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Determination of densities and dielectric polarizabilities of methane at 298.15 k for pressures up to 710 Mpa
Fluid Phase Equilibria,96 (1994) 173-183
DETERMINATION POLARIZABILITIES UP TO 710 MPa.
DIELECTRIC DENSITIES AND OF OF METHANE AT 298.15 K FOR PRESSURES
P. MaIbrunota, J. Vermessd, D. Vidal*, T.K. Bose”>‘,A. Hourrib, J.M. St-Amaudb a Laboratoire d’Ing&rierie des Mat&iaux et des Hautes Pressions (LIMHP), Universitaire Paris-Nord, 93430 Villetaneuse, FRANCE
CNRS, Centre
b Groupe de recherche sur les Dielectriques, Departement de Physique, Universite du Quebec a Trois-Rivieres, C.P. 500, Trois-Rivieres, Quebec, G9A 5H7, CANADA (Received August 20.1993; acceptedin finalform January30,1994) Keywords: Experiment, method, density, dielectric constant, methane. ABSTRACT The dielectric constant of methane has been measured at 298.15 K for pressures up to 710 MPa. The density values determined from the dielectric constant as well as direct weighing measurements are compared with the equation of state developed by Setzmann and Wagner (1991). The deviations from the equation of state are about 0.05% in the low and high pressure regions and up to about 0.1% in the intermediate pressure storage. INTRODUCTION The purpose of this article is to show that the measurement of the dielectric constant combined with the independent determination of the dielectric constant virial coefficients B, and C,, of the ClausiusMossotti expansion leads to very accurate values of the density. In the present article we have carried out dielectric constant measurements of methane up to very high density values corresponding to pressures up to 710 MPa. There exist very few density measurements in the litterature to such high pressures [Setzmann and Wagner (1991)]. The relation between the density p and the dielectric constant E is given by the Clausius-Mossoti relation -=A,
+B,p+C,
pz +...
where Pt is the total polarization and 4, B, and C,, are, respectively, the first, second and third dielectric virial coefficients.The dielectric virial coefficients are independent of pressure and density
1 To whom correspondance should be addressed. 0378-3812/94/$4VSKl@ 1994 - Elsevier Science B.V. All rights reserved SSDI 0378-3812(94)02495-M
174
P. Malbrunot
et al. I Fluid Phase Equdlbria
94 (1994)
173-183
but are fkctions of temperature. The first dielectric virial coefficient 4, describes the isolated molecule contribution to Pt. The second dielectric virial coefficient B, describes the excess contribution to Pl due to the interaction of molecular pairs. The third dielectric virial coefficient C, describes the interaction of molecular triplets.
EXPERIMENTAL PROCEDURES The density measurements corresponding to about 40 MPa have been carried out by independent determination of the higher order virial coefficients B, and C,, in addition to the absolute measurements of the dielectric constant. The high density measurements were calibrated by directly weighing the gas. The matching of the dielectric constant to the corresponding high density values necessitated three dielectric virial coefficients where the first two virial coefficients 4 and B, have been obtained from low density measurements and the third (CL) determined by fi-ee fit corresponds very closely to the experimental value (C,). The purity of methane as determined by the supplier (Linde) was: x (CH,) > 0.9999 where x denotes mole fraction; impurities: x (C, I&J < 35 . 10”; x (C, Hs) < 5 . 104; x (NJ < 20 . 10-6; x (CO,) < 20 . 10-6; x (0,)<5*1O”;x (I!&O)
E
(2)
is associated with eqn. (1) in order to get
CM’
(:)-AC
+(BE---CB)CM’ +AEce-B’;,‘(B2 -C)(CM’)’ +.,,
(3)
where B and C are, respectively, the second and the third thermal virial coefficients and CM’=(E - 1) /(E + 2). A least-squares fit of CM’RT/P as a function of CM’ in the low pressure region gives A, from the intercept of the curve. In order to determine precisely the second (I33 and the third (C,) dielectric virial coefficients, we used an expansion technique developed by Buckingham et al(1970). This method essentially consists of first measuring the sum of capacitances of two similar cells, one of which is filled with gas at density pr and the other is evacuated. The gas is then allowed to fill both cells, the density is almost halved, and the sum of capacitances is measured again. The linear term in density remains the same before and after the expansion, but the quadratic and higher orders change. Therefore, the second and third dielectric virial coefficients are determined by the change in total capacitance. Details of this expansion technique are given by Kirouac and Bose (1976) and Huot and Bose (1991). The working relation for the determination of B, and C, is given by
P. Malbrunot
*+*
et al. I Fluid Phase Equilihrta
=- [$$I
4B;-3A,C,+2B,A;+A; 36A;
94 (1994)
173-183
17.5
[(al--I)+k-01 (4)
1
[krl)+(E2-1)~.
where C, is the mean geometric capacitance of each cell [CL,=( CA +Cn) / 21, D, is the change of capacitance when the gas of dielectric constant or and density pr in cell A is expanded into the evacuated cell B. Similarly, if the expansion takes place from cell B [gas of dielectric constant &2and density p2] to cell A, the change in capacitance is D,. The combination of the two expansion The expansion of capacitors and volumes between the two cells. eliminates mismatch measurements have been carried out in such a way that er z tz2. On plotting the left-hand side against [( or- 1) +( a2 - I)], one can determine B, from the slope and C, from the curvature.
