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The specific heat of saturated liquid para-hydrogen from 15 to 32° K
The Specific
Heat of Saturated para-hydrogen f r o m , 1 5 t o 3 2 ° K
Liquid
B. A. Younglove and D. E.. Diller Cryogenic Engineeriruy Laboratory, National Bureau of Standards, Boulder, Colorado, U.S.A.
Received 26 March 1962
THE specific heat of saturated liquid para-hydrogen C~ has been measured as part of a programme for determining the thermodynamic properties ofpara-hydrogen at high densities and low temperatures. Thermodynamic properties of the compressed liquid have been computed using Co as the baseline.1 The specific heat of argon has been measured for comparison with that of FIubacher, Leadbetter, and Morrison. 2 Also the C,, of para-hydrogen has been measured by others, which provides an additional comparison, a-5 The results obtained in this work serve to evaluate the accuracy that may be expected in future measurements of compressed fluids with this apparatus.
Experimentalprocedure Calibrations. The apparatus has been described by Goodwin. 6 Basically it consists of a spherical, stainlesssteel sample container bearing a heater, platinum resistance thermometer, and adiabatic shield, all mounted in a cryostat (Figure I). Adjustments were made for thermal and pressure expansion of the sample holder, based on gas expansion measurements and thermal expansion data. 7 The relation used is V = Vo+A Y+By2+
C Y 3 + Dy4+Ep
T (vapour pressure) (°K)
T (thermometer) (°K)
(°K)
770"09 770'02 706"23 706"36
20"313 20"313 20'023 20-024
20-316 20"316 20"028 20-028
+ 0-003 + 0-003 +0.005 +0.004
P
The temperature of the adiabatic shield was matched to that of the sample holder to within a millidegree during drift rate measurements. This temperature difference was recorded continuously. The temperature increment corresponding to the energy input was obtained by extrapolating the temperature drift before and after the heating interval to the mid-time interval. During the heating
... (1)
where Y = 1 0 0 - T , with the temperature T in degrees Kelvin, and where P is the pressure in atmospheres. Values of the constants are A = - 2 " 3 5 9 x 10 -3 B = 1.544x 10 -5 C = 6"633 x 10 - s
l-Guard 6ng
D = - 6 " 7 9 6 x 10 - I ° E = 8.0x 10 -4 /Io = 72"74 cm 3
where Vo is the volume of the sample holder at 100 ° K and 1 atm, determined by extrapolation from gas expansion data. The platinum thermometer calibration was checked very near the normal boiling point temperature ofparahydrogen by observing vapour pressures in the sample holder and computing the temperature from the data of Hoge and Arnold, 8 corrected to the NBS-1955 temperature scale. The following are the results of this check. CRYOGENICS • SEPTEMBER 1962
AT
(mm Hg)
~Shleld
. Calorimeter
"/~
~Cose aermometer
Figure 1. Calorimeter region of calorimeter cryostat
283
interval the temperature difference between the shield and the sample container was slightly greater, and in order to calculate the amount of heat exchanged during this time the thermal resistance between the sample holder and adiabatic shield was measured. The adjustment to the heat data for this variation was found to be negligible in all cases. The heat capacity of the sample holder was measured at 35 temperatures between 15 and 100 ° K and fitted with two polynomials by least squares. Below 26 ° K a polynomial of 6 terms was used and above 26 ° K a polynomial of 7 terms was used. The first and second derivatives of the two polynomials at 26 ° K agree to within 0.03 per cent. The energy absorbed by para-to-ortho conversion is quite small and is accounted for in the method used for the determination of temperature increment described above. The variation in para-hydrogen concentration in the course of a data run was assumed to be negligible. 6 The work done by the sample in stretching the sample holder was negligible in measurements of C~. Samples. Electrolytic hydrogen was purified by passing it through a ' molecular sieve' adsorbent at 76 ° K. It was then passed through an iron oxide catalyst at 20°K to obtain equilibrium hydrogen at that temperature. The concentration of chemical impurities in the hydrogen gas is estimated to be less than 0.01 per cent based on analysis furnished by the gas supplier. Isotopic impurity in the hydrogen gas, based on knowledge of the process used in preparation of the gas, is estimated to be the same as that found in naturally occurring hydrogen gas. The commercial argon sample (Linde) was specified 99.995 per cent pure. It was liquefied in the sample holder in the compressed liquid state, cooled slowly to 76°K to avoid forming small crystallites, and then cooled further to 20 ° K. The sample holder was not filled to capacity for specific heat measurements. In the case ofpara-hydrogen, room was allowed for expansion of the liquid, and in the case of argon the filling was limited by time considerations. However, the fact that there were sufficient amounts of sample is clear from the following table.
Sample
Hydrogen Hydrogen Argon
Temperature ratlge
Moles
15 to 2 5 ° K 25 to 3 2 ° K 20 to 6 0 ° K
2'3 1.1 1-7
to be negligible, however. The specific heat of the two phase system was then adjusted for the heat absorbed by the vapour and the heat of vaporization by the method developed by Hoge 9 TdS' Co = C * - - - - N dT
(2)
where C* is the specific heat of the two phase system, C o is the specific heat of the liquid phase, and S' is t h e ' excess' entrrpy as defined by Hoge S'
dP = ~-~(V- Nv¢)
. . . (3)
The volume of the sample holder is V, and vc is the specific volume of the condensed phase. No adjustment was made to the heat measurements for the effect of hydrogen being compressed into the capillary as calculations show this to be of the order of 0.001 cal/mole deg. or less. The vapour pressure forpara-hydrogen was obtained from the work of Hoge and Arnold s and from this laboratory) ° The saturated liquid densities ofpara-hydrogen also are from this laboratory.t Whereas the above adjustment (equation (2)) to the heat data applies also in the determination of the specific heat of solid argon, this adjustment was always less than 0.001 cal/mole deg. The measurements were then adjusted for curvatureS2 and the result fitted with a polynomial. The thermochemical calorie, 4-1840 absolute joules, was used in the calculations.
