- Email: [email protected]
Low temperature PVT data for ethylene by an NMR method
Physica 104A (1980) 255-261 (~) North-Holland Publishing Co.
L O W T E M P E R A T U R E P V T DATA F O R ETHYLENE BY AN NMR METHOD N.J. TRAPPENIERS and B. ARENDS Van der Waals Laboratory, University of Amsterdam, Amsterdam, The Netherlands
(269th publication of the Van der Waals Fund) Received 30 June 1980 Densities of ethylene are reported as a function of pressure in the temperature interval of 123.15 K to 223.15 K and a pressure range of 20 to 2800 bar. The experimental method based on NMR is discussed and a comparison is made with the results of other authors.
1. Introduction In a previous publication t) extensive P V T data on gaseous ethylene were reported, covering a t e m p e r a t u r e range of 273.15 K up to 483.15K and a pressure range up to 3000 bar. Recently, within the f r a m e w o r k of an accurate N M R proton-spin echo investigation into the density d e p e n d e n c e of the coefficient of self-diffusion in gaseous and liquid ethylene2), the need was felt for P V T data covering the liquid phase of ethylene as well as no conventional P V T equipment was available for m e a s u r e m e n t s in the liquid phase, the N M R - t e c h n i q u e itself was e m p l o y e d to determine the densities.
2. Experimental method The density m e a s u r e m e n t s were carried out with a pulsed N M R spectrometer described elsewhere3'4). The principle of density m e a s u r e m e n t s is based on the fact that the proton spin equilibrium magnetization M0, m e a s u r e d f r o m the free induction d e c a y amplitude after a single 90°-pulse, is proportional to the n u m b e r of nuclei contributing to the magnetization according to Curie's law: -N 7 2 h 2 I ( I + 1)Ho. Mo = 3kT
(1)
H e r e N is the n u m b e r of nuclei per unit volume, H0 the static magnetic field and 3, the gyromagnetic ratio. For the density m e a s u r e m e n t s the s p e c t r o m e t e r was modified in two respects: (i) A Bruker B-SN15 field lock was added to the equipment in order 255
256
N.J. T R A P P E N I E R S A N D B. A R E N D S
to improve the long-time stability of the magnetic field. As the relaxation times encountered sometimes exceeded 40 s a precise measurement of an equilibrium magnetization could take as much as two hours. With the field lock a long-time stability of better than l:107 could be achieved during a period of eight hours. (ii) In order to minimize the influence of changing densities inside the resonance coil on the tuning of the resonance circuit, the resonance circuit was damped with a resistance of about 800 ~, giving rise to a poor signal to noise ratio and hence necessitating long signal averaging.
3. Outline of the experimental procedure 3.1.
Relative isothermal density measurements
First, measurements were performed on the 223.15 K isothermal, where densities are known up to 500 barS). Determining the density at pressure P amounts to measuring the equilibrium magnetization at for instance 500 bar and at P bar and then applying, according to (1), the relation
Mo(P ) o(P) - Mo(P = 500 bar) × p(P = 500 bar).
(2)
The reliability of this method, for which a linear input-output characteristic of the receiving system is imperativeS), was thoroughly tested by first trying it out on known densities at several temperatures. At high temperatures the very accurate data of Trappeniers et al. t) were used (273.15 K and 298.15 K) and at low temperatures the 213 K-isothermal of methane 6) was employed to
E} pressure-intervo[: I00- 533 Eg2 pressure-intervol :1350-1804 E] 25° isothermal CzHI, 0o !
1.5-%
L~p •
1'0-
®
T
u
,,
~
/
u
0.5
~ 0
Oz
:~ A p
I 500
I I000
I 1500
I 2000
bor
Fig. 1. T h e observed deviation from a linear M0-dependence of p as a function of pressure.