The schematic diagram of the experimental setup for the dielectric constant measurements as a function of the pressure is presented in Fig. 1. The dielectric constant measurements were carried out at 1 kHz using a capacitance bridge [Andeen Hagerling, model 25OOA]. The precision on capacitance measurement is estimated to be 5 - 10” pF. The cells used were of the three-terminal type. Each cell has a set of parallel plates 3.18 cm in diameter. The separation between the plates is adjusted to have a geometrical capacitance of about 10 pF. The plates were made of copper for good thermal conductivity. Details of the construction are given by Huot and Bose (1991). The measuring system (dielectric cells and valves) was immersed into a constant temperature bath. Temperature was controlled by a circulating bath within f 0.01 K and was measured using a thermistor calibrated against a platinum resistance thermometer. The uncertainty in temperature measurements is 0.01 K. The pressures were measured with a quartz pressure transducer calibrated against a pressure standard (Desgranges and Huot model 5213) having a precision of the order of 0.005%. A polynomial for the pressure values in terms of the readings of the transducer was fitted for the purpose of calibration. Considering the fitting and small errors due to the temperature, the final precision is estimated to be of the order of 0.01% for pressures above 1.5 MPa. The dielectric constant of the gas is given by the relation E=Co(P)/Co(O)
(5)
where Co (P) is the geometric capacitance at pressure P and Co (0) is the capacitance under vacuum. Considering the effect of pressure on the geometric capacitance Co, the relation between Co (P) and Co (0) is given by
(6) where y is the compressibility of the plates. Corrections to 4, yRT/3, (7RT/3)(A, +B)and (yRTI3) (A,B+B, -2Az+C) for copper.
B, and C, are, respectively, where y=2.4.10-6Mpa-1
The successive approximations for the density obtained from eqn. (1) are given by p, =CM’/A,
(7)
176
P Malbrunor
N al. I Fluid Phase Equilibria
94 (1994)
173-183
(8) 2B;-A$, P3=P2
+
(9)
4
where pr, pz and pJ are respectively, the first, second and third approximation of the density. High density measurement The measurements of s as a finction of high density were carried out at 1 kHz using a General Radio capacitance bridge [model 16201 and a cylindrical capacitor with guarded electrodes inserted into a very high pressure vessel. The experimental set-up has already been described in a previous paper by Dedit et al(1974). The dielectric constant is obtained in the same way [eqn. (5)] as in the case of the low density measurements. The uncertainty on the E values are of the order of 5 . 105. The pressure was measured with a manganine gauge previously calibrated. The density p, = m/Vu, is directly obtained from the value of the mass of methane included in the volume (V,) of the high pressure vessel. The practical procedure to measure this mass consists of trapping the gas [see Fig. 21 by condensation in liquid nitrogen into a light aluminum vessel, B, fitted to a pressure valve V, which can be disconnected from the remaining high pressure apparatus at the level of the valve V,, in order to be weighed accurately. The high pressure valves V, and V, are alternatively connected to vacuum or to high pressure generator. As for the volume VnP, it is predetermined by successive condensation and weighing, using argon as a standard gas, [Trappeniers et al. (1966)]. A correction is applied to take account of the effect of the pressure on the volume VHP The measurement of a for each value of the gas density requires four steps. At first, the whole apparatus is evacuated; during the second step, the gas is introduced at the desired pressure; and for the third step, the measurement of the capacity is carried out. Finally, for the last step, the gas is recovered into the bottle and weighed on a balance capable of measuring up to 1 kg with an accuracy of 1 mg.
RESULTS AND DISCUSSION The first dielectric virial coefficient (AJ is obtained from the intercept by plotting CM’ RT/P versus CM’ [eqn. (3)]. Figure 3 shows the curve for the determination of 4. Only the low pressure points have been considered for determining 4. Our value of 4 is compared with the value obtained by Bose et al.( 1972) in Table 1. The agreement is within the experimental uncertainty. The second (B.J and the third (CJ dielectric virial coefficients are obtained by using the expansion technique [see eqn. (4)]. Figure 4 shows a plot of the left-hand side of eqn. (4) as a t?mction of [( EI- 1) +(EZ - l)] for CH, at 298.15 K. Our values of B, and C, are also compared in Table 1, with the values obtained by Bose et al(1972) at 322.5 K and Straty and Goodwin (1973) at 280 K and a new value of C, obtained by a free fit of all the data keeping the measured values of A, and B, fixed. The present measured values of 4, B, and C, in Table 1 are given with three standard deviations.
P. Malbrurwt
CI al. I Fluid Phase Equilibria
94 (1994)
173-183
177
CT6
Fig. 1: Schematic diagram of the experimental set-up for the measurement of the dielectric constant at low densities where A and B: dielectric cells; C: gas compressor; CB: capacitance bridge; CTB: controlled temperature bath; G: gas cylinder; GN: generator; PG: pressure gage; V: vent; VP: vacuum pump.
Mn
1 I
Vhp
Fig. 2: Schematic diagram of the experimental set-up for the measurement of the dielectric constant at very high densities where B: balance; C: capacitance; Mn: manometer; Vhp: high pressure cell; Vi: valve.
178
P. Malbrum
-
e! al. I Fluid Phase Equilibria
94 (1994)
CH4
6.0
T =
0.01
0.00
173-183
0.02
0.03
298.15
0.04
K
0.05
CM' Fig. 3 : Plot of CM’ RT/P as a fbnction of CM’= [( E - 1) / (a + 2)] for the determination of A, of CH, at 298.15 K.
TABLE 1 Values of the first three dielectric virial coefficients for methane.