Experimental results The experimental values of C o for para-hydrogen, after
15
Ratio of sample-tototal heat capacity 0.95 to 0.94 0.91 to 0.90 0-89 to 0.67
""
ii --~)1! ' ii
10 9 " d
76
-
' -
Method of computation The specific heat of the two phase system was computed first by subtracting the heat capacity of the sample holder from that of the total measured and dividing the result by the number of moles of sample N, as determined from gasometry, less the amount contained in the nuisance or capillary volume. This adjustment in Nis sufficiently small 284
14 16 11520 22 2T4--26--2B-~O32 Temperoture ~ (%,) Figure 2. The specific heat of saturated liquid para-hydrogen CRYOGENICS
. SEPTEMBER 1962
a d j u s t m e n t for c u r v a t u r e , are listed in T a b l e 1 a n d s h o w n in F i g u r e 2. T h e c a l c u l a t e d v a l u e s o f Ca are f r o m
Table 2. Values of the Constants for equation (4)
AT C o = ( T c - T),~ ~-B + C T + D T 2 + E T 3 + F T 4 + G T s . .. (4)
A B C D E F G
w h e r e Tc = 3 2 " 9 8 4 ° K is t h e v a l u e f r o m H o g e a n d Lassiter 13 a d j u s t e d by s u b t r a c t i n g 0.01 ° K , w h i c h c o n v e r t s the N B S - 1 9 3 9 t e m p e r a t u r e scale to N B S - 1 9 5 5 . T h e f o r m o f
1-6815742 -- 3"2802789 x 6.8169871 --7"3194341 x 3'3574357 x - 7"6829740 x 6.9029224 x
I01 10 -1 10 -2 10 -4 10- 6
Table 1. A Comparison of the Experimental and Calculated Values of the Specific Heat of Saturated Liquid para-hydrogen T(exp) (°K)
14.830 15.817 16-753 17.735 18.701 20-010 21.046 21.987 22"986 23.952 24"615 24.773 24.913 25.418 25.563 26-124 26.350 26"981 27'110 27-614 27.920 28.186 28-455 28"749 28.988 29"170 29.484 29-925 30"413 30-922 31"228 31-539
Ce(exp) (call moledeg.)
Ca (calc) ] Difference i Difference (call (per cent) .... mole deg.)
3-277 3.519 3.747 3.976 4-209 4.551 4-835 5'113 5.433 5.775 6.032 6-116 6'163 6-398 6.470 6.739 6.863 7"214 7-298 7'614 7-841 8'065 8.281 8.546 8-824 9-003 9"385 10"017 10.847 11"896 12-693 13"792
3'274 3'524 3-746 3.977 4"210 4"545 4"832 5"114 5"439 5"785 6-044 6"109 6"168 6-390 6"457 6"732 6'850 7-211 7-291 7"627 7-852 8'064 8"295 8"569 8'812 9"011 9'386 9-994 10"814 1 l "906 12-734 13"770
0"003 - 0.005 0.001 -0.001 -0.001 0.006 0.003 - 0.001 - 0.006 -0.010 -0-012 0-007 -- 0.005 0.008 0'013 0.007 0'013 0.003 0.007 -0"013 -0-011 0.001 --0'014 - 0-023 0"012 - - 0.008 --0.001 0-023 0.033 -0.010 --0"041 0"022
0"09 --0.14 0"02 -- 0"02 --0"01 0-14 0"06 -- 0"02 --0"12 --0'17 -- 0"20 0"11 - - 0"08 0"12 0"20 0'11 0-18 0"04 0"10 --0"17 --0-14 0"02 --0"16 -- 0"27 0"13 - - 0-09 -- 0"01
0-23 0"31 - - 0'09 --0"32 0'16
flenlifi?ation umber
108 104 109 105 110 Ill 101 106 102 107 301 201 103 307 206 302 202 308 207 303 203 210 309 208 211 304 204 310 209 305 205 311
AT (°K)
1.901
1.903 1.952 1.941
1.970 1-960 1-921
1.966 1-959 1.966 1.551
1-558 1.896 1.650 1.583 1.471
1-602 1-482 1.517 1-564 1.569 0.778 1.473 1.658 0.831 1.565 1.566 1.473 1.679 1.944 1.929 1.759
e q u a t i o n (4) is s u c h t h a t t h e i n t e g r a l o f ColT a n d t h e i n t e g r a l o f Co w i t h respect to t e m p e r a t u r e , f r o m s o m e t e m p e r a t u r e T o f t h e s a t u r a t e d l i q u i d to t h e critical t e m p e r a t u r e , is finite w h e n 0 < n < 1, w h e r e a s Co b e c o m e s infinite at T c ) 4 E q u a t i o n (4) was a p p l i e d to the d a t a a n d t h e coefficients w e r e c o m p u t e d to give m i n i m u m least s q u a r e s d e v i a t i o n f o r t h e case n = 0.10. T h e v a l u e selected f o r n will also affect t h e r.m.s, d e v i a t i o n a n d t h e a b o v e v a l u e was c h o s e n a m o n g several t h a t w e r e tried, to m i n i m i z e this d e v i a t i o n . T h e coefficients a r e listed in T a b l e 2. T h e i n t e g r a l s o f Ca/T a n d C,, w i t h r e s p e c t to t e m p e r a t u r e f r o m the triple p o i n t t e m p e r a t u r e Tt to selected v a l u e s o f T a r e c o m p u t e d by e q u a t i o n (4) a n d are s h o w n in T a b l e CRYOGENICS
- SEPTEMBER
1962
