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Proton-spin-lattice relaxation and self-diffusion in methanes
Physica
51 (1971) 418-431
0 North-Holland
Pzlblishing
PROTON-SPIN-LATTICE
RELAXATION
AND SELF-DIFFUSION IV.
IN METHANES
SELF-DIFFUSION
P. H. OOSTINGt l’an
der Waals-laboratorium, (204th
publication
Co.
IN METHANE
and N. J. TRAPPENIEKS Universiteit
van Amsterdam,
of the Van der Waals
Received
1 July
Nederland
Fund)
1970
Synopsis The self-diffusion temperature are in almost mation.
is compared nearly
agreement
200 amagat
with
range
Enskog’s
covered,
the same:
I) of methane
using the spin-echo
quantitative
Above
density
coefficient
and density,
with the theoretical
Dp* decreases theory
has been measured technique.
with increasing
for a dense
i.e. up to 500 amagat,
over a wide range of
Up to 200 amagat zero-order density
gas of hard
approxi-
PA. This behaviour
spheres.
the temperature
the results
density Over
dependence
the total is very
D cc To.9.
1. Introduction. The present article, which is the latter in a series of four i), dealing with proton-spin-lattice relaxation and self-diffusion in methane, describes the results obtained on measuring the self-diffusion coefficient D. Of all transport phenomena the self-diffusion is the most simple, yet little experimental work has been done, to elucidate its behaviour as a function of temperature and density. This lack of data must be ascribed to the relative complicated technique of measurement and to the difficulty in obtaining results with an accuracy comparable to that obtained in experiments on the other classical transport coefficients e.g. the shear viscosity and the thermal conductivity. The development of the NMK spin-echo technique233) has provided a new and rather accurate method for measuring the self-diffusion coefficient in a uniform fluid. It does not need the use of a binary mixture, but allows for a direct measurement in a one-component system, which apart from the spin variables, is in thermodynamic equilibrium. Changing the thermodynamic variables, temperature and density, becomes very easy. t Present fabrieken,
address: Eindhoven,
Philips
Research
Laboratories,
Nederland. 418
N.V.
Philips’
Gloeilampen-
PROTON-SPIN-LATTICE
RELAXATION
AND
SELF-DIFFUSION.
IV
419
In this paper the results are presented of an investigation on the behaviour of D in gaseous and liquid methane. The measurements are carried out over a wide range of temperature (90 < T < 300 K) and density < 500 amagat), using the spin-echo spectrometer and methods
(10 < PA < as described
mainly in I and partly in II. The next section is devoted to a brief summary of the theory. In section 3 the main features of the experimental procedure are summarized. Section 4 deals with the presentation of the experimental results and in section 5 a comparison is made of experimental data with the theoretical expressions. statistical mechanics it 2. Theoretical exfwessio~cs. From non-equilibrium is well known that in a two-component molecular system the binary diffusion coefficient, defined by Fick’s law, can be expressed as475) :
Nk is the number of molecules of component k in the volume V ; vzk = Nk/V is the number density, pk = ??%kmk is the mass density, and mk is the molecular mass of component K ; pi is the momentum of molecule j. The average in the time integral has to be taken over an equilibrium ensemble. From (1) the familiar expression for the self-diffusion coefficient in an isotropic fluid (WZ= ~21 = nts) can easily be obtained6) :
To evaluate the time-correlation Zwanzig7) introduced the so-called work and the subsequent is obtained
paper by Kawasaki
that D can be written
D~z=CO+
function of binary-collision
molecular momentum expansion. From his
and Oppenheims)
in terms of the density
Cin+C&logn.+Csnz+
... .
evidence
as: (3)
Zwanzig showed that the first (constant) term in this expansion is identical with the Chapman-Enskog solution of Boltzmann’s equationa). In firstorder Sonine polynomial approximation, Ca is commonly written as 3
co = __ am
(xmkT)h xo2~n(l,l)* 3
where sZ(‘,‘)* is one of the reduced Boltzmann collision integrals as defined in ref. 9. The collisional integral depends on the intermolecular potential and on temperature. Numerical values for Qcl*s)* over a wide range of
420
P. H. OOSTlNG
reduced temperature (6, 12) potential:
N. J. TKAPPENIEKS
T* = kT/c are given by Lileyra)
V(r) = 4&[(o/r)r2 When the potential
AND
for a Lennard-Jones
(0/~)6].
parameters
(5) c/k and u are known,
a comparison
can be
made between the theoretical zero-order density approximation and experimental values. Comparison of theory and experiment at higher densities is more complicated. Until now the coefficients of the first- and higher-order density terms have not been calculated. Estimation of the higher-order coefficients from a fit of the experimental data to (3) depends on which units are chosen for n. Besides, such an estimation seems rather questionable as long as nothing is known about the form of higher-order terms in the expansion (3). The only theoretical prediction for high densities which can be verified by experiment is the Enskog modification of the Chapman-Enskog theory for a dense gas of hard spheres. In the Enskog theory the finite molecular size is taken into account in the Boltzmann equation, which has a double effect. The first one is the introduction of collisional transfer, which, however, does not influence the transport of mass. The second effect is a relative increase in the number of collisions with a factor x. Enskog’s result for diffusion can be written ass): Drt = (Dn)(O)/x,
(6)
where x is the radial distribution function at distance hard-sphere molecular diameter. For a hard-sphere gas x is given by the virial expansion x =
1 + 0.625(bp)
+ 0.2869(bp)s
+ O.l15(bp)3 +
. ...
0h.s.;
Gh.s. is the
(7)
where bp is the molecular covolume with b = 2xa~.,,/3m. x can be calculated if 0h.s. is known. A variety of methods has been proposed to estimate x for a real gas. Usually, the equation of state for a hard-sphere gas PV
= RT(1
+ bpx)
is written for a real gas by substituting pressure T(aP/aT)v. This leads to
(8) instead of the pressure P the thermal
(9) with b=-
1
dB
R
d7.’
where B is the second virial coefficient.
(10)
PROTON-SPIN-LATTICE
3. Experimental
RELAXATION
procedure.
liquid CHJ has been measured
AND SELF-DIFFUSION.