T(K)
A, (cm3 . mol-r)
B, (cm3 . mol-I)2
C, (cm3 . mol-1)s
Authors
298.15
6.551 f 0.002
7.23 f 0.93
-246*54
this worka
298.15
6.551 f 0.002
7.23 f 0.93
-314
this workb
322.5
6.541 f 0.003
7.29 f 0.32”
- 268 f 50
Bose et al(1972)a
288
6.538
- 252.7
Straty and Goodwin (1973)
6.07
a): B, and C,: measured from the expansion technique. b): C,: obtained by a free fit of all the data keeping the measured values of G, and B, fixed. The experimental values of the dielectric constant (E) and the molar density (p,J as a function of the pressure P are presented in Tables 2 and 3. We also present in the same tables, the calculated values of the density (pcp3 from the Setzmann and Wagner (1991) EOS and the deviations A p = [(p, - p,) / pcA ] %. In Table 2 for the low density region measurements [7 MPa
P. Malbrunor
et al. 1 Fluid Phase Equilibria
94 (1994)
179
173.183
0.00
=
-
c
298.15
K
-0.02
‘;
hl U
z
-0.04
c? + 0” h
-0.06
i G-
-0.08
E \ 2
-0.10
-0.12
0.0
0.1
0.3
0.2 (&I
0.4
0.5
0.6
0.7
-1)+(+%2-l)
Fig. 4: Plot of eq. (4): differential measurement for the independant determination of the second (B,) and the third (C,) dielectric virial coefficients of CH, at 298.15 K. In Fig. 5, we plot CM as a function of p, where the closed circles (0) are low pressure [Pc40 MPa] values and open circles (0) are related to high pressure measurements. From the least-squares fit we obtain the value of C’, by a free fit ofCMasatbnctionofp,where CM=(a-1)/(~+2)p,, using the measured values of A, and B, from low density measurements. The value of the third dielectric virial coefficient, C’, determined by free fit is - 3 14 ( cm3 ,mol-1)3 which agrees well with the experimentally determined value of - 246 + 54 ( cm3 - mall’
)’ and that
obtained previously [- 268
f 50 (cm3 . mol-1)3] by Bose et al. (1972). Our values of CM are also compared with values obtained at 280 K (A) and at 300 K (0) by Straty and Goodwin (1973). It should be noted that the three measured dielectric virial coefficients, 4, B, and C, do not fit the Claussius Mossotti function at very high densities (see curve 1, Fig. 5). We believe that the deviation at intermediate and high densities is mainly due to the large uncertainty in the measured value of the third dielectric virial coefficient. It seems that the expansion technique mearurement was not carried out to high enough pressure in order to obtain a precise value of C,. The third dielectric virial coefficient (CL) determined by a free fit of all the data using the measured values of A, and B, gives a value of - 3 14 cm9 mole-3 which agrees reasonably well with other measured values (see curve 2, Fig. 5). It is interesting to note to what extent the precision in the values of the higher dielectric virial coefficients is needed in order to determine precisely high density values. The deviations in density between pcA, the calculated value obtained from Setzmann and Wagner (1991) EOS and the measured value pm, are plotted against pressure in Fig. 6. In Table 2, the deviation in density is about 0.05% in the low and high pressure regions and up to about 0.1% in the intermediate pressure range.
180
P. Malbrunor
et al. I Fluid Phase Equilibria
94 (1994)
173-183
In the low density region, the uncertainty in the determination of the density is related to the uncertainties in CM’, 4, B, and C,. A close examination of the terms involved in density determination shows that the main contribution to the density comes from CM’/&. The remaining contribution which involves higher dielectric virial coefficients, becomes more important at high densities. In our case, the uncertainty of the order of 1 x 10” for the measurement of the dielectric constant leads to uncertainties of the order of 1 x 10-5for the determination of the density.
TABLE 2 The measured dielectric constant (E), the molar density (p,)
obtained from the measured dielectric
virial coefficients (4, B, and Ch), the calculated density (pc,)
from Setzmamr and Wagner (1991)
EOS and the deviation Ap (O/o)as a fbnction of the pressure (P) for CH, at 298.15 K for pressures up to 40 MPa. E
MPa
mol - mq3
7.00327
1.064088
0.3 1828
0.3 1806
0.06755
8.00616
1.074663
0.36937
0.36916
0.05405
9.01104
1.085562
0.42162
0.42143
- 0.04370
10.0365
1.096905
0.47578
0.47564
- 0.02921
11.8975
1.118101
0.57526
0.57522
- 0.00557
13.5176
1.136727
0.66168
0.66178
0.01607
15.0412
1.154159
0.74159
0.74167
0.01084
16.5429
1.170924
0.81759
0.81783
0.02976
18.0965
1.187644
0.89257
0.89301
0.05036
19.5697
1.202768
0.95972
0.96036
0.06646
21.1919
1.218497
1.02890
1.02973
0.08108
23.2668
1.237143
1.11000
1.11111
0.09942
25.2265
1.253247
.17941
1.18080
0.12508
29.1558
1.281675
.30227
1.30229
0.15570
33.7353
1.309415
.41634
1.41885
0.17746
40.0423
1.340580
.54463
1.54781
0.20591
In the high density region the measurement of a and p are independant, therefore the relative error A (CM)/CM on the CM function is
P. Malbrunot et al. I Fluid Phase Equilibria 94 (1994) 173-183
A(CM) CM=
3~ (E-1)(&+2)
181
dc+& E p
(10)
TABLE 3 The measured dielectric constant (a), the measured molar density (p,,J, the calculated density (pck) from Setzmann and Wagner (1991) EOS and the deviation Ap (%) as a tknction of the pressure (P) for CH, at 298.15 K for pressures up to 710 MPa. P
E
MPa
Pm’
10”
&A.
