Thermal conductivity of gaseous hydrogen and of gaseous deuterium

Physica XIV, no 2-3

April

1948

THERMAL CONDUCTIVITY OF GASEOUS HYDROGEN AND OF GASEOUS DEUTERIUM b y J. B. U B B I N K Communication No. 273b from the Kamerlingh Onnes Laboratory, Lei¢lel~

Summary Paragraph 1 shows the importance of the experimental values of ;tn, and ~o, for a theory of tim heat conductivity. In par. 2 the measurements are discussed and that paragraph gives also the results between 273°K and 14°K, while in par. 3 the comparison between the theoretical and experimental values of ] is carried out. From this comparison it appears that the formula of E u c k e n is insufficient. Par. 4 gives the comparison between ~H, and ~.Ot.

1. Introduction. In a previous publication 1) an a p p a r a t u s was described which allowed the m e a s u r e m e n t of the specific heat cond u c t i v i t y 2 of gases at low t e m p e r a t u r e s , in such a w a y n o t w i t h standing the high densities no convection occurs, as can be seen, for instance, from a test m e a s u r e m e n t on hydrogen. In this p a p e r t h e complete e x p e r i m e n t a l d a t a concerning 2u, and 2o, are given a n d the theoretical formula for the connection between 2 and the viscosity r/viz. 2 = [.~/.c o as well as the formula for the ratio b e t w e e n ;tH2 .and ;tD~ n a m e l y : ~n2/2D2 = ~/2 are checked. We observe here t h a t these gases are v e r y i m p o r t a n t for testing the theory, while, moreover, the t h e o r y of the t r a n s p o r t p h e n o m e n a for particles w i t h only translation energy ( / = 2.5) and the t h e o r y for particles w i t h translation- and r o t a t i o n energy / = 1/4(9y--5 ) according to E u c ken can also be tested. This complicated test is m a d e possible b y the fact t h a t for b o t h gases c~ decreases from S/2R a t 273°K to 3/2R in the liquid h y d r o g e n region. This takes place, moreover, for the two gases in different ways, as can be seen f r o m graphs 4 and 5.

2. Measurements. We o b t a i n e d gaseous h y d r o g e n of high p u r i t y b y e v a p o r a t i n g liquid hydrogen. T h e gaseous d e u t e r i u m was made, with e x t r e m e care, b y Prof. M i c h e 1 s of the University of A m s t e r d a m b y electrolysis of h e a v y water. --

165

I

166

j . B . UBBINK

In most measurements the pressure of the gases was a few cm Hg, only once in the liquid hydrogen region, it was higher (see the test measurement 1)). One measurement on para-hydrogen at 17.5°K was carried out, and compared with a measurement on normal hydrogen in the same circumstances. The two measurements were carried out, one immediately after the other and with the same cryostate filling. The 2's 1ooocaLJsec

de~

/

750

/

J |%g

I Fig.

1.

[,+ 1' --

C

--

R as a function

of X.

turned out to be the same within the limits of accuracy. Moreover, a certain adjustment remained the same when parahydrogen was pumped of and normal hydrogen (same pressure) was introduced. Besides the separate measurements on H~ and D~, one measure-

T H E R M A L C O N D U C T I V I T Y OF G A S E O U S H Y D R O G E N

167

ment at 15°K was carried out under the same conditions. Fig. I gives for this case the straight lines ---R-(for the meaning of this formula see i)). The results for H 2 are tabulated in I and for D 2 in I f .

I

cvro J tal. c'm J ,~ec'~dcq '~

J

f x

J

J

J

J

r I t4

-

T

tG

to

20

°K

F i g . 2. Jl a s a f u n c t i o n o f T.

3. Comparison between the theoretical and experimental values o//. For this comparison we consider three temperatures regions: I. The measurements at 273°K. For both gases practically full rotation energy. II. The measurements in the liquid hydrogen region (22~14°K). For both gases no rota/ion energy. III. The measurements in the liquid oxygen-nitrogen region (90-60°K). Rapid change of rotation energy (see the figures 4 and 5). The values of r/had to be recalculated.in most cases because the

168

J.B.