The self-diffusion
coefficient
with the spin-echo
IV
421
of gaseous
and
spectrometer
which has
been described in detail in I. The diffusion constant was estimated from the height of the spin echo as a function of the time interval between a 90”-180” pulse sequence at constant field gradient. The field gradient was generated by two coils on the pole surfaces of the magnet, designed according to Berger and Butterweckii). The gradient was calculated from the geometry of the coils and measured from the position of the first minima of the echo after a 90”-180” pulse sequence, using a cylindrical water sample. Calculated and measured values were in good agreement: 12.62 and 12.50 + 0.02 G cm-i, respectively, for a current of 1 A; the measured value was assumed to be the most reliable. In all cases 1’s_effects could be avoided by choosing the gradient between 3 and 10 G cm-i. As described in I, two different techniques were used for varying temperature and density. In the first one a liquid-nitrogen cryostat was used. The temperature could be varied continuously between 77 I< and room temperature, and was measured within 0.01 K. The cryostat contained the samples in sealed-off glas tubes with constant over-all CH4 density. Maximum pressure at room temperature obtained in this way was about 200 atm corresponding with an over-all density of about 235 amagat which is just above the critical density (226 amagat). For the measurements in the cryostat five samples were used: CH4(49), from which the amagat densiCH4(75), CH4( 108), CH4( l26), and CH&35), ty, as given in the brackets, was estimated as described in I and II.
loa
Lo.105 +
I
’ 33OK
I
I 250
I
II 2’JoTk
I
I 150
I
I I
100T,
Fig. 1. Self-diffusion coefficient of CH4 in the coexistence region and along the isochores; o experimental points; l obtained from interpolation of the isotherms; v obtained from interpolation of the coexistence line at PA = 300, 400 and 500.
I'.H. OOSTlNG
422
ANI> N. J. TRAPPENIEKS
A second technique, developed for measurements up to higher densities, made use of a high-pressure vessel designed to withstand pressures up to 3000 atm. With the high-pressure vessel measurements were carried out at three different temperatures : 25.00 & O.O2”C, 0.00 & O.O5”C, and -78.40
f
0.05”C. All measurements were carried out on very pure methane (Matheson; 99.997; CH4). Oxygen was removed from it to less than 0.1 ppm, especially for Tr-measurements,
which
were carried
out simultaneously.
report of part of the results pre4. Experimental results. A preliminary sented here is published earlier 12). Measured values of D in the liquid-gas coexistence region and along the isochores (PA = 49,75, 108, 126,235) are shown in fig. 1. Moreover, a number of points is given which is obtained from interpolation of the isothermal measurements at corresponding densities and at three higher densities (PA = 300, 400, 500). Interpolated values on the coexistence curve at 300, 400 and 500 amagat are also shown. Measurements carried out with the different techniques are in good agreement. TABLE Self-diffusion
coefficient Liquid
1000/T (K-l)
PA
I
of CH4 in the coexistence
region Vapour
branch
105D
lOsD(‘J)
(cm2 s-r)
(cm2 s-l)
(exp.)
(theor.)
PA
branch 1030(O)
1030 (cm2 s-l)
(cm2 s-l)
(exp.)
(theor.)
90.92
11.00
631
95.94
10.50
623
3.01
100.00
10.00
614
3.61
105.26 111.11
9.50
604
4.35
9.00
593
5.23
117.65
8.50
579
6.34
125.00
8.00
562
7.76
133.33
7.50
545
9.56
137.93
7.25
13.2
7.76
7.32
142.86 148.15
7.00 6.75
522
11.8
16.8 21.3
6.15 4.99
5.96 4.87
153.85
6.50
492
15.3
160.00 166.67
6.25 450
21.2
25.9
27.6 35.8 47.0
3.95 3.06 2.46
3.90 3.13 2.48
55
2.18
2.16
420
24.9
28.9
65 77
1.87 1.71
1.87 1.61
170.21 173.91
6.00 5.875 5.75
177.78
5.625
2.52
181.82
5.50
379
30.3
33.4
95
1.45
1.33
186.05
5.375
350
35.8
37.0
1.15
1.08
190.48
5.25
-
58.5
120 -
0.67
PROTON-SPIN-LATTICE
KELAXATlON
TABLE Self-diffusion a =
coefficient
1030 (cm2 s-l)
,,A = 49
1000/T tKml)
a
b
170.21
5.875
2.45
2.42
173.91 181.82 186.05 190.48 195.12 200.00 210.53 222.22 235.29 250.00 266.67 285.71 299.20 307.69
5.75 5.50 5.375 5.25 5.125 5.00 4.75 4.50 4.25 4.00 3.75 3.50 3.343 3.25
2.46
2.48
2.60
SELF-DIFFUSION.
(exp.);
423
II
b =
1030(O) (cm2 s-l)
p* = 75
PA = 108
b
a
b
2.58
1.72
1.69
2.80
2.70
1.84
1.76
1.25
1.20
1.29
2.88
2.83
1.91
1.85
3.06
2.97 3.12
2.03 2.11
3.28 3.45
3.28
3.67
(theor.)
pa = 126 a
b
1.22
1.12
1.05
1.36
1.28
1.18
1.10
1.94 2.04
1.40
1.35
1.23
1.47
1.41
2.29
2.14
1.52
3.46
2.39
2.26
3.77
3.66
2.58
4.11
3.89
4.34 4.39
Pi = 235 a
b
0.61
0.58
0.62
0.59
1.15
0.63
0.62
1.29
1.21
0.67
0.65
1.49
1.35
1.28
0.70
0.68
1.65
1.57
1.44
1.35
0.73
0.72
2.39
1.73
1.66
1.51
1.42
0.76
0.76
2.62
2.54
1.87
1.76
1.62
1.51
0.79
0.81
4.05
2.75
2.64
1.96
1.83
1.69
1.57
0.84
0.84
4.14
2.90
2.70
1.96
1.88
1.72
1.61
0.87
0.86
TABLE coefficient
25.OO”C DPA
IV
of CH4 along the isochores;
a
Self-diffusion
PA
AND
III
of CH4 along the isotherms
- 78.4”C
o.oo”c PA
(cm2 s-l)
DPA
PA
DPA (cm2 s-l)
110 124 242 282 322 338 356 387 424 447 480 501
0.146 0.144 0.136 0.135 0.127 0.127, 0.121 0.115 0.108 0.103 0.103 0.090
(cm2 s-l)
137
0.207
50
167
0.201
75
194
0.199
100
220
0.201
125
263
0.196
150
300
0.191
175
324
0.183
200
347
0.181
225
382 431
0.171
250
0.155
275
465
0.147
300
501 520
0.133 0.124
350
549
0.116
375
325
400 425 450 475 500 525 550 575
0.202 0.198 0.200 0.200 0.196 0.192 0.193 0.191 0.186 0.179 0.175 0.171 0.163 0.155 0.152 0.145 0.140 0.135 0.122 0.114 0.103 0.095
424
P. H. OOSTING
AND N. J. TKAPPENlEKS
0.05 -
0
Fig. 2. Self-diffusion
‘\\_
I
I 200
I
‘\
‘I_ I 400
..__
-.-P._/
A
1 600
coefficient of CH4 along the isotherms; o experimental l obtained from interpolation of the isochores.