P V T DATA FOR ETHYLENE BY AN NMR METHOD
257
verify whether the assumption M 0 - P at constant temperature was justified. In all cases we found that the M0 vs. p - p l o t showed a slight positive deviation from a straight line. These deviations were independent of temperature and depended linearly on the pressure difference between the points 1 and 2, for which the ratio Mol[Mo2was determined. This is illustrated in fig. 1. All measured ratios Mo(P1)/Mo(P2) (P1 > P2) were, therefore, corrected by a factor of -0.60%/100 bar. The cause of the non-linear behaviour of the M0-p plot probably lies in a pressure dependence of the sample holder. The M0-ratios were measured independently three or four times; the values reproduced well within one percent. At the lowest temperatures, where no density data were available except near the coexistence lineS), relative isothermal density measurements were carried out. All measured densities were related to the density at 70 bar. 3.2. The determination of absolute densities In order to get absolute density values from the relative measurements again use was made of Curie's law (1); one value of M0 was measured at a known density at 223.15 K, another value at the same pressure but at a density which was previously determined by the relative method at 173.15 K or 123.15K, respectively. Here again, several experiments were done at known densities to check the Mo-pIT relation. No systematic deviations from this relation were found. The 800 12 damping resistance across the resonance circuit apparently also suppressed the influence of temperature changes on the properties of the resonance circuit.
4. Results
Table I gives the results of our density measurements. The accuracy of these data is estimated to be of the order of -+ 1%, although the reproducibility is slightly better. The relation between density and pressure can be described very well by means of a so-called Tait equation 7.8), i.e., a relation of the form p0
P ( P ) = 1 - C log [(B + P)/(B + Po)]'
where p(P) is P0, and B and obtained from For p0 and P0,
(3)
the density at pressure P, p0 is a reference density at pressure C are the parameters of the equation. The values for B and C, our experimentally determined densities, are given in table II. the low pressure values were taken from ref. 5. The agreement
N.J. T R A P P E N I E R S A N D B. A R E N D S
258
TABLE I
Experimental values for the density of ethylene T = 223.15 K
T = 173.15 K
T = 123.15 K
P
p
P
p
P
p
(bar)
(g/cm 3)
(bar)
(g/cm 3)
(bar)
(g/cm 3)
15.00 25.00 100.00 200.00 300.00 400.00 500.00 709.28 851.13 992.99 1134.84 1276.70 1418.55 1560.41 1702.26 1985.97 2340.61 2695.25
0.4823 a 0.4848 a 0.5007 a 0.5169 a 0.5298 a 0.5408 a 0.5503 a 0.5649 0.5750 0.5855 0.5927 0.6025 0.6091 0.6164 0.6223 0.6321 0.6415 0.6499
1.26 70.94 141.86 212.78 283.71 425.57 567.42 709.28 851.13 992.99 1134.84 1276.70 1418.55 1560.41 1702.26 1985.97 2340.61 2695.25
0.5625 a 0.5712 0.5763 0.5838 0.5892 0.5996 0.6072 0.6152 0.6237 0.6303 0.6352 0.6409 0.6466 0.6512 0.6563 0.6666 0.6769 0.6823
0.02 70.94 141.86 212.78 283.71 425.57 567.42 709.28 851.13 992.99 1200.70
0.6307 a 0.6393 0.6434 0.6486 0.6534 0.6597 0.6658 0.6704 0.6743 0.6800 0.6893
Data taken from ref. 5.
TABLE II
Parameters for Tait equation Pressure range (bar)
T (K)
Po (bar)
#o (g/cm 3)
B (bar)
C
15-2700 1-2700 0-1200
233.15 173.15 123.15
15.0 1.259 0.0202
0.4823 0.5625 0.6307
164.5 438.8 310.5
0.09467 0.09049 0.05127
between the empirical Tait formula and our results is illustrated in fig. 2 for the 223.15 isothermal. An attempt was also made to fit our density data to a Carnahan-Starling 9) equation of state. This equation, however, turned out to be inadequate for describing the data over the full pressure range. Our data may be compared with the results of Matthias et al)°), who found an expression for the rectilinear diameter of ethylene, and combined with those of Ligthart et al.N'12), who determined relative densities along the
P V T DATA FOR ETHYLENE BY AN NMR METHOD
259
P
0.65 g/cm3
T=
.
0.60 --
0.55
x.