UBBINK

TABLE

l

Hydrogen 1" °h"

c a l s~.'c - I

273.6 88.8 77.9 77.4 65.1 20.7 20. I

h. l0 +~ c n l -'1 d e g r e e -I

T °N

41.2 13.6 1211 11.9 10.4 4.10 3.86

19.3 19.3 19. I 17.5 15.1 14.5 15.0

A. 10 +~ degree -I

c a ] s ~ c - 1 Clll - I

3.75 3.75 3.67 3.36 *) 2.93 2.75 2.87

" F A B L E II I)euterium

7" ~K

cal sec - I c m - ' (l?gret ' - I

A. 10 +s

T °K

89.1 84.2 79.3 75.8 72.4 68.3

12.5 12.1 II.0 1"0.7 11.8 10.8

64.8 19.9 18.5 16.5 15.1 14.9

cal 5ec -I

A. 10 +b e l l ) - I d e g r e e -I 10.6 2.27 2.09 1.81 1.57

1.53

authors, had calibrated their apparatuses relatively to air on He and these values were determined with greater accuracy after their measurements had been carried out. A complete and concise discussion of the values of ~ has been given in part A to facilitate the reading. In part B the comparison follows. Part A. All measurements concerning the viscosity of air have been recalculated with the value 172.06 /~P at 273°K, found by B e a r d e n 2). See for the arguments for this value a). All measurements relative to He have been recalculated with the value ~ 184.2 /~P at 273°K found by S t a t e s 29). The mean value of all available measurements however, is 186.2 # P 3), so that the use of the mean value would raise these values 1.2°/,. The values On, and ~o~. are given in table III. " F A B L E I11 lit.

author

I'qH, in /~1'

Vogel

4

Markowski

5 6 7

Gille Trautz Trautz Trautz Yen

and Zink and Binkele and

.Melster

8

9 I0

84.9 84.1 84.2 83.4 83.4 83.5 82.1

THERMAL

CONDUCTIVITY

OF GASEOUS

HYDROGEN

169

The mean value for ;tn2 is 83.6/zP. Only the measurements i)y V e n (rotating cylinders) show a large deviation. The mean value for D 2 is 117.2 ,uP. For the comparison however, the value 118.2/zP is used, which theoretically is to the value for flu,. as v'2 to 1; and fits in with the other data within the limits of accuracy. In the region below 273°K the following measurements have been carried out. I) Measurements by K a m e r l i n g h Onnes, Dorsman and W e b e r 16) on H , capillary flow method. Calibration at 273°K against /~n~ = 84. l /~P. Recalculation with /~n, = 83.6/~P. In all likelihood their measurements below 22°K are not reliable because they found a pressure delJendence, not found by other investigators and theoretically not acceptable. See for a possible explanation Vogel4). 2) Both V o g e l 4 ) and G u n t h e r S ) used the oscillating disk method, relative to air at 273°K (172.4 ,uP), only H 2. Recalculation with the value 172.1 /eP. 3) The same method was used by V a n I t t e r b e e k and C l a e s 15) for H 2 a n d D 2 ; by V a n I t t e r b e e k and P a e mel*4) forH 2andD2;by Keesom and K e e s o m l ~ ) . These authors calibrated against He gas at different temperatures, values chiefly from K a m e r l i n g h Onnes and W e b e r . Thevalues of ~,: were calculated in a). Part B. C o m p a r i s o n . I. Comparison at 273°K. The data for 2 are given in table IV (7) YABLE

IV

Values of A for H 2 a n d I) 2 a t 2 7 3 ° K author W e b e r

Schneider Grcgor.v

a,ld

Archcr

K ann uluik and Gregory Archer Nothdurft oIIr illeilsUrelllellt

Cleave and Ka.nuluik Nothdurft Archer

31ar

t in

lit.

)tH..

18 19 2O 21 22 23 24

41.6,5

41.80 40.6 41.3 42.0 41.8 42.4.5 42.1

Maass 26

27 28

32.94 30.31 30.80

170

j . B . UBBINK

(;t in 10 -s c a l c m - S s e c - l d e g r e e - I ) . T h e v a l u e for 2 n " f o u n d in t h i s p a p e r is 42. l × 10 -s cal. c m -1 sec -l d e g r e e - l , i t s m e a n v a l u e 41.7 x X 10-Scal c m - l s e c -1 d e g r e e -I. T h e m e a n v a l u e for ;tD2is 3 0 . 2 x 10 -s ~-

L

to-sc,i. ~ sealde9"~

"

,o _ _

12

~ D ~

+

P T

14

SO

"'

Ig

18

10

t

ZZ°K

Fig. 3. Comparison of the directly measured ), (drawn line)with tile values of Jl calculated from the values of p. []

Own measurements.