points
PROTON-SPIN-LATTICE
RELAXATION
AND SELF-DIFFUSI~N.~~
425
The experimental values are tabulated in tables I and 11. In the liquid D increases with a factor 25 from the triple point up to the critical point as a result of increasing temperature and decreasing density. From a log(Bz) ZIS.log T plot it was found that along the liquid coexistence line Dn cc T2-77. A similar behaviour with nearly the same exponent of T is observed in ethaners) and xenoni4). It is probable that this behaviour is caused by the simultaneously varying density and temperature along the coexistence curve. In the coexisting gas phase increasing density results in spite of increasing temperature in a decrease of D. Along the isochores D increases with temperature. A quantitative discussion will be given in the next section. Results of the isothermal measurements are shown in fig. 2. Values obtained by interpolation of the isochores at corresponding temperatures are also given. DPA seems to be constant up to rather high densities. At 200 amagat deviations from the zero-density value are about 5%. The lowdensity values of DPA are in reasonable agreement with results obtained by Winni57 16). Th e experimental values are tabulated in table III. 5. Discztssion. 5.1. Diffusion at low densities. From the isothermal measurements it may be seen that Dn is nearly constant up to rather high densities. Evidently the constant term in the expansion (3) is dominant in this density region. Therefore, one may attempt a quantitative comparison between experimental results at lower densities and the zero-order density approximation D(O)for D. It is easily derived from (4) that for methane (molecular weight 16.04; PNTP = 0.717 g CEle3) :
(11) where 0 has to be taken in A and PA is the density in amagat units. From second virial coefficients which are obtained from recent PVT measurements between 0” and 15O”C17) c/k and (T are estimated for a Lennard-Jones (6, 12) potential (5) to be 147.0 K and 3.828 A respectively. These values are in good agreement with those given by Hirschfelder et al. 9). With these numbers and Liley’sro) values for Q(‘*‘)*, D(O) is calculated from (11) along part of the coexistence line, along the isochores and at the three temperatures at which the isotherms are measured. Results are given in tables I and II and fig. 2. Comparison with experimental values shows a good agreement up to rather high densities (PA < 200). Theoretical values are low by 0 to 8%. Optimal agreement between D(O)and experimental lowdensity values is obtained by taking the Lennard- Jones molecular diameter cr = 3.71 A. To compare the theoretical and experimental temperature dependence at
426
I’. H. OOSTING
,Li 2.25
Fig. 3. Temperature o experimental
points;
AND
N. J. TRAPPENIERS
2.30
2.35
dependence
l obtained
2.40
of D(CH4)
2.45
at constant
from interpolation
LogT 2.50
density;
of the isotherms.
low densities, log Go,‘)* was plotted IX. log T in the temperature range of interest. Within 0.50/ the points lay on a straight line with slope -0.38, from which the temperature dependence of the zero-order density approximation is found to be D(O) x T0.88.
(12)
The isochoric measurements provide good material to check this temperature dependence. From fig. 3 where the isochores are plotted vs. temperature on a log-log scale, the experimental behaviour appears to be in good agreement with (12). Moreover, the temperature dependence varies negligibly over a wide range of density as may be seen from the 300, 400, and 500 amagat isochores in fig. 3 which are obtained from interpolation of the Dp* isotherms. This will be discussed in the next section. Summarizing the above considerations it can be concluded that up to about 200 amagat the experimental behaviour of the self-diffusion coef-
PROTON-SPIN-LATTICE
RELAXATION
AND SELF-DIFFUSION.
250gcds-1
0
427
.
?.,Z
---+P
________7"' 100
I\
I
100
I
200
I
A 300
Fig. 4. Shear viscosity 7 of CH4 at 25°C.
ficient can be predicted within a few per cent by the theoretical zero-order density approximation. This behaviour is dissimilar with that of the other classical transport coefficients : the shear viscosity and the thermal conductivity. To demonstrate this fact provisional values for the shear viscosity ~jrof methane at 25”Crs) are plotted against density in fig. 4. The constant, zero-order, density approximation is also shown. The experimental values deviate much more rapidly from ~(0) than was the case in the behaviour of DPA. This is mainly due to the fact that the contribution of collisional
transfer
plays an important
role in the viscosity.
5.2. Diffusion at elevated densities. At the higher densities Dn is no longer constant but decreases with increasing density (fig. 2). The measurements along the coexistence curve show a similar behaviour, as may be seen from comparison of experimental data with D(O) (table I). At higher densities experimental values are smaller than D(O); the deviations increase with increasing density. The verification of Enskog’s theory for high densities will be restricted to the isothermal measurements. The radial distribution function x is calculated with the two methods described in section 2. Within the first method _ which is the more consistent with Enskog’s theory for hard spheres x can be calculated if the hard-sphere diameter Gh.S. is known. Gh.S. is estimated by assuming that at zero density D equals the zero-order density approximation for hard spheres, i.e. is given by (4) with Q(‘,‘)* = 1 and
428
P. Hf. OOSTING
AND
N. J. TRAPPENIERS
From extrapolation of the experimental isotherms to PA -= 0, 0h.s. is found to be 3.85 A at 298.15 K, 3.85 A at 273.15 K and 4.16 A at 194.8 K. From these values x is calculated with (7) and Dn. from (6). Results are shown in fig. 2: Dpt' (I). The agreement with the experimental be0
=
u’h.8..
haviour is poor. Results from the second method for calculating x using (9), (10) and PVT data from the literature, are also shown in fig. 2: Dpf)(II). These curves are drawn in such a way as to give the best fit with experimental data in the region between 50 and 100 amagat. 4t the high temperatures, the agreement with experiment is favourable with regard to the results of method I, though at high densities the deviations are still of the order of 20%. The agreement for the 194.8 K isotherm is much poorer. For comparison, the Enskog predictions for the shear viscosity q(e) (I) and q(e) (II) are also calculated, using the expression for q(e) as given in ref. 9. Results are shown in fig. 4. The agreement of the experimental values with q(e) (II) is of the same order as in the case of D, while q(e) (I) agrees much better with experiment than did L)(e) (I). Summarizing, it can be said that the use of method I would seem to lead to the conclusion that Enskog’s theory gives a reasonable account only for
Fig. 5. T-Q.QD~A as a function o coexistence
region
; A, isochores ;
q
l 194.8 K isotherm;
298.15
of density K isotherm
solid curve:
in CH*;
; x 273.15 K isotherm ;
eq. (13).
