8 253.2 261.2 263.6 273.2 281.2 290.7
302.2
5.92 5.41 4.88 4.44 4.53 3.62 3.17 3.30 26.5 26.1 24.2 23.6 21.5 20.1 18.1 16.6 15.0 13.6
0.93 0.59 1.12 0.34 0.69 0.74 0.75 0.40 0.76 0.24
24.2 23.8 22.3 21.8 19.9 18.7 16.9 15.6 14.2 13.0
1+3 581 568 547 534 507 514 480 457 433 400
18 17 15 12 15 11 10 13
141 130 118 116 112 105 93 88
6 4 6 4 4 5
37 31
192 187 180 176 166 168 156 148 139 127
13 11 14 9 10 11 10 8 10 6
40
26 :: 29 2: 18
0.992 0.992
0.990 0.998 0.992 1.008 1.008 0.998
0.970 0.966 0.968 0.966 0.968 0.984 0.984 0.986 0.990 0.986
EXCESS
ENTHALPIES
OF
GASEOUS
TABLE
T ii
P J mol-1
wm J mol-’
HE J mol-1
MIXTURES
OF
n-ALKANES
5
2-wnrinued
-ha cm3 mol-1
WlP cm3 mol-’
-&a cm3 mol-1
S&S cma mol-l
e
-
1+4 277.0 284.2 289.0 303.2 306.5 314.2 328.7 373.4 383.2 394.3
48.9 45.5 42.9 36.9 36.0 31.3 30.5 20.9 19.3 17.9
1.7 1.1 1.3 0.9 0.6 2.3 0.7 2.0 1.0 1.4
44.3 41.6 39.4 34.4 33.6 29.3 28.9 20.1 18.6 17.4
318.5 333.2 343.4 353.2 363.2 373.2 383.4 393.2 403.5
67.6 58.0 53.0 48.7 46.0 40.7 37.2 33.7 32.3
1.1 1.1 1.0 0.8 1.1 0.9 1.0 1.3 1.0
61.8 53.7 49.5 45.7 43.5 38.6 35.5 32.2 31.0
343.2 363.2 373.2 383.2 389.2 407.7
97.7 82.2 74.5 67.8 65.3 56.8
389.2 390.2 405.7 408.2 413.3
104.8 104.3 93.4 90.0 86.4
410.2 418.3
611 570 557 506 490 z 332 322 305
63 48 51 40 35 70 33 60 iit
198 184 179 161 155 162 124 97 93 87
21 17 18 14 12 25 12 22 I3 14
0.970 0.960 0.964 0.962 0.956 1.000 0.928 0.948 0.956 0.956
58 52 48 43 45 40 41 47 37
178 163 150 139 121 125 121 120 105
20 18 17 15 16 14 15 17 14
0.950 0.954 0.946 0.942 0.912 0.952 0.962 0.984 0.958
79 69 66 60 89 56
180
27
:z 134 124 108
2”: 21 32 20
0.940 0.904 0.914 0.924 0.908 0.900
74 75 Ii 91
162 157 127 135 131
26 26 26 35 33
0.944 0.936 0.892 0.920 0.918
1+5 567 524 486 458 408 420 407 405 366
I+6 88.5 75.9 69.3 63.5 61.3 53.9
583 483 468 455 426 383
;: 2:6
96.1 95.8 86.8 83.8 80.7
1+7 542 530 447 469 456
137.7 127.5
2.5 1.6
125.4 116.7
525 521
109 90
151 150
38 32
0.910 0.924
304.5 333.2 363.2
19.0 13.8 10.2
0.31 0.40 0.65
17.3 12.8 9.6
2+4 1026 857 728
33 28 28
332 333 233
10 9 9
0.984 0.987 0.992
