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The speed of sound in gases II. Acoustic virial coefficients and perfect-gas heat capacities for 2,2-dimethylpropane obtained using a cylindrical interferometer
o-159 J. Chem. Thermodynamics 1986, 18. 511-517
The speed
of sound
in gases
I I. Acoustic virial coefficients and perfect-gas heat capacities for 2,2-dimethylpropane obtained using cylindrical interferometer
a
M. B. EWING, M. L. McGLASHAN. and J. P. M. TRUSLER Department of’ Chemistry, 20 Gordon Street, London
University College London, WCIH OAJ. U.K.
(Received II September 1985) The speed of sound in 2,2-dimethylpropane has been measured between 250 and 340 K and 7 and 100 kPa, or 0.8 of the vapour pressure, using a fixed-pathlength variable-frequency ultrasonic interferometer with a cylindrical cavity. Second and third acoustic virial coefficients and perfect-gas heat capacities are reported.
1. Introduction The speed of sound in 2,2-dimethylpropane has been determined at 19 temperatures between 250 and 340 K using the fixed-pathlength variable-frequency cylindrical interferometer described previously. (l) Measurements were made at ultrasonic frequencies over the pressure range 20 to 100 kPa at the higher temperatures and between about 7 kPa and 0.8 of the vapour pressure for isotherms below 285 K. In a non-relaxing gas the speed of sound u may be represented by u2 = A 0 +A I p+A,p2+.
. .
(1)
Measurements with a gas of known molar mass M over a suitable pressure range give the ratio yp* of the perfect-gas heat capacities: ypg = l/( 1 -R/C;:,)
= A, M/R T,
(2)
and the second Pa and third ya acoustic virial coefficients: 8, = (M/ypg)A1 = RTAJA,.
(3)
;‘, = (MlrPg)A2
(4)
= RTA,/A,,
which may be related”.2’ to the (p, V,, T) virial coefficients. Since the experimental quantity is (u/L), where L is the pathlength, the coefficients of equation (1) obtained from an analysis of the results are in the form (Ai/L2). Consequently, although /?, 0021-9614/86/060511+07
$02.00:0
tc 1986 Academic Press Inc. (London) Limited
512
M. B. EWING,
M. L. McGLASHAN.
AND
J. P. M. TRUSLER
and ya may be determined directly from a regression analysis, the pathlength must be known before ypg or the heat capacities can be obtained from (&/L’) and equation (2). For the results reported here, our previous work”’ with argon and krypton has been used to define the pathlength. We note that, despite the presence of RT in equation (2), any systematic errors in the temperature are eliminated when heat capacities are determined by this procedure.
2. Experimental The 2,2-dimethylpropane was research-grade material supplied by Cambrian Chemicals. Analysis by g.1.c. using a 46 m x 0.5 mm S.C.O.T. carbowax coiumn with a hydrogen-flame ionization detector indicated a single impurity with mole fraction (5+ 1) x 10e4. Since the impurity was probably a hydrocarbon with molar mass and thermophysical properties similar to those of 2,2-dimethylpropane, the relative effect on the measured heat capacity and acoustic virial coefficients was unlikely to exceed 0.05 per cent. Before use, the 2,2-dimethylpropane was degassed by vacuum sublimation and dried over 4A molecular sieves that had been baked previously under vacuum. The equipment and experimental procedure were the same as described for the measurements with argon and krypton”’ except that the 2,2-dimethylpropane was condensed into an ampoule as the pressure along an isotherm was reduced. Analysis of samples recovered from the resonator showed that there was no contamination and, consequently, it was possible to use the same material (repeatedly dried and degassed) for many isotherms. Vibrational relaxation in polyatomic gases may result in dispersion and anomalous absorption of acoustic energy but Lambert and Slater@) observed no dispersion in 2,2-dimethylpropane at frequencies and pressures such that Since for the results reported here (f/p) was less than (f/p) < 500 Hz.Pa-‘. 12 Hz. Pa- l, dispersion was always negligible.
