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Acoustic determination of ideal gas heat capacity and second virial coefficient of small hydrocarbons
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I I ELSEVIER
Fluid Phase Equilibria 105 (1995) 173-192
Acoustic determination of ideal gas heat capacity and second virial coefficient of small hydrocarbons Gfinter Esper, W o l f g a n g L e m m i n g , W i l h e l m B e c k e r m a n n , F r i e d r i c h K o h l e r * Institute of Thermo- and Fluid Dynamics, Ruhr University Bochum, D-44780 Bochum, Germany Received 3 May 1994; accepted in final form 8 September 1994
Abstract
A spherical resonator was constructed to measure the resonance frequencies of sound of various gases. The measurement of the resonance curve was automated to take place at intervals of a tenth of the half-width on both sides of the maximum, and was taken forwards and backwards so that any lack of equilibrium could be seen immediately on the screen. The resonator was placed in a thermostatted autoclave. The diameter of the resonator was determined by calibration with argon. In general, 5-10 isotherms were measured between 250 and 350 K and at pressures between 0.025 and 0.5 MPa for various resonance modes. The evaluation of the sound speeds was done in two ways. First, the acoustic virial coefficients were evaluated for each isotherm, and the ideal gas heat capacity and second thermal virial coefficient derived from these isothermal fits. The second way was a simultaneous fit for all points, assuming temperature functions for the acoustic virial coefficients, but calculating the coefficient of the square term in pressure consistently from the thermal virial coefficients. Results are presented for nitrogen, methane, chlorodifluoromethane (R22), ethane and propane.
Keywords: Experiments; Method; Data; Sound speeds; Ideal gas heat capacities; Second virial coefficients; Hydrocarbons: Nitrogen; Chlorodifluoromethane
I. I n t r o d u c t i o n
The measurements of sound speeds in gases serves two purposes. At pressures below 0.5 MPa, the square of the sound speed c can be developed into a power series in pressure up to the square term: c 2 =Ao + Alp +A2p 2
* Corresponding author. 0378-3812/95/$09.50 @, 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 3 8 1 2 ( 9 4 ) 02608-4
(1)
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
174
where Ai are called acoustic virial coefficients, and where Ao and A1 have important physical significance. A0 is connected with the ratio of ideal gas heat capacity at constant pressure to that at constant volume k0,
Ao = k o R T / M
(2)
where M is the molar mass, so that the ideal gas heat capacity can be evaluated:
C°p/R = ko/(ko - 1)
(3)
The coefficient of the term linear in pressure, A1, is connected to the second thermal virial coefficient B by
A, = (ko/M)(2B + 2B'(ko - l) + B"(ko - 1)2/ko)
4)
where B' stands for T d B / d T and B" stands for T 2 d2B/dT 2. With the construction of spherical resonators, and with the develoment of the small correction terms to the ideal resonator (Mehl and Moldover, 1981; Moldover et al., 1986; Ewing et al., 1986) the sound speed can be measured with an accuracy to better than 0.01%. With k0 given 0 can be evaluated even for polyatomic molecules (where k0 might to an accuracy 0.01-0.02%, Cp be as low as 1.1) to an accuracy of 0.1-0.2%, which is better than possible by direct calorimetric measurement. The situation is not so clear-cut for the second (thermal) virial coefficient. Compared with other experimental methods, the sound speed has the big advantage that systematic errors due to adsorption are practically absent, but the difficulty is caused by the B' and B" terms in Eq. (4). The usual procedure is to assume a certain temperature function B(T) and to determine the parameters of this function such that Am(T) is given in an optimal way. First, the question is how to get the best values of A~(T). We have chosen two possibilities. One is to fit the c 2 values along each isotherm to Eq. (1). The disadvantage of this isothermal fit is a partial compensation of errors in A~ and A2. The other possibility is to fit the c 2 values of all isotherms to temperature functions Ao(T) and A~(T). Here the A2(T) values were determined via correlation of the second and third (thermal) virial coefficients according to the equations
A2
(a2B + a z c ) / ( m R T )
=
a2B
=
- ( k o + l)B 2 + (k0 - 1)(2ko - 1)2B '2 + (ko + (ko - 1)2(2BB ' + BB" + (4ko - 2)B'B")
a2c = (2ko + 1)C + (ko2 - 1)C' + (ko - 1)2C"/2
(5) 1)3B
''2
(5a) (5b)
where again C ' = T d C / d T and C " = T 2 d2C/dT 2. The simultaneous fit has the advantage that it smooths small inaccuracies in the temperature measurements, and that it ensures consistency between A~ and A2. It has the disadvantage that the temperature functions Ao(T ) and A~(T) may not be optimal. The method of calculation was first to assume values of B(T) and C(T), to calculate the A2 values from them, then to fit the coefficients of the Ao(T) and A~(T) functions, then to calculate from A~(T) improved values of B(T), and to repeat the iteration loop. The functions used here for Ao(T) and A~(T) were
175
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173 192
Ao = aol T + ao2 T 2 + ao3 T 3 4- a04 T 4
(6a)
A1 = a~o + a ~ / T + a l 3 / T 3
(6b)
The second question is w h a t properties the function B ( T ) should have for obtaining the best results in B. Test calculations have shown t h a t the function B(T) should n o t c o n t a i n m o r e t h a n three p a r a m e t e r s when A ~ ( T ) is given for a t e m p e r a t u r e interval n o t larger t h a n 100 K. In addition - - to ensure t h a t B ( T ) has a reasonable slope and c u r v a t u r e on the upper end o f the interval - - we have always calculated the Boyle t e m p e r a t u r e (usually quite a way above o u r
1 2
3
/.
5 6 7 8 1
9
10 11
Fig. 1. Spherical resonator and autoclave: 1, electrical connection; 2, pressure connection; 3, connection for cooling system; 4, opening for Pt25 thermometer; 5, sound generator; 6, outer pressurized vessel; 7, thermostat; 8, spherical resonator; 9, microphone; 10, inner autoclave; 11, heating; 12, cooling.
176
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
temperature interval) and ensured that it is within reasonable limits. With respect to the kind of function suitable for B ( T ) , we have had good results from a polynomial in inverse temperature containing the zeroth, first and third power: (7)
B = bo + b l / T + b s / T 3
Another reasonable choice has been used by Ewing et al. (1987) and Colgate and Sivaranam (1993), which is the form generated by a square-well potential: (8)
B = bo - b, e x p ( b 2 / T )
It is difficult to estimate the uncertainty in B. Comparing the results of different calculations, it is probably about 1% in the middle of the temperature interval investigated, and may go up to 5% at the borders of the interval.
2. Experimental (Esper, 1987; Lemming, 1989) The experimental task was to measure the resonance frequencies of the spherical modes in a spherical resonator. In order to measure them with sufficient accuracy, the detailed resonance curves had to be determined, i.e. 40 points per resonance frequency. As temperature, pressure, and frequency variation demand about 200 300 resonance frequencies per substance, it was clear that the determination of the resonance curves had to be automated.
u(f) v(f)
EB Ps]
l
A = u(f) + iv(f)
T = konst.
Fig. 2. Schematic arrangement for automated measurements: PC, computer; FS, frequency synthesizer; LIA, lock-in amplifier; DV, digital voltmeter; B, bridge for temperature measurement; RE, recorder; SB, semiconductor bridge; PS, Paroscientific pressure indicator; S, sound generator; Pt, thermometer; M, microphone.
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
177
The main piece of the apparatus is shown in Fig. 1. The spherical resonator, made of nimonic, is set together from two half-spheres. The deviation from ideal sphericity is of the order of 0.01 mm. The wall thickness is 12.5 mm, and the inner diameter is about 80 m m (the exact value given by calibration with argon, see later). One of the half-spheres has three openings (at right angles), one for the sound generator, one for the microphone, and the third for allowing circulation between resonator and autoclave, in order to remove any pressure difference in the sound generator or microphone. The sound generator is a piezo-disc (Valvo, type PXE 52), kept in position by small Teflon discs, and inserted into the half-sphere with an adapter. The microphone (Conrad Electronic, type MCE-2000, with incorporated amplifier) is also inserted with an adapter. Care was taken that the resonances occurring in the openings interfered with the resonance frequencies as little as possible. Pressure is measured by two devices: up to 150 kPa a semiconductor bridge (Natec & Schulthei8, type POCR 100/w) with a digital voltmeter is used, and between 150 and 500 kPa a piezoelectric device (Paroscientific, model 2200-AS) with a frequency counter. Both devices are calibrated against a piston-type dead-weight gauge, and installed such that the zero can be controlled after each measurement in vacuum. Uncertainty in pressure is smaller than 0.15%. Around the autoclave is a thermostatted copper vessel with provisions for heating and cooling. The temperature is measured by a Pt25 thermometer in close contact with the spherical resonator, the calibration being made according to the IPTS-68. The uncertainty in the temperature is smaller than 10 mK. The copper vessel is placed in an outer pressurized vessel, which usually contains some excess pressure of dry nitrogen. The automation is indicated in Fig. 2. The complex signal from the microphone goes to a lock-in amplifier, where it is separated into the real and imaginary parts. All measured quantities go to the computer (Commodore, PC 10). The input is the approximate value of the frequency of the first non-trivial mode. The computer then looks for the maximum and approximate half-width g of this mode, and determines the amplitudes of 20 points on both sides of the maximum in intervals of g/lO. The resonance curve is displaced on a screen. It is determined forwards and backwards so that equilibrium is ensured when both curves are identical. Then from the 40 points the resonance curve is (automatically) calculated and the exact maximum f , and half-width go given. Then the computer goes to the next mode and repeats the procedure. The calculation offn and g,, is done
Table 1 The acoustic virial coefficients A t a n d T (K)
Of a r g o n from the isothermal as well as from the simultaneous fit
Isothermal fit A l (m 2 s 2 MPa
260.001 279.409 299.811 310.159 331.838 350.165
A2
68.8 291.0 480.5 545.1 701.4 873.3
± 13.1 -+- 6.1 + 22.6 -t-23.7 ± 14.6 ± 9.9
Simultaneous tit ')
A2 ( m 2 s 2 M P a 2)
A 1 (m 2 s 2 MPa
54.7 ± 24.9 52.7 ± 11.6 70.4 ± 43.0 120.4-I-45.1 142.3 ± 27.9 27.2 mr 18.8
62.2 ± 2.9 283.3 + 1.5 482.8 + 5.0 573.6_+5.9 743.2 ± 5.3 859.3 ± 2.6
')
A 2 ( m 2 s 2 M P a 2) 67.6 67.6 65.9 64.5 60.1 54.8
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
178
by fitting the experimental amplitudes ,4 = u + iv to be generalized Lorentz function (Mehl, 1978; Moldover et al., 1986): A(f)
= u(f)
+ iv(f) = O f / ( F 2 _ f 2 ) q_ 1~ q- C f
(9)
where F =fn + ign. The other complex paramerters give small corrections: Parameter L) compensates the phase shift between sound generator and detector, and/~ and C take care of the background noise. The values o f f n and gn need to be corrected for the non-ideality of the spherical resonator. These corrections, listed below, are done automatically by the computer. The output is temperature, pressure, resonance frequency corrected to the ideal resonator f0L, and the correction terms, and the calculated half-width g0L for the L t h spherical mode due to the non-idealities. The resulting resonance frequencies were accurate to 0.003%. The frequency range was 3-22 kHz, usually involving four modes, L = 2 to L = 5.