372.2 383.2 403.2
51.8 46.6 39.5
1.21 0.72 0.35
47.5 43.0 36.9
2+6 1049 987 878
61 51 42
330 310 274
18 15 13
0.980 0.981 0.976
1.1 1.2
I+8
6
C. J. WORMALD,
E. J. LEWIS, TABLE
T ic
-- HE Jmol-1
403.2 413.2
120.5 110.3
344.2 363.2 396.9
47.4 36.5 27.2
383.2 403.2 413.2
o(H=l Jmol-1
HE Jmol-’
AND
D. J. HUTCHINGS
2-contiwed
-ha cmamol-’
Wla ems mol-l
--Baa ana mol-1
6&Z cm amol-l
112 99
354 331
64 51
0.950 0.944
e
2+8
2;
107.9 99.7
1153 1077
0.7 0.5 0.5
41.7 32.6 25.1
1872 1682 1347
68 57 46
574 519 418
19 16 14
0.988 0.997 0.989
62.9 49.2 47.6
1.0 1.0 1.5
56.0 44.5 43.6
3+7 1759 1600 1457
76 67 71
537 490 448
21 20 21
0.989 1.ooo 0.984
403.2 410.2 413.2
94.9 89.9 86.5
1.75 1.5 1.7
84.0 80.1 77.4
1787 1690 1673
101 93 95
541 512 508
28 27 27
0.977 0.971 0.974
363.2 373.2 383.6 393.2
19.6 17.7 16.4 14.0
1.0 i-Y 1:4
17.3 15.7 14.7 12.7
2309 2157 2010 1902
64 61 62 61
694 651 610 579
17 17 17 17
0.996 0.996 0.994 0.997
403.2 410.5 413.2
70.1 62.7 61.9
1.4 1.3 1.2
61.2 55.2 54.5
2362
98 92 89
735 714 700
343.2 363.2 383.2 403.2
6.9 4.7 4.0 4.3
1.3 1.2 1.2 1.1
5.7 4.2 3.7 4.0
3563 3059 2647 2305
84 74 68 57
1023 893 783 689
21 19 18 15
1.ooo 1.000 1.000 0.997
393.2 403.2 413.2
18.8 16.2 15.0
1.3 1.4 1.3
16.5 14.2 13.3
5+7 2991 2812 2630
80 77 72
871 823 774
20 20 19
0.997 0.999 0.998
403.2 413.2
42.6 38.5
1.5 1.0
37.5 34.5
5+8 3320 3096
101 90
953 895
403.2 413.2
21.7 19.0
1.0 1.2
18.6 16.3
4201 3926
101 96
1173 1105
26 25
0.998 0.999
403.2 413.2
6.2 4.9
1.0 1.0
5.6 4.4
5168 4815
100 95
1407 1322
30 28
0.999 1.000
3+6
3+8
4+6
4+8
E
0.985 0.991 0.989
5-i-6
0.995 0.994
6+8
7+8
EXCESS ENTHALPIES
7
OF GASEOUS htIXTURI?S OFn-ALKANES
because of heat leaks cannot be discounted. reference 3 we obtain
From the first term of equation (6) in
(4) 6412 = P.ewB11x(l -xl)” +@41 II2 + @~22)2~I”2. The uncertainties 6+,, and 6+22 were taken to be 2 per cent of q511 and c$~~, and a(H’) is the standard deviation of the measurements listed in table 1 obtained from HE/4x(l -x) against x. The uncertainty 6B,, in B12 was calculated from the equation : &2
(5)
= S~,,(dB,,/dT)(d~iz/dT)-'.