3. I n t e g r a t i n g to the critical t e m p e r a t u r e i n v o l v e s an extrap o l a t i o n t h a t is s t r o n g l y d e P e n d e n t on the v a l u e o f n in e q u a t i o n (4), w h i c h c o u l d be d e t e r m i n e d f r o m the v a l u e o f an e n t r o p y difference b e t w e e n the critical p o i n t a n d s o m e o t h e r p o i n t such as the triple p o i n t . A n a c c u r a t e v a l u e for this e n t r o p y difference will be a v a i l a b l e later. T h e difference a n d p e r c e n t a g e difference b e t w e e n Co (calculated) a n d C,~ ( m e a s u r e d ) are s h o w n in T a b l e 1, as well as the t e m p e r a t u r e rise d u r i n g the h e a t i n g i n t e r v a l AT. T h e i d e n t i f i c a t i o n n u m b e r i n d i c a t e s the run n u m b e r by the first digit a n d p o i n t n u m b e r in t h a t r u n by the next t w o digits. T h e c h a n g e in i n t e r n a l e n e r g y o f the s a t u r a t e d liquid w i t h t e m p e r a t u r e c a n be c o m p u t e d f r o m T
f CodT r, by s u b t r a c t i n g V(T)
f
P(v)dV
Vt
Table 3. Calculated Values at Selected Temperatures Using Equation (4) to Represent Specific Heat T
T (°K) 13.803 14.000 15-000 16.000 17.000 18-000 19.000 20-000 21-000 22-000 23.000 24-000 25.000 26.000 27-000 28-000 29.000 30.000 31-000 31.500
C~ (calc)
f C~dT
Tt
Tt
(cal/mole deg.) (cal/mole deg.) 3.048 3 "094 3.324 3-557 3.795 4.038 4-289 4-552 4"828 5-124 5.444 5"799 6-200 6-665 7.223 7'919 8.829 10.105 12.095 13-634
T
:%, 0.000 0.044 0'265 0-487 0-710 0'933 1.158 1.385 1"614
1"845 2.080 2-319 2-563 2-815 3'077 3"352 3.645 3.964 4"325 4"530
(cal/mole) 0.000 0.605 3"814 7-254 10.930 14'846 19-009 23.429 28"117 33.091 38.373 43.992 49.987 56.413 63 '348 70"905 79.256 88"683 99'697 106.107
285
where Vt is the specific volume at the triple point, a n d P(v) is v a p o u r pressure o f the s a t u r a t e d liquid. Values o f P(v) are to be published by this l a b o r a t o r y , t° Accuracy of measurements
The m e a s u r e m e n t s o f F l u b a c h e r et al. 2 on argon were c o m p a r e d with those o f this w o r k (see F i g u r e 3 a n d T a b l e 4) b y graphically s m o o t h i n g b o t h sets o f d a t a a n d c o m p u t i n g the percentage difference at every 5 ° K f r o m 20 to 60 ° K. The r.m.s, percentage difference was 0.35, which is an indication o f the accuracy o f the measurements. A n estimate o f the precision o f the para-hydrogen d a t a is readily o b t a i n e d from the deviation p l o t (Figure 4) where the curves indicate the location o f + 0-3 p e r cent deviation. R u n 1 has a precision o f + 0.2 per cent or less a n d runs 2 a n d 3 have a b o u t + 0.3 per cent. R u n s 2 a n d 3 w o u l d naturally have less precision, as the a m o u n t o f sample was less and the correction for the v a p o u r was much greater, as shown in Table 5. T h e runs overlap at .7.0 i
i-
i
-5 6 . 5 - 6.0
f
- ; S e~ L T - -', I
;
..."
i
....
,
I 4.5
Table 5. The Correction to Specific Heat Due to Vapour Corrections, Expressed as a Percentage of the Adjusted Specific Heat Identification number
Correction (per cent)
Identification number
108 105 I01 107 301 201 103
1-3 1"5 0-5 --2"5 24-6 28 '6 --4'2
302 207 309 204 305 205 311
Correction (per cent) 23-2 26-1 16.9 15.3 -4-I -4-4 -16.1
a b o u t 2 5 ° K where points 3 0 1 , 2 0 1 , a n d 103 have quite different corrections a l t h o u g h the m a x i m u m deviation is only 0.3 per cent. This gives a g o o d evaluation o f the accuracy to which the v a p o u r correction can be m a d e . Next to inaccuracies i n t r o d u c e d by the v a p o u r correction, the m o s t i m p o r t a n t source o f inaccuracy is m e a s u r e m e n t o f the t e m p e r a t u r e interval, which is estim/tted to be from 0-I to 0-2 p e r cent. N e a r the critical t e m p e r a t u r e the curvature correction increases sharply, f r o m I p e r cent o f C~ at the 30.413°K d a t u m p o i n t to 5.8 per cent at the 31.530°K d a t u m point. In this range uncertainties in C~ resulting f r o m the curvature correction could be as m u c h as a few tenths o f I p e r cent.