3. Corrections for non-ideality of the resonator (cf. Ewing et al., 1987) (1) Elasticity of the wall: (10)
Afe, = Q p c 2 f / ( 1 -- f / f r e s ) 2
where p is the density and Q depends on the ratio of wall thickness to radius of the sphere and on properties of the material (density and sound velocity), in our case nimonic alloy 90. This gives Q = 8 TPa -I. The resonance frequency of the sphere, fres, can be calculated (Mehl, 1985) to be 27.9 kHz. For gaseous densities corresponding to 0.5 MPa or less this correction is very small (0.001% or less). There is no contribution to the half-width.
Table 2 The acoustic virial coefficients A~ (m 2 s 2 MPa ~) for nitrogen T (K)
Fit
Ao
AI
A2
n
6 (%)
249.971
isoth simult
103829.67 103837.8
339.95 284.8
63.45 100.25
18
0.0012
269.831
isoth simult
112099.49 112097.2
663.17 647.4
80.73 88.29
24
0.0006
289.641
isoth simult
120332.12 120337.7
959.85 939.5
50.57 76.45
25
0.0005
310.229
isoth simult
128900.02 128900.5
1212.97 1188.7
57.70 64.20
28
0.0017
331.879
isoth simult
137892.05 137899.0
1438.33 1405.9
46.47 51.13
25
0.0015
350.809
isoth simult
145751.75 145757.5
1587.70 1566.8
72.27 39.28
24 144
0.0013 0.0019
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
179
(2) Openings (tubes): Aft - igt = (cr~/8rc2a 3) c o t [ k / + ~KHl + 6 + i(~KHI + 7]
(1 1)
This is for a cylindrical tube of radius ro and length l; a is the radius of the spherical resonator, ~KH is the K i r c h h o f f - H e l m h o l t z attenuation constant, and k is the wavevector k = 2rcf/c. For an open tube (as used for pressure equilibration) 7 = (kro)2/2
3 = 8kro/3rc
(12a)
For a tube which ends in a closed volume VHR (the linear dimension of the Helmholtz resonator H R being small in comparison with the wavelength), as is the case for the sound generator,
Table 3 Ideal gas heat capacities a n d second virial coefficients for n i t r o g e n T(K)
o (cp/R)isoth
o (Cp/R)simul t
B ( c m 3 m o l 1)
249.971 269.831 289.641 310.229 331.879 350.809
3.5013 3.4997 3.4995 3.4985 3.4987 3.4991
3.5006 3.4999 3.4991 3.4985 3.4983 3.4987
-15.5 10.3 - 5.9 -2.1 1.3 3.9
Table 4 T h e acoustic virial coefficients A i ( m 2 s -2 M P a i) for m e t h a n e
6 (%)
//
T (K)
Fit
Ao
A1
A2
232.095
isoth simult
159316.49 159323.04
- 6729.9 - 6735.9
162.0 116.3
9
0.0010
250.000
isoth simult
171067.48 171066.33
- 5485.4 - 5439.7
260.9 164.0
9
0.0007
270.420
isoth simult
184187.05 184166.14
- 4349.2 - 4244.0
315.7 185.2
9
0.0006
290.016
isoth simult
196451.50 196436.84
- 3396.9 - 3305.5
285.9 188.3
9
0.0005
309.718
isoth simult
208484.87 208479.77
2527.3 - 2516.8
175.1 183.2
9
0.0005
331.357
isoth simult
221385.84 221379.56
- 1794.6 1787.6
144.7 173.3
8
0.0004
350.366
isoth simult
232443.84 232451.79
- 1222.2 - 1239.7
177.3 163.4
8 61
0.0005 0.0014
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
180
7 = 0
(12b)
c5 =- --arctan(rcr~/kVHR)
The K i r c h h o f f - H e l m h o l t z tube attenuation constant is given by
~K. = [0zf)'i21crol[D~i2
+ (ko -
1)D~/2]
(13)
where the viscous diffusivity Ds is given in terms of the shear viscosity q and the thermal d i f f u s i v i t y Oth in terms of heat conductivity and specific heat gp: Ds = q Ip
( 14)
D r . = 21pgp
For the input into the correction terms it is usually sufficient to give either t/ or 2 and to calculate the other quantity by means of the so-called Eucken equation: 2 = t/~p(9k0- 5/4ko
(15)
This equation gives the relation between ~/and ). for simple polyatomic molecules to an accuracy of a few per cent (Hirschfelder et al., 1964). As recent measurements of ). on chlorodiflu o r o m e t h a n e ( H a m m e r s c h m i d t , 1994) show, this is also true for the polar R22. The order of magnitude of the correction (Eq. (1 l)) is 0.005%. (3) Thermal b o u n d a r y layer: Afth - - igth =
(k0 - 1)[ - ( 1 +
i ) ( D t ~ C / / z ) ' / 2 / 2 a + iDth/(2rca 2) + f t . / a ]
(16)
where la is the accomodation length l, = ( 2 / p ) ( r c M T / 2 R ) ' I 2 ( c v / R + 1/2)
1(2 - h ) / h
(17)
with h being the a c c o m m o d a t i o n coefficient sensitive to surface properties. As this term is inversely proportional to pressure, h can be estimated by experiments at very small pressures. Ewing et al. (1986) gave h = 0.9 for argon and nitrogen on stainless steel. For larger molecules, h should rather tend to unity. We have set h = 0.9 for all substances. The corresponding term of Eq. (16) contributes 0.001% or less to the frequency, whereas the first term of Eq. (16) constitutes a larger correction, between 0.04% or less for argon and less than 0.004% for R22.
Table 5 Ideal gas heat capacities and second virial coefficients for methane
T (K)
.o (~p/R)isoth
(¢op/R)simult
B (cm3 mol 1)
232.095 250.000 270.420 290.016 309.718 331.357 350.366
4.0820 4.1221 4.1826 4.2572 4.3464 4.4586 4.5703
4.0815 4.1222 4.1841 4.2583 4.3467 4.4590 4.5697
- 77.7 - 65.7 -54.6 -45.9 38.4 - 31.5 -26.3
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
181
(4) Sound absorption by the gas: This influences only the half-width but not the resonance frequency: gb = r t ( f / c ) 2 [ 4 D s / 3 + (ko - 1)Dth + qb/P]
(18)
with qb being the bulk viscosity, arising mainly from vibrational relaxations. For the substances described here, r/b could be neglected, with the exception of methane. The resonance frequency of the ideal resonator (Rayleigh model) is then f o c = VolC/(2rca) =j~ - Afe, - Aft - Afth
(19)
with v0L the Lth solution of tan(ka)= ka. Eq. (19) gives the connection between resonance frequency and sound velocity. The half-width, due to the non-idealities, is given theoretically by g0L =
(20)
g t -q- gth + g b
which is then compared with the experimental value gn. This comparison shows the absence of disturbing effects, e.g. too strong interference of the Helmholtz resonators in the openings (tubes) with the resonance in the main sphere. If the comparison was unsatisfactory, the point was Table 6 The acoustic virial coefficients Ai ( m 2 s 2 M P a - i ) for c h l o r o d i f l u o r o m e t h a n e (R22) T (K)
Fit
Ao
AI
A2
n
6 (%)
269.893
isoth simult
30764.9 30765.9
-8484.6 -8498.0
- 1472.0 - 1465.2
21
0.0017
289.703
isoth simult
32796.4 32804.9
- 7131.2 -7169.2
-978.6 -970.6
30
0.0026
309.620
isoth simult
34854.2 34848.5
-6059.6 - 6131.9
-618.6 - 547.1
28
0.0032
331.933
isoth simult a
37137.8 37130.2
- 5435.3 - 5233.6
-37.7 -230.8
29
0.0050
349.656
isoth simult
38936.0 38938.4
-4626.9 -4669.8
170.3 -154.1
25 108
0.0020 0.0093
Only the lowest pressure points are included in " s i m u l t " .