] Heat / conductivity measurements

I -[]I

A ~7 (D

E u c k e n. KamerlinghOnnes, Dorsman Keesom and K e e s o m . Vogel.

t C)

and W e b e r .

Viscosity measurements.

G ii n t 11 e r.

I

(~

Van

Itterbeek

and C l a e s .

Itterbeek

and P a e m e l .

I

-<~-Van I

c a l c m - ' s e c -ldegree -1 , if the value found by K a n n u l u i k is left out. This value lies outside the limits of accuracy of the other values and may possibly be explained by assuming a contamination by H 2.

T H E R M A L C O N D U C T I V I T Y OF G A S E O U S H Y D R O G E N

171

Comparison of these values for H~_ give [ = 2.07 x ;trio = 42. l X × 10-s) against a theoretical value derived from the "formula of Eucken: ] = 1.92. Use of the value found by Y e n for /XH_ "

t caL ,tool -t t

//

: I

J"

100

I

lO0

$O0OK

F i g . 4. C o n n e c t i o n b e t w e e n c v a n d ~. for H 2.

,-o

s caL "tooL"~

" .......

-Jr

~/ •o i-,--

'~

X

/

-/~""

"r

/I

I

/'71I

,Joo

ioo

F i g . 5. C o n n e c t i o n b e t w e e n c v a n d ;t f o r D 2.

300 "~.

172

j.B. UBBINK

increases this discrepancy, while use of the mean value for ~;~m. only slightly decreases the difference. Comparison of the data for D 2 gives an experimental value of ] ~. 2.05 against a theoretical value / = 1.90. x,o~cat, cm'seC' clecj'

4 o

i/

// 12

./

I tO

,

T

"70

I

80

90"K

Fig. 6. Comparison of ).n, with the values of ;tH,, calculated from /~n,[ ] Our measurements. ~ ).measurements I ~ according t o t h e - [ ] - E tl c k e n ( hot wire I J method. t -G--Spencer, Gregory and D o c k . I K a m e r I ingh Onnes, Dorsman ~(~ V a n

ltterbeek

and W e b e r .

and P a e m e l .

Viscosity measurements.

~), V a n I t t e r b e e k and C l a e s . (~ Vogel. Trautz and Z i m m e r m a n . o"caL

Gin' sec" u=~'

~2

/ /

t2

I I I

*o ....

r

70

ao

s~K

Fig. 7. Comparison of ;to, with the values of •D,, calculated from /zD,.

[]

Our measurements

0

Van Van

&

Itterbeek Itterbeek

and C l a e s . and P a e m e l .

/ } /

Viscosity measurements.

THERMAL CONDUCTIVITY OF GASEOUS HYDROGEN

173

II. Comparison below 22°K is carried out by plotting (see figure 3) the directly measured values of An2 and ADo - against the values of Ano- and ADo.calculated with / = 2.5 from the viscosity data. This value fits in with the d a t a for H 2 though a lower value down to / = 2.45 would also do. This value, however, does not, g i v e a good agreement for D 2 and should be changed to / = 2.3. This discrepancy from the theoretical value cannot be explained by assuming a contamination of the D 2 b y H 2, neither in the '7 nor in the A measurements, because such a c o n t a m i n a t i o n would raise ADo - or lower/*Do-. I IIa. Qualitative comparison in the liquid oxygen-nitrogen region The connection between c,. and / can be seen from fig. 4 for H 2 and from fig. 5 for DA. In these figures are drawn, besides the values of 4, also those of 2.5 cv./~ and 2.07 cv./~ (resp. 2.05 cv/~). In order to show the connection with (c~),o, the latter is drawn in the upper parts. For H 2 the (c,.),o, at 273°K is lower t h a n 2 cal/mol and in consequence thereof I lies between 2.07 c,, u and 2.5 c , / , .and gradually increases to 2.5 c,. ~ as (c~),o, decreases. For D2, the (c~),o, increases fit the beginning and A decreases as a consequence thereof to below 2.05 c~/~. At the temperature, however, at which (c,.),o, passes the value 2 cal/mol, A passes again the value 2.05 c,, ,u and remains further between these curves. IIIb. Q u a n t i t a t i v e comparison has been carried out in the same way as has been done in the region below 22°K. / has been derived from the formula of E u c k e n a n d gives the full-drawn line. Fig. 5 shows the result for HA: the theoretical value is too large. Fig. 6 shows the result for D 2 : the theoretical value is not sensitive enough to represent the changes in (c~),o,. These figures show a close connection between (c,.), a a n d / . But the t h e o r y of E u c k e n is too simple to account accurately for this connection.

