Inverted forms of the new helium vapour pressure equations

Equations relating the vapour pressures of He 3 and He 4 to temperature on the "Echelle Provisoire de Temp#rature de 1976" have been approved by the Comite International des Poids et Mesures. Using these definitive equations temperatures can only be calculated from measured pressures by iteration. Moreover, the equations are not amenable for use at less than full precision. Alternative equations expressing temperature as a function of pressure over somewhat reduced ranges are now presented in forms which are more easily solved for temperature and are capable of further simplification if reduced accuracy is acceptable.

Inverted forms of the n e w helium vapour pressure equations R.L. Rusby and M. Durieux Kay words: cryogenics, helium vapour, pressure equation

Background The helium vapour pressure scales of 1958 (for He~)~ and 1962 (for He3) 2 were effectively superceded by the introduction of the 'lEchelle Provisoire de Temprrature de 1976 ~ntre 0.5 K et 30 K', the EPT-767 ,~ This is a provisional scale introduced by the Comit6 Consultatif de Thermom~trie (CCT) with the approval of the Comit6 International des Poids et Mesures (CIPM) in order to overcome the inconsistencies which exist between the International Practical Temperature Scale of 1968 (IPTS-68) 5 in the range from 13.81 K to 30 K and the Ts8 and T62 scales below 5.2 IC In particular it had become apparent that the helium scales were low, with respect to thermodynamic temperature, by about 0.2%, and that smooth interpolation between 5.2 K and 13.81 K, for example with a magnetic thermometer, could not be achieved without applying corrections to the established scales, see Fig. 1. One of the tasks associated with formulating the EPT-76 was to tabulate these corrections accurately. Although it is labelled 'provisional', the EPT-76 is not expected to be amended until the IPTS-68 itself is revised, possibly at the end of the 1980s. 8--

6-4--

Z

o Vopour pressure

-h

-8--

o

I

5

I IO

I

.1 ....

15

2O

25

30

T,K Fig. 1 Differences between IPTS-68 (as realized at NPL), T5s, T62 and T76, from 4

0011-2275/84/007363-04 CRYOGENICS . JULY 1 9 8 4

Once the scale had been defined it was desirable for practical purposes to relate vapour pressures, P, to T76 analytically. Vapour pressure equations have been derived and published with the approval of the CIPM, 6,7 as follows He3:0.2 K to 3.3162 K 4

ln(e/Pa) =

~

ai(T76/K)i + asln(T76/K)

(1)

l=-- 1

He4:0.5 K to 2.1768 K 6

ln(P/pa) =

~

bi(T,6/K) i

(2)

t=-- 1

He~: 2.1768 K to 5.1953 K 8

ln(P/Pa) =

~

ct t i + c g ( 1 - t ) 1"9

(3)

i = -- 1

where t = T~6/5.1953 and the coefficients are as in Table 1. Equation (1) has the same form as that derived for T62;2 indeed in the lower part of the range it is based on the same thermodynamic data recalculated for consistency with the new scale. The lower limit, 0.2 K, is also the same as in T62. Strictly this represents a thermodynamic extrapolation of EPT-76, which is only defined down to 0.5 IC For He ~ it was necessary to divide the range into two parts, above and below the lambda point, now taken s to be 2.1768 K at 5041.8 Pa or 37.817 Torr: c f t h e T58 values, 2.172 K at 37.80 Torr. The value at the l a m b d a point, and the first derivative (dlnP/dT = 2.461 K-~), were forced in the fits to give continuity, but it was not possible to take account of the singularity in d=P/dT ~ without undue complication. The equation above the lambda point is of the form used by McCart3P which includes a term to give a divergence in d2P/dT 2 at the

$ 0 3 . 0 0 © 1 9 8 4 Butterworth f~ Co (Publishers) Ltd. 363

Table 1.