Table 7 Ideal gas heat capacities a n d second virial coefficients for c h l o r o d i f l u o r o m e t h a n e (R22) T (K)
(c~) ]R)i~oth
(C,op/R)simult
B (cm 3 mol i)
269.893 289.703 309.620 331.933 349.656
6.3915 6.6389 6.8585 7.1133 7.3258
6.3892 6.6315 6.8642 7.1213 7.3326
461.0 -387.4 - 329.3 -277.7 244.3 -
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
182
Table 8 T h e a c o u s t i c virial coefficients Ai ( m 2 s 2 M P a T (K)
Fit
223.408
isoth simult
239.770
Ao
i) for e t h a n e A1
A2
rt
6 (%)
75964.4 75968.0
- 18054.7 - 17892.9
- 3422.2 -3922.3
13
0.0022
isoth simult
80909.0 80909.0
- 15677.7 - 15626.6
-2126 -2417.7
22
0.0014
259.871
isoth simult
86872.9 86875.4
- 13425.1 - 13385.5
- 1021 - 1217.5
27
0.0011
280.448
isoth simult
92893.3 92881.1
- 11564.6 11526.1
-526 -485.4
21
0.0015
300.475
isoth simult
98640.5 98647.1
-9995.1 - 10030.0
-288 - 178.7
18
0.0009
312.446
isoth simult
102062.3 102065.9
-9228.2 -9251.6
- 131 - 133.4
19
0.0018
330.166
isoth slmult
107088.9 107103.1
- 8160.4 - 8228.5
-64 - 53.3
23
0.0015
350.762
lsoth simult
112949.7 112947.2
- 7215.0 -7198.5
-28 - 13.3
20 163
0.0021 0.0074
Table 9 Ideal g a s h e a t c a p a c i t i e s a n d s e c o n d virial coefficients f o r e t h a n e T (K)
o (cp/R)isoth
0 (cp/R)simul t
B ( c m 3 tool 1)
223.408 239.770 259.871 280.448 300.475 312.446 330.166 350.762
5.3533 5.5376 5.7851 6.0528 6.3407 6.5136 6.7797 7.0764
5.3521 5.5376 5.7842 6.0567 6.3384 6.5125 6.7744 7.0774
-322.5 -281.0 -240.2 -206.8 - 180.2 - 166.6 - 148.7 - 130.9
T a b l e 10 T h e a c o u s t i c virial coefficients Ai ( m 2 s -2 M P a -i) for p r o p a n e
(%)
T (K)
Fit
Ao
A1
Ae
n
230.134
isoth simult
50264.7 50273.2
-27288.6 -27098.8
-4576.3 -8369.5
28
0.0022
239.968
isoth simult
52216.2 52195.7
-25231.7 -24588.1
3137.8 -7136.8
40
0.0037
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
183
Table 10 (continued) T (K)
Fit
Ao
At
A2
n
6 (%)
249.832
isoth simult
54117.8 54114.8
-22442.4 -22437.0
-5659.8 -6039.3
44
0.0041
259.779
isoth simult
56063.8 56041.8
-20793.6 -20573.2
-4170.1 -5057.3
36
0.0045
270.292
isoth simult
58057.7 58071.4
- 18695.0 -18877.7
-4441.8 -4140.4
44
0.0043
279.878
isoth simult
59904.8 59915.9
- 17438.4 - 17535.9
-3579.3 -3401.1
38
0.0054
290.248
isoth simult
61886.2 61906.3
- 16244.4 - 16267.3
-2668.4 -2694.2
38
0.0041
300.004
isoth simult
63774.8 63775.6
- 15271.0 - 15220.4
- 1963.0 -2108.6
35
0.0029
310.114
isoth simult
65714.0 65710.4
- 14278.7 - 14262.3
- 1501.5 -1575.2
37
0.0039
319.838
isoth simult
67582.4 67570.2
- 13559.1 - 13444.9
-981.7 - 1126.0
38
0.0060
330.157
isoth simult
69550.1 69543.8
- 12685.0 -12673.0
-715.7 711.7
38
0.0038
340.064
isoth simult
71424.1 71439.7
- 11829.5 -12011.6
-678.1 -368.9
38
0.0029
350.034
isoth simult
73329.1 73350.5
- 11182.0 -11414.3
-475.0 -73.7
38 482
0.0026 0.0079
Table 11 Ideal gas heat capacities and second virial coefficients for p r o p a n e T(K)
o (cp/R)isoth
o (cp/R)simult
B (cm 3 mol i)
230.134 239.968 249.832 259.799 270.292 279.878 290.248 300.004 310.114 319.838 330.157 340.064 350.034
7.3128 7.4906 7.7172 7.9193 8.1829 8.3964 8.6424 8.8465 9.0728 9.2862 9.5274 9.7769 10.0024
7.3053 7.5101 7.7204 7.9373 8.1699 8.3857 8.6219 8.8455 9.0779 9.3010 9.5359 9.7580 9.9647
-730.8 -656.0 -592.6 -538.2 -489.2 -450.7 -414.4 -384.5 -357.1 - 333.7 -311.6 -292.5 -275.2
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
184
deleted. Usually the difference between g0c and gn was less than 0.002% of the resonance frequency, which shows that the acoustic model is essentially correct.
4. Calibration o f the resonator radius
Calibration was done with argon between 260 and 350 K. Argon (Messer-Griesheim) had a purity better than 99.999%. The pressure varied between 0.025 and 0.5 MPa. The basic equation for the calibration was
a = Voe/(2rcfoL)(koRT/M + A,p + A2p 2) 1/2
(21)
where
k o R / M = 346.8923 m 2 s 2 K-1
loo
(c;
-
(22)
) / []
1.0
[]
[]
0
[]
0
0.5
0
0 0
0
250
o
I
•
+
•
300 I
350 ×
t
I
,
T/K
+
-0.5
-1.0
-
Fig. 3. Percentage deviation of heat capacity from the values of the simultaneous fit for R22: • , isothermal fit; ~ Barho (1965) (empirical anharmonicity correction); ×, harmonic oscillator approximation with frequencies of Bohn (1984); + , harmonic oscillator approximation with frequencies of Chen et al. (1976); ~ , calorimetric measurements of Ernst and Biisser (1970); + , calorimetric measurements of Ernst et al. (1991).
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173 192 loo
-
185
) /
0.5 X
X x X
0
0
•
x
~
[]
X
w
T./K
[]
X
-0.5
0 I
I
I
I
220
250
300
350
Fig. 4. Percentage deviation of heat capacity from the values of the simultaneous fit for ethane: • , isothermal fit; ×, Sychev et al. (1987); [2, Ernst and Hochberg (1989); ~ , Bier et al. (1976).
a n d A 2 was calculated via Eq. (5) with the following correlations for B(T) (Moldover et al.,
1986) and C(T) (correlated from values of D y m o n d and Smith 1980): B ( T ) / m 3 m o l - ' = (0.3454 - 119.1 l I T - 9475.6/T 2) × 10 -4
(23)
C(T)/m 6 mo1-2 = (623580 - 3994T + 10.702T 2 - 0.01005T 3) × 10 -4
(24)
The values of AI(T) and A 2 ( T ) a r e given in Table 1, and the details of the input for the corrections and the experimental results are given by Esper (1987). The result of the calibration is a/mm = 39.9774 [1 + ~ o ( T - 300) + ~ I ( T - 300) 2]
(25)
with ~o = 12.0 × 10 -6 K - I (the manufacturer gives 12.2 x 10 6 K 1). For the determination of the second coefficient ~1 the temperature interval was too small. Therefore the value given by the manufacturer, ~i = 5.67 x 10 -9 K -2, was taken.
186
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192 1 0 0 (B - Bsim) /IBsim I
0
220
250
300
I
I
I
350
d
:- T / K
o X
[]
•
X
-1.0
X
0 []
X
[]
-2.0
-
[]
Fig. 5. Percentage deviation of second virial coefficient from the values of the simultaneous fit for ethane: • , recommended values of Dymond and Smith (1980); ©, Young (1978); ×, Pope et al. (1973); L2, Douslin and Harrison (1973).
5. Results and discussion
The details of the input and the experimental results on nitrogen, methane, ethane, and chlorodifluoromethane are given by Lemming (1989). The corresponding details for propane (Beckermann, 1988) are documented in a report which is available on request 1. Tables 2, 4, 6, 8 and 10 give the values of the acoustic virial coefficients A o ( T ) , A I ( T ) and A2(T) from the isothermal as well as from the simultaneous fit, together with the relative standard deviation of the sound speed c (in %): (26) i = 1 \
Ci, e x p
/
where n is the number of experimental points. In the simultaneous fit, the temperature functions chosen for A o ( T ) and A~ (T) were given by Eq. (6), whilst Az(T ) was calculated iteratively from 1From Bibliothek, Inst. f. Thermo-& Fluiddynamik, Ruhr-Univ. Bochum, D-44780.
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192 loo
-
187
) /
I
A []
[]
oo
O
0.5 0
X
[]
0 × •
•
x
-0.5 I
250
300
~T/K
35O
Fig. 6. Percentage deviation of heat capacity from the values of the simultaneous fit for propane: O, isothermal fit; ©, statistical mechanical values of Chao et al. (1973); x, statistical mechanical determation of Kistiakowsky and Rice, 1940; ~, calorimetric determination by Ernst and Biisser (1970); O, calorimetric determination by Daily and Felsing (1943): A, calorimetric determination by Kistiakowsky and Rice (1940).