10
4
mol mmm3
mol - me3
AP %
10.359
1.1006980
0.49262
0.49283
0.0403 1
20.235
1.2096252
0.98923
0.98942
0.01950
30.399
12900252
1.33593
1.33629
0.02737
40.610
1.3433226
1.55789
1.55804
0.00935
50.290
1.3799848
1.70725
1.70719
- 0.00295
60.430
1.4100489
1.82755
1.82754
- 0.00048
70.410
1.4343868
1.92348
1.92435
- 0.00140
80.660
1.4555756
2.00634
2.00625
- 0.00355
90.830
1.4738263
2.07732
2.07711
- 0.00998
100.0
1.4885223
2.13448
2.13372
- 0.03549
150.0
1.5501796
2.36542
2.36793
0.10584
200.0
1.5944552
2.53185
2.53312
0.04995
250.0
1.6304259
2.65932
2.66252
0.12025
300.0
1.6600129
2.76691
2.76985
0.10609
350.0
1.6856698
2.86016
2.86210
0.06769
399.9
1.7081561
2.94026
2.94320
0.10003
450.0
1.7287640
3.01431
3.01622
0.06330
499.9
1.7473400
3.08082
3.08219
0.04453
549.8
1.7644634
3.14228
3.14305
0.02441
599.7
1.7806280
3.20019
3.19932
- 0.02717
656.3
1.7977076
3.26003
3.25862
- 0.04318
709.4
1.8125309
3.3 1208
3.31054
- 0.04628
As mentioned above, the relative error on the dielectric constant We does not exceed 0.005%. As firr as the relative error on the density Aplp is concerned, it is evaluated to be 0.05% taking into account the precision on weighing, the recovered gas and the precision on volwne calibration. The error A (CM)/(CM), a decreasing tkction with pressure, varies in the present case from 0.1% at moderate pressures to 0.05% at higher pressures.
182
P. Malbrunor
CI al. I Fluid Phase Equilibria
94 (1994)
173-183
6.62 :
6.60
t
6.56
mi
6.56
,u
6.54
qE h =
6.52
e
6.46
0
7
6.46
A
Straty
and
Coodwln
(1175):
2110 K
E
6.44
0
Stroty
and
Coodwln
(1073):
500 K
20
25
6.50 0
Thlm work
: 2911.15
K
6.42 0
5
10
15
30
Pm ( mol/l
35
40
1
,,, as a fimction of pm where (0) are low density experimental Fig. 5: Plot of CM=(a-l)/(a+2)p values and 0 are high density experimental values. The curve 1 is the least-squares fit with three measured dielectric virial coefficients (4, B, and CJ. The curve 2 is a fit using the measured values of G, and B, and C’, as a free fitting parameter.
0.6
T :
CH4
T=298.15K
0.4
s
0.2 B
3 -E
P I
00*
I -0.2
s” -0.4
-0.6 0
300
400 P
(MPa)
Fig. 6: Deviations in density for high density measurements.
500
600
700
P. Malbrunot
et al. I Fluid Phase Equilibria
94 (1994)
173-183
183
CONCLUSION This article clearly shows that the dielectric constant method is a powerful technique for precision PVT measurements. It also proves that even up to 710 MPa for CH, at 298.15 K, only three dielectric virial coefficients are necessary to reproduce accurate values of the density. Since the dielectric technique is relatively simple and inexpensive, it has great potential for industrial application. In fact, the dielectric constant method has already been applied to the density measurement for gas adsorption at high pressures by Vidal et al. (1990) and Malbnmot et al. (1992). ACKNOWLEDGMENTS The authors would like to thank the referees for their suggestions which have improved the paper substantially. We would also like to thank Mr. J. Hamelin for his help in computer programming and Mrs. L. Bellemare for typing the text. REFERENCES Bose, T.K., Sochanski, J.S. and Cole, R.H., 1972. Dielectric and pressure virial coefficients of imperfect gases. Octupole moments of CH, and CF.,. J. Chem. Phys., 57: 3592-3595. Buckingham,.A.D., Cole, R.H. and Sutter, H., 1970. Direct determination of the imperfect gas contribution to dielectric polarization. J. Chem. Phys., 52: 5960-5961. Dedit, A., Brielles, J., Lallemand, M. and Vidal, D., 1974. Condensateur pour la mesure precise de la constante di&ctrique des gaz cornprimes jusqu’a 12 kbar: application a l’helium. High Temp. High Press., 6: 189-193. Huot, J. and Bose, T.K., 1991. Determination of the quadropole moment of nitrogen from the dielectric second virial coefficient. J. Chem. Phys., 94: 3849-3854. Kirouac, S. and Bose, T.K., 1976. Polarizability and dielectric properties of helium. J. Chem. Phys., 64: 1580-1582. Malbrunot, P., Vidal, D., Vermesse, J., Chahine, R. and Bose, T.K., 1992. Adsorption Measurements of Argon, Neon, Krypton, Nitrogen and Methane on Activated Carbon up to 650 MPa., Langmuir, 8: 577-580. Setzmann, U. and Wagner, W., 1991. A New Equation of State and Tables of Thermodynamics Properties for Methane Covering the Range from Melting Line to 625 K at Pressures up to 1000 MPa. J. Phys. Chem. Ref. Data, 20: 1061-l 155. Straty, G.C. and Goodwin, R.D., 1973. Dielectric constant and polarizability of saturated and compressed fluid methane. Cryogenics, 13: 7 12-7 15. Trappeniers, N.J., Wassenaar, T. and Walkers, G.J., 1966. Isotherms and Thermodynamic Properties of Krypton at Temperatures between 0 “C and 150 “C and at Densities up to 620 Amagat. Physica, 32: 1503-1520. Vidal, D., Malbrunot, P., Guengant, L., Bose, T.K. and Chahine, R., 1990. Measurement of physical adsorption of gases at high pressure. Rev. Sci. Instrum., 6 1: 13 14- 13 18.