Coefficients of the definitive vapour pressure equations (1), (2) and (3)

He3: (1) 0.2 Kto 3.3162 K

He4: (2) 0.5 Kto 2.1768 K

He4: (3) 2.1768 Kto 5.1953 K

a_1 ao a1 a2 a3 a4 as

b-1 bo bl b2 b3 b4 bs b6

c_1 Co Cl c2 c3 c4 cs c6 c7 cs c9

= = = = = = =

Table 2.

-2.50943 9.70876 -0.304433 0.210429 -0.0545145 0.0056067 2.25484

= = = = = = = =

-7.41816 5.42128 9.903203 -9.617095 6.804602 -3.0154606 0.7461357 -0.0791791

-30.93285 392.47361 -2328.04587 8111.30347 - 1 7 809.80901 25 766.52747 - 2 4 601.4 14 944.65142 -5240.36518 807.93168 14.53333

Input data for the fits to (4) and (5)

He3 data In(PAPa)

T/K

11.52946 11.42939 11.31911 11.20487 11.08625 10.96278 10.83395 10.69918 10.55786 10.40925 10.25253 10.08675 9.91079 9.72334 9.52279 9.30724 9.07432 8.82109 8.54384 8.23776 7.89661 7.51198 7.07233 6.56131 5.95489 5.21614 4.28486 3.02090 2.27116 1.32133 0.13904 -0.72493 -1,38550 -2.13023 -2.97762 -3.95223 -5.08775 -5.72961 -6.52045

3.20000 3.10627 3.00606 2.90584 2.80562 2,70540 2.60517 2.50495 2.40473 2.30452 2.20434 2.10418 2.00405 1.90395 1.80388 1.70383 1.60377 1.50371 1.40365 1.30358 1.20352 1.10346 1.00338 0.90326 0.80310 0.70291 0.60266 0.50000 0.45223 0.40207 0.35189 0.32176 0.30168 0.28160 0.26152 0.24142 0.22132 0.21128 0.20000

He4 ata above 2,1768 K In(P/Pa) T/K

He4 data below 2.1768 K In(P/Pa) T/K

12.19157 12.11378 12.03418 11.95268 11.86915 11.78345 11.69545 11.60501 11.52609 11.51195 11.41595 11.31689 11.21457 11.10875 10.99913 10.88541 10.76726 10.64438 10.51635 10.38281 10.24326 10.09718 9.94393 9.78276 9.61283 9.43322 9.24296 9~04072 8.82484 8.59322 8.525518

8.525518 8.458572 8.328485 8.191814 8.048416 7.898043 7.740380 7.575047 7.401596 7.219515 7.028215 6.827025 6.615178 6.391802 6.155903 5.906346 5.641829 5.360856 5.061700 4.742361 4.400507 4.033411 3.637856 3.210028 2.745370

critical point, 5.1953 IC The boiling point at 10 1325 Pa was forced to be 4.2221 K, as defined in EPT-76 (compared with 4.215 K in Tss). Overall these equations are consistent with the differences from Tss and T~2 given in the definition of EPT-76 (Table 3 of Reference 3) to within :t: 0.3 mK. The thermodynamic accuracy of

364

= = = = = = = = = = =

5.00732 4.90732 4.80729 4.70723 4.60717 4.50712 4.40710 4.30710 4.22210 4.20710 4,10710 4.00710 3.90709 3.80707 3.70704 3.60700 3.50694 3.40686 3.30676 3.20664 3.10650 3.00633 2.90613 2.80591 2.70567 2.60542 2.50518 2.40495 2.30473 2.20463 2.17680

2.17680 2.15000 2.10000 2.05000 2.00000 1.95000 1.90000 1.85000 1.80000 1.75000 1.70000 1.65000 1.60000 1.55000 1.50000 1.45000 1.40000 1.35000 1.30000 1.25000 1.20000 1,15000 1.10000 1.05000 1.00000

EPT-76 in this range is estimated to be ___ 1 mK. 4 The forms of (1), (2) and (3) were chosen partly because of similarities with the thermodynamic equations and with earlier equations, but the main requirement was naturally to achieve good fits to the experimental and calculated data over the complete

CRYOGENICS. JULY 1984

Table 3.