Eq. (5) with C(T) taken either from the literature or from a correlation to B(T) according to Nguyen Van N h u et al. (1989) (cf. also Iglesias-Silva and Kohler, 1992). Tables 3, 5, 7, 9 and 11 give then the c°p/R values calculated from A0, and B(T) coming from the iterative solution of the simultaneous fit. 0 For nitrogen and methane, the results are a good confirmation of our calibration. The Cp values are well within 0.1% of the theoretical values (for methane cf. McDowell and Kruse, 1963), and the second virial coefficients agree with the best literature values (for nitrogen, Jacobsen et al., 1986, for methane, Kleinrahm et al., 1988) within 1 cm 3 mo1-1. For methane, there is an extra contribution to the half-width of the resonance lines due to vibrtational relaxation. This extra contribution is (at 290 and 310 K) in line with the value of G a m m o n and
188
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
100 (B - Bsim) /Igsiml I
I
I
0
6
0 _
0 0 2
-
0 •
0
0
0
"2
0
QON
O
O
O
-
L 250
I 300
I 350
~'T/K
Fig. 7. Percentage deviation of second virial coefficient from the values of the simultaneous fit for propane: O, recommended values of Dymond and Smith (1980); ~, correlation by Goodwin and Haynes (1982).
Douslin (1976) for the relaxation time at 298.15 K and 0.1 MPa (r = 1.4 x 10 -+ s) within 6"/0 and 15% respectively. This value agrees also with a recent determination of Ewing and Goodwin (1992). For chlorodifluoromethane (R22), the errors in the sound speed are somewhat larger (Table 6). This may have to do with the small values of the sound speed and with the lower purity (99.97% compared with nitrogen and methane, which had a purity of at least 99.999%. Nevertheless the results for Cp 0 and B(T) seem to be quite consistent. The ideal gas heat capacity compares well with the statistical-mechanical calculation (harmonic oscillator approximation), especially when a new determination of fundamental frequencies is used (Bohn, 1984). An empirical correction for anharmonicity applied by Barho (1965) is not suitable (Fig. 3). The second virial coefficients lie comfortably between the high temperature results of Demiriz et al. (1993) and the low temperature results of Natour et al. (1989). o values show a slightly different temperature behavior to For ethane (purity 99.995%), the Cp the tabulated results of Sychev et al. (1987), practically identical with the values of Goodwin et al. (1976) (Fig. 4), but agree with a recent calorimetric determination (Ernst and Hochberg, 1989). The second virial coefficients are in line with the literature (Fig. 5). Finally, the results on propane (purity 99.95%) cover a wider temperature interval. Conse0 quently, the second virial coefficient has been correlated with four coefficients (Table 11). The Cp values are a little bit lower than the literature (Fig. 6) but probably within the error limits. The second virial coefficients tend to somewhat more negative values at low temperatures (Fig. 7).
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
189
Acknowledgments The authors are grateful for the advice of Mike Moldover, which was essential for the construction of the apparatus. They thank also M.B. Ewing and J.P.M. Trusler for helpful discussions. Part of the work has been supported by a grant from the Deutsche Forschungsgemeinschaft, which is gratefully acknowledged.
List of symbols a xzl
Ai aij
B bi C .0 (~p Cv Cp
C Ds Dth
f F
g h k k0 l M /7
P
Q
R /'0
T /,/ U
V.R
radius of resonator (complex) amplitude acoustic virial coefficients (cf. Eq. (1)) coefficients in the correlation of acoustic virial coefficients second (thermal) virial coefficient coefficients in the correlation of second virial coefficients (complex)coefficient in the generalized Lorentz function speed of sound ideal gas heat capacity at constant pressure isochoric heat capacity specific heat at constant pressure third (thermal) virial coefficient (complex) coefficient in the generalized Lorentz function viscous diffusivity thermal diffusivity (complex) coefficient in the generalized Lorentz function frequency (complex) resonance frequency half-width of resonance curve accommodation coefficient wave vector ratio
0t
0
Cp/C v
length of a tube accommodation length molar mass number of experimental points pressure factor in the correction for elasticity of the wall gas constant radius of a tube temperature real part of amplitude imaginary part of amplitude volume of Helmholtz resonator
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173 192
190
Greek letters ~KH
7,6 qb
YOL
P
Kirchhoff-Helmholtz attenuation constant temperature coefficients of resonator radius parameters for the tube correction relative standard deviation (%) bulk viscosity shear viscosity heat conductivity L t h solution of tan x = x density relaxation time
Subscripts b el isoth OL n res
sim(ult) t
th
bulk elasticity isothermal (fit) L t h mode of spherical resonance experimental value resonance simultaneous (fit) tube thermal boundary layer
References Barho, W., 1965. Die Molwfirme der Fluor-Chlor-Derivate des Methans im Zustand idealer Gase. Kfilte-Technik, 17: 219 222. Beckermann, W., 1988. Messung der Schallgeschwindigkeiten in Propan im Temperaturebereich yon 230 K bis 370 K. Diplomarbeit, Rhur-Universitfit Bochum. Bier, K., Kunze, J., Maurer, G. and Sand, H., 1976. Experimental results for heat capacity and Joule-Thomson coefficient of ethane at zero pressure. J. Chem. Eng. Data, 21:5 7. Bohn, M.A., 1984. Untersuchung der intermolekularen Wechselwirkung und der Dynamik yon C H C I F 2 bis zu hohen Temperaturen und Driicken mit Infrarotspektroskopie zwischen 9200 und 10 cm -1. Dissertation, Universitfit Karlsruhe. Chao, J., Wilhoit, R.C. and Zwolinski, B.J., 1973. Ideal gas thermodynamic properties of ethane and propane, J. Phys. Chem. Ref. Data, 2:427 437. Chen, S.S., Wilhoit, R.C. and Zwolinski, B.J., 1976. Ideal gas thermodynamic properties of six chlorottuoromethanes. J. Phys. Chem. Ref. Data, 5: 571-580. Colgate, S.O. and Sivaraman, A., 1993. Thermophysical properties of natural gas mixtures derived from acoustic cavity measurements. In: E. Kiran and J.F. Brennecke (Eds.), Supercritical Fluid Engineering Science. ACS Symp. Set. 514, American Chemical Society, Washington, DC, pp. 121-132. Dailey, B.P. and Felsing, W.A., 1943. The heat capacity at higher temperatures of ethane and propane. J. Am. Chem. Soc., 65: 42-44.
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
191
Demiriz, A.M., Kohlen, R., Koopmann, C., Moeller, D., Sauermann, P., lglesias-Silva, G.A. and Kohler, F., 1993. The virial coefficients and the equation of state behavior of the polar components chlorodifluoromethane, fluoromethane and ethanenitrile. Fluid Phase Equilibria, 85: 313-333. Douslin, D.R. and Harrison, R.H., 1973. Pressure, volume, temperature relations of ethane. J. Chem. Thermodyn., 5: 491-512. Dymond, J.H. and Smith, E.B., 1980. The Virial Coefficients of Pure Gases and Mixtures. A Critical Compilation. Clarendon Press, Oxford. Ernst, G. and Biisser, J., 1970. Ideal and real gas state heat capacities Cp of C3H8, i-C4H w, CzFsCI, CH2CICF3, CF2C1CFC12 and CHF2CI. J. Chem. Thermodyn., 2: 787-791. Ernst, G. and Hochberg, U.E., 1989. Flow-calorimetric results for the specific heat capacity % of CO> of C2H6, and of (0.5CO 2 4-0.5C2H6) at high pressures. J. Chem. Thermodyn., 21: 407-414. Ernst, G., Brfiuning, G. and Giirtner, J., 1991. Personal communication. Esper, G.J., 1987, Direkte und indirekte p - v - T Messungen an Fluiden. Fortschrittber. VDI, Reihe 3, Nr. 148. VDI-Verlag, Dtisseldorf. Ewing, M.B. and Goodwin, A.R.H., 1992. Speeds of sound, perfect-gas heat capacities, and acoustic virial coefficients for methane determined using a spherical resonator at temperatures between 255 K and 300 K and pressures in the range 171 kPa to 7.1 MPa. J. Chem. Thermodyn., 24: 1257-1274. Ewing, M.B., Goodwin, A.R.H. and Trusler, J.P.M., 1986. The temperature-jump effect and the theory of the thermal boundary layer for a spherical resonator. Speeds of sound in argon at 273.16 K. Metrologia, 22:93 102. Ewing, M.B., Goodwin, A.R.H., McGlashan, M.L. and Trusler, J.P.M., 1987. Thermophysical properties of alkanes from speeds of sound determined using a spherical resonator. 1. Apparatus, acoustic model, and results for dimethylpropane, J. Chem. Thermodyn., 19: 721-737. Gammon, B.E. and Douslin, D.R,, 1976. The velocity of sound and heat capacity in methane from near-critical to subcritical conditions and equation-of-state implications. J. Chem. Phys., 64:203 218. Goodwin, R.D. and Haynes, W.M., 1982. Thermophysical properties of propane from 85 to 700 K at pressures to 70 MPa. Natl. Bur. Stand. (U.S.) Monogr. 170. Goodwin, R.D., Roder, H.M. and Straty, G.C., 1976. Thermophysical properties of ethane from 90 to 600 K at pressures to 700 bar. Natl. Bur. Stand. (U.S.) Tech. Note 684. Hammerschmidt, U., 1994. thermal conductivity of a wide range of alternative refrigerants measured with an improved guarded hot-plate apparatus. Presented at the 12th Symp. on Thermophysical Properties, Boulder, CO. Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B., 1964. Molecular Theory of Gases and Liquids, 2nd printing. Wiley, New York. Iglesias-Silva, G.A. and Kohler, F., 1991. A simple equation of state for non-polar gases. Fluid Phase Equilibria, 67: 87 98. Jacobsen, R.T., Stewart, R.B. and Jahangiri, M,, 1986. Thermodynamic properties of nitrogen from the freezing line to 2000 K at pressures to 1000 MPa. J. Phys. Chem. Ref. Data 15: 735-909. Kistiakowsky, G.B. and Rice, W.W., 1940. Gaseous heat capacities. II. J. Chem. Phys., 8: 610-618. Kleinrahm, R., Duschek, W., Wagner, W. and Jaeschke, M., 1988. Measurement and correlation of the (pressure, density, temperature) relation of methane in the temperature range from 273.15 K to 323.15 K at pressures up to 8 MPa. J. Chem. Thermodyn., 20: 621-631. Lemming, W., 1989. Experimentelle Bestimmung akustischer und thermischer Virialkoeffizienten von Arbeitsstoffen der Energietechnik. Fortschrittber. VDI Reihe 19, Nr. 32, VDI-Verlag, Diisseldorf. McDowell, R.S. and Kruse, F.H., 1963. Thermodynamic functions of methane. J. Chem. Eng. Data, 8: 547-548. Mehl, J.B., 1978. Analysis of resonance standing-wave measurements. J. Acoust. Soc. Am., 64: 1523-1525. Mehl, J.B., 1985. Spherical acoustic resonator: Effects of shell motion. J. Acoust. Soc. Am., 78: 782. Mehl, J.B. and Moldover, M.R., 1981. Precision acoustic measurements with a spherical resonator: Ar and C2H 4. J. Chem. Phys., 74: 4062-4077. Moldover, M.R., Waxman, M. and Greenspan, M., 1979. Spherical acoustic resonators for temperature and thermophysical property measurements. High Temp. High Pressures, 11:75 86. Moldover, M.R., Mehl, J.B. and Greenspan, M., 1986. Gas-filled spherical resonators; theory and experiment. J. Acoust. Soc. Am., 79: 253-272.