3. Results Between 66 and 367 resonance frequenciesf,, depending on the pressure range, were determined for each isotherm and the results are summarized in table 1 as mean values of (u/J3 = u&+4, for each temperature and pressure; 1 is the order of the longitudinal mode and 6 is the phase shift.‘” The experimental (u/L)s are sensitive to the value of S used in the analysis; a change in the phase shift of 0.01 (a typical uncertainty for an isotherm) alters (u/L) by about 0.25 s- ’ and the derived values of C,P,,,, /I,, and ya by about O.O6R, 3 cm3 .mol-‘, and 0.02 cm3. kPa-’ . mol-‘, respectively. While such changes in the acoustic virial coefficients are not particularly significant, the pathlength also depends on 6 and meaningful values of the heat capacities will be obtained only when the same phase shift has been applied to the measurements used
SPEED OF SOUND IN C(CH,),(g) TABLE
513
1. Mean values of (u/L) and standard deviations of the results from the means for 2.2-dimethylpropane at various temperatures 7’ and pressures p
T/K
@Pa
251.232
22.151 20.015 25.443 22.873 19.723 33.056 29.270 24.696 40.855 36.450 32.510 48.070 44.362 40.340 36.642 61.398 55.195 48.869 42.513 38.458 74.216 66.915 58.844 51.363 79.544 70.492 99.214 90.781 82.230 73.693 78.163 70.832 62.107 97.538 86.860 75.779 65.779 101.042 92.034 83.203 74.736 96.319 86.166 74.893 67.170 102.144 91.005 82.218 73.438
255.045
260.030
264.967 270.040
275.086
280.241
284.050 290.039
290.293
294.963
300.280
304.176
309.980
(ulL),‘s
’
1744.35kO.22 1746.48 kO.23 1754.75kO.16 1757.22+0.21 1760.15~0.18 1765.43kO.20 1768.75iO.23 1772.72 iO.26 1776.13+0.13 1779.84kO.22 1783.15kO.19 1788.13&0.16 1791.12&0.15 1794.32kO.15 1797.26kO.15 1795.22kO.13 1800.08f0.13 1804.88+0.13 1809.72+0.13 1812.72iO.14 1803.91 kO.14 1so9.39+0.14 1815.32kO.15 1820.79+0.14 1813.75iO.12 1820.27kO.17 1822.19+0.13 1827.63+0.12 1833.32+0.13 1839.11~0.14 1836.63+0.13 1841.93+0.11 1847.68_+0.12 1840.28+0.17 1847.31 kO.14 1854.24~0.15 1860.55+0.18 1857.37+0.14 1862.96kO.14 1868.35+0.13 1873.5370.13 1874.32_+0.17 1880.24kO.15 1886.86+0.16 1891.36+_0.16 1890.60+0.11 1896.82+0.12 1901.76kO.13 1906.62+0.12
PlkPa
(U’L)!SK’
p!kPa
(u/L)K
’
17.362 15.183 16.833 13.900
1749.19kO.25 1751.34kO.27 1762.76+0.21 1765.41 iO.24
13.184 10.825 10.940 7.970
1753.08+0.20 1755.39kO.30 1768.06kO.31 1770.59+0.35
20.800 16.548
1776.15kO.27 1779.79kO.39
13.089 9.394
1782.84+0.41 1785.91 kO.32
28.403 24.473 20.147 32.262 28.314 24.068
1786.55+0.19 1789.73kO.17 1793.40+0.19 1800.69+0.14 1803.72+0.17 1806.99kO.21
15.073 10.212
1797.44+0.17 1801.43+0.20
19.701 15.408 10.263
1810.49kO.28 1813.56+0.18 1817.49+0.20
34.496 30.406 26.533 22.624
1815.64kO.17 1818.72+0.15 1821.57~0.16 1824.33+0.23
18.717 14.884 10.926 7.111
1827.32&0.32 1829.91 kO.19 1832.73kO.26 1835.53+0.14
43.574 35.854 28.005 20.252 53.250 44.485 63.386 54.052 44.899
1826.34_+0.15 1831.77*0.15 1837.26+0.16 1842.67+0.15 1832.37&0.11 1838.46+0.13 1846.03_+0.14 1852.11+0.11 1858.06_+0.13