,3 4.0 O.O5 _.o o 0.04 20 25
30
3S 40
45 " S0 55
Temperature ~
• Flubacher et al. 2
6'
o-01
• NBS, CEL
Table 4. Specific Heat of Solid Argon q (cal/mole deg.)
I_o °
•
"d -0.02 -0.03 -0.04 i
<2
28 6
3"320 3"699 3"977 4-228 4.462 4"680 4"880 5 '074 5"213 5"362 6'012 6"525
- - ~ _ ~ -
.,.
.
-o.os
14
16
© Run 1 21"979 24"272 26"175 28"094 30-059 32-067 34.129 36"213 37"993 39"998 50"025 59.267
/
0-02
(OK)
Figure 3. The specific heat of solid argon
T (°K)
,¢
0.03
15
20 22 24 26 Temperature ---'-~ V Run 2
25
30
32 (*K)
[] Run 3
Figure 4. Deviations of measured vahtes of Ca(NBS ) for parahydrogen from those calcMated by equation (4). The curves indicate 0.3 per cent deviation
C o m p a r i s o n o f o u r m e a s u r e m e n t s for C o o f parah y d r o g e n with those f r o m o t h e r sources 3-5 shows w h a t a p p e a r s to be a systematic deviation ( F i g u r e 5). T h e reason for this is n o t k n o w n . T h e t e m p e r a t u r e scale used in the m e a s u r e m e n t s o f J o h n s t o n et al. 4 was b a s e d on a v a p o u r pressure relation which differs systematically f r o m CRYOGENICS ' SEPTEMBER 1962
the vapour pressures of Hoge and Arnold, the maximum difference being 0.040°K at 15.5°K. The shape of the curve relating the temperatures of Hoge and Arnold to those of Johnston affects the temperature differences used in specific heat measurements by as much as 1.4 per cent with an r.m.s, average of about 1 per cent. N o attempt was
0.5(2 0"40 ; 2 ~E 0.3C o 0.20 L 0.IC. . . . . 0 -0.1(:3 -0.2C .~! {o
'_ : ! .i ....
L~ I
-
:
-0"3C
i
--
~ -o-4o -o.
,
.
; !
, A:
Smith et
20 22 24 Temperot.ure ~
~.,. I"~
~
:T-!-T 18
•
:
,
t 16
! I -
• i
I
!
2_.~_.....~2_
26
!
:
•
28
I
-:-, 30
32 (°K)
al. 3 O Johnston et al. 4 [] Clusius and Hiller5
Figure5. Deviations o f reported values o f C a f o r para-hydrogen fi.om smoothed vahtes o f C a ( N B S )
made to adjust the magnitude of the specific heat. The temperatures of Johnston et al. below the normal boiling point were adjusted using the above comparison in addition to correcting the NBS-1939 scale to the NBS-1955. The measurements of Smith, Hallett, and Johnston 3 were based on a different temperature scale from the above.4 The relation between this temperature scale and NBS-1955 is not apparent and no correction was made. It was noticed
CRYOGENICS
° SEPTEMBER
1962
This report stems from an integrated programme on thermodynamic properties of para-hydrogen, to which R. D. Goodwin, H. M. Roder, and L. A. Weber have made various experimental, and computational contributions. REFERENCES
._
~__+ _~
--"--=~-w~,-T~
o
14
'. L--_~__,_
L_ ....
~
:~-
I ,
...........
that the critical temperature used in that paper 3 was 33"24 ° K which is the same as that determined by White, Friedman, and Johnston 16 for normal hydrogen.
1. RODER, H. M., and GOODWIN, R. D. 'Provisional Thermodynamic Functions for Parahydrogen.' N B S Technical Note 130 2. FLUBACHER,P., LEADBETTER,A, J., and MORRISON,J. A. Proc. phys. Soc. Lond. To b e p u b l i s h e d 3. SMrrH, A. L., HALLE'rT, N. C., and JOHNSTON,H. L. J. Amer. chem. Soc. 76, 1486 (1954) 4. JOHNSTON,H. L., CLARKE,J. T., R[FKIN,E. B., and KERR, E. C. J. Amer. chem. Soc. 72, 3933 (1950) 5. CLUS[US,K., and HtLLER, K. Z. Phys. Chem. !}4, 158 (1929) 6. GOODWrN,R. D. J. Res. nat. Bur. Stand. 65C, 231 (1961) 7. CORRUCCINI,R. J., and GNIEWEK,J. J. N B S Monograph 29 (1961) 8. HOGE, H. J., and ARNOLD, R. D. J. Res. nat. Bur. Stand. 47, 64
(1951) 9. HOGE, H. J. J. Res. nat. Bur. Stand. 36, I l l (1946) 10. WEBER,L. A. To b e p u b l i s h e d 11. GOODWXN,R. D., DILLER,D. E., RODER,H. M., and WEBER,L. A. Cryogenics 2, 81 (1961) 12. OSBORNE,N. S., S'nMSON,H. F., SUGH, T. S., and CRAGOE,C. S. N B S Sci. Pap. 20, 65 (1925) 13. HOGE, H. J.,'and LASSITER,J. W . J. Res. nat. Bur. Stand. 47, 75 (1951) 14. OSBORNE,N. S., and VAN DUSEN, M. S. J. Amer. chem. Soc. 40, 1 (1918) 15. RUB[N, T., JOHNSTON, H. L., and ALTMAN, H. J. Amer. chem. Soc. 73, 3401 (1951) 16. WHn-E, D., FRIEDMAN, A. S., and JOHNSTON, H. L. J. Amer. chem. Soc. 72, 3565 (1950) ( T h i s work was supported by Administration.)
the National
Aeronautics and Space
287
Heat of Saturated para-hydrogen f r o m , 1 5 t o 3 2 ° K
Liquid
B. A. Younglove and D. E.. Diller Cryogenic Engineeriruy Laboratory, National Bureau of Standards, Boulder, Colorado, U.S.A.
