- Email: [email protected]
The isothermal Joule-Thomson coefficient of n-hexane, n-heptane, and n-octane
The isothermal Joule-Thomson coefficient of n-hexane, n-heptane, and n-octane NABIL
AL-BIZREH
and CHRISTOPHER
J. WORMALDO
Schoolof Chemistry,The University, Bristol BS8 ITS, U.K. (Received1 April 1977) Usinga flow-calorimetricapparatus,measurements have beenmadeof the isothermal Joule-Thomson coefficientof n-hexane,n-heptane,and n-octanein the range313.15to 393.15K. Valuesof (B-TU/dT) obtainedfrom the measurements have beentestedfor thermodynamic consistency with measurements of the secondvirial coefficient B, and with flow-calorimetricmeasurements of Tadag/dTa. Agreementbetweenthe measurements of B and(B-TdB/dT) isfound, but the valuesof T2d2B/dT2 above0.75T, appearto bein error. Thecorrelationof McGlashanandPottergivesvaluesof (B- TdB/dT) which are 3 per cent more negative than the measurements on n-hexane and 1 per cent more negative than the
measurements on n-octane.A modificationof the correlationwhichgivesbetteragreement with the measurements is proposed.
1. Introduction A flow-calorimetric apparatus for the measurementof the isothermal Joule-Thomson coefficient &, of vapours was described recently. @)Measurements of deviations from ideal-gas behaviour by this technique have the advantage over compression experiments that the results are free from errors due to adsorption. The coefficient $p is simply related to the secondvirial coefficient B and yields the sameinformation about the pair potential. The deviations from the principle of corresponding statesshown by the B’s for the n-alkanes have been correlated by McGlashan and Potter.@) We estimate that the average uncertainty on the measurementsof B to which the correlation was fitted is approximately f3 per cent, but the uncertainty on the n-hexane, n-heptane, and n-octane measurementsis greater than this. In the absenceof accurate measurementsof &-, for the n-alkanes there can be no certainty that differentiation of this empirical correlation will give reliable temperature derivatives. We have made measurementsof 4, for n-hexane, n-heptane, and n-octane in the temperature range 313.15 to 393.15 K. The apparatus and experimental technique were the sameas previously described.01 The hydrocarbons were supplied by British Drug Houses Ltd. The n-hexane was 99 moles per cent pure, while the purities of the n-heptane and n-octane were 99.5 moles per cent. The U.V. spectrum of the n-hexane showed that benzene was present as an impurity. The n-alkanes were distilled using a o To whom correspondence shouldbe addressed.
N. AL-BIZREH
232
AND C. J. WORMALD