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Natour, G., Schuhmacher, H. and Schramm, B., 1989. Second virial coefficients and intermolecular forces in the system CC12F2/CHC1F2, Fluid Phase Equilibria, 49: 67-74. Nguyen Van Nhu, Iglesias-Silva, G.A. and Kohler, F., 1989. Correlation of third virial coefficients to second virial coefficients. Ber. Bunsenges. Phys. Chem., 93: 526-531. Pope, G.A., Chappelear, P.S. and Kobayashi, R., 1973. Virial coefficients of argon, methane and ethane at low reduced temperature. J. Chem. Phys., 59: 423-434. Shields, F.D. and Faughn, J., 1969. Sound velocity and absorption in low-pressure gases confined to tubes of circular cross section. J. Acoust. Soc. Am., 46: 158-163. Sychev, V.V., Vasserman, A.A., Kozlov, A.D., Zagurochenko, V.A., Spiridonov, G.A. and Tsymarny, V.A., 1987. Thermodynamic Properties of Ethane. Springer-Verlag, Berlin. Young, J.G., 1978. Determination of the interaction second virial coefficient for ethane, carbon dioxide mixtures between 250 and 300 K. M.S. Thesis, Texas A&M University.
5
7
~
I I ELSEVIER
Fluid Phase Equilibria 105 (1995) 173-192
Acoustic determination of ideal gas heat capacity and second virial coefficient of small hydrocarbons Gfinter Esper, W o l f g a n g L e m m i n g , W i l h e l m B e c k e r m a n n , F r i e d r i c h K o h l e r * Institute of Thermo- and Fluid Dynamics, Ruhr University Bochum, D-44780 Bochum, Germany Received 3 May 1994; accepted in final form 8 September 1994
Abstract
A spherical resonator was constructed to measure the resonance frequencies of sound of various gases. The measurement of the resonance curve was automated to take place at intervals of a tenth of the half-width on both sides of the maximum, and was taken forwards and backwards so that any lack of equilibrium could be seen immediately on the screen. The resonator was placed in a thermostatted autoclave. The diameter of the resonator was determined by calibration with argon. In general, 5-10 isotherms were measured between 250 and 350 K and at pressures between 0.025 and 0.5 MPa for various resonance modes. The evaluation of the sound speeds was done in two ways. First, the acoustic virial coefficients were evaluated for each isotherm, and the ideal gas heat capacity and second thermal virial coefficient derived from these isothermal fits. The second way was a simultaneous fit for all points, assuming temperature functions for the acoustic virial coefficients, but calculating the coefficient of the square term in pressure consistently from the thermal virial coefficients. Results are presented for nitrogen, methane, chlorodifluoromethane (R22), ethane and propane.
Keywords: Experiments; Method; Data; Sound speeds; Ideal gas heat capacities; Second virial coefficients; Hydrocarbons: Nitrogen; Chlorodifluoromethane
I. I n t r o d u c t i o n
The measurements of sound speeds in gases serves two purposes. At pressures below 0.5 MPa, the square of the sound speed c can be developed into a power series in pressure up to the square term: c 2 =Ao + Alp +A2p 2
* Corresponding author. 0378-3812/95/$09.50 @, 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 3 8 1 2 ( 9 4 ) 02608-4
(1)
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
174
where Ai are called acoustic virial coefficients, and where Ao and A1 have important physical significance. A0 is connected with the ratio of ideal gas heat capacity at constant pressure to that at constant volume k0,
Ao = k o R T / M
(2)
where M is the molar mass, so that the ideal gas heat capacity can be evaluated:
C°p/R = ko/(ko - 1)
(3)
The coefficient of the term linear in pressure, A1, is connected to the second thermal virial coefficient B by
A, = (ko/M)(2B + 2B'(ko - l) + B"(ko - 1)2/ko)
4)
where B' stands for T d B / d T and B" stands for T 2 d2B/dT 2. With the construction of spherical resonators, and with the develoment of the small correction terms to the ideal resonator (Mehl and Moldover, 1981; Moldover et al., 1986; Ewing et al., 1986) the sound speed can be measured with an accuracy to better than 0.01%. With k0 given 0 can be evaluated even for polyatomic molecules (where k0 might to an accuracy 0.01-0.02%, Cp be as low as 1.1) to an accuracy of 0.1-0.2%, which is better than possible by direct calorimetric measurement. The situation is not so clear-cut for the second (thermal) virial coefficient. Compared with other experimental methods, the sound speed has the big advantage that systematic errors due to adsorption are practically absent, but the difficulty is caused by the B' and B" terms in Eq. (4). The usual procedure is to assume a certain temperature function B(T) and to determine the parameters of this function such that Am(T) is given in an optimal way. First, the question is how to get the best values of A~(T). We have chosen two possibilities. One is to fit the c 2 values along each isotherm to Eq. (1). The disadvantage of this isothermal fit is a partial compensation of errors in A~ and A2. The other possibility is to fit the c 2 values of all isotherms to temperature functions Ao(T) and A~(T). Here the A2(T) values were determined via correlation of the second and third (thermal) virial coefficients according to the equations
A2
(a2B + a z c ) / ( m R T )
=
a2B
=
- ( k o + l)B 2 + (k0 - 1)(2ko - 1)2B '2 + (ko + (ko - 1)2(2BB ' + BB" + (4ko - 2)B'B")
a2c = (2ko + 1)C + (ko2 - 1)C' + (ko - 1)2C"/2
(5) 1)3B
''2
(5a) (5b)
where again C ' = T d C / d T and C " = T 2 d2C/dT 2. The simultaneous fit has the advantage that it smooths small inaccuracies in the temperature measurements, and that it ensures consistency between A~ and A2. It has the disadvantage that the temperature functions Ao(T ) and A~(T) may not be optimal. The method of calculation was first to assume values of B(T) and C(T), to calculate the A2 values from them, then to fit the coefficients of the Ao(T) and A~(T) functions, then to calculate from A~(T) improved values of B(T), and to repeat the iteration loop. The functions used here for Ao(T) and A~(T) were
175
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173 192
Ao = aol T + ao2 T 2 + ao3 T 3 4- a04 T 4
(6a)
A1 = a~o + a ~ / T + a l 3 / T 3
(6b)
The second question is w h a t properties the function B ( T ) should have for obtaining the best results in B. Test calculations have shown t h a t the function B(T) should n o t c o n t a i n m o r e t h a n three p a r a m e t e r s when A ~ ( T ) is given for a t e m p e r a t u r e interval n o t larger t h a n 100 K. In addition - - to ensure t h a t B ( T ) has a reasonable slope and c u r v a t u r e on the upper end o f the interval - - we have always calculated the Boyle t e m p e r a t u r e (usually quite a way above o u r
1 2
3
/.
5 6 7 8 1
9
10 11
Fig. 1. Spherical resonator and autoclave: 1, electrical connection; 2, pressure connection; 3, connection for cooling system; 4, opening for Pt25 thermometer; 5, sound generator; 6, outer pressurized vessel; 7, thermostat; 8, spherical resonator; 9, microphone; 10, inner autoclave; 11, heating; 12, cooling.
176
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
temperature interval) and ensured that it is within reasonable limits. With respect to the kind of function suitable for B ( T ) , we have had good results from a polynomial in inverse temperature containing the zeroth, first and third power: (7)
B = bo + b l / T + b s / T 3
Another reasonable choice has been used by Ewing et al. (1987) and Colgate and Sivaranam (1993), which is the form generated by a square-well potential: (8)
B = bo - b, e x p ( b 2 / T )
It is difficult to estimate the uncertainty in B. Comparing the results of different calculations, it is probably about 1% in the middle of the temperature interval investigated, and may go up to 5% at the borders of the interval.