Received 26 March 1962
THE specific heat of saturated liquid para-hydrogen C~ has been measured as part of a programme for determining the thermodynamic properties ofpara-hydrogen at high densities and low temperatures. Thermodynamic properties of the compressed liquid have been computed using Co as the baseline.1 The specific heat of argon has been measured for comparison with that of FIubacher, Leadbetter, and Morrison. 2 Also the C,, of para-hydrogen has been measured by others, which provides an additional comparison, a-5 The results obtained in this work serve to evaluate the accuracy that may be expected in future measurements of compressed fluids with this apparatus.
Experimentalprocedure Calibrations. The apparatus has been described by Goodwin. 6 Basically it consists of a spherical, stainlesssteel sample container bearing a heater, platinum resistance thermometer, and adiabatic shield, all mounted in a cryostat (Figure I). Adjustments were made for thermal and pressure expansion of the sample holder, based on gas expansion measurements and thermal expansion data. 7 The relation used is V = Vo+A Y+By2+
C Y 3 + Dy4+Ep
T (vapour pressure) (°K)
T (thermometer) (°K)
(°K)
770"09 770'02 706"23 706"36
20"313 20"313 20'023 20-024
20-316 20"316 20"028 20-028
+ 0-003 + 0-003 +0.005 +0.004
P
The temperature of the adiabatic shield was matched to that of the sample holder to within a millidegree during drift rate measurements. This temperature difference was recorded continuously. The temperature increment corresponding to the energy input was obtained by extrapolating the temperature drift before and after the heating interval to the mid-time interval. During the heating
... (1)
where Y = 1 0 0 - T , with the temperature T in degrees Kelvin, and where P is the pressure in atmospheres. Values of the constants are A = - 2 " 3 5 9 x 10 -3 B = 1.544x 10 -5 C = 6"633 x 10 - s
l-Guard 6ng
D = - 6 " 7 9 6 x 10 - I ° E = 8.0x 10 -4 /Io = 72"74 cm 3
where Vo is the volume of the sample holder at 100 ° K and 1 atm, determined by extrapolation from gas expansion data. The platinum thermometer calibration was checked very near the normal boiling point temperature ofparahydrogen by observing vapour pressures in the sample holder and computing the temperature from the data of Hoge and Arnold, 8 corrected to the NBS-1955 temperature scale. The following are the results of this check. CRYOGENICS • SEPTEMBER 1962
AT
(mm Hg)
~Shleld
. Calorimeter
"/~
~Cose aermometer
Figure 1. Calorimeter region of calorimeter cryostat
283
interval the temperature difference between the shield and the sample container was slightly greater, and in order to calculate the amount of heat exchanged during this time the thermal resistance between the sample holder and adiabatic shield was measured. The adjustment to the heat data for this variation was found to be negligible in all cases. The heat capacity of the sample holder was measured at 35 temperatures between 15 and 100 ° K and fitted with two polynomials by least squares. Below 26 ° K a polynomial of 6 terms was used and above 26 ° K a polynomial of 7 terms was used. The first and second derivatives of the two polynomials at 26 ° K agree to within 0.03 per cent. The energy absorbed by para-to-ortho conversion is quite small and is accounted for in the method used for the determination of temperature increment described above. The variation in para-hydrogen concentration in the course of a data run was assumed to be negligible. 6 The work done by the sample in stretching the sample holder was negligible in measurements of C~. Samples. Electrolytic hydrogen was purified by passing it through a ' molecular sieve' adsorbent at 76 ° K. It was then passed through an iron oxide catalyst at 20°K to obtain equilibrium hydrogen at that temperature. The concentration of chemical impurities in the hydrogen gas is estimated to be less than 0.01 per cent based on analysis furnished by the gas supplier. Isotopic impurity in the hydrogen gas, based on knowledge of the process used in preparation of the gas, is estimated to be the same as that found in naturally occurring hydrogen gas. The commercial argon sample (Linde) was specified 99.995 per cent pure. It was liquefied in the sample holder in the compressed liquid state, cooled slowly to 76°K to avoid forming small crystallites, and then cooled further to 20 ° K. The sample holder was not filled to capacity for specific heat measurements. In the case ofpara-hydrogen, room was allowed for expansion of the liquid, and in the case of argon the filling was limited by time considerations. However, the fact that there were sufficient amounts of sample is clear from the following table.
Sample
Hydrogen Hydrogen Argon
Temperature ratlge
Moles
15 to 2 5 ° K 25 to 3 2 ° K 20 to 6 0 ° K
2'3 1.1 1-7
to be negligible, however. The specific heat of the two phase system was then adjusted for the heat absorbed by the vapour and the heat of vaporization by the method developed by Hoge 9 TdS' Co = C * - - - - N dT
(2)
where C* is the specific heat of the two phase system, C o is the specific heat of the liquid phase, and S' is t h e ' excess' entrrpy as defined by Hoge S'
dP = ~-~(V- Nv¢)
. . . (3)
The volume of the sample holder is V, and vc is the specific volume of the condensed phase. No adjustment was made to the heat measurements for the effect of hydrogen being compressed into the capillary as calculations show this to be of the order of 0.001 cal/mole deg. or less. The vapour pressure forpara-hydrogen was obtained from the work of Hoge and Arnold s and from this laboratory) ° The saturated liquid densities ofpara-hydrogen also are from this laboratory.t Whereas the above adjustment (equation (2)) to the heat data applies also in the determination of the specific heat of solid argon, this adjustment was always less than 0.001 cal/mole deg. The measurements were then adjusted for curvatureS2 and the result fitted with a polynomial. The thermochemical calorie, 4-1840 absolute joules, was used in the calculations.