Inverse coefficients for He 4, (4) and (5)

a) 2.1768 K to 5.0 K A = 1 0 . 3 , 8 = 1.9, n = 7 i E

0 1 2 3 4 5 6 7

2

o

t

0

I

I

2

I 3

I

4

I

ai

6.716744 1.424445 0.213109 0.021728 0.001233 -0.000430 -0.000135 -0.000078

bi

3.146631 1.357655 0.413923 0.091159 0.016349 0.001826 -0.004325 -0.004973

b) 1.0 K t o 2 . 1 7 6 8 K

5

T,K

A = 5 . 6 , B = 2.9, n = 8 Fig. 2 Differences T76(He 4) ~ T58 and T76(He 3) - - T62 ). The dashed line indicates (T76 - - Tvp) taken from 3

ranges. Unfortunately it was not possible to express T76 as a function of lnP over such wide ranges with adequate accuracy at an acceptable order of fit. Nevertheless such equations would be more convenient for calculating temperatures from measured vapour pressures. We have now derived two equations for each isotope, covering reduced ranges, in the belief that they will widen the practical implementation of the EPT-76. Before proceeding, we should emphasize that the definitive equations are those given above. The following are approximations, albeit ones which are consistent with the definitive equations to about 0.0001 K. They are more easily used with small calculators and can moreover be further simplified if reduced accuracy is acceptable. The input data, Table 2, are essentially the same as those used previously? and the differences from the Tss and T6~ scales are plotted in Fig. 2. In both sets of equations the unit of pressure is the pascal. The conversion from m m H g to pascal is set out in Appendix A. Inverse

equations

The equations are all given as the sum of Chebyshev series, Ti(x), and as the equivalent normalised power series: n

T76/K =

ai Ti (x)

(4)

i=0

and

i

ai

bi

0 1 2 3 4 5 6 7 8

2.967474 0.569587 0.094010 0.015139 0.002844 0.000715 0.000267 0.000085 0.000104

1.392408 0.527153 0.166756 0.050988 0.026514 0.001975 -0.017976 0.005409 0.013259

Table 4.

Inverse coefficients for He 3 (4) and (5)

a) 0.5 K to 3.2 K

A = 7.3, B = 4 . 3 , n = 9 i

ai

0 1 2 3 4 5 6 7 8 9

2.898530 1.279822 0.415023 0.103630 0.019150 0.001863 --0.000092 -0.000541 -0.000037 -0.000215

bi

1.053447 0.980106 0.676380 0.372692 0.151656 --0.002263 0.006596 0.088966 -0.004770 -0.054943

b) 0.2 K to 2.0 K n

T76/K = i ~= bi xi

(5)

where x = (ln(P/Pa) - - A)/B and the constants A and 8 were chosen so that the new variable, x, lies within (or only marginally outside) the range -1 to + l . Equations for He 4 were derived for the ranges 1.0 K to the lambda point and from the lambda point to 5.0 K, with the value and first derivative at the l a m b a point forced to be continuous as before, and the normal boiling point forced to be 4.2221 IC For the upper range, termination at 5.0 K saved several orders of fit, and at order 7 the root m e a n square (RMS) residual was 0.03 mK. The m a x i m u m residual was 0.06 mIC For