DETERMINATION POLARIZABILITIES UP TO 710 MPa.
DIELECTRIC DENSITIES AND OF OF METHANE AT 298.15 K FOR PRESSURES
P. MaIbrunota, J. Vermessd, D. Vidal*, T.K. Bose”>‘,A. Hourrib, J.M. St-Amaudb a Laboratoire d’Ing&rierie des Mat&iaux et des Hautes Pressions (LIMHP), Universitaire Paris-Nord, 93430 Villetaneuse, FRANCE
CNRS, Centre
b Groupe de recherche sur les Dielectriques, Departement de Physique, Universite du Quebec a Trois-Rivieres, C.P. 500, Trois-Rivieres, Quebec, G9A 5H7, CANADA (Received August 20.1993; acceptedin finalform January30,1994) Keywords: Experiment, method, density, dielectric constant, methane. ABSTRACT The dielectric constant of methane has been measured at 298.15 K for pressures up to 710 MPa. The density values determined from the dielectric constant as well as direct weighing measurements are compared with the equation of state developed by Setzmann and Wagner (1991). The deviations from the equation of state are about 0.05% in the low and high pressure regions and up to about 0.1% in the intermediate pressure storage. INTRODUCTION The purpose of this article is to show that the measurement of the dielectric constant combined with the independent determination of the dielectric constant virial coefficients B, and C,, of the ClausiusMossotti expansion leads to very accurate values of the density. In the present article we have carried out dielectric constant measurements of methane up to very high density values corresponding to pressures up to 710 MPa. There exist very few density measurements in the litterature to such high pressures [Setzmann and Wagner (1991)]. The relation between the density p and the dielectric constant E is given by the Clausius-Mossoti relation -=A,
+B,p+C,
pz +...
where Pt is the total polarization and 4, B, and C,, are, respectively, the first, second and third dielectric virial coefficients.The dielectric virial coefficients are independent of pressure and density
1 To whom correspondance should be addressed. 0378-3812/94/$4VSKl@ 1994 - Elsevier Science B.V. All rights reserved SSDI 0378-3812(94)02495-M
174
P. Malbrunot
et al. I Fluid Phase Equdlbria
94 (1994)
173-183
but are fkctions of temperature. The first dielectric virial coefficient 4, describes the isolated molecule contribution to Pt. The second dielectric virial coefficient B, describes the excess contribution to Pl due to the interaction of molecular pairs. The third dielectric virial coefficient C, describes the interaction of molecular triplets.
EXPERIMENTAL PROCEDURES The density measurements corresponding to about 40 MPa have been carried out by independent determination of the higher order virial coefficients B, and C,, in addition to the absolute measurements of the dielectric constant. The high density measurements were calibrated by directly weighing the gas. The matching of the dielectric constant to the corresponding high density values necessitated three dielectric virial coefficients where the first two virial coefficients 4 and B, have been obtained from low density measurements and the third (CL) determined by fi-ee fit corresponds very closely to the experimental value (C,). The purity of methane as determined by the supplier (Linde) was: x (CH,) > 0.9999 where x denotes mole fraction; impurities: x (C, I&J < 35 . 10”; x (C, Hs) < 5 . 104; x (NJ < 20 . 10-6; x (CO,) < 20 . 10-6; x (0,)<5*1O”;x (I!&O)
E
(2)
is associated with eqn. (1) in order to get
CM’
(:)-AC
+(BE---CB)CM’ +AEce-B’;,‘(B2 -C)(CM’)’ +.,,
(3)
where B and C are, respectively, the second and the third thermal virial coefficients and CM’=(E - 1) /(E + 2). A least-squares fit of CM’RT/P as a function of CM’ in the low pressure region gives A, from the intercept of the curve. In order to determine precisely the second (I33 and the third (C,) dielectric virial coefficients, we used an expansion technique developed by Buckingham et al(1970). This method essentially consists of first measuring the sum of capacitances of two similar cells, one of which is filled with gas at density pr and the other is evacuated. The gas is then allowed to fill both cells, the density is almost halved, and the sum of capacitances is measured again. The linear term in density remains the same before and after the expansion, but the quadratic and higher orders change. Therefore, the second and third dielectric virial coefficients are determined by the change in total capacitance. Details of this expansion technique are given by Kirouac and Bose (1976) and Huot and Bose (1991). The working relation for the determination of B, and C, is given by
P. Malbrunot
*+*
et al. I Fluid Phase Equilihrta
=- [$$I
4B;-3A,C,+2B,A;+A; 36A;
94 (1994)
173-183
17.5
[(al--I)+k-01 (4)
1
[krl)+(E2-1)~.
where C, is the mean geometric capacitance of each cell [CL,=( CA +Cn) / 21, D, is the change of capacitance when the gas of dielectric constant or and density pr in cell A is expanded into the evacuated cell B. Similarly, if the expansion takes place from cell B [gas of dielectric constant &2and density p2] to cell A, the change in capacitance is D,. The combination of the two expansion The expansion of capacitors and volumes between the two cells. eliminates mismatch measurements have been carried out in such a way that er z tz2. On plotting the left-hand side against [( or- 1) +( a2 - I)], one can determine B, from the slope and C, from the curvature.
