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Viscosity of binary mixtures of hydrogen isotopes and mixtures of He and Ne
Riet-veld, A . O . Van I~erbeek, A. Velds, C. A. 1959
Physica 25 205-216
VISCOSITY OF BINARY MIXTURES OF HYDROGEN ISOTOPES AND MIXTURES OF He AND Ne b y A. O. R I E T V E L D , A. VAN I T T E R B E E K and C. A. VELDS Communication No. 314b from the Kamerlingh Onnes Laboratorium, Leiden, Nederland
Synopsis The coefficient of viscosity of mixtures H~-HD, H~-D~, HD-D~ aud He-Ne has been determined, using the method of the oscillating disc at 293, 230, 195, 90, 70, 20 and 14°K and at different concentrations. The results are compared with the results calculated from a Lennard-Jones potential and the appearing differences are discussed.
1. Introduction. In a previous paper 1) a series of measurements of the
viscosity of binary mixtures made with the components Hz, H D and Dg. has been described. The results obtained for one of these mixtures (Hz--HD) have already been published. The series has now been finished and is published here in its entirety, together with measurements on H e - - N e mixtures. The temperatures are obtained with liquid propane, liquid nitrogen, liquid oxygen and liquid hydrogen baths. In these new measurements the purity of the H D was checked again b y means of a mass-spectrometer. It appeared
to be 93 4-. 2%. 2. Measurements. All the measurements have been' carried out with an oscillating disc, as described earlier 2) 3). For the method of measuring we refer to our previous paper 1). An extra measurement has been made to check the importance of the damping of the mirror. This extra damping, which is proportional to the viscosity of the gas in which the mirror oscillates, will be eliminated in relative measurements when the mirror and the disc are at the same temperature. At the low temperatures at which we measured this is not the case, the mirror is always at room temperature. In this case the extra damping does not cancel completely. If there is a noticeable influence then it would be at the lowest temperature. Therefore we repeated the determination of the viscosity of some of the pure components at 20°K after a second mirror, similar to the first one, had been attached on the suspension bar. The results did not differ from the first measurements within the accuracy of the measurement.
-- 205 --
206
A . O . R I E T V E L D , A. VAN I T T E R B E E K A N D C. A. V E L D S
The zero damping was determined as accurately as possible and the precision was 10%. A good determination of this zero damping was necessary, especially for the measurements at liquid hydrogen temperature, since the damping of the disc is small at that temperature. In the most unfavourable case the zero damping amounts to 25% of the total damping and in the most favourable case to 1%. The accuracy of the measurements at low temperatures must be limited at 2 or 3 % ; at higher temperatures at 1 or 2%. The pressures, at which the mixtures of the hydrogen isotopes have been measured were between 0.4 and 1.1 cm Hg at liquid hydrogen temperatures and between 1.5 and 4 cm Hg at the other temperatu.res. The pressures of the He-Ne mixtures are given in table III. They were chosen so, that in one measuring series the depth of penetration of the viscous perturbation is kept constant. 3. Calibration. In a previous paper 1) we gave a table for the coefficients of viscosity of He, which were taken from values given in the literature. Now we give a corrected table for the viscosity of He, for the reason that many of the literature values have been based on an incorrect value for the viscosity of air (see table I). TABLE I The coefficient of viscosity of helium T °K
'r/ `UP
T °K
r/ ,uP
293.1 229.0 196.0 194.0 90.2
196.1 166.4 180.4 149.3 91.2
90.1 71.5 65.8 20.4 14.4
91.1 78.6 74.5 35.0 28.8
4. Experimental results. Tables II and I I I give the measured results. They are plotted in figs 1, 2, 3 and 4. From these graphs we determined the smoothed values. They are listed in table IV, together with the ratio of the coefficients of viscosity of the hydrogen isotopes. These ratio's ought to be temperature independent. At lower temperatures they do not satisfy this criterion owing to quantum mechanical influences (see section 5). 5. Discussion. a. We assume that the potential fields of the isotopes are the same, e. g. a Lennard-Jones 12-6 potential with the parameters ~/k and a. In this case the ratio of two coefficients of viscosity (determined at the same temperature) only depends on the molecular masses. Thus we find the well-