PROTON-SPIN-LATTICE
RELAXATION
AND
SELF-DIFFUSION.
the collisional transfer, which affords an important shear viscosity, but does not influence the transport cation of Enskog’s theory according to method II gives agreement at the high temperatures, though at the pirical fit of the radial distribution function x.
IV
429
contribution to the of mass. The appliin all cases moderate cost of a rather em-
Contrary to the Enskog theory, the more rigorous theoretical treatment of self-diffusion at higher densities, which led to the expansion (3), is much more difficult to compare with the experimental results. Especially an experimental confirmation of the occurrence of a logarithmic term in the expansion cannot be obtained from the present measurements. To demonstrate this, all experimental points are drawn in fig. 5, where T-o.gDp~ is plotted vs. density, such a plot being justified by the fact that the temperature dependence was found to be the same over the total density range covered. With the method of least squares four different expansions are fitted to the experimental points. The results of this calculation are TABLE IV Different Fit to T-O.gDpa
fits to T-“.gDpA lo%1
tow0
1osc2
CO f
&PA
+
C2PiInPA
0.1181
& 0.0007
0.785 5 0.056
-0.0520
f
CO +
+‘A
+
CZP;
0.1170
f
0.0007
1.040 f
0.064
-0.371
+ 0.010
CO +
&PA
f
C2p%lnpA
0.1198
f
0.0013
0.359 f
0.269
-0.137
& 0.053
CO +
&PA+
0.1197
i
0.0010
0.532 * 0.158
-0.164
& 0.060
c2P;
Fit to
+
+
&Pi
c3P;
co +
&PA
+
C2Piln
CO +
&PA
+
c2Pi
ClPA
+
C2&
st. dev.
x 104
0.31
PA
0.32
CO + Clpa + Czpi ln PA + C& CO +
1O’OC3
T-“,gDpA
0.0014
+ c3Pi
6 1.O 0.0224
* 37.6
0.31
f
0.31
0.0064
summarized in table IV. From the tabulated standard deviations, which are all smaller than the experimental inaccuracy, it is clear that no discrimination can be made between the different expansions. The results only suggest that third- and higher-order density terms may be neglected in the density range covered. One of the calculated curves, viz. T-O.gDp* = lo-a(O.1181
+ 0.785
x 10-4~~ - 0.520
x IO-7pilnpa)
(13)
is drawn in fig. 5, and will be used to discuss the temperature dependence of DPA along the liquid branche of the coexistence curve, which was mentioned in section 4 to be Dp* a T 2.77. It will be shown that this “strong tempera-
P. H. OOSTING AND N. J. TRAPPENIERS
430
015 right hand
side
0.10 0.00 -
0.06 -
slope:
2.56
-1
I 100
1 90
0.01
Fig. 6. Log-log
I
I
110
120
I
I
I
140
I 160
I
I
I
160
I 200°K
plot of right-hand side of eq. (15) vs. temperature.
ture dependence” is essentially due to the varying density along the coexistence line, in contradiction with conclusions drawn by Noble and Bloom 13). To prove this statement, use is made of the Guggenheim descriptionis) of the density pa along the liquid branch of the coexistence curve: PLIP cr =
1 +
$(l
-
T/T,,)
+
p(1 -
T/T,,)+,
where per and T cr are the critical density and When (14) together with the critical constants = 190.6 Ii, are put into eq. (13), one obtains an the liquid branch of the coexistence line, which only. This expression can be written as: DPA (coexistence
0.520
=
1 + a(1 -
temperature, respectively. per = 226 amagat, T,, = expression for DPA along contains the temperature
line)
= 10-V-“.9[0.1181 -
(14)
+ 0.785
x 226 x 10-4/(T)
x 2262 x lo-7{f(T)}2
In 226/(T)],
(15)
where f(T)
T/190.6)
+ p(l -
T/190.6)*.
(16)
In fig. 6 a log-log plot is made of the right-hand side of eq. (15) VS. temperature T. The points lie on a straight line with slope 2.58, which is very near to the experimental coefficient 2.77.
PROTON-SPIN-LATTICE
RELAXATION
Acknowledgement.
AND
This investigation
SELF-DIFFUSION.
IV
431
is part of the research program
of the “Stichting
voor Fundamenteel Onderzoek der Materie (F.O.M.)“, “Organisatie voor Zuiver-Wetenschappelijk Onderzoek supported by the der Materie (Z.W.O.)“. The authors are much indebted to Mr. G. L. Boekhout, Mr. J. A. Langendijk, and Mr. J. C. A. Offringa for assistance with the measurements.
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C. J. and Trappeniers,
Gerritsma, Oosting,
C. J., Oosting,
P. H. and Trappeniers,
2)
Hahn,
3)
Carr, H. Y. and Purcell,
4)
Green,
E. L., Phys.
Rev.
5)
Oosting, Ernst,
P. H., thesis,
7)
Zwanzig,
Phys.
8)
Kawasaki,
9)
Hirschfelder,
J. O., Curtiss,
John Wiley
10)
Liley,
11)
Berger,
12)
Trappeniers,
129 (1963)
K. and Oppenheim,
P. E., TPRC
Report
Noble,
Carr, H. Y., private
J. D. and Bloom,
94 (1954)
physics
Inc.
630.
(1968).
IX C (1967)
Rev.
139 (1965)
15)
Winn,
E. B. and Ney,
Winn,
E. B., Phys.
(New York,
H. J., Arch.
theory
of gases and
1954).
Elektrotechnik
P. H., Phys.
M., Phys. E. P., Phys.