,
nil
/
0.50 /
_~> p I
500
I
1000
I
1500
I
2000
I
2500 bar
Fig. 2. Experimental densities of ethylene compared to the Tait equation.
ordered solid-liquid and the disordered solid-liquid melting line of ethylene. The highest pressure of our measurements at 123.15 K is 1200.7 bar. According to the Simon-Giatzel equation, which was used to represent the transition line"), the pressure at the melting line at this temperature is 1203.39bar. Extrapolating our density data over a very small pressure range therefore enables us to make a comparison between our results and those of refs. l l , 12. The formula for the rectilinear diameter of ethylene reads~°): Y = 0.22179- 0.00061277 t g/cm 3,
(4)
where Y = ~(#~iq-p~as) and t is the temperature in degrees Celsius. This expression yields a molar volume for the liquid at the melting point 003.98 K, 0 bar) of 43.10 cm 3. Now, using the relative density measurements of Ligthart"), one arrives at a value of 42.11 cm 3 for the molar volume at the triple point (468 bar, 110.36 K, see the phase diagram of ethylene, fig. 3), while the molar volume along the disordered solid-liquid melting line varies as Vmol,li
q :
42.11(1 - (P - 468) x 2.845 x 10-5) cm 3.
(5)
Substitution of P = 1203.39 bar in (5) yields a molar volume of 41.23 cm 3 and, hence, a density of 0.6804 g/cm 3. On the other hand, inserting P = 1203.39 bar in the Tait equation for T = 123.15 K gives a density of 0.6865 g/cm 3, i.e., 0.9% higher. The agreement is, thus, fairly good. It is also possible to find an expression for the rectilinear diameter using ref. 2, where a table is given of liquid and gas densities along the vapour pressure line. This table was prepared using the data of Matthias et al) °) together with the more recent data of Menes et al)3). Applying a least squares
260
N.J. TRAPPENIERS AND B. ARENDS
2500
Ethylene
D
P
bar
/
f
/ dpsordered /
2000--
ordered
/
sotidn /
1500-/
1000--
/
/
~
Liquid
D m~tti~Line5oL,dt
//
®meLti~,nesotid~
500
--
[]
O
~
100
I
110
I
120
i
130
~" T
i
140K
Fig. 3. Phase-diagram of ethylene.
fit to these data, one finds for the rectilinear diameter: Y = 0.22116 - 0.00061523 t g/cm 3,
(6)
where the temperature t is again in degrees Celsius. The expression yields a density of 0.6800g/cm 3 at 1203.39bar and 123.15K. The agreement of our results with those of other authors is, therefore, within one percent. The comparison shows that it is very well possible to determine densities with the NMR-technique when an extremely high accuracy is not required.
P V T DATA FOR ETHYLENE BY AN NMR METHOD
261
Acknowledgement T h e w o r k p r e s e n t e d in this p a p e r w a s s u p p o r t e d b y the " S t i c h t i n g v o o r F u n d a m e n t e e l O n d e r z o e k der M a t e r i e ( F O M ) " . T h e a u t h o r s wish to t h a n k Mr. D. v.d. P u t t e n for his v a l u a b l e c o n t r i b u t i o n to the e x p e r i m e n t a l work.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
N.J. Trappeniers, T. Wassenaar and G.J. Wolkers, Physica 82A (1976) 305. B. Arends, KO. Prins and N.J. Trappeniers, to be published. K.O. Prins, N.J. Trappeniers and B. Arends, to be published. B. Arends, thesis, University of Amsterdam (1979). S. Angus, B. Armstrong and K.M. de Reuck, International Thermodynamic Tables of the Fluid State, Ethylene (Butterworths, London, 1972). A.J. Vennix and R. Kobayashi, Am. Inst. Chem. Eng. J. 15 (1969) 926. P.G. Tait, Reports on some of the physical properties of fresh water and sea water, Physics and Chemistry of the voyage of H.M.S. Challenger, vol. II, part IV (1888). H.J. Parkhurst, Jr. and J. Jonas, J. Chem. Phys. 63 (1975) 2698. N.F. Carnahan and K.E. Starling, J. Chem. Phys. 51 (1969) 635. E. Matthias, C.A. Crommelin and H.G. Watts, Comm. Phys. Lab. Univ. Leiden 189A (1927). F.A.S. Ligthart, Thesis, University of Amsterdam (1975). F.A.S. Ligthart, N.J. Trappeniers and K.O. Prins, Physica 97B (1979) 172. F. Menes, T. Dorfmiiller and J. Bigeleisen, J. Chem. Phys. 53 (1970) 2869.