2. Experimental (Esper, 1987; Lemming, 1989) The experimental task was to measure the resonance frequencies of the spherical modes in a spherical resonator. In order to measure them with sufficient accuracy, the detailed resonance curves had to be determined, i.e. 40 points per resonance frequency. As temperature, pressure, and frequency variation demand about 200 300 resonance frequencies per substance, it was clear that the determination of the resonance curves had to be automated.
u(f) v(f)
EB Ps]
l
A = u(f) + iv(f)
T = konst.
Fig. 2. Schematic arrangement for automated measurements: PC, computer; FS, frequency synthesizer; LIA, lock-in amplifier; DV, digital voltmeter; B, bridge for temperature measurement; RE, recorder; SB, semiconductor bridge; PS, Paroscientific pressure indicator; S, sound generator; Pt, thermometer; M, microphone.
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
177
The main piece of the apparatus is shown in Fig. 1. The spherical resonator, made of nimonic, is set together from two half-spheres. The deviation from ideal sphericity is of the order of 0.01 mm. The wall thickness is 12.5 mm, and the inner diameter is about 80 m m (the exact value given by calibration with argon, see later). One of the half-spheres has three openings (at right angles), one for the sound generator, one for the microphone, and the third for allowing circulation between resonator and autoclave, in order to remove any pressure difference in the sound generator or microphone. The sound generator is a piezo-disc (Valvo, type PXE 52), kept in position by small Teflon discs, and inserted into the half-sphere with an adapter. The microphone (Conrad Electronic, type MCE-2000, with incorporated amplifier) is also inserted with an adapter. Care was taken that the resonances occurring in the openings interfered with the resonance frequencies as little as possible. Pressure is measured by two devices: up to 150 kPa a semiconductor bridge (Natec & Schulthei8, type POCR 100/w) with a digital voltmeter is used, and between 150 and 500 kPa a piezoelectric device (Paroscientific, model 2200-AS) with a frequency counter. Both devices are calibrated against a piston-type dead-weight gauge, and installed such that the zero can be controlled after each measurement in vacuum. Uncertainty in pressure is smaller than 0.15%. Around the autoclave is a thermostatted copper vessel with provisions for heating and cooling. The temperature is measured by a Pt25 thermometer in close contact with the spherical resonator, the calibration being made according to the IPTS-68. The uncertainty in the temperature is smaller than 10 mK. The copper vessel is placed in an outer pressurized vessel, which usually contains some excess pressure of dry nitrogen. The automation is indicated in Fig. 2. The complex signal from the microphone goes to a lock-in amplifier, where it is separated into the real and imaginary parts. All measured quantities go to the computer (Commodore, PC 10). The input is the approximate value of the frequency of the first non-trivial mode. The computer then looks for the maximum and approximate half-width g of this mode, and determines the amplitudes of 20 points on both sides of the maximum in intervals of g/lO. The resonance curve is displaced on a screen. It is determined forwards and backwards so that equilibrium is ensured when both curves are identical. Then from the 40 points the resonance curve is (automatically) calculated and the exact maximum f , and half-width go given. Then the computer goes to the next mode and repeats the procedure. The calculation offn and g,, is done
Table 1 The acoustic virial coefficients A t a n d T (K)
Of a r g o n from the isothermal as well as from the simultaneous fit
Isothermal fit A l (m 2 s 2 MPa
260.001 279.409 299.811 310.159 331.838 350.165
A2
68.8 291.0 480.5 545.1 701.4 873.3
± 13.1 -+- 6.1 + 22.6 -t-23.7 ± 14.6 ± 9.9
Simultaneous tit ')
A2 ( m 2 s 2 M P a 2)
A 1 (m 2 s 2 MPa
54.7 ± 24.9 52.7 ± 11.6 70.4 ± 43.0 120.4-I-45.1 142.3 ± 27.9 27.2 mr 18.8
62.2 ± 2.9 283.3 + 1.5 482.8 + 5.0 573.6_+5.9 743.2 ± 5.3 859.3 ± 2.6
')
A 2 ( m 2 s 2 M P a 2) 67.6 67.6 65.9 64.5 60.1 54.8
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
178
by fitting the experimental amplitudes ,4 = u + iv to be generalized Lorentz function (Mehl, 1978; Moldover et al., 1986): A(f)
= u(f)
+ iv(f) = O f / ( F 2 _ f 2 ) q_ 1~ q- C f
(9)
where F =fn + ign. The other complex paramerters give small corrections: Parameter L) compensates the phase shift between sound generator and detector, and/~ and C take care of the background noise. The values o f f n and gn need to be corrected for the non-ideality of the spherical resonator. These corrections, listed below, are done automatically by the computer. The output is temperature, pressure, resonance frequency corrected to the ideal resonator f0L, and the correction terms, and the calculated half-width g0L for the L t h spherical mode due to the non-idealities. The resulting resonance frequencies were accurate to 0.003%. The frequency range was 3-22 kHz, usually involving four modes, L = 2 to L = 5.
3. Corrections for non-ideality of the resonator (cf. Ewing et al., 1987) (1) Elasticity of the wall: (10)
Afe, = Q p c 2 f / ( 1 -- f / f r e s ) 2
where p is the density and Q depends on the ratio of wall thickness to radius of the sphere and on properties of the material (density and sound velocity), in our case nimonic alloy 90. This gives Q = 8 TPa -I. The resonance frequency of the sphere, fres, can be calculated (Mehl, 1985) to be 27.9 kHz. For gaseous densities corresponding to 0.5 MPa or less this correction is very small (0.001% or less). There is no contribution to the half-width.
Table 2 The acoustic virial coefficients A~ (m 2 s 2 MPa ~) for nitrogen T (K)
Fit
Ao
AI
A2
n
6 (%)
249.971
isoth simult
103829.67 103837.8
339.95 284.8
63.45 100.25
18
0.0012
269.831
isoth simult
112099.49 112097.2
663.17 647.4
80.73 88.29
24
0.0006
289.641
isoth simult
120332.12 120337.7
959.85 939.5
50.57 76.45
25
0.0005
310.229
isoth simult
128900.02 128900.5
1212.97 1188.7
57.70 64.20
28
0.0017
331.879
isoth simult
137892.05 137899.0
1438.33 1405.9
46.47 51.13
25
0.0015
350.809
isoth simult
145751.75 145757.5
1587.70 1566.8
72.27 39.28
24 144
0.0013 0.0019
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
179
(2) Openings (tubes): Aft - igt = (cr~/8rc2a 3) c o t [ k / + ~KHl + 6 + i(~KHI + 7]
(1 1)
This is for a cylindrical tube of radius ro and length l; a is the radius of the spherical resonator, ~KH is the K i r c h h o f f - H e l m h o l t z attenuation constant, and k is the wavevector k = 2rcf/c. For an open tube (as used for pressure equilibration) 7 = (kro)2/2
3 = 8kro/3rc
(12a)
For a tube which ends in a closed volume VHR (the linear dimension of the Helmholtz resonator H R being small in comparison with the wavelength), as is the case for the sound generator,
Table 3 Ideal gas heat capacities a n d second virial coefficients for n i t r o g e n T(K)
o (cp/R)isoth
o (Cp/R)simul t
B ( c m 3 m o l 1)
249.971 269.831 289.641 310.229 331.879 350.809
3.5013 3.4997 3.4995 3.4985 3.4987 3.4991
3.5006 3.4999 3.4991 3.4985 3.4983 3.4987
-15.5 10.3 - 5.9 -2.1 1.3 3.9
Table 4 T h e acoustic virial coefficients A i ( m 2 s -2 M P a i) for m e t h a n e
6 (%)
//
T (K)
Fit
Ao
A1
A2
232.095
isoth simult
159316.49 159323.04
- 6729.9 - 6735.9
162.0 116.3
9
0.0010
250.000
isoth simult
171067.48 171066.33
- 5485.4 - 5439.7
260.9 164.0
9
0.0007
270.420
isoth simult
184187.05 184166.14
- 4349.2 - 4244.0
315.7 185.2
9
0.0006
290.016
isoth simult
196451.50 196436.84
- 3396.9 - 3305.5
285.9 188.3
9
0.0005
309.718
isoth simult
208484.87 208479.77
2527.3 - 2516.8
175.1 183.2
9
0.0005
331.357
isoth simult
221385.84 221379.56
- 1794.6 1787.6
144.7 173.3
8
0.0004
350.366
isoth simult
232443.84 232451.79
- 1222.2 - 1239.7
177.3 163.4
8 61
0.0005 0.0014
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
180
7 = 0
(12b)
c5 =- --arctan(rcr~/kVHR)
The K i r c h h o f f - H e l m h o l t z tube attenuation constant is given by
~K. = [0zf)'i21crol[D~i2
+ (ko -
1)D~/2]
(13)
where the viscous diffusivity Ds is given in terms of the shear viscosity q and the thermal d i f f u s i v i t y Oth in terms of heat conductivity and specific heat gp: Ds = q Ip
( 14)
D r . = 21pgp
For the input into the correction terms it is usually sufficient to give either t/ or 2 and to calculate the other quantity by means of the so-called Eucken equation: 2 = t/~p(9k0- 5/4ko
(15)
This equation gives the relation between ~/and ). for simple polyatomic molecules to an accuracy of a few per cent (Hirschfelder et al., 1964). As recent measurements of ). on chlorodiflu o r o m e t h a n e ( H a m m e r s c h m i d t , 1994) show, this is also true for the polar R22. The order of magnitude of the correction (Eq. (1 l)) is 0.005%. (3) Thermal b o u n d a r y layer: Afth - - igth =
(k0 - 1)[ - ( 1 +
i ) ( D t ~ C / / z ) ' / 2 / 2 a + iDth/(2rca 2) + f t . / a ]
(16)
where la is the accomodation length l, = ( 2 / p ) ( r c M T / 2 R ) ' I 2 ( c v / R + 1/2)
1(2 - h ) / h
(17)
with h being the a c c o m m o d a t i o n coefficient sensitive to surface properties. As this term is inversely proportional to pressure, h can be estimated by experiments at very small pressures. Ewing et al. (1986) gave h = 0.9 for argon and nitrogen on stainless steel. For larger molecules, h should rather tend to unity. We have set h = 0.9 for all substances. The corresponding term of Eq. (16) contributes 0.001% or less to the frequency, whereas the first term of Eq. (16) constitutes a larger correction, between 0.04% or less for argon and less than 0.004% for R22.