Rev.
A 1763.
R. B., Molecular
Rev.
Letters
Letters
80 (1950)
17)
Trappeniers,
N. J. and Wassenaar,
18)
Trappeniers,
N. J. and Botzen,
19)
Guggenheim,
E. A., J. them.
Rev.
14 (1965) 250.
72 (1947)
77.
1024. T., to be published.
A., to be published. Phys.
13 (1945)
42 (1956)
23 (1966)
communication.
16)
417.
486.
I., Phys.
N. J. and Oosting,
13)
Rev.
15 (1963).
W. and Butterweck,
14)
51 (1971) 381. (II)
395. (III).
344.
C. F. and Bird,
and Sons,
51 (1971)
of Amsterdam
in theoretical
Rev.
365. (I);
580.
2 (1961)
University
M. H., Lectures R., Phys.
80 (1950)
51 (1971)
N. J., Physica
N. J., Physica
E. M., Phys.
H. S., J. math.
6)
liquids,
N. J.. Physica
P. H. and Trappeniers,
253.
445.
216.
;
51 (1971) 418-431
0 North-Holland
Pzlblishing
PROTON-SPIN-LATTICE
RELAXATION
AND SELF-DIFFUSION IV.
IN METHANES
SELF-DIFFUSION
P. H. OOSTINGt l’an
der Waals-laboratorium, (204th
publication
Co.
IN METHANE
and N. J. TRAPPENIEKS Universiteit
van Amsterdam,
of the Van der Waals
Received
1 July
Nederland
Fund)
1970
Synopsis The self-diffusion temperature are in almost mation.
is compared nearly
agreement
200 amagat
with
range
Enskog’s
covered,
the same:
I) of methane
using the spin-echo
quantitative
Above
density
coefficient
and density,
with the theoretical
Dp* decreases theory
has been measured technique.
with increasing
for a dense
i.e. up to 500 amagat,
over a wide range of
Up to 200 amagat zero-order density
gas of hard
approxi-
PA. This behaviour
spheres.
the temperature
the results
density Over
dependence
the total is very
D cc To.9.
1. Introduction. The present article, which is the latter in a series of four i), dealing with proton-spin-lattice relaxation and self-diffusion in methane, describes the results obtained on measuring the self-diffusion coefficient D. Of all transport phenomena the self-diffusion is the most simple, yet little experimental work has been done, to elucidate its behaviour as a function of temperature and density. This lack of data must be ascribed to the relative complicated technique of measurement and to the difficulty in obtaining results with an accuracy comparable to that obtained in experiments on the other classical transport coefficients e.g. the shear viscosity and the thermal conductivity. The development of the NMK spin-echo technique233) has provided a new and rather accurate method for measuring the self-diffusion coefficient in a uniform fluid. It does not need the use of a binary mixture, but allows for a direct measurement in a one-component system, which apart from the spin variables, is in thermodynamic equilibrium. Changing the thermodynamic variables, temperature and density, becomes very easy. t Present fabrieken,
address: Eindhoven,
Philips
Research
Laboratories,
Nederland. 418
N.V.
Philips’
Gloeilampen-
PROTON-SPIN-LATTICE
RELAXATION
AND
SELF-DIFFUSION.
IV
419
In this paper the results are presented of an investigation on the behaviour of D in gaseous and liquid methane. The measurements are carried out over a wide range of temperature (90 < T < 300 K) and density < 500 amagat), using the spin-echo spectrometer and methods
(10 < PA < as described
mainly in I and partly in II. The next section is devoted to a brief summary of the theory. In section 3 the main features of the experimental procedure are summarized. Section 4 deals with the presentation of the experimental results and in section 5 a comparison is made of experimental data with the theoretical expressions. statistical mechanics it 2. Theoretical exfwessio~cs. From non-equilibrium is well known that in a two-component molecular system the binary diffusion coefficient, defined by Fick’s law, can be expressed as475) :
Nk is the number of molecules of component k in the volume V ; vzk = Nk/V is the number density, pk = ??%kmk is the mass density, and mk is the molecular mass of component K ; pi is the momentum of molecule j. The average in the time integral has to be taken over an equilibrium ensemble. From (1) the familiar expression for the self-diffusion coefficient in an isotropic fluid (WZ= ~21 = nts) can easily be obtained6) :
To evaluate the time-correlation Zwanzig7) introduced the so-called work and the subsequent is obtained
paper by Kawasaki
that D can be written
D~z=CO+
function of binary-collision
molecular momentum expansion. From his
and Oppenheims)
in terms of the density
Cin+C&logn.+Csnz+
... .
evidence
as: (3)
Zwanzig showed that the first (constant) term in this expansion is identical with the Chapman-Enskog solution of Boltzmann’s equationa). In firstorder Sonine polynomial approximation, Ca is commonly written as 3
co = __ am
(xmkT)h xo2~n(l,l)* 3
where sZ(‘,‘)* is one of the reduced Boltzmann collision integrals as defined in ref. 9. The collisional integral depends on the intermolecular potential and on temperature. Numerical values for Qcl*s)* over a wide range of
420
P. H. OOSTlNG
reduced temperature (6, 12) potential:
N. J. TKAPPENIEKS
T* = kT/c are given by Lileyra)
V(r) = 4&[(o/r)r2 When the potential
AND
for a Lennard-Jones
(0/~)6].
parameters
(5) c/k and u are known,
a comparison
can be
made between the theoretical zero-order density approximation and experimental values. Comparison of theory and experiment at higher densities is more complicated. Until now the coefficients of the first- and higher-order density terms have not been calculated. Estimation of the higher-order coefficients from a fit of the experimental data to (3) depends on which units are chosen for n. Besides, such an estimation seems rather questionable as long as nothing is known about the form of higher-order terms in the expansion (3). The only theoretical prediction for high densities which can be verified by experiment is the Enskog modification of the Chapman-Enskog theory for a dense gas of hard spheres. In the Enskog theory the finite molecular size is taken into account in the Boltzmann equation, which has a double effect. The first one is the introduction of collisional transfer, which, however, does not influence the transport of mass. The second effect is a relative increase in the number of collisions with a factor x. Enskog’s result for diffusion can be written ass): Drt = (Dn)(O)/x,
(6)
where x is the radial distribution function at distance hard-sphere molecular diameter. For a hard-sphere gas x is given by the virial expansion x =
1 + 0.625(bp)
+ 0.2869(bp)s
+ O.l15(bp)3 +
. ...