VISCOSITY OF MIXTURES OF H2, H D , D2, H e AND N e
207
known values: ~D2
~ = ~H2
~/2,
~
~HD
°
~H2
If we calculate in the same way the ratio of the coefficient of viscosity of an isotopic mixture and the coefficient of viscosity of one of the components (at the same temperature), we get a value, which only depends on the masses and the concentrations. We can justify this on the basis of the formulae given in H i r s c h f e l d e r ' s TABLE II The coefficient of viscosity of mixtures of hydrogen isotopes H~-HD
H2-D~
HD-D~
T oK %HD
/~P
% D2
/~P
% D~
/~P
293.1
0 24.1 49.8 74.8 I00
88.3 92.8 98.0 102.0 106.9
0 24.6 50.7 75.3 lO0
88.6 98.4 107.8 115.6 123.0
0 25.8 50.9 73.6 I00
107.5 111.7 116.0 I19.9 124.0
229.0
0 19.6 49.7 74.8 100
74.5 78.4 83.1 87.2 91.0
0 24.8 50.5 75.5 lO0
75.7 83.8 91.5 97.8 104.3
0 24.9 49.5 75.5 I00
91.0 94.6 98.0 101.6 104.8
196.0
0 23.6 49.6 74.6 100
67.0 70.7 74.8 78.1 81.6
0 25,1 49,7 75,3 I00
67.5 75. I 81.7 88.0 93.6
0 24.9 50.0 75.0 I00
82.2 85.2 88.3 91.2 94.0
90. I
0 25.3 49.9 74.1 lO0
39,2 41.7 43.6 45.3 47.5
0 26.2 50.2 74.5 100
38.6 43. I 46.8 50.0 53.3
0 23.8 49.2 74.9 I00
47.4 49.0 50.7 52.5 54.0
71.5
0 25.0 49.9 74.9 I00
32.6 34.5 36.2 37.9 39.5
0 24.8 50.2 74.9 100
32.4 35.8 39.0 41.6 44.4
0 25.4 50.7 75.5 l O0
39.3 40.6 42.0 43.4 44.8
20.4
0 24.0 50.5 75.4 I00
II.I II.5 II.8 12.1 12.5
0 33.4 67.7 I00
10.8 11.9 12.9 13.7
0 24.2 50.3 75.1 100
12.7 13.1 13.4 13.8 14.1
14.4
0 25.4 50.1 75.7 I00
7.9 8.2 8.4 8.7 8.8
0 26.9 50.4 76.0 100
7.9 8.5 9.0 9,4 I0,0
0 26. I 49.7 71.6 I00
9.1 9.4 9.7 9.9 10.0
208
A. O. R I E T V E L D ,
A. V A N
ITTERBEEK
AND
C. -A. V E L D S
TABLE III The coefficient of viscosity of m i x t u r e s of He-Ne T OK
P cm Hg
x %Ne
~7 /zP
293.t
5.8 3.8 2.8 2.2 1.9
0 26.2 49.8 75.2 100
196.1 .247.6 277.2 297.3 309.7
194.0
5.7 3.7 2.7 2.1 1.8
0 24.4 48.2 75.9 100
149.3 188.2 211.0 227.3 236.0
90.2
4.0 2.5 1.8 1.5 1.2
0 25.1 49.1 75.5 100
91.2 113.5 125.1 131.9 135.0
65.8
5.8 3.6 2.6 2.1 1.7
0 25.8 50.9 76.1 100
74.5 91.5 99.6 103.2 104.5
20.4
4.0 1.9 1.3 0.9 0.7
0 25.6 49.2 72.0 100
35.0 36.7 36.9 36.1 35.1
1 O O ~
80 7
5O 4
3d 2O
O x b2s H2
so
7s
loo%
H0
F i g . 1. T h e c o e f f i c i e n t of v i s c o s i t y of t - I ~ - H D m i x t u r e s .
,,9
T = 293.1°K T = 229.0°K
T T = 90.1°K A T = 71.5°K
T =
[]
196.0°K
T = 20.4°K
O
T =
14.4°K
VISCOSITY OF MIXTURES OF Hg, H D , D2, He AND N e
120 t10
tOO ~'~
90, 80
60 50-
4o,/
/
36-
~ll2° | 101 ~ O
~
25~
50
7S
H2
D2
F i g . 2. T h e c o e f f i c i e n t • T = 293.1°K T A T ~) T = 2 2 9 . 0 ° K [] T O T = 196.0°K
0 HD
IOO°~o
of = = =
x 25~
v i s c o s i t y of H 2 - D 2 m i x t u r e s . 0 T = 14.4°K 90.1°K 71.5°K 20.4°K
50
75
100°1¢ D2
F i g . 3. T h e c o e f f i c i e n t of v i s c o s i t y of H D - D 2 m i x t u r e s . 0 T = 14.4°K T=293.1°K ~ T= 90.1°K A T = 71.5°K ~) T = 2 2 9 . 0 ° K [] T = 2 0 . 4 ° K o T = 196.0°K
209
210
A. O. R I E T V E L D , A. V A N
ITTERBEEK
AND
C. A. V E L D S
350
~P
300
2sc
//
/
Z
/
150¢/ ...._...~ ,-------~
100
.~
/,
.ql5c
d , 2sj,
o~
He
5o
75
1oOqo N¢
•
Fig. 4. T h e coefficient of v i s c o s i t y of H e - N e m i x t u r e s . T = 293.1°K V T = 90.2°K [] T = 20.4°K
o
T=
194.0°K
A
T----65.8°K
TABLE I V Smoothed values of the viscosity obtained from figs I, 2, 3, 4 and ratio's of the viscosity of hydrogen isotopes T mixture
concentration in % of the heaviest element
I
2S
I
5O
I
75
I
lO0
r/~/rh
oK
0
H2-HD
293. l 229,0 196.0 90. I 71.5 20.4 14.4
88.3 74.5 67.0 39.2 32.6 II.I 7.9
93. I 78.9 70.8 41.4 34.5 II.5 8. I
97.9 83.2 74.6 43.6 36.2 II.8 8.4
I02.4 87.2 78.2 45.6 37.9 12.1 8.6
I06.9 91.0 81.6 47.5 39.5 12.4 8.8
1.21 1.22 1.22 1.21 1.21 1.12 l.l I
H2-D2
293. I 229.0 196.0 90. I 71.5 20.4 14.4
88.6 75.7 67.6 38.6 32.5 10.9 8.0
98.4 83.8 75.0 42.8 35.8 I 1.7 8.5
107.8 91.4 81.8 46.7 38.9 12.4 9.0
I 15.5 98.2 87.9 50. I 41.7 13. I 9.5
123.0 I04.3 93.6 53.3 44.4 13.7 I0.0
1.39 1.38 1.38 1.38 1.37 1.26 1.25
HD~D~
293. I 229.0 196.0 90. I 71.5 20.4 14.4
107.5 91.0 82.2 47.4 39.3 12.7 9.1
I 11.7 94.6 85.2 49.0 40.6 13.1 9.4
I 15.9 98. I 88.3 50.8" 42.0 13.4 9,7
l 19.9 I01.5 91.2 52.4 43.4 13.8 9.9
124.0 I04.8 94.0 54.0 44.7 14.1 I0.0
I. 15 I. 15 I. 14 I. 14 I. 14 I.I I l.lO
Ne-He
293. I 194.0 90.2 65.8 20.4
196. I 149.3 91.2 74.5 35.0
246.0 188.7 113.5 9 I. I 36.7
277.3 212.2 125.4 99,3 36.9
297. I 226.8 131.8 103.2 36. l
309.7 236.0 135.0 104.5 35.1
VISCOSITY OF MIXTURES OF H2, HD, D2, He AND Ne book
211
4):
r/z = c/(Ti*)
T½ ais.Q2~*(Tt*)
(Ml) ~
T'
(
r/il = c/(Tt/*) ai/2f229.,(Ttt.)