CRYOGENICS. JULY 1984

A=1.8, B=8.2, n= 8 i

ai

bi

0 1 2 3 4 5 6 7 8

1.459515 0.775061 0.347291 0.136233 0.047137 0.014218 0.003611 0.000696 0.000062

0.426055 0.432581 0.380500 0.299547 0.213673 0.149533 0.099716 0,044546 0,007914

365

the lower range a fit of order 8 was chosen, with RMS and m a x i m u m residuals of 0.05 m K and 0.12 m K respectively. Once again, the required order of fit increased rapidly for even minor extensions of the range, this time to lower temperatures. The coefficients for both these He* equations are given in Table 3. For He 3 two overlapping ranges were selected. The first, from 0.5 K to 3.2 K (again, just short of the critical point) was fitted by a curve of order 9 with an RMS deviation of 0.02 m K and a m a x i m u m residual of 0.06 mK. The second range was from 0.2 K to 2.0 K for which a curve of order 8 gave RMS and m a x i m u m deviations of 0.02 m K and 0.03 inK. No forcing conditions were applied in either case. The coefficients are given in Table 4, The normalisation of (4) and (5) to the reduced variable, x, ensures that no term in either equation is larger than its coefficient. The series may therefore be truncated with errors no greater than the sum of the moduli of the discarded coefficients, and the coefficients themselves m a y be rounded to just one place beyond what is required in the value of T~6. Inspection of Tables 3 and 4 shows that truncation will in practice only be acceptable for the Chebyshev series. Thus in Table 3(a) omitting the last three or four Chebyshev coefficients will cause less than 0.7 or 2.0 m K error, respectively. Appendix B shows how a Chebyshev series, truncated or otherwise, can be evaluated directly or converted to a power series.

However, this does not lend itself to economical evaluation, and the following routine is recommended:For i = n, n-1, n-2 . . . . . 0, calculate ci = 2xci+i - ci+2 + ai The value of the series, ie the temperature, is (co - - c,)/2. The power series (5) can be evaluated most efficiently by nested multiplication starting with b n, then alternately multiplying by x and adding the next lower coefficient, thus:T / K = {[(bn x + bn-~)x + bn-2] x + . . . . + bo}

The conversion from Chebyshev coefficients to power series coefficients can be made using the following relations:To(X) = 1/2

T,(x) = x

T2(x) = 2 x 2 - - 1

T3(x) = 4 x 3 - - 3 x

or generally (for i > 2)

Ti(x) = ~Ti_,(x) - Ti_2(x) Thus if it is desired to derive a power series equivalent of a truncated Chebyshev series, this can be simply accomplished.

Appendix A The conversion of the readings of mercury manometers to pressure (in pascal) is made using the following relation:P = h g P2o(1 - - 0.00018(t - - 20))

where h is the height of the mercury column in metres, I~ is the acceleration of free fall in m.s -2, P~o is the density of mercury at 20°C, 13545.87 kg.m -3 and t is the mercury temperature in °C. The term in brackets, the correction to the density of mercury, is a simplified form of that given in the IPTS-68, 5 but it will be sufficiently accurate for the present purpose. For instruments calibrated in Torr, ie m m H g at 0°C and at standard gravity, the following relation should be used (P/Pa) = (P/Torr) (101325/760)

The Chebyshev series of degree n may be written as n

366

R L R is from the Division of Quantum Metrology, National Physical Laboratory, Teddington, Middlesex, UIC M D is from Kamerlingh Onnes Laboratorium, Leiden University, Leiden, The Netherlands. Paper received l0 February 1984.

References 1 Brickwedde, F.G., van Dijk, H., Durieux, M., Clement, J.R.,

Logan, J.K. J Res NBS 64A (1960) 1 2 Sydoriak, S.G., Sherman, R.H., Roberts, T.R. J Res NBS 68A

(1964) 547-588 (4 papers) 3 Bureau International des Poids et Mesures Metrologia 15 (1979) 65 4 Durieux, M., Astrov, D.N., Kemp, W.R.G., Swenson, C.A. Metrologia 15 (1979) 57 5 Preston-Thomas, H. Metrologia 12 (1976) 7

Appendix B

T [ K = ao/2 +

Authors

~=

a i cos(/arcosx)

6 Durieux, M., van Dijk, J.E., ter Harmsel, H., Rem, P.C., Rusby,

R.L. Temperature, its measurement and control in science and industry, J.F. Schooley (Ed) Vol 5 Part 1 American Institute Of Physics, New York (1982) 145-153 7 Durieax, M., Rasby, R.L Metmlogia 19 (1983) 67 8 Rusby,R.L, Swenson, C.A. Metrologia 16 (1980) 73 9 MeCarty, ILD. J P h y s Chem RefData 2 (1973) 923

CRYOGENICS . J U L Y 1 9 8 4