40-plate column and were dried over sodium wire. A g.1.c. analysis of the purified materials showed the n-hexane, n-heptane, and n-octane to be 99.97, 99.98, and 99.97 moles per cent pure. No trace of benzene could be detected in the purified n-hexane. The second virial coefficient B is related to +,, and to the pressure derivative of the heat capacity x0 through the equations?) 4,, = lim,,,,,(~H/~p), = B- TdB/dT, (1) x0 = lim~,,o,T(X,/ap), = - TZ(dZB/dT2). (2) In an isothermal throttling experiment at mean pressure (p} = (pr +p,)/2 the quantity obtained is 4,. It was shown previously”’ that 4, = ci)O-t (p)(C’- TdC’/dT), (3) where C’- TdC’/dT = -2B&/RT +(2C- TdC/dT)IRT. (4) Here C is the third virial coefficient and C’ = (C- B’)/RT. 2. Results The results of our measurements on n-hexane, n-heptane, and n-octane are listed in table 1. Where two consecutive runs were done under similar conditions the mean value was taken. These runs are marked in the table, and were given a statistical weight of 2 in the linear regression which was used to obtain the intercepts $. and the slopes (C’- TdC’/dT). The results of the linear regression are listed in table 2 together with the standard deviations of the calculated quantities. As the measurements made at low temperature were not spread over a sufficiently wide range of mean pressure for a meaningful value of the slope to be determined, the slopes obtained at higher temperatures were fitted to a quadratic equation, extrapolated to lower temperatures, and lines of these extrapolated slopes were drawn through the low-temperature points. As before 3or the experimental error was estimated to be + 1 per cent. TABLE 1. Results of measurementson n-alkanes. Runs marked ’ in table 1 were experiments in which two consecutive runs were done under nearly the same conditions, and the mean value of the measurements is reported. These runs were given a statistical weight of 2 in the least-squares programme used to obtain the slopes and intercepts listed in table 2. P mW
----
f iiiizFp2
P1-Pa
kPa
7.138
20.000 23.559 29.309 32.223 24.569
0.4570 0.4874 0.5731 0.6231 0.5699
7.862 8.282 8.317 6.914
49.389 52.968 56.312 60.291 60.961
0.7253 0.7143 0.8155 0.8651 0.8921
12.143 12.158 12.224 12.283 12.029
ISi
14.625 14.827 18.604 21.211 21.783 19.965 22.333 24.329 26.842 28.890
~- 4,
cm3 mol-’
P
my
n-hexane T = 313.15 K -6131 33.480 -6147 27.386 -6175 37.034 -6218 30.862 -6236 T = 323.15 K -5608 a 62.169 -5626 a 64.643 -5649” 68.825 -5673 a 69.205 -5682
f
(P>
4,
mmols-’
PI-P2
kPa
kPa
cm3 mol-1
0.6643 0.6312 0.7204 0.7013
8.062 6.923 8.151 6.956
24.394 25.883 27.764 30.884
-6250 - 6266 -6306 -6326
0.9081 0.9383 0.9926 0.9831
12.037 12.055 12.072 12.267
29.125 31.547 34.740 36.405
-5688 -5714 -5744 -5738
ISOTHERMAL
JOULE-THOMSON TABLE
P a’
f s
PG kPa
k=
cd
233
COEFFICIENTS
l-cuntinued
h mol-l
P
f
mW
mmols-‘kPa
39.442 43.774 49.484 53.450 63.910 69.623
0.6457 0.7338 0.8101 0.8660 0.9421 1.0025
11.985 11.556 11.936 11.996 13.068 13.329
17.235 22.182 25.595 28.578 30.834 33.584
T = 333.15 K -5097 75.819 -5162 80.589 -5117 84.408 -5145 89.047 -5191 90.954 -5210
56.950 59.777 66.793 71.776 72.164 75.701
0.8645 0.9024 1.0302 1.0946 1.1003 1.1466
14.235 14.276 13.904 13.980 13.987 14.006
25.196 27.051 35.010 38.789 39.133 41.899
T = 343.15 K -4628 76.819 -4640 79.567 -4663 83.321 -4690 83.906 -4689 92.225 -4714 93.061
47.629 50.678 51.462 47.830 54.101 57.556 60.518