2MIM/
)'
Ml + M.~
in which
r/il
a, E
the viscosity of a hypothetical gas with a mass 2MtMj/(Mi+Mj) and a potential field given b y ¢t1 and a 0 = a slowly varying function of T* which gives the transition from the first to a higher approximation. = potential parameters
T*
= kT/E
/(T*)
SQ22*(T*) = a tabulated collision integral c = 26.693 × 10-6 . At the same temperature and same parameters (isotopes) T l * = T / * = Tit*. And so :
r/_! - - V Mt
r/O = V
r/t
r/t
~
2M/ Mt + M/
(1)
The viscosity of a mixture can now be written: r/mixt.
(c~ MI~-~z+
:
1)x12+L~/F](Mlq-M2)2(r/ \r/1 12.q r/2/ r/12)_l}.JFl~2XlX2.q_(ccM2~ Mll -~-|) x22 •
r/1
4M1M2
r/lr/2
r/12
If we divide the two members of this equation b y r/1 or b y r/2, then a formula arises which (with application of (1)) mainly depends only on M1, M2, Xl and x2 and is nearly independent of temperature. A little temperature dependence remains because ~ is a function of T*.o~=O.6A*(T*). According to the tables in H i r s c h f e l d e r e.a. 4) A* varies b y 4% if T* goes from 8 till 0.4 (as in our experiments). The influence of this on the ratio tlmixt./r/1 appears to be less than 1%o and can hence be neglected. These considerations are limited to the case that quantum effects can be neglected, otherwise these ratio's become temperature dependent. To show the increasing influence of quantum effects with lower temperatures, we plotted the calculated temperature independent ratio's r/mlxt.H2-HD/r/H2, r/mixt.H2-D2/r/H2 and r/mixt.HD--D2]~'/HDfor the classical case together with the measured ones in fig. 5. We point out, that until 70°K these measured points fall, in fact, on one line as can be expected (somewhat below the theoretical calculated curves). The measured points at 20°K and 14°K di-
(2)
212
A.O. RIETVELD, A. VAN ITTERBEEK AND C. A. VELDS
v e r g e f r o m this curve. One c a n clearly see here t h e increasing influence of q u a n t u m effects. T h e larger t h e m a s s r a t i o of t h e isotopes the earlier t h e effect is visible as c a n be seen f r o m t h e v i s c o s i t y of H~.--D~, w h e r e d e v i a t i o n s occur a l r e a d y at 70°K. 1.3 I 1.2
~
*
~_n~:_|, _ I~ / "~O H2 t.2
HD ~ 1.5
__..~ m , ~ - ~ x
m ~, 25
SO
. 7S
, IOO~o HD
D2
1,4
H2
D2
Fig. 5. Ratio's of the viscosity of a mixture and the viscosity of its lightest component as a function of the concentration, at different temperatures for H2-HD, HD-D~ and H2-D2 mixtures. ............... theoretical classical curve. H2-HD and HD De: (D T = 293.1°K-71.5°K [] T = 20.4°K and 14.4°K H2-D2: (D T = 293.1°K-90.1°K A T= 71.5°K [] T = 20.4°K . T = 14.4°K b. I t follows f r o m f o r m u l a (2) t h a t we c a n d e t e r m i n e f r o m t h e e x p e r i m e n tal values of 7mlxt. (at c e r t a i n c o n c e n t r a t i o n ) a n d 71 a n d 7z (the coefficients of viscosity of t h e t w o c o m p o n e n t s ) t h e q u a n t i t y 712, if a t least ~ is known. Since the f o r m u l a is o n l y h t t l e d e p e n d e n t on ~, we c a n use t h e ~ v a l u e for t h e L e n n a r d - J o n e s 12-6 potential. W e like to stress h e r e the fact t h a t f o r m u l a (2) is also v a l i d in the case t h a t q u a n t u m corrections are needed. W e carried out this calculation for H z - D , . a n d H e - N e m i x t u r e s . E a c h 71~. has b e e n c a l c u l a t e d t h r e e t i m e s (25%, 5 0 % arid 75%) a n d we f o u n d a l m o s t identical values as p r e d i c t e d b y t h e o r y . T h e y are listed in t a b l e V. I n figures 6 a n d 7 one finds t h e coefficients of v i s c o s i t y of t h e c o m p o n e n t s t o g e t h e r w i t h t h e q u a n t i t y 712 p l o t t e d as a f u n c t i o n of t h e t e m p e r a t u r e , for tile m i x t u r e s H 2 - D 2 a n d H e - N e .