The temperature derivatives were calculated from the motied equation of McGlashan and Potter. The &,‘s listed in table 2 show no trend with temperature. The mean value of <,, for each mixture is listed in table 3. The average uncertainty on the ca’s is +0.007. It is TABLE 3. Comparison of < obtained from experiment with equation (7). c. is obtained from our HE
measurements, eb from virial-coefficient measurements, COfrom critical-temperature measurements, and & is calculated from equation (7) -.-1+2 l-i-3 1+4 1+5 1+6 1+7
I+8
2+3 2+4 2+5 2+6 2+7 2+8 3+4
e. 0.997 0.997 0.960
0.951 0.917 0.922 0.917 0.987 0.979
1.012
0.991
0.995 0.969 0.949 0.918
1.002
0.992 0.978 0.959
0.947 1.001
0.976 0.958
0.941 0.925 1.009
0.990 0.988 0.982
l.ooo
:E
O:!B6 0.987 0.976 0.965 0.953 0.942 0.997
3+5 3+6 3+7 3+8 4+5 4+6 4+7 4+8 5+6 5+7 5+8 6+7 648 7+8
0.991 0.991 0.974 0.996 0.988 0.999 0.998 0.994 0.998 0.999
0.994 0.983 0.965
0.938 l.ooo
0.994 0.999
0.982 0.988 0.968
0.963 0.999
1.002 1.002 1.003
l.ooo
0.997 0.986
1.001 1.002 1.001
0.992 0.984 0.976
0.967
0.998
0.994 0.989 0.983 0.999 0.996
0.992 0.999 0.997 0.999
interesting to compare these 5,‘s with e’s obtained from other experiments. In a recent compilation of measurements of the second virial coefficients of n-alkane mixtures made by Pompe and Spurling c5) the most extensive set of measurements is that of Dantzler, Knobler, and Windsor (‘) who measured the quantity E = B, 2- +(B, 1+ Bz2) for all 15 binary mixtures of the nalkanes from methane to n-hexane. Using a similar iterative procedure to the above, and using the modified equation of McGlashan and Potter to obtain B, 1 and Bz2,we calculated values of g which we refer to as &,. As the measurements on each mixture were made at 4 temperatures the value of &, included in table 3 is the mean. The only measurements on mixtures containing higher n-alkanes are those of BM made by McGlashan and Potter(‘) on propane + n-heptane and propane + n-octane, and &,‘s obtained from these measurements are also included in table 3. Values of t obtained from measurements of the critical temperature Ts of binary mixtures of n-alkanes including the mixtures from ethane to n-octane are listed by Hicks and Young. (‘I Their value 0.892 for ethane + n-heptane should be
8
C. J. WORMALD,
E. J. LEWIS,
AND D. J. HUTCHINOS
0.982.t’) Values of < obtained from Tb measurements will be referred to as <,, and these are listed in table 3. Inspection of the three sets of 5’s shows the overall agreement to be remarkably good. While a few of the es obtained from both E and TM measurements are greater than unity this is probably experimental error. As the uncertainty in the critical temperatures of these mixtures is often about 1 K, and as the measurements are usually all close to one end of the composition range, the cc’s are probably of no better accuracy than our &,‘s. Agreement between the three sets of r’s is to within the combined experimental error. Indeed the &,‘s obtained from our HE’s deviate on average by less than 1 per cent from the corresponding cc’s. 3. Comparison of 5 with combining rules We have investigated 6 combining rules for T”,, from which we have obtained values of < by dividing through by (T;1T&)“2. The equations for c are t, = (K,~o,2>1’2(~~2>-‘, t2 = 2~1(~1~2)1’2(~, +I,)l, +zJ1, 53 = 2~:(z~z,)1’2(z, C4 = 2(T;,T;,)1’2(T;,+T;,)-1, & = 2(T;,T;2)1’6((T;,)“3+(T~2)1’3}-1,
(6)
(7) (8)
(9) w-3 56 = 2(TflT”,2)“25~~f(V”,2)2~la2~Tfl(~C,~a2)2+T”,2(~‘,2al)2)-‘. (11) Here c1is the polarisability, Z the ionization potential, and VC,, is calculated from the Lorentz rule. c3) Equation (8) i s the combining rule of Hudson and McCoubrey.(lo) Equation (7) is a modification of (8) made following our observation(‘) that HE for methane + n-alkane lies midway between values calculated using c = 1 and equation (8). Equation (6) may be used instead of (7) when the ionization potentials Zr and I2 are similar. Equation (9) is the harmonic-mean rule of Fender and Halsey,“‘) equation (10) is a similar rule suggested byHicks and Young, (*) and equation (11) was suggested Munn.