20.261 14.771 9.686
1842.68kO.23 1846.46kO.22 1849.84+0.20
26.580 17.770 35.361 26.390 17.519
1850.59-tO.15 1856.4320.24 1864.25+0.13 1869.91 kO.18 1875.65kO.24
54.172 43.153 34.673 56.528 47.424 40.116
1852.84kO.14 1859.99kO.16 1865.51+0.14 1866.38kO.14 1872.05kO.16 1876.66kO.20
26.220 20.364
1870.92+0.10 1874.68+0.16
31.370 23.365 16.049
1881.96kO.19 1886.85~021 1891.13~0.21
65.967 57.259 48.117
1878.75&0.14 1884.19kO.14 1889.59+0.13
39.603 30.357 20.266
1894.54kO.14 1899.97+0.14 1905.74kO.19
57.177 48.776 39.486
1897.23kO.16 1902.02+0.16 1907.21 to.16
29.730 20.176 11.145
1912.67~0.19 1918.04kO.24 1922.86kO.42
64.668 55.734 46.514
1911.40~0.13 1916.31&0.13 1921.26+0.13
37.530 28.078 19.267
1926.04+0.15 1931.22_+0.17 1935.90*0.24
514
M. B. EWING.
M. L. McGLASHAN, TABLE
AND
J. P. M. TRUSLER
I--rontinurd
TlK
plkpa
(u/LjJs-’
320.075
99.153 91.488 83.951 76.870
1925.78+0.14 1929.69+0.12 1933.53kO.16 1937.12+0.12
68.714 61.322 52.418
1941.20k0.13 1944.91 kO.14 1949.36kO.15
44.088 35.026 26.205
1953.45+0.14 1957.92kO.15 1962.27+0.17
325.225
97.035 88.367 79.141 70.417
1943.65k0.13 1947.83kO.13 1952.19kO.14 1956.57+0.11
61.457 51.756 42.536
1960.87+0.11 1965.47+0.13 1969.88k0.13
33.502 24.650 15.803
1974.20+0.14 1978.29iO.10 1982.75 + 0.27
329.955
100.232 90.830 81.914 75.538
1957.31+0.15 1961.76kO.12 1965.87+0.12 1970.27f0.11
64.591 58.036 50.228 41.062
1973.90+0.14 1976.92+0.15 1980.49 i 0.12 1984.60+0.13
32.093 24.043 19.001
1988.67f0.13 1992.45kO.14 1994.81 iO.11
334.934
101.089 91.734 82.393 73.590
1972.88k0.16 1977.06&0.13 1981.24kO.16 1985.18kO.14
67.717 59.042 49.116
1987.77&0.13 1991.67kO.11 1995.95*0.11
39.444 30.996 21.045
2000.13~0.14 2004.46+0.17 2008.17~0.10
340.170
98.180 85.201 73.533 64.264
1990.34*0.14 1996.16kO.13 2001.15~0.13 2005.1 l&O.14
55.223 46.195 39.729
2009.06&0.10 2012.88&0.11 2015.58kO.14
33.514 27.136 20.237
2018.38&0.10 2020.95+0.18 2023.76kO.21
@Pa
(IilL)&
’
PlkPa
(u,:L)is
’
to determine L. Consequently, as for the previous work with argon,“’ the isotherms above 293 K were analysed with 6 = 0.128 and those below with 6 = 0.158 and the pathlength was calculated from L/mm = 100.532+0.00221{(T/K)-300) where d, = 0.063 for T < 293 K and zero otherwise. The N resonance frequencies for each isotherm previously”’ and analysed in terms of the equation:
+d,,
(5)
were weighted as described
fl* = j~~+~~/~)2j~~,/~Z~+~~~/~Z~p+~~tl~2~~Z)(6) Table 2 lists the coefficients obtained from these three-term regressions expressed as CpPf,,,lR A, and YA the uncertainties given in the table refer to the 0.99 confidence interval and for Cj?m include contributions, combined in quadrature, from the molar mass and the pathlength. There was some difficulty in determining ya at low temperatures where the pressure range is restricted by the vapour pressure and, to a lesser extent, at the higher temperatures where the third virial coefficient makes a Consequently, although the coefficients are comparatively small contribution. generally significant at a probability of 0.99, the term in A, is significant at a probability of only 0.95 for the isotherms at 251.233 and 329.955 K while it is not significant at 260.030 K. However, the third acoustic virial coefficients of 2,2-dimethylpropane obtained from our work with spherical resonators(4) can be represented by the empirical equation: y.J(cm3. kPa-’
.mol-‘)
= -5.002 x lo-* exp(2100 K/T).
(7)
SPEED OF SOUND IN C(CH,),(g)
515
TABLE 2. Perfect-gas heat capacities Ci:,,,, second p. and third ya acoustic virial coefficients, and standard deviations s(f;*) obtained from N resonance frequencies ,f, by regressions with equation (6). The uncertainties refer to the 0.99 confidence interval T it --
251.232 255.045 260.030 264.961 270.040 275.086 280.241 284.050 290.039 290.293 294.963 300.280 304.176 309.980 320.075 325.225 329.955 334.934 340.170
N
66 94 146 203 260 342 307 231
324 186 360 352 325 327 367 317 361 319 277
cp* 2 R
12.63+0.30 12.70+0.11 12.77 kO.09 12.96 + 0.05 13.25 kO.05 13.47kO.05 13.67kO.05 13.82kO.05 13.93+0.05 13.94+0.06 14.25+0.05 14.55+0.06 14.65 +0.06 14.91+0.06 15.36_fO.O7 15.47kO.06 15.67+0.06 15.85kO.07 15.97+0.07
B, cm3,molm’
2’. cms.kPa-‘.mcr
three-term regressions - 1900+516 -17*15 -196Ok176 -5.654.9 -2050f116 -0.5 f 2.5 -1933*40 -1.00*0.75 -1782133 - 1.77kO.53 -1734*15 - 1.24+0.20 -1670+ 15 -1.06+0.16 -1628+17 -0.90+0.16 -1610+15 -0.29+0.1 I -1601 I20 -0.37*0.20 -1533+13 -0.43+0.11 -1474+13 -0.49*0.10 -1427+15 -0.48+0.12 -1383+ 12 -0.32+0.09 -1288+14 -0.30~0.11 -1274+12 -0.10~0.10 -1234+10 -0.07+0.09 -0.17&0.12 -1185+15 -1152+15 -0.24kO.12
-L SC/3
-I kHz’ 1.12 1.00 1.55 0.95 0.97 0.93 0.92 0.79 0.93 0.69 1.03 0.84 1.00 0.80 0.79 0.75 0.75 0.97 0.70
constrained regressions 12.45kO.07 -2210+47 12.63kO.05 -209Ok26 12.8OkO.05 -1998k 18 12.98&0.04 -1912+7 13.21 kO.04 -1817+6 13.46+0.04 -1749+3 13.66+0&I - 1684+3 13.8OkO.05 - 1637+3 14.04&0.05 -1559*3 13.99+0.05 --1569k3 14.29+0.05 - 1512+3 14.56+0.05 - 1467+2 14.66+0.05 - 1425_+3 14.95+0.05 -1368+2 15.38+0.06 -1282k2 15.52kO.06 -1248+2 15.72f0.06 -1209&Z 15.87+0.06 -1175+3 15.97rt:O.O6 --1152k3
1.14 1.02 1.56 0.95 0.98 0.94 0.93 0.79 1.05 0.73 1.06 0.84 1.00 0.81 0.79 0.78 0.79 0.97 0.70
to within 0.009 cm3. kPa-’ . mol-’ and figure 1 shows that the results given in table 2, including that at 260.030 K, are in good agreement with equation (7) although the deviations are rather irregular at low temperatures. The differences between the second acoustic virial coefficients obtained from the three-term regressions with equation (6) and those determined with the spherical
FIGURE 1. Deviations Aya of the third acoustic virial coefficient ;‘a for C(CH,),, obtained using the cylindrical resonator, from equation (7). which is based on results obtained with a spherical resonator.“”
516
M. B. EWING, M. L. McGLASHAN,
AND J. P. M. TRUSLER
T/K FIGURE 2. Differences A& between the second acoustic virial coefficients /I, for C(CH,), obtained using the cylindrical and a spherical resonator. 14’ 0 Unconstrained three-term regressions; 0, results for the cylindrical resonator constrained using the y. irotn equation (7).