VISCOSITY OF M I X T U R E S OF
H2, H D , D2, H e AND Ne
213
TABLE V Values of U12 of H~-D~ and H e - N e mixtures at different t e mpe ra t ure s T oK
71
Hs-D2
293.1 229.0 196.0 90.1 71.5 20.4 14.4
He-Ne
293.1 194.0 90.2 65.8 20.4
mixture
~,, (~P) /~P
from 25%
from
from
50%
75%
123.0 104.3 93.6 53.3 44.4 13.7 10.0
88.6 75.7 67.6 38.6 32.5 10.9 8.0
101.0 85.8 76.8 44.0 36.5 11.6 8.4
101.8 86.1 76.9 44.1 36.5 11.6 8.4
101.0 86.2 76.8 43.9 36.5 11.6 8.4
101.3 86.0 76.8 44.0 36.5 11.6 8.4
309.7 236.0 135.0 104.5 35.1
196.1 149.3 91.2 74.5 35.0
212.9 165.1 99.9 80.2 30.3
213.3 164.4 99.6 80.2 30.8
213.3 164.4 99.6 80.4 30.8
213.2 164.5 99.7 80.3 30.6
15C
I~F
/
10(
average
/ /
// 1 5O
T ~00
200
4 O0°K
Fig. 6. The viscosity of H~ and D~ as a function of temperature, together with the quantity ~1~ for H2-D~ mixtures. [] deuterium A h3~drogen O ~12 for H~-D~ mixtures c. W h e n we plot the r e d u c e d viscosity ~* ----~a2/(mE) ~ as a function of T*, t h e n we get in the classical case the same curve for different gases. T h e deviations from this curve owing to q u a n t u m mechanical influences can be expressed, within certain limits of T*, b y a t e r m which is p r o p o r t i o n a l to A*% (A*, the r e d u c e d de Broglie w a v e l e n g t h = h/a(mE)½). F o r this we refer to the papers of D e B o e r 5) a n d C o r e m a n s e.a. 8). Now we also reduce the q u a n t i t y ~12 according to GI2 2
214
A.o.
RIETVELD,
A. VAN ITTERBEEK
A N D C. ~ . V E L D S
F o r t h e . h y d r o g e n i s o t o p i c m i x t u r e s o13 a n d Elz/k a r e e q u a l t o t h o s e of t h e c o m p o n e n t s : a l s = 2.928, ¢12/k = 37. F o r t h e m i x t u r e H e - N e one c a n d e t e r m i n e ~19. a n d a l s w i t h t h e c o m b i n a t i o n laws a l g . = ( a l + a9.)/2 a n d ~19. = (~lEs) ~. I f w e t a k e f o r :
~/k =
He: a = 2.556A Ne:o=2.78 A t h e n H e - N e : a ---- 2 . 6 6 8 A
I0.22°K ¢/k = 34.9°K
¢/k=
18.9°K.
/
350
v.P
250
//
150
,I O
O
100
T ,z~oo
3o0
400°K
Fig. 7. The viscosity of He and Ne as a function of.temperature, together with the quantity ~1~ for He--Ne mixtures. [] neon A helium O ~l~ for He-Ne mixtures -0.6
f
--0.8
/ -I.2 / --0.6
laI T " .
0
+O~
41.2
Fig. 8. Reduced viscosity as a function of reduced temperature. 1 hydrogen 3 deuterium 2 hydrogen-deuterium (~1~*/%/T*) 4 classical curve
VISCOSITY OF MIXTURES OF Hg.,
HD, Dg., l-Ie AND N e
215
In figures 8 and 9 log ~I~.*/~/T--~ is plotted against log T* for these mixtures in comparison with log ~*]~/T-~ for their components. The shape of the Hz-D9 curve is normal. The H e - N e curve appears to be shifted with regard to the He- and t h e : N e curve. This can be due to uncertainties in the reduction factor of the mixtures; e.g. if we take the potential parameter alZ about 1% less t h a n (al + az)/2 or ¢19 about 9 ~ less than (¢1¢9.)t the curve comes into agreement with the others. -0.5
f
-O]
-C~9
/
÷C 9
.1
Fig. 9. Reduced viscosity as a function of reduced temperature. 1 helium 3 neon 2 helium-neon (~19.*]~/T-;) 4 classical curve -0.6--
f
Y
-O.|
-I£
- 1:'0~
/
IoqTt _
O
*O&
÷12
Fig. 10, Reduced viscosity as a function of reduced temperature for H D and the mixtures H~-D~ and He-Ne. O He-Ne [] H D & H~-D~ classical curve.
The ti~eoretical, classic .al curve for the reduced viscosity does not quite coincide with our measured points in the classical range. The deviation is 2 or 3~o. This can be caused by: A) the inaccuracy of the measurements (1 or 2°/o),
216
VISCOSITY OF MIXTURES OF
H2,
HD, De, He A~D Ne
B) wrong calibration values (viscosity of He), C) wrong potential parameters, D) deviations of the Lennard-Jones 12-6 model. A more detailed conclusion does not seem possible. d. If the quantum deviation of ~7*/~/T * is determined by A .2, then it is interesting to compare the gas HD and the mixtures H2-D2 and He-Ne since t h e y have almost the same value of A .9, namely 2.00, 2.25 and 2.13 resp. In fig. 10 the measured points of this gas and these mixtures are plotted. As one should expect t h e y actually lie (within 3%) on one curve. Within the accuracy of the measurem~n.ts the deviation from classical behaviour agrees with that found by C o r e m a n s e.a. 8). This work is a part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)" and has been made possible by financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)". We are much indebted to Dr J. J. M. B e e n a k k e r for his valuable advice and to Mr P. Z a n d b e r g e n , nat.phil, cand., for his help during the first part of the measurements and the calculations. Received 31-12-58
REFERENCES I) R i e t v e l d , A. O. and V a n I t t e r b e e k , A., Commun. Kamerlingh Onnes Lab., Leiden No. 309c; Physiea 23 (1957) 838. 2) R i e t v e l d , A. O . , V a n I t t e r b e e k , A. a n d V a n d e n Berg, G.J.,Commun. No. 292a; Physica 19 (1953) 517. 3) R i e t v e l d , A. O. and Van I t t e r b e e k , A., Commun. No. 304e; Physica 22 (1956) 785. 4) H i r s c h f e l d e r , J. O., C u r t i s s , C. F. and Bird, R. B., Molecular theory of gases and liquids (John Wiley & Sons, New York 1954). 5) De Boer, J. and Bird, R. B., Physica 20 (1954) 185. 6) C o r e m a n s , J. M. J., Van I t t e r b e e k , A., B e e n a k k e r , J. J. M., K n a a p , H. F. P. and Z a n d b e r g e n , P., Commun. No. 311a; Physica 24 (1958) 557.