0h.s.;
Gh.s. is the
(7)
where bp is the molecular covolume with b = 2xa~.,,/3m. x can be calculated if 0h.s. is known. A variety of methods has been proposed to estimate x for a real gas. Usually, the equation of state for a hard-sphere gas PV
= RT(1
+ bpx)
is written for a real gas by substituting pressure T(aP/aT)v. This leads to
(8) instead of the pressure P the thermal
(9) with b=-
1
dB
R
d7.’
where B is the second virial coefficient.
(10)
PROTON-SPIN-LATTICE
3. Experimental
RELAXATION
procedure.
liquid CHJ has been measured
AND SELF-DIFFUSION.
The self-diffusion
coefficient
with the spin-echo
IV
421
of gaseous
and
spectrometer
which has
been described in detail in I. The diffusion constant was estimated from the height of the spin echo as a function of the time interval between a 90”-180” pulse sequence at constant field gradient. The field gradient was generated by two coils on the pole surfaces of the magnet, designed according to Berger and Butterweckii). The gradient was calculated from the geometry of the coils and measured from the position of the first minima of the echo after a 90”-180” pulse sequence, using a cylindrical water sample. Calculated and measured values were in good agreement: 12.62 and 12.50 + 0.02 G cm-i, respectively, for a current of 1 A; the measured value was assumed to be the most reliable. In all cases 1’s_effects could be avoided by choosing the gradient between 3 and 10 G cm-i. As described in I, two different techniques were used for varying temperature and density. In the first one a liquid-nitrogen cryostat was used. The temperature could be varied continuously between 77 I< and room temperature, and was measured within 0.01 K. The cryostat contained the samples in sealed-off glas tubes with constant over-all CH4 density. Maximum pressure at room temperature obtained in this way was about 200 atm corresponding with an over-all density of about 235 amagat which is just above the critical density (226 amagat). For the measurements in the cryostat five samples were used: CH4(49), from which the amagat densiCH4(75), CH4( 108), CH4( l26), and CH&35), ty, as given in the brackets, was estimated as described in I and II.
loa
Lo.105 +
I
’ 33OK
I
I 250
I
II 2’JoTk
I
I 150
I
I I
100T,
Fig. 1. Self-diffusion coefficient of CH4 in the coexistence region and along the isochores; o experimental points; l obtained from interpolation of the isotherms; v obtained from interpolation of the coexistence line at PA = 300, 400 and 500.
I'.H. OOSTlNG
422
ANI> N. J. TRAPPENIEKS
A second technique, developed for measurements up to higher densities, made use of a high-pressure vessel designed to withstand pressures up to 3000 atm. With the high-pressure vessel measurements were carried out at three different temperatures : 25.00 & O.O2”C, 0.00 & O.O5”C, and -78.40
f
0.05”C. All measurements were carried out on very pure methane (Matheson; 99.997; CH4). Oxygen was removed from it to less than 0.1 ppm, especially for Tr-measurements,
which
were carried
out simultaneously.
report of part of the results pre4. Experimental results. A preliminary sented here is published earlier 12). Measured values of D in the liquid-gas coexistence region and along the isochores (PA = 49,75, 108, 126,235) are shown in fig. 1. Moreover, a number of points is given which is obtained from interpolation of the isothermal measurements at corresponding densities and at three higher densities (PA = 300, 400, 500). Interpolated values on the coexistence curve at 300, 400 and 500 amagat are also shown. Measurements carried out with the different techniques are in good agreement. TABLE Self-diffusion
coefficient Liquid
1000/T (K-l)
PA
I
of CH4 in the coexistence
region Vapour
branch
105D
lOsD(‘J)
(cm2 s-r)
(cm2 s-l)
(exp.)
(theor.)
PA
branch 1030(O)
1030 (cm2 s-l)
(cm2 s-l)
(exp.)
(theor.)
90.92
11.00
631
95.94
10.50
623
3.01
100.00
10.00
614
3.61
105.26 111.11
9.50
604
4.35
9.00
593
5.23
117.65
8.50
579
6.34
125.00
8.00
562
7.76
133.33
7.50
545
9.56
137.93
7.25
13.2
7.76
7.32
142.86 148.15
7.00 6.75
522
11.8
16.8 21.3
6.15 4.99
5.96 4.87
153.85
6.50
492
15.3
160.00 166.67
6.25 450
21.2
25.9
27.6 35.8 47.0
3.95 3.06 2.46
3.90 3.13 2.48
55
2.18
2.16
420
24.9
28.9
65 77
1.87 1.71
1.87 1.61
170.21 173.91
6.00 5.875 5.75
177.78
5.625
2.52
181.82
5.50
379
30.3
33.4
95
1.45
1.33
186.05
5.375
350
35.8
37.0
1.15
1.08
190.48
5.25
-
58.5
120 -
0.67
PROTON-SPIN-LATTICE
KELAXATlON
TABLE Self-diffusion a =
coefficient
1030 (cm2 s-l)
,,A = 49
1000/T tKml)
a
b
170.21
5.875
2.45
2.42
173.91 181.82 186.05 190.48 195.12 200.00 210.53 222.22 235.29 250.00 266.67 285.71 299.20 307.69
5.75 5.50 5.375 5.25 5.125 5.00 4.75 4.50 4.25 4.00 3.75 3.50 3.343 3.25
2.46
2.48
2.60
SELF-DIFFUSION.