resonatorc4) are shown as the open circles in figure 2. As with ya we note that at low temperatures the deviations are irregular although within the 0.99 confidence interval. The errors resulting from the truncation of an infinite series such as equation (1) are well documented(5,6’ and restrict useful comparisons between virial coefficients obtained from different orders of fit. The true /I, may be obtained if the term in pz is considered but, in common with most experimental studies of virial coefficients near the normal boiling temperature, our results do not unambiguously justify such an analysis for all isotherms. However, this difficulty can be resolved by systematically adjusting the measured resonance frequencies using equation (7) to give the new dependent variable: h2 - {~(l+6)/2)~(A,lL’)(l’alRT), which is then analysed in terms of equation (6) truncated at the term in p. The results of these constrained regressions are given in the second half of table 2 and the final values of /I, are represented by the solid symbols in figure 2. As expected there is little change in & at high temperatures, where ya is small and the full pressure range was accessible. However, below 265 K the fl, now deviate systematically from the results obtained in the spherical resonator. The heat capacities of gaseous 2,2-dimethylpropane were determined by Hossenlopp and Scott (‘) between 298.15 and 523.15 K using vapour-flow calorimetry and their C,P,“,s J‘oin smoothly (to O.OlR) with those obtained with the spherical resonator c4) between 250 and 323.15 K. The close agreement between results obtained using two such different experimental techniques will be discussed
517
SPEED OF SOUND IN C(CH,),(g) I
0.10 0
5 5: 4
l l
-0.
I
l
-(I.?()-
l
I
I
l e
ee 0
0 Oo
I
l
l
0
0
l
l
O 0
I
l ee
l ee
II1 3
I
I
0
0
0
-b 0
0
0
0
0 0 ,
-0.30I 750
I
170
I
I
I
190
,j 0
I
310
330
T/K
FIGURE 3. Deviations AC;*, from equation (8) of the perfect-gas heat capacities Cg,gmof C(CH3), obtained when measurements with the cylindrical resonator are constrained with the sa from equation (7). 0. No boundary-layer corrections; 0, pathlengths L increased by 43 pm to account for the boundary-layer correction in the argon measurements that were used to determine L.
elsewheret4’ but at present it is sufficient to note that C;f,/R = O.O45lO(T/K)+323(K/T),
(8) describes the combined results between 250 and 373 K to O.OlR. Figure 3 shows that the heat capacities calculated from the constrained regressions are systematically lower than equation (8) by 1 per cent which is too large to arise from impurities. However, equation (5) for the pathlength is based on argon measurements which were not corrected for the boundary layer. As noted before”’ these boundary-layer losses are important to the extent that, when they are taken into account, the pathlengths are decreased by (43 + 4) f.rrn which is large compared with the standard deviation of 3 urn of the results from equation (5). Unfortunately, our argon measurements were not sufficiently precise to allow separation of the contributions toS,’ arising from /I, and the boundary layer, which are proportional to p and l/p”‘, respectively. When boundary-layer corrections are applied to the argon measurements (the corrections for 2,2-dimethylpropane are a factor of 5 smaller and may be safely neglected), CpP:,,/Ris increased by about 1 per cent and. as figure 3 indicates, the root-mean-square deviation has been reduced from 0.16 to 0.06R which is of the same size as the estimated uncertainties given in table 2. (p. V’,, T) virial coefficients may be obtained from values of pa and ~1~over a temperature range but this is postponed until the much more precise results obtained with the spherical resonator are presented.
REFERENCES I. Ewing, M. B.: McGlashan, M. L.; Trusler, J. P. M. J. Chem. Thermodynamics 1985, 17, 549. van Dael, W. Experimenfal Thermodynamics. Vol. 2. Le Neidre. B.: Vodar. B.: editors. Butterworths: London. 195, p. 542. 3. Lambert, J. D.; Slater, R. Proc. Roy. Sot. A 1959, 253. 277. 4. Ewing, M. B.; Goodwin, A. R. H.; McGlashan, M. L.; Trusler. J. P. M. to be published. 5. ScottTR. L.; Dunlap, R. D. J. Chem. Phys. 1962, 66. 639. 6. Knobler. C. M. Pure Appl. Chem. 1983, 55,455. 1981, 13. 415. 7. Hossenlopp, 1. A.; Scott, D. W. J. Chem. Thermo&namics 2.