Physica 25 205-216
VISCOSITY OF BINARY MIXTURES OF HYDROGEN ISOTOPES AND MIXTURES OF He AND Ne b y A. O. R I E T V E L D , A. VAN I T T E R B E E K and C. A. VELDS Communication No. 314b from the Kamerlingh Onnes Laboratorium, Leiden, Nederland
Synopsis The coefficient of viscosity of mixtures H~-HD, H~-D~, HD-D~ aud He-Ne has been determined, using the method of the oscillating disc at 293, 230, 195, 90, 70, 20 and 14°K and at different concentrations. The results are compared with the results calculated from a Lennard-Jones potential and the appearing differences are discussed.
1. Introduction. In a previous paper 1) a series of measurements of the
viscosity of binary mixtures made with the components Hz, H D and Dg. has been described. The results obtained for one of these mixtures (Hz--HD) have already been published. The series has now been finished and is published here in its entirety, together with measurements on H e - - N e mixtures. The temperatures are obtained with liquid propane, liquid nitrogen, liquid oxygen and liquid hydrogen baths. In these new measurements the purity of the H D was checked again b y means of a mass-spectrometer. It appeared
to be 93 4-. 2%. 2. Measurements. All the measurements have been' carried out with an oscillating disc, as described earlier 2) 3). For the method of measuring we refer to our previous paper 1). An extra measurement has been made to check the importance of the damping of the mirror. This extra damping, which is proportional to the viscosity of the gas in which the mirror oscillates, will be eliminated in relative measurements when the mirror and the disc are at the same temperature. At the low temperatures at which we measured this is not the case, the mirror is always at room temperature. In this case the extra damping does not cancel completely. If there is a noticeable influence then it would be at the lowest temperature. Therefore we repeated the determination of the viscosity of some of the pure components at 20°K after a second mirror, similar to the first one, had been attached on the suspension bar. The results did not differ from the first measurements within the accuracy of the measurement.
-- 205 --
206
A . O . R I E T V E L D , A. VAN I T T E R B E E K A N D C. A. V E L D S
The zero damping was determined as accurately as possible and the precision was 10%. A good determination of this zero damping was necessary, especially for the measurements at liquid hydrogen temperature, since the damping of the disc is small at that temperature. In the most unfavourable case the zero damping amounts to 25% of the total damping and in the most favourable case to 1%. The accuracy of the measurements at low temperatures must be limited at 2 or 3 % ; at higher temperatures at 1 or 2%. The pressures, at which the mixtures of the hydrogen isotopes have been measured were between 0.4 and 1.1 cm Hg at liquid hydrogen temperatures and between 1.5 and 4 cm Hg at the other temperatu.res. The pressures of the He-Ne mixtures are given in table III. They were chosen so, that in one measuring series the depth of penetration of the viscous perturbation is kept constant. 3. Calibration. In a previous paper 1) we gave a table for the coefficients of viscosity of He, which were taken from values given in the literature. Now we give a corrected table for the viscosity of He, for the reason that many of the literature values have been based on an incorrect value for the viscosity of air (see table I). TABLE I The coefficient of viscosity of helium T °K
'r/ `UP
T °K
r/ ,uP
293.1 229.0 196.0 194.0 90.2
196.1 166.4 180.4 149.3 91.2
90.1 71.5 65.8 20.4 14.4
91.1 78.6 74.5 35.0 28.8
4. Experimental results. Tables II and I I I give the measured results. They are plotted in figs 1, 2, 3 and 4. From these graphs we determined the smoothed values. They are listed in table IV, together with the ratio of the coefficients of viscosity of the hydrogen isotopes. These ratio's ought to be temperature independent. At lower temperatures they do not satisfy this criterion owing to quantum mechanical influences (see section 5). 5. Discussion. a. We assume that the potential fields of the isotopes are the same, e. g. a Lennard-Jones 12-6 potential with the parameters ~/k and a. In this case the ratio of two coefficients of viscosity (determined at the same temperature) only depends on the molecular masses. Thus we find the well-
VISCOSITY OF MIXTURES OF H2, H D , D2, H e AND N e