L O W T E M P E R A T U R E P V T DATA F O R ETHYLENE BY AN NMR METHOD N.J. TRAPPENIERS and B. ARENDS Van der Waals Laboratory, University of Amsterdam, Amsterdam, The Netherlands
(269th publication of the Van der Waals Fund) Received 30 June 1980 Densities of ethylene are reported as a function of pressure in the temperature interval of 123.15 K to 223.15 K and a pressure range of 20 to 2800 bar. The experimental method based on NMR is discussed and a comparison is made with the results of other authors.
1. Introduction In a previous publication t) extensive P V T data on gaseous ethylene were reported, covering a t e m p e r a t u r e range of 273.15 K up to 483.15K and a pressure range up to 3000 bar. Recently, within the f r a m e w o r k of an accurate N M R proton-spin echo investigation into the density d e p e n d e n c e of the coefficient of self-diffusion in gaseous and liquid ethylene2), the need was felt for P V T data covering the liquid phase of ethylene as well as no conventional P V T equipment was available for m e a s u r e m e n t s in the liquid phase, the N M R - t e c h n i q u e itself was e m p l o y e d to determine the densities.
2. Experimental method The density m e a s u r e m e n t s were carried out with a pulsed N M R spectrometer described elsewhere3'4). The principle of density m e a s u r e m e n t s is based on the fact that the proton spin equilibrium magnetization M0, m e a s u r e d f r o m the free induction d e c a y amplitude after a single 90°-pulse, is proportional to the n u m b e r of nuclei contributing to the magnetization according to Curie's law: -N 7 2 h 2 I ( I + 1)Ho. Mo = 3kT
(1)
H e r e N is the n u m b e r of nuclei per unit volume, H0 the static magnetic field and 3, the gyromagnetic ratio. For the density m e a s u r e m e n t s the s p e c t r o m e t e r was modified in two respects: (i) A Bruker B-SN15 field lock was added to the equipment in order 255
256
N.J. T R A P P E N I E R S A N D B. A R E N D S
to improve the long-time stability of the magnetic field. As the relaxation times encountered sometimes exceeded 40 s a precise measurement of an equilibrium magnetization could take as much as two hours. With the field lock a long-time stability of better than l:107 could be achieved during a period of eight hours. (ii) In order to minimize the influence of changing densities inside the resonance coil on the tuning of the resonance circuit, the resonance circuit was damped with a resistance of about 800 ~, giving rise to a poor signal to noise ratio and hence necessitating long signal averaging.
3. Outline of the experimental procedure 3.1.
Relative isothermal density measurements
First, measurements were performed on the 223.15 K isothermal, where densities are known up to 500 barS). Determining the density at pressure P amounts to measuring the equilibrium magnetization at for instance 500 bar and at P bar and then applying, according to (1), the relation
Mo(P ) o(P) - Mo(P = 500 bar) × p(P = 500 bar).
(2)
The reliability of this method, for which a linear input-output characteristic of the receiving system is imperativeS), was thoroughly tested by first trying it out on known densities at several temperatures. At high temperatures the very accurate data of Trappeniers et al. t) were used (273.15 K and 298.15 K) and at low temperatures the 213 K-isothermal of methane 6) was employed to
E} pressure-intervo[: I00- 533 Eg2 pressure-intervol :1350-1804 E] 25° isothermal CzHI, 0o !
1.5-%
L~p •
1'0-
®
T
u
,,
~
/
u
0.5
~ 0
Oz
:~ A p
I 500
I I000
I 1500
I 2000
bor
Fig. 1. T h e observed deviation from a linear M0-dependence of p as a function of pressure.