(12) We compare these rules with the T’s listed in table 3. As the ionization energies(13) for the n-alkanes from ethane (1.87 x lo-‘s J) to n-octane (1.64 x IO-‘s J) are much the same, whereas that for methane is 2.11 x 10-r* J, it is informative to examine the mixtures containing methane separately from the rest. For each mixture 5’s were calculated from equations (6) to (ll), the standard percentage deviations from the experimentally determined r’s were obtained, and these are listed for the two groups of mixtures in table 4. Comparison with the &,‘s obtained from compression experiments was restricted to the measurements on the 15 mixtures reported in reference 6. The measurements of Z?, for propane + n-heptane(‘) give a value of &, which is a little low though still in agreement with that calculated from PM but &, for propane + n-octane(‘) is in poor agreement with other values. Table 4 shows that for each mixture both <, and c2 agree equally well with experiment, which suggests that the ionization potentials in equation (7) are unimportant. This term is largest for methane + n-octane for which t1 = 0.901 and r2 = 0.894, a difference of only 0.8 per cent. As the comparison between the r’s obtained from
EXCESS ENTHALPIES
OF GASEOUS MIXTURES
OF n-ALKAIWS
9
TABLE 4. Standard percentage deviations of values of e calculated from equations (6) to (11) from values obtained experimentally Mixtures containing methane e, Cb f: ii 4
Mixtures without methane t-e Cb c
0.95 1.23 6.93 3.97
1.45 1.65 4.89 4.22
0.63 0.68 0.46 2.79
0.37 0.46 1.44
0.89 0.87 0.85 2.28
5.10 10.6
3.92 7.63
1.72 4.10
1.48 2.30
1.60 3.32
experiment and equation (7) is worth making in detail, values of & are included in table 3. While equations (8) to (11) all appear to give better agreement with the results containing no methane, this is deceptive. Most of the mixtures in this group contain n-alkanes of not very different chain length so that the mean value of < is 0.985 whereas for the methane mixtures it is 0.825. For similar molecules even poor combining rules give reasonable values of r. Of equations (8) to (11) the harmonic-mean rule, equation (9), is the best. The Hudson-McCoubrey rule is not as good as earlier work suggested.(14) A good test of equation (7) is the comparison shown in figure 1 of the experimental HE’s for methane + n-alkane with HE calculated using the combining rule as described previously (‘) This figure should be compared with figure 2 of reference 1. In view of the wide difference in pair potentials which we might expect for methane + ethane, methane + n-octane, and n-heptane + n-octane, it is surprising to find a combining rule which fits the measurements on all mixtures to within experimental error. The possibility that our measurements (2) of 4 for n-hexane, n-heptane, and n-octane may be in error must be considered. We therefore repeated the analysis of our HE’s using the original equation of McGlashan and Potter(‘) rather than the modified equation. (2) Agreement between the &,‘s and tc’s is not now so good. The ca’s for the 7 methane mixtures are increased by an average of 3.0 per cent, and the 5;s for the 13 remaining mixtures are increased by an average of 0.45 per cent. For the mixtures of methane with the n-alkanes from ethane to n-octane the ga’s calculated using the original equation are 1.001, 0.999, 0.987,0.986,0.959,0.961, and 0.945. These figures should be compared with the /&‘s and &,‘s listed in table 3. When the HE and E@) results are analysed using the modified rather than the original equation of McGlashan and Potter, closer agreement between the ra’s and &,‘s is found. This suggests that the modification made to the equation so as to fit the 4’s is an improvement, and that the 4’s are not seriously in error. 4.
Cross-term second virial coetllcients
As cross-term second virial coefficients are a primary source of information about pair potentials between unlike molecules, it is important to compare the B,,‘s listed in table 2 with those obtained by other methods. It is convenient to do this by plotting the measurements on a reduced scale using the parameters T; 2 = (T,“, Ti2)1’2 and V; 2
10
C. J. WORMALD,
E. J. LEWIS, AND
D. J. HUTCHINGS
125
I \ \ 100
\ \ \ \ \ \ - \\ \ \
\\
50
\
‘\
\ \ \ \ \
\ \ \ \ \ \ \ \ \
h \ \
\
\\
\
\
\ Q
\ a\ \ b\
\\
\
‘s \
25
250
5 T/K
FIGURE 1. Excess enthalpies of methane + each of the n-alkanes from ethane to n-octane. The dashed Iines are calculated by the method described previously@’ using the modified correspondingstates equation’“) of McGlashau and Potter together with the combining rule T,Oa= &(Z’i’lT&)l/a, where & is given by equation (7). 0, reference 1.