207
known values: ~D2
~ = ~H2
~/2,
~
~HD
°
~H2
If we calculate in the same way the ratio of the coefficient of viscosity of an isotopic mixture and the coefficient of viscosity of one of the components (at the same temperature), we get a value, which only depends on the masses and the concentrations. We can justify this on the basis of the formulae given in H i r s c h f e l d e r ' s TABLE II The coefficient of viscosity of mixtures of hydrogen isotopes H~-HD
H2-D~
HD-D~
T oK %HD
/~P
% D2
/~P
% D~
/~P
293.1
0 24.1 49.8 74.8 I00
88.3 92.8 98.0 102.0 106.9
0 24.6 50.7 75.3 lO0
88.6 98.4 107.8 115.6 123.0
0 25.8 50.9 73.6 I00
107.5 111.7 116.0 I19.9 124.0
229.0
0 19.6 49.7 74.8 100
74.5 78.4 83.1 87.2 91.0
0 24.8 50.5 75.5 lO0
75.7 83.8 91.5 97.8 104.3
0 24.9 49.5 75.5 I00
91.0 94.6 98.0 101.6 104.8
196.0
0 23.6 49.6 74.6 100
67.0 70.7 74.8 78.1 81.6
0 25,1 49,7 75,3 I00
67.5 75. I 81.7 88.0 93.6
0 24.9 50.0 75.0 I00
82.2 85.2 88.3 91.2 94.0
90. I
0 25.3 49.9 74.1 lO0
39,2 41.7 43.6 45.3 47.5
0 26.2 50.2 74.5 100
38.6 43. I 46.8 50.0 53.3
0 23.8 49.2 74.9 I00
47.4 49.0 50.7 52.5 54.0
71.5
0 25.0 49.9 74.9 I00
32.6 34.5 36.2 37.9 39.5
0 24.8 50.2 74.9 100
32.4 35.8 39.0 41.6 44.4
0 25.4 50.7 75.5 l O0
39.3 40.6 42.0 43.4 44.8
20.4
0 24.0 50.5 75.4 I00
II.I II.5 II.8 12.1 12.5
0 33.4 67.7 I00
10.8 11.9 12.9 13.7
0 24.2 50.3 75.1 100
12.7 13.1 13.4 13.8 14.1
14.4
0 25.4 50.1 75.7 I00
7.9 8.2 8.4 8.7 8.8
0 26.9 50.4 76.0 100
7.9 8.5 9.0 9,4 I0,0
0 26. I 49.7 71.6 I00
9.1 9.4 9.7 9.9 10.0
208
A. O. R I E T V E L D ,
A. V A N
ITTERBEEK
AND
C. -A. V E L D S
TABLE III The coefficient of viscosity of m i x t u r e s of He-Ne T OK
P cm Hg
x %Ne
~7 /zP
293.t
5.8 3.8 2.8 2.2 1.9
0 26.2 49.8 75.2 100
196.1 .247.6 277.2 297.3 309.7
194.0
5.7 3.7 2.7 2.1 1.8
0 24.4 48.2 75.9 100
149.3 188.2 211.0 227.3 236.0
90.2
4.0 2.5 1.8 1.5 1.2
0 25.1 49.1 75.5 100
91.2 113.5 125.1 131.9 135.0
65.8
5.8 3.6 2.6 2.1 1.7
0 25.8 50.9 76.1 100
74.5 91.5 99.6 103.2 104.5
20.4
4.0 1.9 1.3 0.9 0.7
0 25.6 49.2 72.0 100
35.0 36.7 36.9 36.1 35.1
1 O O ~
80 7
5O 4
3d 2O
O x b2s H2
so
7s
loo%
H0
F i g . 1. T h e c o e f f i c i e n t of v i s c o s i t y of t - I ~ - H D m i x t u r e s .
,,9
T = 293.1°K T = 229.0°K
T T = 90.1°K A T = 71.5°K
T =
[]
196.0°K
T = 20.4°K
O
T =
14.4°K
VISCOSITY OF MIXTURES OF Hg, H D , D2, He AND N e
120 t10
tOO ~'~
90, 80
60 50-
4o,/
/
36-
~ll2° | 101 ~ O
~
25~
50
7S
H2
D2
F i g . 2. T h e c o e f f i c i e n t • T = 293.1°K T A T ~) T = 2 2 9 . 0 ° K [] T O T = 196.0°K
0 HD
IOO°~o
of = = =
x 25~
v i s c o s i t y of H 2 - D 2 m i x t u r e s . 0 T = 14.4°K 90.1°K 71.5°K 20.4°K
50
75
100°1¢ D2
F i g . 3. T h e c o e f f i c i e n t of v i s c o s i t y of H D - D 2 m i x t u r e s . 0 T = 14.4°K T=293.1°K ~ T= 90.1°K A T = 71.5°K ~) T = 2 2 9 . 0 ° K [] T = 2 0 . 4 ° K o T = 196.0°K
209
210
A. O. R I E T V E L D , A. V A N
ITTERBEEK
AND
C. A. V E L D S
350
~P
300
2sc
//
/
Z
/
150¢/ ...._...~ ,-------~
100
.~
/,
.ql5c
d , 2sj,
o~
He
5o
75
1oOqo N¢
•
Fig. 4. T h e coefficient of v i s c o s i t y of H e - N e m i x t u r e s . T = 293.1°K V T = 90.2°K [] T = 20.4°K
o
T=
194.0°K
A
T----65.8°K
TABLE I V Smoothed values of the viscosity obtained from figs I, 2, 3, 4 and ratio's of the viscosity of hydrogen isotopes T mixture
concentration in % of the heaviest element
I
2S
I
5O
I
75
I
lO0
r/~/rh
oK
0
H2-HD
293. l 229,0 196.0 90. I 71.5 20.4 14.4
88.3 74.5 67.0 39.2 32.6 II.I 7.9
93. I 78.9 70.8 41.4 34.5 II.5 8. I
97.9 83.2 74.6 43.6 36.2 II.8 8.4
I02.4 87.2 78.2 45.6 37.9 12.1 8.6
I06.9 91.0 81.6 47.5 39.5 12.4 8.8
1.21 1.22 1.22 1.21 1.21 1.12 l.l I
H2-D2
293. I 229.0 196.0 90. I 71.5 20.4 14.4
88.6 75.7 67.6 38.6 32.5 10.9 8.0
98.4 83.8 75.0 42.8 35.8 I 1.7 8.5
107.8 91.4 81.8 46.7 38.9 12.4 9.0
I 15.5 98.2 87.9 50. I 41.7 13. I 9.5
123.0 I04.3 93.6 53.3 44.4 13.7 I0.0
1.39 1.38 1.38 1.38 1.37 1.26 1.25
HD~D~
293. I 229.0 196.0 90. I 71.5 20.4 14.4
107.5 91.0 82.2 47.4 39.3 12.7 9.1
I 11.7 94.6 85.2 49.0 40.6 13.1 9.4
I 15.9 98. I 88.3 50.8" 42.0 13.4 9,7
l 19.9 I01.5 91.2 52.4 43.4 13.8 9.9
124.0 I04.8 94.0 54.0 44.7 14.1 I0.0
I. 15 I. 15 I. 14 I. 14 I. 14 I.I I l.lO
Ne-He
293. I 194.0 90.2 65.8 20.4
196. I 149.3 91.2 74.5 35.0
246.0 188.7 113.5 9 I. I 36.7
277.3 212.2 125.4 99,3 36.9
297. I 226.8 131.8 103.2 36. l
309.7 236.0 135.0 104.5 35.1
VISCOSITY OF MIXTURES OF H2, HD, D2, He AND Ne book
211
4):
r/z = c/(Ti*)
T½ ais.Q2~*(Tt*)
(Ml) ~
T'
(
r/il = c/(Tt/*) ai/2f229.,(Ttt.)