P V T DATA FOR ETHYLENE BY AN NMR METHOD
257
verify whether the assumption M 0 - P at constant temperature was justified. In all cases we found that the M0 vs. p - p l o t showed a slight positive deviation from a straight line. These deviations were independent of temperature and depended linearly on the pressure difference between the points 1 and 2, for which the ratio Mol[Mo2was determined. This is illustrated in fig. 1. All measured ratios Mo(P1)/Mo(P2) (P1 > P2) were, therefore, corrected by a factor of -0.60%/100 bar. The cause of the non-linear behaviour of the M0-p plot probably lies in a pressure dependence of the sample holder. The M0-ratios were measured independently three or four times; the values reproduced well within one percent. At the lowest temperatures, where no density data were available except near the coexistence lineS), relative isothermal density measurements were carried out. All measured densities were related to the density at 70 bar. 3.2. The determination of absolute densities In order to get absolute density values from the relative measurements again use was made of Curie's law (1); one value of M0 was measured at a known density at 223.15 K, another value at the same pressure but at a density which was previously determined by the relative method at 173.15 K or 123.15K, respectively. Here again, several experiments were done at known densities to check the Mo-pIT relation. No systematic deviations from this relation were found. The 800 12 damping resistance across the resonance circuit apparently also suppressed the influence of temperature changes on the properties of the resonance circuit.
4. Results
Table I gives the results of our density measurements. The accuracy of these data is estimated to be of the order of -+ 1%, although the reproducibility is slightly better. The relation between density and pressure can be described very well by means of a so-called Tait equation 7.8), i.e., a relation of the form p0
P ( P ) = 1 - C log [(B + P)/(B + Po)]'
where p(P) is P0, and B and obtained from For p0 and P0,
(3)
the density at pressure P, p0 is a reference density at pressure C are the parameters of the equation. The values for B and C, our experimentally determined densities, are given in table II. the low pressure values were taken from ref. 5. The agreement
N.J. T R A P P E N I E R S A N D B. A R E N D S
258
TABLE I
Experimental values for the density of ethylene T = 223.15 K
T = 173.15 K
T = 123.15 K
P
p
P
p
P
p
(bar)
(g/cm 3)
(bar)
(g/cm 3)
(bar)
(g/cm 3)
15.00 25.00 100.00 200.00 300.00 400.00 500.00 709.28 851.13 992.99 1134.84 1276.70 1418.55 1560.41 1702.26 1985.97 2340.61 2695.25
0.4823 a 0.4848 a 0.5007 a 0.5169 a 0.5298 a 0.5408 a 0.5503 a 0.5649 0.5750 0.5855 0.5927 0.6025 0.6091 0.6164 0.6223 0.6321 0.6415 0.6499
1.26 70.94 141.86 212.78 283.71 425.57 567.42 709.28 851.13 992.99 1134.84 1276.70 1418.55 1560.41 1702.26 1985.97 2340.61 2695.25
0.5625 a 0.5712 0.5763 0.5838 0.5892 0.5996 0.6072 0.6152 0.6237 0.6303 0.6352 0.6409 0.6466 0.6512 0.6563 0.6666 0.6769 0.6823
0.02 70.94 141.86 212.78 283.71 425.57 567.42 709.28 851.13 992.99 1200.70
0.6307 a 0.6393 0.6434 0.6486 0.6534 0.6597 0.6658 0.6704 0.6743 0.6800 0.6893
Data taken from ref. 5.
TABLE II
Parameters for Tait equation Pressure range (bar)
T (K)
Po (bar)
#o (g/cm 3)
B (bar)
C
15-2700 1-2700 0-1200
233.15 173.15 123.15
15.0 1.259 0.0202
0.4823 0.5625 0.6307
164.5 438.8 310.5
0.09467 0.09049 0.05127
between the empirical Tait formula and our results is illustrated in fig. 2 for the 223.15 isothermal. An attempt was also made to fit our density data to a Carnahan-Starling 9) equation of state. This equation, however, turned out to be inadequate for describing the data over the full pressure range. Our data may be compared with the results of Matthias et al)°), who found an expression for the rectilinear diameter of ethylene, and combined with those of Ligthart et al.N'12), who determined relative densities along the
P V T DATA FOR ETHYLENE BY AN NMR METHOD
259
P
0.65 g/cm3
T=
.
0.60 --
0.55
x.