The schematic diagram of the experimental setup for the dielectric constant measurements as a function of the pressure is presented in Fig. 1. The dielectric constant measurements were carried out at 1 kHz using a capacitance bridge [Andeen Hagerling, model 25OOA]. The precision on capacitance measurement is estimated to be 5 - 10” pF. The cells used were of the three-terminal type. Each cell has a set of parallel plates 3.18 cm in diameter. The separation between the plates is adjusted to have a geometrical capacitance of about 10 pF. The plates were made of copper for good thermal conductivity. Details of the construction are given by Huot and Bose (1991). The measuring system (dielectric cells and valves) was immersed into a constant temperature bath. Temperature was controlled by a circulating bath within f 0.01 K and was measured using a thermistor calibrated against a platinum resistance thermometer. The uncertainty in temperature measurements is 0.01 K. The pressures were measured with a quartz pressure transducer calibrated against a pressure standard (Desgranges and Huot model 5213) having a precision of the order of 0.005%. A polynomial for the pressure values in terms of the readings of the transducer was fitted for the purpose of calibration. Considering the fitting and small errors due to the temperature, the final precision is estimated to be of the order of 0.01% for pressures above 1.5 MPa. The dielectric constant of the gas is given by the relation E=Co(P)/Co(O)
(5)
where Co (P) is the geometric capacitance at pressure P and Co (0) is the capacitance under vacuum. Considering the effect of pressure on the geometric capacitance Co, the relation between Co (P) and Co (0) is given by
(6) where y is the compressibility of the plates. Corrections to 4, yRT/3, (7RT/3)(A, +B)and (yRTI3) (A,B+B, -2Az+C) for copper.
B, and C, are, respectively, where y=2.4.10-6Mpa-1
The successive approximations for the density obtained from eqn. (1) are given by p, =CM’/A,
(7)
176
P Malbrunor
N al. I Fluid Phase Equilibria
94 (1994)
173-183
(8) 2B;-A$, P3=P2
+
(9)
4
where pr, pz and pJ are respectively, the first, second and third approximation of the density. High density measurement The measurements of s as a finction of high density were carried out at 1 kHz using a General Radio capacitance bridge [model 16201 and a cylindrical capacitor with guarded electrodes inserted into a very high pressure vessel. The experimental set-up has already been described in a previous paper by Dedit et al(1974). The dielectric constant is obtained in the same way [eqn. (5)] as in the case of the low density measurements. The uncertainty on the E values are of the order of 5 . 105. The pressure was measured with a manganine gauge previously calibrated. The density p, = m/Vu, is directly obtained from the value of the mass of methane included in the volume (V,) of the high pressure vessel. The practical procedure to measure this mass consists of trapping the gas [see Fig. 21 by condensation in liquid nitrogen into a light aluminum vessel, B, fitted to a pressure valve V, which can be disconnected from the remaining high pressure apparatus at the level of the valve V,, in order to be weighed accurately. The high pressure valves V, and V, are alternatively connected to vacuum or to high pressure generator. As for the volume VnP, it is predetermined by successive condensation and weighing, using argon as a standard gas, [Trappeniers et al. (1966)]. A correction is applied to take account of the effect of the pressure on the volume VHP The measurement of a for each value of the gas density requires four steps. At first, the whole apparatus is evacuated; during the second step, the gas is introduced at the desired pressure; and for the third step, the measurement of the capacity is carried out. Finally, for the last step, the gas is recovered into the bottle and weighed on a balance capable of measuring up to 1 kg with an accuracy of 1 mg.
RESULTS AND DISCUSSION The first dielectric virial coefficient (AJ is obtained from the intercept by plotting CM’ RT/P versus CM’ [eqn. (3)]. Figure 3 shows the curve for the determination of 4. Only the low pressure points have been considered for determining 4. Our value of 4 is compared with the value obtained by Bose et al.( 1972) in Table 1. The agreement is within the experimental uncertainty. The second (B.J and the third (CJ dielectric virial coefficients are obtained by using the expansion technique [see eqn. (4)]. Figure 4 shows a plot of the left-hand side of eqn. (4) as a t?mction of [( EI- 1) +(EZ - l)] for CH, at 298.15 K. Our values of B, and C, are also compared in Table 1, with the values obtained by Bose et al(1972) at 322.5 K and Straty and Goodwin (1973) at 280 K and a new value of C, obtained by a free fit of all the data keeping the measured values of A, and B, fixed. The present measured values of 4, B, and C, in Table 1 are given with three standard deviations.
P. Malbrurwt
CI al. I Fluid Phase Equilibria
94 (1994)
173-183
177
CT6
Fig. 1: Schematic diagram of the experimental set-up for the measurement of the dielectric constant at low densities where A and B: dielectric cells; C: gas compressor; CB: capacitance bridge; CTB: controlled temperature bath; G: gas cylinder; GN: generator; PG: pressure gage; V: vent; VP: vacuum pump.
Mn
1 I
Vhp
Fig. 2: Schematic diagram of the experimental set-up for the measurement of the dielectric constant at very high densities where B: balance; C: capacitance; Mn: manometer; Vhp: high pressure cell; Vi: valve.
178
P. Malbrum
-
e! al. I Fluid Phase Equilibria
94 (1994)
CH4
6.0
T =
0.01
0.00
173-183
0.02
0.03
298.15
0.04
K
0.05
CM' Fig. 3 : Plot of CM’ RT/P as a fbnction of CM’= [( E - 1) / (a + 2)] for the determination of A, of CH, at 298.15 K.
TABLE 1 Values of the first three dielectric virial coefficients for methane.