0.8239 0.8784 0.9047 0.9032 0.9739 1.0341 1.0839
13.472 13.449 13.321 12.382 12.863 12.871 12.955
25.137 28.190 29.981 31.936 35.248 39.071 42.205
-4291 -4290 -4270 -4277 -4318 -4324 -4310
36.091 40.269 41.203 44.860 53.238 55.939 59.129 62.586
0.6942 0.7725 0.8068 0.8713 0.9539 1.0112 1.0638 1.1221
13.259 13.250 12.974 13.063 14.074 13.972 14.001 13.996
19.754 23.636 26.033 29.578 32.187 35.989 39.153 43.141
-3921 -3934 -3936 -3941= -3965 -3959 -3969 -3985
22.721 37.030 42.532 46.153 48.831 50.927 54.007
0.5598 0.7326 0.8221 0.8819 0.9711 1.0117 1.0726
11.215 13.941 14.203 14.296 13.738 13.703 13.709
16.889 21.410 25.672 28.696 35.232 38.098 42.284
-3619 -3626 -3642 -3661 -3660’ -3674” -3673
P1-Pa
kPa
4s cm3 mol-1
1.0681 1.1307 1.1816 1.2378 1.2904
13.562 13.627 13.613 13.547 13.308
36.825 40.418 43.842 47,531 50.871
-5234 -5230 -5247 -5310 -5296
1.1653 1.1986 1.2485 1.2602 1.3627 1.3848
13.962 14.095 14.073 14.037 14.173 14.081
43.526 45.289 48.784 49.738 56.623 58.557
-4721 -4709 -4742 -4743 -4775 -4772
1.1325 1.2022 1.2576 1.3101 1.3614 1.4684
12.996 13.287 13.365 13.470 13.461 14.206
45.388 49.536 53.174 56.739 60.757 66.258
-4339 -4348 -4368 -4417 -4433 -4432
1.1806 1.2455 1.2963 1.3480 1.3839 1.3788 1.4543
14.037 14.088 14.164 14.344 14.045 13.233 13.814
46.994 51.521 54.921 58.347 62.522 65.677 69.187
-4002 -4019 -4027 -4039 -4050 -4067 -4076
1.1317 1.1914 1.2283 1.2727 1.3229 1.3698 1.4374
13.729 13.787 13.721 13.724 13.757 13.943 14.201
46.417 50.662 53.706 56.413 60.878 64.047 68.503
-3695 -3701 -3717 -3727 -3732 -3747 -3758
a o
1.0923 1.1450 1.1892 1.2363 1.2827 1.3238 1.3657 1.3830
13.475 13.517 13.462 13.515 13.565 13.607 13.619 13.399
45.823 49.688 53.383 57.118 60.647 64.006 67.395 70.149
-3434 -3452” -3473 -3474 -3482 -3486 -3508 -3506
a
T = 353.15 K a a
e a
63.870 69.467 73.419 77.939 81.242 92.460
a a
T = 363.15 K
a
66.329 70.524 73.943 78.106 78.733 74.217 81.887
o
T = 373.15 K
a
a
57.417 60.790 62.656 65.104 67.937 71.562 76.717
T = 383.15 K 30.959 34.514 37.419 37.113 40.732 40.645 43.204 45.997 48.159
0.6722 0.7435 0.8011 0.8113 0.8652 0.8857 0.9419 0.9956 1.0396
13.567 13.706 13.812 13.408 13.824 13.437 13.437 13.470 13.484
19.654 23.163 26.044 27.395 29.689 31.772 35.403 38.929 41.950
-3395 -3387 -3382 -3412 -3405 -3415 -3414” -3430 -3436
a
a
50.535 53.435 55.594 58.039 60.591 62.799 65.248 64.991
234
N. AL-BIZREH
AND
TABLE P mT
36.033 39.572 41.965 44.397 47.684 50.338
f mmols-’
0.8205 0.8872 0.9480 1.0051 1.0647 1.1193
C. J. WORMALD
l-continued
PI -Pz kPa
_~ kPa
4, cm3 mol-’
13.816 14.003 13.956 13.940 14.065 14.035
27.996 31.632 35.630 39.511 43.522 47.470
T = 393.15 K -3179 52.816 -3185 54.104 -3172 57.760 -3168 60.394 -3183 62.219 -3204
kPa
1.1696 1.2069 1.2657 1.3121 1.3537
14.095 13.901 14.132 14.165 14.167
51.176 54.898 58.560 62.324 65.741
-3204" -3224 -3229 -3249 -3244
0.6761 0.7416 0.8034
11.984 11.939 12.068
21.067 24.734 28.721
-- 7374 -7439 - 7499
0.8543 0.8595 0.9091 0.9747 1.0260
13.369 12.897 12.210 12.233 12.254
29.665 30,858 35.957 40.542 44.163