The speed
of sound
in gases
I I. Acoustic virial coefficients and perfect-gas heat capacities for 2,2-dimethylpropane obtained using cylindrical interferometer
a
M. B. EWING, M. L. McGLASHAN. and J. P. M. TRUSLER Department of’ Chemistry, 20 Gordon Street, London
University College London, WCIH OAJ. U.K.
(Received II September 1985) The speed of sound in 2,2-dimethylpropane has been measured between 250 and 340 K and 7 and 100 kPa, or 0.8 of the vapour pressure, using a fixed-pathlength variable-frequency ultrasonic interferometer with a cylindrical cavity. Second and third acoustic virial coefficients and perfect-gas heat capacities are reported.
1. Introduction The speed of sound in 2,2-dimethylpropane has been determined at 19 temperatures between 250 and 340 K using the fixed-pathlength variable-frequency cylindrical interferometer described previously. (l) Measurements were made at ultrasonic frequencies over the pressure range 20 to 100 kPa at the higher temperatures and between about 7 kPa and 0.8 of the vapour pressure for isotherms below 285 K. In a non-relaxing gas the speed of sound u may be represented by u2 = A 0 +A I p+A,p2+.
. .
(1)
Measurements with a gas of known molar mass M over a suitable pressure range give the ratio yp* of the perfect-gas heat capacities: ypg = l/( 1 -R/C;:,)
= A, M/R T,
(2)
and the second Pa and third ya acoustic virial coefficients: 8, = (M/ypg)A1 = RTAJA,.
(3)
;‘, = (MlrPg)A2
(4)
= RTA,/A,,
which may be related”.2’ to the (p, V,, T) virial coefficients. Since the experimental quantity is (u/L), where L is the pathlength, the coefficients of equation (1) obtained from an analysis of the results are in the form (Ai/L2). Consequently, although /?, 0021-9614/86/060511+07
$02.00:0
tc 1986 Academic Press Inc. (London) Limited
512
M. B. EWING,
M. L. McGLASHAN.
AND
J. P. M. TRUSLER
and ya may be determined directly from a regression analysis, the pathlength must be known before ypg or the heat capacities can be obtained from (&/L’) and equation (2). For the results reported here, our previous work”’ with argon and krypton has been used to define the pathlength. We note that, despite the presence of RT in equation (2), any systematic errors in the temperature are eliminated when heat capacities are determined by this procedure.
2. Experimental The 2,2-dimethylpropane was research-grade material supplied by Cambrian Chemicals. Analysis by g.1.c. using a 46 m x 0.5 mm S.C.O.T. carbowax coiumn with a hydrogen-flame ionization detector indicated a single impurity with mole fraction (5+ 1) x 10e4. Since the impurity was probably a hydrocarbon with molar mass and thermophysical properties similar to those of 2,2-dimethylpropane, the relative effect on the measured heat capacity and acoustic virial coefficients was unlikely to exceed 0.05 per cent. Before use, the 2,2-dimethylpropane was degassed by vacuum sublimation and dried over 4A molecular sieves that had been baked previously under vacuum. The equipment and experimental procedure were the same as described for the measurements with argon and krypton”’ except that the 2,2-dimethylpropane was condensed into an ampoule as the pressure along an isotherm was reduced. Analysis of samples recovered from the resonator showed that there was no contamination and, consequently, it was possible to use the same material (repeatedly dried and degassed) for many isotherms. Vibrational relaxation in polyatomic gases may result in dispersion and anomalous absorption of acoustic energy but Lambert and Slater@) observed no dispersion in 2,2-dimethylpropane at frequencies and pressures such that Since for the results reported here (f/p) was less than (f/p) < 500 Hz.Pa-‘. 12 Hz. Pa- l, dispersion was always negligible.