Experimental results The experimental values of C o for para-hydrogen, after
15
Ratio of sample-tototal heat capacity 0.95 to 0.94 0.91 to 0.90 0-89 to 0.67
""
ii --~)1! ' ii
10 9 " d
76
-
' -
Method of computation The specific heat of the two phase system was computed first by subtracting the heat capacity of the sample holder from that of the total measured and dividing the result by the number of moles of sample N, as determined from gasometry, less the amount contained in the nuisance or capillary volume. This adjustment in Nis sufficiently small 284
14 16 11520 22 2T4--26--2B-~O32 Temperoture ~ (%,) Figure 2. The specific heat of saturated liquid para-hydrogen CRYOGENICS
. SEPTEMBER 1962
a d j u s t m e n t for c u r v a t u r e , are listed in T a b l e 1 a n d s h o w n in F i g u r e 2. T h e c a l c u l a t e d v a l u e s o f Ca are f r o m
Table 2. Values of the Constants for equation (4)
AT C o = ( T c - T),~ ~-B + C T + D T 2 + E T 3 + F T 4 + G T s . .. (4)
A B C D E F G
w h e r e Tc = 3 2 " 9 8 4 ° K is t h e v a l u e f r o m H o g e a n d Lassiter 13 a d j u s t e d by s u b t r a c t i n g 0.01 ° K , w h i c h c o n v e r t s the N B S - 1 9 3 9 t e m p e r a t u r e scale to N B S - 1 9 5 5 . T h e f o r m o f
1-6815742 -- 3"2802789 x 6.8169871 --7"3194341 x 3'3574357 x - 7"6829740 x 6.9029224 x
I01 10 -1 10 -2 10 -4 10- 6
Table 1. A Comparison of the Experimental and Calculated Values of the Specific Heat of Saturated Liquid para-hydrogen T(exp) (°K)
14.830 15.817 16-753 17.735 18.701 20-010 21.046 21.987 22"986 23.952 24"615 24.773 24.913 25.418 25.563 26-124 26.350 26"981 27'110 27-614 27.920 28.186 28-455 28"749 28.988 29"170 29.484 29-925 30"413 30-922 31"228 31-539
Ce(exp) (call moledeg.)
Ca (calc) ] Difference i Difference (call (per cent) .... mole deg.)
3-277 3.519 3.747 3.976 4-209 4.551 4-835 5'113 5.433 5.775 6.032 6-116 6'163 6-398 6.470 6.739 6.863 7"214 7-298 7'614 7-841 8'065 8.281 8.546 8-824 9-003 9"385 10"017 10.847 11"896 12-693 13"792
3'274 3'524 3-746 3.977 4"210 4"545 4"832 5"114 5"439 5"785 6-044 6"109 6"168 6-390 6"457 6"732 6'850 7-211 7-291 7"627 7-852 8'064 8"295 8"569 8'812 9"011 9'386 9-994 10"814 1 l "906 12-734 13"770
0"003 - 0.005 0.001 -0.001 -0.001 0.006 0.003 - 0.001 - 0.006 -0.010 -0-012 0-007 -- 0.005 0.008 0'013 0.007 0'013 0.003 0.007 -0"013 -0-011 0.001 --0'014 - 0-023 0"012 - - 0.008 --0.001 0-023 0.033 -0.010 --0"041 0"022
0"09 --0.14 0"02 -- 0"02 --0"01 0-14 0"06 -- 0"02 --0"12 --0'17 -- 0"20 0"11 - - 0"08 0"12 0"20 0'11 0-18 0"04 0"10 --0"17 --0-14 0"02 --0"16 -- 0"27 0"13 - - 0-09 -- 0"01
0-23 0"31 - - 0'09 --0"32 0'16
flenlifi?ation umber
108 104 109 105 110 Ill 101 106 102 107 301 201 103 307 206 302 202 308 207 303 203 210 309 208 211 304 204 310 209 305 205 311
AT (°K)
1.901
1.903 1.952 1.941
1.970 1-960 1-921
1.966 1-959 1.966 1.551
1-558 1.896 1.650 1.583 1.471
1-602 1-482 1.517 1-564 1.569 0.778 1.473 1.658 0.831 1.565 1.566 1.473 1.679 1.944 1.929 1.759
e q u a t i o n (4) is s u c h t h a t t h e i n t e g r a l o f ColT a n d t h e i n t e g r a l o f Co w i t h respect to t e m p e r a t u r e , f r o m s o m e t e m p e r a t u r e T o f t h e s a t u r a t e d l i q u i d to t h e critical t e m p e r a t u r e , is finite w h e n 0 < n < 1, w h e r e a s Co b e c o m e s infinite at T c ) 4 E q u a t i o n (4) was a p p l i e d to the d a t a a n d t h e coefficients w e r e c o m p u t e d to give m i n i m u m least s q u a r e s d e v i a t i o n f o r t h e case n = 0.10. T h e v a l u e selected f o r n will also affect t h e r.m.s, d e v i a t i o n a n d t h e a b o v e v a l u e was c h o s e n a m o n g several t h a t w e r e tried, to m i n i m i z e this d e v i a t i o n . T h e coefficients a r e listed in T a b l e 2. T h e i n t e g r a l s o f Ca/T a n d C,, w i t h r e s p e c t to t e m p e r a t u r e f r o m the triple p o i n t t e m p e r a t u r e Tt to selected v a l u e s o f T a r e c o m p u t e d by e q u a t i o n (4) a n d are s h o w n in T a b l e CRYOGENICS
- SEPTEMBER
1962