2MIM/
)'
Ml + M.~
in which
r/il
a, E
the viscosity of a hypothetical gas with a mass 2MtMj/(Mi+Mj) and a potential field given b y ¢t1 and a 0 = a slowly varying function of T* which gives the transition from the first to a higher approximation. = potential parameters
T*
= kT/E
/(T*)
SQ22*(T*) = a tabulated collision integral c = 26.693 × 10-6 . At the same temperature and same parameters (isotopes) T l * = T / * = Tit*. And so :
r/_! - - V Mt
r/O = V
r/t
r/t
~
2M/ Mt + M/
(1)
The viscosity of a mixture can now be written: r/mixt.
(c~ MI~-~z+
:
1)x12+L~/F](Mlq-M2)2(r/ \r/1 12.q r/2/ r/12)_l}.JFl~2XlX2.q_(ccM2~ Mll -~-|) x22 •
r/1
4M1M2
r/lr/2
r/12
If we divide the two members of this equation b y r/1 or b y r/2, then a formula arises which (with application of (1)) mainly depends only on M1, M2, Xl and x2 and is nearly independent of temperature. A little temperature dependence remains because ~ is a function of T*.o~=O.6A*(T*). According to the tables in H i r s c h f e l d e r e.a. 4) A* varies b y 4% if T* goes from 8 till 0.4 (as in our experiments). The influence of this on the ratio tlmixt./r/1 appears to be less than 1%o and can hence be neglected. These considerations are limited to the case that quantum effects can be neglected, otherwise these ratio's become temperature dependent. To show the increasing influence of quantum effects with lower temperatures, we plotted the calculated temperature independent ratio's r/mlxt.H2-HD/r/H2, r/mixt.H2-D2/r/H2 and r/mixt.HD--D2]~'/HDfor the classical case together with the measured ones in fig. 5. We point out, that until 70°K these measured points fall, in fact, on one line as can be expected (somewhat below the theoretical calculated curves). The measured points at 20°K and 14°K di-
(2)
212
A.O. RIETVELD, A. VAN ITTERBEEK AND C. A. VELDS
v e r g e f r o m this curve. One c a n clearly see here t h e increasing influence of q u a n t u m effects. T h e larger t h e m a s s r a t i o of t h e isotopes the earlier t h e effect is visible as c a n be seen f r o m t h e v i s c o s i t y of H~.--D~, w h e r e d e v i a t i o n s occur a l r e a d y at 70°K. 1.3 I 1.2
~
*
~_n~:_|, _ I~ / "~O H2 t.2
HD ~ 1.5
__..~ m , ~ - ~ x
m ~, 25
SO
. 7S
, IOO~o HD
D2
1,4
H2
D2
Fig. 5. Ratio's of the viscosity of a mixture and the viscosity of its lightest component as a function of the concentration, at different temperatures for H2-HD, HD-D~ and H2-D2 mixtures. ............... theoretical classical curve. H2-HD and HD De: (D T = 293.1°K-71.5°K [] T = 20.4°K and 14.4°K H2-D2: (D T = 293.1°K-90.1°K A T= 71.5°K [] T = 20.4°K . T = 14.4°K b. I t follows f r o m f o r m u l a (2) t h a t we c a n d e t e r m i n e f r o m t h e e x p e r i m e n tal values of 7mlxt. (at c e r t a i n c o n c e n t r a t i o n ) a n d 71 a n d 7z (the coefficients of viscosity of t h e t w o c o m p o n e n t s ) t h e q u a n t i t y 712, if a t least ~ is known. Since the f o r m u l a is o n l y h t t l e d e p e n d e n t on ~, we c a n use t h e ~ v a l u e for t h e L e n n a r d - J o n e s 12-6 potential. W e like to stress h e r e the fact t h a t f o r m u l a (2) is also v a l i d in the case t h a t q u a n t u m corrections are needed. W e carried out this calculation for H z - D , . a n d H e - N e m i x t u r e s . E a c h 71~. has b e e n c a l c u l a t e d t h r e e t i m e s (25%, 5 0 % arid 75%) a n d we f o u n d a l m o s t identical values as p r e d i c t e d b y t h e o r y . T h e y are listed in t a b l e V. I n figures 6 a n d 7 one finds t h e coefficients of v i s c o s i t y of t h e c o m p o n e n t s t o g e t h e r w i t h t h e q u a n t i t y 712 p l o t t e d as a f u n c t i o n of t h e t e m p e r a t u r e , for tile m i x t u r e s H 2 - D 2 a n d H e - N e .