4. Comparison o~ An, and 4o2. TheoreticaUy # ,~ m 6 and A ,-~ m -i (4 ---./. cv/M. p). This gives/*n,. = ' 1/ %/2./~Oo- and ;tn0' -- x/2. ADO-. At 273°K this relation is confirmed within the limits of accuracy (see § 3, sub A). At lower t e m p e r a t u r e s a discrepancy arises, as can be seen from a comparison of Ano-/ADo- with %/2 = 1.414 (I) and from a comparison of these values calculated from ~/ measurements by

174

T H E R M A L C O N D U C T I V I T Y OF GASEOUS H Y D R O G E N

Van Itterbeek and C l a e s (II) aS) and V a n Itterb e e k and P a e m e I (III) ~4) w i t h / H , = 2.5 a n d / n , = 2.3. Table V shows that the theoretical value, the experimental value derived from ;t measurements and the experimental value derived from # measurements do not agree. Note, for example, the change in the values of column I against the constancy in those of columns II and III. TABLE V T

1

II

Ill

20°K 17°K 15°K

1.67 1.75

1.75 1.77

1.75 1.75

1.86

1.75

1.75

Received October 21st, 1947. REFERENCES 1) J. B. U b b i n k and W. J. d e H a a s , Commun. Kamerlingh Onnes Lab., Leiden No. 266c; Physica, 's-Gray. 10, 451, 1943. 2) J . A . B e a r d e n , Phys. Rev.(2) 56, 1023, 1939. 3) J . B . U b b i n k , 4) H. V o g e l , Ann. Physik (4) 4:|, 1235, 1914. 5) H. M a r k o w s k i , Ann. Physik (4) 14, 742, 1904. 6) A. G i l l e , Ann. Physik (4) 48, 799, 1915. 7) M. T r a u t z and R. Z i n k, Ann. Physik (5) 7, 427, 1930. 8) M. T r a u t z and H. E. B i n k e l e , Ann. P h y s i k ( 5 ) 5 , 561, 1930. 9) M. T r a u t z andA. Melster, Ann. P h y s i k ( 5 ) 7 , 4 0 9 , 1930. I0) K i a L o k Y e n, Phil. Mag. (7) :38, 582, 1919. 11) H . C . T o r r e y , Phys. Rev.(2) 4 7 , 6 4 4 , 1935. 12) A. B. v a n C l e a v e and O. M a a s s , Canad. J. Research. t ° , 5 7 , 1935. 13) A . B . v a n C l e a v e andO. Maass, Canad. J. Research 1:3,384, 1935. 14) A . v . I t t e r b e e k a n d O . v. P a e m e l , P h y s i c a , ' s - G r a v . 7 , 2 6 5 , 1940. 15) A . v . I t t e r b e e k and Miss A. C l a e s, Physiea, 's-Gray. ~, 938, 1938. 16} H. K a m e r l i n g h Onnes, C. D o r s m a n en S. W e b e r , Commun. No. 134a, 1913; Proe. kon. Akad. A m s t e r d a m I~i, 1386, 1913. 17) W . H . I 4 e e s o m and P. H. K e e s o m , Commun. No. 257c; Physica, 's-Gray. 7, 29, 1940. 18) S. W e b e r , Ann. Physik (4) .%4, 325, 1917. 19) E. S c h n e i d e r , Ann. Physik (4) 80, 215, 1927. 20) H. G r e g o r y and C. T. A r c h e r , Phil. Mag. (7) I, 593, 1926. 21) W. G. K a n n u l u i k and L. H. M a r t i n , Proc. roy. Sot., London A 144~ 496, 1934. 22 H. G r e g o r y, Proc. roy. Soc., London A 14!1, 35, 1934. 23 C.T . A r c h e r, Nature, London 138, 286, 1936. 24 W. N o t l l d u r f t , Ann. P h y s i k ( 5 ) °8, 137, 1937. 25 A. B. v a n C l e a v e and-D. M a a s s , Canad. J. Research I 2 , 3 7 2 , 1935. 26 W . G . K a n u u 1 u i k, Nature, London i:]7, 741, 1936. 27 W. N o t h d u r f t, Ann. Physik (5) °8, 157, 19,37. 28 C . T . A r c h e r, Nature, London 1"18, 286, 1~36. 29 M. N. S t a r e s , Phys. Rev. (2) ° ! , 662, 1923.