3. Results Between 66 and 367 resonance frequenciesf,, depending on the pressure range, were determined for each isotherm and the results are summarized in table 1 as mean values of (u/J3 = u&+4, for each temperature and pressure; 1 is the order of the longitudinal mode and 6 is the phase shift.‘” The experimental (u/L)s are sensitive to the value of S used in the analysis; a change in the phase shift of 0.01 (a typical uncertainty for an isotherm) alters (u/L) by about 0.25 s- ’ and the derived values of C,P,,,, /I,, and ya by about O.O6R, 3 cm3 .mol-‘, and 0.02 cm3. kPa-’ . mol-‘, respectively. While such changes in the acoustic virial coefficients are not particularly significant, the pathlength also depends on 6 and meaningful values of the heat capacities will be obtained only when the same phase shift has been applied to the measurements used
SPEED OF SOUND IN C(CH,),(g) TABLE
513
1. Mean values of (u/L) and standard deviations of the results from the means for 2.2-dimethylpropane at various temperatures 7’ and pressures p
T/K
@Pa
251.232
22.151 20.015 25.443 22.873 19.723 33.056 29.270 24.696 40.855 36.450 32.510 48.070 44.362 40.340 36.642 61.398 55.195 48.869 42.513 38.458 74.216 66.915 58.844 51.363 79.544 70.492 99.214 90.781 82.230 73.693 78.163 70.832 62.107 97.538 86.860 75.779 65.779 101.042 92.034 83.203 74.736 96.319 86.166 74.893 67.170 102.144 91.005 82.218 73.438
255.045
260.030
264.967 270.040
275.086
280.241
284.050 290.039
290.293
294.963
300.280
304.176
309.980
(ulL),‘s
’
1744.35kO.22 1746.48 kO.23 1754.75kO.16 1757.22+0.21 1760.15~0.18 1765.43kO.20 1768.75iO.23 1772.72 iO.26 1776.13+0.13 1779.84kO.22 1783.15kO.19 1788.13&0.16 1791.12&0.15 1794.32kO.15 1797.26kO.15 1795.22kO.13 1800.08f0.13 1804.88+0.13 1809.72+0.13 1812.72iO.14 1803.91 kO.14 1so9.39+0.14 1815.32kO.15 1820.79+0.14 1813.75iO.12 1820.27kO.17 1822.19+0.13 1827.63+0.12 1833.32+0.13 1839.11~0.14 1836.63+0.13 1841.93+0.11 1847.68_+0.12 1840.28+0.17 1847.31 kO.14 1854.24~0.15 1860.55+0.18 1857.37+0.14 1862.96kO.14 1868.35+0.13 1873.5370.13 1874.32_+0.17 1880.24kO.15 1886.86+0.16 1891.36+_0.16 1890.60+0.11 1896.82+0.12 1901.76kO.13 1906.62+0.12
PlkPa
(U’L)!SK’
p!kPa
(u/L)K
’
17.362 15.183 16.833 13.900
1749.19kO.25 1751.34kO.27 1762.76+0.21 1765.41 iO.24
13.184 10.825 10.940 7.970
1753.08+0.20 1755.39kO.30 1768.06kO.31 1770.59+0.35
20.800 16.548
1776.15kO.27 1779.79kO.39
13.089 9.394
1782.84+0.41 1785.91 kO.32
28.403 24.473 20.147 32.262 28.314 24.068
1786.55+0.19 1789.73kO.17 1793.40+0.19 1800.69+0.14 1803.72+0.17 1806.99kO.21
15.073 10.212
1797.44+0.17 1801.43+0.20
19.701 15.408 10.263
1810.49kO.28 1813.56+0.18 1817.49+0.20
34.496 30.406 26.533 22.624
1815.64kO.17 1818.72+0.15 1821.57~0.16 1824.33+0.23
18.717 14.884 10.926 7.111
1827.32&0.32 1829.91 kO.19 1832.73kO.26 1835.53+0.14
43.574 35.854 28.005 20.252 53.250 44.485 63.386 54.052 44.899
1826.34_+0.15 1831.77*0.15 1837.26+0.16 1842.67+0.15 1832.37&0.11 1838.46+0.13 1846.03_+0.14 1852.11+0.11 1858.06_+0.13