VISCOSITY OF M I X T U R E S OF
H2, H D , D2, H e AND Ne
213
TABLE V Values of U12 of H~-D~ and H e - N e mixtures at different t e mpe ra t ure s T oK
71
Hs-D2
293.1 229.0 196.0 90.1 71.5 20.4 14.4
He-Ne
293.1 194.0 90.2 65.8 20.4
mixture
~,, (~P) /~P
from 25%
from
from
50%
75%
123.0 104.3 93.6 53.3 44.4 13.7 10.0
88.6 75.7 67.6 38.6 32.5 10.9 8.0
101.0 85.8 76.8 44.0 36.5 11.6 8.4
101.8 86.1 76.9 44.1 36.5 11.6 8.4
101.0 86.2 76.8 43.9 36.5 11.6 8.4
101.3 86.0 76.8 44.0 36.5 11.6 8.4
309.7 236.0 135.0 104.5 35.1
196.1 149.3 91.2 74.5 35.0
212.9 165.1 99.9 80.2 30.3
213.3 164.4 99.6 80.2 30.8
213.3 164.4 99.6 80.4 30.8
213.2 164.5 99.7 80.3 30.6
15C
I~F
/
10(
average
/ /
// 1 5O
T ~00
200
4 O0°K
Fig. 6. The viscosity of H~ and D~ as a function of temperature, together with the quantity ~1~ for H2-D~ mixtures. [] deuterium A h3~drogen O ~12 for H~-D~ mixtures c. W h e n we plot the r e d u c e d viscosity ~* ----~a2/(mE) ~ as a function of T*, t h e n we get in the classical case the same curve for different gases. T h e deviations from this curve owing to q u a n t u m mechanical influences can be expressed, within certain limits of T*, b y a t e r m which is p r o p o r t i o n a l to A*% (A*, the r e d u c e d de Broglie w a v e l e n g t h = h/a(mE)½). F o r this we refer to the papers of D e B o e r 5) a n d C o r e m a n s e.a. 8). Now we also reduce the q u a n t i t y ~12 according to GI2 2
214
A.o.
RIETVELD,
A. VAN ITTERBEEK
A N D C. ~ . V E L D S
F o r t h e . h y d r o g e n i s o t o p i c m i x t u r e s o13 a n d Elz/k a r e e q u a l t o t h o s e of t h e c o m p o n e n t s : a l s = 2.928, ¢12/k = 37. F o r t h e m i x t u r e H e - N e one c a n d e t e r m i n e ~19. a n d a l s w i t h t h e c o m b i n a t i o n laws a l g . = ( a l + a9.)/2 a n d ~19. = (~lEs) ~. I f w e t a k e f o r :
~/k =
He: a = 2.556A Ne:o=2.78 A t h e n H e - N e : a ---- 2 . 6 6 8 A
I0.22°K ¢/k = 34.9°K
¢/k=
18.9°K.
/
350
v.P
250
//
150
,I O
O
100
T ,z~oo
3o0
400°K
Fig. 7. The viscosity of He and Ne as a function of.temperature, together with the quantity ~1~ for He--Ne mixtures. [] neon A helium O ~l~ for He-Ne mixtures -0.6
f
--0.8
/ -I.2 / --0.6
laI T " .
0
+O~
41.2
Fig. 8. Reduced viscosity as a function of reduced temperature. 1 hydrogen 3 deuterium 2 hydrogen-deuterium (~1~*/%/T*) 4 classical curve
VISCOSITY OF MIXTURES OF Hg.,
HD, Dg., l-Ie AND N e
215
In figures 8 and 9 log ~I~.*/~/T--~ is plotted against log T* for these mixtures in comparison with log ~*]~/T-~ for their components. The shape of the Hz-D9 curve is normal. The H e - N e curve appears to be shifted with regard to the He- and t h e : N e curve. This can be due to uncertainties in the reduction factor of the mixtures; e.g. if we take the potential parameter alZ about 1% less t h a n (al + az)/2 or ¢19 about 9 ~ less than (¢1¢9.)t the curve comes into agreement with the others. -0.5
f
-O]
-C~9
/
÷C 9
.1
Fig. 9. Reduced viscosity as a function of reduced temperature. 1 helium 3 neon 2 helium-neon (~19.*]~/T-;) 4 classical curve -0.6--
f
Y
-O.|
-I£
- 1:'0~
/
IoqTt _
O
*O&
÷12
Fig. 10, Reduced viscosity as a function of reduced temperature for H D and the mixtures H~-D~ and He-Ne. O He-Ne [] H D & H~-D~ classical curve.
The ti~eoretical, classic .al curve for the reduced viscosity does not quite coincide with our measured points in the classical range. The deviation is 2 or 3~o. This can be caused by: A) the inaccuracy of the measurements (1 or 2°/o),
216
VISCOSITY OF MIXTURES OF
H2,
HD, De, He A~D Ne
B) wrong calibration values (viscosity of He), C) wrong potential parameters, D) deviations of the Lennard-Jones 12-6 model. A more detailed conclusion does not seem possible. d. If the quantum deviation of ~7*/~/T * is determined by A .2, then it is interesting to compare the gas HD and the mixtures H2-D2 and He-Ne since t h e y have almost the same value of A .9, namely 2.00, 2.25 and 2.13 resp. In fig. 10 the measured points of this gas and these mixtures are plotted. As one should expect t h e y actually lie (within 3%) on one curve. Within the accuracy of the measurem~n.ts the deviation from classical behaviour agrees with that found by C o r e m a n s e.a. 8). This work is a part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)" and has been made possible by financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)". We are much indebted to Dr J. J. M. B e e n a k k e r for his valuable advice and to Mr P. Z a n d b e r g e n , nat.phil, cand., for his help during the first part of the measurements and the calculations. Received 31-12-58
REFERENCES I) R i e t v e l d , A. O. and V a n I t t e r b e e k , A., Commun. Kamerlingh Onnes Lab., Leiden No. 309c; Physiea 23 (1957) 838. 2) R i e t v e l d , A. O . , V a n I t t e r b e e k , A. a n d V a n d e n Berg, G.J.,Commun. No. 292a; Physica 19 (1953) 517. 3) R i e t v e l d , A. O. and Van I t t e r b e e k , A., Commun. No. 304e; Physica 22 (1956) 785. 4) H i r s c h f e l d e r , J. O., C u r t i s s , C. F. and Bird, R. B., Molecular theory of gases and liquids (John Wiley & Sons, New York 1954). 5) De Boer, J. and Bird, R. B., Physica 20 (1954) 185. 6) C o r e m a n s , J. M. J., Van I t t e r b e e k , A., B e e n a k k e r , J. J. M., K n a a p , H. F. P. and Z a n d b e r g e n , P., Commun. No. 311a; Physica 24 (1958) 557.