3. I n t e g r a t i n g to the critical t e m p e r a t u r e i n v o l v e s an extrap o l a t i o n t h a t is s t r o n g l y d e P e n d e n t on the v a l u e o f n in e q u a t i o n (4), w h i c h c o u l d be d e t e r m i n e d f r o m the v a l u e o f an e n t r o p y difference b e t w e e n the critical p o i n t a n d s o m e o t h e r p o i n t such as the triple p o i n t . A n a c c u r a t e v a l u e for this e n t r o p y difference will be a v a i l a b l e later. T h e difference a n d p e r c e n t a g e difference b e t w e e n Co (calculated) a n d C,~ ( m e a s u r e d ) are s h o w n in T a b l e 1, as well as the t e m p e r a t u r e rise d u r i n g the h e a t i n g i n t e r v a l AT. T h e i d e n t i f i c a t i o n n u m b e r i n d i c a t e s the run n u m b e r by the first digit a n d p o i n t n u m b e r in t h a t r u n by the next t w o digits. T h e c h a n g e in i n t e r n a l e n e r g y o f the s a t u r a t e d liquid w i t h t e m p e r a t u r e c a n be c o m p u t e d f r o m T
f CodT r, by s u b t r a c t i n g V(T)
f
P(v)dV
Vt
Table 3. Calculated Values at Selected Temperatures Using Equation (4) to Represent Specific Heat T
T (°K) 13.803 14.000 15-000 16.000 17.000 18-000 19.000 20-000 21-000 22-000 23.000 24-000 25.000 26.000 27-000 28-000 29.000 30.000 31-000 31.500
C~ (calc)
f C~dT
Tt
Tt
(cal/mole deg.) (cal/mole deg.) 3.048 3 "094 3.324 3-557 3.795 4.038 4-289 4-552 4"828 5-124 5.444 5"799 6-200 6-665 7.223 7'919 8.829 10.105 12.095 13-634
T
:%, 0.000 0.044 0'265 0-487 0-710 0'933 1.158 1.385 1"614
1"845 2.080 2-319 2-563 2-815 3'077 3"352 3.645 3.964 4"325 4"530
(cal/mole) 0.000 0.605 3"814 7-254 10.930 14'846 19-009 23.429 28"117 33.091 38.373 43.992 49.987 56.413 63 '348 70"905 79.256 88"683 99'697 106.107
285
where Vt is the specific volume at the triple point, a n d P(v) is v a p o u r pressure o f the s a t u r a t e d liquid. Values o f P(v) are to be published by this l a b o r a t o r y , t° Accuracy of measurements
The m e a s u r e m e n t s o f F l u b a c h e r et al. 2 on argon were c o m p a r e d with those o f this w o r k (see F i g u r e 3 a n d T a b l e 4) b y graphically s m o o t h i n g b o t h sets o f d a t a a n d c o m p u t i n g the percentage difference at every 5 ° K f r o m 20 to 60 ° K. The r.m.s, percentage difference was 0.35, which is an indication o f the accuracy o f the measurements. A n estimate o f the precision o f the para-hydrogen d a t a is readily o b t a i n e d from the deviation p l o t (Figure 4) where the curves indicate the location o f + 0-3 p e r cent deviation. R u n 1 has a precision o f + 0.2 per cent or less a n d runs 2 a n d 3 have a b o u t + 0.3 per cent. R u n s 2 a n d 3 w o u l d naturally have less precision, as the a m o u n t o f sample was less and the correction for the v a p o u r was much greater, as shown in Table 5. T h e runs overlap at .7.0 i
i-
i
-5 6 . 5 - 6.0
f
- ; S e~ L T - -', I
;
..."
i
....
,
I 4.5
Table 5. The Correction to Specific Heat Due to Vapour Corrections, Expressed as a Percentage of the Adjusted Specific Heat Identification number
Correction (per cent)
Identification number
108 105 I01 107 301 201 103
1-3 1"5 0-5 --2"5 24-6 28 '6 --4'2
302 207 309 204 305 205 311
Correction (per cent) 23-2 26-1 16.9 15.3 -4-I -4-4 -16.1
a b o u t 2 5 ° K where points 3 0 1 , 2 0 1 , a n d 103 have quite different corrections a l t h o u g h the m a x i m u m deviation is only 0.3 per cent. This gives a g o o d evaluation o f the accuracy to which the v a p o u r correction can be m a d e . Next to inaccuracies i n t r o d u c e d by the v a p o u r correction, the m o s t i m p o r t a n t source o f inaccuracy is m e a s u r e m e n t o f the t e m p e r a t u r e interval, which is estim/tted to be from 0-I to 0-2 p e r cent. N e a r the critical t e m p e r a t u r e the curvature correction increases sharply, f r o m I p e r cent o f C~ at the 30.413°K d a t u m p o i n t to 5.8 per cent at the 31.530°K d a t u m point. In this range uncertainties in C~ resulting f r o m the curvature correction could be as m u c h as a few tenths o f I p e r cent.
,3 4.0 O.O5 _.o o 0.04 20 25
30
3S 40
45 " S0 55
Temperature ~
• Flubacher et al. 2
6'
o-01
• NBS, CEL
Table 4. Specific Heat of Solid Argon q (cal/mole deg.)