- 6768 -6804 -6845 -6890 a -6936
0.9628 1.0135 1.0694 1.1253 1.1754 1.2320
12.878 12.833 12.816 12.964 12.956 13.087
39.271 43.117 47.422 51.357 55.345 59.632
-6248 -6236” -6279 -6334 -6394 -6446
K 43.482 64.612 49.428 70.815 75.584 58.624 68.603
0.7443 0.8888 0.8222 0.9660 1.0305 0.9441 1.0179
10.285 12.754 10.511 12.774 12.834 10.746 11.659
32.090 35.675 37.503 41.254 45.968 47.032 49.389
- 5679 - 5699 -5720 -5734 -5715 -5778 -5781
K 55.481 58.481 55.481 58.202 59.429 62.316 64.684
0.9208 0.9435 0.9408 0.9834 1.0016 1.0416 1.0835
11.306 11.676 11.093 11.066 11.050 11.117 I 1.025
44.175 44.665 46.567 50.513 52.226 55.729 60.162
-5329 -5308 -5316” -5348 -5369 -5381 -5415
K 48.085 49.940 54.361 51.568 58.200 60.917 49.168
0.8488 0.8877 0.9393 0.9265 0.9943 1.0367 0.9785
11.656 11.507 11.724 11.268 11.853 11.854 10.170
37.877 41.347 44.877 46.082 49.061 52.741 56.046
-4861 -4889 -4936 -4939 -4938 -4957 -4941
28.588 53.664 55.283
0.4207 0.5896 0.6233
9.248 12.324 12.038
12.160 16.324 18.325
n-heptane T = 343.15 K -7347 59.754 -7385 65.873 -7368 72.719
44.806 52.333 64.103 60.763 70.243
0.5423 0.6348 0.7309 0.7351 0.7975
12.361 12.286 13.078 12.222 13.015
14.630 19.118 23.003 24.715 26.877
T = 353.15 K -6684 a 77.307 -6710 a 75.430 -6705" 75.987 -6763" 82.161 -6768" 87.206
38.882 47.343 52.529 54.233 59.503 66.129 69.491
0.4896 0.5913 0.6583 0.6742 0.7397 0.8350 0.8922
12.989 13.073 13.091 13.085 13.091 12.776 12.539
12.084 16.592 19.662 20.741 24.383 30.826 35.280
27.281 37.862 43.876 48.151 47.740 58.344 59.185
0.4438 0.5541 0.6373 0.7066 0.7401 0.8076 0.8242
11.024 12.189 12.275 12.122 11.432 12.838 12.644
12.542 16.523 20.746 25.061 28.809 30.103 31.488
30.070 35.182 39.538 44.299 45.400 44.847 50.492 53.057
0.4957 0.5794 0.6471 0.7171 0.7427 0.7825 0.8523 0.8898
11.730 11.681 11.755 11.831 11.665 10.889 11.172 11.204
14.609 19.161 23.013 27.215 29.405 34.360 38.964 41.822
23.572 30.850 34.775 37.047 36.080 38.013 39.435 42.000
0.4709 0.5598 0.6278 0.6785 0.6821 0.7130 0.7489 0.7741
10.407 11.518 11.538 11.339 10.999 11.011 10.843 11.204
15.681 18.899 22.885 26.556 27.713 29.843 32.637 33.768
T= -6113 -6124' -6095 -6147 -6145 -6198 -6211" T= -5576 -5606 -5608 -5621 -5643" -5627 -5679
363.15
K 77.466 81.107 86.058 92.401 97.387 103.943
T= -5171 -5198 -5198 -5221 -5240 -5263 -5303 -5322 T= -4810" -4784 -4801 -4815 -4809 -4842' -4856" -4843" -____-~
383.15
373.15
393.15
4, cm3 mol-’
a
a
q
ISOTHERMAL
JOULE-THOMSON
COEFFICIENTS
235
TABLE 1-continued P f mwiGGiPkpa
Pr-Pa
kpa
30.323 36.140
0.3738 0.4426
9.017 9.029
11.855 15.765
36.002 67.623 72.675
0.4136 0.6374 0.6912
10.553 12.665 12.517
12.611 22.017 25.695
43.874 53.490 62.209 57.611 73.063 64.061 69.779 79.324
0.4672 0.5406 0.6181 0.6052 0.6869 0.6695 0.6945 0.7461
12.475 12.970 13.129 12.500 13.966 12.511 13.039 13.904
13.591 16.791 20.827 20.988 23.660 24.976 25.647 27.385
32.836 41.368 48.212 53.523 56.575 64.059 67.130
0.4019 0.4901 0.5671 0.5988 0.6671 0.7358 0.7613
11.666 12.088 12.123 12.839 11.982 12.239 12.431