calculated from the Lorentz rule. With this choice the B12’s for different mixtures lie almost on a single curve, in marked contrast to graphs in which either the geometric mean or Hudson-McCoubrey rules for TE, are used. In figure 2 we compare our B12’sfor methane + ethane to methane + n-hexane with reference 6. The solid lines in figure 2 were calculated from the modified equation@) of McGlashan and Potter, and to be consistent we recalculated the B,,‘s reported in reference 6 using this equation. Recalculation makes a significant difference only for methane + n-pentane and methane + n-hexane for which the new B,2’~ are on average 7 per cent less negative. The upper part of figure 2 shows that our B,z’s agree as well with the calculated curves as do the reference 7 measurements in the lower part. A similar comparison for mixtures containing no methane is made in figure 3 where BIZ’s for mixtures of propane + higher n-alkanes are plotted. Both the BIZ’s from reference 6 for mixtures of propane with n-butane, n-pentane, and n-hexane, and our B, i’s for mixtures with n-hexane, n-heptane, and n-octane are in very close agreement
U.i
1.?5
09
1.0
1.1
1.2
1.3
1.4
1.5
16
T/T&
FIGURE 2. Cross-term second virial coefficients for methane + each of the n-alkanes from ethane to n-hexane. The solid lines are calculated as described in the text for the mixtures methane + ethane, methane f n-butane,andmethane + n-hexane.The reducingparametersare T& = &(IIJ&)‘la and VfSgiven by the Lorentz rule. The Bla’s listed in table 2 are plotted on the upper curves and those of reference 6 are plotted on the lower curves. I:), methane + ethane; A, methane + propane; V, methane + n-butane; 0. methane + n-pentane; +, methane -t n-hexane.
b" 1 .A'--L' 3 ,o L0.7
iwe--I_0.8
..- I--L... 0.9 T/T,]
i
FIGURE 3. Cross-term second virial coetlicients for propane + n-alkane from n-butane to n-octane. The solid lines arc calculated as described in the text for propane + n-butane, propane + n-hexane, and propane + n-octane. The reducing parameters are the same as for figure 2. 0, propane + n-hexane (this work); q , propane + n-heptane (this work); 0, propane + n-octane (this work); A, propane + n-butane (reference 6); V, propane + n-pentane (reference 6); 8, propane + n-hexane (reference 6); x, propane + n-heptane (reference 7); +, propane + n-octane (reference 7).
12
C. J. WORMALD,
E. J. LEWIS, AND
D. J. HUTCHINGS
with the calculated lines. The reference 7 results are too positive, and conclusions”4) about combining rules made on the basis of these measurements must be revised. For the remaining 11 mixtures on which we have made HE measurements the B12’s are, on average, in as close agreement with the calculated HE’s as are the B12’s for the propane mixtures. These comparisons suggest that the B,,‘s obtained from our HE measurements are no less accurate than those obtained from compression measurements, and that the limits of error 6B,, in table 2 are probably too wide. REFERENCES 1. Hutchings, D. J.; Lewis, E. J.; Wormald, C. J. J. Chem. Thermodynamics 1978, 10, 559. 2. Al-Bizreh, N. ; Wormald, C. J. J. Chem. Thermodynamics 1978, 10,231. 3. Wormald, C. J. J. Chem. Thermodynamics 1977, 9, 901. 4. Chueh, P. L.; Prausnitz, J. M. AZChEJ 1967, 13, 896. 5. Pompe, A.; Spurling, T. H. C.S.I.R.O. Aust. Div. Appl. Organic Gem. Tech. Pap. No. 3 1976, 1. 6. Dantzler, E. M. ; Knobler, C. M.; Windsor, M. L. J. Phys. Chem. 1968, 72, 676. 7. McGlashan, M. L.; Potter, D. J. B. Proc. Roy. Sot. A 1%2,267,478. 8. Hicks, C. P.; Young, C. L. Chem. Rev. 1975,75, 119. 9. Hicks, C. P. Personal communication. 10. Hudson, G. H.; McCoubrey, J. C. Trans. Faraday Sot. 1960,56,761. 11. Fender, B. E. F.; Halsey, G. D. J. Chem. Phys. 1962, 36, 1881. 12. Munn, R. J. Trans. Faraday Sot. 1961,57, 187. 13. Reed, R. J. Zon Production by Electron Impact. Academic Press: London. 1962. 14. Cruickshank, A. J. B.; Windsor, M. L.; Young, C. L. Trans. Faraday Sot. 1966,62,2341.