20.261 14.771 9.686
1842.68kO.23 1846.46kO.22 1849.84+0.20
26.580 17.770 35.361 26.390 17.519
1850.59-tO.15 1856.4320.24 1864.25+0.13 1869.91 kO.18 1875.65kO.24
54.172 43.153 34.673 56.528 47.424 40.116
1852.84kO.14 1859.99kO.16 1865.51+0.14 1866.38kO.14 1872.05kO.16 1876.66kO.20
26.220 20.364
1870.92+0.10 1874.68+0.16
31.370 23.365 16.049
1881.96kO.19 1886.85~021 1891.13~0.21
65.967 57.259 48.117
1878.75&0.14 1884.19kO.14 1889.59+0.13
39.603 30.357 20.266
1894.54kO.14 1899.97+0.14 1905.74kO.19
57.177 48.776 39.486
1897.23kO.16 1902.02+0.16 1907.21 to.16
29.730 20.176 11.145
1912.67~0.19 1918.04kO.24 1922.86kO.42
64.668 55.734 46.514
1911.40~0.13 1916.31&0.13 1921.26+0.13
37.530 28.078 19.267
1926.04+0.15 1931.22_+0.17 1935.90*0.24
514
M. B. EWING.
M. L. McGLASHAN, TABLE
AND
J. P. M. TRUSLER
I--rontinurd
TlK
plkpa
(u/LjJs-’
320.075
99.153 91.488 83.951 76.870
1925.78+0.14 1929.69+0.12 1933.53kO.16 1937.12+0.12
68.714 61.322 52.418
1941.20k0.13 1944.91 kO.14 1949.36kO.15
44.088 35.026 26.205
1953.45+0.14 1957.92kO.15 1962.27+0.17
325.225
97.035 88.367 79.141 70.417
1943.65k0.13 1947.83kO.13 1952.19kO.14 1956.57+0.11
61.457 51.756 42.536
1960.87+0.11 1965.47+0.13 1969.88k0.13
33.502 24.650 15.803
1974.20+0.14 1978.29iO.10 1982.75 + 0.27
329.955
100.232 90.830 81.914 75.538
1957.31+0.15 1961.76kO.12 1965.87+0.12 1970.27f0.11
64.591 58.036 50.228 41.062
1973.90+0.14 1976.92+0.15 1980.49 i 0.12 1984.60+0.13
32.093 24.043 19.001
1988.67f0.13 1992.45kO.14 1994.81 iO.11
334.934
101.089 91.734 82.393 73.590
1972.88k0.16 1977.06&0.13 1981.24kO.16 1985.18kO.14
67.717 59.042 49.116
1987.77&0.13 1991.67kO.11 1995.95*0.11
39.444 30.996 21.045
2000.13~0.14 2004.46+0.17 2008.17~0.10
340.170
98.180 85.201 73.533 64.264
1990.34*0.14 1996.16kO.13 2001.15~0.13 2005.1 l&O.14
55.223 46.195 39.729
2009.06&0.10 2012.88&0.11 2015.58kO.14
33.514 27.136 20.237
2018.38&0.10 2020.95+0.18 2023.76kO.21
@Pa
(IilL)&
’
PlkPa
(u,:L)is
’
to determine L. Consequently, as for the previous work with argon,“’ the isotherms above 293 K were analysed with 6 = 0.128 and those below with 6 = 0.158 and the pathlength was calculated from L/mm = 100.532+0.00221{(T/K)-300) where d, = 0.063 for T < 293 K and zero otherwise. The N resonance frequencies for each isotherm previously”’ and analysed in terms of the equation:
+d,,
(5)
were weighted as described
fl* = j~~+~~/~)2j~~,/~Z~+~~~/~Z~p+~~tl~2~~Z)(6) Table 2 lists the coefficients obtained from these three-term regressions expressed as CpPf,,,lR A, and YA the uncertainties given in the table refer to the 0.99 confidence interval and for Cj?m include contributions, combined in quadrature, from the molar mass and the pathlength. There was some difficulty in determining ya at low temperatures where the pressure range is restricted by the vapour pressure and, to a lesser extent, at the higher temperatures where the third virial coefficient makes a Consequently, although the coefficients are comparatively small contribution. generally significant at a probability of 0.99, the term in A, is significant at a probability of only 0.95 for the isotherms at 251.233 and 329.955 K while it is not significant at 260.030 K. However, the third acoustic virial coefficients of 2,2-dimethylpropane obtained from our work with spherical resonators(4) can be represented by the empirical equation: y.J(cm3. kPa-’
.mol-‘)
= -5.002 x lo-* exp(2100 K/T).
(7)
SPEED OF SOUND IN C(CH,),(g)
515
TABLE 2. Perfect-gas heat capacities Ci:,,,, second p. and third ya acoustic virial coefficients, and standard deviations s(f;*) obtained from N resonance frequencies ,f, by regressions with equation (6). The uncertainties refer to the 0.99 confidence interval T it --
251.232 255.045 260.030 264.961 270.040 275.086 280.241 284.050 290.039 290.293 294.963 300.280 304.176 309.980 320.075 325.225 329.955 334.934 340.170
N
66 94 146 203 260 342 307 231
324 186 360 352 325 327 367 317 361 319 277
cp* 2 R
12.63+0.30 12.70+0.11 12.77 kO.09 12.96 + 0.05 13.25 kO.05 13.47kO.05 13.67kO.05 13.82kO.05 13.93+0.05 13.94+0.06 14.25+0.05 14.55+0.06 14.65 +0.06 14.91+0.06 15.36_fO.O7 15.47kO.06 15.67+0.06 15.85kO.07 15.97+0.07
B, cm3,molm’
2’. cms.kPa-‘.mcr
three-term regressions - 1900+516 -17*15 -196Ok176 -5.654.9 -2050f116 -0.5 f 2.5 -1933*40 -1.00*0.75 -1782133 - 1.77kO.53 -1734*15 - 1.24+0.20 -1670+ 15 -1.06+0.16 -1628+17 -0.90+0.16 -1610+15 -0.29+0.1 I -1601 I20 -0.37*0.20 -1533+13 -0.43+0.11 -1474+13 -0.49*0.10 -1427+15 -0.48+0.12 -1383+ 12 -0.32+0.09 -1288+14 -0.30~0.11 -1274+12 -0.10~0.10 -1234+10 -0.07+0.09 -0.17&0.12 -1185+15 -1152+15 -0.24kO.12
-L SC/3
-I kHz’ 1.12 1.00 1.55 0.95 0.97 0.93 0.92 0.79 0.93 0.69 1.03 0.84 1.00 0.80 0.79 0.75 0.75 0.97 0.70
constrained regressions 12.45kO.07 -2210+47 12.63kO.05 -209Ok26 12.8OkO.05 -1998k 18 12.98&0.04 -1912+7 13.21 kO.04 -1817+6 13.46+0.04 -1749+3 13.66+0&I - 1684+3 13.8OkO.05 - 1637+3 14.04&0.05 -1559*3 13.99+0.05 --1569k3 14.29+0.05 - 1512+3 14.56+0.05 - 1467+2 14.66+0.05 - 1425_+3 14.95+0.05 -1368+2 15.38+0.06 -1282k2 15.52kO.06 -1248+2 15.72f0.06 -1209&Z 15.87+0.06 -1175+3 15.97rt:O.O6 --1152k3
1.14 1.02 1.56 0.95 0.98 0.94 0.93 0.79 1.05 0.73 1.06 0.84 1.00 0.81 0.79 0.78 0.79 0.97 0.70
to within 0.009 cm3. kPa-’ . mol-’ and figure 1 shows that the results given in table 2, including that at 260.030 K, are in good agreement with equation (7) although the deviations are rather irregular at low temperatures. The differences between the second acoustic virial coefficients obtained from the three-term regressions with equation (6) and those determined with the spherical
FIGURE 1. Deviations Aya of the third acoustic virial coefficient ;‘a for C(CH,),, obtained using the cylindrical resonator, from equation (7). which is based on results obtained with a spherical resonator.“”
516
M. B. EWING, M. L. McGLASHAN,
AND J. P. M. TRUSLER
T/K FIGURE 2. Differences A& between the second acoustic virial coefficients /I, for C(CH,), obtained using the cylindrical and a spherical resonator. 14’ 0 Unconstrained three-term regressions; 0, results for the cylindrical resonator constrained using the y. irotn equation (7).