(exp.);
423
II
b =
1030(O) (cm2 s-l)
p* = 75
PA = 108
b
a
b
2.58
1.72
1.69
2.80
2.70
1.84
1.76
1.25
1.20
1.29
2.88
2.83
1.91
1.85
3.06
2.97 3.12
2.03 2.11
3.28 3.45
3.28
3.67
(theor.)
pa = 126 a
b
1.22
1.12
1.05
1.36
1.28
1.18
1.10
1.94 2.04
1.40
1.35
1.23
1.47
1.41
2.29
2.14
1.52
3.46
2.39
2.26
3.77
3.66
2.58
4.11
3.89
4.34 4.39
Pi = 235 a
b
0.61
0.58
0.62
0.59
1.15
0.63
0.62
1.29
1.21
0.67
0.65
1.49
1.35
1.28
0.70
0.68
1.65
1.57
1.44
1.35
0.73
0.72
2.39
1.73
1.66
1.51
1.42
0.76
0.76
2.62
2.54
1.87
1.76
1.62
1.51
0.79
0.81
4.05
2.75
2.64
1.96
1.83
1.69
1.57
0.84
0.84
4.14
2.90
2.70
1.96
1.88
1.72
1.61
0.87
0.86
TABLE coefficient
25.OO”C DPA
IV
of CH4 along the isochores;
a
Self-diffusion
PA
AND
III
of CH4 along the isotherms
- 78.4”C
o.oo”c PA
(cm2 s-l)
DPA
PA
DPA (cm2 s-l)
110 124 242 282 322 338 356 387 424 447 480 501
0.146 0.144 0.136 0.135 0.127 0.127, 0.121 0.115 0.108 0.103 0.103 0.090
(cm2 s-l)
137
0.207
50
167
0.201
75
194
0.199
100
220
0.201
125
263
0.196
150
300
0.191
175
324
0.183
200
347
0.181
225
382 431
0.171
250
0.155
275
465
0.147
300
501 520
0.133 0.124
350
549
0.116
375
325
400 425 450 475 500 525 550 575
0.202 0.198 0.200 0.200 0.196 0.192 0.193 0.191 0.186 0.179 0.175 0.171 0.163 0.155 0.152 0.145 0.140 0.135 0.122 0.114 0.103 0.095
424
P. H. OOSTING
AND N. J. TKAPPENlEKS
0.05 -
0
Fig. 2. Self-diffusion
‘\\_
I
I 200
I
‘\
‘I_ I 400
..__
-.-P._/
A
1 600
coefficient of CH4 along the isotherms; o experimental l obtained from interpolation of the isochores.
points
PROTON-SPIN-LATTICE
RELAXATION
AND SELF-DIFFUSI~N.~~
425
The experimental values are tabulated in tables I and 11. In the liquid D increases with a factor 25 from the triple point up to the critical point as a result of increasing temperature and decreasing density. From a log(Bz) ZIS.log T plot it was found that along the liquid coexistence line Dn cc T2-77. A similar behaviour with nearly the same exponent of T is observed in ethaners) and xenoni4). It is probable that this behaviour is caused by the simultaneously varying density and temperature along the coexistence curve. In the coexisting gas phase increasing density results in spite of increasing temperature in a decrease of D. Along the isochores D increases with temperature. A quantitative discussion will be given in the next section. Results of the isothermal measurements are shown in fig. 2. Values obtained by interpolation of the isochores at corresponding temperatures are also given. DPA seems to be constant up to rather high densities. At 200 amagat deviations from the zero-density value are about 5%. The lowdensity values of DPA are in reasonable agreement with results obtained by Winni57 16). Th e experimental values are tabulated in table III. 5. Discztssion. 5.1. Diffusion at low densities. From the isothermal measurements it may be seen that Dn is nearly constant up to rather high densities. Evidently the constant term in the expansion (3) is dominant in this density region. Therefore, one may attempt a quantitative comparison between experimental results at lower densities and the zero-order density approximation D(O)for D. It is easily derived from (4) that for methane (molecular weight 16.04; PNTP = 0.717 g CEle3) :
(11) where 0 has to be taken in A and PA is the density in amagat units. From second virial coefficients which are obtained from recent PVT measurements between 0” and 15O”C17) c/k and (T are estimated for a Lennard-Jones (6, 12) potential (5) to be 147.0 K and 3.828 A respectively. These values are in good agreement with those given by Hirschfelder et al. 9). With these numbers and Liley’sro) values for Q(‘*‘)*, D(O) is calculated from (11) along part of the coexistence line, along the isochores and at the three temperatures at which the isotherms are measured. Results are given in tables I and II and fig. 2. Comparison with experimental values shows a good agreement up to rather high densities (PA < 200). Theoretical values are low by 0 to 8%. Optimal agreement between D(O)and experimental lowdensity values is obtained by taking the Lennard- Jones molecular diameter cr = 3.71 A. To compare the theoretical and experimental temperature dependence at
426
I’. H. OOSTING
,Li 2.25
Fig. 3. Temperature o experimental
points;
AND
N. J. TRAPPENIERS
2.30
2.35
dependence
l obtained
2.40
of D(CH4)
2.45
at constant
from interpolation
LogT 2.50
density;
of the isotherms.
low densities, log Go,‘)* was plotted IX. log T in the temperature range of interest. Within 0.50/ the points lay on a straight line with slope -0.38, from which the temperature dependence of the zero-order density approximation is found to be D(O) x T0.88.
(12)
The isochoric measurements provide good material to check this temperature dependence. From fig. 3 where the isochores are plotted vs. temperature on a log-log scale, the experimental behaviour appears to be in good agreement with (12). Moreover, the temperature dependence varies negligibly over a wide range of density as may be seen from the 300, 400, and 500 amagat isochores in fig. 3 which are obtained from interpolation of the Dp* isotherms. This will be discussed in the next section. Summarizing the above considerations it can be concluded that up to about 200 amagat the experimental behaviour of the self-diffusion coef-
PROTON-SPIN-LATTICE
RELAXATION
AND SELF-DIFFUSION.
250gcds-1
0
427
.
?.,Z
---+P
________7"' 100
I\
I
100
I
200
I
A 300
Fig. 4. Shear viscosity 7 of CH4 at 25°C.
ficient can be predicted within a few per cent by the theoretical zero-order density approximation. This behaviour is dissimilar with that of the other classical transport coefficients : the shear viscosity and the thermal conductivity. To demonstrate this fact provisional values for the shear viscosity ~jrof methane at 25”Crs) are plotted against density in fig. 4. The constant, zero-order, density approximation is also shown. The experimental values deviate much more rapidly from ~(0) than was the case in the behaviour of DPA. This is mainly due to the fact that the contribution of collisional
transfer
plays an important
role in the viscosity.