I_o °
•
"d -0.02 -0.03 -0.04 i
<2
28 6
3"320 3"699 3"977 4-228 4.462 4"680 4"880 5 '074 5"213 5"362 6'012 6"525
- - ~ _ ~ -
.,.
.
-o.os
14
16
© Run 1 21"979 24"272 26"175 28"094 30-059 32-067 34.129 36"213 37"993 39"998 50"025 59.267
/
0-02
(OK)
Figure 3. The specific heat of solid argon
T (°K)
,¢
0.03
15
20 22 24 26 Temperature ---'-~ V Run 2
25
30
32 (*K)
[] Run 3
Figure 4. Deviations of measured vahtes of Ca(NBS ) for parahydrogen from those calcMated by equation (4). The curves indicate 0.3 per cent deviation
C o m p a r i s o n o f o u r m e a s u r e m e n t s for C o o f parah y d r o g e n with those f r o m o t h e r sources 3-5 shows w h a t a p p e a r s to be a systematic deviation ( F i g u r e 5). T h e reason for this is n o t k n o w n . T h e t e m p e r a t u r e scale used in the m e a s u r e m e n t s o f J o h n s t o n et al. 4 was b a s e d on a v a p o u r pressure relation which differs systematically f r o m CRYOGENICS ' SEPTEMBER 1962
the vapour pressures of Hoge and Arnold, the maximum difference being 0.040°K at 15.5°K. The shape of the curve relating the temperatures of Hoge and Arnold to those of Johnston affects the temperature differences used in specific heat measurements by as much as 1.4 per cent with an r.m.s, average of about 1 per cent. N o attempt was
0.5(2 0"40 ; 2 ~E 0.3C o 0.20 L 0.IC. . . . . 0 -0.1(:3 -0.2C .~! {o
'_ : ! .i ....
L~ I
-
:
-0"3C
i
--
~ -o-4o -o.
,
.
; !
, A:
Smith et
20 22 24 Temperot.ure ~
~.,. I"~
~
:T-!-T 18
•
:
,
t 16
! I -
• i
I
!
2_.~_.....~2_
26
!
:
•
28
I
-:-, 30
32 (°K)
al. 3 O Johnston et al. 4 [] Clusius and Hiller5
Figure5. Deviations o f reported values o f C a f o r para-hydrogen fi.om smoothed vahtes o f C a ( N B S )
made to adjust the magnitude of the specific heat. The temperatures of Johnston et al. below the normal boiling point were adjusted using the above comparison in addition to correcting the NBS-1939 scale to the NBS-1955. The measurements of Smith, Hallett, and Johnston 3 were based on a different temperature scale from the above.4 The relation between this temperature scale and NBS-1955 is not apparent and no correction was made. It was noticed
CRYOGENICS
° SEPTEMBER
1962
This report stems from an integrated programme on thermodynamic properties of para-hydrogen, to which R. D. Goodwin, H. M. Roder, and L. A. Weber have made various experimental, and computational contributions. REFERENCES
._
~__+ _~
--"--=~-w~,-T~
o
14
'. L--_~__,_
L_ ....
~
:~-
I ,
...........
that the critical temperature used in that paper 3 was 33"24 ° K which is the same as that determined by White, Friedman, and Johnston 16 for normal hydrogen.
1. RODER, H. M., and GOODWIN, R. D. 'Provisional Thermodynamic Functions for Parahydrogen.' N B S Technical Note 130 2. FLUBACHER,P., LEADBETTER,A, J., and MORRISON,J. A. Proc. phys. Soc. Lond. To b e p u b l i s h e d 3. SMrrH, A. L., HALLE'rT, N. C., and JOHNSTON,H. L. J. Amer. chem. Soc. 76, 1486 (1954) 4. JOHNSTON,H. L., CLARKE,J. T., R[FKIN,E. B., and KERR, E. C. J. Amer. chem. Soc. 72, 3933 (1950) 5. CLUS[US,K., and HtLLER, K. Z. Phys. Chem. !}4, 158 (1929) 6. GOODWrN,R. D. J. Res. nat. Bur. Stand. 65C, 231 (1961) 7. CORRUCCINI,R. J., and GNIEWEK,J. J. N B S Monograph 29 (1961) 8. HOGE, H. J., and ARNOLD, R. D. J. Res. nat. Bur. Stand. 47, 64
(1951) 9. HOGE, H. J. J. Res. nat. Bur. Stand. 36, I l l (1946) 10. WEBER,L. A. To b e p u b l i s h e d 11. GOODWXN,R. D., DILLER,D. E., RODER,H. M., and WEBER,L. A. Cryogenics 2, 81 (1961) 12. OSBORNE,N. S., S'nMSON,H. F., SUGH, T. S., and CRAGOE,C. S. N B S Sci. Pap. 20, 65 (1925) 13. HOGE, H. J.,'and LASSITER,J. W . J. Res. nat. Bur. Stand. 47, 75 (1951) 14. OSBORNE,N. S., and VAN DUSEN, M. S. J. Amer. chem. Soc. 40, 1 (1918) 15. RUB[N, T., JOHNSTON, H. L., and ALTMAN, H. J. Amer. chem. Soc. 73, 3401 (1951) 16. WHn-E, D., FRIEDMAN, A. S., and JOHNSTON, H. L. J. Amer. chem. Soc. 72, 3565 (1950) ( T h i s work was supported by Administration.)
the National
Aeronautics and Space
287