11.600 15.668 19.933 20.821 26.776 31.235 32.652
P iis
4P cm8 mo+
f
.Pl-Pa kPa
ZiGi
0.5084 0.5629
9.225 9.097
19.514 23.453
-9093 -9169
0.7747 0.8277 0.8857
13.048 12.863 12.787
30.006 34.036 38.607
-8454 -8556 -8610
0.7704 0.8278 0.8057 0.8931 0.8715 0.8888 0.9443 1.0062
12.995 14.487 12.232 14.394 12.706 12.936 13.101 13.980
30.863 31.564 35.267 36.202 39.071 39.758 43.759 45.975
-7762 -7704 -7807 -7777 -7841 -7831 -7886 -7904
0.8192 0.8794 0.8989 0.9230 0.9482 0.9777
12.458 12.628 12.405 12.393 12.324 12.535
37.032 41.495 43.757 45.849 48.214 50.255
-7143 -7172 -7198” - 7229 -7281 -7289
mmols-’
n-octane T = 363.15 K -8994 42.645 -9044 46.956 T = 373.15 K -8249 85.465 -8376 91.087 -8399 97.522 T = 383.15 K -7527 77.703 -7628 92.399 -7665 76.940 -7616 99.971 -7615 86.818 -7647 90.042 - 7705 97.564 -7647 111.190 T = 393.15 K -7004 72.904 -6982 79.647 -7013 80.260 -6962 82.694 -7077 85.078 -7113 89.347 -7093
4. ems mol-1
TABLE 2. Values of (B-TdB/dT) and of (C’-TdC’/dT) with their standard deviations o calculated as described in the text from the results listed in table 1. The values in parentheses were obtained by extrapolation as described in the text T K-
B-TdB/dT cm5 mol-X
d cma mol-l
Q
cm3 mol-l kPa-I
n-hexane
333.15 323.15 333.15 343.15 353.15 363.15 373.15 383.15 393.15
16
C’-TdC’IdT cm3 mol-l kPa-’ (-8.62) ‘I.82 -4156 -3.77 -3.24 -2.69 -2.40 -2.09
0.21 0.30 0.21 0.13 0.11 0.10 0.17
n-heptane
343.15 353.15 363.15 373.15 383.15 393.15
‘I;;:? -5995 -5495 -5086 -4711
363.15 373.15 383.15 393.15
(-8813) -8082 -7420 -6851
6-y; -6:60 -5.52 -5.25 -4.36
0.15 0.13 0.10 0.25 0.27
n-octane ‘~;‘;;I
ii 23
-10:43 -8.18
1.01 0.82 0.64
236
N. AL-BIZREH
AND C. J. WORMALD
3. Third virial coeilicients The values of (Cl- TdC’/dT) listed in table 2 are plotted in figure 1. The solid lines in the figure are calculated from equation (4) using the correlation of McGlashan and Potter to obtain the term - 2B&/RT and the correlation of Chueh and Prausnitzc3’ to obtain the term in C. In this latter correlation we used values of the parameter d of 3.45, 3.90, and 4.25 for n-hexane, n-heptane, and n-octane. Agreement between the experimental and calculated slopes is within experimental error. At the bottom of figure 1 the terms in C in equation (4) for the three n-alkanes are plotted; these terms are only about 15 to 20 per cent of the terms in B. As no measurements of C below T/T, = 0.8 were available to Chueh and Prausnitz when they constructed their correlation they were unable to recommend its use below this temperature. Our measurements are all below T/T, = 0.8 and extend down to T/T, = 0.64. Within the experimental errors of our slopes the correlation of Chueh and Prausnitz gives a satisfactory representation of the C term.
FIGURE 1. Experimental and calculated values of (C’-TdC’/dT). The upper curves (a, n-hexane; b, n-heptane; c, n-octane) are calculated from equation (4). The lower curves are the terms in C in this equation. The error bars on the points are the standard deviations u listed in table 2.