Table 5 Ideal gas heat capacities and second virial coefficients for methane
T (K)
.o (~p/R)isoth
(¢op/R)simult
B (cm3 mol 1)
232.095 250.000 270.420 290.016 309.718 331.357 350.366
4.0820 4.1221 4.1826 4.2572 4.3464 4.4586 4.5703
4.0815 4.1222 4.1841 4.2583 4.3467 4.4590 4.5697
- 77.7 - 65.7 -54.6 -45.9 38.4 - 31.5 -26.3
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
181
(4) Sound absorption by the gas: This influences only the half-width but not the resonance frequency: gb = r t ( f / c ) 2 [ 4 D s / 3 + (ko - 1)Dth + qb/P]
(18)
with qb being the bulk viscosity, arising mainly from vibrational relaxations. For the substances described here, r/b could be neglected, with the exception of methane. The resonance frequency of the ideal resonator (Rayleigh model) is then f o c = VolC/(2rca) =j~ - Afe, - Aft - Afth
(19)
with v0L the Lth solution of tan(ka)= ka. Eq. (19) gives the connection between resonance frequency and sound velocity. The half-width, due to the non-idealities, is given theoretically by g0L =
(20)
g t -q- gth + g b
which is then compared with the experimental value gn. This comparison shows the absence of disturbing effects, e.g. too strong interference of the Helmholtz resonators in the openings (tubes) with the resonance in the main sphere. If the comparison was unsatisfactory, the point was Table 6 The acoustic virial coefficients Ai ( m 2 s 2 M P a - i ) for c h l o r o d i f l u o r o m e t h a n e (R22) T (K)
Fit
Ao
AI
A2
n
6 (%)
269.893
isoth simult
30764.9 30765.9
-8484.6 -8498.0
- 1472.0 - 1465.2
21
0.0017
289.703
isoth simult
32796.4 32804.9
- 7131.2 -7169.2
-978.6 -970.6
30
0.0026
309.620
isoth simult
34854.2 34848.5
-6059.6 - 6131.9
-618.6 - 547.1
28
0.0032
331.933
isoth simult a
37137.8 37130.2
- 5435.3 - 5233.6
-37.7 -230.8
29
0.0050
349.656
isoth simult
38936.0 38938.4
-4626.9 -4669.8
170.3 -154.1
25 108
0.0020 0.0093
Only the lowest pressure points are included in " s i m u l t " .
Table 7 Ideal gas heat capacities a n d second virial coefficients for c h l o r o d i f l u o r o m e t h a n e (R22) T (K)
(c~) ]R)i~oth
(C,op/R)simult
B (cm 3 mol i)
269.893 289.703 309.620 331.933 349.656
6.3915 6.6389 6.8585 7.1133 7.3258
6.3892 6.6315 6.8642 7.1213 7.3326
461.0 -387.4 - 329.3 -277.7 244.3 -
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
182
Table 8 T h e a c o u s t i c virial coefficients Ai ( m 2 s 2 M P a T (K)
Fit
223.408
isoth simult
239.770
Ao
i) for e t h a n e A1
A2
rt
6 (%)
75964.4 75968.0
- 18054.7 - 17892.9
- 3422.2 -3922.3
13
0.0022
isoth simult
80909.0 80909.0
- 15677.7 - 15626.6
-2126 -2417.7
22
0.0014
259.871
isoth simult
86872.9 86875.4
- 13425.1 - 13385.5
- 1021 - 1217.5
27
0.0011
280.448
isoth simult
92893.3 92881.1
- 11564.6 11526.1
-526 -485.4
21
0.0015
300.475
isoth simult
98640.5 98647.1
-9995.1 - 10030.0
-288 - 178.7
18
0.0009
312.446
isoth simult
102062.3 102065.9
-9228.2 -9251.6
- 131 - 133.4
19
0.0018
330.166
isoth slmult
107088.9 107103.1
- 8160.4 - 8228.5
-64 - 53.3
23
0.0015
350.762
lsoth simult
112949.7 112947.2
- 7215.0 -7198.5
-28 - 13.3
20 163
0.0021 0.0074
Table 9 Ideal g a s h e a t c a p a c i t i e s a n d s e c o n d virial coefficients f o r e t h a n e T (K)
o (cp/R)isoth
0 (cp/R)simul t
B ( c m 3 tool 1)
223.408 239.770 259.871 280.448 300.475 312.446 330.166 350.762
5.3533 5.5376 5.7851 6.0528 6.3407 6.5136 6.7797 7.0764
5.3521 5.5376 5.7842 6.0567 6.3384 6.5125 6.7744 7.0774
-322.5 -281.0 -240.2 -206.8 - 180.2 - 166.6 - 148.7 - 130.9
T a b l e 10 T h e a c o u s t i c virial coefficients Ai ( m 2 s -2 M P a -i) for p r o p a n e
(%)
T (K)
Fit
Ao
A1
Ae
n
230.134
isoth simult
50264.7 50273.2
-27288.6 -27098.8
-4576.3 -8369.5
28
0.0022
239.968
isoth simult
52216.2 52195.7
-25231.7 -24588.1
3137.8 -7136.8
40
0.0037
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
183
Table 10 (continued) T (K)
Fit
Ao
At
A2
n
6 (%)
249.832
isoth simult
54117.8 54114.8
-22442.4 -22437.0
-5659.8 -6039.3
44
0.0041
259.779
isoth simult
56063.8 56041.8
-20793.6 -20573.2
-4170.1 -5057.3
36
0.0045
270.292
isoth simult
58057.7 58071.4
- 18695.0 -18877.7
-4441.8 -4140.4
44
0.0043
279.878
isoth simult
59904.8 59915.9
- 17438.4 - 17535.9
-3579.3 -3401.1
38
0.0054
290.248
isoth simult
61886.2 61906.3
- 16244.4 - 16267.3
-2668.4 -2694.2
38
0.0041
300.004
isoth simult
63774.8 63775.6
- 15271.0 - 15220.4
- 1963.0 -2108.6
35
0.0029
310.114
isoth simult
65714.0 65710.4
- 14278.7 - 14262.3
- 1501.5 -1575.2
37
0.0039
319.838
isoth simult
67582.4 67570.2
- 13559.1 - 13444.9
-981.7 - 1126.0
38
0.0060
330.157
isoth simult
69550.1 69543.8
- 12685.0 -12673.0
-715.7 711.7
38
0.0038
340.064
isoth simult
71424.1 71439.7
- 11829.5 -12011.6
-678.1 -368.9
38
0.0029
350.034
isoth simult
73329.1 73350.5
- 11182.0 -11414.3
-475.0 -73.7
38 482
0.0026 0.0079
Table 11 Ideal gas heat capacities and second virial coefficients for p r o p a n e T(K)
o (cp/R)isoth
o (cp/R)simult
B (cm 3 mol i)
230.134 239.968 249.832 259.799 270.292 279.878 290.248 300.004 310.114 319.838 330.157 340.064 350.034
7.3128 7.4906 7.7172 7.9193 8.1829 8.3964 8.6424 8.8465 9.0728 9.2862 9.5274 9.7769 10.0024
7.3053 7.5101 7.7204 7.9373 8.1699 8.3857 8.6219 8.8455 9.0779 9.3010 9.5359 9.7580 9.9647
-730.8 -656.0 -592.6 -538.2 -489.2 -450.7 -414.4 -384.5 -357.1 - 333.7 -311.6 -292.5 -275.2
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
184
deleted. Usually the difference between g0c and gn was less than 0.002% of the resonance frequency, which shows that the acoustic model is essentially correct.
4. Calibration o f the resonator radius
Calibration was done with argon between 260 and 350 K. Argon (Messer-Griesheim) had a purity better than 99.999%. The pressure varied between 0.025 and 0.5 MPa. The basic equation for the calibration was
a = Voe/(2rcfoL)(koRT/M + A,p + A2p 2) 1/2
(21)
where
k o R / M = 346.8923 m 2 s 2 K-1
loo
(c;
-
(22)
) / []
1.0
[]
[]
0
[]
0
0.5
0
0 0
0
250
o
I
•
+
•
300 I
350 ×
t
I
,
T/K
+
-0.5
-1.0
-
Fig. 3. Percentage deviation of heat capacity from the values of the simultaneous fit for R22: • , isothermal fit; ~ Barho (1965) (empirical anharmonicity correction); ×, harmonic oscillator approximation with frequencies of Bohn (1984); + , harmonic oscillator approximation with frequencies of Chen et al. (1976); ~ , calorimetric measurements of Ernst and Biisser (1970); + , calorimetric measurements of Ernst et al. (1991).