resonatorc4) are shown as the open circles in figure 2. As with ya we note that at low temperatures the deviations are irregular although within the 0.99 confidence interval. The errors resulting from the truncation of an infinite series such as equation (1) are well documented(5,6’ and restrict useful comparisons between virial coefficients obtained from different orders of fit. The true /I, may be obtained if the term in pz is considered but, in common with most experimental studies of virial coefficients near the normal boiling temperature, our results do not unambiguously justify such an analysis for all isotherms. However, this difficulty can be resolved by systematically adjusting the measured resonance frequencies using equation (7) to give the new dependent variable: h2 - {~(l+6)/2)~(A,lL’)(l’alRT), which is then analysed in terms of equation (6) truncated at the term in p. The results of these constrained regressions are given in the second half of table 2 and the final values of /I, are represented by the solid symbols in figure 2. As expected there is little change in & at high temperatures, where ya is small and the full pressure range was accessible. However, below 265 K the fl, now deviate systematically from the results obtained in the spherical resonator. The heat capacities of gaseous 2,2-dimethylpropane were determined by Hossenlopp and Scott (‘) between 298.15 and 523.15 K using vapour-flow calorimetry and their C,P,“,s J‘oin smoothly (to O.OlR) with those obtained with the spherical resonator c4) between 250 and 323.15 K. The close agreement between results obtained using two such different experimental techniques will be discussed
517
SPEED OF SOUND IN C(CH,),(g) I
0.10 0
5 5: 4
l l
-0.
I
l
-(I.?()-
l
I
I
l e
ee 0
0 Oo
I
l
l
0
0
l
l
O 0
I
l ee
l ee
II1 3
I
I
0
0
0
-b 0
0
0
0
0 0 ,
-0.30I 750
I
170
I
I
I
190
,j 0
I
310
330
T/K
FIGURE 3. Deviations AC;*, from equation (8) of the perfect-gas heat capacities Cg,gmof C(CH3), obtained when measurements with the cylindrical resonator are constrained with the sa from equation (7). 0. No boundary-layer corrections; 0, pathlengths L increased by 43 pm to account for the boundary-layer correction in the argon measurements that were used to determine L.
elsewheret4’ but at present it is sufficient to note that C;f,/R = O.O45lO(T/K)+323(K/T),
(8) describes the combined results between 250 and 373 K to O.OlR. Figure 3 shows that the heat capacities calculated from the constrained regressions are systematically lower than equation (8) by 1 per cent which is too large to arise from impurities. However, equation (5) for the pathlength is based on argon measurements which were not corrected for the boundary layer. As noted before”’ these boundary-layer losses are important to the extent that, when they are taken into account, the pathlengths are decreased by (43 + 4) f.rrn which is large compared with the standard deviation of 3 urn of the results from equation (5). Unfortunately, our argon measurements were not sufficiently precise to allow separation of the contributions toS,’ arising from /I, and the boundary layer, which are proportional to p and l/p”‘, respectively. When boundary-layer corrections are applied to the argon measurements (the corrections for 2,2-dimethylpropane are a factor of 5 smaller and may be safely neglected), CpP:,,/Ris increased by about 1 per cent and. as figure 3 indicates, the root-mean-square deviation has been reduced from 0.16 to 0.06R which is of the same size as the estimated uncertainties given in table 2. (p. V’,, T) virial coefficients may be obtained from values of pa and ~1~over a temperature range but this is postponed until the much more precise results obtained with the spherical resonator are presented.
REFERENCES I. Ewing, M. B.: McGlashan, M. L.; Trusler, J. P. M. J. Chem. Thermodynamics 1985, 17, 549. van Dael, W. Experimenfal Thermodynamics. Vol. 2. Le Neidre. B.: Vodar. B.: editors. Butterworths: London. 195, p. 542. 3. Lambert, J. D.; Slater, R. Proc. Roy. Sot. A 1959, 253. 277. 4. Ewing, M. B.; Goodwin, A. R. H.; McGlashan, M. L.; Trusler. J. P. M. to be published. 5. ScottTR. L.; Dunlap, R. D. J. Chem. Phys. 1962, 66. 639. 6. Knobler. C. M. Pure Appl. Chem. 1983, 55,455. 1981, 13. 415. 7. Hossenlopp, 1. A.; Scott, D. W. J. Chem. Thermo&namics 2.