4. Analysis of the +O measurements Following our previous procedure”’ we set out to fit measurements of B, &, and x0 by the method of least squares both to polynomial equations in powers of the reciprocal temperature and to equations of square-well form. In performing the analysis previously on benzene we were fortunate to have measurements of B, I$~, and x0 which were thermodynamically consistent, so that changing the statistical weight on any one set of measurements had little effect on the resulting least-squares equation. This was found not to be the case for the nalkanes.
ISOTHERMAL
JOULE-THOMSON
COEFFICIENTS
237
The heat capacity of n-hexane vapour has been measured by Waddington and Douslin,(4’ and that of n-heptane by Waddington, Todd, and Huffmann; no measurements on n-octane have been made. The two sets of measurements are each at five temperatures but are mostly at two pressures only. The precision of the n-hexane measurements is +0.2 per cent and that of the n-heptane measurements is +O.l per cent. By drawing lines of maximum and minimum slope through the points at the two experimental pressures with the above limits of error on each point we obtained maximum and minimum values of dC,ldp. For n-hexane the limits of error are + 18 per cent at 333.84 K and + 166 per cent at 468.19 K, while for n-heptane they are + 12 per cent at 357.09 K and & 67 per cent at 466.09 K. A preliminary calculation showed that for both n-hexane and n-heptane no equation which would fit the B,&, and x0 measurements to within experimental error could be found, but when the x0’s were omitted a good simultaneous fit of the B and 4. measurements was obtained, and the computed values of x0 agreed with the experimental values to well within the above wide limits of error. Furthermore the uneven distribution of the B and +. measurements made it difficult to obtain satisfactory equations of polynomial form. As no such difficulties were experienced using the square-well form we continued the analysis using this equation, fitting the B and #o measurements only, and giving equal weight to all points. For n-hexane we selected the B’s measured by McGlashan and Patter,(2) Bottomley and Reeves,@’ Di Zio, Abbott, Zibello, and Van Ness,(‘) and the values calculated by Pompe and Spurling (‘1 from the p,V,T measurements of Grisky and Canjarcg) and Kelso and Felsing. (lo) The 39 values of B together with the 9 values of 4. listed in table 2 are fitted by the equation B/cm3 mol-’
= 337.4-206.8
exp(703.8 K/T).
(5)
The standard percentage deviations of the B, $I~, and x0 measurements are 4.2,0.6, and 41.4. For n-heptane we selected the B’s measured by McGlashan and Patter,(2) and those calculated by Pompe and Spurling (*) from the p, V,T measurements of Smith, Beattie, and Kay. (11)The 14 values of B together with the 6 values of c/J,,listed in table 2 are fitted by the equation: B/cm3 mol-’
= 362.0-209.0
exp(813.5 K/T).
(6)
The standard percentage deviations of the B, 40, and x0 measurements are 1.8, 0.2, and 25.4 For n-octane we selected the B’s measured by McGlashan and Potter(‘) and by Connolly and Kandalic. (r2) The 15 values of B together with the 4 values of +. listed in table 2 are fitted by the equation: B/cm3 mol-r
= 478.9-246.8
exp(872.7 K/T).
(7)
The standard percentage deviations of the B and $. measurements are 1.2 and 0.1. The values of $. listed in table 2 are plotted as reduced quantities in figure 2. The diameter of the circles in the figure is approximately equal to the + 1 per cent error bar on the measurements. The broken lines in figure 2 are calculated from the correlation
238
N. AL-BIZREH
AND C. J. WORMALD
I
0.62
0.66
.~I
0.70
0.74
0.78
T/T, FIGURE 2. The reduced zero-pressure isothermal Joule-Thomson coefficients of a, n-hexane; b, n-heptane; and c, n-octane. The broken lines are calculated from the correlation of McGlashan and Potter.(2’ The solid lines are calculated from a modification of this correlation in which the exponent 4.5 is replaced by equation (11).