,
nil
/
0.50 /
_~> p I
500
I
1000
I
1500
I
2000
I
2500 bar
Fig. 2. Experimental densities of ethylene compared to the Tait equation.
ordered solid-liquid and the disordered solid-liquid melting line of ethylene. The highest pressure of our measurements at 123.15 K is 1200.7 bar. According to the Simon-Giatzel equation, which was used to represent the transition line"), the pressure at the melting line at this temperature is 1203.39bar. Extrapolating our density data over a very small pressure range therefore enables us to make a comparison between our results and those of refs. l l , 12. The formula for the rectilinear diameter of ethylene reads~°): Y = 0.22179- 0.00061277 t g/cm 3,
(4)
where Y = ~(#~iq-p~as) and t is the temperature in degrees Celsius. This expression yields a molar volume for the liquid at the melting point 003.98 K, 0 bar) of 43.10 cm 3. Now, using the relative density measurements of Ligthart"), one arrives at a value of 42.11 cm 3 for the molar volume at the triple point (468 bar, 110.36 K, see the phase diagram of ethylene, fig. 3), while the molar volume along the disordered solid-liquid melting line varies as Vmol,li
q :
42.11(1 - (P - 468) x 2.845 x 10-5) cm 3.
(5)
Substitution of P = 1203.39 bar in (5) yields a molar volume of 41.23 cm 3 and, hence, a density of 0.6804 g/cm 3. On the other hand, inserting P = 1203.39 bar in the Tait equation for T = 123.15 K gives a density of 0.6865 g/cm 3, i.e., 0.9% higher. The agreement is, thus, fairly good. It is also possible to find an expression for the rectilinear diameter using ref. 2, where a table is given of liquid and gas densities along the vapour pressure line. This table was prepared using the data of Matthias et al) °) together with the more recent data of Menes et al)3). Applying a least squares
260
N.J. TRAPPENIERS AND B. ARENDS
2500
Ethylene
D
P
bar
/
f
/ dpsordered /
2000--
ordered
/
sotidn /
1500-/
1000--
/
/
~
Liquid
D m~tti~Line5oL,dt
//
®meLti~,nesotid~
500
--
[]
O
~
100
I
110
I
120
i
130
~" T
i
140K
Fig. 3. Phase-diagram of ethylene.
fit to these data, one finds for the rectilinear diameter: Y = 0.22116 - 0.00061523 t g/cm 3,
(6)
where the temperature t is again in degrees Celsius. The expression yields a density of 0.6800g/cm 3 at 1203.39bar and 123.15K. The agreement of our results with those of other authors is, therefore, within one percent. The comparison shows that it is very well possible to determine densities with the NMR-technique when an extremely high accuracy is not required.
P V T DATA FOR ETHYLENE BY AN NMR METHOD
261
Acknowledgement T h e w o r k p r e s e n t e d in this p a p e r w a s s u p p o r t e d b y the " S t i c h t i n g v o o r F u n d a m e n t e e l O n d e r z o e k der M a t e r i e ( F O M ) " . T h e a u t h o r s wish to t h a n k Mr. D. v.d. P u t t e n for his v a l u a b l e c o n t r i b u t i o n to the e x p e r i m e n t a l work.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
N.J. Trappeniers, T. Wassenaar and G.J. Wolkers, Physica 82A (1976) 305. B. Arends, KO. Prins and N.J. Trappeniers, to be published. K.O. Prins, N.J. Trappeniers and B. Arends, to be published. B. Arends, thesis, University of Amsterdam (1979). S. Angus, B. Armstrong and K.M. de Reuck, International Thermodynamic Tables of the Fluid State, Ethylene (Butterworths, London, 1972). A.J. Vennix and R. Kobayashi, Am. Inst. Chem. Eng. J. 15 (1969) 926. P.G. Tait, Reports on some of the physical properties of fresh water and sea water, Physics and Chemistry of the voyage of H.M.S. Challenger, vol. II, part IV (1888). H.J. Parkhurst, Jr. and J. Jonas, J. Chem. Phys. 63 (1975) 2698. N.F. Carnahan and K.E. Starling, J. Chem. Phys. 51 (1969) 635. E. Matthias, C.A. Crommelin and H.G. Watts, Comm. Phys. Lab. Univ. Leiden 189A (1927). F.A.S. Ligthart, Thesis, University of Amsterdam (1975). F.A.S. Ligthart, N.J. Trappeniers and K.O. Prins, Physica 97B (1979) 172. F. Menes, T. Dorfmiiller and J. Bigeleisen, J. Chem. Phys. 53 (1970) 2869.