T(K)
A, (cm3 . mol-r)
B, (cm3 . mol-I)2
C, (cm3 . mol-1)s
Authors
298.15
6.551 f 0.002
7.23 f 0.93
-246*54
this worka
298.15
6.551 f 0.002
7.23 f 0.93
-314
this workb
322.5
6.541 f 0.003
7.29 f 0.32”
- 268 f 50
Bose et al(1972)a
288
6.538
- 252.7
Straty and Goodwin (1973)
6.07
a): B, and C,: measured from the expansion technique. b): C,: obtained by a free fit of all the data keeping the measured values of G, and B, fixed. The experimental values of the dielectric constant (E) and the molar density (p,J as a function of the pressure P are presented in Tables 2 and 3. We also present in the same tables, the calculated values of the density (pcp3 from the Setzmann and Wagner (1991) EOS and the deviations A p = [(p, - p,) / pcA ] %. In Table 2 for the low density region measurements [7 MPa
P. Malbrunor
et al. 1 Fluid Phase Equilibria
94 (1994)
179
173.183
0.00
=
-
c
298.15
K
-0.02
‘;
hl U
z
-0.04
c? + 0” h
-0.06
i G-
-0.08
E \ 2
-0.10
-0.12
0.0
0.1
0.3
0.2 (&I
0.4
0.5
0.6
0.7
-1)+(+%2-l)
Fig. 4: Plot of eq. (4): differential measurement for the independant determination of the second (B,) and the third (C,) dielectric virial coefficients of CH, at 298.15 K. In Fig. 5, we plot CM as a function of p, where the closed circles (0) are low pressure [Pc40 MPa] values and open circles (0) are related to high pressure measurements. From the least-squares fit we obtain the value of C’, by a free fit ofCMasatbnctionofp,where CM=(a-1)/(~+2)p,, using the measured values of A, and B, from low density measurements. The value of the third dielectric virial coefficient, C’, determined by free fit is - 3 14 ( cm3 ,mol-1)3 which agrees well with the experimentally determined value of - 246 + 54 ( cm3 - mall’
)’ and that
obtained previously [- 268
f 50 (cm3 . mol-1)3] by Bose et al. (1972). Our values of CM are also compared with values obtained at 280 K (A) and at 300 K (0) by Straty and Goodwin (1973). It should be noted that the three measured dielectric virial coefficients, 4, B, and C, do not fit the Claussius Mossotti function at very high densities (see curve 1, Fig. 5). We believe that the deviation at intermediate and high densities is mainly due to the large uncertainty in the measured value of the third dielectric virial coefficient. It seems that the expansion technique mearurement was not carried out to high enough pressure in order to obtain a precise value of C,. The third dielectric virial coefficient (CL) determined by a free fit of all the data using the measured values of A, and B, gives a value of - 3 14 cm9 mole-3 which agrees reasonably well with other measured values (see curve 2, Fig. 5). It is interesting to note to what extent the precision in the values of the higher dielectric virial coefficients is needed in order to determine precisely high density values. The deviations in density between pcA, the calculated value obtained from Setzmann and Wagner (1991) EOS and the measured value pm, are plotted against pressure in Fig. 6. In Table 2, the deviation in density is about 0.05% in the low and high pressure regions and up to about 0.1% in the intermediate pressure range.
180
P. Malbrunor
et al. I Fluid Phase Equilibria
94 (1994)
173-183
In the low density region, the uncertainty in the determination of the density is related to the uncertainties in CM’, 4, B, and C,. A close examination of the terms involved in density determination shows that the main contribution to the density comes from CM’/&. The remaining contribution which involves higher dielectric virial coefficients, becomes more important at high densities. In our case, the uncertainty of the order of 1 x 10” for the measurement of the dielectric constant leads to uncertainties of the order of 1 x 10-5for the determination of the density.
TABLE 2 The measured dielectric constant (E), the molar density (p,)
obtained from the measured dielectric
virial coefficients (4, B, and Ch), the calculated density (pc,)
from Setzmamr and Wagner (1991)
EOS and the deviation Ap (O/o)as a fbnction of the pressure (P) for CH, at 298.15 K for pressures up to 40 MPa. E
MPa
mol - mq3
7.00327
1.064088
0.3 1828
0.3 1806
0.06755
8.00616
1.074663
0.36937
0.36916
0.05405
9.01104
1.085562
0.42162
0.42143
- 0.04370
10.0365
1.096905
0.47578
0.47564
- 0.02921
11.8975
1.118101
0.57526
0.57522
- 0.00557
13.5176
1.136727
0.66168
0.66178
0.01607
15.0412
1.154159
0.74159
0.74167
0.01084
16.5429
1.170924
0.81759
0.81783
0.02976
18.0965
1.187644
0.89257
0.89301
0.05036
19.5697
1.202768
0.95972
0.96036
0.06646
21.1919
1.218497
1.02890
1.02973
0.08108
23.2668
1.237143
1.11000
1.11111
0.09942
25.2265
1.253247
.17941
1.18080
0.12508
29.1558
1.281675
.30227
1.30229
0.15570
33.7353
1.309415
.41634
1.41885
0.17746
40.0423
1.340580
.54463
1.54781
0.20591
In the high density region the measurement of a and p are independant, therefore the relative error A (CM)/CM on the CM function is
P. Malbrunot et al. I Fluid Phase Equilibria 94 (1994) 173-183
A(CM) CM=
3~ (E-1)(&+2)
181
dc+& E p
(10)
TABLE 3 The measured dielectric constant (a), the measured molar density (p,,J, the calculated density (pck) from Setzmann and Wagner (1991) EOS and the deviation Ap (%) as a tknction of the pressure (P) for CH, at 298.15 K for pressures up to 710 MPa. P
E
MPa
Pm’
10”
&A.