5.2. Diffusion at elevated densities. At the higher densities Dn is no longer constant but decreases with increasing density (fig. 2). The measurements along the coexistence curve show a similar behaviour, as may be seen from comparison of experimental data with D(O) (table I). At higher densities experimental values are smaller than D(O); the deviations increase with increasing density. The verification of Enskog’s theory for high densities will be restricted to the isothermal measurements. The radial distribution function x is calculated with the two methods described in section 2. Within the first method _ which is the more consistent with Enskog’s theory for hard spheres x can be calculated if the hard-sphere diameter Gh.S. is known. Gh.S. is estimated by assuming that at zero density D equals the zero-order density approximation for hard spheres, i.e. is given by (4) with Q(‘,‘)* = 1 and
428
P. Hf. OOSTING
AND
N. J. TRAPPENIERS
From extrapolation of the experimental isotherms to PA -= 0, 0h.s. is found to be 3.85 A at 298.15 K, 3.85 A at 273.15 K and 4.16 A at 194.8 K. From these values x is calculated with (7) and Dn. from (6). Results are shown in fig. 2: Dpt' (I). The agreement with the experimental be0
=
u’h.8..
haviour is poor. Results from the second method for calculating x using (9), (10) and PVT data from the literature, are also shown in fig. 2: Dpf)(II). These curves are drawn in such a way as to give the best fit with experimental data in the region between 50 and 100 amagat. 4t the high temperatures, the agreement with experiment is favourable with regard to the results of method I, though at high densities the deviations are still of the order of 20%. The agreement for the 194.8 K isotherm is much poorer. For comparison, the Enskog predictions for the shear viscosity q(e) (I) and q(e) (II) are also calculated, using the expression for q(e) as given in ref. 9. Results are shown in fig. 4. The agreement of the experimental values with q(e) (II) is of the same order as in the case of D, while q(e) (I) agrees much better with experiment than did L)(e) (I). Summarizing, it can be said that the use of method I would seem to lead to the conclusion that Enskog’s theory gives a reasonable account only for
Fig. 5. T-Q.QD~A as a function o coexistence
region
; A, isochores ;
q
l 194.8 K isotherm;
298.15
of density K isotherm
solid curve:
in CH*;
; x 273.15 K isotherm ;
eq. (13).
PROTON-SPIN-LATTICE
RELAXATION
AND
SELF-DIFFUSION.
the collisional transfer, which affords an important shear viscosity, but does not influence the transport cation of Enskog’s theory according to method II gives agreement at the high temperatures, though at the pirical fit of the radial distribution function x.
IV
429
contribution to the of mass. The appliin all cases moderate cost of a rather em-
Contrary to the Enskog theory, the more rigorous theoretical treatment of self-diffusion at higher densities, which led to the expansion (3), is much more difficult to compare with the experimental results. Especially an experimental confirmation of the occurrence of a logarithmic term in the expansion cannot be obtained from the present measurements. To demonstrate this, all experimental points are drawn in fig. 5, where T-o.gDp~ is plotted vs. density, such a plot being justified by the fact that the temperature dependence was found to be the same over the total density range covered. With the method of least squares four different expansions are fitted to the experimental points. The results of this calculation are TABLE IV Different Fit to T-O.gDpa
fits to T-“.gDpA lo%1
tow0
1osc2
CO f
&PA
+
C2PiInPA
0.1181
& 0.0007
0.785 5 0.056
-0.0520
f
CO +
+‘A
+
CZP;
0.1170
f
0.0007
1.040 f
0.064
-0.371
+ 0.010
CO +
&PA
f
C2p%lnpA
0.1198
f
0.0013
0.359 f
0.269
-0.137
& 0.053
CO +
&PA+
0.1197
i
0.0010
0.532 * 0.158
-0.164
& 0.060
c2P;
Fit to
+
+
&Pi
c3P;
co +
&PA
+
C2Piln
CO +
&PA
+
c2Pi
ClPA
+
C2&
st. dev.
x 104
0.31
PA
0.32
CO + Clpa + Czpi ln PA + C& CO +
1O’OC3
T-“,gDpA
0.0014
+ c3Pi
6 1.O 0.0224
* 37.6
0.31
f
0.31
0.0064
summarized in table IV. From the tabulated standard deviations, which are all smaller than the experimental inaccuracy, it is clear that no discrimination can be made between the different expansions. The results only suggest that third- and higher-order density terms may be neglected in the density range covered. One of the calculated curves, viz. T-O.gDp* = lo-a(O.1181
+ 0.785
x 10-4~~ - 0.520
x IO-7pilnpa)
(13)
is drawn in fig. 5, and will be used to discuss the temperature dependence of DPA along the liquid branche of the coexistence curve, which was mentioned in section 4 to be Dp* a T 2.77. It will be shown that this “strong tempera-
P. H. OOSTING AND N. J. TRAPPENIERS
430
015 right hand
side
0.10 0.00 -
0.06 -
slope:
2.56
-1
I 100
1 90
0.01
Fig. 6. Log-log
I
I
110
120
I
I
I
140
I 160
I
I
I
160
I 200°K
plot of right-hand side of eq. (15) vs. temperature.
ture dependence” is essentially due to the varying density along the coexistence line, in contradiction with conclusions drawn by Noble and Bloom 13). To prove this statement, use is made of the Guggenheim descriptionis) of the density pa along the liquid branch of the coexistence curve: PLIP cr =
1 +
$(l
-
T/T,,)
+
p(1 -
T/T,,)+,
where per and T cr are the critical density and When (14) together with the critical constants = 190.6 Ii, are put into eq. (13), one obtains an the liquid branch of the coexistence line, which only. This expression can be written as: DPA (coexistence
0.520
=
1 + a(1 -
temperature, respectively. per = 226 amagat, T,, = expression for DPA along contains the temperature
line)
= 10-V-“.9[0.1181 -
(14)
+ 0.785
x 226 x 10-4/(T)
x 2262 x lo-7{f(T)}2
In 226/(T)],
(15)
where f(T)
T/190.6)
+ p(l -
T/190.6)*.
(16)
In fig. 6 a log-log plot is made of the right-hand side of eq. (15) VS. temperature T. The points lie on a straight line with slope 2.58, which is very near to the experimental coefficient 2.77.
PROTON-SPIN-LATTICE
RELAXATION
Acknowledgement.
AND
This investigation
SELF-DIFFUSION.
IV
431
is part of the research program
of the “Stichting
voor Fundamenteel Onderzoek der Materie (F.O.M.)“, “Organisatie voor Zuiver-Wetenschappelijk Onderzoek supported by the der Materie (Z.W.O.)“. The authors are much indebted to Mr. G. L. Boekhout, Mr. J. A. Langendijk, and Mr. J. C. A. Offringa for assistance with the measurements.
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