of McGlashan and Potter.@’ The &,‘s for n-hexane, n-heptane, and n-octane are on the average 3.0, 1.0, and 0.6 per cent less negative than the lines calculated from this correlation, and this agreement is about as good as can be expected. The above systematic trend in the deviation of the&,‘s from the correlation suggests that a slight modification of the equation might secure better agreement with the &,‘s. To investigate this possibility we expressed the deviation of the reduced second virial coefficient from the principle of corresponding states by an equation of the form (B--&J/K
= m-
uccmb,
(8)
where B,, is the second virial coefficient of a fluid which obeys the principle, a and b are adjustable parameters, and N has its previous meaning.(2) When B/V, = 0.430-0.886(TC/T)-0.694(TC/T)2,
and with a = -0.0375 and b = 4.5, equation (8) is that of McGlashan From equations (1) and (8) it follows that
(9)
and Potter.@)
(40 - (b,N’c = 41 +bW-- WT,/~>b, (10) where 4,, is obtained from equations (1) and (9). The coefficients a and b in equation (10) were obtained by fitting the values of (be listed in table 2, with N = 6,7, and 8, by the method of least squares. For n-hexane we obtained a = -0.0359, b = 4.44; for n-heptane a = -0.0365, b = 4.47; and for n-octane a = -0.0403, b = 4.32. The mean value of a is -0.0375, and confirms the original t2) choice of this constant. Using this value of a, the value of b which gives the best overall fit to the &O’s is 4.44. While equation (10) with b = 4.44 fits all the &,‘s to within the f 1 per cent error on the
ISOTHERMAL
JOULE-THOMSON
COEFFICIENTS
239
measurements, it lies parallel to the best line through the n-hexane &,‘s but is 1 per cent too negative. Best agreement between equation (10) and experiment can be secured by letting b be a function of N, such that b = 4.18+0.04(N-1).
(11)
The solid lines in figure 2 were calculated using equations (10) and (11). Using equation (11) for b in place of 4.5 slightly alters the fit to the B’s for the n-alkanes. The mid-point temperature of the B’s for n-hexane, n-heptane, and n-octane reported in reference 2 is approximately at T/T, = 0.7. At this reduced temperature the use of equation (11) instead of 4.5 for b diminishes the calculated B’s by the following percentages: n-octane, 0.5; n-heptane, 0.6; n-hexane, 0.9; n-pentane, 1.3; n-butane, 1.4; propane, 1.1; ethane, 0.8. As the accuracy of the virial-coefficient measurements themselves is no better than f 3 per cent the use of equation (11) makes only a marginal difference to the fit. Inspection of graphs of B/V, against T/T, for all the above n-alkanes shows that while equation (11) for b fits the measurements as well as does b = 4.5, the uncertainty in the measurement does not permit a choice between these alternatives. The difference between the values of & calculated using b = 4.5 or equation (11) is largest for n-butane. Neither the measurements of the isenthalpic Joule-Thomson coe5cient of n-butane made by Kennedy, Sage, and Lace~,“~) nor their measurements on propane, are of sufficient accuracy to distinguish between these alternatives for b.
5. The pressure derivative of the heat capacity Values of x0 calculated from the heat-capacity measurements reported in references 4 and 5 are plotted on a reduced scale in figure 3. The error bars shown are the maximum limits of error obtained as described above. The solid lines are calculated from equation (10) using b given by (11) while the broken lines are calculated from equations (5), (6), and (7). Also shown in the figure are values of x0 for benzene obtained from the measurements of Scott, Waddington, Smith, and Huffmann,04) and of Douslin, Todd, and Hossenlopp. (15)As the benzene heat-capacity measurements were each made at 4 or 5 pressures the error in dCddp is smaller than that for the n-hexane and n-heptane measurements by a factor of between 3 and 4. The solid line through the benzene x0’s is calculated from equation (10) using N = 4 and b = 4.5. It was shown previouslycg 16) that with these parameters equation (10) fits the B, 40, and x0 measurements on benzene to within experimental error on all three quantities. While the x0’s for n-hexane and n-heptane at the three temperatures below 0.75T, are in good agreement with equation (10) using b given by (1 l), at temperatures above 0.75T, they appear to be too positive, and even cross the line through the benzene points. As our #o’s for benzene, n-hexane, n-heptane, and n-octane form a family of curves which show no sign of crossing, it is improbable that the x0 results should behave differently, and we suggest that the x0’s for n-hexane and n-heptane above 0.75T, are in error. Although equation (10) does not pass through the points, it does lie within the limits of experimental error. Equations (5), (a), and (7) form a family of curves which, on graphs of B/V,, t#@,, and even x0/V, against T/T,, are almost indistinguishable from equation (10) with b
240
N. AL-BIZREH
0.6
AND C. J. WORMALD
0.7
0.8
0.9
1.0
T/ T,
FIGURE 3. Reduced values of x0 obtained from measurements of the pressure dependence of the heat capacity. The solid lines (a, n-hexane; b, n-heptane; c, n-octane) are calculated from a modification of the correlation of McGlashan and Potter(a) in which the exponent 4.5 is replaced by equation (11). The broken lines are calculated from equations (5), (6), and (7). The upper solid line d is for benzene, and is calculatedcB) from the unmodified correlation of McGlashan and Potter. The error bars on the n-hexane and n-heptane points are calculated as described in the text. The error bars on the benzene points are indicated in reference 16. A, benzene, reference 14; V , benzene, reference 15; 0, n-hexane, reference 4; Cl, ?r-heptane, reference 5.