10
4
mol mmm3
mol - me3
AP %
10.359
1.1006980
0.49262
0.49283
0.0403 1
20.235
1.2096252
0.98923
0.98942
0.01950
30.399
12900252
1.33593
1.33629
0.02737
40.610
1.3433226
1.55789
1.55804
0.00935
50.290
1.3799848
1.70725
1.70719
- 0.00295
60.430
1.4100489
1.82755
1.82754
- 0.00048
70.410
1.4343868
1.92348
1.92435
- 0.00140
80.660
1.4555756
2.00634
2.00625
- 0.00355
90.830
1.4738263
2.07732
2.07711
- 0.00998
100.0
1.4885223
2.13448
2.13372
- 0.03549
150.0
1.5501796
2.36542
2.36793
0.10584
200.0
1.5944552
2.53185
2.53312
0.04995
250.0
1.6304259
2.65932
2.66252
0.12025
300.0
1.6600129
2.76691
2.76985
0.10609
350.0
1.6856698
2.86016
2.86210
0.06769
399.9
1.7081561
2.94026
2.94320
0.10003
450.0
1.7287640
3.01431
3.01622
0.06330
499.9
1.7473400
3.08082
3.08219
0.04453
549.8
1.7644634
3.14228
3.14305
0.02441
599.7
1.7806280
3.20019
3.19932
- 0.02717
656.3
1.7977076
3.26003
3.25862
- 0.04318
709.4
1.8125309
3.3 1208
3.31054
- 0.04628
As mentioned above, the relative error on the dielectric constant We does not exceed 0.005%. As firr as the relative error on the density Aplp is concerned, it is evaluated to be 0.05% taking into account the precision on weighing, the recovered gas and the precision on volwne calibration. The error A (CM)/(CM), a decreasing tkction with pressure, varies in the present case from 0.1% at moderate pressures to 0.05% at higher pressures.
182
P. Malbrunor
CI al. I Fluid Phase Equilibria
94 (1994)
173-183
6.62 :
6.60
t
6.56
mi
6.56
,u
6.54
qE h =
6.52
e
6.46
0
7
6.46
A
Straty
and
Coodwln
(1175):
2110 K
E
6.44
0
Stroty
and
Coodwln
(1073):
500 K
20
25
6.50 0
Thlm work
: 2911.15
K
6.42 0
5
10
15
30
Pm ( mol/l
35
40
1
,,, as a fimction of pm where (0) are low density experimental Fig. 5: Plot of CM=(a-l)/(a+2)p values and 0 are high density experimental values. The curve 1 is the least-squares fit with three measured dielectric virial coefficients (4, B, and CJ. The curve 2 is a fit using the measured values of G, and B, and C’, as a free fitting parameter.
0.6
T :
CH4
T=298.15K
0.4
s
0.2 B
3 -E
P I
00*
I -0.2
s” -0.4
-0.6 0
300
400 P
(MPa)
Fig. 6: Deviations in density for high density measurements.
500
600
700
P. Malbrunot
et al. I Fluid Phase Equilibria
94 (1994)
173-183
183
CONCLUSION This article clearly shows that the dielectric constant method is a powerful technique for precision PVT measurements. It also proves that even up to 710 MPa for CH, at 298.15 K, only three dielectric virial coefficients are necessary to reproduce accurate values of the density. Since the dielectric technique is relatively simple and inexpensive, it has great potential for industrial application. In fact, the dielectric constant method has already been applied to the density measurement for gas adsorption at high pressures by Vidal et al. (1990) and Malbnmot et al. (1992). ACKNOWLEDGMENTS The authors would like to thank the referees for their suggestions which have improved the paper substantially. We would also like to thank Mr. J. Hamelin for his help in computer programming and Mrs. L. Bellemare for typing the text. REFERENCES Bose, T.K., Sochanski, J.S. and Cole, R.H., 1972. Dielectric and pressure virial coefficients of imperfect gases. Octupole moments of CH, and CF.,. J. Chem. Phys., 57: 3592-3595. Buckingham,.A.D., Cole, R.H. and Sutter, H., 1970. Direct determination of the imperfect gas contribution to dielectric polarization. J. Chem. Phys., 52: 5960-5961. Dedit, A., Brielles, J., Lallemand, M. and Vidal, D., 1974. Condensateur pour la mesure precise de la constante di&ctrique des gaz cornprimes jusqu’a 12 kbar: application a l’helium. High Temp. High Press., 6: 189-193. Huot, J. and Bose, T.K., 1991. Determination of the quadropole moment of nitrogen from the dielectric second virial coefficient. J. Chem. Phys., 94: 3849-3854. Kirouac, S. and Bose, T.K., 1976. Polarizability and dielectric properties of helium. J. Chem. Phys., 64: 1580-1582. Malbrunot, P., Vidal, D., Vermesse, J., Chahine, R. and Bose, T.K., 1992. Adsorption Measurements of Argon, Neon, Krypton, Nitrogen and Methane on Activated Carbon up to 650 MPa., Langmuir, 8: 577-580. Setzmann, U. and Wagner, W., 1991. A New Equation of State and Tables of Thermodynamics Properties for Methane Covering the Range from Melting Line to 625 K at Pressures up to 1000 MPa. J. Phys. Chem. Ref. Data, 20: 1061-l 155. Straty, G.C. and Goodwin, R.D., 1973. Dielectric constant and polarizability of saturated and compressed fluid methane. Cryogenics, 13: 7 12-7 15. Trappeniers, N.J., Wassenaar, T. and Walkers, G.J., 1966. Isotherms and Thermodynamic Properties of Krypton at Temperatures between 0 “C and 150 “C and at Densities up to 620 Amagat. Physica, 32: 1503-1520. Vidal, D., Malbrunot, P., Guengant, L., Bose, T.K. and Chahine, R., 1990. Measurement of physical adsorption of gases at high pressure. Rev. Sci. Instrum., 6 1: 13 14- 13 18.