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173 192 loo
-
185
) /
0.5 X
X x X
0
0
•
x
~
[]
X
w
T./K
[]
X
-0.5
0 I
I
I
I
220
250
300
350
Fig. 4. Percentage deviation of heat capacity from the values of the simultaneous fit for ethane: • , isothermal fit; ×, Sychev et al. (1987); [2, Ernst and Hochberg (1989); ~ , Bier et al. (1976).
a n d A 2 was calculated via Eq. (5) with the following correlations for B(T) (Moldover et al.,
1986) and C(T) (correlated from values of D y m o n d and Smith 1980): B ( T ) / m 3 m o l - ' = (0.3454 - 119.1 l I T - 9475.6/T 2) × 10 -4
(23)
C(T)/m 6 mo1-2 = (623580 - 3994T + 10.702T 2 - 0.01005T 3) × 10 -4
(24)
The values of AI(T) and A 2 ( T ) a r e given in Table 1, and the details of the input for the corrections and the experimental results are given by Esper (1987). The result of the calibration is a/mm = 39.9774 [1 + ~ o ( T - 300) + ~ I ( T - 300) 2]
(25)
with ~o = 12.0 × 10 -6 K - I (the manufacturer gives 12.2 x 10 6 K 1). For the determination of the second coefficient ~1 the temperature interval was too small. Therefore the value given by the manufacturer, ~i = 5.67 x 10 -9 K -2, was taken.
186
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192 1 0 0 (B - Bsim) /IBsim I
0
220
250
300
I
I
I
350
d
:- T / K
o X
[]
•
X
-1.0
X
0 []
X
[]
-2.0
-
[]
Fig. 5. Percentage deviation of second virial coefficient from the values of the simultaneous fit for ethane: • , recommended values of Dymond and Smith (1980); ©, Young (1978); ×, Pope et al. (1973); L2, Douslin and Harrison (1973).
5. Results and discussion
The details of the input and the experimental results on nitrogen, methane, ethane, and chlorodifluoromethane are given by Lemming (1989). The corresponding details for propane (Beckermann, 1988) are documented in a report which is available on request 1. Tables 2, 4, 6, 8 and 10 give the values of the acoustic virial coefficients A o ( T ) , A I ( T ) and A2(T) from the isothermal as well as from the simultaneous fit, together with the relative standard deviation of the sound speed c (in %): (26) i = 1 \
Ci, e x p
/
where n is the number of experimental points. In the simultaneous fit, the temperature functions chosen for A o ( T ) and A~ (T) were given by Eq. (6), whilst Az(T ) was calculated iteratively from 1From Bibliothek, Inst. f. Thermo-& Fluiddynamik, Ruhr-Univ. Bochum, D-44780.
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192 loo
-
187
) /
I
A []
[]
oo
O
0.5 0
X
[]
0 × •
•
x
-0.5 I
250
300
~T/K
35O
Fig. 6. Percentage deviation of heat capacity from the values of the simultaneous fit for propane: O, isothermal fit; ©, statistical mechanical values of Chao et al. (1973); x, statistical mechanical determation of Kistiakowsky and Rice, 1940; ~, calorimetric determination by Ernst and Biisser (1970); O, calorimetric determination by Daily and Felsing (1943): A, calorimetric determination by Kistiakowsky and Rice (1940).
Eq. (5) with C(T) taken either from the literature or from a correlation to B(T) according to Nguyen Van N h u et al. (1989) (cf. also Iglesias-Silva and Kohler, 1992). Tables 3, 5, 7, 9 and 11 give then the c°p/R values calculated from A0, and B(T) coming from the iterative solution of the simultaneous fit. 0 For nitrogen and methane, the results are a good confirmation of our calibration. The Cp values are well within 0.1% of the theoretical values (for methane cf. McDowell and Kruse, 1963), and the second virial coefficients agree with the best literature values (for nitrogen, Jacobsen et al., 1986, for methane, Kleinrahm et al., 1988) within 1 cm 3 mo1-1. For methane, there is an extra contribution to the half-width of the resonance lines due to vibrtational relaxation. This extra contribution is (at 290 and 310 K) in line with the value of G a m m o n and
188
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173-192
100 (B - Bsim) /Igsiml I
I
I
0
6
0 _
0 0 2
-
0 •
0
0
0
"2
0
QON
O
O
O
-
L 250
I 300
I 350
~'T/K
Fig. 7. Percentage deviation of second virial coefficient from the values of the simultaneous fit for propane: O, recommended values of Dymond and Smith (1980); ~, correlation by Goodwin and Haynes (1982).
Douslin (1976) for the relaxation time at 298.15 K and 0.1 MPa (r = 1.4 x 10 -+ s) within 6"/0 and 15% respectively. This value agrees also with a recent determination of Ewing and Goodwin (1992). For chlorodifluoromethane (R22), the errors in the sound speed are somewhat larger (Table 6). This may have to do with the small values of the sound speed and with the lower purity (99.97% compared with nitrogen and methane, which had a purity of at least 99.999%. Nevertheless the results for Cp 0 and B(T) seem to be quite consistent. The ideal gas heat capacity compares well with the statistical-mechanical calculation (harmonic oscillator approximation), especially when a new determination of fundamental frequencies is used (Bohn, 1984). An empirical correction for anharmonicity applied by Barho (1965) is not suitable (Fig. 3). The second virial coefficients lie comfortably between the high temperature results of Demiriz et al. (1993) and the low temperature results of Natour et al. (1989). o values show a slightly different temperature behavior to For ethane (purity 99.995%), the Cp the tabulated results of Sychev et al. (1987), practically identical with the values of Goodwin et al. (1976) (Fig. 4), but agree with a recent calorimetric determination (Ernst and Hochberg, 1989). The second virial coefficients are in line with the literature (Fig. 5). Finally, the results on propane (purity 99.95%) cover a wider temperature interval. Conse0 quently, the second virial coefficient has been correlated with four coefficients (Table 11). The Cp values are a little bit lower than the literature (Fig. 6) but probably within the error limits. The second virial coefficients tend to somewhat more negative values at low temperatures (Fig. 7).
G. Esper et al./ Fluid Phase Equilibria 105 (1995) 173-192
189
Acknowledgments The authors are grateful for the advice of Mike Moldover, which was essential for the construction of the apparatus. They thank also M.B. Ewing and J.P.M. Trusler for helpful discussions. Part of the work has been supported by a grant from the Deutsche Forschungsgemeinschaft, which is gratefully acknowledged.
List of symbols a xzl
Ai aij
B bi C .0 (~p Cv Cp
C Ds Dth
f F
g h k k0 l M /7
P
Q
R /'0
T /,/ U
V.R
radius of resonator (complex) amplitude acoustic virial coefficients (cf. Eq. (1)) coefficients in the correlation of acoustic virial coefficients second (thermal) virial coefficient coefficients in the correlation of second virial coefficients (complex)coefficient in the generalized Lorentz function speed of sound ideal gas heat capacity at constant pressure isochoric heat capacity specific heat at constant pressure third (thermal) virial coefficient (complex) coefficient in the generalized Lorentz function viscous diffusivity thermal diffusivity (complex) coefficient in the generalized Lorentz function frequency (complex) resonance frequency half-width of resonance curve accommodation coefficient wave vector ratio
0t
0
Cp/C v
length of a tube accommodation length molar mass number of experimental points pressure factor in the correction for elasticity of the wall gas constant radius of a tube temperature real part of amplitude imaginary part of amplitude volume of Helmholtz resonator
G. Esper et al. / Fluid Phase Equilibria 105 (1995) 173 192
190
Greek letters ~KH
7,6 qb
YOL
P
Kirchhoff-Helmholtz attenuation constant temperature coefficients of resonator radius parameters for the tube correction relative standard deviation (%) bulk viscosity shear viscosity heat conductivity L t h solution of tan x = x density relaxation time
Subscripts b el isoth OL n res
sim(ult) t
th
bulk elasticity isothermal (fit) L t h mode of spherical resonance experimental value resonance simultaneous (fit) tube thermal boundary layer
References Barho, W., 1965. Die Molwfirme der Fluor-Chlor-Derivate des Methans im Zustand idealer Gase. Kfilte-Technik, 17: 219 222. Beckermann, W., 1988. Messung der Schallgeschwindigkeiten in Propan im Temperaturebereich yon 230 K bis 370 K. Diplomarbeit, Rhur-Universitfit Bochum. Bier, K., Kunze, J., Maurer, G. and Sand, H., 1976. Experimental results for heat capacity and Joule-Thomson coefficient of ethane at zero pressure. J. Chem. Eng. Data, 21:5 7. Bohn, M.A., 1984. Untersuchung der intermolekularen Wechselwirkung und der Dynamik yon C H C I F 2 bis zu hohen Temperaturen und Driicken mit Infrarotspektroskopie zwischen 9200 und 10 cm -1. Dissertation, Universitfit Karlsruhe. Chao, J., Wilhoit, R.C. and Zwolinski, B.J., 1973. Ideal gas thermodynamic properties of ethane and propane, J. Phys. Chem. Ref. Data, 2:427 437. Chen, S.S., Wilhoit, R.C. and Zwolinski, B.J., 1976. Ideal gas thermodynamic properties of six chlorottuoromethanes. J. Phys. Chem. Ref. Data, 5: 571-580. Colgate, S.O. and Sivaraman, A., 1993. Thermophysical properties of natural gas mixtures derived from acoustic cavity measurements. In: E. Kiran and J.F. Brennecke (Eds.), Supercritical Fluid Engineering Science. ACS Symp. Set. 514, American Chemical Society, Washington, DC, pp. 121-132. Dailey, B.P. and Felsing, W.A., 1943. The heat capacity at higher temperatures of ethane and propane. J. Am. Chem. Soc., 65: 42-44.
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