given by (11). As equations (5), (6), and (7) were obtained using approximately three times as many B’s as (boys,the very close agreement with equation (10) suggests that the use of equation (11) for b, in place of 4.5, improves the fit to the B’s as well as to the 4cl’s. While equation (11) best fits the measurements of r& for n-hexane, n-heptane, and n-octane, the choice b = 4.44 is almost as good. Only accurate measurements of &, for propane or n-butane, for which equation (11) gives b = 4.26 and 4.30 respectively, will confirm that the use of this equation rather than b = 4.44 is justified. For the calculation of the second virial coefficients of the n-alkanes the use of equation (11) is an unnecessary complication, and the replacement of the exponent 4.5 by 4.44 is a sufficient improvement.
ISOTHERMAL
JOULE-THOMSON
COEFFICIENTS
241
REFERENCES 1. Al-Bizreh, N.; Wormald, C. J. J. Chem. Thermodynamics 1977, 9, 749. 2. McGlashan, M. L.; Potter, D. J. B. Proc. Roy. Sot. A. 1%2,267,478. 3. Chueh, P. L.; Prausnitz, J. M. AZChE.J. 1%7, 13, 896. 4. Waddington, G.; Douslin, D. R. J. Am. Chem. Sot. 1947, 69, 2275. 5. Waddington, G. ; Todd, S. S. ; Huffmann, H. M. J.A.C.S. 1947, 69, 22. 6. Bottomley, G. A.; Reeves, C. G. ; J. Chem. Sot. 1958,4, 3794. 7. Di Zio, S. F.; Abbott, M. M.; Zibello, D.; Van Ness, H. C. Znd. Eng. Chem. Fund. 1966, 5, 569. 8. Pompe, A. ; Spurling, T. H. Virial coefficients for gaseous hydrocarbons. Paper No. 1. Commonwealth Scientific and Industrial Research Organisation; Australia 1974. 9. Griskey, R. G.; Canjar, L. N. AZChE. J. 1959, 5, 29. 10. Kelso, E. A.; Felsing, W. A. J. Am. Chem. Sot. 1940, 62, 3132. 11. Smith, L. B.; Beattie, J. A.; Kay, W. C. J. Am. Chem. Sot. 1937, 59, 1587. 12. Connolly, J. F.; Kandalic, G. A. Physics Fluids 1960, 3, 463. 13. Kennedy, G. C. ; Sage, B. H. ; Iacey, W. N. Znd. Eng. Chem. 1936,28,718. 14. Scott, D. W.; Waddington, G.; Smith, J. C.; Huffmann, H. M. J. Chem. Phys. 1947, 15, 565. 15. Douslin, D. R.; Todd, S. S.; Hossenlopp, I. A. Bartlesville Petroleum Research Centre, U.S. Bureau of Mines, Oklahoma, personal communication. 16. Wormald, C. J. J. Chem. Sot. Faraday Trans. Z 1975, 71, 726.